Source document: https://docs-lawep.s3.us-east-2.amazonaws.com/1698844703282.pdf (Saint Kitts and Nevis)
Even if a price for a narrowly deﬁned item is collected by the statistical agency, it may not be equal to the theoretically appropriate time-aggregated unit value price.

This use of an inappropriate price at the very lowest level of aggregation gives rise to time aggre- gation error.10 The statistical agency may classify certain distinct products as being essentially equivalent and this may result in item aggregation error.
For example, when the same product is sold in diﬀerent package sizes, only the per unit price may be collected over the dif- ferent package sizes.

As another example, small quality diﬀerences between products may be ignored.

 The unit value for a particular item may be construc- ted by aggregating over all households in a region or a certain demographic class or by aggregating over all outlets or shops that sell the item in a particular region.
This may give rise to an aggregation over agents or entities error.

20.21 The problems of aggregation and classiﬁca- tion are discussed in more detail in paragraphs 20.23 to 20.37.

20.22 The ﬁve main elementary index number for- mulae are deﬁned in paragraphs 20.30 to 20.45, and in paragraphs 20.46 to 20.57 various numerical rela- tionships between these ﬁve indices are developed.
Paragraphs 20.58 to 20.86 develop the axiomatic and economic approaches to elementary indices, and the ﬁve main elementary formulae used in practice will be evaluated in the light of these approaches.

9 Hausman (2002, p. 14) also noted the importance of collecting quantity data along with price data at the elementary level so that more accurate quality change adjustments can be made by statistical agencies.
10 Many statistical agencies send price collectors to various outlets on certain days of the month to collect list prices of individual items.

Usually, price collectors do not work on weekends, when many sales take place.

Thus the collected prices may not be fully representative of all transactions that occur.

These collected prices can be regarded as approximations to the time-aggregated unit values for those items, but they are only approximations.
357 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Aggregation and classiﬁcation problems for elementary aggregates 20.23 Hawkes and Piotrowski (2003) note that the deﬁnition of an elementary aggregate involves aggrega- tion over four possible dimensions:11 a time dimension; i.e., the item unit value could be item transactions for a year, a calculated for all month, a week, or a day; a spatial dimension; i.e., the item unit value could be calculated for all item transactions in the country, pro- vince or state, city, neighbourhood, or individual location; a product dimension; i.e., the item unit value could be calculated for all item transactions in a broad general category (e.g., food), in a more speciﬁc category (e.g., margarine), for a particular brand (ignoring package size) or for a particular narrowly deﬁned item (e.g., a particular AC Nielsen universal product code); a sectoral (or entity or economic agent) dimension; i.e., the item unit value could be calculated for a particular class of households or a particular class of outlets.
20.24 Each of the above dimensions for choosing the domain of deﬁnition for an elementary aggregate will be discussed in turn.

20.25 As the time period is compressed, several problems emerge: Purchases (by households) and sales (by outlets) become erratic and sporadic.
Thus the frequency of unmatched purchases or sales from one period to the next increases and in the limit (choose the time period to be one minute), nothing will be matched and bilateral index number theory fails.12 As the time period becomes shorter, chained indices exhibit more ‘‘drift’’; i.e., if the value at the end of a chain of periods reverts to the value in the initial period, the chained index does not revert to unity.
As is discussed in paragraphs 15.76 to 15.97 of Chapter 15, it is only appropriate to use chained indices when the underlying price and quantity data exhibit relatively smooth trends.

When the time period is short, seasonal ﬂuctuations13 and periodic sales and advertising campaigns14 can cause prices and quantities to oscillate (or ‘‘bounce’’, to use Szulc’s (1983, p. 548) term), and hence it is not appropriate to use chained indices under these circumstances.
If ﬁxed base indices are used in this short time period situation, 11 Hawkes and Piotrowski (2003, p. 31) combine the spatial and sec- toral dimensions into the spatial dimension.

They also acknowledge the pioneering work of Theil (1954), who identiﬁed three dimensions of aggregation: aggregation over individuals, aggregation over com- modities, and aggregation over time.

12 This point is noted in paragraphs 15.65 to 15.71 of Chapter 15 in relation to the Divisia index.
David Richardson (2003, p. 51) also made the point: ‘‘Deﬁning items with a ﬁner granularity, as is the case if quotes in diﬀerent weeks are treated as separate items, results in more missing data and more imputations.’’ 13 See Chapter 22 for a monthly seasonal example where chained month-to-month indices are useless.

14 See Feenstra and Shapiro (2003) for an example of a weekly superlative index that exhibits massive chain drift.

Richardson (2003, pp.


50–51) discusses the issues involved in choosing weekly unit values versus monthly unit values.

358 then the results will usually depend very strongly on the choice of the base period.

In the seasonal context, not all commodities may even be in the marketplace during the chosen base period.15 All these problems can be mitigated by choosing a longer time period so that trends in the data will tend to dominate the short-term ﬂuctuations.
As the time period contracts, virtually all goods become durable in the sense that they yield services not only for the period of purchase but for subsequent periods.

Thus the period of purchase or acquisition becomes diﬀerent from the periods of use, leading to many complications.16 As the time period contracts, users will not be parti- cularly interested in the short-term ﬂuctuations of the resulting index and there will be demands for smooth- ing the necessarily erratic results.
Put another way, users will want the many, say, weekly or daily move- ments in the index to be summarized as monthly or quarterly movements in prices.

Hence from the viewpoint of meeting the needs of users, there will be relatively little demand for high-frequency indices.

In view of the above considerations, it is recommended that the index number time period be at least four weeks or a month.17 It 20.26 is also necessary to choose the spatial dimension of the elementary aggregate.
Should item prices in each city or region be considered as separate aggregates or should a national item aggregate be con- structed?

Obviously, if it is desired to have regional CPIs which aggregate up to a national CPI, then it will be necessary to collect item prices by region.

It is not clear, however, how ﬁne the ‘‘regions’’ should be.
They could be as ﬁne as a grouping of households in a postal code area or as individual outlets across the country.18 There does not seem to be a clear consensus on what the optimal degree of spatial disaggregation should be.19 15 See Chapter 22 for suggested solutions to these seasonality problems.

16 See Chapter 23 for more material on the possible CPI treatment of durable goods.
17 If there is very high inﬂation in the economy (or even hyperinﬂa- tion), then it may be necessary to move to weekly or even daily indices.
Also, it should be noted that some index number theorists feel that new theories of consumer behaviour should be developed that could use weekly or daily data: ‘‘Some studies have endorsed unit values to reduce high frequency price variation, but this implicitly assumes that the high frequency variation represents simply noise in the data and is not meaningful in the context of a COLI.

That is debatable.
That is debatable.

We need to develop a theory that confronts the data, not truncate the data to ﬁt the theory’’ (Triplett (2003, p. 153)).

Until such new theories are adequately developed, however, a pragmatic approach is to deﬁne the item unit values over months or quarters rather than days or weeks.

18 Iceland no longer uses regional weights but uses individual outlets as the primary geographical unit; see Gudnason (2003, p. 18).
19 William J. Hawkes and Frank W. Piotrowski note that it is quite acceptable to use national elementary aggregates when making inter- national comparisons between countries: When we try to compare egg prices across geography, however, we ﬁnd that lacing across outlets won’t work, because the eyelets on one side of the shoe (or outlets on one side of the river) don’t match up with those on the other side.
Thus, in making interspatial comparisons, we have no choice but to aggregate outlets all the way up to the regional (or, in the case of purchasing power parities, national) level.

We have no hesitation about doing this for interspatial comparisons, but we are reluctant to do so for intertemporal ones.

Why is this?

(Hawkes and Piotrowski (2003, pp.

31–32)).


Each statistical agency will have to make its own judge- ments on the matter of the optimal degree of spatial disaggregation, taking into account the costs of data col- lection and the demands of users for a spatial dimension for the CPI.

20.27 How detailed should the product dimension be?
The possibilities range from regarding all com- modities in a general category as being equivalent to regarding only commodities in a particular package size made by a particular manufacturer or service provider as being equivalent.

All things being equal, Triplett (2002) stresses the advantages of matching products at the most detailed level possible, since this will prevent quality diﬀerences from clouding the period-to-period price comparisons.
This is sensible advice, but then what are the drawbacks to working with the ﬁnest possible commodity classiﬁcation?

The major drawback is that the ﬁner the classiﬁcation, the more diﬃcult it will be to match the item purchased or sold in the base period to the same item in the current period.

Hence, the ﬁner the product classiﬁcation, the smaller will be the number of matched price comparisons that are possible.
This would not be a problem if the unmatched prices followed the same trend as the matched ones in a particular elementary aggregate; but in at least some circumstances, this will not be the case.20 The ﬁner the classiﬁcation system, the more work (in principle) there will be for the statistical agency to adjust for quality or impute the prices that do not match.
Choosing a rela- tively coarse classiﬁcation system leads to a very cost- eﬃcient system of quality adjustment (i.e., essentially no explicit quality adjustment or imputation is done for the prices that do not exactly match), but it may not be very accurate.

Thus all things considered, it seems preferable to choose the ﬁnest possible classiﬁcation system.
20.28 The ﬁnal issue in choosing a classiﬁcation scheme is the issue of choosing a sectoral dimension; i.e., should the unit value for a particular item be calculated for a particular outlet or a particular household, or for a class of outlets or households?

20.29 Before the above question can be answered, it is necessary to ask whether the individual outlet or the individual household is the appropriate ﬁnest level of entity classiﬁcation.
If the economic approach to the CPI is taken, then the individual household is the classiﬁcation.21 appropriate ﬁnest level of entity An answer to their question is that it is preferable to match like with like as closely as possible.

This leads statisticians to prefer the ﬁnest possible level of aggregation, which, in the case of intertemporal comparisons, would be the individual household or the individual outlet.
In making cross-region comparisons, however, matching is not possible unless regional item aggregates are formed, as Hawkes and Piotrowski point out above.

20 Silver and Heravi (2001a; 2001b; 2002; 2003, p. 286) and Koskima¨ ki and Vartia (2001) stressed this point and presented empirical evidence to back up their point.

Feenstra (1994) and Balk (2000b) developed some economic theory based methods to deal with the introduction of new items.
21 This point has been made emphatically by two authors in the recent book on scanner data and price indices: In any case, unit values across stores are not the prices actually faced by households and do not represent the per period price in the COLI, even if the unit values are grouped by type of retail outlet (Triplett (2003, pp.

153–154)).
153–154)).

ELEMENTARY INDICES is not usual Obviously, a single household will not work very well as the basic unit of entity observation because of the sporadic nature of many purchases by an individual household; i.e., there will be tremendous diﬃculties in matching prices across periods for individual house- holds.

For a grouping of households that is suﬃciently large, however, it does become feasible in theory to use the household as the entity classiﬁcation, rather than the outlet as is usually done.
It to use households because of the costs and diﬃculties involved in collecting individual household data on prices and expenditures.22 Price information is usually collected from retail establishments or outlets that sell mainly to households.

Matching problems are mitigated (but not eliminated) using this strategy because the retail outlet generally sells the same items on a continu- ing basis.
20.30 If expenditures by all households in a region are aggregated together, will they equal sales by the retail outlets in the region?

Under certain conditions, the answer to this question is ‘‘yes’’.
The conditions are that the outlets do not sell any items to purchasers who are not local households (no regional exports or sales to local businesses or governments) and that the regional households do not make any purchases of consumption items other than from the local outlets (no household imports or transfers of commodities to local households by governments).

Obviously, these restrictive conditions will not be met in practice, but they may hold as a ﬁrst approximation.
20.31 The eﬀects of regional aggregation and product aggregation can be examined, thanks to a recent study by Koskima¨ ki and Yla¨ -Jarkko (2003).

This study used scanner data for the last week in September 1998 and September 2000 on butter, margarine and other vege- table fats, vegetable oils, soft drinks, fruit juices and detergents that were provided by the AC Nielsen com- pany for Finland.
At the ﬁnest level of item classiﬁcation (the AC Nielsen Universal Product Code), the number of individual items in the sample was 1,028.

The total number of outlets in the sample was 338.

Koskima¨ ki and Yla¨ -Jarkko then considered four levels of spatial disaggregation: – the entire country (1 level); – provinces (4 levels); – AC Nielsen regions (15 levels); – individual outlets (338 levels).
They also considered four aggregation: – the COICOP 5-digit classiﬁcation (6 levels); – the COICOP 7-digit classiﬁcation (26 levels); levels of product dis- Furthermore, note that the relationship being estimated is not a proper consumer demand function but rather an ‘establishment sales function’.

Only after making further assumptions – for example, ﬁxing the dis- tribution of consumers across establishments – is it permissible to jump to demand functions (Ley (2003, p. 380)).
22 However, it is not impossible to collect accurate household data in certain circumstances; see Gudnason (2003), who pioneered a receipts methodology for collecting household price and expenditure data in Iceland.
359 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Table 20.1 Proportion of transactions in 2000 that could be matched to 1998 Table 20.2 Laspeyres price indices by type of classiﬁcation, September 1998–September 2000 COICOP 5-digit COICOP 7-digit AC Nielsen brand AC Nielsen Universal Product Code COICOP 5-digit COICOP 7-digit AC Nielsen brand AC Nielsen Universal Product Code Country Province AC Nielsen region Individual outlet 1.000 1.000 1.000 0.904 1.000 1.000 1.000 0.904 0.982 0.975 0.969 0.846 0.801 0.774 0.755 0.617 Country Province AC Nielsen region Individual outlet 1.079 1.078 1.078 1.086 1.031 1.031 1.031 1.040 1.046 1.048 1.048 1.060 1.023 1.023 1.025 1.028 – the AC Nielsen brand classiﬁcation (266 levels); – the AC Nielsen individual Universal Product Code (1,028 distinct products).
20.32 In order to illustrate the ability to match products over the two-year period as a function of the degree of ﬁneness of the classiﬁcation, Koskima¨ ki and Yla¨ -Jarkko (2003, p. 10) presented a table showing that the proportion of transactions that could be matched across the two years fell steadily as the ﬁneness of the classiﬁcation scheme increased.
At the highest level of aggregation (the national and COICOP 5-digit), all transactions could be matched over the two-year period, but at the ﬁnest level of aggregation (338 outlets times 1,028 individual products or 347,464 classiﬁcation cells in all), only 61.7 per cent of the value of transactions in 2000 could be matched back to their 1998 counterparts.

Koskima¨ ki and Yla¨ -Jarkko’s Table 7 is reproduced as Table 20.1.
20.33 For each of the above 16 levels of product and regional disaggregation, for the products that were available in September 1998 and September 2000, Kos- kima¨ ki and Yla¨ -Jarkko (2003, p. 9) calculated Laspeyres and Fisher price indices.

Their results are reproduced below as Tables 20.2 and 20.3.

20.34 Some of the trends in Tables 20.2 and 20.3 can be explained.
As the product classiﬁcation is made more ﬁne, the indices tend to fall.23 This indicates that the new products entering the sample tend to be more expensive than the continuing products.

The diﬀerences in the COICOP 5-digit results and the AC Nielsen Universal Product Code results are very big indeed and indicate that it is probably best to work at the ﬁnest level of product disaggregation, even if there is the possibility of bias because of neglecting new products.
This possible bias would have to be very substantial to overturn a recommendation to work at the ﬁnest level of product disaggregation.

20.35 As the regional classiﬁcation is made ﬁner, there is a tendency for the Laspeyres indices to become larger.

This can be explained by purchasers switching to the lowest-cost outlets so that the item unit values will be smaller the higher the degree of aggregation.
Put another way, the Laspeyres indices calculated at the outlet level are subject to a certain amount of outlet substitution bias (if one is willing to regard this phe- nomenon as a bias).

23 The results at the AC Nielsen brand level are a counter-example to this general assertion.
360 Table 20.3 Fisher price indices by type of classiﬁcation, September 1998–September 2000 COICOP 5-digit COICOP 7-digit AC Nielsen brand AC Nielsen Universal Product Code Country Province AC Nielsen region Individual outlet 1.080 1.079 1.079 1.089 1.032 1.031 1.030 1.034 1.048 1.048 1.047 1.049 1.015 1.014 1.014 1.011 20.36 What is striking in the Tables 20.1 to 20.3 are the diﬀerences between the Laspeyres and Fisher indices at the ﬁner levels of aggregation.
For the very ﬁnest level of aggregation, the Fisher at 1.011 is 1.7 percentage points below the corresponding Laspeyres at 1.028.

Thus at the ﬁnest level of aggregation, the Laspeyres for this Finnish data set has a representativity or elementary substitution bias of about 0.85 percentage points per year.

20.37 Note that the above index number compar- isons are free of chain drift problems since they make direct comparisons across the two years.
They should also be free of seasonal problems, since the last week in September 1998 is compared with the last week of September 2000.

Elementary indices used in practice 20.38 Suppose that there are M lowest-level items or speciﬁc commodities in a chosen elementary category.

Denote the period t price of item m by pt m for t=0,1 and for items m=1,2, .

.

.

, M. Deﬁne the period t price vector as pt [ pt M] for t=0,1.

2, .

.

.
2, .

.

.

, pt 20.39 The ﬁrst widely used elementary index number formula is attributable to the French economist Dutot (1738): 1, pt PD( p0, p1) M P m=1 M P m=1 1 M 1 M p1 m p0 m = M P m=1 M P m=1 p1 m p0 m (20:1) Thus the Dutot elementary price index is equal to the arithmetic average of the M period 1 prices divided by the arithmetic average of the M period 0 prices.
20.40 The second widely used elementary index number formula is attributable to the Italian economist Carli (1764): PC( p0, p1) P M m=1 1 M p1 m p0 m (20:2) fail this test.
In fact, these last two indices fail the test in the following biased manner: ELEMENTARY INDICES Thus the Carli elementary price index is equal to the arithmetic average of the M item price ratios or price relatives, p1 m=p0 m. 20.41 The third widely used elementary index num- ber formula is attributable to the English economist Jevons (1863): PJ( p0, p1) Q M m=1 s M ﬃﬃﬃﬃﬃﬃ p1 m p0 m (20:3) Thus the Jevons elementary price index is equal to the geometric average of the M item price ratios or price relatives, p1 m=p0 m. 20.42 The fourth elementary index number formula PH is the harmonic average of the M item price rela- tives.
It was ﬁrst suggested in passing as an index number formula by Jevons (1865, p. 121) and Cogge- shall (1887): " M PH( p0, p1) P m=1 # 1 1 M 1 p1 m p0 m (20:4) 20.43 Finally, the ﬁfth elementary index number formula is the geometric average of the Carli and har- monic formulae; i.e., it is the geometric mean of the arithmetic and harmonic means of the M price relatives: PCSWD( p0, p1) p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PC( p0, p1)PH( p0, p1) (20:5) This index number formula was ﬁrst suggested by Fisher (1922, p. 472) as his formula 101.
Fisher also observed that, empirically for his data set, PCSWD was very close to the Jevons index, PJ, and these two indices were his ‘‘best’’ unweighted index number formulae.

In more recent times, Carruthers, Sellwood and Ward (1980, p. 25) and Dale´ n (1992, p. 140) also proposed PCSWD as an elementary index number formula.

20.44 Having deﬁned the most commonly used elementary formulae, the question now arises: which formula is ‘‘best’’?
Obviously, this question cannot be answered until desirable properties for elementary indices are developed.

This will be done in a systematic manner in paragraphs 20.46 to 20.57, but one desirable property for an elementary index will be noted in the present section.

This is the time reversal test, which was noted in Chapter 15.
In the present context, this test for the elementary index P( p0, p1) becomes: P( p0, p1) P( p1, p0)=1 (20:6) This test says that if the prices in period 2 revert to the initial prices of period 0, then the product of the price change going from period 0 to 1, P( p0, p1), times the price change going from period 1 to 2, P( p1, p0), should equal unity; i.e., under the stated conditions, we should end up where we started.
It can be veriﬁed that the Dutot, Jevons, and Carruthers–Sellwood–Ward–Dale´ n indices, PD, PJ and PCSWD, all satisfy the time reversal test, but that the Carli and harmonic indices, PC and PH, PC ( p0, p1) PC( p1, p0) 1 PH ( p0, p1) PH( p1, p0) 1 (20:7) (20:8) with strict inequalities holding in (20.7) and (20.8) pro- vided that the period 1 price vector p1 is not propor- tional to the period 0 price vector p0.24 Thus the Carli index will generally have an upward bias, while the harmonic index will generally have a downward bias.
Fisher (1922, pp.

66 and 383) seems to have been the ﬁrst to establish the upward bias of the Carli index,25 and he made the following observations on its use by statistical agencies: ‘‘In ﬁelds other than index numbers it is often the best form of average to use.

But we shall see that the simple arithmetic average produces one of the very worst of index numbers.
And if this book has no other eﬀect than to lead to the total abandonment of the simple arithmetic type of index number, it will have served a useful purpose’’ (Fisher (1922, pp.

29–30).

20.45 The following section establishes some numer- ical relationships between the ﬁve elementary indices deﬁned in this section.
Then in the subsequent section, a more comprehensive list of desirable properties for elementary indices is developed and the ﬁve elementary formulae are evaluated in the light of these properties or tests.
Numerical relationships between the frequently used elementary indices 20.46 It can be shown26 that the Carli, Jevons and harmonic elementary price indices satisfy the following inequalities: PH( p0, p1) PJ( p0, p1) PC ( p0, p1) (20:9) i.e., the harmonic index is always equal to or less than the Jevons index, which in turn is always equal to or less than the Carli index.
In fact, the strict inequalities in (20.9) will hold provided that the period 0 vector of prices, p0, is not proportional to the period 1 vector of prices, p1.

20.47 The inequalities (20.9) do not tell us by how much the Carli index will exceed the Jevons index and by how much the Jevons index will exceed the harmonic index.
Hence, in the remainder of this section, some approximate relationships between the ﬁve indices 24 These inequalities follow from the fact that a harmonic mean of M positive numbers is always equal to or less than the corresponding arithmetic mean; see Walsh (1901, p. 517) or Fisher (1922, pp.

383– 384).

This inequality is a special case of Schlo¨ milch’s inequality; see Hardy, Littlewood and Po´ lya (1934, p. 26).

25 See also Pigou (1920, pp.

59 and 70), Szulc (1987, p. 12) and Dale´ n (1992, p. 139).
Dale´ n (1994, pp.

150–151) provides some nice intuitive explanations for the upward bias of the Carli index.

26 Each of the three indices PH, PJ and PC is a mean of order r where r equals 1, 0 and 1, respectively, and so the inequalities follow from Schlo¨ milch’s inequality; see Hardy, Littlewood and Po´ lya (1934, p. 26).
361 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE deﬁned in the previous section are developed that pro- vide some practical guidance on the relative magnitudes of each of the indices.
20.48 The ﬁrst approximate relationship to be derived is between the Carli index PC and the Dutot index PD.27 For each period t, deﬁne the arithmetic mean of the M prices pertaining to that period as fol- lows: M pt* P m=1 1 M pt m; t=0, 1 (20:10) Now deﬁne the multiplicative deviation of the mth price in period t relative to the mean price in that period, et m, as follows: m=pt*(1+et pt m); m=1, .

.

.
.

.

, M; t=0, 1 (20:11) Note that equations (20.10) and (20.11) imply that the deviations et m sum to zero in each period; i.e.

M P m=1 et m=0; t=0, 1 (20:12) 20.49 Note that the Dutot index can be written as the ratio of the mean prices, p1*/p0*; i.e.
PD( p0, p1)= p1* p0* (20:13) 20.50 Now substitute equation (20.11) into the deﬁnition of the Jevons index (20.3): PJ( p0, p1)= Q M : M s m=1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p1*(1+e1 m) p0*(1+e0 m) ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (1+e1 m) (1+e0 m) =PD( p0, p1) f (e0, e1) M Q m=1 p1* p0* s = M using equation (20:13) (20:14) where et [et 1, .

.

.
.

.

, et function f is deﬁned as follows: M] for t=0 and 1, and where the PJ ( p0, p1) PD( p0, p1)[1+(1=2M)e0e0 (1=2M)e1e1] =PD( p0, p1)[1+(1=2)var(e0) (1=2)var(e1)] (20:16) where var(et) is the variance of the period t multi- plicative deviations.
Thus, for t=0, 1: var(et) 1 M M P m=1 (et m et* m)2 = = 1 M 1 M M P m=1 etet (et m)2 since et* m=0 using equation ð20:12Þ (20:17) 20.52 Under normal conditions,29 the variance of the deviations of the prices from their means in each period is likely to be approximately constant and so, under these conditions, the Jevons price index will approximate the Dutot price index to the second order.
20.53 Note that with the exception of the Dutot formula, the remaining four elementary indices deﬁned in paragraphs 20.23 to 20.37 are functions of the relative prices of the M items being aggregated.

This fact is used in order to derive some approximate relationships between these four elementary indices.

Thus deﬁne the mth price relative as rm p1 m p0 m ; m=1, .

.

.
.

.

, M (20:18) 20.54 Deﬁne the arithmetic mean of the m price relatives as r* 1 M M P m=1 rm=PC( p0, p1) (20:19) where the last equality follows from the deﬁnition (20.2) of the Carli index.

Finally, deﬁne the deviation em of the mth price relative rm from the arithmetic average of the M price relatives r* as follows:30 rm=r*(1+em); m=1, .

.

.
.

.

, M (20:20) M f (e0, e1) Q m=1 s M ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (1+e1 m) (1+e0 m) (20:15) 20.55 Note that equations (20.19) and (20.20) imply that the deviations em sum to zero: 20.51 Expand f (e0, e1) by a second-order Taylor series approximation around e0=0M and e1=0M.
Using equation (20.12), it can be veriﬁed28 that the following second-order approximate relationship between PJ and PD is obtained: i =p0 i =p0 i=1( p1 j=1p0 j=1p0 i =Pn i =Pn 27 It should be noted that the Dutot index can also be written as a weighted average of the price relatives; i.e., PD( p0, p1) Pn j=1 j =Pn j =Pn i )p0 i=1( p1 p0 i , where the ith weight is deﬁned as wi p0 j .
Thus if the commodities in the elementary aggregate are heterogeneous, the commodities that are more expensive in the chosen units of measurement will get a large weight, which may not be warranted from the viewpoint of expenditures on the com- modity.

28 This approximate relationship was ﬁrst obtained by Carruthers, Sellwood and Ward (1980, p. 25).
i =Pn i=1p1 i )w0 M P m=1 em=0 (20:21) 20.56 Now substitute equation (20.20) into the deﬁnitions (20.2)–(20.5) of PC, PJ, PH and PCSWD in order to obtain the following representations for these the vector of deviations, e: indices in terms of [e1, .

.

.

, eM]: 29 If there are signiﬁcant changes in the overall inﬂation rate, some studies indicate that the variance of deviations of prices from their means can also change.
Also if M is small, then there will be sampling ﬂuctuations in the variances of the prices from period to period.

30 Note that the ratio-type deviations em, deﬁned by equation (20.20), are diﬀerent from the level-type deviations et m, deﬁned by equation (20.11).

362 PC( p0, p1)= P M m=1 1 M rm=r*1 r*fC(e) (20:22) indices, with the degree of divergence growing as the variance of the M price relatives grows.
ELEMENTARY INDICES PJ( p0, p1)= Q M m=1 Mp ﬃﬃﬃﬃﬃrm M =r* Q m=1 Mp ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1+em r*fJ(e) (20:23) PH(p0, p1)= P M 1 M m=1 r*fH(e) (rm) 1 1 M =r* P m=1 1 M 1 (1+em) 1 (20:24) (20:25) PCSWP( p0, p1)= p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PC( p0, p1)PH( p0, p1) p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ fC(e)fH(e) r*fCSWD(e) =r* where the last identity in each of equations (20.22)– (20.25) serves to deﬁne the deviation functions, fC(e), fJ(e), fH(e) and fCSWD(e).
The second-order Taylor these functions31 series approximations to each of around the point e=0M are: fC(e) 1 (20:26) fJ (e) 1 (1=2M)ee=1 (1=2)var(e) (20:27) fH(e) 1 (1=M)ee=1 var(e) (20:28) fCSWD(e) 1 (1=2M)ee=1 (1=2)var(e) (20:29) the Carli where repeated use of equation (20.21) is made in deriving the above approximations.32 To the second index PC will exceed the Jevons order, and Carruthers–Sellwood–Ward–Dale´ n indices, PJ and PCSWD, by (1/2) r* var(e), which is r* times half the variance of the M price relatives p1 m. Similarly, to the second order, the harmonic index PH will lie below the Jevons and Carruthers–Sellwood–Ward–Dale´ n indices, PJ and PCSWD, by r* times half the variance of the M price relatives p1 m=p0 m=p0 m. 20.57 Empirically, it is expected that the Jevons and Carruthers–Sellwood–Ward–Dale´ n indices will be very close to each other.
Using the previous approximation result (20.16), it is expected that the Dutot index PD will also be fairly close to PJ and PCSWD, with some ﬂuc- tuations over time as a result of changing variances of the period 0 and 1 deviation vectors, e0 and e1.

Thus it is expected that these three elementary indices will give much the same numerical answers in empirical applica- tions.
In contrast, the Carli index can be expected to be substantially above these three indices, with the degree of divergence growing as the variance of the M price rela- tives grows.

Similarly, the harmonic index can be expected to be substantially below the three middle 31 From equation (20.22), it can be seen that fC(e) is identically equal to 1 so that the expression (20.26) will be an exact equality rather than an approximation.
32 These second-order approximations are attributable to Dale´ n (1992, p. 143) for the case r*=1 and to Diewert (1995a, p. 29) for the case of a general r*.

The axiomatic approach to elementary indices 20.58 Recall the axiomatic approach to bilateral price indices P( p0, p1, q0, q1) developed in Chapter 16.
In the present chapter, the elementary price index P( p0, p1) depends only on the period 0 and 1 price vectors, p0 and p1, respectively, so that the elementary price index does not depend on the period 0 and 1 quantity vectors, q0 and q1.
One approach to obtaining new tests or axioms for an elementary index is to look at the 20 or so axioms listed in the Fisher axiomatic approach in Chapter 16 for bilateral price indices P( p0, p1, q0, q1) and adapt those axioms to the present context; i.e., use the old bilateral tests for P( p0, p1, q0, q1) that do not depend on the quantity vec- tors q0 and q1 as tests for an elementary index P( p0, p1).33 This is the approach taken in the present section.
20.59 The ﬁrst eight tests or axioms are reasonably straightforward and uncontroversial.

T1: Continuity: P( p0, p1) is a continuous function of the M positive period 0 prices p0 [ p0 1, .

.

.

, p0 M] and the M positive period 1 prices p1 [ p1 1, .

.

.

, p1 M].

Identity: P( p, p)=1; i.e., if the period 0 price T2: vector equals the period 1 price vector, then the index is equal to unity.
T3: Monotonicity in current period prices: P( p0, p1)< P( p0, p) if p1<p; i.e., if any period 1 price increases, then the price index increases.

T4: Monotonicity in base period prices: P( p0, p1)> P( p, p1) if p0<p; i.e., if any period 0 price increases, then the price index decreases.
T5: Proportionality in current period prices: P( p0, lp1)=lP( p0, p1) if l>0; i.e., if all period 1 prices are multiplied by the positive number l, then the initial price index is also multiplied by l. T6: Inverse proportionality in base period prices: P(lp0, p1)=l1 P( p0, p1) if l>0; i.e., if all period 0 prices are multiplied by the positive number l, then the initial price index is multiplied by 1/l.

T7: Mean value test: minm f p1 m: m=1, .

.

.


, Mg Pðp0, p1Þ maxm f p1 the price index lies between the smallest and largest price relatives.

treatment of outlets: P( p0, p1)= T8: Symmetric P( p0*, p1*), where p0* and p1* denote the same per- mutation of the components of p0 and p1; i.e., if we change the ordering of the outlets (or households) from which we obtain the price quotations for the two pe- riods, then the elementary index remains unchanged.
Eichhorn (1978, p. 155) showed that tests T1, T2, T3 and T5 imply test T7, so that not all of the above tests are logically independent.

m=p0 m:m=1, .

.

.

, Mg; m=p0 i.e., 20.60 The following tests are more controversial and are not necessarily accepted by all price statisticians.

T9: The price bouncing test: P( p0, p1)=P( p0*, p1**) where p0* and p1** denote possibly diﬀerent permu- 33 The approach was used by Diewert (1995a, pp.

5–17), who drew on the earlier work of Eichhorn (1978, pp.
152–160) and Dale´ n (1992).

363 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE tations of the components of p0 and p1; i.e., if the ordering of the price quotes for both periods is changed in possibly diﬀerent ways, then the elemen- tary index remains unchanged.

20.61 Obviously, test T8 is a special case of test T9 where the two permutations of the initial ordering of the prices are restricted to be the same.

Thus test T9 implies test T8.

Test T9 is attributable to Dale´ n (1992, p. 138).


He justiﬁed this test by suggesting that the price index should remain unchanged if outlet prices ‘‘bounce’’ in such a manner that the outlets are just exchanging prices with each other over the two periods.

While this test has some intuitive appeal, it is not consistent with the idea that outlet prices should be matched to each other in a one-to- one manner across the two periods.
This outlet price matching is preferable to not matching prices across outlets in case there are quality diﬀerences across outlets.

20.62 The following test was also proposed by Dale´ n (1992) in the elementary index context: T10: Time reversal: P( p1, p0)=1/P( p0, p1); i.e., if the data for periods 0 and 1 are interchanged, then the resulting price index should equal the reciprocal of the original price index.
Since many price statisticians approve of the Laspeyres price index in the bilateral index context and this index does not satisfy the time reversal test, it is obvious that not all price statisticians would regard the time reversal test in the elementary index context as being a funda- mental test that must be satisﬁed.
Nevertheless, many other price statisticians do regard this test as a funda- mental one since it is diﬃcult to accept an index that gives a diﬀerent answer if the ordering of time is reversed.

20.63 The following test is a strengthening of the time reversal test: T11: Circularity: P( p0, p1)P( p1, p2)=P( p0, p2); i.e., the price index going from period 0 to 1 times the price index going from period 1 to 2 equals the price index going from period 0 to 2 directly.
The circularity and identity tests imply the time reversal test (just set p2=p0).

Thus the circularity test is essen- tially a strengthening of the time reversal test, and so price statisticians who do not accept the time reversal test are unlikely to accept the circularity test.

In general, however, the circularity test seems to be a very desirable property: it is a generalization of a property that holds for a single price relative.

M)=P( p0 20.64 The following test is a very important one: 1, .

.

.
.

.

, lMp0 T12: Commensurability: P(l1p0 1, .

.

.

, lMp1 M)=P( p0, p1) for all 1, .

.

.

, p1 1, .

.

.

, p0 l1>0, .

.

.

,lM>0; i.e., if the units of measurement for each commodity are changed, then the elementary index remains unchanged.

M; l1p1 M; p1 In the bilateral index context, virtually every price stat- istician accepts the validity of this test.

In the elementary context, however, this test is more controversial.
If the M items in the elementary aggregate are all homo- geneous, then it makes sense to measure all the items in the same units.

Hence, if the unit of measurement of the homogeneous commodity is changed, then a modiﬁed version of test T12 should restrict all the lm to be the same number (say l) and the modiﬁed test T12 becomes 364 P(lp0, lp1)=P( p0, p1); l > 0 (20:30) Note that this modiﬁed test T12 will be satisﬁed if tests T5 and T6 are satisﬁed.
Thus if the items in the ele- mentary aggregate are homogeneous, then there is no need for the original (unmodiﬁed) test T12.

20.65 In actual practice, there will usually be thou- sands of individual items in each elementary aggregate and the hypothesis of item homogeneity is not warranted.
Under these circumstances, it is important that the ele- mentary index satisfy the commensurability test, since the units of measurement of the heterogeneous items in the elementary aggregate are arbitrary, and hence the price statistician can change the index simply by changing the units of measurement for some of the items.

20.66 This completes the listing of the tests for an elementary index.
There remains the task of evaluating how many tests are passed by each of the ﬁve elementary indices deﬁned in paragraphs 20.38 to 20.45.

20.67 Straightforward computations show that the Jevons elementary index PJ satisﬁes all the tests, and hence emerges as being ‘‘best’’ from the viewpoint of this particular axiomatic approach to elementary indices.

20.68 The Dutot index PD satisﬁes all the tests with the important exception of the commensurability test T12, which it fails.
If there are heterogeneous items in the elementary aggregate, this is a rather serious failure and hence price statisticians should be careful in using this index under these conditions.

20.69 The geometric mean of the Carli and harmo- nic elementary indices, PCSWD, fails only the price bouncing test T9 and the circularity test T11.
The failure of these two tests is probably not a disqualifying con- dition, and so this index could be used by price statis- ticians if, for some reason, it was decided not to use the Jevons formula.

As was observed in paragraphs 20.38 to 20.45, numerically, PCSWD will be very close to PJ.

20.70 The Carli and harmonic elementary indices, PC and PH, fail the price bouncing test T9, the time reversal test T10 and the circularity test T11, and pass the other tests.
The failure of tests T9 and T11 is again not a disqualifying condition, but the failure of the time reversal test T10 is a rather serious matter and so price statisticians should be cautious in using these indices.

The economic approach to elementary indices 20.71 Recall the notation and discussion in para- graphs 20.38 to 20.45.

Suppose that each purchaser of the items in the elementary aggregate has preferences over a vector of purchases q:[q1, .

.

.
.

.

, qM] that can be represented by the linearly homogeneous aggregator (or utility) function f (q).

Further assume that each pur- chaser engages in cost-minimizing behaviour in each period.

Then, as seen in Chapter 17, it can be shown that certain speciﬁc functional forms for the aggregator or utility function f (q) or its dual unit cost function c( p)34 lead to speciﬁc functional forms for the price index 34 The unit cost f (q)=1g.
function is deﬁned as c( p) minq PM m=1pmqm: P( p0, p1, q0, q1) with P( p0, p1, q0, q1) c( p1) c( p0) (20:31) 20.72 Suppose that the purchasers have aggregator functions f deﬁned as follows:35 f (q1, .

.

.

, qM) minm fqm=am: m=1, .

.

.

, Mg (20:32) where the am are positive constants.
Then under these it can be shown that equation (20.31) assumptions, becomes:36 c( p1) c( p0) p1q0 p0q0 = and the quantity vectors of purchases during the two periods must be proportional; i.e., p1q1 p0q1 (20:33) = q1=lq0 for some l > 0 (20:34) 20.73 From the ﬁrst equation in (20.33), it can be seen that the true cost of living index, c( p1)/c( p0), under assumptions (20.32) about the aggregator function f, is equal to the Laspeyres price index, PL( p0, p1, q0, q1): p1q0/p0q0.
It is shown below how various elementary formulae can estimate this Laspeyres formula under alternative assumptions about the sampling of prices.

20.74 In order to provide a justiﬁcation for the use of the Dutot elementary formula, write the Laspeyres index number formula as follows: PL( p0, p1, q0, q1) M P m=1 M P m=1 mq0 p1 m mq0 p0 m = M P m=1 M P m=1 m p1 r0 m m p0 r0 m (20:35) where the base period item probabilities r0 follows: m are deﬁned as r0 m q0 m M P m=1 q0 m ; m=1, .

.

.
.

.

, M (20:36) Thus the base period probability for item m, r0 m, is equal to the purchases of item m in the base period relative to total purchases of all items in the commodity class in the base period.
Note that these deﬁnitions require that all items in the commodity class have the same units.37 20.75 Now it is easy to see how formula (20.35) could be turned into a rigorous sampling framework for sampling prices in the particular commodity class under consideration.38 If item prices in the commodity class were sampled proportionally to their base period prob- abilities r0 m, then the Laspeyres index deﬁned by the ﬁrst equality in (20.35) could be estimated by a probability- weighted Dutot index deﬁned by the second equality in 35 The preferences which correspond to this f are known as Leontief (1936) or no substitution preferences.
36 See Pollak (1983).

Notation: p1q0 is deﬁned as Pn 37 The probabilities deﬁned by equation (20.36) are meaningless unless the items are homogeneous.

38 For the details, see Balk (2002, pp.

8–10).

i , etc.

i=1p1 i q0 ELEMENTARY INDICES (20.35).

In general, with an appropriate sampling scheme, the use of the Dutot formula at the elementary level of aggregation for homogeneous items can be per- fectly consistent with a Laspeyres index concept.
20.76 The Dutot formula can also be consistent with a Paasche index concept.

If the Paasche formula is used at the elementary level of aggregation, then the follow- ing formula is obtained: PP( p0, p1, q0, q1) M P m=1 M P m=1 mq1 p1 m mq1 p0 m = M P m=1 M P m=1 m p1 r1 m m p0 r1 m (20:37) where the period 1 item probabilities r1 follows: m are deﬁned as r1 m q1 m M P m=1 q1 m ; m=1, .

.

.
.

.

, M (20:38) Thus the period 1 probability for item m, r1 m, is equal to the quantity purchased of item m in period 1 relative to total purchases of all items in the commodity class in that period.

20.77 Again, it is easy to see how formula (20.37) could be turned into a rigorous sampling framework for sampling prices in the particular commodity class under consideration.
If item prices in the commodity class were sampled proportionally to their period 1 probabilities r1 m, then the Paasche index deﬁned by the ﬁrst equality in (20.37) could be estimated by the probability-weighted Dutot index deﬁned by the second equality in (20.37).

In general, with an appropriate sampling scheme, the use of the Dutot formula at the elementary level of aggregation (for a homogeneous elementary aggregate) can be per- fectly consistent with a Paasche index concept.
20.78 Rather than use the ﬁxed basket representa- tions for the Laspeyres and Paasche indices, it is possible to use the expenditure share representations for the Laspeyres and Paasche indices, and to use the expendi- ture shares s0 m as probability weights for price relatives.
Thus if the relative prices of items in the commodity class under consideration are sampled using weights that are proportional to their base period expenditure shares in the commodity class, then the following probability-weighted Carli index m or s1 PC( p0, p1, s0) P M m=1 s0 m p1 m p0 m (20:39) 20.79 will be equal to the Laspeyres index.39 Of course, for- mula (20.39) does not require the assumption of homogeneous items as did equations (20.35) and (20.37).
If the relative prices of items in the commodity class under consideration are sampled using weights that are proportional to their period 1 expenditure shares in the commodity class, then the following probability- weighted harmonic index 39 For a rigorous derivation of a sampling framework, see Balk (2002, pp.

13–14).

365 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE PH( p0, p1, s1) P M m=1 !

 1 s1 m 1 p1 m p0 m (20:40) will be equal to the Paasche index.
20.80 The above results show that the Dutot ele- mentary index can be justiﬁed as an approximation to an underlying Laspeyres or Paasche price index for a homogeneous elementary aggregate under appropriate price sampling schemes.

The above results also show that the Carli and harmonic elementary indices can be justiﬁed as approximations to an underlying Laspeyres or Paasche price index for a heterogeneous elementary aggregate under appropriate price sampling schemes.
20.81 Recall that assumption (20.32) on f justiﬁed the Laspeyres and Paasche indices as being the ‘‘true’’ elementary aggregate from the viewpoint of the eco- nomic approach to elementary indices.

Suppose now that assumption (20.32) is replaced by the following assumption of Cobb–Douglas (1928) preferences:40 f (q1, .

.

.

, qM) Q M m=1 M and P m=1 bm=1 qbm; bm > 0 for m=1, .

.

.
.

.

, M (20:41) 20.82 Under assumption (20.41), the true economic elementary price index is:41 c( p1) c( p0) M = Q m=1 bm p1 m p0 m (20:42) 20.83 It turns out that if purchasers have the above Cobb–Douglas preferences, then item expenditures will be proportional over the two periods so that: mq1 p1 m=lp0 mq0 m for m=1, .

.

.
.

.

, M and for some l > 0: (20:43) Under these conditions, the base period expenditure shares s0 m will equal the corresponding period 1 expen- diture shares s1 m, as well as the corresponding bm; i.e., assumption (20.41) implies: m=s1 s0 m bm; m=1, .

.

.
, M (20:44) Thus if the relative prices of items in the commodity class under consideration are sampled using weights that are proportional to their base period expenditure shares in the commodity class, then the following probability- weighted Jevons index ln PJ( p0, p1, s0) P M s0 m ln m=1 p1 m p0 m (20:45) will be equal to the logarithm of the true elementary price aggregate deﬁned by equation (20.42).42 20.84 The above results show that the Jevons ele- mentary index can be justiﬁed as an approximation to 40 These preferences were introduced slightly earlier by Konu¨ s and Byushgens (1926).
41 See Pollak (1983).

42 See Balk (2002, pp.

11–12) for a rigorous derivation.

366 an underlying Cobb–Douglas price index for a hetero- geneous elementary aggregate under an appropriate price sampling scheme.

20.85 The assumption of Leontief preferences implies that the quantity vectors pertaining to the two periods under consideration will be proportional; recall equation (20.34).
In contrast, the assumption of Cobb– Douglas preferences implies that expenditures will be proportional over the two periods; recall equation (20.43).

Index number theorists have been debating the relative merits of the proportional quantities versus proportional expenditures assumption for a long time.

Authors who thought that the proportional expendi- tures assumption was empirically more likely include Jevons (1865, p. 295) and Ferger (1931, p. 39; 1936, p. 271).
These early authors did not have the economic approach to index number theory at their disposal but they intuitively understood, along with Pierson (1895, p. 332), that substitution eﬀects occurred and hence the proportional expenditures assumption was more plau- sible than the proportional quantities assumption.
20.86 The results in the previous section gave some support for the use of the unweighted Jevons elemen- tary index over the use of the unweighted Dutot, Carli and harmonic indices, provided that the propor- tional expenditures assumption is more likely than the proportional quantities assumption.

This support is very weak, however, since an appropriate item price sampling scheme is required in order to justify the results.
Thus, using an unweighted Dutot, Carli or harmonic index (without the appropriate sampling scheme) cannot really be justiﬁed from the viewpoint of the economic approach.
The results in this section nevertheless give considerable support to the use of an appropriately weighted Jevons index over the other weighted indices, since from the economic per- spective, cross-item elasticities of substitution are much more likely to be close to unity (this corres- ponds to the case of Cobb–Douglas preferences) than to zero (this corresponds to the case of Leontief pre- ferences).
If the probability weights in the weighted Jevons index are taken to be the arithmetic average of the period 0 and 1 item expenditure shares and narrowly deﬁned unit values are used as the price concept, then the weighted Jevons index becomes an ideal type of elementary index discussed in paragraphs 20.11 to 20.22.
The sampling approach to elementary indices it 20.87 In the previous section, is shown that appropriately weighted elementary indices are capable of approximating various economic population ele- mentary indices, with the approximation becoming exact as the sampling approaches complete coverage.
Conversely, it can be seen that, in general, it is impos- sible for an unweighted elementary price index of the type deﬁned in paragraphs 20.38 to 20.45 to approach the theoretically ideal elementary price index deﬁned in paragraphs 20.11 to 20.22, even if all item prices in the elementary aggregate are sampled.43 Hence, rather than just sampling prices, it will be necessary for the price statistician to collect information on the transaction values (or quantities) associated with the sampled prices in order to form sample elementary aggregates that will approach the target ideal elementary aggregate as the sample size becomes large.
Thus, instead of just col- lecting a sample of prices, it will be necessary to collect corresponding sample quantities (or values) so that a sample Fisher, To¨ rnqvist or Walsh price index can be constructed.

This sample-based superlative elementary price index will approach the population ideal elemen- tary index as the sample size becomes large.

This approach to the construction of elementary indices in a sampling context was recommended by Pigou (1920, pp.
66–67), Fisher (1922, p. 380), Diewert (1995a, p. 25) and Balk (2002).44 In particular, Pigou (1920, p. 67) sug- gested that the sample-based Fisher ideal price index be used to deﬂate the value ratio for the aggregate under consideration in order to obtain an estimate of the quantity ratio for the aggregate under consideration.
The use of scanner data in constructing elementary aggregates 20.88 Until fairly recently, it was not possible to determine how close an unweighted elementary index of the type deﬁned in paragraphs 20.38 to 20.45 was to an ideal elementary aggregate.
With the availability of scanner data (i.e., of detailed data on the prices and quantities of individual items that are sold in retail outlets), it has now become possible to compute ideal elementary aggregates for some item strata and compare the results with statistical agency estimates of price change for the same class of items.

Of course, the sta- tistical agency estimates of price change are usually based on the use of the Dutot, Jevons or Carli formulae.
The following quotations reﬂect the results of many of the scanner data studies: A second major recent development is the willingness of statistical agencies to experiment with scanner data, which are the electronic data generated at the point of sale by the retail outlet and generally include transactions prices, quantities, location, date and time of purchase and the product described by brand, make or model.
Such detailed data may prove especially useful for constructing better indices at the elementary level.

Recent studies that use scanner data in this way include Silver (1995), Reinsdorf (1996), Bradley, Cook, Leaver and Moulton (1997), Dale´ n (1997), de Haan and Opperdoes (1997) and Hawkes (1997).
Some estimates of elementary index bias (on an annual basis) that emerged from these studies were: 1.1 percentage points for television sets in the United King- dom; 4.5 percentage points for coﬀee in the United States; 1.5 percentage points for ketchup, toilet tissue, milk and tuna in the United States; 1 percentage point for fats, detergents, breakfast cereals and frozen ﬁsh in Sweden; 1 43 The numerical example given in paragraphs 20.91 to 20.99 illustrates this point.
44 Balk (2002) provides the details for this sampling framework.

Hausman (2002) is another recent author who stressed the importance of collecting quantity information along with price information at the elementary level.

ELEMENTARY INDICES percentage point for coﬀee in the Netherlands and 3 percentage points for coﬀee in the United States respec- tively.
These bias estimates incorporate both elementary and outlet substitution biases and are signiﬁcantly higher than our earlier ballpark estimates of .255 and .41 per- centage points.

On the other hand, it is unclear to what extent these large bias estimates can be generalized to other commodities (Diewert (1998a, pp.

54–55)).

Before considering the results it is worth commenting on some general ﬁndings from scanner data.
It is stressed that the results here are for an experiment in which the same data were used to compare diﬀerent methods.

The results for the U.K. Retail Prices Index can not be fairly compared since they are based on quite diﬀerent practices and data, their data being collected by price collectors and having strengths as well as weaknesses (Fenwick, Ball, Silver and Morgan (2003)).
Yet it is worth following up on Diewert’s (2002c) comment on the U.K. Retail Prices Index electrical appliances section, which includes a wide variety of appliances, such as irons, toasters, refrigerators, etc.

which went from 98.6 to 98.0, a drop of 0.6 percentage points from January 1998 to December 1998.

He compares these results with those for washing machines and notes that ‘‘ .

.

.
.

.

it may be that the non washing machine components of the electrical appliances index increased in price enough over this period to cancel out the large apparent drop in the price of washing machines but I think that this is somewhat unlikely.’’ A number of studies on similar such products have been conducted using scanner data for this period.
Chained Fisher indices have been calculated from the scanner data, (the RPI (within year) indices are ﬁxed base Laspeyres ones), and have been found to fall by about 12 per cent for televisions (Silver and Heravi, 2001a), 10 per cent for washing machines (Table 7 below), 7.5 per cent for dishwashers, 15 per cent for cameras and 5 per cent for vacuum cleaners (Silver and Heravi, 2001b).
These results are quite diﬀerent from those for the RPI section and suggest that the washing machine disparity, as Diewert notes, may not be an anomaly.

Traditional methods and data sources seem to be giving much higher rates for the CPI than those from scanner data, though the reasons for these discrepancies were not the subject of this study (Silver and Heravi (2002, p. 25)).
20.89 The above quotations summarize the results of many elementary aggregate index number studies that are based on the use of scanner data.
These studies indicate that when detailed price and quantity data are used in order to compute superlative indices or hedonic indices for an expenditure category, the resulting mea- sures of price change are often below the corresponding oﬃcial statistical agency estimates of price change for that category.45 Sometimes the measures of price change based on the use of scanner data are considerably below the corresponding oﬃcial measures.46 These results 45 Recall also the results obtained by Koskima¨ ki and Yla¨ -Jarkko (2003) that showed the Laspeyres index considerably above the cor- responding Fisher index using Finnish scanner data.
46 Scanner data studies do not, however, always show large potential biases in oﬃcial CPIs.

Masato Okamoto has informed us that a large- scale comparative study in Japan is under way.

Using scanner data for about 250 categories of processed food and daily necessities collected over the period 1997 to 2000, it was found that the indices based on scanner data averaged only about 0.2 percentage points below the corresponding oﬃcial indices per year.
Japan uses the Dutot formula at the elementary level in its oﬃcial CPI.

367 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE indicate that there may be large gains in the precision of elementary indices if a weighted sampling framework is adopted.

20.90 Is there a simple intuitive explanation for the above empirical results?

A partial explanation may be possible by looking at the dynamics of item demand.
In any market economy, there are ﬁrms and outlets that sell items that are either declining or increasing in price.

Usually, the items that decline in price experience an increase in their volume of sales.

Thus the expenditure shares that are associated with items that are declining in price usually increase, and conversely for the items that increase in price.
Unfortunately, elementary indices cannot pick up the eﬀects of this negative correlation between price changes and the induced changes in expenditure shares, because elementary indices depend only on prices and not on expenditure shares.

20.91 An example can illustrate the above point.

Suppose that there are only three items in the elemen- tary aggregate and that in period 0, the price of each item is p0 m=1 and the expenditure share for each item is equal so that s0 m=1=3 for m=1, 2, 3.
Suppose that in period 1, the price of item 1 increases to p1 1=1+i, the price of item 2 remains constant at p1 2=1 and the price of 3=ð1+iÞ 1 where the item 1 rate of item 3 decreases to p1 increase in price is i>0.

Suppose further that the expenditure share of item 1 decreases to s1 1=ð1=3Þ s where s is a small number between 0 and 1/3 and the 3=ð1=3Þ+s.47 expenditure share of item 3 increases to s1 expenditure share of item 2 remains constant at s1 2=1=3.
The ﬁve elementary indices deﬁned in para- graphs 20.23 to 20.37 can all be written as functions of the item 1 inﬂation rate i (which is also the item 3 deﬂation rate) as follows: The PJ( p0, p1)=[(1 i)(1+i) 1]1=3=1 fJ(i) (20:46) PC( p0, p1)=(1=3)(1+i)+(1=3)+(1=3)(1+i) 1 fC(i) (20:47) PH( p0, p1)=[(1=3)(1+i) 1+(1=3)+(1=3)(1+i)] 1 fH(i) (20:48) PCSW ( p0, p1)=[PC( p0, p1)PH( p0, p1)]1=2 fCSW (i) (20:49) 47 The parameter s is a measure of the degree of substitutability between the various items in the elementary aggregate.
It is not pre- cisely equal to the elasticity of substitution parameter s which appeared in the Lloyd–Moulton formula explained in paragraphs 17.61 to 17.64 of Chapter 17.

However, the bigger is the elasticity of substitution, the bigger will be the s parameter which appears in this section.
David E. Lebow and Jeremy B. Rudd note that the marketing literature ﬁnds that the elasticity of substitution between brands in a narrowly deﬁned elementary aggregate is around 2.5 (which is much higher than the Cobb–Douglas case where the elasticity of substitution is 1): ‘‘And, Gerard Tellis (1988) analyzed the results from a large number of papers in the marketing literature that estimate cross brand elasticities and found a mean elasticity (after adjusting for certain biases in the results) of 2.5’’ (Lebow and Rudd (2003, pp.
167–168).

368 PD( p0, p1)=(1=3)(1+i)+(1=3)+(1=3)(1+i) 1 fD(i) (20:50) 20.92 Note that in this particular example, the Dutot index fD(i) turns out to equal the Carli index fC(i).
The second-order Taylor series approximations to the ﬁve elementary indices (20.46)–(20.50) are given by the approximations (20.51)–(20.55): fJ(i) 1 (20:51) fC(i) 1+(1=3)i 2 fH(i) 1 (1=3)i 2 fCSW (i) 1 (20:52) (20:53) (20:54) (20:55) fD(i) 1+(1=3)i 2 Thus for small i, the Carli and Dutot indices will be slightly greater than 1,48 the Jevons and the Carruthers– Sellwood–Ward indices will be approximately equal to 1 and the harmonic index will be slightly less than 1.
Note that the ﬁrst-order Taylor series approximation to all ﬁve indices is 1.

Thus, to the accuracy of a ﬁrst-order approximation, all ﬁve indices equal unity.
20.93 Now calculate the Laspeyres, Paasche and Fisher indices for the elementary aggregate: PL=(1=3)(1+i)+(1=3)+(1=3)(1+i) 1 fL(i) (20:56) PP=f[(1=3) s](1+i)+(1=3)+[(1=3)+s](1+i) 1g 1 fP(i) (20:57) PF =(PLPP)1=2 fF (i) 20.94 First-order Taylor series approximations to the above indices (20.56)–(20.58) around i=0 are given by the approximations (20.59)–(20.61): (20:58) fL(i) 1 fP(i) 1 2si fF (i) 1 si (20:59) (20:60) (20:61) 20.95 An ideal elementary index for the three items index fF(i).
The approximations is the Fisher ideal (20.51)–(20.55) and (20.61) show that the Fisher index will lie below all ﬁve elementary indices by the amount si, taking ﬁrst-order approximations to all six indices.

Thus all ﬁve elementary indices will have an approx- imate upward bias equal to si compared to an ideal elementary aggregate.
20.96 Suppose that the annual item inﬂation rate for the item rising in price is equal to 10 per cent so that i=0.10 (and hence the rate of price decrease for the item 48 Recall the approximate relationship (20.16) in paragraph 20.51 between the Dutot and Jevons indices.

In the present numerical exam- ple, var(e0)=0 whereas var(e1)>0.

This explains why the Dutot index is not approximately equal to the Jevons index in this numerical example.

decreasing in price is approximately 10 per cent as well).
If the expenditure share of the increasing price item declines by 5 percentage points, then s=0.05 and the annual approximate upward bias in all ﬁve elementary indices is si=0.05 0.10=0.005 or half of a percentage point.

If i increases to 20 per cent and s increases to 10 per cent, then the approximate bias increases to si= 0.10 0.20=0.02 or 2 per cent.

Note, however, if prices in period 2 revert to the prices prevailing in period 0, then the bias will reverse itself.
Hence elementary bias of the type modelled above can only cumulate over suc- cessive periods if there are long-run trends in prices and market shares.49 20.97 The above example is highly simpliﬁed.

More sophisticated models are capable of explaining at least some of the discrepancy between oﬃcial elementary indices and superlative indices calculated by using scan- ner data for an expenditure class.
Basically, elementary indices deﬁned without using associated quantity or value weights are incapable of picking up shifts in expen- diture shares that are induced by ﬂuctuations in item prices.50 In order to eliminate the problem of an inability to pick up shifts in expenditure shares that are induced by ﬂuctuations in item prices, it will be necessary to sample values along with prices in both the base and comparison periods.

20.98 A few words of caution are, however, in order at this point.
The use of chained superlative indices can lead to very biased results if there are large period-to- period ﬂuctuations in prices and quantities compared to longer-run trends in prices.

In long runs, large ﬂuctua- tions can be induced by seasonal factors51 or by tem- porary sales.52 49 White’s (2000) research into Canadian outlet substitution bias indicated that not only did discount outlets have lower prices for the same items, but they also had lower inﬂation rates over time.
50 Put another way, elementary indices are subject to substitution or representativity bias.

In the case of Cobb–Douglas preferences, how- ever, the parameter s in this section would be equal to zero and the Jevons elementary aggregate would be unbiased.

But the results from the marketing literature (recall Tellis (1988)) indicate that s will be greater than zero and hence that the Jevons elementary index will have an upward bias.
Thus Lebow and Rudd’s (2003, p. 167) estimate that elementary substitution bias is only around 0.05 percentage points per year if the Jevons formula is used seems rather low.

51 For an example where the use of chained superlative indices leads to a tremendous downward bias induced by seasonal ﬂuctuations, see Chapter 22.
52 For an example where the use of chained superlative indices leads to a tremendous upward bias induced by periodic sales, see Robert C. Feenstra and Matthew D. Shapiro (2003): The reason for this is that periods of low prices (i.e., sales) attract high purchases only when they are accompanied by advertising, and this tends to occur in the ﬁnal weeks of a sale.
Thus, the initial price decline, when the sale starts, does not receive as much weight in the cumulative index as the ﬁnal price increase when the sale ends.

The demand behavior that leads to this upward bias of the chained To¨ rnqvist – with higher purchases at the end of a sale – means that consumers are very likely purchasing goods for inventory accumulation.
The only theoretically correct index to use in this type of situation is a ﬁxed base index, as demonstrated in section 5.3 (Feenstra and Shapiro (2003, p. 125)).

The use of a ﬁxed base index in these circumstances may, however, lead to results that are highly dependent on the choice of the base period.
Other solutions that could be tried in this type of circumstance are either lengthening the period of time (as discussed in paragraphs 20.23 to 20.37) or using the moving year idea explained in Chapter 22 below.

ELEMENTARY INDICES 20.99 In the following section, a simple regression- based approach to the construction of elementary indi- ces is outlined.

The importance of weighting the price quotes will again emerge from the analysis.
A simple stochastic approach to elementary indices 20.100 Recall the notation used in paragraphs 20.38 to 20.45 above.

Suppose the prices of the M items for period 0 and 1 are approximately equal to the right- hand sides of equations (20.62) and (20.63): p0 m bm; m=1, .

.

.

, M (20:62) (20:63) p1 m abm; m=1, .

.

.

, M where a and the bm are positive parameters.


Note that there are 2M prices on the left-hand sides of equations (20.62) and (20.63), but only M+1 parameters on the right-hand sides of these equations.

The basic hypothesis in the model of price behaviour deﬁned by equations (20.62) and (20.63) is that the two price vectors p0 and p1 are proportional (with p1=ap0 so that a is the factor of proportionality) random multiplicative for errors.
Hence a represents the value of the underlying elementary price aggregate.

Taking logarithms of both sides of equations (20.62) and (20.63), and adding some random errors e0 m to the right-hand sides of the resulting equations, the following linear regression model is obtained: m and e1 except ln p0 m=dm+e0 m; m=1, .

.

.

, M ln p1 m=g+dm+e1 m; m=1, .

.

.

, M (20:64) (20:65) where g ln a and dm ln m; m=1, .

.

.
.

.

, M (20:66) 20.101 Note that equations (20.64) and (20.65) can be interpreted as a highly simpliﬁed hedonic regression model.53 The only characteristic of each commodity is the commodity itself.

This model is also a special case of the country product dummy method for making interna- tional comparisons between the prices of diﬀerent coun- tries.54 A major advantage of this regression method for constructing an elementary price index is that standard errors for the index number a can be obtained.
This advantage of the stochastic approach to index number theory was stressed by Selvanathan and Rao (1994).

20.102 It can be veriﬁed that the least squares esti- mator for g is: M g* P m=1 1 M ln p1 m p0 m (20:67) 53 See Chapters 7, 8 and 21 for discussion of hedonic regression models.

54 See Summers (1973).

In our special case, there are only two ‘‘countries’’ which are the two observations on the prices of the ele- mentary aggregate for two periods.
369 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE 20.103 If g* is exponentiated, then the following M s PJ( p0, p1) M a* Q m=1 estimator for the elementary aggregate a is obtained: ﬃﬃﬃﬃﬃﬃ p1 m p0 m where PJ ( p0, p1) is the Jevons elementary price index deﬁned in paragraphs 20.38 to 20.45 above.

Thus the simple regression model deﬁned by equations (20.64) and (20.65) leads to a justiﬁcation for the use of the Jevons elementary index.
(20:68) 20.104 Consider the following unweighted least squares model: min g, d’s M P m=1 ( ln p0 m dm)2+ P M m=1 ( ln p1 m g dm)2 (20:69) It can be veriﬁed that the g solution to the unconstrained minimization problem (20.69) is the g* deﬁned by equa- tion (20.67).
20.105 There is a problem with the unweighted least squares model deﬁned by equation (20.69), namely, that the logarithm of each price quote is given exactly the same weight in the model no matter what the expendi- ture on that item was in each period.

This is obviously unsatisfactory since a price that has very little economic importance (i.e., a low expenditure share in each period) is given the same weight in the regression model as a very important item.
Thus it is useful to consider the following weighted least squares model:55 min g, d’s m( ln p0 s0 m dm)2+ P M P m=1 M m=1 m( ln p1 s1 m g dm)2 (20:70) where the period t expenditure share on commodity m is deﬁned in the usual manner as: mqt pt m M P m=1 t=0, 1; m=1, .

.

.

, M (20:71) st m mqt pt m ; In the model (20.70), the logarithm of each item price quotation in each period is weighted by its expenditure share in that period.
Note that weighting prices by their economic importance is consistent with Theil’s (1967, pp.

136 –138) stochastic approach to index number theory.56 20.106 The g solution to the minimization problem (20.70) is p1 m p0 m (20:72) M P m=1 h(s0 m, s1 m) ln M P m=1 h(s0 i , s1 i ) g** where 55 Balk (1980c) considers a similar weighted least squares model for many periods but with diﬀerent weights.
56 Theil’s approach is also pursued by Rao (1995), who considered a generalization of equation (20.70) to cover the case of many time periods.

370 h(a, b) [(1=2)a 1+(1=2)b 1] 1=2ab=(a+b) (20:73) m=p0 and h(a, b) is the harmonic mean of the numbers a and b.

Thus g** is a share-weighted average of the logarithms of the price ratios p1 m. If g** is exponentiated, then an estimator a** for the elementary aggregate a is obtained.
20.107 How does a** compare to the three ideal elementary price indices deﬁned in paragraphs 20.11 to 20.22?

It can be shown57 that a** approximates those three indices to the second order around an equal price and quantity point; i.e., for most data sets, a** will be very close to the Fisher, To¨ rnqvist and Walsh elemen- tary indices.
20.108 In fact, a slightly diﬀerent weighted least squares problem that is similar to the minimization problem (20.70) will generate exactly the To¨ rnqvist ele- mentary index.
Consider the following weighted least squares model: ming, d’s M + P m=1 M P m=1 1 2 1 2 (s0 m+s1 m)( ln p0 m dm)2 (s0 m+s1 m)( ln p1 m g dm)2 (20:74) Thus in the model (20.74), the logarithm of each item price quotation in each period is weighted by the arithmetic average of its expenditure shares in the two periods under consideration.
20.109 The g solution to the minimization problem (20.74) is M g***= P m=1 1 2 (s0 m+s1 m) ln p1 m p0 m (20:75) which is the logarithm of the To¨ rnqvist elementary index.

Thus the exponential of g*** is precisely the To¨ rnqvist price index.

20.110 The results in this section provide some weak support for the use of the Jevons elementary index, but they provide much stronger support for the use of weighted elementary indices of the type deﬁned in paragraphs 20.11 to 20.22.
20.111 The results in this section also provide sup- port for the use of value-based weights in hedonic regressions.

Conclusion 20.112 The main results in this chapter can be summarized as follows: In order to deﬁne a ‘‘best’’ elementary index number formula, it is necessary to have a target index number concept.
In paragraphs 20.11 to 20.22, it is suggested that normal bilateral index number theory applies at the elementary level as well as at higher levels and hence the target concept should be one of the Fisher, To¨ rnqvist or Walsh formulae.

 When aggregating the prices of the same narrowly deﬁned item within a period, the narrowly deﬁned unit value is a reasonable target price concept.

57 Using the techniques in Diewert (1978).


ELEMENTARY INDICES If The axiomatic approach to traditional elementary indices (i.e., no quantity or value weights are avail- able) supports the use of the Jevons formula under all circumstances.58 in the elementary the items aggregate are homogeneous (i.e., they have the same unit of measurement), then the Dutot formula can be used.
In the case of a heterogeneous elementary aggregate (the usual case), the Carruthers–Sellwood– Ward formula can be used as an alternative to the Jevons formula, but both will give much the same numerical answers.

 The Carli index has an upward bias and the harmonic index has a downward bias.

 The economic approach to elementary indices weakly supports the use of the Jevons formula.

 None of the ﬁve unweighted elementary indices is really satisfactory.
A much more satisfactory ap- proach would be to collect quantity or value infor- mation along with price information, and form sample superlative indices as the preferred elementary indices.

If a chained superlative index is calculated, however, it should be examined for chain drift; i.e., a chained index should only be used if the data are relatively smooth and subject to long-term trends rather than short-term ﬂuctuations.
A simple hedonic regression approach to elementary indices supports the use of the Jevons formula, but a weighted hedonic regression approach is more satis- factory.

The resulting index will closely approximate the ideal indices deﬁned in paragraphs 20.11 to 20.22.

58 One exception to this advice is when a price can be zero in one period and positive in another comparison period.
In this situation, the Jevons index will fail and the corresponding item will have to be ignored in the elementary index, or the technique outlined in para- graphs 17.90 to 17.94 of Chapter 17 could be used.

371 QUALITY CHANGE AND HEDONICS 21 Introduction 21.1 Chapters 15 to 20 cover theoretical issues relating to the choice of index number formula and are based on a simplifying assumption: that the aggregation is over the same i=1, .

.

.

n matched items in the two peri- ods being compared.
A comparison of prices between two periods requires the quality of each item to remain the same between the periods.

Price collectors are asked to match items with the same quality speciﬁcation in each month, so that only ‘‘pure’’ price changes are measured, not price changes tainted by changes in the quality of what is consumed.

In practice, the quality of what is consumed does change.
Furthermore, new goods and services appear on the market, and their relative price changes may diﬀer from the price changes of existing ones.

In addition, the expenditure share of these new goods and services may be substantial.

Paragraphs 21.2 to 21.60 outline a theoretical framework which extends the deﬁnition of items to include their quality char- acteristics.
It helps to provide a background for the practical implementation of quality adjustment, dis- cussed in Chapter 7, and for ways of dealing with item substitution and new goods, covered in Chapter 8.

New and disappearing items 21.2 The assumption in the previous chapters was that the same set of items was being compared in each period.

This can be considered as sampling from all the matched items available in periods 0 and t – the inter- section universe1 which includes only matched items.
Yet in many product areas old items disappear and new items appear.

Constraining the sample to be drawn from this intersection universe is unrealistic.

Outlets may sell an item in period 0, but it may not be sold in subsequent periods t.2 New items may be introduced after period 0 which cannot be compared with a corresponding item in period 0.

These items may be variants of the old existing ones, or provide totally new services which cannot be directly compared with anything that previously existed.
items in periods 0 and t is the This universe of all dynamic double universe.

21.3 There is a third universe from which prices might be sampled: a replacement universe.

The prices of 1 The terminology is attributable to Dale´ n (1998a); see also Appendix 8.1.

2 Its absence may be temporary, being say a seasonal item; ways of treating such temporarily unavailable items are considered in Chapter 8.

The concern here, however, is with items that disappear on a permanent basis.
a sample of items in period 0 are determined and their prices are monitored in subsequent periods.

If the item is discontinued and there are no longer prices to record for the particular item, prices of a comparable replace- ment item may be used to continue the series of prices.

This universe is a replacement universe that starts with the base period universe, but also includes one-to-one replacements when an item from the sample in the base period is missing in the current period.
21.4 Ideally the replacement item is comparable in quality to the item it replaces.

When a comparable replacement is unavailable, a non-comparable one may be selected.

Two approaches are possible.

An explicit adjustment may be made to the price of either the old or the replacement item for the quality diﬀerence, since the two items are of diﬀerent quality.
Alternatively, an assumption may be made that the price change of the ‘‘old’’ item, had it continued to exist, would have been the same as that of the overall price change of a targeted group of items from the matched universe.
In this sec- ond case, an implicit adjustment is being made about the eﬀect on price of the quality change, namely that there is no diﬀerence between the price change for the matched group and the quality-adjusted price change of the old item, had it continued to exist.3 Here, the prob- lem of missing items is considered from the point of view of adjusting prices for quality diﬀerences.

21.5 Three practical diﬃculties emerge.


First, there is the problem of explicit quality adjustment between a replacement and old item.

The item is no longer con- sumed, a replacement is found which is not strictly comparable in quality, the diﬀerences in quality are identiﬁed, and a price has to be put on these diﬀerences if the series of prices for the replacement ‘‘new’’ item is to be used to continue the series of prices for the ‘‘old’’ item.
21.6 Second, in markets where the turnover of items is high, the sample space selected from the matched universe is going to become increasingly unrepresenta- tive of the dynamic universe, as argued in Chapter 8.

Even the replacement universe may be inappropriate, as it will comprise series which only replenish themselves when an item needs replacing.

Given the rapidly chang- ing technology in many product areas, this universe may be unrepresentative.
In such cases, it may be preferable to collect prices, not from matched samples, but from a sample in each period of the main (or a representative sample of) items available, even though they are of a diﬀerent quality.

A comparison between the average prices of such items would be biased if, say, the quality 3 Such methods and their assumption are discussed in detail Chapter 7. in 373 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE of the items were improving.
The need for, and details of, mechanisms to remove the eﬀects of such changes from the average price comparisons are discussed in Chapter 7.

21.7 Finally, there is the problem of new and dis- appearing goods and services.

These are ones where the new item is not a variant of the old but provides a com- pletely new service.

It is not possible to use it as a replacement for an old item by adjusting a price for the quality diﬀerential because what it provides is, by deﬁ- nition, something new.
21.8 There are a number of approaches to quality adjustment and these are considered in Chapter 7.

One of the approaches is to make explicit adjustments to prices for the quality diﬀerence between the old and replace- ment items using the coeﬃcients or predicted values from hedonic regression equations.
Hedonic regressions are regressions of the prices of individual models of a pro- duct on their characteristics, for example the prices of television sets on screen size, the possession of stereo or text-retrieval features, etc.

The coeﬃcients of such vari- ables provide estimates of the marginal values of diﬀer- ent quantiﬁable characteristics of the product.
They can be used to adjust the price of a non-comparable replace- ment item for diﬀerences in its quality as compared with the old item – for example, the replacement television set may have text-retrieval facilities when the previous ver- sion did not.
It is important that a clear understanding exists of the meaning of such estimated coeﬃcients if they are to be used for quality adjustment, especially given that their use is being promoted.4 To understand what these estimated parameters mean, it is ﬁrst necessary to conceive of items as aggregates of their characteristics.

Unlike items, characteristics have no separate price attached to them.

The price of the item, however, is the price of a ‘‘tied’’ bundle of its characteristics.
Then it is necessary to consider what might determine the prices of these characteristics: economic theory points towards examining the demand and supply for characteristics (discussed in paragraphs 21.13 to 21.21) and the inter- action of the two to determine an equilibrium price (see paragraphs 21.22 and 21.23).
Having developed the analytical framework for such prices, it is then necessary to see what interpretation the economic theory frame- work allows us to put on these calculated coeﬃcients (see paragraphs 21.24 to 21.28).

21.9 In Chapter 7, paragraphs 7.125 to 7.158, two main approaches were advised for the treatment of product areas with rapid turnover of items.
If the sample in period 0 is soon outdated, the matched universe and even replacement universe will become increasingly unrepresentative of the double universe, and repeated sampling from the double universe is required.

In this case, either chained indices were advised (see paragraphs 7.153 to 7.158), or one of a number of hedonic indices (see paragraphs 7.132 to 152).

Such indices diﬀer from the use of hedonic regression for adjusting prices for quality diﬀerences for a missing item.
These indices use hedonic regressions, say by including a dummy variable 4 See Boskin, Dullberger, Gordon, Griliches and Jorgenson (1996 and 1998) and Schultze and Mackie (2002).

374 for time on the right-hand side of the equation, to estimate the quality-adjusted price change, as outlined below and in Chapter 7, and build upon the theory in Chapter 17 and paragraphs 21.13 to 21.36.
The eco- nomic theory of price indices outlined in Chapter 17 is developed to include those tied bundles of good that can be deﬁned in terms of their characteristics as an item in the consumer’s utility function.

Theoretical consumer price indices are deﬁned which include changes in the prices of characteristics.
As with the price indices for goods considered in Chapter 17, there are many for- mulations that such indices can take, and analogous issues and formulae arise when discussing alternative approaches in paragraphs 21.40 to 21.60.

21.10 The estimation of hedonic regressions and the testing of their statistical properties are facilitated by the availability of user-friendly, yet powerful, statistical and econometric software.
There are many standard prob- lems in the estimation of regression equations, which can be resolved using the diagnostic tests available in such software, as discussed by Kennedy (1998) and Maddala (1988).

There are, however, matters of func- tional form, the use of weighted least squares estimators and speciﬁcation that are speciﬁc to the estimation of hedonic equations.
Many of these are illustrated in Chapter 7, while Appendix 21.1 to this chapter considers some of the theoretical aspects; see also Gordon (1990), Griliches (1990) and Triplett (1990).

21.11 Finally, in paragraphs 21.61 to 21.68, eco- nomic theory is used to advise on the problem of new and disappearing goods and services.
This problem arises where diﬀerences between existing goods and services and the new goods and services are substantive in nature and cannot be meaningfully compared, even with a quality adjustment.

The economic theory of reser- vation prices is considered and some concern about its practical implementation is expressed.
Hedonic prices and implicit markets Items as tied bundles of characteristics 21.12 A hedonic regression is a regression equation which relates the prices of items, p, to the quantities of characteristics, given by the vector z=(z1, z2, .

.

.

, zn), i.e.

p(z)=p(z1, z2, .

.

.

, zn) (21:1) where the items are deﬁned in terms of varying amounts of their characteristics.
In practice, what will be observed for each item (variant of the product) is its price, its characteristics, and possibly the quantity and thus value sold.

Empirical work in this area has been concerned with two issues: estimating how the price of an item changes as a result of unit changes in each characteristic, that is, the estimated coeﬃcients of equation (21.1); and estimating the demand and supply functions for each characteristic.
The depiction of an item as a basket of characteristics, each characteristic having it own implicit (shadow) price, requires in turn the speciﬁcation of a market for such characteristics, since prices result from the workings of markets.

Houthakker (1952), Becker (1965), Lancaster (1966) and Muth (1966) have all identiﬁed the demand for items in terms of their char- acteristics.
The sale of an item is the sale of a tied bundle of characteristics to a consumer whose economic behav- iour in choosing between items is depicted as one of choosing between bundles of characteristics.5 Rosen (1974) further developed the analysis by providing a structural market framework in terms of both producers and consumers.

There are two sides: demand and supply.


How much of each characteristic is supplied and con- sumed is determined by the interaction of the demand for characteristics by consumers and the supply of char- acteristics by producers.

These are considered in turn.

The consumer or demand side j ).

21.13 Consider Figure 21.1, from Triplett (1987, p. 634), which shows a simpliﬁed characteristics-space between only two characteristics.
The hedonic surfaces p1 and p2 trace out all the combinations of the two characteristics z1 and z2 that can be purchased at prices p1 and p2.

An indiﬀerence curve q* j maps the combina- tions of z1 and z2 that the consumer is indiﬀerent about purchasing; that is, the consumer will derive the same utility from any point on the curve.
The tangency of q* j with p1 at A is the solution to the utility maximization problem for a given budget (price p1) and given tastes (reﬂected in q* 21.14 The slope of the hedonic surface is the mar- ginal cost to the consumer of acquiring the combination of characteristics and the slope of the utility function is the marginal utility gained from their purchase; the tangency at A is the utility-maximizing combination of characteristics to be purchased at that price.
If con- sumers purchased any other combination of character- istics in the space of Figure 21.1, it would either cost them more to do so or lead to a lower level of utility.

Position A0, for example, has more of both z1 and z2, and the consumer receives a higher level of utility, being on qj, but the consumer also has to have a higher budget and pays more, p2, for being there.
Figure 17.1 in Chapter 17 illustrates in goods-space how the consumer would choose between diﬀerent combinations of outputs, q1 and q2.

The characteristics-space problem in Figure 21.1 is analogous to the goods-space one, with consumers choosing between combinations of characteristics z1 and z2.

Note that the hedonic surface depicted in Figure 21.1 is non-linear, so relative characteristic prices are not ﬁxed.

The consumer with tastes q* k chooses characteristic set B at p1.
Thus the data observed in the market depend on the set of tastes.

Triplett (2002) has argued that if tastes were all the same then only one model of, say, a personal computer would be purchased.

But in the real world more than one model does exist, reﬂecting het- erogeneous tastes and income levels.
Rosen (1974) shows that, of all the characteristic combinations and prices at which they may be oﬀered, the hedonic surface traces out 5 The range of items is assumed to be continuous in terms of the combinations of characteristics that deﬁne them.

A non-continuous case can be depicted where the price functions are piece-wise linear and an optimal set of characteristics is obtained by combining the pur- chases of diﬀerent items; see Lancaster (1971) and Gorman (1980).
QUALITY CHANGE AND HEDONICS Figure 21.1 Consumption and production decisions for combinations of characteristics z1 z1A A′ qj A * qj S*G B S*H q*k z2B p1 p2 z2 j and q* an envelope6 of tangencies, including q* k on p1 in Figure 21.1.

This envelope is simply a description of the locus of the points chosen.
Since these points are what economic theory tells us will be observed in the market, assuming rational utility-maximizing con- sumers, these are the points that will be used to estimate the hedonic regression.

Note further that points A and B alone will not allow the regression to determine the price of z1 relative to z2 since the observed data will be two combinations of outputs at the same price.

The locus of points on an expansion path AA0 would, however, allow this to be determined.
Of course, there may be expansion paths for consumers with diﬀerent tastes, such as B, and this may give rise to conﬂicting valuations.

The overall parameter estimates determined by the regression from transactions observed in the market are an amalgam of such data.

Of course, this is just a reﬂection of the reality of economic life.

What arises from this exposition is the fact that the form of the hedonic function is determined in part by the distribution of buyers and their tastes in the market.
21.15 The exposition is now formalized to include parameters for tastes and a numeraire commodity7 against which combinations of other aggregates are 6 Note that an envelope is more formally deﬁned by letting f (x, y, k)=0 be an implicit function of x and y.

The form of the function is assumed to depend on k, the tastes in this case.

A diﬀerent curve corresponds to each value of k in the xy plane.
The envelope of this family of curves is itself a curve with the property that it is tangent to each member of the family.

The equation of the envelope is obtained by taking the partial derivative of f (x, y, k) with respect to k and eliminating k from the two equations f (x, y, k)=0 and @f (x, y, k)/@k=0; see Osgood (1925).

7 The numeraire commodity represents all other goods and services consumed – it represents the ‘‘normal’’ non-hedonic commodities.
The price of x is set equal to unity, and p(z) and income are measured using that unit.

375 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE selected, following Rosen (1974).

The hedonic function p(z) describes variation in the market price of the items in terms of their characteristics.
The consumer purchase decision is assumed to be based on utility maximization behaviour, the utility function being given by U(z, x; a) where x is a numeraire commodity, the maximization of utility being subject to a budget constraint given by income y measured as y=x+p(z) (the amount spent on the numeraire commodity and the hedonic products) and a is a vector of the features of individual consumers which describe their tastes.
Naturally x in the utility function can be described in terms of y p(z).

Con- sumers maximize their utility by selecting a combination of quantities of x and characteristics z subject to a budget constraint.

The market is assumed to be com- petitive and consumers are described as ‘‘price takers’’; they only purchase the one item, so their purchase decision does not inﬂuence the market price.

The price they pay for a combination of characteristics, vector z, is given by p(z).
Since they are optimizing consumers, the combination chosen is such that: [@U(z, y p(z); a)=@zi]=[@U(z, y p(z); a)=@x] =@p(z)=@zi pi(z) (21:2) where @p(z)/@zi is the ﬁrst derivative of the hedonic function (21.1) with respect to each z characteristic.

The coeﬃcients of the hedonic function are equal to their shadow prices, pi, which equal the utility derived from that characteristic relative to the numeraire good for given budgets and tastes.
21.16 A value function y can be deﬁned as the value of expenditure a consumer with tastes a is willing to pay for alternative values of z at a given utility u and income y, represented by y(z; u, y, a).

It deﬁnes a family of indiﬀerence curves relating the zi to forgone x, ‘‘money’’.

For an individual characteristic z1, y is the marginal rate of substitution between zi and money, or the implicit marginal valuation that the consumer with tastes a puts on zi at a given utility level and income.
It is an indication of the reservation demand8 price for an additional unit of zi.9 The price in the market is p(z) and utility is max- imized when y(z; u, y, a)=p(z), i.e.

the purchase takes place where the surface of the indiﬀerence curve y is tangent to the hedonic price surface.

If diﬀerent buyers have diﬀerent value functions (tastes), some will buy more of a characteristic than others for a given price function, as illustrated in Figure 21.1.
21.17 The joint distribution function of tastes and income sets out a family of value functions each of which, when tangential to the price function, depicts a purchase and simultaneously deﬁnes the price function whose envelope is the market hedonic price function.

The points of purchase traced out by the hedonic function thus depend on the budget and the tastes of the individ- ual consumer purchasing an individual set of character- istics.
If demand functions are to be traced out, the joint probability distribution of consumers with particular budgets and tastes occurring in the market needs to be speciﬁed, i.e., F( y, a).

This function, along with equation (21.1) allows the demand equations to be represented for each characteristic.

The producer or supply side 21.18 Figure 21.1, reproduced from Triplett (1987), also shows the production side.
Consider a revenue- maximizing producer whose revenue maximization problem is given by:10 R( p, v) max q pnqn: (q) belongs to S(v) (21:3) N P n=1 is where R( p, v) the maximum value of output, PN n=1pnqn, that the establishment can produce, given that it faces the vector of output prices p and given that the vector of inputs v is available for use, using the period t technology.
Recall that Figure 17.1 illustrates in goods-space how the producer would choose between diﬀerent combinations of outputs, q1 and q2.

In Figure 21.1 the characteristics-space problem is one of produc- ers choosing between combinations of z1 and z2 to produce for a particular level of technology and inputs S(v).

For a particular producer with level of inputs and technology S* G facing a price surface p1, the optimal production combination is at A.
However, a diﬀerent producer with technology and inputs S* H facing a price surface p1 would produce at B.

At these points, the mar- ginal cost of z1 with respect to z2 is equal to its marginal price from the hedonic surface, as depicted by the tan- gency of the point.

Production under these circum- stances at any other combination would not be optimal.
The envelope of tangencies such as S* H trace out the production decisions that would be observed in the market from optimizing, price-taking producers and are used as data for estimating the hedonic regressions.

The hedonic function can be seen to be determined, in part, by the distribution of technologies of producers, includ- ing their output scale.
G and S* 21.19 Rosen (1974) formalizes the producer side whereby price-taking producers are assumed to have cost functions described by C(Q, z; t),11 where Q=Q(z) is the output scale – the number of units produced by an establishment oﬀering speciﬁcations of an item with characteristics z.

The producers have to decide which items to produce, i.e., which package of z.
To do this, a cost minimization problem is solved which requires t, equivalent to S(v) above, a vector of the technology of each producer that describes the output combinations each producer can produce with given input costs, using that producer’s factors of production and the factor prices.
It is the variation in t across producers that 8 This is the hypothetical price which just makes the demand for the commodity equal to zero; that is, it is the price which, when inserted into the demand function, sets demand to zero.

The utility function is assumed to be strictly concave, so that y is concave in z and the value function is increasing in zi at a decreasing rate.
9 The utility function is assumed strictly concave so that y is concave in z and the value function is increasing in zi at a decreasing rate.

10 The time superscripts are not relevant in this context.

11 The cost function is assumed to be convex with no indivisibilities, and the marginal cost of producing one more item of a given combi- nation of characteristics is assumed to be positive and increasing.
Similarly, the marginal cost of increasing production of each compo- nent characteristic is positive and non-decreasing.

376 distinguishes producer A’s decision as to which combi- nation of z to produce from that of producer B in Figure 21.1.

Producers are optimizers who seek to maximize proﬁts, given by: Qp(z)C(Q, z; t) (21:4) by selecting Q and z optimally.
The supplying market is assumed to be competitive, and producers are price takers so the producer cannot inﬂuence price by a pro- duction decision.

The producer’s decision as to how much to produce of each z is determined by the price of z, assuming that the producer can vary Q and z in the short run.12 Dividing the expression (21.4) by Q and setting it equal to zero, the ﬁrst-order proﬁt-maximizing condi- tions are given by: @p @zi =pi= Czi(Q, z; t) Q (21:5) where p=p(z1, z2, .

.

.
.

.

, zn) from equation (21.1).

21.20 The marginal unit revenue from producing characteristic zi is given by its shadow price in the price function and its marginal cost of production.

In the producer case, the probability distribution of the tech- nologies of ﬁrms G(t) is necessary if the overall quantity supplied of items with given sets of characteristics is to be revealed.
Since it is a proﬁt maximization problem to select the optimal combination of characteristics to produce, marginal revenue from the additional attri- butes must equal their marginal cost of production per unit sold.

Quantities are produced up to the point where unit revenues p(z) equal their marginal production cost, evaluated at the optimum bundle of characteristics supplied.

21.21 While a value function was considered for consumers, producers require an oﬀer function f(z; p, t).
The oﬀer price is the price the seller is willing to accept for various values of z at constant proﬁt level p, when quantities produced are optimally chosen, while p(z) is the maximum price obtainable from those models in the market.

Producer equilibrium is characterized by a tan- gency between a proﬁt characteristics indiﬀerence sur- face and the market characteristics price surface where pi (zi)=jzi (z; pz; p) and pi (zi)=jzi (z; pz; p).
Since there is a distribution of the producer equilibrium is characterized by a family of oﬀer functions that envelop the market hedonic price function.

The varying t will depend on diﬀerent factor prices for items produced in diﬀerent countries, multi-product ﬁrms with economies of scale and diﬀerences in the technology, be it the quality of capital, labour or intermediate inputs and their organization.
A family of production surfaces is deﬁned for diﬀerent values of t. technologies G(t), 12 Rosen (1974) considered two other supply characterizations: the short run in which only M is variable and the long run in which plants can be added and retired.

The determination of equilibrium supply and demand is not straightforward.

A function p(z) is required such that market demand for all z will equate to market supply and clear the market.
But demand and supply depend on the whole p(z), since any adjustment to prices to equate demand and supply for one combina- tion of items will induce substitutions and changes for others.

Rosen (1974, pp.

44–48) discusses this in some detail.

QUALITY CHANGE AND HEDONICS Equilibrium 21.22 The theoretical framework ﬁrst deﬁnes each item as a point on a plane of several dimensions made up by the z1, z2, .

.

.

, zn quality characteristics; each item is a combination of values z1, z2, .

.

.

, zn.
.

.

, zn.

If only two characteristics deﬁne the item, then each point in the positive space of Figure 21.1 would deﬁne an item.

The characteristics are not bought individually, but as bun- dles of characteristics tied together to make up an item.
It is assumed that the markets are diﬀerentiated so that there is a wide range of choices to be made.13 The market is also assumed to be perfectly competitive, with consumers and producers as price takers undertaking optimizing behaviour to decide which items (tied sets of characteristics) to buy and sell.

Competitive markets and optimizing behaviour are assumed so that the quan- tity demanded of an item equals the quantity supplied.
Since the items are made up of characteristics z, the quantity demanded of all characteristics z must equal their quantity supplied.

It has been shown that con- sumers’ and producers’ choices or ‘‘locations’’ on the plane will be dictated by consumer tastes and producer technology.
Tauchen and Witte (2001, p. 4) show that the hedonic price function will diﬀer across markets in accordance with the means and variances (and in some cases also higher moments) of the distributions of household and ﬁrm characteristics.

21.23 Rosen (1974, p. 44) notes that a buyer and seller are perfectly matched when their respective value and oﬀer functions are tangential to each other.

The common gradient at that point is given by the gradient of the market clearing implicit price function p(z).
The consumption and production decisions are seen in the value and oﬀer functions to be jointly determined, for given p(z), by F( y, a) and G(t).

In competitive markets there is a simultaneity in the determination of the hedonic equation, as the distribution of F( y, a) and G(t) helps determine the quantities demanded and supplied, as well as the slope of the function.

Although the deci- sions made by consumers and producers are as price takers, the prices taken are those from the hedonic function.
There is a sense in which the hedonic function and its shadow prices emerge from the operations of the market.

The product markets implicitly reveal the hedonic function.

Since consumers and producers are optimizers in competitive markets, the hedonic function, in principle, gives the minimum price of any bundle of characteristics.

Given all of this, Rosen (1974, p. 44) asked: What do hedonic prices mean?
What hedonic prices mean 21.24 It would be convenient if, for CPI construc- tion, the estimated coeﬃcients from hedonic regres- sions were estimates of the marginal utility based on a characteristic or user value.

But theory tells us that this is not the case and that the interpretation is not clear.
21.25 There used to be an erroneous perception, in the 1960s, that the coeﬃcients from hedonic methods 13 So that choices among combinations of z are continuous, assume further that z possesses continuous second-order derivatives.

377 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE represented user values as opposed to resource costs.

Rosen (1974), as has been shown, found that hedonic coeﬃcients generally reﬂect both user values and resource costs, both supply and demand situations.
The ratios of these coeﬃcients may reﬂect consumers’ mar- ginal rates of substitution or producers’ marginal rates of substitution (transformation) for characteristics.

There is what is referred to in econometrics as an ‘‘identiﬁcation’’ problem in which the observed prices and quantities are jointly determined by supply and demand considera- tions, and their underlying sources cannot be separated.
The data collected on prices jointly arise from variations in demand by diﬀerent consumers with diﬀerent tastes and preferences, and from variations in supply by pro- ducers with diﬀerent technologies.

21.26 First, it is necessary to come to terms with this simultaneity problem.
Hedonic regressions are an increas- ingly important analytical tool, one implicitly promoted by the attention given to it in this manual, but also pro- moted in manuals by organizations such as the OECD (see Triplett (2002)), Eurostat (2001a) and widely used by the United States Bureau of Labor Statistics; see Kokoski, Waehrer and Rozaklis (2001) and Moulton (2001).

So how do economists writing on the subject shrug their intellectual shoulders in the light of these ﬁndings?
Rosen (1974, p. 43) refers to the hedonic function as: .

.

.

a joint envelope of a family of value functions and another family of oﬀer functions.

An envelope function by itself reveals nothing about the underlying members that generate it; and they in turn constitute the generating structure of the observations.
Griliches (1988, p. 120) notes the following: My own view is that what the hedonic approach tries to do is to estimate aspects of the budget constraint facing consumers, allowing thereby the estimation of ‘‘missing’’ prices when quality changes.

It is not in the business of estimating utility functions per se, though it can also be useful for these purposes .

.

.
.

.

what is being estimated is the actual locus of intersection of the demand curves of dif- ferent consumers with varying tastes and the supply curves of diﬀerent producers with possible varying technologies of production.

One is unlikely, therefore, to be able to recover the underlying utility and cost functions from such data alone, except in very special circumstances.
Triplett (1987) states: It is well-established – but still not widely understood – that the form of h() [the hedonic function] cannot be derived from the form of Q() and t() [utility and pro- duction functions], nor does h() represent a ‘‘reduced form’’ of supply and demand functions derived from Q() and t().
Diewert (2003a, p. 320), with his focus on the con- sumer side, says: Thus I am following Muellbauer’s (1974, p. 977) example where he says that his ‘‘approach is una- shamedly one-sided; only the demand side is treated .

.

.

’’.

Its subject matter is therefore rather diﬀerent from that of the recent paper by Sherwin Rosen.

The supply side and simultaneity problems which may arise are ignored.
21.27 Diewert’s (2003a) approach is of interest since the derivation of estimates of hedonic coeﬃcients is one that follows the demand side only.

It is a useful exercise to consider the conditions under which the hedonic 378 coeﬃcients are determined by only demand-side or supply-side factors – the circumstances under which clear explanations would be valid.
The problem is that because the coeﬃcients of a hedonic function are the outcome of the interaction of consumer and producer optimizing conditions, it is not possible to interpret the function only in terms of, say, producer marginal costs or consumer marginal values.

Suppose, however, that the production technology t is the same for each produc- ing establishment.

Buyers diﬀer but sellers are identical.


Then instead of there being a confusing family of oﬀer functions, there is a unique oﬀer function with the hedonic function describing the prices of characteristics the ﬁrm will supply with the given ruling technology to the current mixture of tastes.

The function p(z) becomes the oﬀer function since there is no distribution of t to confuse it.
There are diﬀerent tastes on the consumer side, so what appears in the market is the result of ﬁrms trying to satisfy consumer preferences, all for a constant technology and proﬁt level; the structure of supply is revealed by the hedonic price function.

In Figure 21.1 only the expansion path traced out by, say, S* H akin to AA0, would be revealed.

Now suppose that sellers diﬀer, but that buyers’ tastes a are identical.
Here the family of value functions collapses to be revealed as the hedonic function p(z) which identiﬁes the structure of demand, in Figure 21.1.14 Triplett (1987, p. 632) such as AA0 notes that of these possibilities, uniformity of technolo- gies is the most likely, especially when access to tech- nology is unrestricted in the long run, while uniformity of tastes is unlikely.
There may, of course, be segmented markets where tastes are more uniform, to which speciﬁc sets of items are tailored and for which hedonic equa- tions can be estimated for individual segments.15 21.28 The analysis in paragraphs 21.12 to 21.27 demonstrates the ambiguity surrounding the interpreta- tion of hedonic coeﬃcients.
This analysis is needed since issues relating to the estimation of the underlying supply and demand functions for characteristics have implica- tions for the estimation of hedonic functions.

Appendix 21.1 considers identiﬁcation and estimation issues in this light.

The next section provides an alternative theoretical derivation, based on Diewert (2003a), which shows the assumptions required for a demand-based (consumer- based) interpretation.
14 Correspondingly, if the supply curves were perfectly inelastic, so that a change in price would not aﬀect the supply of any of the diﬀerentiated products, then the variation in prices underlying the data and feeding the hedonic estimates would be determined by demand factors.

The coeﬃcients would provide estimates of user values.

Similarly, if the supplying market were perfectly competitive, the estimates would be of resource costs.
None of the price diﬀerences between diﬀerentiated items would be attributable to, say, novel conﬁgurations of character- istics, and no temporary monopoly proﬁt would be achieved as a reward for this, or as a result of the exercise of market power; see Berndt (1991).

15 Berry, Levinsohn and Pakes (1995) provide a detailed and interesting example for cars, in which makes are used as market segments.
Tau- chen and Witte (2001) provide a systematic theoretical study of esti- mation issues for supply, demand and hedonic functions, where con- sumers and producers and their transactions are indexed across communities.

An alternative, consumer-based hedonic theoretical formulation 21.29 This section takes a consumer-based approach to deriving theoretical hedonic functions.

It assumes: that every consumer has the same separable sub-utility function, f(z1, .

.

.
.

.

, zN) that gives the consumer the sub- utility Z=f(z) from the purchase of one unit of the complex hedonic commodity that has the vector of characteristics z:(z1, .

.

.

, zN);16 the sub-utility that the consumer gets from consuming Z units of the hedonic commodity is combined with the consumption of X units of a composite ‘‘other’’ commodity to give the consumer an overall utility of u=Ut(X, Z) in period t, where Ut is the period t ‘‘macro’’ utility function.
Rosen (1974, p. 38) nor- malized the price of X to be unity.

This is not required in the present approach.

Instead, there is an explicit period t price, pt, for one unit of the general con- sumption commodity X.

21.30 The approach starts by considering the set of X and Z combinations that can yield the consumer’s period t utility level, ut.
This is the set {(X, Z): Ut(X, Z)=ut}, which is the consumer’s period t indiﬀerence curve over equivalent combinations of the general consumption commodity X and the hedonic commodity Z.
The equation Ut(X, Z)=ut for X is solved as a function of ut and Z; i.e.17 X=gt(ut, Z) (21:6) It is assumed that the indiﬀerence curve slopes down- ward, and the stronger assumption is made that gt is diﬀerentiable with respect to Z and @gt(ut, Z)=@Z < 0 (21:7) Let pt and Pt be the prices for one unit of X and Z, respectively, in period t. The consumer’s period t expen- diture minimization problem may be deﬁned as follows: QUALITY CHANGE AND HEDONICS Pt=pt@gt(ut, Z)=@Z > 0 (21:10) where the inequality follows from assumption (21.7).
The right-hand side of the equation (21.10) can now be interpreted as the consumer’s period t willingness to pay price function: wt(Z, ut, pt) pt@gt(ut, Z)=@Z (21:11) 21.31 Thus, for each point (indexed by Z) on the consumer’s period t indiﬀerence curve, equation (21.11) gives the amount of money the consumer would be willing to pay per unit of Z in order to stay on the same indiﬀerence curve, which is indexed by the utility level ut.
The period t willingness to pay value function vt can now be deﬁned as the product of the quantity of Z consumed times the corresponding per unit willingness to pay price, wt(Z, ut, pt): vt(Z, ut, pt) Zwt(Z, ut, pt)=Zpt@gt(ut, Z)=@Z (21:12) where the last equality follows using equation (21.11).

The function vt is the counterpart to Rosen’s (1974, p. 38) value or bid function; it gives us the amount of money the consumer is willing to pay in order to con- sume Z units.
All the above algebra has an interpreta- tion that is independent of the hedonic model; it is simply an exposition of how to derive a willingness to pay price and value function using a consumer’s pre- ferences deﬁned over two commodities.

21.32 It is assumed now that the consumer has a separable sub-utility function, f(z1, .

.

.

, zN), which gives the consumer the sub-utility Z=f(z) from the purchase of one unit of the complex hedonic commodity18 that has the vector of characteristics z:(z1, .

.

.
.

.

, zN).

Note that it has been assumed that the function f is time invariant.

Let the consumer’s period t utility function be Ut(X, f(z)).

The above algebra on willingness to pay is still valid.

In particular, the new period t willingness to pay price function, for a particular model with char- acteristics z=(z1, .

.

.
, zN), is: minX, Z f ptX+PtZ : X=gt(ut, Z)g = minZ f ptgt(ut, Z)+PtZg (21:8) wt( f (z), ut, pt) pt@gt(ut, f (z))=@Z (21:13) The ﬁrst-order necessary condition for Z to solve equation (21.8) is: The new period t willingness to pay value function (which is the amount of money the consumer is willing pt@gt(ut, Z)=@Z+Pt=0 (21:9) Equation (21.9) can be rearranged to give the price of the hedonic aggregate Pt as a function of the period t utility level ut and the price of general consumption pt: 16 It is not assumed that all possible models exist in the marketplace.
In fact, we will assume that only a ﬁnite set of models exist in each period.

It is assumed, however, that the consumer has preferences over all possible models, where each model is indexed by its vector of char- acteristics, z=(z1, .

.

.

, zN).

Thus each consumer will prefer a potential model with characteristics vector z1=(z1 N ) over another potential model with the characteristics vector z2=(z2 N ) if and only if f(z1)>f(z2).
17 If the period t indiﬀerence curve intersects both axes, then gt(ut, Z) will only be deﬁned for a range of non-negative Z up to an upper bound.

1, .

.

.

, z1 1, .

.

.

, z2 18 If a consumer purchases, say, two units of a model at price P that has characteristics z1, .

.

.

, zN, then we can model this situation by introdu- cing an artiﬁcial model that sells at price 2P and has characteristics 2z1, .

.

.

, 2zN.
.

.

, 2zN.

Thus the hedonic surface, Z=f(z), consists of only the most eﬃcient models including the artiﬁcial models.

We do not assume that f(z) is a quasi-concave or concave function of z.

In normal consumer demand theory, f(z) can be assumed to be quasi-concave without loss of generality because linear budget constraints and the assumption of perfect divisibility will imply that ‘‘eﬀective’’ indiﬀerence curves enclose convex sets.

As Rosen (1974, pp.
As Rosen (1974, pp.

37–38) points out, however, in the case of hedonic commodities, the various characteristics cannot be untied.

Moreover, perfect divisibility cannot be assumed and not all possible combinations of characteristics will be available in the marketplace.

Thus the usual assumptions made in ‘‘normal’’ consumer demand the- ory are not satisﬁed in the hedonic context.
Note also that while we place a smoothness assumption on the macro functions gt(u, Z), namely the existence of the partial derivative @gt(u, Z)/@Z), we do not place any smoothness restrictions on the hedonic sub-utility function f(z).
379 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE to pay to have the services of a model with character- istics vector z) is: vt( f (z), ut, pt) f (z)wt( f (z), ut, pt) =f (z)pt@gt(ut, f (z))=@Z (21:14) k (zt 21.33 Now suppose that there are Kt models avail- able to the consumer in period t, where model k sells at the per unit price of Pt k and has the vector of char- acteristics zt 1k, .

.

.

, zt Nk) for k=1, 2, .

.

.


, K. If the consumer purchases a unit of model k in period t, then the model price Pt k can be equated to the appropriate willingness to pay value deﬁned by equation (21.14), where z is replaced by zt k; i.e., the following equation should hold: k=f (zt Pt k)pt@gt(ut, f (zt k))=@Z for t=1, .

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.

, T; k=1, .

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.

, K t (21:15) What is the meaning of the separability assumption?
Suppose the hedonic commodity is a car and suppose that there are only three characteristics: number of seats in the vehicle, fuel economy and horsepower.

The separability assumption means that the consumer can trade oﬀ these three characteristics and determine the utility of any car with any mix of these three characteristics, independently of his or her other choices of commodities.
In particular, the utility ranking of car models is independent of the number of children the consumer might have or what the price of petrol might be.

Obviously, the separability assumption is not likely to be exactly satisﬁed in the real world, but this somewhat restrictive assumption is re- quired to make our model tractable.

21.34 Another aspect of our model needs some fur- ther explanation.

It is being explicitly assumed that con- sumers cannot purchase fractional units of each model.
; they can purchase only a non-negative integer amount of each model.

That is, indivisibilities are being explicitly assumed on the supply side of our model.

Thus, in each period, there is only a ﬁnite number of models of the hedonic commodity available.

While the consumer is assumed to have continuous preferences over all possible combinations of characteristics (z1, .

.

.

, zN) in each per- iod, there is only a ﬁnite number of isolated models available on the market.
21.35 At this point, the model is further specialized.

It is assumed that every consumer has the same hedonic sub-utility function19 f(z) and consumer i has the fol- lowing linear indiﬀerence curve macro utility function in period t: 19 The sameness assumption is very strong and needs some justiﬁca- tion.

This assumption is entirely analogous with the assumption that consumers have the same homothetic preferences over, say, food.
Although this assumption is not justiﬁed for some purposes, it suﬃces for the purpose of constructing a price index for food, since we are mostly interested in capturing the substitution eﬀects in the aggregate price of food as the relative prices of food components vary.

In a similar fashion, we are interested in determining how the ‘‘average’’ consumer values a faster computer speed against more memory; i.e., we are primarily interested in hedonic substitution eﬀects.
380 i, Z) atZ+bt i(ut gt and i=1, .

.

.

, I i ut i for t=1, .

.

.

, T (21:16) where at and bt i are positive constants.

For each period t and each consumer i, the period t indiﬀerence curve between combinations of X and Z is linear, with the constant slope at being the same for all consumers.20 Note that this slope is allowed to change over time.

Now diﬀerentiate equation (21.16) with respect to Z and substitute this partial derivative into equation (21.15).
The resulting equation is:21 k=ptatf (zt Pt k) for t=1, .

.

.

, T and k=1, .

.

.

, K t for rt ptat t=1, .

.

.

, T (21:17) Deﬁne the aggregate price of one unit of Z in period t as:22 (21:18) Now substitute equation (21.18) into equation (21.17) in order to obtain our basic system of hedonic equations.23 for t=1, .

.

.

, T and k=1, .

.

.


, K t (21:19) k=rt f (zt Pt k) 21.36 All that is needed is to postulate a functional form for the hedonic sub-utility function f and add a stochastic speciﬁcation to equation (21.19) to yield a basic hedonic regression model.
The unknown para- meters in f along with the period t hedonic price para- meters rt can then be estimated.24 It is possible to generalize the above model, but get the same model i , f (zt 20 We do not require a linear indiﬀerence curve globally, but only locally over a certain range of purchases.

Alternatively, we can view the linear indiﬀerence curve as providing a ﬁrst-order approximation to a non-linear indiﬀerence curve.
21 Comparing equation (21.17) with equation (21.15), it can be seen that the simplifying assumptions (21.16) enable us to get rid of the terms @gt(ut k))=@Z, which depend on individual consumer indif- ference curves between the hedonic commodity and other commod- ities.

If we had individual household data on the consumption of hedonic and other commodities, then we could use normal consumer demand techniques in order to estimate the parameters that char- acterized these indiﬀerence curves.
22 There has been a switch to subscripts from superscripts in keeping with the conventions for parameters in regression models; i.e., the constants rt will be regression parameters in what follows.

Note also that rt is the product of the price of the ‘‘other’’ commodity pt times the period t slope parameter at.
We need to allow this slope parameter to change over time in order to be able to model the demand for high- technology hedonic commodities, which have been falling in price relative to ‘‘other’’ commodities; i.e., we think of at as decreasing over time for high-technology commodities.

23 The basic model ends up being very similar to one of Muellbauer’s (1974, pp.

988–989) hedonic models; see in particular his equation (32).
24 It is possible to rework the above theory and give it a producer theory interpretation.
The counterpart to the expenditure minimization pro- blem (21.8) is now the following proﬁt maximization problem: maxx,z {PtZ wtX : X=gt(kt, Z)} where Z is hedonic output and Pt is a period t price for one unit of the hedonic output, wt is the period t price of a variable input and X is the quantity used of it, kt is the period t quantity of a ﬁxed factor (capital, say) and gt is the ﬁrm’s factor requirements function.
Assuming that Z=f (z), we end up with the following pro- ducer theory counterpart to equation (21.15): Pt k))= @Z.

The counterpart to assumption (21.16) is, for ﬁrm i, gt i , Z) i kt atZbt i and the counterpart to equation (21.17) becomes Pt k= wtatf (zt k).

The producer theory model assumptions are, however, not as plausible as the corresponding consumer theory model assumptions.
In particular, it is not very likely that each producer will have the same period t aggregate price for a unit of variable input wt and it is not very likely that each ﬁrm producing in the hedonic market will have the same technology parameter at.

The key assumption that will not generally be k) @gt(kt, f (zt k=f (zt i (kt if the composite ‘‘other’’ commodity X is (21.19) replaced by h(x), where x is a consumption vector and h is a linearly homogeneous, increasing and concave aggregator function.
Instead of equation (21.17), under these new assumptions, the following equation results: for t=1, .

.

.

, T and k=1, .

.

.

, K t k=c( pt)at f (zt Pt k) (21:20) where pt is now the vector of prices for the x com- modities in period t and c is the unit cost or expenditure function that is dual to h.25 Now redeﬁne rt as c( pt)at and the basic system of hedonic equations (21.19) is still obtained.
Equation (21.19) has one property that is likely to be present in more complex and realistic models of consumer choice.

This property is that the model prices in period t are homogeneous of degree one in the general price level pt.
Thus if pt is replaced by lpt for any l>0 (think of a sudden hyperinﬂation where l is large), then equations (21.17) and (21.19) imply that the model prices should become lPt k. Note that this homogeneity property will not hold for the following additive hedonic model: k=rt+f (zt Pt k) for t=1, .

.

.

, T and k=1, .

.

.

, K t (21:21) Thus hedonic regressions based on the linear model (21.21) may be ruled out on a priori grounds.
Note that hedonic models that take the logarithm of the model price Pt k as the dependent variable will tend to be con- sistent with basic hedonic equations (21.19), whereas linear models like (21.21) will not be consistent with the normal implied by microeconomic theory.

linear homogeneity properties Hedonic indices 21.37 It was noted above that hedonic functions are required for two purposes with regard to a quality adjustment.
The ﬁrst is when an item is no longer avail- able and the replacement item, of which the price is used to continue the series, is of a diﬀerent quality to the original price basis.

The diﬀerences in quality can be established in terms of diﬀerent values of a subset of the z price-determining variables.
The coeﬃcients from the hedonic regressions, as estimates of the monetary value of additional units of each quality component z, can then be used to adjust the price of, say, the old item so that it is comparable with the price of the new item26 – so that, again, like is compared with like.
This process could be satisﬁed in the producer context is that each producer is able to pro- duce the entire array of hedonic models; whereas, in the consumer context, it is quite plausible that each consumer has the possibility of purchasing and consuming each model.

25 Deﬁne c as c( pt):minx { ptx : h(x)=1}, where ptx denotes the inner product between the vectors pt and x.

26 Various mechanisms for such adjustments are outlined in para- graphs 7.103 to 7.109, and Triplett (2002).
They include using the coeﬃcients from the salient set of characteristics or using the predicted values from the regression as a whole and, in either case, making the adjustment to the old for comparison with the new, or to the new for comparison with the old, or some eﬀective average of the two.
QUALITY CHANGE AND HEDONICS described as ‘‘patching’’, in that an adjustment is needed to the price of the old (or new replacement) series for the quality diﬀerences, to enable the new series to be patched onto the old.

A second use of hedonic functions is for estimating hedonic indices.
These are suitable when the pace and scale of replacements of items are substantial and when an extensive use of patching might lead to extensive errors if there were some error or bias in the quality adjustment process and to sampling from a biased replacement universe (as outlined in paragraphs 21.12 to 21.36).
Hedonic indices use data in each period from a refreshed sample of items, which should include those with a substantial share of sales expenditure – with sampling in each period from the double universe.

There is no need to establish a preselected set of items for matching and for price collectors to keep collecting pri- ces for that set.

What is required are samples of items to be redrawn in each month along with information on their prices, characteristics and, possibly, quantities/ values.
The ‘‘partialling out’’ in the hedonic regressions controls for quality diﬀerences, as opposed to the match- ing of price by the price collectors.

There are a number of procedures for estimating hedonic indices and these are brieﬂy considered below.

Theoretical characteristics price indices 21.38 Theoretical cost of living indices are deﬁned in Chapter 17 and practical index number formulae are considered as estimates of these indices.
Theoretical cost of living index numbers are deﬁned here not just on the goods produced, but also on their characteristics.

The Konu¨ s (1924) family of true cost of living indices per- taining to two periods, where the consumer faces the strictly positive price vectors p0 ( p0 N) and p1 ( p1 N) in periods 0 and 1 respectively, was deﬁned in Chapter 17 as the ratio of the minimum costs of achieving the same utility level u:f(q), where q: (q1, .

.

.
.

.

, qN) is a positive reference quantity vector; i.e., PK ( p0, p1, q) C[u, p1]=C[u, p0]=C[ f (q), p1]=C[ f (q), p0] (21:22) 1, .

.

.

, p0 1, .

.

.


, p1 For theoretical indices in characteristic space, the rev- enue functions are also deﬁned over goods made up of bundles of characteristics represented by the hedonic function:27 PK ( p0, p1, q) C[u, p1, p(z1)]=C[u, p0, p(z0)] =C[ f (q), p1, p(z1)]=C[ f (q), p0, p(z0)] (21:23) 21.39 The theoretical price index deﬁned by equation (21.23) is a ratio of the period 1 to period 0 hypothetical costs to consumers of achieving a given utility.
Equation 27 Triplett (1987) and Diewert (2002d), following Pollak (1975), con- sider a two-stage budgeting process whereby that portion of utility concerned with items deﬁned as characteristics has its theoretical index deﬁned in terms of a cost-minimizing selection of characteristics, conditioned on an optimum output level for composite and hedonic commodities.

These quantities are then fed back into the second-stage maximization of overall revenue.
381 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE (21.23) incorporates substitution eﬀects: if the prices of some characteristics increase more than others, then utility-maximizing consumers can switch their output mix of characteristics in favour of such characteristics.
The numerator in equation (21.23) is the cost of the maximum utility that the consumer could attain if faced with the commodity prices and implicit hedonic shadow prices of period 1, p1 and p(z1), while the denominator in equation (21.23) is the maximum utility that the con- sumer could attain if faced with the commodity and characteristic prices of period 0, p0 and p(z0).
Note that all the variables in the numerator and denominator functions are exactly the same, except that the com- modity price and characteristic price vectors diﬀer.

This is a deﬁning characteristic of a price index.

As with the economic indices in Chapter 15, there is of course an entire family of indices depending on which reference utility level is chosen.
Some explicit formulations are considered in paragraphs 21.48 to 21.58, including a base period 0 reference level and a current period 1 reference level analogous to the derivation of the Laspeyres and Paasche indices in Chapter 17.

Before considering such hedonic indices, two simpler formulations are ﬁrst con- sidered: hedonic regressions using dummy variables on time (paragraphs 21.40 to 21.42), and hedonic imputa- tion indices (paragraphs 21.43 to 21.47).
They are simple, and widely used because they require no information on quantities or weights.

They also do not require matched data, so can be used when resampling all the data.

Yet their interpretation from economic theory is therefore more limited on account of this.

As will be shown in Appendix 21.1, however, weighted formulations are possible using a weighted least squares estimator.
(21.24) for t=1, 2; the index for period 3, I1,3, would use equation (21.24) for t=1, 3; the index for period 4, I1,4, using equation (21.24) for t=1, 4 and so forth.

The dummy variable in such cases would take values of 1 for the current period and 0 for the price reference period.

For example, for I1,4 it would be 1 for observations in period 4 and 0 otherwise, i.e.

for observations in period 1.

Only data in periods 1 and 4 are used for the estimated equation.
The coeﬃcient of the dummy vari- able is an estimate of quality-adjusted price change.

In each case, the index constrains the estimated coeﬃcients on the quality characteristics to be the same for the current and price reference periods, periods 1 and 4.

A ﬁxed base bilateral comparison using equation (21.24) makes use of the constrained parameter estimates over the two periods of the price comparison.
A chained for- mulation would estimate I1,4, for example, as the product of a series of links: I1,4=I1,2 I2,3 I3,4.

Each successive binary comparison, or link, is combined by successive multiplication.

The index for each link is estimated using equation (21.24).
Because the periods of time being compared are close, it is generally more likely that the constraining of parameters required by chained time dummy hedonic indexes are considered to be less severe than that required of their ﬁxed base counterparts.

21.42 There is no explicit weighting in these for- mulations, and this is a serious disadvantage.

In prac- tice, cut-oﬀ sampling might be employed to include only the most important items.
If sales data are available, a weighted (by relative sales shares; see Appendix 21.1) least squares estimator, as opposed to an ordinary least squares estimator (OLS), should be used.28 Matched data are not required for this method; items can be resampled in each period to include new technologies.
Hedonic regressions and dummy variables of time Hedonic imputation indices 21.40 Let there be K characteristics of a product and let model or item i of the product in period t have the vector of characteristics zt i [zt iK ] for i=1, .

.

.

, I and t=1, .

.

.

, T. Denote the price of model i in period t by pt i.

A hedonic regression of the price of model i in period t on its characteristics set zt i1, .

.

.
.

.

, zt i is given by: ln pt i=g0+ P T t=2 K gtDt+ P k=1 bkzt ik+et i (21:24) where Dt are dummy variables for the time periods, D2 being 1 in period t=2, zero otherwise; D3 being 1 in period t=3, zero otherwise, etc.

The coeﬃcients gt are estimates of quality-adjusted price changes, having con- trolled for the eﬀects of variation in quality (via PK k=1gkzt ki) – but see Goldberger (1968) and Teekens and Koerts (1972) for the adjustment for estimation bias.
21.41 The above approach uses the dummy variables on time to compare prices in period 1 with prices in each subsequent period.

In doing so, the bk parameters are constrained to be constant over the period t=1, .

.

.

, T. Such an approach is ﬁne retrospectively, but in real time the index may be estimated as a ﬁxed base or chained base formulation.
The ﬁxed base formulation would estimate the index for period 1 and 2, I1,2, using equation 382 21.43 An alternative approach for a comparison between period 1 and t is to estimate a hedonic regression for period t, and insert the values of the characteristics of each model existing in period 1 into the period t regres- sion to predict, for each item, its price ^ppt i ).

This would generate predictions of the price of items existing in period 1, at period t shadow prices, ^ppt i ), i=1, .

.

.
.

.

, N. These prices (or an average) can be compared with (the average of) the actual prices of models i=1, .

.

.

, N models in period 1.

The averages may be arithmetic, as in a Dutot index, or geometric, as in a Jevons index.
The arithmetic formulation is deﬁned as follows: i(z1 i(z1 N P i=1 N P i=1 (1=N)^ppt i(z1 i ) (1=N)p1 i (z1 i ) (21:25a) 21.44 Alternatively, the characteristics of models existing in period 1 can be inserted into a regression for period t. Predicted prices of period t items generated at 28 Ioannidis and Silver (1999) and Bode and van Dale´ n (2001) com- pared the results from these diﬀerent estimators, ﬁnding notable dif- ferences, but not in all cases; see also Silver and Heravi (2003).
period 1 shadow prices (or an average) can be compared with (the average of) the actual prices in period t: N P i=1 N P i=1 (1=N)pt i(zt i) (1=N)^pp1 i (zt i) (21:25b) 21.45 For a ﬁxed base bilateral comparison using either formula (21.25a) or (21.25b), the hedonic equation need only be estimated for one period.
The denominator in formula (21.25a) is the average observed price in period 1, which should be equal to the average price that a hedonic regression based on period 1 data will predict using period 1 characteristics.

The numerator, however, requires an estimated hedonic regression to predict per- iod 1 characteristics at period t hedonic prices.

Similarly, in formula (21.25b) a hedonic regression is required only for the denominator.
For reasons analogous to those explained in Chapter 15, a symmetric average of these indices should have some theoretical support.

21.46 Note that all the indices described in para- graphs 21.40 to 21.45 can use matched data or all the data available in each period.

If there is a new item in, say, period 4, it is included in the data set and its quality diﬀerences controlled for by the regression.

Similarly, if old items drop out, they are still included in the indices in the periods in which they exist.
This is part of the natural estimation procedure and diﬀers from the use of hedonic regressions only for price adjustments to non- comparable items.

21.47 As with the dummy variable approach, there is no need for matched data.

Yet there is also no explicit use of quantity weighting in these formulations, and this is a serious disadvantage.

Were data on quantities or values available, it is immediately apparent that such weights could be attached to the individual i=1, .

.

.
.

.

, N prices or their estimates.

This is considered in the next section.

Superlative and exact hedonic indices 21.48 In Chapter 17, Laspeyres and Paasche bounds were deﬁned on a theoretical basis, as were superlative indices, which treat both periods symmetrically.

These superlative formulae included the Fisher index, which was seen in Chapter 16 to have desirable axiomatic properties.
Furthermore, the Fisher index was supported from economic theory as a symmetric average of the Laspeyres and Paasche bounds, and was found to be the most suitable such average of the two on axiomatic grounds.

The To¨ rnqvist index also possessed desirable axiomatic properties, seemed to be best from the sto- chastic viewpoint, and also did not require strong assumptions for its derivation from the economic ap- proach as a superlative index.
The Laspeyres and Paasche indices were found to correspond to (be exact for) underlying (Leontief) aggregator functions with no substitution possibilities, while superlative indices were exact for ﬂexible functional forms, including the quad- ratic and translog forms for the Fisher and To¨ rnqvist indices respectively.
If data on prices, characteristics and quantities are available, then analogous approaches and QUALITY CHANGE AND HEDONICS ﬁndings arise for hedonic indices; see Fixler and Zie- schang (1992) and Feenstra (1995).

Exact bounds on such an index were deﬁned by Feenstra (1995).

Consider the theoretical index in equation (21.23), but now only deﬁned over items in terms of their characteristics.

The prices are still of items, but they are wholly deﬁned through p(z).
An arithmetic aggregation for a linear hedonic equation ﬁnds that a Laspeyres upper bound (as quantities supplied decrease with increasing relative pri- ces) is given by: N P i=1 N P i=1 xt1 i ^ppt i i pt1 xt1 i N =P i=1 st1 i ^ppt i pt1 i C(ut1, p(z)t) C(ut1, p(z)t1) (21:26a) where the right-hand side expression is the ratio of the cost of achieving a period t 1 level of utility (ut 1), where utility is a function of the vector of quantities; i.e., ut 1=f(xt 1); the price comparison is evaluated at a ﬁxed level of period t 1 quantities and st1 are the shares in the total value of expenditure on product i in period t 1: i i =xt1 st1 i pt1 i N .P i=1 xt1 i pt1 i 21.49 The diﬀerence between a Laspeyres formula and the left-hand side of equation (21.26a) is that the price in the numerator of the left-hand side of equation (21.26a) is a predicted price: i ^ppt ^ppt i(zt1 i K )= P k=0 bt k zt1 ik (21:26b) or, if a non-comparable replacement is used, then the predicted price adjusts for the diﬀerence in quality between the old and new items.
That is, the predicted price N i pt ^ppt iP i=1 bt k(zt ikzt1 ik ) (21:26c) is the price in period t adjusted for the sum of the changes in each quality characteristic weighted by their coeﬃcients derived from a linear hedonic regression.
Note that the summation is over the same i in both periods since replacements are included when an item is missing, and (21.26c) adjusts the prices in period t for quality diﬀerences via PN i=1bt 21.50 A Paasche lower bound is estimated as: ikzt1 ik ).
k(zt N = P i=1 st i ^ppt1 i pt i 1 C(ut, p(z)t) C(ut, p(z)t1) (21:27a) ipt xt i N P i=1 N P i=1 where st i ^ppt1 xt i i=xt ipt i= PN i=1 xt ipt i and k i P ^ppt1 k=0 bt1 k zt ik (21:27b) 383 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE i pt1 ^ppt1 i + P bt1 k (zt ikzt1 ik ) (21:27c) N k=0 which are the imputation and replacement adjustment, respectively.
The latter are the prices in periods t 1 adjusted for the sum of the changes in each quality characteristic weighted by their respective coeﬃcients derived from a linear hedonic regression.
21.51 Following from the inequalities in (17.5), where the Laspeyres PL and Paasche PP indices form bounds (17.8) on their ‘‘true’’ PK economic theoretic indices: PL PK PP or PP PK PL (21:28) a suitable index is thus a Fisher geometric mean of the Laspeyres PL and Paasche PP indices, which incorporate hedonic adjustments for quality diﬀerences.
21.52 Thus, the approach based on using superlative and exact hedonic indices ﬁrst applies the coeﬃcients from hedonic regressions to changes in the character- istics to adjust observed prices for quality changes.

Sec- ond, it incorporates a weighting system using data on the quantities sold of each model and their characteristics, rather than treating each model as equally important.

Finally, it has a direct correspondence to the formulation deﬁned using economic theory.
21.53 Semi-logarithmic hedonic regressions would supply a set of b coeﬃcients suitable for use with the base and current period geometric bounds: st i N Q i=1 pt i ^ppt1 i C(ut, p(z)t) C(ut, p(z)t1) N Q i=1 ^ppt i pt1 i st1 i (21:29a) bt1 k zt k k i exp P ^ppt1 i exp P ^ppt k k=0 k=0 (21:29b) bt kzt1 k i pt1 ^ppt1 i N exp P i=1 bt1 k (zt ikzt1 ik ) i pt ^ppt i exp P N ikzt1 ik ) bt k(zt (21:29c) i=1 21.54 In the inequality (21.29a), the two bounds on the respective theoretical indices have been shown to be brought together.
The calculation of such indices is rela- tively straightforward for matched data, but for un- matched data is no small task.

For an example of its application for unmatched comparisons over time, see Silver and Heravi (2002; 2003) and Chapter 7, para- graphs 7.132 to 7.152, and see Kokoski, Moulton and Zieschang (1999) for matched price comparisons across regions of a country.
21.55 Exact hedonic indices can also be deﬁned using the theoretical framework outlined by Diewert (2003a).29 Recall the basic hedonic equation (21.19).

29 The assumptions are quite diﬀerent from those made by Fixler and Zieschang (1992), who took yet another approach to the construction of exact hedonic indices.
384 Assume that the price Pt k is the average price for all the models of type k sold in period t and let qt k be the number of units sold of model k in period t. Recall that the number of models in the marketplace during period t is Kt.

Assume that there are K models in the marketplace over all T periods in our sample period.

If a particular model k is not sold at all during period t, then it will be assumed that Pt k are both zero.
With these con- ventions in mind, the total value of consumer purchases during period t is equal to: k and qt K P k=1 K kqt Pt k= P k=1 rt f (zk)qt k for t=1, .

.

.

, T (21:30) 21.56 The hedonic sub-utility function f has done all the hard work in the model by converting the utility yielded by model k in period t into a ‘‘standard’’ utility f(zk) that is cardinally comparable across models.
For each model type k, it is only necessary to multiply by the total number of units sold in period t, qt k, in order to obtain the total period t market quantity of the hedonic commodity, Qt say.

This yields:30 K Qt P k=1 f (zk)qt k for t=1, .

.

.

, T (21:31) 21.57 The aggregate price for the hedonic commod- ity corresponding to Qt is rt.
Thus in the highly simpliﬁed model outlined in paragraphs 21.29 to 21.36, the aggre- gate exact period t price and quantity for the hedonic commodity are rt and Qt deﬁned by equation (21.31), which can readily be calculated provided that the pa- rameters in the hedonic regression have been estimated and provided that data on quantities sold are available k.31 Once rt and Qt have been determined each period qt for t=1, .

.

.
.

.

, T, then these aggregate price and quantity estimates for the hedonic commodity can be combined with the aggregate prices and quantities of non-hedonic commodities using normal index number theory.

Any of the index number formulae considered in Chapter 17, including Laspeyres, Paasche and Fisher, can be accord- ingly deﬁned based on the use of quantity information.
21.58 The above illustrates how weighted quality- adjusted price index number formulae might be con- structed using data on prices,quantities and characteristics of an item.

The method using dummy variables of time, described in paragraphs 21.40 to 21.42, does not require matched data.

Appendix 21.1 discusses a weighting sys- tem.

The use of weighted superlative indices for matched data is outlined above.
Weighted superlative indices may also be applied to unmatched data, using a method 30 This is a counterpart to the quantity index deﬁned by Muellbauer (1974, p. 988) in one of his hedonic models; see his equation (30).

Of course, treating rt as a price for the hedonic commodity quantity aggregate deﬁned by equation (21.31) can be justiﬁed by appealing to Hicks’ (1946, pp.

312–313) aggregation theorem, since the model prices pt k=rt f (zt) all have the common factor of proportionality rt.
31 If data are available for the qt k, then it is best to run sales-weighted regressions, as discussed in Appendix 21.1.

If we do not have complete market data on individual model sales but we do have total sales in each period, then the hedonic regression model can be run using a sample of model prices, and period t sales can be divided by our estimated rt parameter in order to obtain an estimator for Qt.

outlined in Chapter 7 and in Silver and Heravi (2001a; 2001b; 2003).
But what of unweighted indices, which was the concern of the initial section of this chapter?

What correspondence does the unweighted hedonic time dummy index (outlined in paragraphs 21.40 to 21.42), which uses all the data, have to the matched unweighted index number formulae?

This is a critical question for product areas where there is a rapid turnover of items.

It was suggested above that the time dummy variable method be used instead of the matched method.
So how do they diﬀer for unweighted indices?

The eﬀect and use of weights is considered in Appendix 21.1.

Unweighted hedonic indices and unweighted matched index number formulae 21.59 Triplett (2002) and Diewert (2003a) have argued that an unweighted geometric mean Jevons index (see equation (20.3)) for matched data gives the same result as a logarithmic dummy variable hedonic index run on the same data.
An index from a dummy variable hedonic regression such as equation (21.24), but in double-logarithmic (log-log) form, for matched models can be shown (see Aizcorbe, Corrado and Doms (2001)) to be equal to: (ln pt mZt m)=Mt ln ( pt=pt1)= P m 2 Mt P m 2 Mt1 (ln pt1 m Zt1 m )=Mt1 (21:32) k=1bkzt where m is the matched sample and Zt and Zt 1 are the quality adjustments to the dummy variables for time in equation (21.24), that is, PK k. Equation (21.32) is simply the diﬀerence between two geometric means of quality-adjusted prices.
The sample space m=Mt=Mt 1 is the same model in each period.

Con- sider the introduction of a new model n introduced in period t with no counterpart in t 1 and the demise of an old model o so it has no counterpart in t. So in period t, the sample space Mt is composed of the period t matched items m and the new items n, and in period t 1 Mt 1 is composed of the period t 1 matched items m and the old items.
Silver and Heravi (2002) have shown the dummy variable hedonic comparison to be: ln ( pt=pt1)=[m=(m+n)P m ( ln pt mZm)=m +n=(m+n)P n ( ln pt nZn)=n] m Zm)=m ( ln pt1 [m=(m+o)P m +o=(m+o)P ( ln pt1 o =[m=(m+n)P m ( ln pt o Zo)=o] mZm)=m ( ln pt1 m Zm)=m] ( ln pt nZn)=n m=(m+o)P m +[n=(m+n)P n o=(m+o)P o ( ln pt1 o Zo)=o] (21:33) QUALITY CHANGE AND HEDONICS 21.60 Consider the second expression in equation (21.33).
First there is the price change for m matched observations.

This is the change in mean prices of matched models m in period t and t 1 adjusted for quality.

Note that the weight in period t for this matched component is the proportion of matched observations to all observations in period t, both matched and un- matched new (n).

Similarly, for period t 1, the matched weight depends on how many matched and unmatched old observations (o) are in the sample.
In the last line of equation (21.33) the change is between the unmatched new and the unmatched old mean (quality-adjusted) prices in periods t and t 1.

Thus, the matched methods can be seen to ignore the last line in equation (21.33) and will thus diﬀer from the hedonic dummy variable approach.
It can be seen from equation (21.33) that the hedonic dummy variable approach, in its inclusion of unmatched old and new observations, may possibly diﬀer from a geometric mean of matched prices changes, in this un- the extent of any diﬀerence depending, weighted formulation, on the proportions of old and new items leaving and entering the sample and on the price changes of old and new items relative to those of matched items.
If the market for products is one in which old quality-adjusted prices are unusually low while new quality-adjusted prices are unusually high, then the matched index will understate price changes; see Silver and Heravi (2002) and Berndt, Ling and Kyle (2003) for examples.

Diﬀerent market behaviour will lead to diﬀerent forms of bias.

There is a second way in which the results will diﬀer.

Index number formulae provide weights for the price changes.
The Carli index, for example, weights each observation equally, while the Dutot index weights each observation according to its relative price in the base period.

The Jevons index, with no assumptions as to economic behaviour, weights each observation equally.

Silver (2002) has argued, however, that the weight given to each observation in an ordinary least squares regression also depends on the character- istics of the observations, some observations with unusual characteristics having more leverage.
In this way, the results from the two approaches may diﬀer even more.

New goods and services 21.61 This section brieﬂy highlights theoretical issues relating to the incorporation of new goods into the index.

Practical issues are outlined in Chapter 8, paragraphs 8.36 to 8.60.
The term ‘‘new goods’’ will be used here to refer to those that provide a substantial and substantive change in what is provided, as opposed to more of a currently available set of service ﬂows, such as a new model of a car that has a bigger engine.

In this latter instance, there is a continuation of a service and production ﬂow, and this may be linked to the service ﬂow and production technology of the existing model.
The practical concern with the deﬁnition of new goods as against quality changes is that the former cannot be easily linked to existing items as a continuation of an existing resource base and service ﬂow, because of the very nature of their ‘‘newness’’.

There are alternative 385 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE deﬁnitions; Oi (1997) directs the problem of deﬁning ‘‘new’’ goods to that of deﬁning a monopoly.

If there is no close substitute, the good is new.


A monopoly sup- plier may be able to supply an item with new combi- nations of the hedonic z characteristics resulting from a new technology and have a monopoly power in doing so, but in practice the new good can be linked via the set of hedonic characteristics to the existing goods.

In this practical sense, such goods are not considered ‘‘new’’ for the purposes of the manual.
21.62 The terminology adopted here is that used by Merkel (2000) for the measurement of producer price indices, but considered in the context of consumer price indices (CPIs).

The aim is to distinguish between evo- lutionary and revolutionary goods.

Evolutionary goods are replacement or supplementary models which con- tinue to provide a similar service ﬂow, perhaps in new ways or to diﬀerent degrees.
In contrast, revolutionary goods are entirely new goods not closely tied to a pre- viously available good.

Although revolutionary goods may satisfy a long-standing consumer need in a novel way, they do not ﬁt into any established CPI item category.

Thus, in principle, the underlying theory of what is meant by new goods applies as much to evolu- tionary as to revolutionary goods.
However, the prac- ticalities of index number construction imply a need to consider a new good in terms of something that is not an extension or a modiﬁcation of an existing one.

Evolu- tionary goods can be incorporated into an index by means of the methods discussed in Chapter 7, even though utility gains from their introduction are ignored.

This procedure has further problems.


Because the item is by its nature unique, it is unlikely to be incorporated into the sample as a replacement for an existing one.

It would be neither comparable nor amenable to explicit adjustments to its price for quality diﬀerences with existing goods.

Since it is not replacing an item, it does not have an existing weight and its introduction implies a need to reweight the index.
21.63 The main concern regarding the incorporation of new goods into the CPI is the decision on the need and timing for their inclusion.

Waiting for a new good to be established or waiting for the rebasing of an index before incorporating new products may lead to errors in the measurement of price changes if the unusual price movements at critical stages in the product life cycles are ignored.
There are practical approaches to the early adoption of both evolutionary and revolutionary goods, as outlined in Chapter 8.

For evolutionary goods, such strategies include the rebasing of the index, resampling of items and introduction of new goods as directed sample substitutions; see Merkel (2000).
Also of use are hedonic quality adjustments and indices outlined in Chapter 7, paragraphs 7.103 to 7.109 and 7.153 and 7.158, and paragraphs 21.37 to 21.60, which facilitate the incorporation of evolutionary goods.

These goods possess a similar characteristic set to existing goods, but deliver diﬀerent quantities of these characteristics.

Short-run or chained frameworks outlined in para- graphs 7.153 to 7.173 may also be more appropriate for product areas with a high turnover of items.
These approaches can incorporate the price change of new 386 goods into the index as soon as prices are available for two successive periods, though issues relating to the proper weighting of such changes may remain.

21.64 For revolutionary goods, however, substitu- tion may not be appropriate.

First, revolutionary goods may not be able to be deﬁned within the existing classi- ﬁcation systems.
Second, they may be primarily pro- duced by a new outlet, which will require extending the sample to such outlets.

Third, there will be no previous items to match them against to make a quality adjust- ment to prices, since by deﬁnition they are substantially diﬀerent from pre-existing goods.

Finally, there is no weight to attach to the new outlet or item.

Sample aug- mentation is appropriate for revolutionary goods, as opposed to sample substitution for evolutionary goods.
It is necessary to bring the new revolutionary goods into the sample in addition to what exists.

This may involve extending the classiﬁcation, the sample of out- lets, and the item list within new or existing outlets (Merkel, 2000).

21.65 The second measurement issue with respect to new products is the incorporation of the welfare eﬀect of those products at introduction.
The preceding dis- cussion has been concerned with the incorporation of price changes into the index once two successive quo- tations are available.

Yet there is a gain to the consumer when comparing the price in the ﬁrst of these periods with the price in the period that preceded its introduc- tion had it existed.
In the context of the CPI, the appropriate period 1 shadow price for the new good is that price that just induces the consumer of the new good to consume zero quantities in the preceding period.

This is a hypothetical price.

If it is relatively high in the period before the introduction of the good, but the actual price in the period of introduction is much lower, then the introduction of the new good is clearly of some beneﬁt to the consumer.
To ignore this beneﬁt, and the change from the virtual price to the actual price in its period of introduction, is to ignore something of the price movements that give rise to expenditure changes.

21.66 The sample augmentation procedures miss the eﬀects on price between the period preceding the intro- duction of a new good and its introduction.

There exist in economic theory and practice the tools for estimating such eﬀects; see Hicks (1940) and Diewert (1980, pp.

498–503).
498–503).

This involves setting a virtual price in the period before introduction.

This price is the one at which supply is set to zero.

The virtual price is compared with the actual price in the period of introduction and this is used to estimate the welfare gain from the introduction of the good.

Hausman (1997) provides some estimates of consumer welfare for the introduction of a new brand of breakfast cereals, Apple-Cinnamon Cheerios.
He con- cludes: The correct economic approach to the evaluation of new goods has been known for over ﬁfty years, since Hicks’ pioneering contribution.

However, it has not been implemented by government statistical agencies, perhaps because of its complications and data requirements.

Data are now available.


The impact of new goods on consumer welfare appears to be signiﬁcant according to the demand QUALITY CHANGE AND HEDONICS estimates of this paper, the CPI for cereal may be too high by about 25 per cent because it does not account for new cereal brands.

An estimate this large seems worth worrying about.

21.67 Shapiro and Wilcox (1997b, p. 144) share the same concerns: .

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the rare new item that delivers services radically diﬀerent from anything previously available.
For exam- ple, even the earliest generation of personal computers allowed consumers to undertake tasks that previously would have been prohibitively expensive.

This problem can be solved only by estimating the consumer surplus created by the introduction of each new item.

Hausman (1997) argues that this must involve explicit modeling of the demand for each new item.

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Although explicit modeling of demand may be of dubious practicality for widespread implementation in the CPI, strategic application in a few selected cases might be worthwhile.

21.68 The expertise required for such estimates is considerable, and even when applied, is not beyond dispute; see Bresnahan (1997) on this last point.
An alternative approach is outlined for the CPI by Balk (2000b), with empirical estimates provided by de Haan (2001), the details being provided in Chapter 8 and Appendix 8.2.

While this approach is simpler than that undertaken by Hausman (1997), both require consider- able statistical and econometric expertise.
The inclusion of such eﬀects on a routine basis is not something being actively considered, even by statistical oﬃces with well- developed systems.32 32 Even if virtual prices were estimated, there would still be problems with including new goods in indices such as the Laspeyres index because of the absence of weights in the base period.

387 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Appendix 21.1 Some econometric issues 1.
Hedonic regression estimates are seen in Chapter 7 to have a potential use for quality adjustment to prices.

A number of issues arise from the speciﬁcation and estimation of hedonic regressions, the use of diagnostic statistics, and courses of action when the standard assumptions of ordinary least squares (OLS) are seen to break down.

Many of these issues are stan- dard econometric ones and not the subject of this manual.

This is not to say they are unimportant.


The use of hedonic regres- sions requires some econometric and statistical expertise, but suitable texts are generally available; see Berndt (1991) – particularly the chapter on hedonic regressions – and Maddala (1988) and Kennedy (1998) amongst many others.

Modern statistical and econometric software has adequate diagnostic tests for testing when OLS assumptions break down.
There remain, however, some speciﬁc aspects which merit attention; these points are over and above the important standard econometric considerations dealt with in econometric texts.

Identiﬁcation and appropriate estimators 2.

Wooldridge (1996, pp.

400–401) has shown on standard econometric grounds that the estimation of supply and demand functions by OLS is biased and this bias carries over to the estimation of the hedonic function.
It is ﬁrst useful to consider estimation issues regarding demand and supply functions.

The demand and supply functions are rarely esti- mated in practice.
The more common approach is to estimate oﬀer functions, with the marginal price oﬀered by the ﬁrm dependent upon chosen attributes (product characteristics) and ﬁrm characteristics, and bid or value functions, with the marginal prices paid by a consumer dependent upon chosen attributes and consumer characteristics.33 As noted earlier, the observed prices and quantities are the result of the interaction between structural demand and supply equations and the distributions of producer technologies and consumer tastes; they cannot reveal the parameters of the oﬀer and value functions.
Rosen (1974, pp.

50–51) suggested a procedure for determining these parameters.

Since these estimates are con- ditioned on tastes (a) and technologies (t), the estimation procedure needs to include empirical measures or ‘‘proxy variables’’ of a and t. For the tastes a of consumers, the empiri- cal counterparts may be socio-demographic and economic variables, which may include age, income, education and geographical region.
For technologies t, variables may include types of technologies, scale and factor prices.

First, the hedonic equation is estimated in the normal manner, without these variables, using the best-ﬁtting functional form.

This is to represent the price function that consumers and producers face when making their decisions.

Then, an implicit marginal price function is computed for each characteristic as @p(z)= @zi=^ppi(z), where ^pp(z) is the estimated hedonic equation.
Bear in mind that in normal demand/supply studies for products, the prices are observed in the market.

Prices for characteristics are unobserved; this ﬁrst stage is to estimate the parameters from the hedonic regression.

The actual values of each zi bought and sold are then inserted into each implicit marginal price function to yield a numerical value for each character- istic.
These marginal values are used in the second stage34 of 33 These are equivalent to inverse demand (or supply) functions, with the prices dependent upon the quantities demanded (or supplied) and the individual consumer (or producer) characteristics.

34 This two-stage approach is common in the literature, though Wooldridge (1996) discusses the joint estimation of the hedonic, demand-side and supply-side functions as a system.
388 estimation as endogenous variables for the estimation of the demand side: ^ppi(z)=F(z1, .

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.

, zK , a*) (A21:1) where a* are the proxy variables for tastes; and the supply side: ^ppi(z)=F(z1, .

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, zK , t*) (A21:2) where t* are the proxy variables for technologies.

The variables t* drop out when there is no variation in technologies and ^ppi(z) is an estimate of the oﬀer function.
Similarly, the variables a* drop out when sellers diﬀer and buyers are identical, and cross-section estimates trace out compensated demand functions.

3.

Epple (1987) has argued that Rosen’s modelling strategy is likely to give rise to inappropriate estimation procedures of the demand and supply parameters.
In the hedonic approach to estimating the demand for characteristics, a diﬃculty arises from the fact that marginal prices are likely to be endogenous – they depend on the amount of each characteristic consumed and must be estimated from the hedonic function rather than observed directly.

There are two resulting problems.

First, there is an identiﬁcation problem (see Epple (1987)) because both the marginal price of a characteristic and the inverse bid depend on the levels of characteristics consumed.
Second, if important characteristics are unmeasured and they are corre- lated with measured characteristics, the coeﬃcients of mea- sured characteristics will be biased.

This applies to all econo- metric models, but is particularly relevant to hedonic models; on this point, see Wooldridge (1996, pp.

400–401) in particu- lar.

The equilibrium conditions for characteristic prices imply functional relationships among the characteristics of deman- ders, suppliers and products.
This in turn reduces the like- lihood that important excluded variables will be uncorrelated with the included variables of the model; see also Bartik (1988) on this point.
The bias arises because buyers are diﬀerentiated by characteristics (y, a) and sellers by technologies t. The type of item buyers will purchase is related to (y, a) and the type sellers provide to t. On the plane of combinations of z trans- acted, the equilibrium ones chosen may be systematically related; the characteristics of buyers are related to those of sellers.
Epple (1987) uses the example of stereo equipment: the higher income of some buyers leads to purchases of high- quality equipment, and the technical competence of sellers leads them to provide it.

The consumer and producer char- acteristics may be correlated.

4.

Wooldridge (1996, pp.

400–401) suggests that individual consumer and ﬁrm characteristics, such as income, education and input prices, should be used as instruments in estimating hedonic functions.
In addition, variables other than a good’s characteristics should be included as instruments if they are price-determining, such as geographical location (proximity to ports, good road systems, climate and so on).

Communities of economic agents are assumed, within which consumers con- sume and producers produce for each other at prices that vary across communities for identical goods.
Variables of the characteristics of the communities will not in themselves enter the demand and supply equation, but are price-determining for observed prices that are recorded across communities.

Tau- chen and Witte (2001) provide a systematic investigation of the conditions under which the characteristics of consumers, pro- ducers and communities will aﬀect the hedonic parameter estimates for a single regression equation estimated across all communities.
A key concern is whether the error term of the hedonic price function represents factors that are unobserved by both the economic agents and the researcher, or only by the researcher.

In the latter case, the error term may be correlated with the product attributes; instrumental variable estimation is required.
If the error term is not correlated with the product characteristics – preferences are quasi-linear – then a properly speciﬁed hedonic regression, including community-speciﬁc characteristics or appropriate slope dummies, can be estimated using ordinary least squares.
In other cases, depending on the correlation between consumer and producer character- istics, assumptions about the error term and the method of incorporating community characteristics into the regression, instrumental variables, including consumer, producer or com- munity dummy or characteristics, may need to be used.

Functional form 5.
Functional form 5.

Triplett (1987 and 2002) argues that neither classical utility theory nor production theory can specify the functional form of the hedonic function.35 This point dates back to Rosen (1974, p. 54), who describes the observations as being ‘‘ .

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a joint-envelope function and cannot by themselves identify the structure of consumer preferences and producer technologies that generate them’’.
A priori judgements as to what the form should look like may be based on ideas as to how consumers and production technologies respond to price changes.

These judgements are diﬃcult to make when the observations are jointly determined by demand and supply factors, but not impossible in rare instances.

They are, however, complicated when pricing is with a mark-up, the extent of which may vary over the life cycle of a product.

Some tied combinations of characteristics will have higher mark-ups than others.
New item introductions are likely to be attracted to these areas of characteristic space, and this will have the eﬀect of increasing supply and thus lowering the mark-up and price; see Cockburn and Anis (1998), Feenstra (1995, p. 647) and Triplett (1987).

This again must be taken into account in any a priori reasoning – not an easy or straightforward matter.

6.

It may be that in some cases the hedonic function’s functional form will be straightforward.
For example, prices on the web sites for options for products are often additive.

The underlying cost and utility structures are unlikely jointly to generate such linear functions, but the producer or con- sumer is also paying for the convenience of selling in this way and is willing to bear losses or make gains if the cost or utility at higher values of z are priced lower or worth more than the price set.
In general, the data should convey what the func- tional form should look like; imposing artiﬁcial structures simply leads to speciﬁcation bias.

For examples of econometric testing of hedonic functional form, see Cassel and Mendelsohn (1985), Cropper, Deck and McConnell (1988), Rasmussen and Zuehlke (1990), Bode and van Dale´ n (2001) and Curry, Morgan and Silver (2001).

7.

The three forms prevalent in the literature are linear, semi-logarithmic and double-logarithmic (log-log).
A number of studies have used econometric tests, in the absence of a clear theoretical statement, to choose between them.

There have been a large number of hedonic studies and, as illustrated by Curry, Morgan and Silver (2001), in many of these the quite simple forms do well, at least in terms of the RR2 presented36 and the parameters according with a priori reasoning, usually on the consumer side.
Of the three popular forms, some are QUALITY CHANGE AND HEDONICS found that the three functional favoured in testing; for example, Murray and Sarantis (1999) favoured the semi-logarithmic form, while others, for example Hoﬀmann (1998), forms scarcely diﬀered in terms of their explanatory power.

That the parameters from these simple forms accord with a priori rea- soning, usually from the consumer side, is promising, but researchers should be aware that such matters are not assured.
There is much that may happen on the supply side to aﬀect parameter values.

Indeed Pakes (2001) has argued that no intuitive sign can be given to the parameters of the variables, since producers may vary their price mark-ups on character- istics in ways that would result in counter-intuitive negative signs on some desirable characteristics.

8.

Of the three forms, the semi-logarithmic form has much to commend it.
The interpretation of its coeﬃcients is quite straightforward, as proportionate changes in prices arise from a unit change in the value of the characteristic37 (see Chapter 7, paragraphs 7.39 and 7.40).

This is a useful for- mulation since quality adjustments are usually undertaken by making multiplicative as opposed to additive adjustments.

9.

The semi-logarithmic form, unlike the log-log model, can incorporate dummy variables for characteristics which are either present, zi=1, or not, zi=0.
Furthermore, Diewert (2002e) has argued that it is more likely that the errors from a semi-logarithmic hedonic equation are homoskedastic (have a constant variance) compared to the errors from a linear hedonic equation, since items with very large characteristic values will have high prices and are very likely to have rela- tively large error terms.
On the other hand, models with very small amounts of characteristics will have small prices and small means, and the deviation of a model price from its mean will necessarily be small.

Since an assumption of OLS is that the residuals are homoskedastic, the semi-logarithmic equation is preferred to the linear one.

10.

More complicated forms are, of course, possible.

Simple forms have the virtue of parsimony and allow more eﬃcient estimates to be made for a given sample.
However, parsimony is not something to be achieved at the cost of misspeciﬁcation bias.

First, if the hedonic function is estimated across multiple independent markets, then interaction terms are required (see Mendelsohn (1984) for ﬁshing sites).

Excluding them is tanta- mount to omitting variables and inappropriately constraining the estimated coeﬃcients of the regression.

Tauchen and Witte (2001) have outlined the particular biases that can arise for such omitted variables in hedonic studies.
Second, it may be argued that the functional form should correspond to the aggregator for the index – linear for a Laspeyres index, logarithmic for a geometric Laspeyres index, translogarithmic for a To¨ rnqvist index, and quadratic for a Fisher index (see Chapter 17).
As Triplett (2002) notes, however, the purpose of estimating hedonic regressions is to adjust prices for quality diﬀerences; imposing a functional form on the data which is inconsistent with the data might create an error in the quality adjustment procedure.
Yet, as Diewert (2003a) notes, ﬂexible functional forms encompass these simple forms, the log-log form being a special case of the translog form given in equation (17.42) and the semi-log form being a special case of the semi-log quadratic form given in equation (17.49).
If there are a priori reasons 35 Although Arguea, Haseo and Taylor (1994) propose a linear form on the basis of arbitrage for characteristics, held to be likely in com- petitive markets, Triplett (2002) argues that this is unlikely to be a realistic scenario in most commodity markets.
36 While the use of RR2 as a criterion for deciding between the ﬁt of semi-logarithmic and log-log models has some validity, its use is not advised for comparing linear models with either of these logarithmic formulations, a number of tests being appropriate for such compar- isons; see Maddala (1988).

37 There are two caveats: ﬁrst, eb^1 is required for the interpretation of the coeﬃcients, where b^ is the estimated coeﬃcient.
Second, the anti- logarithms of the OLS estimated coeﬃcients are not unbiased – the estimation of semi-logarithmic functions as transformed linear regres- sions requires an adjustment to provide minimum variance unbiased estimates of parameters of the conditional mean.

A standard adjust- ment is to subtract half of the coeﬃcient’s squared standard error from the estimated coeﬃcient; see Goldberger (1968) and Teekens and Koerts (1972).
389 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE to expect interaction terms for speciﬁc characteristics, as illus- trated in the example in paragraph 7.99, then these more general forms allow this.

The theory of hedonic functions neither dictates the form of the hedonic form nor restricts it.

the use of a single period’s characteristics set, say the current period, should be treated with caution.
More regular updating of the hedonic regressions is likely to reduce spread because the periods being compared will be closer and the characteristics of the items in the periods compared more similar.

Changing tastes and technologies the coeﬃcients 11.

The estimates of from a hedonic regression may change over time.

Some of this change will be attributed to sampling error, especially if multicollinearity is present, as discussed below.
But in other cases it may be a genuine reﬂection of changes in tastes and technologies.

If a subset of the estimated coeﬃcients from a hedonic regression is to be used to make a quality adjustment to a non-comparable replacement price, then the use of estimated out-of-date coef- ﬁcients from some previous period to adjust the prices of the new replacement model may be inappropriate.
There is a need to update the indices as regularly as the changes demanded.38 Estimating hedonic imputation indices is more complicated.

Silver (1999), using a simple example, showed how the estimate of quality-adjusted price changes requires a reference basket of characteristics.

This is apparent for the hedonic imputation indices in paragraphs 21.37 to 21.60, where separate indices using base and current period characteristics are estimated.
A symmetric average of such indices is considered appropriate.

A hedonic index based on a time dummy variable implicitly constrained the estimated coeﬃcients from the base and cur- rent periods to be the same.

Diewert (2003a) formalized the problem of choosing the reference characteristics when com- paring prices over time, when the parameters of the hedonic function may themselves be changing over time.
He found the results of hedonic indices not to be invariant to the choice of reference period characteristic vector set z.
He considered the use of a sales-weighted average vector of characteristics, as proposed by Silver (1999), but he notes that over long time periods this may become unrepresentative.39 Of course, if a chained formulation is used, the weighted averages of char- acteristics remain reasonably up to date, although chaining has its own pros and cons (see paragraph 17.44 to 17.49 of Chapter 17).
A ﬁxed base alternative noted by Diewert (2003a) is to use a Laspeyres type comparison with the base period parameter set, and a Paasche type current period index with the current period parameter set, and take the geometric mean of the two indices for reasons similar to those given in Chapter 15, paragraphs 15.18 to 15.32.
The resulting Fisher type index is akin to a geometric mean of Laspeyres and Paasche indices – given in equations (21.26) and (21.27) – based on Feenstra (1995).40 A feature of the time dummy approach in paragraphs 21.40 to 21.42 is that it implicitly takes a symmetric average of the coeﬃcients by constraining them to be the same.

But what if, as is more likely the case, only base period hedonic regres- sion coeﬃcients are available?
Since hedonic indices based on a symmetric average of the coeﬃcients are desirable, the ‘‘spread’’ or diﬀerence between estimates based on either a current or a reference period characteristic set is an indication of potential bias and estimates of such spread may be under- taken retrospectively.

If the spread is large, estimates based on Weighting 12.
Ordinary least squares estimators implicitly treat each item as being of equal importance, even though some items will have quite substantial sales, while sales of others will be minimal.

It is axiomatic that an item with sales of over 5,000 in a month should not be accorded the same inﬂuence in the regression estimator as one with a few transactions.

Items with very low sales may be at the end of their life cycles or be custom-made.
Either way, their (quality-adjusted) prices and price changes may be unusual.41 Observations with unusual prices should not be allowed unduly to inﬂuence the index.42 13.

The estimation of hedonic regression equations by a weighted least squares (WLS) estimator is preferable.
This estimator minimizes the sum of weighted squared deviations between the actual prices and the predicted prices from the regression equation, as opposed to ordinary least squares (OLS), which uses an equal weight for each observation.

There is a question as to whether to use quantity (volume) or expenditure weights.

The use of quantity weights can be sup- ported by considering the nature of their equivalent ‘‘price’’.

Such prices are the average (usually the same) price over a number of transactions.
The underlying sampling unit is the individual transaction, so there is a sense that the data may be replicated as being composed of, say, 12 individual observa- tions using an OLS estimator, as opposed to a single obser- vation with a weight of 12 using a WLS estimator.

Both would yield the same result.

Diewert (2002e) has argued on the grounds of representativity that sales values are the appro- priate weights.
Quantity weighting gives too little weight to models with high prices and too much weight to cheap models that have relatively low amounts of useful characteristics.

The need to equate the weights with relative expenditure or sales value arises from a prime concern with index numbers: that they serve to decompose changes in value into their price and quantity components.
Silver (2002) has shown that a WLS estimator using value weights will not necessarily give each observation a weight equal to its relative value.

The estimator will give more weight to those observations with high leverage eﬀects and residuals.

Observations with values of character- istics with large deviations from their means, say very old or new models, have relatively high leverage.
New and old models are likely to be priced at quite diﬀerent prices than those predicted from the hedonic regression, even after taking into account their diﬀerent characteristics.

Such prices result, for example, from a pricing strategy designed to skim segments of the market willing to pay a premium for a new model, or from a strategy to charge relatively low prices for an old model to dump it to make way for a new one.
In such cases, the inﬂu- ence these models have on deriving the estimated coeﬃcients will be over and above that attributable to their value weights.

38 Adjusting the base versus the current period price entails diﬀerent data demands; see Chapter 7, paragraph 7.49.

39 Other averages may of course be proposed; for example, the needs of an index representative of the ‘‘typical’’ establishment would be better met by a trimmed mean or median.
40 Diewert (2002e) also suggests matching items where possible, and using hedonic regressions to impute the prices of the missing old and new ones.

Diﬀerent forms of weighting systems, including superlative ones, can be applied to this set of price data in each period for both matched and unmatched data.

41 Such observations would have higher variances of their error terms, leading to imprecise parameter estimates.
This would argue for the use of weighted least squares estimators with quantity sold as the weight.

This is one of the standard treatments for heteroskedastic errors; see Berndt (1991).

42 Silver and Heravi (2002) show that old items have above-average leverage eﬀects and below-average residuals.

Not only are they dif- ferent, but they exert undue inﬂuence for their size (number of observations).

See Berndt, Ling and Kyle (2003), Cockburn and Anis (1998) and Silver and Heravi (2002) for examples.
390 Silver (2002) suggests that leverage eﬀects should be calculated for each observation, and those with high leverage and low weights should be deleted, and the regression rerun.

Thus, while quantity or value weights are preferable to no weights (i.e., OLS), value weights are more appropriate than quantity ones and, even so, account should be taken of observations with undue inﬂuence.

14.
14.

Diewert (2002e) has also considered the issue of weighting with respect to the dummy time hedonic indices outlined in paragraphs 21.40 to 21.42.

The use of WLS by value involves weights being applied to observations in both periods.

However, if, for example, there is high inﬂation then the sales values for a model in the current period will generally be larger than those of the corresponding model in the base period and the assumption of homoskedastic residuals is unlikely to be met.
Diewert (2002e) suggests the use of expenditure shares in each period, as opposed to values, as weights for WLS for time dummy hedonic indices.

He also suggests that an average of expenditure shares in the periods being compared be used for matched models.

15.

Data on sales are not always available for weights, but the major selling items can generally be identiﬁed.
In such cases, it is important to restrict the number of observations of items with relatively low sales, the extent of the restriction depending on the number of observations and the skewness of the sales distribution.

In some cases, items with few sales provide the variability necessary for eﬃcient estimates of the regression equation.

In other cases, their low sales may be due to factors that make them unrepresentative of the hedonic surface, their residuals being unusually high.
An example is low-selling models about to be dumped to make way for new models.

Unweighted regressions may thus suﬀer from a sampling problem – even if the prices are perfectly quality adjusted, the index can be biased because it is unduly inﬂu- enced by low-selling items with unrepresentative price–char- acteristic relationships.
In the absence of weights, regression diagnostics have a role to play in helping to determine whether the undue variance in some observations belongs to such unusual low-selling items.43 Multicollinearity 16.

There are a priori reasons to expect, for some products, that the variation in the value of one quality characteristic is not independent of one quality characteristic or a linear combination of more than one such characteristic.

As a result, imprecise.


estimates will be unbiased yet parameter To illustrate this, a plot of the conﬁdence interval for one para- meter estimate against another collinear one is often described as elliptical, since the combinations of possible values they may take can easily drift from, say, high values of b1 and low values of b2 to high values of b2 and low values of b1.
Since the sample size for the estimates is eﬀectively reduced, additions to and deletions from the sample may aﬀect the parameter estimates more than would be expected.
These are standard statistical issues, dealt with by Maddala (1988) and Kennedy 43 A less formal procedure is to take the standardized residuals from the regression and plot them against model characteristics that may denote low sales, such as certain brands (makes) or vintage (if not directly incorporated), or some technical feature which makes it un- likely that the item is being bought in quantity.

Higher variances may be apparent from the scatter plot.
If certain features are expected to have, on average, low sales, but seem to have high variances, leverages and residuals (see Silver and Heravi (2002)), a case exists for at least down-playing their inﬂuence.

Bode and van Dale´ n (2001) use formal statistical criteria to decide between diﬀerent weighting systems and compare the results of OLS and WLS, ﬁnding, as with Ioannidis and Silver (1999), that diﬀerent results can arise.

QUALITY CHANGE AND HEDONICS (1998).


In a hedonic regression, multicollinearity might be expected, as some characteristics may be technologically tied to others.

Producers including one characteristic may need to include others for the product to work, while consumer pur- chasing, say, an up-market brand may expect a certain bundle of features to come with it.

Triplett (2002) argues strongly for the researcher to be aware of the features of the product and the consumer market.
There are standard, though not completely reliable, indicators of multicollinearity (such as variance inﬂation factors), but an exploration of its nature is greatly aided by an understanding of the market along with exploration of the eﬀects of including and excluding individual variables on the signs and coeﬃcients and on other diagnostic test statistics; see Maddala (1988).44 17.
If a subset of the estimated coeﬃcients from a hedonic regression is to be used to quality-adjust a non-comparable replacement price, and if there is multicollinearity between variables in this subset and other independent variables, then the estimates of the coeﬃcients to be used for the adjustment will be imprecise.

The multicollinearity eﬀectively reduces the sample size, and some of the eﬀects of the variables in the subset may be wrongly ascribed to the other independent variables.
The extent of this error will be determined by the strength of the multiple correlation coeﬃcient between all such ‘‘independent’’ variables (the multicollinearity), the standard error or ﬁt of the regression, the dispersion of the independent variable concerned and the sample size.

These all aﬀect the estimates since they are components in the standard error of the t-statistics.
Even if multicollinearity is expected to be quite high, large sample sizes and a well-ﬁtting model may reduce the standard errors on the t-statistics to acceptable levels.

If multicollinearity is expected to be severe, the predicted value for an item’s price may be computed using the whole regression and an adjust- ment made using this predicted value, as explained in Chapter 17, paragraphs 17.103 to 17.109.
There is a sense in which it does not matter whether the variation that, for example, should have been attributed to b1 was wrongly attributed to b2, or vice versa if the predicted price based on both b1 and b2 is used.

the precision of Omitted variable bias 18.

The exclusion of tastes, technology and community characteristics has already been discussed.

The concern here is with product characteristics.


Consider again the use of a subset of the estimated coeﬃcients from a hedonic regression to quality-adjust a non-comparable replacement price.

It is well established that multicollinearity of omitted variables with included variables leads to bias in the estimates of the coeﬃcients of included ones.

If omitted variables are inde- pendent of the included variables, then the estimates of the coeﬃcients on the included variables are unbiased.
This is acceptable in the present instance, the only caveat being that the quality adjustment for the replacement item may also require an adjustment for these omitted variables and this, as noted by Triplett (2002), has to be undertaken using a separate method and data.

But what if the omitted variable is multi- collinear with a subset of included variables which are to be used to quality-adjust a non-comparable item?
In this case, the coeﬃcient of the subset of the included variables may wrongly pick up some of the eﬀects of the omitted variables.

The subset of included variables will be used to quality-adjust prices for items which diﬀer only with regard to this subset, and the price comparison will be biased if the characteristics of included and 44 Triplett (2002) stresses the point that RR2 alone is insuﬃcient for this purpose.
391 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE omitted variables have diﬀerent price changes.

For hedonic indices using a dummy time trend, the estimates of quality- adjusted price changes will suﬀer from a similar bias if omitted variables that are multicollinear with the time change are excluded from the regression.

What are picked up as quality- adjusted price changes over time may, in part, be changes attributable to the prices of these excluded variables.
This happens when the prices of the omitted characteristics follow a diﬀerent trend.

Such eﬀects are most likely when there are gradual improvements in the quality of items, such as the reliability and safety of consumer durables,45 which are diﬃcult to measure, at least for the sample of items in real time.

The quality-adjusted price changes will thus overstate price changes in such instances.
392 45 There are, of course, some commodity areas, such as airline comfort, which have been argued to have overall patterns of decreasing quality.

THE TREATMENT OF SEASONAL PRODUCTS 22 Introduction 22.1 The existence of seasonal commodities poses some signiﬁcant challenges for price statisticians.
Sea- sonal commodities are commodities which are either: (a) not available in the marketplace during certain seasons of the year, or (b) are available throughout the year, but there are regular ﬂuctuations in prices or quantities that are synchronized with the season or the time of the year.1 A commodity that satisﬁes (a) is termed a strongly seasonal commodity, whereas a commodity that satisﬁes (b) is called a weakly seasonal commodity.
It is strongly seasonal commodities that create the biggest problems for price statisticians in the context of producing a monthly or quarterly consumer price index (CPI) because if a commodity price is available in only one of the two months (or quarters) being compared, then obviously it is not possible to calculate a relative price index for the commodity and traditional bilateral number theory breaks down.
In other words, if a com- modity is present in one month but not the next, how can the month-to-month amount of price change for that commodity be computed?2 In this chapter, a solu- tion to this problem is presented which ‘‘works’’, even if the commodities consumed are entirely diﬀerent for each month of the year.3 22.2 There are two main sources of seasonal ﬂuc- tuations in prices and quantities: (a) climate, and (b) custom.4 In the ﬁrst category, ﬂuctuations in tempera- ture, precipitation and hours of daylight cause ﬂuctu- ations in the demand or supply for many commodities; for example, summer versus winter clothing, the demand for light and heat, holidays, etc.
With respect to custom and convention as a cause of seasonal ﬂuctuations, consider the following quotation: 1 This classiﬁcation of seasonal commodities corresponds to Balk’s narrow and wide sense seasonal commodities; see Balk (1980a, p. 7; 1980b, p. 110; 1980c, p. 68).

Diewert (1998b, p. 457) used the terms type 1 and type 2 seasonality.
2 Victor Zarnowitz (1961, p. 238) was perhaps the ﬁrst to note the importance of this problem: ‘‘But the main problem introduced by the seasonal change is precisely that the market basket is diﬀerent in the consecutive months (seasons), not only in weights but presumably often also in its very composition by commodities.
This is a general and complex problem which will have to be dealt with separately at later stages of our analysis.’’ 3 The same commodities must, however, reappear each year for each separate month.

4 This classiﬁcation dates back to Wesley C. Mitchell (1927, p. 236) at least: ‘‘Two types of seasons produce annually recurring variations in economic activity – those which are due to climates and those which are due to conventions’’.
Conventional seasons have many origins – ancient religious observances, folk customs, fashions, business practices, statute law .

.

.

Many of the conventional sea- sons have considerable eﬀects on economic behaviour.
We can count on active retail buying before Christmas, on the Thanksgiving demand for turkeys, on the ﬁrst of July demand for ﬁreworks, on the preparations for June weddings, on heavy dividend and interest payments at the beginning of each quarter, on an increase in bankruptcies in January, and so on (Mitchell (1927, p. 237)).
22.3 Examples of important seasonal commodities are: many food items; alcoholic beverages; many cloth- ing and footwear items; water; heating oil; electricity; ﬂowers and garden supplies; vehicle purchases; vehicle operation; many entertainment and recreation expendi- tures; books; insurance expenditures; wedding expendi- tures; recreational equipment; toys and games; software; air travel and tourism expenditures.
For a ‘‘typical’’ country, seasonal expenditures will often amount to one-ﬁfth to one-third of all consumer expenditures.5 22.4 In the context of producing a monthly or quarterly CPI, it must be recognized that there is no completely satisfactory way of dealing with strongly seasonal commodities.
If a commodity is present in one month but missing from the marketplace in the next month, then none of the index number theories that were considered in Chapters 15 to 20 can be applied because all these theories assumed that the dimension- ality of the commodity space was constant for the two periods being compared.
However, if seasonal com- modities are present in the market during each season, then, in theory, traditional index number theory can be applied in order to construct month-to-month or quar- ter-to-quarter price indices.

This ‘‘traditional’’ approach to the treatment of seasonal commodities will be fol- lowed in paragraphs 22.78 to 22.90.
The reason why this straightforward approach is deferred to the end of the chapter is twofold: The approach that restricts the index to commodities that are present in every period often does not work well in the sense that systematic biases can occur.

 The approach is not fully representative; i.e., it does not make use of information on commodities that are not present in every month or quarter.
5 Alterman, Diewert and Feenstra (1999, p. 151) found that over the 40 months between September 1993 and December 1996, somewhere between 23 and 40 per cent of United States imports and exports exhibited seasonal variations in quantities, whereas only about 5 per cent of United States export and import prices exhibited seasonal ﬂuctuations.

393 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE 22.5 In the next section, a modiﬁed version of Tur- vey’s (1979) artiﬁcial data set is introduced.
This data set will be used in order to evaluate numerically all the index number formulae suggested in this chapter.

It will be seen in paragraphs 22.63 to 22.77 that very large seasonal ﬂuctuations in volumes, combined with systematic sea- sonal changes in price, can make month-to-month or quarter-to-quarter price indices behave rather poorly.
22.6 Even though existing index number theory cannot deal satisfactorily with seasonal commodities in the context of constructing month-to-month indices of consumer prices, it can deal satisfactorily with seasonal commodities if the focus is changed from month-to- month CPIs to CPIs that compare the prices of one month with the prices of the same month in a previous year.

Thus, in paragraphs 22.16 to 22.34, year-over-year monthly CPIs are studied.
Turvey’s seasonal data set is used to evaluate the performance of these indices and they are found to perform quite well.

22.7 In paragraphs 22.35 to 22.44, the year-over- year monthly indices deﬁned in paragraphs 23.16 to 23.34 are aggregated into an annual index that compares all the monthly prices in a given calendar year with the corresponding monthly prices in a base year.
In para- graphs 22.45 to 22.54, this idea of comparing the prices of a current calendar year with the corresponding prices in a base year is extended to annual indices that compare the prices of the last 12 months with the corresponding prices in the 12 months of a base year.

The resulting rolling year indices can be regarded as seasonally adjusted price indices.

The modiﬁed Turvey data set is used to test out these year-over-year indices, and they are found to work very well on this data set.
22.8 The rolling year indices can provide an accu- rate gauge of the movement of prices in the current rolling year compared to the base year.

This measure of price inﬂation can, however, be regarded as a measure of inﬂation for a year that is centred around a month six months prior to the last month in the current rolling year.
Hence for some policy purposes, this type of index is not as useful as an index that compares the prices of the current month to the previous month, so that more up-to-date information on the movement of prices can be obtained.
In paragraphs 22.55 to 22.62, it will nevertheless be shown that under certain conditions, the year-over-year monthly index for the current month, along with the year-over-year monthly index for last month, can successfully predict or forecast a rolling year index that is centred around the current month.
22.9 The year-over-year indices deﬁned in para- graphs 22.16 to 22.34, and their annual averages studied in paragraphs 22.35 to 22.54, oﬀer a theoretically satis- factory method for dealing with strongly seasonal com- modities; i.e., commodities that are available only during certain seasons of the year.
These methods rely on the year-over-year comparison of prices and hence cannot be used in the month-to-month or quarter-to-quarter type of index, which is typically the main focus of a consumer price programme.

Thus there is a need for another type of index, which may not have very strong theoretical foun- dations, but which can deal with seasonal commodities in the context of producing a month-to-month index.
In 394 paragraphs 22.63 to 22.77, such an index is introduced and it is implemented using the artiﬁcial data set for the commodities that are available during each month of the year.

Unfortunately, because of the seasonality in both prices and quantities of the always available commod- ities, this type of index can be systematically biased.

This bias shows up for the modiﬁed Turvey data set.
22.10 Since many CPIs are month-to-month indices that use annual basket quantity weights, this type of index is studied in paragraphs 22.78 to 22.84.

For months when the commodity is not available in the marketplace, the last available price is carried forward and used in the index.

In paragraphs 22.85 and 22.86, an annual quantity basket is again used but instead of carrying forward the prices of seasonally unavailable items, an imputation method is used to ﬁll in the missing prices.
The annual basket type indices deﬁned in paragraphs 22.78 to 22.84 are implemented using the artiﬁcial data set.

Unfortu- nately, the empirical results are not satisfactory in that the indices show tremendous seasonal ﬂuctuations in prices, so they would not be suitable for users who wanted up-to-date information on trends in general inﬂation.
22.11 In paragraphs 22.87 to 22.90, the artiﬁcial data set is used in order to evaluate another type of month-to-month index that is frequently suggested in the literature on how to deal with seasonal commodities; namely the Bean and Stine Type C (1924) or Rothwell (1958) index.

Again, this index does not get rid of the tremendous seasonal ﬂuctuations that are present in the modiﬁed Turvey data set.
22.12 Paragraphs 22.78 to 22.84 show that the annual basket type indices with carry forward of missing prices or imputation of missing prices do not get rid of seasonal ﬂuctuations in prices.

However, in paragraphs 22.91 to 22.96, it is shown how seasonally adjusted ver- sions of these annual basket indices can be used suc- cessfully to forecast rolling year indices that are centred on the current month.
In addition, the results show how these annual basket type indices can be seasonally adjusted (using information obtained from rolling year indices from prior periods or by using traditional sea- sonal adjustment procedures), and hence these seasonally adjusted annual basket indices could be used as suc- cessful indicators of general inﬂation on a timely basis.

22.13 Paragraph 23.97 outlines some conclusions.


A seasonal commodity data set 22.14 It is useful to illustrate the index number for- mulae deﬁned in subsequent sections by computing them for an actual data set.

Turvey (1979) constructed an artiﬁcial data set for ﬁve seasonal commodities (apples, peaches, grapes, strawberries and oranges) for four years by month so that there are 5 4 12=240 observations in all.
At certain times of the year, peaches and straw- berries (commodities 2 and 4) are unavailable, so in Tables 22.1 and 22.2 the prices and quantities for these two commodities are entered as zeros.6 The data in 6 The corresponding prices are not zeros, but they are entered as zeros for convenience in programming the various indices.
THE TREATMENT OF SEASONAL PRODUCTS Table 22.1 An artiﬁcial seasonal data set: Prices Table 22.2 An artiﬁcial seasonal data set: Quantities Year t Month m pt, m pt , m pt, m pt , m pt, m Year t Month m qt, m 1 3086 3765 4363 4842 4439 5323 4165 3224 4025 5784
qt, m qt, m 252
202 3269 6111 5964 2824 731 250 102 30 25 220 252 266 4813 8803 8778 4517 1073 qt, m qt, m Tables 22.1 and 22.2 are essentially the same as the data set constructed by Turvey except that a number of adjustments have been made to it in order to illustrate various points.
The two most important adjustments are: The data for commodity 3 (grapes) have been adjus- ted so that the annual Laspeyres and Paasche indices (deﬁned in paragraphs 22.35 to 22.44) would diﬀer more than in the original data set.7 7 After the ﬁrst year, the price data for grapes has been adjusted downward by 30 per cent each year and the corresponding volume has been adjusted upward by 40 per cent each year.
In addition, the quantity of oranges (commodity 5) for November 1971 has been changed from 3,548 to 8,548 so that the seasonal pattern of change for this commodity is similar to that of other years.

For similar reasons, the price of oranges in December 1970 has been changed from 1.31 to 1.41 and in January 1971 from 1.35 to 1.45.
After the above adjustments were made, each price in the last year of data was escalated by the monthly inﬂation factor 1.008 so that month-to-month inﬂation for the last year of data would be at an approximate monthly rate of 1.6 per cent per month compared to about 0.8 per cent per month for the ﬁrst three years of data.8 8 Pierre Duguay of the Bank of Canada, while commenting on a pre- liminary version of this chapter, observed that rolling year indices would not be able to detect the magnitude of systematic changes in the month-to-month inﬂation rate.
The original Turvey data set was roughly consistent with a month-to-month inﬂation rate of 0.8 per cent per month; i.e., prices grew roughly at the rate of 1.008 each month over the four years of data.

This second major adjustment of the Turvey data was introduced to illustrate Duguay’s observation, which is quite cor- rect: the centred rolling year indices pick up the correct magnitude of the new inﬂation rate only after a lag of half a year or so.
They do, however, quickly pick up the direction of change in the inﬂation rate.

395 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE 22.15 Ralph Turvey sent his artiﬁcial data set to statistical agencies around the world, asking them to use their normal techniques to construct monthly and annual average price indices.
About 20 countries replied, and Turvey (1979, p. 13) summarized the responses as follows: ‘‘It will be seen that the monthly indices display very large diﬀerences, e.g., a range of 129.12–169.50 in June, while the range of simple annual means is much smaller.

It will also be seen that the indices vary as to the peak month or year.’’ The above (modiﬁed) data are used to test out var- ious index number formulae in subsequent sections.
Year-over-year monthly indices 22.16 It can be seen that the existence of seasonal commodities that are present in the marketplace in one month but not the next causes the accuracy of a month- to-month index to fall.9 A way of dealing with these strongly seasonal commodities is to change the focus from short-term month-to-month price indices and instead focus on making year-over-year price compar- isons for each month of the year.
In the latter type of comparison, there is a good chance that seasonal com- modities that appear, say, in February will also appear in subsequent Februarys so that the overlap of com- modities will be maximized in these year-over-year monthly indices.

22.17 For over a century, it has been recognized that making year-over-year comparisons10 provides the simplest method for making comparisons that are free from the contaminating eﬀects of seasonal ﬂuctuations.
According to W. Stanley Jevons (1884, p. 3): In the daily market reports, and other statistical pub- lications, we continually ﬁnd comparisons between num- bers referring to the week, month, or other parts of the year, and those for the corresponding parts of a previous year.

The comparison is given in this way in order to avoid any variation due to the time of the year.

And it is obvious to everyone that this precaution is necessary.
Every branch of industry and commerce must be aﬀected more or less by the revolution of the seasons, and we must allow for what is due to this cause before we can learn what is due to other causes.

22.18 The economist A.W.

Flux and the statistician G. Udny Yule also endorsed the idea of making year- over-year comparisons to minimize the eﬀects of sea- sonal ﬂuctuations: Each month the average price change compared with the corresponding month of the previous year is to be computed.

.

.

.
.

.

.

The determination of the proper seasonal variations of weights, especially in view of the liability of seasons to vary from year to year, is a task from which, I imagine, most of us would be tempted to recoil (Flux (1921, pp.

184–185)).

My own inclination would be to form the index number for any month by taking ratios to the corre- sponding month of the year being used for reference, the year before presumably, as this would avoid any diﬃ- culties with seasonal commodities.
I should then form the annual average by the geometric mean of the monthly ﬁgures (Yule (1921, p. 199)).

In more recent times, Victor Zarnowitz (1961, p. 266) also endorsed the use of year-over-year monthly indices: There is of course no diﬃculty in measuring the aver- age price change between the same months of successive years, if a month is our unit ‘‘season’’, and if a constant seasonal market basket can be used, for traditional methods of price index construction can be applied in such comparisons.
22.19 In the remainder of this section, it is shown how year-over-year Fisher indices and approximations to them can be constructed.11 For each month m= 1, 2, .

.

.

,12, let S(m) denote the set of commodities that are available in the marketplace for each year t=0, 1, .

.

.

, T. For t=0, 1, .

.

.

, T and m=1, 2, .

.

.

,12, let pt, m and qt, m denote the price and quantity of commodity n that is in the marketplace in month m of year t, where n belongs to S(m).
Let pt, m and qt, m denote the month m and year t price and quantity vectors, respectively.

Then the year- over-year monthly Laspeyres, Paasche and Fisher indices going from month m of year t to month m of year t+1 can be deﬁned as follows: n n PL( pt, m, pt+1, m, qt, m)= P pt+1, m n qt, m n n 2 SðmÞ P n qt, m pt, m n m=1, 2, .

.

.

, 12 (22:1) n 2 SðmÞ PP( pt, m, pt+1, m, qt+1, m)= P pt+1, m n qt+1, m n n 2 SðmÞ P n qt+1, m pt, m n m=1, 2, .

.

.
.

.

, 12 (22:2) n 2 SmÞ PF ( pt, m, pt+1, m, qt, m, qt+1, m) p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PL( pt, m, pt+1, m, qt, m)PP( pt, m, pt+1, m, qt+1, m) m=1, 2, .

.

.

, 12: (22:3) 22.20 The above formulae can be rewritten in price relative and monthly expenditure share form as follows: PL( pt, m, pt+1, m, st, m)= P st, m n n 2 SðmÞ ( pt+1, m n =pt, m n ) m=1, 2, .

.

.
.

.

, 12 (22:4) 9 At the limit, if each commodity appeared in only one month of the year, then a month-to-month index would break down completely.

10 In the seasonal price index context, this type of index corresponds to Bean and Stine’s (1924, p. 31) Type D index.

11 Diewert (1996b, p. 17–19; 1999a, p. 50) noted various separability restrictions on consumer preferences that would justify these year- the economic over-year monthly indices from the viewpoint of approach to index number theory.
396 PP(pt, m,pt+1, m, st+1,m)= P " st+1, m n n 2 SðmÞ THE TREATMENT OF SEASONAL PRODUCTS #1 (pt+1, m n =pt,m n )1 PAP( pt, m, pt+1, m, s0, m)= P " s0, m n n 2 SðmÞ #1 ( pt+1, m n =pt, m n )1 m=1, 2, ..., 12 (22:5) m=1, 2, .

.

.
.

.

, 12 (22:9) PF ( pt, m, pt+1, m, st, m, st+1, m) p r ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PL( pt, m, pt+1, m, st, m)PP( pt, m, pt+1, m, st+1, m) ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ =pt, m ( pt+1, m ) n n P = st, m n n 2 SðmÞ v u u t P ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ #1 " ( pt+1, m n =pt, m n )1 st, m n n 2 SðmÞ m=1, 2, .

.

.
.

.

, 12 (22:6) where the monthly expenditure share for commodity n [ S(m) for month m in year t is deﬁned as: st, m n = n qt, m pt, m n P i qt, m pt, m i i 2 SðmÞ m=1, 2, .

.

.

, 12 n 2 S(m) t=0, 1, .

.

.

, T (22:7) and st, m denotes the vector of month m expenditure shares in year t, [st, m ] for n [ S(m).

22.21 Current period expenditure shares st, m are not likely to be available.
Hence it will be necessary to approximate these shares using the corresponding expenditure shares from a base year 0. n n 22.22 Use the base period monthly expenditure share vectors s0, m in place of the vector of month m and year t expenditure shares st, m in equation (22.4), and use the base period monthly expenditure share vectors s0, m in place of the vector of month m and year t+1 expenditure shares st+1, m in equation (22.5).
Similarly, replace the share vectors st, m and st+1, m in equation (22.6) by the base period expenditure share vector for month m, s0, m. The resulting approximate year-over-year monthly Las- peyres, Paasche and Fisher indices are deﬁned by equa- tions (22.8) to (22.10):12 PAL( pt, m, pt+1, m, s0, m)= P s0, m n n 2 SðmÞ ( pt+1, m n =pt, m n ) m=1, 2, .

.

.
.

.

, 12 (22:8) n 12 If the monthly expenditure shares for the base year, s0, m , are all equal, then the approximate Fisher index deﬁned by equation (22.10) reduces to Fisher’s (1922, p. 472) formula 101.

Fisher (1922, p. 211) observed that this index was empirically very close to the unweighted geometric mean of the price relatives, while Dale´ n (1992, p. 143) and Diewert (1995a, p. 29) showed analytically that these two indices approximated each other to the second order.
The equally weighted version of equation (22.10) was recommended as an elementary index by Car- ruthers, Sellwood and Ward (1980, p. 25) and Dale´ n (1992, p. 140).
PAF ( pt, m, pt+1, m, s0, m, s0, m) p r ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PAL( pt, m, pt+1, m, s0, m)PP( pt, m, pt+1, m, s0, m) ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ =pt, m ( pt+1, m ) n n P = st, m n n 2 SðmÞ v u u t P ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ #1 " ( pt+1, m n =pt, m n )1 st, m n n 2 SðmÞ m=1, 2, .

.

.
.

.

, 12: (22:10) 22.23 The year-over-year approximate Fisher monthly indices deﬁned by equation (22.10) will provide adequate approximations to their true Fisher counter- parts deﬁned by equation (22.6) only if the monthly expenditure shares for the base year 0 are not too dif- ferent from their current year t and t+1 counterparts.
Hence, it will be useful to construct the true Fisher indices on a delayed basis in order to check the ade- quacy of the approximate Fisher indices deﬁned by equation (22.10).

22.24 The year-over-year monthly approximate Fisher indices deﬁned by equation (22.10) will normally have a certain amount of upward bias, since these indices cannot reﬂect long-term substitution of con- sumers towards commodities that are becoming rela- tively cheaper over time.
This reinforces the case for computing true year-over-year monthly Fisher indices deﬁned by equation (22.6) on a delayed basis so that this substitution bias can be estimated.

22.25 Note that the approximate year-over-year monthly Laspeyres and Paasche indices, PAL and PAP deﬁned by equations (22.8) and (22.9) above, satisfy the following inequalities: PAL( pt, m, pt+1, m, s0, m)PAL( pt+1, m, pt, m, s0, m) 1 m=1, 2, .

.

.
.

.

, 12 PAP( pt, m, pt+1, m, s0, m)PAP( pt+1, m, pt, m, s0, m) 1 m=1, 2, .

.

.

, 12 (22:11) (22:12) that with strict inequalities if the monthly price vectors pt, m and pt+1, m are not proportional to each other.13 The inequality (22.11) says that the approximate year-over- year monthly Laspeyres index fails the time reversal test with an upward bias, while the inequality (22.12) says the approximate year-over-year monthly Paasche index fails the time reversal test with a down- ward bias.
Hence the ﬁxed weight approximate Las- peyres index PAL has a built-in upward bias and the ﬁxed weight approximate Paasche index PAP has a built-in downward bias.

Statistical agencies should avoid the use of these formulae.

The formulae can, however, be 13See Hardy, Littlewood and Po´ lya (1934, p. 26).
397 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE combined as in the approximate Fisher formula (22.10) and the resulting index should be free from any sys- tematic formula bias (but there still could be some sub- stitution bias).

22.26 The year-over-year monthly indices deﬁned in this section are illustrated using the artiﬁcial data set given in Tables 22.1 and 22.2.
Although ﬁxed base indices are not formally deﬁned in this section, these indices have similar formulae to the year-over-year indices except that the variable base year t is replaced by the ﬁxed base year 0.

The resulting 12 year-over-year monthly ﬁxed base Laspeyres, Paasche and Fisher indices are listed in Tables 22.3 to 22.5.
22.27 Comparing the entries in Tables 22.3 and 22.4, it can be seen that the year-over-year monthly ﬁxed base Laspeyres and Paasche price indices do not diﬀer sub- stantially for the early months of the year, but that there are substantial diﬀerences between the indices for the last ﬁve months of the year by the time the year 1973 is reached.

The largest percentage diﬀerence between the Laspeyres and Paasche indices is 12.5 per cent for month 10 in 1973 (1.4060/1.2496=1.125).
However, all the year-over-year monthly series show a smooth year-over-year trend.

22.28 Approximate ﬁxed base year-over-year Las- peyres, Paasche and Fisher indices can be constructed by replacing current month expenditure shares for the ﬁve commodities by the corresponding base year monthly expenditure shares on the ﬁve commodities.
The result- ing approximate Laspeyres indices are equal to the origi- nal ﬁxed base Laspeyres indices so there is no need to present the approximate Laspeyres indices in a table.

The approximate year-over-year Paasche and Fisher indices do, however, diﬀer from the ﬁxed base Paasche and Fisher indices found in Tables 22.4 and 22.5, so these new approximate indices are listed in Tables 22.6 and 22.7.
22.29 Comparing Table 22.4 with Table 22.6, it can be seen that, with a few exceptions, the entries corre- spond fairly closely.

One of the bigger diﬀerences is the 1973 entry for the ﬁxed base Paasche index for month 9, which is 1.1664, while the corresponding entry for the approximate ﬁxed base Paasche index is 1.1920, for a 2.2 per cent diﬀerence (1.1920/1.1664=1.022).
In general, the approximate ﬁxed base Paasche indices are some- what bigger than the true ﬁxed base Paasche indices, as could be expected, since the approximate indices have Table 22.3 Year-over-year monthly ﬁxed base Laspeyres indices Year Month Table 22.4 Year-over-year monthly ﬁxed base Paasche indices Year Month Table 22.5 Year-over-year monthly ﬁxed base Fisher indices Year Month 1.4327
1.1251 1.2383 1.4861 1.0000 1.0837 1.1930 1.4327 Table 22.6 Year-over-year approximate monthly ﬁxed base Paasche indices Year Month some substitution bias built into them as their expen- diture shares are held ﬁxed at the 1970 levels.
22.30 Turning now to the chained year-over-year monthly indices using the artiﬁcial data set, the resulting 12 year-over-year monthly chained Laspeyres, Paasche and Fisher indices, PL, PP and PF, where the month-to- month links are deﬁned by equations (22.4) to (22.6), are listed in Tables 22.8 to 22.10.
22.31 Comparing the entries in Tables 22.8 and 22.9, it can be seen that the year-over-year monthly chained Laspeyres and Paasche price indices have smaller dif- ferences than the corresponding ﬁxed base Laspeyres and Paasche price indices in Tables 22.3 and 22.4.

This is a typical pattern, as found in Chapter 19: the use of chained indices tends to reduce the spread between Paasche and Laspeyres indices compared to their ﬁxed base counterparts.
The largest percentage diﬀerence between corresponding entries for the chained Laspeyres and Paasche indices in Tables 22.8 and 22.9 is 4.1 per for month 10 in 1973 (1.3593/1.3059=1.041).

cent Recall that the ﬁxed base Laspeyres and Paasche indices diﬀered by 12.5 per cent for the same month, so that chaining does tend to reduce the spread between these two equally plausible indices.
THE TREATMENT OF SEASONAL PRODUCTS 22.32 The chained year-over-year Fisher indices lis- ted in Table 22.10 are regarded as the ‘‘best’’ estimates of year-over-year inﬂation using the artiﬁcial data set.

22.33 The year-over-year chained Laspeyres, Paasche and Fisher indices listed in Tables 22.8 to 22.10 can be approximated by replacing current period com- modity expenditure shares for each month by the cor- responding base year monthly commodity expenditure shares.
The resulting 12 year-over-year monthly approxi- mate chained Laspeyres, Paasche and Fisher indices, PAL, PAP and PAF, where the monthly links are deﬁned by equations (22.8) to (22.10), are listed in Tables 22.11 to 22.13.

22.34 The year-over-year chained indices listed in Tables 22.11 to 22.13 approximate their true chained counterparts listed in Tables 22.8 to 22.10 very closely.
For the year 1973, the largest discrepancies are for the Paasche and Fisher indices for month 9: the chained Paasche is 1.2018, while the corresponding approximate chained Paasche is 1.2183 for a diﬀerence of 1.4 per cent, and the chained Fisher is 1.2181, while the corresponding approximate chained Fisher is 1.2305 for a diﬀerence of 1.0 per cent.
It can be seen that for the modiﬁed Turvey data set, the approximate Table 22.7 Year-over-year approximate monthly ﬁxed base Fisher indices Year Month Table 22.8 Year-over-year monthly chained Laspeyres indices Year Month Table 22.9 Year-over-year monthly chained Paasche indices Year Month Table 22.10 Year-over-year monthly chained Fisher indices Year Month 12
indices Year Month CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Table 22.11 Year-over-year monthly approximate chained Laspeyres indices Year Month 9
Laspeyres indices Year Month Table 22.12 Year-over-year monthly approximate chained Paasche indices Year Month Table 22.13 Year-over-year monthly approximate chained Fisher indices Year Month year-over-year monthly approximate Fisher indices lis- ted in Table 22.13 approximate the theoretically pre- ferred (but in practice unfeasible in a timely fashion) Fisher chained indices listed
 in Table 22.10 quite satis- factorily.
Since the approximate Fisher indices are just as easy to compute as the approximate Laspeyres and Paasche indices, it may be useful to ask that statistical agencies make available to the public these approximate Fisher indices along with the approximate Laspeyres and Paasche indices.
Year-over-year annual indices 22.35 Assuming that each commodity in each season of the year is a separate ‘‘annual’’ commodity is the sim- plest and theoretically most satisfactory method for dealing with seasonal commodities when the goal is to construct annual price and quantity indices.
This idea can be traced back to Bruce D. Mudgett in the consumer price context and to Richard Stone in the producer price context: The basic index is a yearly index and as a price or quantity index is of the same sort as those about which books and pamphlets have been written in quantity over the years (Mudgett (1955, p. 97)).
The existence of a regular seasonal pattern in prices which more or less repeats itself year after year suggests very strongly that the varieties of a commodity available at diﬀerent seasons cannot be transformed into one another without cost and that, accordingly, in all cases where seasonal variations in price are signiﬁcant, the varieties available at diﬀerent times of the year should be in principle, as separate commodities (Stone treated, (1956, pp.

74–75)).
74–75)).

400 22.36 Using the notation introduced in the previous section, the Laspeyres, Paasche and Fisher annual (chain link) indices comparing the prices of year t with those of year t+1 can be deﬁned as follows: PL( pt, 1, .

.

.

, pt, 12; pt+1, 1, .

.

.

, pt+1, 12; qt, 1, .

.

.

, qt, 12) 12 P m=1 P pt+1, m n qt, m n n 2 SðmÞ 12 P m=1 P n qt, m pt, m n n 2 SðmÞ (22:13) PP( pt, 1, .

.

.

, pt, 12; pt+1, 1, .

.

.

, pt+1, 12; qt+1, 1, .

.

.


, qt+1, 12) 12 P m=1 P pt+1, m n qt+1, m n n 2 SðmÞ 12 P m=1 P n qt+1, m pt, m n n 2 SðmÞ (22:14) PF (pt, 1, .

.

.

, pt, 12; pt+1, 1, .

.

.

, pt+1, 12; qt, 1, .

.

.

, qt, 12; qt+1, 1, .

.

.

, qt+1, 12) p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PL(pt, 1, .

.

.

, pt, 12; pt+1, 1, .

.

.

, pt+1, 12; qt, 1, .

.

.


, qt, 12) p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PP(pt, 1, .

.

.

, pt, 12; pt+1, 1, .

.

.

, pt+1, 12; qt+1;1, .

.

.

, qt+1, 12) (22:15) 22.37 The above formulae can be rewritten in price relative and monthly expenditure share form as follows: PL( pt, 1, .

.

.

, pt, 12; pt+1, 1, .

.

.

, pt+1, 12; st 1st, 1, .

.

.


, st 12st, 12) 12 P m=1 P mst, m st n n 2 SðmÞ ( pt+1, m n =pt, m n ) 12 = P m=1 mPL( pt, m, pt+1, m, st, m) st (22:16) PP( pt, 1, .

.

.

, pt, 12; pt+1, 1, .

.

.

, pt+1, 12; st+1 1 " 12 P m=1 P m st+1, m st+1 n ( pt+1, m n =pt, m n )1 n 2 SðmÞ st+1, 1, .

.

.
.

.

, st+1 #1 12 st+1, 12) " 12 = P m=1 st+1 m P st+1, m n n 2 SðmÞ ( pt+1, m n =pt, m n )1 12 = P m=1 m [PP( pt, m, pt+1, m, st+1, m)]1 st+1 #1 1 (22:17) where the expenditure share for month m in year t is deﬁned as: PF ( pt, 1, .

.

.

, pt, 12; pt+1, 1, .

.

.

, pt+1, 12; st 1st, 1, .

.

.

, st 12st, 12; st+1 1 st+1, 1, .

.

.
, st+1 12 st+1, 12) v u u u u u u u t ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 12 P m=1 P mst, m st n n 2 SðmÞ ( pt+1, m n =pt, m n ) 12 P m=1 P m st+1, m st+1 n ( pt+1, m n =pt, m n )1 n 2 SðmÞ v u u u u u u t = st n ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ m[PL( pt, m, pt+1, m, st, m)] st 12 P m=1 (22:18) 12 P m=1 m [PP( pt, m, pt+1, m, st+1, m)]1 st+1 P n qt, m pt, m n n 2 SðmÞ 12 P i=1 P j qt, i pt, i j j 2 SðiÞ m=1, 2, .
.

.

, 12; t=0, 1, .

.

.

, T (22:19) and the year-over-year monthly Laspeyres and Paasche (chain link) price indices PL(pt, m, pt+1, m, st, m) and PP( pt, m, ptþ1, m, stþ1, m) are deﬁned by equations (22.4) and (22.5), respectively.
As usual, the annual chain link Fisher index PF deﬁned by equation (22.18), which compares the prices in every month of year t with the corresponding prices in year t+1, is the geometric mean of the annual chain link Laspeyres and Paasche indices, PL and PP, deﬁned by equations (22.16) and (22.17).
The last equations in (22.16), (22.17) and (22.18) show that these annual indices can be deﬁned as (monthly) share-weighted averages of the year-over-year monthly chain link Laspeyres and Paasche indices, PL(pt, m, pt+1, m, st, m) and PP(pt, m, pt+1, m, st+1, m), deﬁned by equations (22.4) and (22.5).

Hence once the year-over- year monthly indices deﬁned above have been calculated numerically, it is easy to calculate the corresponding annual indices.
THE TREATMENT OF SEASONAL PRODUCTS 22.38 Fixed base counterparts to the formulae deﬁned by equations (22.16) to (22.18) can readily be deﬁned: simply replace the data pertaining to period t by the corresponding data pertaining to the base period 0.
22.39 The annual ﬁxed base Laspeyres, Paasche and Fisher indices, as calculated using the data from the artiﬁcial data set tabled in paragraphs 22.14 and 22.15, are listed in Table 22.14, which shows that by 1973, the annual ﬁxed base Laspeyres index exceeds its Paasche counterpart by 4.5 per cent.

Note that each series increases steadily.

22.40 The annual ﬁxed base Laspeyres, Paasche and Fisher indices can be approximated by replacing any current shares by the corresponding base year shares.
The resulting annual approximate ﬁxed base Laspeyres, Paasche and Fisher indices are listed in Table 22.15.

Also listed in the last column of Table 22.15 is the ﬁxed base geometric Laspeyres annual index, PGL.

This is the weighted geometric mean counterpart to the ﬁxed base Laspeyres index, which is equal to a base period weighted arithmetic average of the long-term price relatives; see Chapter 19.
It can be shown that PGL approximates the approximate ﬁxed base Fisher index, PAF, to the second order around a point where all the long-term price relatives are equal to unity.14 It can be seen that the entries for the Laspeyres price indices are exactly the same in Tables 22.14 and 22.15.

This is as it should be, because the ﬁxed base Laspeyres price index uses only expenditure shares from the base year 1970; hence the approximate ﬁxed base Laspeyres index is equal to the true ﬁxed base Laspeyres index.
Comparing the columns labelled PP and PF in Table 22.14 with the columns PAP and PAF in Table 22.15 shows that the approximate Paasche and approximate Fisher indices are quite close to the corresponding annual Paasche and Fisher indices.
Hence, for the artiﬁcial data set, the true annual ﬁxed base Fisher index can be very closely approximated by the corresponding approximate Fisher index, PAF (or the geometric Laspeyres index, PGL), which, of course, can be computed using the same information set that is normally available to statistical agencies.
22.41 Using the artiﬁcial data set in Tables 22.1 and 22.2, the annual chained Laspeyres, Paasche and Fisher indices can readily be calculated, using the formulae (22.16) to (22.18) for the chain links.

The resulting indices are listed in Table 22.16, which shows that the use of chained indices has substantially narrowed the gap between the Paasche and Laspeyres indices.
The diﬀerence between the chained annual Laspeyres and Paasche indices in 1973 is only 1.5 per cent (1.3994 versus 1.3791), whereas from Table 22.14, the diﬀerence between the ﬁxed base annual Laspeyres and Paasche indices in 1973 is 4.5 per cent (1.4144 versus 1.3536).

Thus the use of chained annual indices has substantially reduced the substitution (or representativity) bias of the Laspeyres and Paasche indices.
Comparing Tables 22.14 and 22.16, it can be seen that for this particular artiﬁcial data set, the annual ﬁxed base Fisher indices are very 14 See footnote 12.
401 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Table 22.14 Annual ﬁxed base Laspeyres, Paasche and Fisher price indices Table 22.16 Annual chained Laspeyres, Paasche and Fisher price indices Year 1970 1971 1972 1973 PL 1.0000 1.1008 1.2091 1.4144 PP 1.0000 1.0961 1.1884 1.3536 PF 1.0000 1.0984 1.1987 1.3837 Year 1970 1971 1972 1973 PL 1.0000 1.1008 1.2052 1.3994 PP 1.0000 1.0961 1.1949 1.3791 PF 1.0000 1.0984 1.2001 1.3892 Table 22.15 Annual approximate ﬁxed base Laspeyres, Paasche, Fisher and geometric Laspeyres indices Table 22.17 Annual approximate chained Laspeyres, Paasche and Fisher price indices Year 1970 1971 1972 1973 PAL 1.0000 1.1008 1.2091 1.4144 PAP 1.0000 1.0956 1.1903 1.3596 PAF 1.0000 1.0982 1 1.1996 1.3867 PGL 1.0000 .0983 1.2003 1.3898 Year 1970 1971 1972 1973 PAL 1.0000 1.1008 1.2051 1.3995 PAP 1.0000 1.0956 1.1952 1.3794 PAF 1.0000 1.0982 1.2002 1.3894 close to their annual chained Fisher counterparts.
The annual chained Fisher indices should, however, nor- mally be regarded as the more desirable target index to approximate, since this index will normally give better results if prices and expenditure shares are chang- ing substantially over time.15 and st+1 22.42 Obviously, the current year weights, st, m and st m and st+1, m m , which appear in the chain link formulae (22.16) to (22.18), can be approximated by the corresponding base year weights, s0, m m. This leads to the annual approximate chained Laspeyres, Paasche and Fisher indices listed in Table 22.17. and s0 n n n 22.43 Comparing the entries in Tables 22.16 and 22.17 shows that the approximate chained annual Laspeyres, Paasche and Fisher indices are extremely close to the cor- responding true chained annual Laspeyres, Paasche and Fisher indices.
Hence, for the artiﬁcial data set, the true annual chained Fisher index can be very closely approxi- mated by the corresponding approximate Fisher index, which can be computed using the same information set that is normally available to statistical agencies.
22.44 The approach to computing annual indices outlined in this section, which essentially involves taking monthly expenditure share-weighted averages of the 12 year-over-year monthly indices, should be contrasted with the approach that simply takes the arithmetic mean of the 12 monthly indices.
The problem with the latter approach is that months where expenditures are below the average (e.g., February) are given the same weight in the unweighted annual average as months where expenditures are above the average (e.g., December).

Rolling year annual indices 22.45 In the previous section, the price and quantity data pertaining to the 12 months of a calendar year were 15 The gap between the Laspeyres and Paasche indices will normally be reduced using chained indices under these circumstances.
Of course, if there are no substantial trends in prices, so that prices are just changing randomly, then it will generally be preferable to use the ﬁxed base Fisher index.

402 compared to the 12 months of a base calendar year.
There is, however, no need to restrict attention to calendar-year comparisons: any 12 consecutive months of price and quantity data could be compared to the price and quantity data of the base year, provided that the January data in the non-calendar year are compared to the January data of the base year, the February data of the non-calendar year are compared to the February data of the base year, and so on, up to the December data of the non-calendar year being compared to the December data of the base year.16 Alterman, Diewert and Feenstra (1999, p. 70) called the resulting indices rolling year or moving year indices.17 22.46 In order to theoretically justify the rolling year indices from the viewpoint of the economic approach to index number theory, some restrictions on preferences are required.
The details of these assumptions can be found in Diewert (1996b, pp.

32–34; 1999a, pp.

56–61).

22.47 The problems involved in constructing rolling year indices for the artiﬁcial data set are now considered.

For both ﬁxed base and chained rolling year indices, the ﬁrst 13 index number calculations are the same.

For the year that ends with the data for December of 1970, the index is set equal to 1 for the Laspeyres, Paasche and Fisher moving year indices.
The base year data are the 44 non-zero price and quantity observations for the calen- dar year 1970.

When the data for January 1971 become available, the three non-zero price and quantity entries for January of calendar year 1970 are dropped and replaced by the corresponding entries for January 1971.
The data for the remaining months of the comparison year remain the same; i.e., for February to December of the comparison year, the data for the rolling year are set equal to the corresponding entries for February to December 1970.

Thus the Laspeyres, Paasche or Fisher 16 Diewert (1983c) suggested this type of comparison and termed the resulting index a ‘‘split year’’ comparison.
17 Crump (1924, p. 185) and Mendershausen (1937, p. 245), respec- tively, used these terms in the context of various seasonal adjustment procedures.

The term ‘‘rolling year’’ seems to be well established in the business literature in the United Kingdom.

rolling year index value for January 1971 compares the prices and quantities of January 1971 with the corre- sponding prices and quantities of January 1970.
For the remaining months of this ﬁrst moving year, the prices and quantities of February to December 1970 are simply compared with exactly the same prices and quantities of February to December 1970.

When the data for Feb- ruary 1971 become available, the three non-zero price and quantity entries for February for the last rolling year (which are equal to the three non-zero price and quantity entries for February 1970) are dropped and replaced by the corresponding entries for February 1971.
The result- ing data become the price and quantity data for the second rolling year.

The Laspeyres, Paasche or Fisher rolling year index value for February 1971 compares the prices and quantities of January and February 1971 with the corresponding prices and quantities of January and February 1970.

For the remaining months of this ﬁrst moving year, the prices and quantities of March to December 1970 are compared with exactly the same prices and quantities of March to December 1970.
This process of exchanging the price and quantity data of the current month in 1971 with the corresponding data of the same month in the base year 1970 in order to form the price and quantity data for the latest rolling year continues until December 1971 is reached, when the cur- rent rolling year becomes the calendar year 1971.
Thus the Laspeyres, Paasche and Fisher rolling year indices for December 1971 are equal to the corresponding ﬁxed base (or chained) annual Laspeyres, Paasche and Fisher indices for 1971, listed in Tables 22.14 or 22.16.
22.48 Once the ﬁrst 13 entries for the rolling year the indices have been deﬁned as indicated above, remaining ﬁxed base rolling year Laspeyres, Paasche and Fisher indices are constructed by taking the price and quantity data of the last 12 months and rearranging the data so that the January data in the rolling year are compared to the January data in the base year, the February data in the rolling year are compared to the February data in the base year, and so on, up to the December data in the rolling year being compared to the December data in the base year.
The resulting ﬁxed base rolling year Laspeyres, Paasche and Fisher indices for the artiﬁcial data set are listed in Table 22.18.

22.49 Once the ﬁrst 13 entries for the ﬁxed base rolling year indices have been deﬁned as indicated above, the remaining chained rolling year Laspeyres, Paasche and Fisher indices are constructed by taking the price and quantity data of the last 12 months and comparing these data to the corresponding data of the rolling year of the 12 months preceding the current rolling year.
The resulting chained rolling year Laspeyres, Paasche and Fisher indices for the artiﬁcial data set are listed in the last three columns of Table 22.18.

Note that the ﬁrst 13 entries of the ﬁxed base Laspeyres, Paasche and Fisher indices are equal to the corresponding entries for the chained Laspeyres, Paasche and Fisher indices.
It will also be noted that the entries for December (month 12) of 1970, 1971, 1972 and 1973 for the ﬁxed base rolling year Laspeyres, Paasche and Fisher indices are equal to the corresponding ﬁxed base annual Laspeyres, Paasche and Fisher indices listed in Table 22.14.
Similarly, the entries THE TREATMENT OF SEASONAL PRODUCTS in Table 22.18 for December (month 12) of 1970, 1971, 1972 and 1973 for the chained rolling year Laspeyres, Paasche and Fisher indices are equal to the corre- sponding chained annual Laspeyres, Paasche and Fisher indices listed in Table 22.16.

22.50 Table 22.18 shows that the rolling year indices are very smooth and free from seasonal ﬂuctuations.
For the ﬁxed base indices, each entry can be viewed as a seasonally adjusted annual consumer price index that compares the data of the 12 consecutive months that end with the year and month indicated with the corre- sponding price and quantity data of the 12 months in the base year, 1970.
Thus rolling year indices oﬀer sta- tistical agencies an objective and reproducible method of seasonal adjustment that can compete with existing time series methods of seasonal adjustment.18 22.51 Table 22.18 shows that the use of chained indices has substantially narrowed the gap between the ﬁxed base moving year Paasche and Laspeyres indices.
The diﬀerence between the rolling year chained Las- peyres and Paasche indices in December 1973 is only 1.5 per cent (1.3994 versus 1.3791), whereas the diﬀerence between the rolling year ﬁxed base Laspeyres and Paasche indices in December 1973 is 4.5 per cent (1.4144 versus 1.3536).

Thus, the use of chained indices has substantially reduced the substitution (or representativ- ity) bias of the Laspeyres and Paasche indices.
As in the previous section, the chained Fisher rolling year index is regarded as the target seasonally adjusted annual index when seasonal commodities are in the scope of the CPI.

This type of index is also a suitable index for central banks to use for inﬂation targeting purposes.19 The six series in Table 22.18 are charted in Figure 22.1.
The ﬁxed base Laspeyres index is the highest one, followed by the chained Laspeyres, the two Fisher indices (which are virtually indistinguishable), and the chained Paasche.

Finally, the ﬁxed base Paasche is the lowest index.
An increase in the slope of each graph can clearly be seen for the last eight months, reﬂecting the increase in the month-to-month inﬂation rates that was built into the data for the last 12 months of the data set.20 22.52 As in the previous section, the current year m , which appear m and st+1, m and st+1 weights, st, m and st n n 18 For discussions on the merits of econometric or time series methods versus index number methods of seasonal adjustment, see Diewert (1999a, pp.
61–68) and Alterman, Diewert and Feenstra (1999, pp.

78– 110).

The basic problem with time series methods of seasonal adjust- ment is that the target seasonally adjusted index is very diﬃcult to specify in an unambiguous way; i.e., there is an inﬁnite number of possible target indices.

For example, it is impossible to identify a temporary increase in inﬂation within a year from a changing seasonal factor.
Hence diﬀerent econometricians will tend to generate diﬀerent seasonally adjusted series, leading to a lack of reproducibility.

19 See Diewert (2002c) for a discussion of the measurement issues involved in choosing such an index.

20 The arithmetic average of the 36 month-over-month inﬂation rates for the rolling year ﬁxed base Fisher indices is 1.0091; the average of these rates for the ﬁrst 24 months is 1.0076, for the last 12 months is 1.0120 and for the last 2 months is 1.0156.
Hence, the increased month- to-month inﬂation rates for the last year are not fully reﬂected in the rolling year indices until a full 12 months have passed.

However, the fact that inﬂation has increased for the last 12 months of data com- pared to the earlier months is picked up almost immediately.
403 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Table 22.18 Rolling year Laspeyres, Paasche and Fisher price indices Year 1970 1971 1972 1973 Month PL (ﬁxed) PP (ﬁxed) PF (ﬁxed) PL (chain) PP (chain) 1.0000 1.0087 1.0170 1.0274 1.0364 1.0448 1.0537 1.0635 1.0706 1.0740 1.0792 1.0901 1.0961 1.1040 1.1147 1.1260 1.1342 1.1415 1.1505 1.1622 1.1709 1.1730 1.1771 1.1869 1.1949 1.2047 1.2172 1.2310 1.2442 1.2579
 1.2758 1.2961 1.3169 1.3268 1.3384 1.3609 1.3791 PF (chain) Figure 22.1 Rolling year ﬁxed base and chained Las- peyres, Paasche and Fisher indices 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 La Pa Fi La Ch Pa Ch Fi Ch 404 in the chain link formulae (22.16) to (22.18) or in the corresponding ﬁxed base formulae, can be approxi- mated by the corresponding base year weights, s0, m and s0 m. This leads to the annual approximate ﬁxed base and chained rolling year Laspeyres, Paasche and Fisher indices listed in Table 22.19. n 22.53 Comparing the indices in Tables 22.18 and 22.19, it can be seen that the approximate rolling year ﬁxed base and chained Laspeyres, Paasche and Fisher indices listed in Table 22.19 are very close to their true rolling year counterparts listed in Table 22.18.
In par- ticular, the approximate chain rolling year Fisher index (which can be computed using just base year expendi- ture share information, along with current information on prices) is very close to the preferred target index, the rolling year chained Fisher index.

In December 1973, these two indices diﬀer by only 0.014 per cent (1.3894/ 1.3892=1.00014).

The indices in Table 22.19 are charted in Figure 22.2.
It can be seen that Figures 22.1 and 22.2 are very similar; in particular, the Fisher ﬁxed base and chained indices are virtually identical in both ﬁgures.
Table 22.19 Rolling year approximate Laspeyres, Paasche and Fisher price indices THE TREATMENT OF SEASONAL PRODUCTS Year 1970 1971 1972 1973 Month PAL (ﬁxed) PAP (ﬁxed) Figure 22.2 Rolling year approximate ﬁxed base and chained Laspeyres, Paasche and Fisher indices 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 La Ap Fb Pa Ap Fb Fi Ap Fb La Ap Ch Pa Ap Ch Fi Ap Ch PAF (ﬁxed) PAL (chain) PAP (chain) 1.0000
1.3386 1.3538 1.3782 1.3995 PAP (chain) PAF (chain) From the above tables, it can be seen that year- over-year monthly indices and their generalizations to rolling year indices perform very well using the modiﬁed Turvey data set; like is compared to like, and the existence of seasonal commodities does not lead to erratic ﬂuctua- tions in the indices.
The only drawback to the use of these indices is that it seems that they cannot give any infor- mation on short-term, month-to-month ﬂuctuations in prices.

This is most evident if seasonal baskets are totally diﬀerent for each month since in this case, there is no possibility of comparing prices on a month-to-month basis.

In the following section, it is shown how a current period year-over-year monthly index can be used to predict a rolling year index that is centred on the current month.
Predicting a rolling year index using a current period year- over-year monthly index 22.55 It might be conjectured that under a regime where the long-run trend in prices is smooth, changes in the year-over-year inﬂation rate for a particular month compared to the previous month could give valuable 405 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE information about the long-run trend in price inﬂation.

For the modiﬁed Turvey data set, this conjecture turns out to be true, as seen below.
22.56 The basic idea is illustrated using the ﬁxed base Laspeyres rolling year indices listed in Table 22.18 and the year-over-year monthly ﬁxed base Laspeyres indices listed in Table 22.3.

In Table 22.18, the ﬁxed base Laspeyres rolling year entry for December of 1971 compares the 12 months of price and quantity data pertaining to 1971 with the corresponding prices and quantities pertaining to 1970.

This index number, PL, is the ﬁrst entry in Table 22.20.
Thus the PLRY column of Table 22.20 shows the ﬁxed base rolling year Laspeyres index, taken from Table 22.18, starting at December 1971 and carrying through to December 1973, which is 24 observations in all.

Looking at the ﬁrst entry of this column, it can be seen that the index is a weighted average of year-over-year price rela- tives over all 12 months in 1970 and 1971.
Thus this index is an average of year-over-year monthly price changes, centred between June and July of the two years for which prices are being compared.
Hence, an approximation to this annual index could be obtained by taking the arith- metic average of the June and July year-over-year monthly indices pertaining to the years 1970 and 1971 (see the entries for months 6 and 7 for the year 1971 in Table 22.3, 1.0844 and 1.1103).21 The next rolling year ﬁxed base Laspeyres index corresponds to the January 1972 entry in Table 22.18.
An approximation to this rolling year index, PARY, could be obtained by taking the arith- metic average of the July and August year-over-year monthly indices pertaining to the years 1970 and 1971 (see the entries for months 7 and 8 for the year 1971 in Table 22.3, 1.1103 and 1.0783).

These arithmetic averages of the two year-over-year monthly indices that are in the middle of the corresponding rolling year are listed in the PARY column of Table 22.20.
From Table 22.20, it can be seen that the PARY column does not approximate the PLRY column particularly well, since the approximate indices in the PARY column are seen to have some pronounced seasonal ﬂuctuations, whereas the rolling year indices in the PLRY column are free from seasonal ﬂuctuations.

22.57 Some seasonal adjustment factors (SAF ) are listed in Table 22.20.
For the ﬁrst 12 observations, the entries in the SAF column are simply the ratios of the entries in the PLRY column, divided by the correspond- ing entries in the PARY column; i.e., for the ﬁrst 12 observations, the seasonal adjustment factors are simply the ratio of the rolling year indices starting at December 1971, divided by the arithmetic average of the two year- over-year monthly indices that are in the middle of the corresponding rolling year.22 The initial 12 seasonal 21 Obviously, if an average of the year-over-year monthly indices for May, June, July and August were taken, a better approximation to the annual index could be obtained, and if an average of the year-over- year monthly indices for April, May, June, July, August and Sep- tember were taken, an even better approximation could be obtained to the annual index, and so on.
22 Thus if SAF is greater than one, this means that the two months in the middle of the corresponding rolling year have year-over-year rates of price increase that average out to a number below the overall average of the year-over-year rates of price increase for the entire rolling year, and above the overall average if SAF is less than one.
406 Table 22.20 Rolling year ﬁxed base Laspeyres and seasonally adjusted approximate rolling year price indices Year 1971 1972 1973 Month PLRY PSAARY PARY SAF adjustment remaining entries for the SAF column.
factors are then just repeated for the 22.58 Once the seasonal adjustment factors have been deﬁned, then the approximate rolling year index PARY can be multiplied by the corresponding seasonal adjustment factor SAF in order to form a seasonally adjusted approximate rolling year index, PSAARY, as lis- ted in Table 22.20.
22.59 Comparing the PLRY and PSAARY columns in Table 22.20, the rolling year ﬁxed base Laspeyres index, PLRY, and the seasonally adjusted approximate rolling year index, PSAARY, are identical for the ﬁrst 12 observations, which follows by construction since PSAARY equals the approximate rolling year index, PARY, multiplied by the seasonal adjustment factor SAF which in turn is equal to the rolling year Laspeyres index, PLRY, divided by PARY.
However, starting at December 1972, the rolling year index, PLRY, diﬀers from the corresponding seasonally adjusted approx- imate rolling year index, PSAARY.

It can be seen that for these last 13 months, PSAARY is surprisingly close to PLRY.23 Figure 22.3 shows PLRY, PSAARY and PARY gra- phically.
Because of the acceleration in the monthly inﬂation rate for the last year of data, it can be seen that the seasonally adjusted approximate rolling year series, PSAARY, does not pick up this accelerated inﬂa- tion rate for the ﬁrst few months of the last year (it lies well below PLRY for February and March 1973) but, 23 The means for the last 13 observations in columns PLRY and PARY of Table 22.20 are 1.2980 and 1.2930.
A regression of PL on PSAARY leads to an R2 of 0.9662 with an estimated variance of the residual of 0.000214.

Figure 22.3 Fixed base Laspeyres, seasonally adjusted approximate and approximate rolling year indices 1 3 5 7 9 11 13 15 17 19 21 23 PLRY PSAARY PARY in general, it predicts the corresponding centred year quite well.

22.60 The above results for the modiﬁed Turvey data set are quite encouraging.
If these results can be replicated for other data sets, then it means that statis- tical agencies can use the latest information on year- over-year monthly inﬂation to predict reasonably well the (seasonally adjusted) rolling year inﬂation rate for a rolling year that is centred around the last two months.
Thus policy-makers and other interested users of the CPI can obtain a reasonably accurate forecast of trend inﬂation (centred around the current month) some six months in advance of the ﬁnal estimates being calculated.

22.61 The method of seasonal adjustment used in this section is rather crude compared to some of the sophisticated econometric or statistical methods that are available.

Thus, these more sophisticated methods could be used in order to improve the forecasts of trend inﬂation.
If improved forecasting methods are used, however, it is useful to use the rolling year indices as targets for the forecasts, rather than using a statis- tical package that simultaneously seasonally adjusts current data and calculates a trend rate of inﬂation.
What is being suggested here is that the rolling year concept can be used in order to eliminate the lack of reproducibility in the estimates of trend inﬂation that existing statistical methods of seasonal adjustment generate.24 22.62 In this section and the previous ones, all the suggested indices have been based on year-over-year monthly indices and their averages.
In the subsequent sections of this chapter, attention will be turned to more traditional price indices that attempt to compare the prices in the current month with the prices in a previous month.

24 The operator of a statistical seasonal adjustment package has to make somewhat arbitrary decisions on many factors.

For example, are the seasonal factors additive or multiplicative?

How long should the moving average be and what type of average should be calculated?
Thus diﬀerent operators of the seasonal adjustment package will tend to produce diﬀerent estimates of the trend and the seasonal factors.

THE TREATMENT OF SEASONAL PRODUCTS Maximum overlap month-to-month price indices 22.63 A reasonable method for dealing with sea- sonal commodities in the context of picking a target index for a month-to-month CPI is the following:25 Determine the set of commodities that is present in the marketplace in both months of the comparison.
For this maximum overlap set of commodities, cal- culate one of the three indices recommended in pre- vious chapters; i.e., calculate the Fisher, Walsh or To¨ rnqvist–Theil index.26 Thus the bilateral index number formula is applied only to the subset of commodities that is present in both periods.27 22.64 The question now arises: should the compar- ison month and the base month be adjacent months (thus leading to chained indices) or should the base month be ﬁxed (leading to ﬁxed base indices)?
It seems reasonable to prefer chained indices over ﬁxed base indices for two reasons: The set of seasonal commodities which overlaps dur- ing two consecutive months is likely to be much larger than the set obtained by comparing the prices of any given month with a ﬁxed base month (such as January of a base year).

Hence the comparisons made using chained indices will be more comprehensive and accurate than those made using a ﬁxed base.
In many economies, on average 2 or 3 per cent of price quotes disappear each month because of the introduction of new commodities and the dis- appearance of older ones.

This rapid sample attrition means that ﬁxed base indices rapidly become un- representative.

Hence it seems preferable to use chained indices which can more closely follow market- place developments.28 22.65 It will be useful to review the notation at this point and deﬁne some new notation.
Let there be N commodities that are available in some month of some year and let pt, m n denote the price and quantity of commodity n that is in the marketplace29 in month m of year t (if the commodity is unavailable, deﬁne pt, m and qt, m n n 25 For more on the economic approach and the assumptions on con- sumer preferences that can justify month-to-month maximum overlap indices, see Diewert (1999a, pp.

51–56).

26 For simplicity, only the Fisher index is considered in detail in this chapter.
27 Keynes (1930, p. 95) called this the highest common factor method for making bilateral index number comparisons.

Of course, this target index drops those strongly seasonal commodities that are not present in the marketplace during one of the two months being compared.

Thus the index number comparison is not completely comprehensive.
Mudgett (1955, p. 46) called the ‘‘error’’ in an index number com- parison that is introduced by the highest common factor method (or maximum overlap method) the ‘‘homogeneity error’’.

28 This rapid sample degradation essentially forces some form of chaining at the elementary level in any case.

29 As was seen in Chapter 20, it is necessary to have a target concept for the individual prices and quantities pt, m at the ﬁnest level of aggregation.
Under most circumstances, these target concepts can be taken to be unit values (for prices) and total quantities consumed (for quantities).

and qt, m n n 407 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE 1 1 , .

.

.

, pt, m , .

.

.

, qt, m to be 0).

Let pt, m [ pt, m , qt, m 2 , pt, m and qt, m N ] and n 2 qt, m [qt, m N ] be the month m and year t price and quantity vectors, respectively.
Let S(t,m) be the set of commodities that is present in month m of year t and the following month.

Then the maximum overlap Laspeyres, Paasche and Fisher indices going from month m of year t to the following month can be deﬁned as follows:30 PL( pt, m, pt, m+1, qt, m, S(t, m)) pt, m+1 n n 2 Sðt, mÞ qt, m n P = P n qt, m pt, m n n 2 Sðt, mÞ m=1, 2, .

.

.

, 11 (22:20) PP( pt, m, pt, m+1, qt, m+1, S(t, m)) P pt, m+1 n n 2 Sðt, mÞ qt, m+1 n P n qt, m+1 pt, m n = n 2 Sðt, mÞ m=1, 2, .

.

.
.

.

, 11 (22:21) PF ( pt, m, pt, m+1, qt, m, qt, m+1, S(t, m)) p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PL( pt, m, pt, m+1, qt, m, S(t, m))PP( pt, m, pt, m+1, qt, m+1, S(t, m)) m=1, 2, .

.

.
.

.

, 11: (22:22) Note that PL, PP and PF depend on the two (complete) price and quantity vectors pertaining to months m and m+1 of year t, pt, m, pt, mþ1, qt, m, qt, mþ1, but they also depend on the set S(t, m), which is the set of commod- ities that are present in both months.
Thus, the com- modity indices n that are in the summations on the right- hand sides of equations (22.20) to (22.22) include indices n that correspond to commodities that are present in both months, which is the meaning of n [ S(t, m); i.e., n belongs to the set S(t, m).

22.66 In order to rewrite the deﬁnitions (22.20) to (22.22) in expenditure share and price relative form, some additional notation is required.
Deﬁne the expen- diture shares of commodity n in month m and m+1 of year t, using the set of commodities that are present in month m of year t and the subsequent month, as follows: st, m n (t, m)= n qt, m pt, m n P i qt, m pt, m i i 2 Sðt, mÞ n 2 S(t, m) m=1, 2, .

.

.

, 11 st, m+1 n (t, m)= pt, m+1 n P qt, m+1 n pt, m+1 i i 2 Sðt, mÞ qt, m+1 i (22:23) n 2 S(t, m) m=1, 2, .

.

.
.

.

, 11 (22:24) The notation in equations (22.23) and (22.24) is rather messy because st, m+1 (t, m) has to be distinguished from n 30 The formulae are slightly diﬀerent for the indices that go from December to January of the following year.

408 n (t, m+1).
408 n (t, m+1).

The expenditure share st, m+1 st, m+1 (t, m) is the n share of commodity n in month m+1 of year t where n is restricted to the set of commodities that are present in month m of year t and the subsequent month, whereas st, m+1 (t, m+1) is the share of commodity n in month n m+1 of year t but where n is restricted to the set of commodities that are present in month m+1 of year t and the subsequent month.
Thus, the set of superscripts t, m+1 in st, m+1 (t, m) indicates that the expenditure share is calculated using the price and quantity data of month m+1 of year t and (t, m) indicates that the set of admissible commodities is restricted to the set of com- modities that are present in both month m of year t and the subsequent month.

n n n 22.67 Now deﬁne vectors of expenditure shares.
If commodity n is present in month m of year t and the following month, deﬁne st, m (t, m) using equation (22.23); if this is not the case, deﬁne st, m (t, m)=0.

Similarly, if commodity n is present in month m of year t and the following month, deﬁne st, m+1 (t, m) using equation (22.24); if this is not the case, deﬁne st, m+1 (t, m)=0.

Now deﬁne the N-dimensional vectors st, m(t, m) [st, m (t, m), st, m N (t, m)] and st, m+1(t, m) [st, m+1 (t, m), 2 (t, m), .
.. , st, m+1 st, m+1 (t, m)].

Using these share deﬁni- 2 tions, the month-to-month Laspeyres, Paasche and Fisher formulae (22.20) to (22.22) can also be rewritten in expenditure share and price form as follows: (t, m), .

.

.

, st, m N 1 n n 1 PL( pt, m, pt, m+1, st, m(t, m)) st, m n n 2 Sðt, mÞ (t, m)( pt, m+1 X n =pt, m n ) m=1, 2, .

.

.

, 11 (22:25) PP( pt, m, pt, m+1, st, m+1(t, m)) 2 4 X st, m+1 n n 2 Sðt, mÞ 3 1 (t, m)( pt, m+1 n =pt, m n )1 5 m=1, 2, .

.

.
.

.

, 11 (22:26) PF (pt, m, pt, m+1, st, m(t, m), st, m+1(t, m)) ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (t, m)(pt, m+1 =pt, m n P ) n v u u u u u t st, m n n 2 SðmÞ P st, m+1 n n 2 SðmÞ (t, m)(pt, m+1 n =pt, m n )1 m=1, 2, .

.

.
.

.

, 11: (22:27) n 22.68 It is important to recognize that the expendi- ture shares st, m (t, m) that appear in the maximum overlap month-to-month Laspeyres index deﬁned by equation (22.25) are not the expenditure shares that could be taken from a consumer expenditure survey for month m of year t: instead, they are the shares that result after expendi- tures on seasonal commodities that are present in month m of year t, but are not present in the following month, are dropped.
Similarly, the expenditure shares st, m+1 (t, m) that appear in the maximum overlap month- n to-month Paasche index deﬁned by equation (22.26) are Table 22.21 Month-to-month maximum overlap chained Laspeyres, Paasche and Fisher price indices Table 22.22 Month-to-month chained Laspeyres, Paasche and Fisher price indices THE TREATMENT OF SEASONAL PRODUCTS Year 1970 1971 1972 1973 Month PL PP PF Year Month PL(3) PP (3) PF (3) PL(2) PP (2) PF (2) 0.4684
0.9784
1.2792
not the expenditure shares that could be taken from a consumer expenditure survey for month m+1 of year t: instead, they are the shares that result after expenditures on seasonal commodities that are present in month m+1 of year t, but are not present in the preceding month, are dropped.31 The maximum overlap month-to-month Fisher index
 deﬁned by equation (22.27) is the geometric mean of the Laspeyres and Paasche indices deﬁned by equations (22.25) and (22.26).
22.69 Table 22.21 lists the maximum overlap chained month-to-month Laspeyres, Paasche and Fisher price 31 It is important that the expenditure shares used in an index number formula add up to unity.

The use of unadjusted expenditure shares from a household expenditure survey would lead to a systematic bias in the index number formula.

indices for the data listed in Tables 22.1 and 22.2.

These indices are deﬁned by equations (22.25), (22.26) and (22.27).
22.70 The chained maximum overlap Laspeyres, Paasche and Fisher indices for December 1973 are 1.0504, 0.1204 and 0.3556, respectively.

Comparing these results to the year-over-year results listed in Tables 22.3, 22.4 and 22.5 (page 398) indicates that the results in Table 22.21 are not at all realistic.

These hugely diﬀerent direct indices compared with the last row of Table 22.21 indicate that the maximum overlap indices suﬀer from a serious downward bias for the artiﬁcial data set.
22.71 What are the factors that can explain this downward bias?

It is evident that part of the problem has to do with the seasonal pattern of prices for peaches and strawberries (commodities 2 and 4).

These are the com- modities that are not present in the marketplace for each 409 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE month of the year.
When these commodities ﬁrst become available, they come into the marketplace at relatively high prices and then, in subsequent months, their prices drop substantially.

The eﬀects of these initially high prices (compared to the relatively low prices that pre- vailed in the last month that the commodities were available in the previous year) are not captured by the maximum overlap month-to-month indices, so the resulting indices build up a tremendous downward bias.
The downward bias is most pronounced in the Paasche indices, which use the quantities or volumes of the cur- rent month.

Those volumes are relatively large compared to the volumes in the initial month when the commod- ities become available, reﬂecting the eﬀects of lower prices as the quantity dumped in the market increases.
22.72 Table 22.22 lists the results using chained Laspeyres, Paasche and Fisher indices for the artiﬁcial data set where the strongly seasonal commodities 2 and 4 are dropped from each comparison of prices.

Thus, the indices in Table 22.22 are the usual chained Laspeyres, Paasche and Fisher indices restricted to commodities 1, 3 and 5, which are available in each season.

The indices derived using these three commodities are labelled PL(3), PP(3) and PF (3).
22.73 The chained Laspeyres, Paasche and Fisher indices (using only the three always present commod- ities) for January 1973 are 1.2038, 0.5424 and 0.8081, respectively.

From Tables 22.8, 22.9 and 22.10, the chained year-over-year Laspeyres, Paasche and Fisher indices for January 1973 are 1.3274, 1.3243 and 1.3258, respectively.

Thus the chained indices using the always present commodities which are listed in Table 22.22 evidently suﬀer from substantial downward biases.
22.74 If the data in Tables 22.1 and 22.2 are exam- ined, it can be seen that the quantities of grapes (com- modity 3) on the marketplace varies tremendously over the course of a year, with substantial increases in price for the months when grapes are almost out of season.
Thus the price of grapes decreases substantially as the quantity in the marketplace increases during the last half of each year, but the annual substantial increase in the price of grapes takes place in the ﬁrst half of the year when quantities in the market are small.
This pattern of seaso- nal price and quantity changes will cause the overall index to take on a downward bias.32 To verify that this con- jecture is true, see the last three columns of Table 22.22 where chained Laspeyres, Paasche and Fisher indices are calculated using only commodities 1 and 5.

These indices are labelled PL(2), PP(2) and PF (2), respectively, and for January 1973 they are equal to 1.0033, 0.9408 and 0.9715, respectively.
These estimates based on two always present commodities are much closer to the chained year-over- year Laspeyres, Paasche and Fisher indices for January 1973, which were 1.3274, 1.3243 and 1.3258, respectively, than the estimates based on the three always present commodities.
It can be seen that the chained Laspeyres, 32 Andrew Baldwin (1990, p. 264) used the Turvey data to illustrate various treatments of seasonal commodities and discusses what causes various month-to-month indices to behave badly: ‘‘It is a sad fact that for some seasonal commodity groups, monthly price changes are not meaningful, whatever the choice of formula.’’ 410 Paasche and Fisher indices restricted to commodities 1 and 5 still have very substantial downward biases for the artiﬁcial data set.
Basically, the problems are caused by the high volumes associated with low or declining prices, and the low volumes caused by high or rising prices.
These weight eﬀects make the seasonal price declines bigger than the seasonal price increases using month-to-month index number formulae with variable weights.33 22.75 In addition to the downward biases that show up in Tables 22.21 and 22.22, all of these month-to- month chained indices show substantial seasonal ﬂuc- tuations in prices over the course of a year.

Hence these month-to-month indices are of little use to policy-makers who are interested in short-term inﬂationary trends.
Thus, if the purpose of the month-to-month CPI is to indicate changes in general inﬂation, then statistical agencies should be cautious about including commodities that show strong seasonal ﬂuctuations in prices in the month-to-month index.34 If seasonal commodities are included in a month-to-month index that is meant to indicate general inﬂation, then a seasonal adjustment procedure should be used to remove these strong seasonal ﬂuctuations.
Some simple types of seasonal adjustment procedures are considered in paragraphs 22.91 to 22.96.

22.76 The rather poor performance of the month-to- month indices listed in Tables 22.21 and 22.22 does not always occur in the context of seasonal commodities.
In the context of calculating import and export price indices using quarterly data for the United States, Alterman, Diewert and Feenstra (1999) found that maximum overlap month-to-month indices worked reasonably well.35 Statistical agencies should check that their month- to-month indices are at least approximately consistent with the corresponding year-over-year indices.
22.77 Obviously the various Paasche and Fisher indices computed in this section could be approximated by indices that replaced all current period expenditure shares by the corresponding expenditure shares from the base year.

These approximate Paasche and Fisher 33 This remark has an application to Chapter 20 on elementary indices where irregular sales during the course of a year could induce a similar downward bias in a month-to-month index that used monthly weights.
Another problem with month-to-month chained indices is that pur- chases and sales of individual commodities can become quite irregular as the time period becomes shorter and shorter, and the problem of zero purchases and sales becomes more pronounced.

Feenstra and Shapiro (2003, p. 125) ﬁnd an upward bias for their chained weekly indices for canned tuna compared to a ﬁxed base index; their bias was caused by variable weight eﬀects resulting from the timing of adver- tising expenditures.
In general, these drift eﬀects of chained indices can be reduced by lengthening the time period, so that the trends in the data become more prominent than the high-frequency ﬂuctuations.
34 If the purpose of the index is to compare the prices that consumers actually face in two consecutive months, ignoring the possibility that the consumer may regard a seasonal good as being qualitatively dif- ferent in the two months, then the production of a month-to-month CPI that has large seasonal ﬂuctuations can be justiﬁed.
35 They checked the validity of their month-to-month indices by cumulating them for four quarters and comparing them to the corre- sponding year-over-year indices, and found only relatively small dif- ferences.

However, note that irregular high-frequency ﬂuctuations will tend to be smaller for quarters than for months, and hence chained quarterly indices can be expected to perform better than chained monthly or weekly indices.
indices will not be reproduced here since they resemble their ‘‘true’’ counterparts and hence are also subject to tremendous downward bias.

Annual basket indices with carry forward of unavailable prices 22.78 Recall that the Lowe (1823) index deﬁned in earlier chapters had two reference periods:36 a reference period for the vector of quantity weights; a reference period for the base period prices.
The Lowe index for month m is deﬁned by the following formula: PLo( p0, pm, q) P N n=1 pm n qn N P n=1 p0 nqn (22:28) 1 , .

.

.

, pm 1, .

.

.

, p0 where p0 [ p0 N] is the base month price vector, pm [ pm N] is the current month m price vector, and q:[q1, .

.

.

, qN] is the base year reference quantity vector.

For the purposes of this section, where the modiﬁed Turvey data set is used to illustrate the index numerically, the base year will be taken to be 1970.
The resulting base year quantity vector turns out to be: q [q1, .

.

.

, q5]=[53889, 12881, 9198, 5379, 68653]: (22:29) The base period for the prices will be taken to be December 1970.

For prices that are not available in the current month, the last available price is carried for- ward.

The resulting Lowe index with carry forward of missing prices using the modiﬁed Turvey data set can be found in the PLo column of Table 22.23 on p. 412.
22.79 Andrew Baldwin’s (1990, p. 258) comments on this type of annual basket (AB) index are worth quoting at length: For seasonal goods, the AB index is best considered an index partially adjusted for seasonal variation.

It is based on annual quantities, which do not reﬂect the seasonal ﬂuctuations in the volume of purchases, and on raw monthly prices, which do incorporate seasonal price ﬂuctuations.

Zarnowitz (1961, pp.

256–257) calls it an index of ‘‘a hybrid sort’’.


Being neither of sea nor land, it does not provide an appropriate measure either of monthly or 12 month price change.
The question that an AB index answers with respect to price change from January to February say, or January of one year to January of the next, is ‘‘What would the change in con- sumer prices have been if there were no seasonality in purchases in the months in question, but prices none- theless retained their own seasonal behaviour?’’ It is hard to believe that this is a question that anyone would be interested in asking.
On the other hand, the 12 month ratio of an AB index based on seasonally adjusted prices would be conceptually valid, if one were interested in eliminating seasonal inﬂuences.

THE TREATMENT OF SEASONAL PRODUCTS 22.80 Recall that the Young (1812) index was deﬁned in earlier chapters as follows: PY ( p0, pm, s) P N n=1 sn( pm n =p0 n) (22:30) where s : [s1, .

.

.

, sN] is the base year reference vector of expenditure shares.
For the purposes of this section, where the modiﬁed Turvey data set is used to numeri- cally illustrate the index, the base year will be taken to be 1970.

The resulting base year expenditure share vec- tor turns out to be: s [s1, .

.

.

, s5]=[0:3284, 0:1029, 0:0674, 0:0863, 0:4149]: (22:31) Again, the base period for the prices will be taken to be December 1970.

For prices that are not available in the current month, the last available price is carried for- ward.
The resulting Young index with carry forward of missing prices using the modiﬁed Turvey data set can be found in the PY column of Table 22.23.

22.81 The geometric Laspeyres index is deﬁned in Chapter 19 as follows: PGL( p0, pm, s) Q N n=1 ( pm n =p0 n)sn : (22:32) Thus the geometric Laspeyres index makes use of the same information as the Young index except that a geometric average of the price relatives is taken instead of an arithmetic one.
Again, the base year is taken to be 1970 and the base period for prices is taken to be December 1970.

The index is illustrated using the mod- iﬁed Turvey data set with carry forward of missing pri- ces; see the PGL column of Table 22.23.

22.82 It is of interest to compare the above three indices that use annual baskets to the ﬁxed base Laspeyres rolling year indices computed earlier.

The rolling year index that ends in the current month is centred ﬁve-and-a- half months backwards.
Hence the above three annual basket type indices will be compared with an arithmetic average of two rolling year indices that have their last month ﬁve and six months forward.

This latter centred rolling year index is labelled PCRY and is listed in the last column of Table 22.23.37 Note the zero entries for the last six rows of this column; the data set does not extend six months into 1975, so the centred rolling year indices cannot be calculated for these last six months.
22.83 It can be seen that the Lowe, Young and geometric Laspeyres indices have a considerable amount of seasonality in them and do not at all approximate their rolling year counterparts listed in the last column of Table 22.23.38 Hence, without seasonal adjustment, Despite Baldwin’s somewhat negative comments on the Lowe index, it is the index that is preferred by many statistical agencies, so it is necessary to study its prop- erties in the context of strongly seasonal data.
36 In the context of seasonal price indices, this type of index corre- sponds to Bean and Stine’s (1924, p. 31) Type A index.

37 This series is normalized to equal 1 in December 1970, so that it is comparable to the other month-to-month indices.

38 The sample means of the four indices are 1.2935 (Lowe), 1.3110 (Young), 1.2877 (geometric Laspeyres) and 1.1282 (rolling year).
Of course, the geometric Laspeyres indices will always be equal to or less than their Young counterparts, since a weighted geometric mean is always equal to or less than the corresponding weighted arithmetic mean.
411 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Table 22.23 Lowe, Young, geometric Laspeyres and centred rolling year indices with carry forward prices Figure 22.4 centred rolling year Laspeyres indices Lowe, Young, geometric Laspeyres and Year 1970 1971 1972 1973 Month PLO PY PGL PCRY the Lowe, Young and geometric Laspeyres indices are not suitable predictors for their seasonally adjusted rolling year
 counterparts.39 The four series, PLO, PY, PGL and PCRY, listed in Table 22.23 are also plotted in Figure 22.4.
It can be seen that the Young price index is generally the highest, followed by the Lowe index, while the geometric Laspeyres is the lowest of the three month-to-month indices.
The centred rolling year Las- peyres counterpart index, PCRY, is generally below the other three indices (and of course does not have the strong seasonal movements of the other three series), but it moves in a roughly parallel fashion to the other three indices.40 Note that the seasonal movements of PLO, PY, and PGL are quite regular.

This regularity is exploited in paragraphs 22.91 to 22.96 in order to use these month-to-month indices to predict their rolling year counterparts.
39 In paragraphs 22.91 to 22.96, the Lowe, Young and geometric Laspeyres indices are seasonally adjusted.

40 In Figure 22.4, PCRY is artiﬁcially set equal to the June 1973 value for the index, which is the last month that the centred index can be constructed from the available data.
412 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 PLO PY PGL PCRY 22.84 Part of the problem may be the fact that the prices of strongly seasonal goods have been carried forward for the months when the commodities are not available.

This will tend to add to the amount of sea- sonal movements in the indices, particularly when there is high general inﬂation.
Thus in the following section, the Lowe, Young and geometric Laspeyres indices are computed again using an imputation method for the missing prices rather than simply carrying forward the last available price.

Annual basket indices with imputation of unavailable prices 22.85 Instead of simply carrying forward the last available price of a seasonal commodity that is not sold during a particular month, it is possible to use an imputation method to ﬁll in the missing prices.
Alter- native imputation methods are discussed by Armknecht and Maitland-Smith (1999) and Feenstra and Diewert (2001).

The basic idea is to take the last available price and impute prices for the missing periods, using the trend of another index.

This other index could be an index of available prices for the general category of commodity or higher-level components of the CPI.
For the purposes of this section, the imputation index is taken to be a price index that grows at the multiplicative rate of 1.008, since the ﬁxed base rolling year Laspeyres indices for the modiﬁed Turvey data set grow at approximately 0.8 per cent per month.41 Using this imputation method to ﬁll in the missing prices, the Lowe, Young and geometric Laspeyres indices deﬁned in the previous section can be recomputed.
The resulting indices are listed in Table 22.24, along with the centred rolling year index PCRY for comparison purposes.

22.86 As could be expected, the Lowe, Young and geometric Laspeyres indices that use imputed prices are on average somewhat higher than their counterparts that use carry forward prices, but the variability of the 41 For the last year of data, the imputation index is escalated by an additional monthly growth rate of 1.008.
Table 22.24 Lowe, Young, geometric Laspeyres and centred rolling year indices with imputed prices Figure 22.5 imputed prices and centred rolling year indices Lowe, Young and geometric Laspeyres with THE TREATMENT OF SEASONAL PRODUCTS Year 1970 1971 1972 1973 Month PLOI PYI PGLI PCRY imputed indices is generally a little lower.42 The series listed in Table 22.24 are also plotted in Figure 22.5.
It can be seen that the Lowe, Young and geometric Las- peyres indices that use imputed prices still have a huge amount of seasonality in them and do not closely approximate their rolling year counterparts listed in the last column of Table 22.24.43 Hence, without seasonal adjustment, the Lowe, Young and geometric Laspeyres indices using imputed prices are not suitable predictors for their seasonally adjusted rolling year counterparts.44 42 For the Lowe indices, the mean for the ﬁrst 31 observations increases (with imputed prices) from 1.3009 to 1.3047, but the standard devia- tion decreases from 0.18356 to 0.18319.
For the Young indices, the mean for the ﬁrst 31 observations increases from 1.3186 to 1.3224, but the standard deviation decreases from 0.18781 to 0.18730.

For the geometric Laspeyres indices, the mean for the ﬁrst 31 observations increases from 1.2949 to 1.2994, and the standard deviation also increases slightly from 0.17582 to 0.17599.
The imputed indices are preferred to the carry forward indices on general methodological grounds; in high-inﬂation environments, the carry forward indices will be subject to sudden jumps as the previously unavailable commodities become available.

43 Note also that Figures 22.4 and 22.5 are very similar.

44 In paragraphs 22.91 to 22.96, the Lowe, Young and geometric Laspeyres indices using imputed prices are seasonally adjusted.
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 PLOI PYI PGLI PCRY As these indices stand, they are not suitable as measures of general inﬂation going from month to month.

Bean and Stine Type C or Rothwell indices 22.87 The ﬁnal month-to-month index45 that is considered in this chapter is the Bean and Stine Type C (1924, p. 31) or Rothwell (1958, p. 72) index.46 This index makes use of seasonal baskets in the base year, denoted as the vectors q0, m for the months m=1, 2, .

.

.

, 12.
.

.

, 12.

The index also makes use of a vector of base year unit value prices, p0 [ p0 5], where the nth price in this vec- tor is deﬁned as: 1, .

.

.

, p0 p0 n 12 P m=1 n q0, m p0, m n 12 P m=1 q0, m n n=1, .

.

.

, 5: (22:33) The Rothwell price index for month m in year t can now be deﬁned as follows: PR( p0, pt, m, q0, m) n q0, m pt, m n 5 P n=1 5 P n=1 m=1, .

.

.
.

.

, 12: nq0, m p0 n (22:34) Thus as the month changes, the quantity weights for the index change, and hence the month-to-month move- ments in this index are a mixture of price and quantity changes.47 45 For other suggested month-to-month indices in the seasonal context, see Balk (1980a; 1980b; 1980c; 1981).

46 This is the index favoured by Baldwin (1990, p. 271) and many other price statisticians in the context of seasonal commodities.
47 Rothwell (1958, p. 72) showed that the month-to-month movements in the index have the form of an expenditure ratio divided by a quantity index.
413 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Table 22.25 The Lowe with carry forward prices, Rothwell and normalized Rothwell indices Figure 22.6 The Lowe and Rothwell price indices Year 1970 1971 1972 1973 Month PLO PNR PR Using the modiﬁed Turvey data set, the base year is chosen to be 1970 as usual and the index is started oﬀ at December 1970.
The Rothwell index PR is compared to the Lowe index with carry forward of missing prices, PLO, in Table 22.25.

To make the series slightly more comparable, the normalized Rothwell index PNR is also listed in Table 22.25; this index is simply equal to the original Rothwell index divided by its ﬁrst observation.
index, shows 22.89 Figure 22.6, which plots the Lowe index with the carry forward of the last price and the nor- malized Rothwell the Rothwell index has smaller seasonal movements than the Lowe index, and is less volatile in general.48 It is evident large seasonal movements in the that there are still Rothwell index, and it may not be a suitable index for measuring general inﬂation without some sort of seasonal adjustment.
that 48 For all 37 observations in Table 22.25, the Lowe index has a mean of 1.3465 and a standard deviation of 0.20313, while the normalized Rothwell index has a mean of 1.2677 and a standard deviation of 0.18271.
414 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 P LO P ROTHN 22.90 In the following section, the annual basket type indices (with and without imputation) deﬁned in paragraphs 22.78 to 22.86 will be seasonally adjusted using essentially the same method that was used in paragraphs 22.55 to 22.62.
Forecasting rolling year indices using month-to-month annual basket indices 22.91 Recall Table 22.23 showing the Lowe, Young, geometric Laspeyres (using carry forward prices) and the centred rolling year indices for the 37 observations run- ning from December 1970 to December 1973, PLO, PY, PGL and PCRY, respectively.

For each of the ﬁrst three series, deﬁne a seasonal adjustment factor, SAF, as the centred rolling year index, PCRY, divided by PLO, PY and PGL, respectively, for the ﬁrst 12 observations.
Now for each of the three series, repeat these 12 seasonal adjust- ment factors for observations 13 to 24, and then repeat them again for the remaining observations.

These operations will create three SAF series for all 37 obser- vations (label them SAFLO, SAFY and SAFGL, respec- tively).

Only the ﬁrst 12 observations in the PLO, PY, PGL and PCRY series are used to create the three SAF series.
Finally, deﬁne seasonally adjusted Lowe, Young and geo- metric Laspeyres indices by multiplying each unadjusted index by the appropriate seasonal adjustment factor: PLOSA PLOSAFLO PYSA PY SAFY PGLSA PGLSAFGL: (22:35) These three seasonally adjusted annual basket type indices are listed in Table 22.26 along with the target index, the centred rolling year index, PCRY.
22.92 The four series in Table 22.26 coincide for their ﬁrst 12 observations, which follows from the way the seasonally adjusted series were deﬁned.

Also, the last six observations are missing for the centred rolling year series, PCRY, since data for the ﬁrst six months of 1974 would be required in order to calculate all these index values.
Note that from December 1971 to December 1973, the three seasonally adjusted annual basket type indices can be used Table 22.26 Seasonally adjusted Lowe, Young and geo- metric Laspeyres indices with carry forward prices and the centred rolling year index Figure 22.7 Seasonally adjusted Lowe, Young and geo- metric Laspeyres indices with carry forward prices and the centred rolling year index THE TREATMENT OF SEASONAL PRODUCTS Year 1970 1971 1972 1973 Month PLOSA PYSA PGLSA PCRY 1.2608
1.1690 1.1806 1.1924 1.2049 1.2203 1.2386 1.2608 1.2809 1.2966 1.3176 1.3406 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 to predict the corresponding centred rolling year entries; see Figure 22.7 for plots of these predictions.
What is remarkable in Table 22.26 and Figure 22.7 is that the predicted values of these seasonally adjusted series are fairly close to the corresponding target index values.49 This result is somewhat unexpected since the annual basket indices use price information for only two con- secutive months, whereas the corresponding centred roll- ing year index uses price information for some 25 months.50 It should be noted that the seasonally adjusted 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 P LOSA PYSA P GLSA P CRY geometric Laspeyres index is generally the best predictor of the corresponding rolling year index for this data set.
It can be seen from Figure 22.7 that for the ﬁrst few months of 1973, the three month-to-month indices underestimate the centred rolling year inﬂation rate, but by the middle of 1973, the month-to-month indices are right on target.51 22.93 The above manipulations can be repeated, replacing the carry forward annual basket indices by their imputed counterparts; i.e., use the information in Table 22.24 (instead of Table 22.23) and in Table 22.27 (instead of Table 22.26).
A seasonally adjusted version of the Rothwell index presented in the previous section may also be found in Table 22.27.52 The ﬁve series in Table 22.27 are also represented graphically in Figure 22.8.
22.94 Again, the seasonally adjusted annual basket type indices listed in the PLOSA, PYSA and PGLSA col- umns of Table 22.27 (using imputations for the missing prices) are reasonably close to the corresponding centred rolling year index listed in the last column of Table 22.27.53 The seasonally adjusted geometric Laspeyres index is the closest to the centred rolling year index, and the seasonally adjusted Rothwell index is the furthest away.
The three seasonally adjusted month-to- month indices that use annual weights, PLOSA, PYSA and PGLSA, dip below the corresponding centred rolling year index, PCRY, for the ﬁrst few months of 1973 when the rate of month-to-month inﬂation suddenly increases, but by the middle of 1973, all four indices are fairly close to each other.

The seasonally adjusted Rothwell does not do 49 For observations 13 to 31, the seasonally adjusted series can be regressed on the centred rolling year series.
For the seasonally adjusted Lowe index, an R2 of 0.8816 is obtained; for the seasonally adjusted Young index, an R2 of 0.9212 is obtained; and for the seasonally adjusted geometric Laspeyres index, an R2 of 0.9423 is obtained.

These ﬁts are not as good as the ﬁt obtained in paragraphs 22.55 to 22.62, where the seasonally adjusted approximate rolling year index is used to predict the ﬁxed base Laspeyres rolling year index.

This R2 is 0.9662; recall the discussion of Table 22.20.
50 For seasonal data sets that are not as regular as the modiﬁed Turvey data set, the predictive power of the seasonally adjusted annual basket type indices may be considerably less; i.e., if there are abrupt changes in the seasonal pattern of prices, these month-to-month indices cannot be expected to accurately predict a rolling year index.
51 Recall that the last six months of PCRY have been artiﬁcially held constant; six months of data for 1974 would be required to evaluate these centred rolling year index values, but these data are not available.

52 The same seasonal adjustment technique as was deﬁned by equations (22.35) was used.

53 For observations 13 to 31, the seasonally adjusted series can be regressed on the centred rolling year series.
For the seasonally adjusted Lowe index, an R2 of 0.8994 is obtained; for the seasonally adjusted Young index, an R2 of 0.9294 is obtained; and for the seasonally adjusted geometric Laspeyres index, an R2 of 0.9495 is obtained.

For the seasonally adjusted Rothwell index, an R2 of 0.8704 is obtained, which is lower than the other three ﬁts.

For the Lowe, Young and geometric Laspeyres indices using imputed prices, these R2 are higher than those obtained using carry forward prices.
415 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Table 22.27 Seasonally adjusted Lowe, Young and geo- metric Laspeyres indices with imputed prices, seasonally adjusted Rothwell and centred rolling year indices Year Month PLOSA PYSA PGLSA PROTHSA PCRY 1.1591
1.1065 1.1174 1.1254 1.1313 1.1402 1.1502 1.1591 1.1690 1.1806 1.1924 1.2049 1.2203 1.2386 1.2608 1.2809 1.2966 1.3176 1.3406 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 a very good job of approximating PCRY for this particular data set, although this could be a function of the rather simple method of seasonal adjustment that is used.
22.95 Comparing the results in Tables 22.26 and 22.7, it can be seen that, for the modiﬁed Turvey data set, it did not make a great deal of diﬀerence whether missing prices are carried forward or imputed; the sea- sonal adjustment factors picked up the lumpiness in the unadjusted indices that occurs if the carry forward method is used.
Nevertheless, the three month-to-month indices that used annual weights and imputed prices did predict the corresponding centred rolling year indices somewhat better than the three indices that used carry forward prices.

Hence, the use of imputed prices over carry forward prices is recommended.
22.96 The conclusions that emerge from this section are rather encouraging for statistical agencies that wish to use an annual basket type index as their ﬂagship index.54 It appears that for commodity groups that have 54 Taking into account the results of previous chapters, the use of the annual basket Young index is not encouraged because of its failure of the time reversal test and the resulting upward bias.
416 Figure 22.8 Seasonally adjusted Lowe, Young and geo- metric Laspeyres indices with imputed prices, seasonally adjusted Rothwell and centred rolling year indices 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 P LOSA PYSA PGLSA P ROTSA PCRY strong seasonality, an annual basket type index for this group can be seasonally adjusted55 and the resulting seasonally adjusted index value can be used as a price relative for the group at higher stages of aggregation.
The preferred type of annual basket type index appears to be the geometric Laspeyres index rather than the Lowe index, but the diﬀerences between the two are not large for this data set.

Conclusion 22.97 A number of tentative conclusions can be drawn from the results of the previous sections in this chapter: The inclusion of seasonal commodities in maximum overlap month-to-month indices will frequently lead to substantial biases.
Hence, unless the maximum overlap month-to-month indices using seasonal com- modities cumulated for a year are close to their year- over-year counterparts, the seasonal commodities should be excluded from the month-to-month index or the seasonal adjustment procedures suggested in paragraphs 22.91 to 22.96 should be used.
Year-over-year monthly indices can always be con- structed even if there are strongly seasonal commod- ities.56 Many users will be interested in these indices; moreover, these indices are the building blocks for annual indices and for rolling year indices.

Statistical agencies should compute these indices, which may be in order to prevent user labelled ‘‘analytic series’’ confusion with the primary month-to-month CPI.
55 It is not necessary to use rolling year indices in the seasonal adjustment process, but their use is recommended because they increase the objectivity and reproducibility of the seasonally adjusted indices.

56 There may be problems with the year-over-year indices if shifting holidays or abnormal weather changes ‘‘normal’’ seasonal patterns.
In general, choosing a longer time period will mitigate these types of problems; i.e., quarterly seasonal patterns will be more stable than monthly patterns, which in turn will be more stable than weekly pat- terns.

 Rolling year indices should also be made available as analytic series.

These indices will give the most reliable indicator of annual inﬂation at a monthly frequency.
This type of index can be regarded as a seasonally adjusted CPI, and is the most natural to use as a central bank inﬂation target.

It has the disadvantage of measuring year-over-year inﬂation with a lag of six months; hence it cannot be used as a short-run indi- cator of month-to-month inﬂation.

Nevertheless, the techniques suggested in paragraphs 22.55 to 22.62 and 22.91 to 22.96 could be used, so that timely forecasts of these rolling year indices can be made using current price information.
Annual basket indices can also be successfully used in the context of seasonal commodities.

Most users of the CPI will, however, want to use seasonally adjusted versions of these annual basket type indices.
The seasonal adjustment can be done using the index number methods explained in paragraphs 22.91 to 22.96, or traditional statistical agency seasonal adjust- ment procedures could be used.57 From an a priori point of view, when making a price comparison between any two periods, the Paasche and Laspeyres indices are of equal impor- tance.

Under normal circumstances, the spread between the Laspeyres and Paasche indices will be reduced by using chained indices rather than ﬁxed base indices.
Hence, suggested that when constructing year-over-year monthly or annual indices, the chained Fisher index (or the chained is it THE TREATMENT OF SEASONAL PRODUCTS To¨ rnqvist–Theil index, which closely approximates the chained Fisher) be chosen as the target index that a statistical agency should aim to approximate.

When constructing month-to-month indices, however, chained indices should always be checked against their year-over-year counterparts to check for chain drift.
If substantial drift is found, the chained month- to-month indices must be replaced by ﬁxed base indices or seasonally adjusted annual basket type indices.58 expenditure from base If current period expenditure shares are not all that diﬀerent shares, year approximate chained Fisher indices will normally provide a very close practical approximation to the indices.
Approximate Las- chained Fisher target peyres, Paasche and Fisher indices use base period expenditure shares whenever they occur in the index number formula in place of current period (or lagged current period) expenditure shares.

Approximate Laspeyres, Paasche and Fisher indices can be com- puted by statistical agencies using their normal information sets.
The geometric Laspeyres index is an alternative to the approximate Fisher index; it uses the same informa- tion and will normally be close to the approximate Fisher index.

It is evident that more research needs to be carried out on the problems associated with the index number treatment of seasonal commodities.

There is, as yet, no consensus on what is best practice in this area.
57 There is, however, a problem with using traditional X-11 type sea- sonal adjustment procedures for seasonally adjusting the main CPI because ‘‘ﬁnal’’ seasonal adjustment factors are generally not available until data have been collected for two or three more years.

Since the main CPI cannot be revised, this may preclude using X-11 type sea- sonal adjustment procedures on it.

Note that the index number method of seasonal adjustment explained in this chapter does not suﬀer from this problem.
58 Alternatively, some sort of multilateral index number formula could be used; see, for example, Caves, Christensen and Diewert (1982a) or Feenstra and Shapiro (2003).
417 DURABLES AND USER COSTS 23 Introduction 23.1 When a durable good (other than housing) is purchased by a consumer, national consumer price indices (CPIs) attribute all that expenditure to the period of purchase, even though the use of the good extends beyond the period of purchase.1 By deﬁnition, a durable good delivers services longer than the period under consideration.2 The System of National Accounts 1993 deﬁnes a durable good as follows: In the case of goods, the distinction between acquisition and use is analytically important.
It underlies the dis- tinction between durable and non-durable goods exten- sively used in economic analysis.

In fact, the distinction between durable and non-durable goods is not based on physical durability as such.

Instead, the distinction is based on whether the goods can be used once only for purposes of production or consumption or whether they can be used repeatedly, or continuously.

For example, coal is a highly durable good in a physical sense, but it can be burnt only once.
A durable good is therefore deﬁned as one which may be used repeatedly or continuously over a period of more than a year, assuming a normal or average rate of physical usage.

A consumer durable is a good that may be used for purposes of consumption repeatedly or continuously over a period of a year or more (Commission of the European Communities et al.

(1993, p. 208)).
(1993, p. 208)).

This chapter is mainly concerned with the problems involved in pricing durable goods according to the above deﬁnition.3 Durability is more than the fact that a good can physically persist for more than a year (this is true of most goods): a durable good is distinguished from a non-durable good by its ability to deliver useful services to a consumer through repeated use over an extended period of time.
23.2 Since the beneﬁts of using the consumer dur- able extend over more than one period, it may not be appropriate to charge the entire purchase cost of the 1 This treatment of the purchases of durable goods dates back to Alfred Marshall (1898, pp.
594–595) at least: We have noticed also that though the beneﬁts which a man derives from living in his own house are commonly reckoned as part of his real income, and estimated at the net rental value of his house; the same plan is not followed with regard to the beneﬁts which he derives from the use of his furniture and clothes.
It is best here to follow the common practice, and not count as part of the national income or dividend anything that is not commonly counted as part of the income of the individual.

2 An alternative deﬁnition of a durable good is that the good delivers services to its purchaser for a period exceeding three years: ‘‘The Bureau of Economic Analysis deﬁnes consumer durables as those durables that have an average life of at least 3 years’’ (Katz (1983, p. 422)).
3 In paragraphs 23.136 to 23.145 there is a brief discussion about accounting for the purchase, consumption and inventory holdings of non-durable goods.

durable to the initial period of purchase.

If this point of view is taken, then the initial purchase cost must be distributed somehow over the useful life of the asset.

This is a fundamental problem of accounting.4 Charles R. Hulten (1990, pp.
120–121) explains the consequences for accountants of the durability of a purchase as follows: Durability means that a capital good is productive for two or more time periods, and this, in turn, implies that a distinction must be made between the value of using or renting capital in any year and the value of owning the capital asset.
This distinction would not necessarily lead to a measurement problem if the capital services used in any given year were paid for in that year; that is, if all capital were rented.

In this case, transactions in the rental market would ﬁx the price and quantity of capital in each time period, much as data on the price and quantity of labor services are derived from labor market transactions.
But, unfortunately, much capital is utilized by its owner and the transfer of capital services between owner and user results in an implicit rent typically not observed by the statistician.

Market data are thus inadequate for the task of directly estimating the price and quantity of capital services, and this has led to the development of indirect procedures for inferring the quantity of capital, like the perpetual inventory method, or to the acceptance of ﬂawed measures, like book value.
23.3 There are three main methods for dealing with the durability problem: ignore the problem of distributing the initial cost of the durable over the useful life of the good and allo- cate the entire charge to the period of purchase.

This is known as the acquisitions approach, and is the approach currently used by CPI statisticians for all durables, with the exception of housing; the rental equivalence or leasing equivalence approach.
In this approach, a period price is imputed for the durable which is equal to the rental price or leasing price of an equivalent consumer durable for the same period of time; 4 According to Stephen Gilman (1939) and David Solomons (1961): The third convention is that of the annual accounting period.

It is this convention which is responsible for most of the diﬃcult accounting pro- blems.
Without this convention, accounting would be a simple matter of recording completed and fully realized transactions: an act of primitive simplicity (Gilman (1939, p. 26)).

All the problems of income measurement are the result of our desire to attribute income to arbitrarily determined short periods of time.

Every- thing comes right in the end; but by then it is too late to matter (Solomons (1961, p. 378)).
Note that these authors do not mention the additional complications that arise from the fact that future revenues and costs must be dis- counted to yield values that are equivalent to present dollars.

419 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE the user cost approach.
In this approach, the initial purchase cost of the durable is decomposed into two parts: one part which reﬂects an estimated cost of using the services of the durable for the period, and another part which is regarded as an investment that must earn some exogenous rate of return.

approach, a fourth approach has been used: the pay- ments approach,5 which is a type of cash ﬂow approach.

It is explained in paragraphs 23.134 and 23.135.


23.8 Paragraphs 23.136 to 23.145 outline some of the problems involved in implementing the three main approaches for pricing owner-occupied housing.

These three approaches will be discussed more fully in the following three sections.

23.4 The above three approaches to the treatment of durable purchases can be applied to the purchase of any durable commodity.
Historically, it turns out that the rental equivalence and user cost approaches have only been applied to owner-occupied housing.

In other words, the acquisitions approach to the purchase of con- sumer durables has been universally used by statistical agencies, with the exception of owner-occupied housing.

A possible reason for this is tradition; Marshall set the standard, and statisticians have followed his example for the past century.
Another possible reason is that unless the durable good has a very long useful life, it usually will not make a great deal of diﬀerence in the long run whether the acquisitions approach or one of the two alternative approaches is used.

This fact is demonstrated in paragraphs 23.39 to 23.42.

23.5 a major component of the user cost approach to valuing the services of owner-occupied housing is the depreciation component.
In paragraphs 23.43 to 23.68, a general model of depreciation for a consumer durable is presented and then specialized to the three most com- mon models of depreciation that are in use.

These models assume that homogeneous units of the durable are produced in each period so that information on the prices of the various vintages of the durable at any point in time can be used to determine the pattern of depre- ciation.
However, many durables (like housing) are custom produced and thus the methods for determining the form of depreciation explained in paragraphs 23.43 to 23.68 are not applicable.

The special problems caused by these uniquely produced consumer durables are considered in paragraphs 23.69 to 23.78.

23.6 Subsequent sections treat some of the special problems involved in implementing the user cost and rental equivalence methods for valuing the services provided by owner-occupied housing.
Paragraphs 23.79 to 23.93 present a derivation for the user cost of owner- occupied housing and various approximations to it.

Paragraphs 23.94 to 23.120 consider some of the costs that are tied to home-ownership, while paragraphs 23.121 to 23.133 consider how a landlord’s costs might diﬀer from a home-owner’s costs.
This material is rele- vant if the rental equivalence approach to valuing the services of owner-occupied housing is used: care must be taken to remove some costs that are embedded in market rents that home-owners do not face.

23.7 Following Marshall, statistical agencies have used alternatives to the acquisitions approach when dealing with owner-occupied housing.
In addition to the rental the usual approach by statistical agencies) and the user cost equivalence approach (which is 5 This is the term used by Goodhart (2001, pp.

F350–F351).
F350–F351).

420 The acquisitions approach 23.9 The net acquisitions approach to the treatment of owner-occupied housing is described by Charles Goodhart (2001, p. F350) as follows: The ﬁrst is the net acquisition approach, which is the change in the price of newly purchased owner-occupied dwellings, weighted by the net purchases of the reference population.
This is an asset based measure, and therefore comes close to my preferred measure of inﬂation as a change in the value of money, though the change in the price of the stock of existing houses rather than just of net purchases would in some respects be even better.

It is, moreover, consistent with the treatment of other dur- ables.
A few countries, e.g., Australia and New Zealand, have used it, and it is, I understand, the main contender for use in the Euro-area Harmonized Index of Consumer Prices (HICP), which currently excludes any measure of the purchase price of (new) housing, though it does include minor repairs and maintenance by home-owners, as well as all expenditures by tenants.
23.10 The weights for the net acquisitions approach are the net purchases of the household sector of houses from other institutional sectors in the base period.

Note that, in principle, purchases of second-hand dwellings from other sectors are relevant here; for example, a local government may sell rental dwellings to owner-occupiers.

Typically, however, newly built houses form a major part of these types of transactions.
Thus the long-term price relative for this category of expenditure will be primarily the price of (new) houses (quality adjusted) in the current period relative to the price of new houses in the base period.6 If the net acquisitions approach is applied to other consumer durables, it is extremely easy to imple- ment: the purchase of a durable is treated in the same way as a non-durable or service purchase is treated.
23.11 One additional implication of the net acqui- sitions approach is that major renovations and additions to owner-occupied dwelling units could also be con- sidered as being in scope for this approach.

In practice, these costs typically are not covered in a standard CPI.

The treatment of renovations and additions is con- sidered in more detail in paragraphs 23.107 to 23.117.
23.12 Traditionally, the net acquisitions approach also includes transfer costs relating to the buying and selling of second-hand houses as expenditures that are in scope for an acquisitions type CPI.

These costs are 6 This price index may or may not include the price of the land on which the new dwelling unit is situated.

Thus a new house price con- struction index would typically not include the land cost.
The acqui- sitions approach concentrates on the purchases by households of goods and services that are provided by suppliers from outside the household sector.

If the land on which a new house is situated was previously owned by the household sector, then presumably the cost of this land would be excluded from an acquisitions type new house price index.

23.13 The major advantage of mainly the costs of using a real estate agent’s services and asset transfer taxes.
These transfer costs are further dis- cussed in paragraphs 23.100, 23.101 and 23.118 to 23.120. the acquisitions approach is that it treats durable and non-durable pur- chases in a completely symmetric manner, and thus no special procedures have to be developed by a statistical agency to deal with durable goods.
As will be seen later, the major disadvantage of this approach is that the expenditures associated with this approach will tend to understate the corresponding expenditures on durables that are implied by the rental equivalence and user cost approaches.
23.14 Some diﬀerences between the acquisitions approach and the other approaches are: If rental or leasing markets for the durable exist and the durable has a long useful life, then the expenditure weights implied by the rental equivalence or user cost approaches will typically be much larger than the corresponding expenditure weights implied by the acquisitions approach; see paragraphs 23.34 to 23.42.
If the base year corresponds to a boom year (or a slump year) for the durable, then the base period expenditure weights may be too large or too small.

Put another way, the aggregate expenditures that correspond to the acquisitions approach are likely to be more volatile than the expenditures for the aggre- gate that are implied by the rental equivalence or user cost approaches.
In making comparisons of consumption across coun- tries where the proportion of owning versus renting or leasing the durable varies greatly,7 the use of the acquisitions approach may lead to misleading cross- country comparisons.

The reason for this is that capital costs are excluded in the net acquisitions approach, whereas they are explicitly or implicitly included in the other two approaches.
23.15 More fundamentally, whether the acquisitions approach is the right one or not depends on the overall purpose of the index number.

If the purpose is to mea- sure the price of current period consumption services, then the acquisitions approach can only be regarded as an approximation to a more appropriate approach (which would be either the rental equivalence or user cost approach).
If the purpose of the index is to measure monetary (or non-imputed) expenditures by households during the period, then the acquisitions approach is preferable.

The rental equivalence approach 23.16 The rental equivalence approach simply values the services yielded by the use of a consumer durable good for a period by the corresponding market rental 7 According to Hoﬀmann and Kurz (2002, pp.
3–4), about 60 per cent of German households live in rented dwellings, whereas only about 20 per cent of Spaniards rent their dwellings.

DURABLES AND USER COSTS value for the same durable for the same period of time (if such a rental value exists).
This is the approach taken in the System of National Accounts 1993 (SNA 1993) for owner-occupied housing: As well-organized markets for rented housing exist in most countries, the output of own-account housing ser- vices can be valued using the prices of the same kinds of services sold on the market with the general valuation rules adopted for goods and services produced on own account.
In other words, the output of housing services produced by owner-occupiers is valued at the estimated rental that a tenant would pay for the same accom- modation, taking into account factors such as location, neighbourhood amenities, etc.

as well as the size and quality of the dwelling itself (Commission of the Euro- pean Communities et al.

(1993, p. 134)).
(1993, p. 134)).

23.17 The SNA 1993 nevertheless follows Marshall (1898, p. 595) and does not extend the rental equivalence approach to consumer durables other than housing.
This seemingly inconsistent treatment of durables is explained in the SNA 1993 as follows: The production of housing services for their own ﬁnal consumption by owner-occupiers has always been included within the production boundary in national accounts, although it constitutes an exception to the general exclusion of own-account service production.
The ratio of owner-occupied to rented dwellings can vary signiﬁcantly between countries and even over short periods of time within a single country, so that both international and intertemporal comparisons of the production and con- sumption of housing services could be distorted if no imputation were made for the value of own-account services (Commission of the European Communities et al.

(1993, p. 126)).
(1993, p. 126)).

23.18 Eurostat’s (2001) Handbook on Price and Volume Measures in National Accounts also recommends the rental equivalence approach for the treatment of the dwelling services for owner-occupied housing: ‘‘The output of dwelling services of owner-occupiers at cur- rent prices is in many countries estimated by linking the actual rents paid by those renting similar properties in the rented sector to those of owner-occupiers.
This allows the imputation of a notional rent for the service owner-occupiers receive from their property’’ (Eurostat (2001, p. 99)).

23.19 The United States statistical agencies, the Bureau of Labor Statistics and the Bureau of Economic Analysis, both use the rental equivalence approach to value the services of owner-occupied housing.
Arnold J. Katz (1983, p. 411) describes the Bureau of Economic Analysis (BEA) procedures as follows: the value of Basically, BEA measures the gross rent (space rent) of owner-occupied housing from data on the rent paid for similar housing with the same market value.
To get the service value that is added to GNP (gross housing pro- duct), intermediate goods and services included in this ﬁgure (e. g., expenditures for repair and maintenance, insurance, condominium fees, and closing costs) are subtracted from the space rent.

To obtain a net return (net rental income), depreciation, taxes, and net interest are subtracted from, and subsidies added to, the service value.
23.20 There are some problems with the above treatment of housing and they are discussed in later 421 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE sections, after the user cost approach to durables has been discussed.8 23.21 To summarize the above material, it can be seen that the rental equivalence approach to the treat- ment of durables is conceptually simple: impute a cur- rent period rental or leasing price for a comparable product as the price for the purchase of a unit of a consumer durable.
For existing stocks of used consumer durables, the rental equivalence approach would entail ﬁnding rental prices for comparable used units.9 To date, as noted above, statistical agencies have not used the rental equivalence approach to the treatment of durables, with the single exception of owner-occupied housing.

Note, however, that in order to implement the it is necessary for the rental equivalence approach, relevant rental or leasing markets to exist.
Often this will not be the case, particularly when it is recognized that vintage speciﬁc rental prices are required for all vintages of the durable held by households.10 The user cost approach 23.22 The user cost approach to the treatment of durable goods is in some ways very simple: it calculates the cost of purchasing the durable at the beginning of the period, using the services of the durable during the period and then netting oﬀ from these costs the beneﬁt that could be obtained by selling the durable at the end of the period.
Several details of this procedure are, however, somewhat controversial.

These involve the use of opportunity costs, which are usually imputed costs, the treatment of interest, and the treatment of capital gains or holding gains.
23.23 Another complication with the user cost approach is that it involves making distinctions between current period (ﬂow) purchases within the period under consideration and the holdings of physical stocks of the durable at the beginning and the end of the accounting 8 To anticipate the later results: the main problem is that the rental equivalence approach to valuing the services of owner-occupied housing may give a higher valuation for these services than the user cost approach.
9 Another method for determining rental price equivalents for stocks of consumer durables is to ask households what they think their durables would rent for.

This approach is used by the United States Bureau of Labor Statistics in order to determine expenditure weights for owner- occupied housing; i.e., home-owners are asked to estimate what their house would rent for if it were rented to a third party; see the Bureau of Labor Statistics (1983).
Lebow and Rudd (2003, p. 169) note that these estimates of imputed rents in the United States based on a consumer expenditure survey diﬀer considerably from the corresponding Bureau of Economic Analysis estimates for imputed rents, which are derived by applying a rent-to-value ratio for rented properties to the owner- occupied stock of housing.
Lebow and Rudd feel that the expenditure survey estimates may be less reliable than the ratio of rent-to-value method because of the relatively small size of the consumer expendi- ture survey plus the diﬃculties households may have in recalling or estimating expenditures.

10 If the form of depreciation is of the ‘‘one hoss shay’’ or light bulb type, then the rental price for the durable will be the same for all vintages, and hence a detailed knowledge of market rentals by vintage will not be required.
The light bulb model of depreciation dates back to Bo¨ hm-Bawerk (1891, p. 342).

For more recent material on this model, see paragraphs 23.62 to 23.68, or Hulten (1990) or Diewert (2003b).

422 period.

Up to this point in the manual, all prices and quantity purchases have been thought of as taking place at a single point in time, say the middle of the period under consideration, and consumption has been thought of as taking place within the period as well.
Thus, there has been no need to consider the behaviour (and valuation) of stocks of consumer durables that house- holds may have at their disposal.

The rather complex problems involved in accounting for stocks and ﬂows are unfamiliar to most price statisticians.

23.24 To determine the net cost of using the durable good during, say, period 0, assume that one unit of the durable good is purchased at the beginning of period 0 at the price P0.
The ‘‘used’’ or ‘‘second-hand’’ durable good can be sold at the end of period 0 at the price P1 S. It might seem that a reasonable net cost for the use of one unit of the consumer durable during period 0 is its initial purchase price P0 less its end of period 0 ‘‘scrap value’’ P1 S. However, money received at the end of the period is not as valuable as money received at the beginning of the period.
Thus, in order to convert the end of period value into its beginning of period equivalent value, it is S by the term 1+r0, necessary to discount the term P1 where r0 is the beginning of period 0 nominal interest rate that the consumer faces.
Hence the period 0 user cost u0 for the consumer durable11 is deﬁned as: u0 P 0 P1 S (1+r0) (23:1) 23.25 There is another way to view the user cost formula (23.1): the consumer purchases the durable at the beginning of period 0 at the price P0 and charges himself or herself the rental price u0.
The remainder of the purchase price, I 0, deﬁned as I 0 P 0 u0 (23:2) can be regarded as an investment, which is to yield the appropriate opportunity cost of capital r0 that the con- sumer faces.
At the end of period 0, this rate of return could be realized provided that I 0, r0 and the selling price of the durable at the end of the period P1 S satisfy the following equation: I 0(1+r0)=P1 S (23:3) S and r0, equation (23.3) determines I 0, which in Given P1 turn, given P0, determines the user cost u0 via equa- tion (23.2).12 23.26 It should be noted that some price statisticians object to the user cost concept as a valid pricing concept for a CPI: A suitable price concept for a CPI ought to reﬂect only a ratio of exchange of money for other things, not a ratio at which money in one form or time period can be traded for money in another form or time period.
The ratio at which money today can be traded for money tomorrow 11 This approach to the derivation of a user cost formula was used by Diewert (1974b), who in turn based it on an approach attributable to Hicks (1946, p. 326).

12 This derivation for the user cost of a consumer durable was also made by Diewert (1974b, p. 504).
by paying an interest rate or by enjoying actual or expected holding gains on an appreciating asset has no part in a measure of the current purchasing power of money (Reinsdorf (2003)).
User costs are not like the prices of non-durables or services because the user cost concept involves pricing the durable at two points in time rather than at a single point in time.13 Because the user cost concept involves prices at two points in time, money received or paid out at the ﬁrst point in time is more valuable than money paid out or received at the second point in time, so interest rates creep into the user cost formula.
Further- more, because the user cost concept involves prices at two points in time, expected prices can be involved if the user cost is calculated at the beginning of the period under consideration instead of at the end.

With all these complications, it is no wonder that many price statisti- cians would like to avoid using user costs as a pricing concept.
However, even for price statisticians who would prefer to use the rental equivalence approach to the treatment of durables over the user cost approach, there is some justiﬁcation for considering the user cost approach in some detail, since this approach gives insights into the economic determinants of the rental or leasing price of a durable.

As is seen in paragraphs 23.121 to 23.133, the user cost for a house can diﬀer substantially for a landlord compared to an owner.
Thus adjustments should be made to market rents for dwell- if they are to be used as imputations for ing units, owner-occupied rents.

23.27 The user cost formula (23.1) can be put into a more familiar form if the period 0 economic depreciation rate d and the period 0 ex post asset inﬂation rate i 0 are deﬁned.

Deﬁne d by: (1d) P1 S=P1 where P1 S is the price of a used asset at the end of period 0 and P1 is the price of a new asset at the end of period 0.
The period 0 inﬂation rate for the new asset, i 0, is deﬁned by: (23:4) 1+i 0 P1 (23:5) Eliminating P1 from equations (23.4) and (23.5) leads to the following formula for the end of period 0 used asset price: S=P1 S=(1d)(1+i 0)P 0 P1 (23:6) Substitution of equation (23.6) into equation (23.1) yields the following expression for the period 0 user cost u0: 13 Woolford suggested that interest should be excluded from an ideal price index that measured inﬂation.
In his view, interest is not a con- temporaneous price; i.e., an interest rate necessarily refers to two points in time: a beginning point when the capital is loaned and an ending point when the capital loaned must be repaid.

Thus, if attention is restricted to a domain of deﬁnition that consists of only contem- poraneous prices, interest rates are excluded.
Woolford (1999, p. 535) noted that his ideal inﬂation measure ‘‘would be contemporary in nature, capturing only the current trend in prices associated with transactions in goods and services.
It would exclude interest rates on the grounds that they are intertemporal prices, representing the relative price of consuming today rather than in the future.’’ DURABLES AND USER COSTS u0= [(1+r0)(1d)(1+i 0)]P 0 1+r0 (23:7) Note that r0i 0 can be interpreted as a period 0 real interest rate and d(1+i 0) can be interpreted as an inﬂation-adjusted depreciation rate.

23.28 The user cost u0 is expressed in terms of prices that are discounted to the beginning of period 0.
It is also possible to express the user cost in terms of prices that are ‘‘discounted’’ to the end of period 0.14 Deﬁne the end of period 0 user cost p0 as:15 p 0 (1+r0)u0=[r0i 0+d(1+i 0)]P 0 (23:8) where the last equation follows using equation (23.7).
If the real interest rate r0* is deﬁned as the nominal interest rate r0 less the asset inﬂation rate i 0, and the small term di 0 is neglected, then the end of period user cost deﬁned by equation (23.8) reduces to: p0=(r0*+d)P 0 (23:9) 23.29 Abstracting from transactions costs and inﬂation, it can be seen that the end of period user cost deﬁned by equation (23.9) is an approximate rental cost; i.e., the rental cost for the use of a consumer (or pro- ducer) durable good should equal the (real) opportunity cost of the capital tied up, r0*P0, plus the decline in value of the asset over the period, dP0.
Formulae (23.8) and (23.9) thus cast some light on the economic determinants of rental or leasing prices for consumer durables.

23.30 If the simpliﬁed user cost formula deﬁned by equation (23.9) is used, then forming a price index for the user costs of a durable good is not very much more diﬃcult than forming a price index for the purchase price of the durable good, P0.
The price statistician needs only to: make a reasonable assumption as to what an appro- interest rate r0* priate monthly or quarterly real should be; 14 The beginning of the period user cost u0 discounts all monetary costs and beneﬁts into their dollar equivalent at the beginning of period 0, whereas p0 discounts (or appreciates) all monetary costs and beneﬁts into their dollar equivalent at the end of period 0.
This leaves open how ﬂow transactions that take place within the period should be treated.

Following the conventions used in ﬁnancial accounting sug- gests that ﬂow transactions taking place within the accounting period should be regarded as taking place at the end of the accounting period.

Following this convention, end of period user costs should be used by the price statistician.
15 Christensen and Jorgenson (1969) derived a user cost formula similar to equation (23.7) in a diﬀerent way, using a continuous time optimi- zation model.

If the inﬂation rate i equals 0, then the user cost formula (23.7) reduces to that derived by Walras (1954, p. 269; ﬁrst edition 1874).

This zero inﬂation rate user cost formula was also derived by the industrial engineer A. Hamilton Church (1901, pp.
907–908), who perhaps drew on the work of Ewing Matheson (1910, p. 169, ﬁrst published in 1884): ‘‘In the case of a factory where the occupancy is assured for a term of years, and the rent is a ﬁrst charge on proﬁts, the rate of interest, to be an appropriate rate, should, so far as it applies to the buildings, be equal (including the depreciation rate) to the rental which a landlord who owned but did not occupy a factory would let it for.’’ Additional derivations of user cost formulae in discrete time have been made by Katz (1983, pp.
408–409) and Diewert (2003b).

423 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE make an assumption as to what a reasonable monthly or quarterly depreciation rate d should be;16 collect purchase prices P0 for the durable; make an estimate of the total stock of the durable which was held by the reference population during the base period for quantities.

In order to construct a superlative index, estimates of the stock held will have to be made for each period.
23.31 If it is thought necessary to implement the more complicated user cost formula (23.8) in place of the simpler formula (23.9), then the situation is more complicated.

As it stands, the end of period user cost formula (23.8) is an ex post user cost: the asset inﬂation rate i 0 cannot be calculated until the end of period 0 has been reached.

Formula (23.8) can be converted into an ex ante user cost formula if i 0 is interpreted as an anticipated asset inﬂation rate.
The resulting formula should approximate a market rental rate for the asset under inﬂationary conditions.17 23.33 23.32 Note that in the user cost approach to the treatment of consumer durables, the entire user cost formula (23.8) or (23.9) is the period 0 price.
Thus, in the time series context, it is not necessary to deﬂate each component of the formula separately; the period 0 price p0:[r0i 0+d(1+i 0)]P0 is compared to the correspond- ing period 1 price, p1 [r1i1+d(1+i1)]P1 and so on.

In principle, depreciation rates can be esti- mated using information on the selling prices of used units of the durable good.

This methodology will be explained in more detail in paragraphs 23.43 to 23.68.
Before this is done, however, it will be useful to use the material in this section to explain what the relationship between the user cost and acquisition approaches to the treatment of durables is likely to be.

This topic is dis- cussed in the following section.

The relationship between user costs and acquisition costs 23.34 In this section, the user cost approach to the treatment of consumer durables is compared to the acquisitions approach.
Obviously, in the short run, the value ﬂows associated with each approach could be very diﬀerent.

For example, if real interest rates, r0i 0, are very high and the economy is in a severe recession or depression, then purchases of new consumer durables, Q0 say, could be very low and even approach zero for very long-lived assets, such as houses.
In contrast, using 16 The geometric model for depreciation, explained in more detail in paragraphs 23.43 to 23.68, requires only a single monthly or quarterly depreciation rate.

Other models of depreciation may require the esti- mation of a sequence of vintage depreciation rates.

If the estimated annual geometric depreciation rate is da, then the corresponding monthly geometric depreciation rate d can be obtained by solving the equation (1d)12=1da.
Similarly, the estimated annual real interest rate is ra*, then the corresponding monthly real interest rate r* can be obtained by solving the equation (1+r*)12=1+ra*.
17 Since landlords must set their rent at the beginning of the period (and in fact, they usually set their rent for an extended period of time), if the user cost approach is used to model the economic determinants of market rental rates, then the asset inﬂation rate i 0 should be interpreted as an expected inﬂation rate rather than an ex post actual inﬂation rate.
if 424 the user cost approach zero, existing stocks of consumer durables would be carried over from previous periods and priced out at the appropriate user costs, and the resulting consumption value ﬂow could be quite large.

Thus, in the short run, the monetary values of consumption under the two approaches could be vastly diﬀerent.
Hence, follows, a (hypothetical) longer-run comparison is considered where real interest rates are held constant.18 in what 23.35 Suppose that in period 0, the reference popu- lation of households purchases q0 units of a consumer durable at the purchase price P0.
Then the period 0 value of consumption from the viewpoint of the acquisitions approach is: V 0 (23:10) these hypotheses, A P 0q0 23.36 Recall that the end of period user cost for one new unit of the asset purchased at the beginning of period 0 is p0 deﬁned by equation (23.8).
In order to simplify the analysis, declining balance depreciation is assumed; i.e., at the beginning of period 0, a one- period-old asset is worth (1d)P0; a two-period-old asset is worth (1d)2P0; a t-period-old asset is worth (1d)tP0; etc.
Under the corre- sponding end of period 0 user cost for a new asset purchased at the beginning of period 0 is p0; the end of period 0 user cost for a one-period-old asset at the beginning of period 0 is (1d)p0; the correspond- ing user cost for a two-period-old asset at the beginning of period 0 is (1d)2p0; the corresponding user cost for a t-period-old asset at the beginning of period 0 is (1d)tp0; etc.19 The ﬁnal simplifying assumption is that household purchases of the consumer durable have been growing at the geometric rate g into the indeﬁnite past.
This means that if household purchases of the durable were q0 in period 0, then in the previous period the households purchased q0/(1+g) new units; two periods ago, they purchased q0/(1+g)2 new units; t periods ago, they purchased q0/(1+g)t new units; etc.

Putting all of these assumptions together, it can be seen that the period 0 value of consumption from the view- point of the user cost approach is: V 0 U p0q0+ (1d)p0q0 1+g + (1d)2p0q0 (1+g)2 + .

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.
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.

(23:11) = = (1+g)p0q0 g+d summing the infinite series (1+g)[r0i 0+d(1+i 0)]P 0q0 g+d using equation (23:8) (23:12) 23.37 Equation (23.12) can be simpliﬁed by letting the asset inﬂation rate i 0 be 0 (or by replacing r0i 0 by the real interest rate r0* and by ignoring the small term di 0).

Under these conditions, the ratio of the user cost 18The following material is based on Diewert (2002c).
19 For many consumer durables, the ‘‘one hoss shay’’ assumption for depreciation may be more realistic than the declining balance model; see paragraphs 23.43 to 23.68, or Hulten (1990) or Diewert and Lawrence (2000).
ﬂow of consumption (23.12) to the acquisitions measure of consumption in period 0, (23.10), is: V 0 U V 0 A (1+g)(r0*+d) g+d (23:13) = 23.38 Using formula (23.13), it can be seen that if A will be greater than 1+g>0 and d+g>0, then V 0 unity if U=V 0 r0* > g(1d) 1+g (23:14) a condition that will usually be satisﬁed.20 Thus under normal conditions and over a longer time horizon, household expenditures on consumer durables using the user cost approach will tend to exceed the corresponding money outlays on new purchases of the consumer dura- ble.
The diﬀerence between the two approaches will tend to grow as the life of the asset increases (i.e., as the depreciation rate d decreases).
U=V 0 23.39 To get a rough idea of the possible magnitude of the value ratio for the two approaches, V 0 A, equa- tion (23.13) is evaluated for a ‘‘housing’’ example using annual data where the depreciation rate is 2 per cent (i.e., d=0.02), the real interest rate is 4 per cent (i.e., r0*=0.04) and the growth rate for the production of new houses is 1 per cent (i.e., g=0.01).

In this base case, the ratio of user cost expenditures on housing to the purchases of new housing in the same period, V 0 is 2.02.
If the U=V 0 depreciation rate is increased to 3 per cent, then V 0 A decreases to 1.77; if the depreciation rate is reduced to 1 per cent, then V 0 A increases to 2.53.

Again looking at the base case, if the real interest rate is increased to 5 per cent, then V 0 A increases to 2.36, while if the real U=V 0 interest rate is reduced to 3 per cent, then V 0 A decreases to 1.68.
Finally, if the growth rate for new houses is increased to 2 per cent, then V 0 A decreases to 1.53, while if the growth rate is reduced to 0, then V 0 U=V 0 A increases to 3.00.

Thus an acquisitions approach to housing in the CPI is likely to give about half the expen- diture weight that a user cost approach would give.
U=V 0 A, U=V 0 U=V 0 U=V 0 23.40 For shorter-lived assets, the diﬀerence between the acquisitions approach and the user cost approach indicating that the acquisitions will not be so large, approach is approximately ‘‘correct’’ as a measure of consumption services.21 U =V 0 the durable, then from the 20 Note that if the real interest rate r0 equals g, the real rate of growth in purchases of ratio (23.13), V 0 A=(1+g) and the acquisitions approach will be more or less equivalent to the user cost approach over the long run.
21 The simpliﬁed user cost approach can be used for other consumer durables as well.

In formula (23.13), let r0*=0.04, g=0.01 and d= U =V 0 0.15.

Under these conditions, V 0 A=1:20; i.e., for a declining bal- ance depreciation rate of 15 per cent, the user cost approach leads to an estimated value of consumption that is 20 per cent higher than the acquisitions approach under the conditions speciﬁed.
Thus for con- sumer durable depreciation rates that are lower than 15 per cent, it could be useful for the statistical agency to produce user costs for these goods and for the national accounts division to produce the corre- sponding consumption ﬂows as ‘‘analytic series’’.

It should be noted that this extends the present national accounts treatment of housing to other long-lived consumer durables.
Note also that this revised treat- ment of consumption in the national accounts would tend to make rich countries richer, since poorer countries hold fewer long-lived consumer durables on a per capita basis.

DURABLES AND USER COSTS 23.41 The following is a list of some of the problems and diﬃculties that might arise in implementing a user cost approach to purchases of a consumer durable:22 It is diﬃcult to determine what the relevant nominal interest rate r0 is for each household.
If a consumer has to borrow to ﬁnance the cost of a durable good purchase, then this interest rate will typically be much higher than the safe rate of return that would be the appropriate opportunity cost rate of return for a consumer who had no need to borrow funds to ﬁnance the purchase.23 It may be necessary to simply use a benchmark interest rate that would be deter- mined by either the government, a national statistical agency or an accounting standards board.
It will generally be diﬃcult to determine what the rele- vant depreciation rate is for the consumer durable.24 Ex post user costs based on formula (23.8) will be too volatile to be acceptable to users25 (owing to the volatility of the asset inﬂation rate i 0), and hence an ex ante user cost concept will have to be used.
This cre- ates diﬃculties in that diﬀerent national statistical agencies will generally make diﬀerent assumptions and use diﬀerent methods to construct forecasted struc- tures and land inﬂation rates.

Hence, the resulting ex ante user costs of the durable may not be comparable across countries.26 The user cost formula (23.8) must be generalized to accommodate various taxes that may be associated 22 For additional material on diﬃculties with the user cost approach, see Diewert (1980, pp.
475–479) and Katz (1983, pp.

415–422).

23 Katz (1983, pp.

415–416) comments on the diﬃculties involved in determining the appropriate rate of interest to use: There are numerous alternatives: a rate on ﬁnancial borrowings, on savings, and a weighted average of the two; a rate on nonﬁnancial investments.

e.g.

residential housing, perhaps adjusted for capital gains; and the consumer’s subjective rate of time preference.
Furthermore, there is some controversy about whether it should be the maximum observed rate, the average observed rate, or the rate of return earned on invest- ments that have the same degree of risk and liquidity as the durables whose services are being valued.
24 It is not necessary to assume declining balance depreciation in the user cost approach: any pattern of depreciation can be accommodated, including ‘‘one hoss shay’’ depreciation, where the durable yields a constant stream of services over time until it is scrapped.

See Diewert and Lawrence (2000) for some empirical examples for Canada, using diﬀerent assumptions about the form of depreciation.
For references to the depreciation literature and for empirical methods for estimating depreciation rates, see Hulten and Wykoﬀ (1981a; 1981b; 1996) and Jorgenson (1996).

25 Goodhart (2001, p. F351) comments on the practical diﬃculties of using ex post user costs for housing, as follows: An even more theoretical user cost approach is to measure the cost fore- gone by living in an owner-occupied property as compared with selling it at the beginning of the period and repurchasing it at the end .

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.

But this gives the absurd result that as house prices rise, so the opportunity cost falls; indeed the more virulent the inﬂation of housing asset prices, the more negative would this measure become.

Although it has some academic aﬁcionados, this ﬂies in the face of common sense; I am glad to say that no country has adopted this method.

As will be seen later, Iceland has in fact adopted a simpliﬁed user cost framework.
26 For additional material on the diﬃculties involved in constructing ex ante user costs, see Diewert (1980, pp.

475–486) and Katz (1983, pp.

419–420).

For empirical comparisons of diﬀerent user cost for- mulae, see Harper, Berndt and Wood (1989) and Diewert and Lawrence (2000).
425 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE with the purchase of a durable or with the continuing use of the durable.27 23.42 Some of the problems associated with esti- mating depreciation rates will be discussed in the next section.
Once d0 0 has been deﬁned by equation (23.15), the per- iod 0 cross-sectional depreciation rate for a unit of the durable good that is one period old at the begin- ning of period 0, d0 1, can be deﬁned using the following equation: Alternative models of depreciation A general model of depreciation for (unchanging) consumer durables 23.43 In this subsection, a ‘‘general’’ model of depreciation for durable goods that appear on the mar- ket each period without undergoing quality change will be presented.
In three subsequent subsections, this gen- eral model will be specialized to the three most common models of depreciation that appear in the literature.

In paragraphs 23.69 to 23.78 below, the additional pro- blems that occur when the durable is built as a unique good will be discussed.
23.44 The main tool that can be used to identify depreciation rates for a durable good is the (cross- sectional) sequence of vintage asset prices that units of the good sell for on the second-hand market at any point of time.28 23.45 Some notation is required.

Let P0 be the price of a newly produced unit of the durable good at the beginning of period 0 (this is the same notation as was used earlier).
Let Pt v be the second-hand market price at the beginning of period t of a unit of the durable good that is v periods old.29 Let d0 v be the period 0 deprecia- tion rate for a unit of the durable good that is v periods old at the beginning of period 0.
These depreciation rates can be deﬁned recursively, starting with the period 0 depreciation rate for a brand new unit, d0 0, using the period 0 vintage asset prices P0 0=P0 v as follows: 1=P 0 (23:15) 1d0 27 For example, property taxes are associated with the use of housing services and hence should be included in the user cost formula; see paragraphs 23.100 and 23.101.
As Katz (1983, p. 418) noted, taxation issues also have an impact on the choice of the interest rate: ‘‘Should the rate of return be a before or after tax rate?’’ From the viewpoint of a household that is not borrowing to ﬁnance the purchase of the durable, an after tax rate of return seems appropriate; but from the point of view of a leasing ﬁrm, a before tax rate of return seems appropriate.
This diﬀerence helps to explain why rental equivalence prices for the durable might be higher than user cost prices.

28 Another information source that could be used to identify depre- ciation rates for the durable good is the sequence of vintage rental or leasing prices that might exist for some consumer durables.
In the closely related capital measurement literature, the general framework for an internally consistent treatment of capital services and capital stocks in a set of vintage accounts was set out by Jorgenson (1989) and Hulten (1990, pp.

127–129; 1996, pp.

152–160).

29 Using this notation for vintages, it can be seen that the vintage v=0 price at the beginning of period t=0, P0 0, is equal to the price of a new unit of the good, P0.
If these second-hand vintage prices depend on how intensively the durable good has been used in previous periods, then it will be necessary to further classify the durable good not only by its vintage v, but also according to the intensity of its use.

In this case, think of the sequence of vintage asset prices, P0 v , as corresponding to the prevailing market prices of the various vintages of the good at the beginning of period 0 for assets that have been used at ‘‘average’’ intensities.
426 (1d0 1)(1d0 0)=P0 2=P 0 (23:16) Note that P0 2 is the beginning of period 0 asset price of a unit of the durable good that is two periods old, and it is compared to the price of a brand new unit of the dur- able, P0 (which is equal to P0 0 using the vintage good notation).

23.46 Given that the period 0 cross-sectional depreciation rates for units of the durable that are 0, 1, 2, .

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.

, v1 periods old at the beginning of period 0 are deﬁned (these are the depreciation rates d0 2, .

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, d0 v1), then the period 0 cross-sectional depreciation rate for units of the durable that are v periods old at the beginning of period 0 can be deﬁned using the following equation: 1, d0 0, d0 23.47 (23:17) (1d0 v+1=P 0 1)(1d0 v) .

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(1d0 0)=P0 It should be clear how the sequence of period 0 vintage asset prices P0 v can be converted into a sequence of period 0 vintage depreciation rates.
It should also be clear that the sequence of equations (23.15)–(23.17) can be repeated using the vintage asset price data pertaining to the beginning of period t, Pt v, in order to obtain a sequence of period t vintage depreciation rates, dt v. In the literature, it is usually assumed that the sequence of vintage depreciation rates, dt is independent of the v, period t so that: dt v=dv for all periods t and all vintages v (23:18) 23.48 The above material shows how the sequence of vintage or used durable goods prices at a point in time can be used in order to estimate depreciation rates.
This type of methodology, with a few extra modiﬁcations to account for diﬀering ages of retirement, was pioneered by Beidelman (1973; 1976) and Hulten and Wykoﬀ (1981a; 1981b; 1996).30 23.49 Recall the user cost formula for a new unit of the durable good under consideration deﬁned by equation (23.1).

The same approach can be used in order to deﬁne a sequence of period 0 user costs for all vintages v of the durable.
Thus suppose that P1a v+1 is the anticipated end of period 0 price of a unit of the durable good that is v periods old at the beginning of period 0, and let r0 be the consumer’s opportunity cost of capital.

Then the discounted to the beginning of period 0 user cost of a unit of the durable good that is v periods old at the beginning of period 0, u0 v, is deﬁned as follows: v P0 u0 vP1a v+1=(1+r0) v=0, 1, 2, .

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(23:19) 23.50 It is now necessary to specify how the end of period 0 anticipated vintage asset prices P1a v are related to their counterpart beginning of period 0 vintage asset 30 See also Jorgenson (1996) for a review of the empirical literature on the estimation of depreciation rates.
prices P0 v. The assumption that is made now is that the entire sequence of vintage asset prices at the end of period 0 is equal to the corresponding sequence of asset prices at the beginning of period 0 times a general anticipated period 0 inﬂation rate factor (1+i 0), where i 0 is the anticipated period 0 (general) asset inﬂation rate.

Thus it is assumed that P1a v =(1+i 0)P0 v v=0, 1, 2, .

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(23:20) Substituting equations (23.20) and (23.15)–(23.18) into equation (23.19) leads to the following beginning of period 0 sequence of vintage user costs:31 u0 v=(1dv1)(1dv2) .

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(1d0) [(1+r0)(1+i 0)(1dv)]P 0=(1+r0) =(1dv1)(1dv2) .

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(1d0) [r0i 0+dv(1+i 0)]P 0=(1+r0) v=0, 1, 2, .

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(23:21) 23.51 Note that if v=0, then the u0 0 deﬁned by equation (23.21) agrees with the user cost formula for a new purchase of the durable u0 that was derived earlier in equation (23.7).

23.52 The sequence of vintage user costs u0 v deﬁned by equation (23.21) are expressed in terms of prices that are discounted to the beginning of period 0.
However, as was done in paragraphs 23.22 to 23.33, it is also possible to express the user costs in terms of prices that are ‘‘discounted’’ to the end of period 0.

Thus, deﬁne the sequence of vintage end of period 0 user cost, p0 v, as follows: v (1+r0)u0 p0 v=(1dv1)(1dv2) .

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(1d0) v=0, 1, 2, .

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 [r0i 0+dv(1+i 0)]P 0 (23:22) 23.53 If the real interest rate r0* is deﬁned as the nominal interest rate r0 less the asset inﬂation rate i 0, and the small terms dv i 0 are neglected in equation (23.22), then the sequence of end of period user costs deﬁned by equation (23.22) reduces to: v=(1dv1)(1dv2) .

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(1d0)[r0*+dv]P 0 p0 v=0, 1, 2, .

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(23:23) Thus, if the price statistician has estimates for the vintage depreciation rates dv and the real interest rate r0*, and is able to collect a sample of prices for new units of the durable good P0, then the sequence of vintage user costs deﬁned by equation (23.23) can be calculated.

To complete the model, the price statistician should gather information on the stocks held by the household sector of each vintage of the durable good.
Then normal index number theory can be applied to these p and Q values, with the set of p being vintage user costs and the set of Q being the vintage stocks pertaining to each period.

For some worked examples of this methodology under various assumptions about depreciation rates and the calculation of expected asset 31 When v=0, deﬁne d1:1; i.e., the terms in front of the square brackets on the right-hand side of equation (23.21) are set equal to 1.
DURABLES AND USER COSTS inﬂation rates, see Diewert and Lawrence (2000) and Diewert (2003c).32 23.54 In the following three subsections, the general methodology described above is specialized by making additional assumptions about the form of the vintage depreciation rates dv.
Geometric or declining balance depreciation 23.55 The declining balance method of depreciation dates back to Matheson (1910, p. 55) at least.33 In terms of the algebra presented in the previous subsection, the method is very simple: all the cross-sectional vintage depreciation rates d0 v deﬁned by equations (23.15)– (23.17) are assumed to be equal to the same rate d, where d is a positive number less than one; i.e., for all time periods t and all vintages v, it is assumed that dt v=d v=0, 1, 2, .

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(23:24) Substitution of equation (23.24) into equation (23.22) leads to the following formula for the sequence of period 0 vintage user costs: v=(1d)v[(1+r0)(1+i 0)(1d)]P 0 p0 =(1d)vp0 0 v=0, 1, 2, .

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(23:25) 23.56 The second set of equations in (23.25) says that all the vintage user costs are proportional to the user cost for a new asset.
This proportionality means that it is not necessary to use an index number formula to aggregate over vintages to form a durable services it is useful to calculate the aggregate.

To see this, aggregate value of services yielded by all vintages of the consumer durable at the beginning of period 0.

Let qv be the quantity of the durable purchased by the household sector v periods ago for v=1, 2, .

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.

and let q0 be the new purchases of the durable during period 0.
The beginning of period 0 price for these vintages of age v will be p0 v, deﬁned by equation (23.25) above.

Thus the aggregate services of all vintages of the good, including those purchased in period 0, will have the following value, S 0: 1q1+p0 S0=p0 =p0 0q0+p0 0q0+(1d)p0 using equation (23:25) 2q2+ .

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0q1+(1d)2p0 0q2+ .

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=p0 =p0 0[q0+(1d)q1+(1d)2q2+ .

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0Q0 (23:26) 32 Additional examples and discussion can be found in two recent OECD manuals on productivity measurement and the measurement of capital; see OECD (2001a; 2001b).

33 A case for attributing the method to Walras (1954, pp.

268–269) could be made, but he did not lay out all of the details.

Matheson (1910, p. 91) used the term ‘‘diminishing value’’ to describe the method.
Hotelling (1925, p. 350) used the term ‘‘the reducing balance method’’, while Canning (1929, p. 276) used the term ‘‘the declining balance formula’’.

427 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE where the period 0 aggregate (quality-adjusted) quantity of durable services consumed in period 0, Q0, is deﬁned as Q0 q0+(1d)q1+(1d)2q2+ .

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(23:27) 23.57 Thus the period 0 services quantity aggregate Q0 is equal to new purchases of the durable in period 0, q0, plus one minus the depreciation rate d times the purchases of the durable in the previous period, q1, plus the square of one minus the depreciation rate times the purchases of the durable two periods ago, q2, and so on.
The service price that can be applied to this quantity aggregate is p0 0, and the imputed rental price or user cost for a new unit of the durable purchased in period 0.

23.58 If the depreciation rate d and the purchases of the durable in prior periods are known, then the aggregate service quantity Q0 can readily be calculated using equation (23.27).
Then, using equation (23.26), it can be seen that the value of the services of the durable (over all vintages), St, decomposes into the price term p0 0 times the quantity term Q0.

Hence, it is not necessary to use an index number formula to aggregate over vintages using this depreciation model.
Straight line depreciation 23.59 Another very common model of depreciation is the straight line model.34 In this model, a most prob- able length of life for the durable is somehow deter- mined, say L periods, so that after being used for L periods, the durable is scrapped.
In the straight line depreciation model, it is assumed that the period 0 cross- sectional vintage asset prices P0 v follow the following pattern of linear decline relative to the period 0 price of a new asset P 0: P0 v P 0 = Lv L for v=0, 1, 2, .

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, L1 (23:28) For v=L, L+1, .

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, it is assumed that P0 v=0.


Now substitute equations (23.20) and (23.28) into the begin- ning of the period user cost formula (23.19) in order to obtain the following sequence of period 0 vintage user costs for the durable: v(1+i 0)P0 v=P0 u0 (Lv)P 0 L P 0 1+r0* v+1=(1+r0) (1+i 0)(Lv1)P 0 (1+r0)L 1 L for v=0, 1, 2, .

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, L1 (Lv)r0* L (23:29) + = = where the asset-speciﬁc real interest rate for period 0, r0*, is deﬁned by 1+r0* 1+r0 1+i 0 (23:30) 23.60 The user costs for units of the durable good v 0 for that are older than L periods are zero; i.e., u0 34 This model of depreciation dates back to the late 1800s; see Matheson (1910, p. 55), Garcke and Fells (1893, p. 98) or Canning (1929, pp.

265–266).
265–266).

428 v L. Looking at the terms in square brackets on the right-hand side of equation (23.29), it can be seen that the ﬁrst term is a real interest opportunity cost for holding and using the unit of the durable that is v periods old (and this imputed interest cost declines as the dur- able good ages), and the second term is a depreciation term that is equal to the constant rate 1/L.
23.61 In this model of depreciation, it is necessary to keep track of household purchases of the durable for L periods and weight up each vintage quantity qv of these purchases by the corresponding vintage user cost u0 v, deﬁned by equation (23.29), or the end of period vintage user costs p0 v, deﬁned as (1+r0)u0 v, could be used.35 ‘‘One hoss shay’’ or light bulb depreciation 23.62 The ﬁnal model of depreciation that is in common use is the ‘‘light bulb’’ or ‘‘one hoss shay’’ model of depreciation.36 In this model, the durable delivers the same services for each vintage: a chair is a chair, no matter what its age (until it falls to pieces and is scrapped).
Thus this model also requires an estimate of the most probable life L of the consumer durable.37 In the ‘‘one hoss shay’’ model, it is assumed that the sequence of vintage beginning of the period user costs u0 v, deﬁned by the ﬁrst line of equation (23.29), is con- stant for all vintages younger than the asset lifetime L; i.e., it is assumed that v(1+i 0)P0 for v=0, 1, 2, .

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.

, L1 v+1=(1+r0) u0=u0 =P0 v=P0 vgP0 v+1 (23:31) where the discount factor g is deﬁned as g 1 1+r0* 1+i 0 1+r0 = and the asset-speciﬁc real interest rate r0* was deﬁned by equation (23.30).
Now the second equation in (23.31) can be used to express the vintage v asset price P0 v in terms of the common user cost u0 and the vintage v+1 asset price, P0 (23:32) v+1, so that v=u0+gP0 P0 23.63 Now start out using equation (23.33) with v=0, then substitute out P0 1 using equation (23.33) with v=1, then substitute out P0 2 using equation (23.33) with v=2, etc.

The process ﬁnally ends after L such (23:33) v+1 35 A worked example using this model of depreciation can be found in Diewert (2003b).
36 This model can be traced back to Bo¨ hm-Bawerk (1891, p. 342).

For a more comprehensive exposition, see Hulten (1990, p. 124) or Diewert (2003b).

37 The assumption of a single life L for a durable can be relaxed using a methodology attributable to Charles R. Hulten: We have thus far taken the date of retirement T to be the same for all assets in a given cohort (all assets put in place in a given year).
However, there is no reason for this to be true, and the theory is readily extended to allow for diﬀerent retirement dates.

A given cohort can be broken into components, or subcohorts, according to date of retirement and a sepa- rate T assigned to each.

Each subcohort can then be characterized by its own eﬃciency sequence, which depends among other things on the sub- cohort’s useful life Ti (Hulten (1990, p. 125)).

substitutions, when P0 equals zero.


The following equation is obtained: L is reached and, of course, P0 L P 0=u0+gu0+g2u0+ .

.

.

+gL1u0 =u0[1+g+g2+ .

.

.
+gL1] gL 1g =u0 provided that g < 1 1 1g =u0 1gL 1g : (23:34) 23.64 Now use the last equation in (23.34) in order to solve for the constant over vintages (beginning of the period) user cost for this model, u0, in terms of the period 0 price for a new unit of the durable, P0, and the discount factor g deﬁned by equation (23.32): u0= 1g 1gL 23.65 The end of period 0 user cost, p0, is, as usual, equal to the beginning of period 0 user cost, u0, times the nominal interest rate factor, 1+r0: (23:35) P 0: p0 (1+r0)u0 (23:36) 23.66 The aggregate services of all vintages of the good, including those purchased in period 0, will have the following value, S 0: S0=p0 0q0+p0 1q1+p0 2q2+ .
.

.

=p0[q0+q1+q2+ .

.

.

+q(L1)] =p0Q0 (23:37) where the period 0 aggregate (quality-adjusted) quantity of durable services consumed in period 0, Q0, is deﬁned as follows for this ‘‘one hoss shay’’ depreciation model: Q0 q0+q1+q2+ .

.

.

+q(L1) (23:38) 23.67 Thus in this model of depreciation, the vintage quantity aggregate is the simple sum of household pur- chases over the last L periods.
As was the case with the geometric depreciation model, the ‘‘one hoss shay’’ model does not require index number aggregation over vintages: there is a constant service price p0, and the associated period 0 quantity Q0 is a weighted sum of past purchases for the geometric model and a simple sum over the purchases of the last L periods for the light bulb model.38 23.68 How can the diﬀerent models of depreciation be distinguished empirically?
In principle, information on the prices of second-hand durable goods may be used in order to decide which model of depreciation best ﬁts the empirical facts.

In practice, this is diﬃcult to do, and hence diﬀerent statistical agencies may make diﬀerent assumptions about the ‘‘correct’’ pattern of depreciation (which generates the ‘‘correct’’ pattern of vintage user costs), based on whatever information they have at their disposal.
38 Thus equation (23.38) is the quantity aggregate counterpart to equation (23.27).

DURABLES AND USER COSTS Unique durable goods and the user cost approach 23.69 In the previous sections, it was assumed that a newly produced unit of the durable good remained the same from period to period.

This means that the various vintages of the durable good repeat themselves going from period to period.
Hence, a particular vintage of the good in the current period can be compared with the same vintage in the next period.

In particular, consider the period 0 user cost of a new unit of a durable good, p0 0, deﬁned by equation (23.8).
For convenience, the formula is repeated here: 0=[(1+r0)(1+i 0)(1d0)]P 0=[r0i 0+d0(1+i 0)]P 0 p0 (23:39) 23.70 Recall that P0 is the beginning of period 0 purchase price for the durable, r0 is the nominal oppor- tunity cost of capital that the household faces in period 0, i 0 is the anticipated period 0 inﬂation rate for the durable good, and d0 is the one-period depreciation rate for a new unit of the durable good.
In previous sections, it was assumed that the period 0 user cost p0 0 for a new unit of the durable could be compared with the corresponding period 1 user cost p1 0 for a new unit of the durable pur- chased in period 1.

This period 1 user cost can be deﬁned as follows: 0=[(1+r1)(1+i1)(1d0)]P1=[r1i1+d0(1+i1)]P1 p1 (23:40) 23.71 However, many durable goods are produced as one of a kind models.

For example, a new house may have many features that are speciﬁc to that particular house.
An exact duplicate of it is unlikely to be built in the following period.

Thus, if the user cost for the house is constructed for period 0 using formula (23.39), where the new house price P0 plays a key role, then since there will not necessarily be a comparable new house price for the same type of unit in period 1, it will not be possible to construct the period 1 user cost for a house of the same type, p1 0 deﬁned by equation (23.40), because the comparable new house price P1 will not be available.
23.72 Recall the notation that was introduced in paragraphs 23.43 to 23.54 above, where Pt v was the second-hand market price at the beginning of period t of a unit of a durable good that is v periods old.

Deﬁne dv to be the depreciation rate for a unit of the durable good that is v periods old at the beginning of the period under consideration.
Using this notation, the user cost of the house (which is now one period old) for period 1, p1 1, can be deﬁned as follows: (23:41) 1(1+i1)(1d1)P1 1 (1+r1)P1 p1 1 where P1 1 is the beginning of period 1 price for the house that is now one period old, r1 is the nominal opportunity cost of capital that the household faces in period 1, i1 is the anticipated period 1 inﬂation rate for the durable good and d1 is the one-period depreciation rate for a house that is one period old.
For a unique durable good, there is no beginning of period 1 price for a new unit of 429 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE the durable, P1, but it is natural to impute this price as the potentially observable market price for the used durable, P1 1, divided by one minus the period 0 depre- ciation rate, d0; i.e., deﬁne an imputed period 1 price for a new unit of the unique durable as follows: P1 P1 1 (1d0) (23:42) 23.73 If equation (23.42) is solved for P1 1 and the solution is substituted into the user cost deﬁned by then the following expression is equation (23.41), obtained for p1 1, the period 1 user cost of a one-period-old unique consumer durable: 1 (1d0)[(1+r1)(1+i1)(1d1)]P1 p1 (23:43) 23.74 If it is further assumed that the unique con- sumer durable follows the geometric model of depre- ciation, then d=d0=d1 (23:44) Substituting equation (23.44) into equations (23.43) and (23.40) leads to the following relationship between the imputed rental cost in period 1 of a new unit of the con- sumer durable, p1 0, and the period 1 user cost of the one- period-old consumer durable, p1 1: p1 1 (1d) (23:45) p0 1= 23.75 Thus, in order to obtain an imputed rental price for the unique consumer durable for period 1, p1 0, that is comparable to the period 0 rental price for a new unit of the consumer durable, p0 0, it is necessary to make a quality adjustment to the period 1 rental price for the one-period-old durable, p1 1, by dividing this latter price
 by one minus the one-period geometric depreciation rate, d. This observation has implications for the quality adjustment of observed market rents of houses.
With- out this type of quality adjustment, observed dwelling unit rents will have a downward bias, since the observed rents do not adjust for the gradual lowering of the quality of the unit as a result of depreciation of the unit.39 23.76 Note that in order to obtain an imputed purchase price for the unique consumer durable for period 1, P1, that is comparable to the period 0 purchase price for a new unit of the consumer durable, P0, it is necessary to make a quality adjustment to the period 1 used asset price for the one-period-old durable, P1 1, by dividing this latter price by one minus the period 0 depreciation rate, d0; recall equation (23.42) above.40 39 There is an exception to this general observation: if housing depre- ciation is of the ‘‘one hoss shay’’ type, then there is no need to quality- adjust observed rents for the same unit over time.
However, ‘‘one hoss shay’’ depreciation is empirically unlikely in the housing market since renters are generally willing to pay a rent premium for a new unit over an older unit of the same type.

For empirical evidence of this age premium, see Malpezzi, Ozanne and Thibodeau (1987), and Hoﬀmann and Kurz (2002, p. 19).
40 This type of quality adjustment to the asset prices for unique con- sumer durables will always be necessary; i.e., there is no exception to this rule, as was the case for ‘‘one hoss shay’’ depreciation in the context of quality-adjusting rental prices.

430 23.77 This section concludes with some observations on the diﬃculties for economic measurement that occur when attempting to determine depreciation rates empiri- cally for unique assets.
Consider again equation (23.42), which allows the potentially observable market price of the unique asset at the beginning of period 1, P1 1, to be expressed as being equal to (1d0)P1, where P1 is a hypo- thetical period 1 price for a new unit of the unique asset.
If it is assumed that this hypothetical period 1 new asset price is equal to the period 0 to 1 inﬂation rate factor (1+i 0) times the observable period 0 asset price P0, then the following relationship between the two observable asset prices is obtained: (23:46) 1=(1d0)(1+i 0)P 0 P1 Thus the potentially observable period 1 used asset price 1 is equal to the period 0 new asset price P0 times the P1 product of two factors: (1d0), a quality adjustment factor that takes into account the eﬀects of ageing on the unique asset; and (1+i 0), a period-to-period pure price change factor holding quality constant.
The problem with unique assets is that cross-sectional information on used asset prices at any point in time is no longer available to make it possible to sort out the separate eﬀects of these two factors.
Thus there is a fundamental identiﬁcation problem with unique assets; without extra information or assumptions, it will be impossible to distinguish the separate eﬀects of asset deterioration and asset inﬂation.41 In practice, this identiﬁcation problem is solved by making somewhat arbitrary assumptions about the form of depreciation that the asset is expected to experience.42 23.78 Housing is the primary example of a unique asset.
However, in addition to the problems outlined in this section, there are other major problems associated with this particular form of unique asset.

These pro- blems are discussed in the following sections.

The user cost of owner-occupied housing 23.79 Owner-occupied housing typically an example of a unique consumer durable, so the material on the quality adjustment of both stock and rental prices developed in the previous section applies to this com- modity.
Owner-occupied housing is, however, also an is 41 Special cases of this fundamental identiﬁcation problem have been noted in the context of various econometric housing models, by Martin J. Bailey, Richard F. Muth and Hugh O. Nourse (1963, p. 936): ‘‘For some purposes one might want to adjust the price index for deprecia- tion.

Unfortunately, a depreciation adjustment cannot be readily esti- mated along with the price index using our regression method.

.

.

.
.

.

.

In applying our method, therefore, additional information would be needed in order to adjust the price index for depreciation.’’ 42 For example, if the unique asset is a painting by a master, then the depreciation rate can be assumed to be very close to zero.

As another example, a reasonable guess at the likely length of life L of the unique asset could be made, and then the ‘‘one hoss shay’’ or straight line depreciation models could be implemented.
Alternatively, the length of life L could be converted into an equivalent geometric depreciation rate d using the conversion rule d=n/L, where n is a number between 1 and 2. example of a composite good; i.e., two distinct com- modities are bundled together and sold (or rented) at a single price.

The two distinct commodities are: – the structure; – the land on which the structure is situated.
SQ0 S, where Q0 23.80 To model this situation, consider a particular newly constructed dwelling unit that is purchased at the beginning of period 0.

Suppose that the purchase price is V0.
This value can be regarded as the sum of a cost of producing the structure, say P0 S is the number of square metres of ﬂoor space in the structure and P0 S is the beginning of period 0 price of construction per square metre, and the cost of the land, say P0 LQ0 L, where Q0 L is the number of square metres of the ground on which the structure is situated and the associated land, and P0 L is the beginning of period 0 price of the land per square metre.43 Thus at the beginning of period 0, the value of the dwelling unit, V0, is deﬁned as follows: SQ0 V 0=P0 S+P0 (23:47) LQ0 L 23.81 Suppose that the anticipated price of a unit of a new structure at the beginning of period 1 is P1a S and that the anticipated price of a unit of land at the beginning of period 1 is P1a L .
Deﬁne the period 0 antici- pated inﬂation rates for new structures and land, i 0 S and i 0 L, respectively, as follows: 1+i 0 S 1+i 0 L P1a S P0 S P1a L P0 L (23:48) (23:49) 23.82 Let d0 be the period 0 depreciation rate for the structure.

Then the anticipated beginning of period 1 value for the structure and the associated land is equal to L Q0 L (23:50) S+P1a V 1a=P1a S (1d0)Q0 Note the presence of the depreciation term (1d0) on the right-hand side of equation (23.50).
Should this term be associated with the expected beginning of period 1 price for a new unit of structure P1a S or with the structure quantity term Q0 S?
On the principle that like should be compared with like for prices, it seems preferable to associate (1d0) with the quantity term Q0 S. This is consistent with the treatment of unique assets that was suggested in the previous section; i.e., the initial quantity of structure Q0 S should be quality adjusted downwards to the amount (1d0) Q0 S at the beginning of period 1.
23.83 Now calculate the cost (including the imputed opportunity cost of capital r0) of buying the dwelling unit at the beginning of period 0 and (hypothetically) selling it at the end of period 0.

The following end of period 0 user cost or imputed rental cost R0 for the dwelling unit is obtained using equations (23.47)– (23.50): 43 If the dwelling unit is part of a multiple unit structure, then the land associated with it will be the appropriate share of the total land space.
DURABLES AND USER COSTS =[P0 =[P0 R0 V 0(1+r0)V 1a L](1+r0)[P1a LQ0 S+P0 SQ0 LQ0 SQ0 L](1+r0) S+P0 S )(1d0)Q0 S(1+i 0 [P0 LQ0 S+p0 SQ0 L S+P0 =p0 S (1d0)Q0 S+P1a L Q0 L] L(1+i 0 L)Q0 L] (23:51) where separate period 0 user costs of structure and land, S and p0 p0 L, are deﬁned as follows: S [(1+r0)(1+i 0 p0 S +d0(1+i 0 =[r0i 0 S )(1d0)]P0 S S )]P0 S (23:52) (23:53) L)]P0 L L [(1+r0)(1+i 0 p0 L]P0 =[r0i 0 L 23.84 Note that the above algebra indicates some of the major determinants of market rents for rented properties.
The user cost formulae deﬁned by equations (23.52) and (23.53) can be further simpliﬁed if the same approximations that were made in paragraphs 23.22 to 23.33 are made here (recall equation (23.9); i.e., assume that the terms r0i 0 L can be approximated by a real interest rate r0* and neglect the small term d0 times i 0 S in equation (23.52).
Then the user costs deﬁned by equations (23.52) and (23.53) reduce to: S and r0i 0 S [r0*+d0]P0 p0 S L r0*P0 p0 L (23:54) (23:55) 23.85 Thus the imputed rent for an owner-occupied dwelling unit is made up of three main costs: the real opportunity cost of the ﬁnancial capital tied up in the structure; the real opportunity cost of the ﬁnancial capital tied up in the land; the depreciation cost of the structure.
23.86 The above simpliﬁed approach to the user cost of housing can be even further simpliﬁed by assuming that the ratio of the quantity of land to structure is ﬁxed, and so the aggregate user cost of housing is equal to [r0*+d]P0 H, where PH is a quality-adjusted housing price index that is based on all properties sold in the country to households during the period under consideration and d is a geometric depreciation rate that applies to the composite of household structure and land.
This extremely simpliﬁed approach is used in Iceland; see Gudnason (2003, pp.

28–29).44 A variant of this 44 The real interest rate that is used is approximately 4 per cent per year and the combined depreciation rate for land and structures is assumed to equal 1.25 per cent per year.

The depreciation rate for structures alone is estimated to be 1.5 per cent per year.

Property taxes are accounted for separately in the Icelandic CPI.
Housing price information is provided by the State Evaluation Board, based on property sales data of both new and old housing.

The State Evaluation Board also estimates the value of the housing stock and land in Iceland, using a hedonic regression model based on property sales data.

The value of each household’s dwelling is collected in the household budget survey.
431 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE approach is used by the United States Bureau of Eco- nomic Analysis: Lebow and Rudd (2003, p. 168) note that the United States national accounts imputation for the services of owner-occupied housing is obtained by applying rent-to-value ratios for tenant-occupied hous- ing to the stock of owner-occupied housing.

The rent- to-value ratio can be regarded as an estimate of the applicable real interest rate plus the depreciation rate.
23.87 Returning to the period 0 imputed rental cost model for a new structure deﬁned by equations (23.47)– (23.53), now calculate the cost (including the imputed opportunity cost of capital r1) of buying the used dwell- ing unit at the beginning of period 1 and (hypothetically) selling it at the end of period 1.
Thus at the beginning of period 1, the value of the depreciated dwelling unit is V1 deﬁned as follows: V 1=P1 S(1d0)Q0 S+P1 LQ0 L (23:56) where P1 S is the beginning of period 1 construction price for building a new dwelling unit of the same type and P1 L is the beginning of period 1 price of land for the dwelling unit.
Note that equation (23.56) is an end of period 0 ex post or actual value of the dwelling unit, whereas the similar expression (23.50) deﬁned a beginning of period 0 ex ante or anticipated value of the dwelling unit.

23.88 Suppose that the anticipated price of a unit of a new structure at the beginning of period 2 is P2a S and that the anticipated price of a unit of land at the beginning of period 2 is P2a L .
Deﬁne the period 1 antici- pated inﬂation rates for new structures and land, i1 S and i1 L, respectively, as follows: 1+i1 S P2a S =P1 S 1+i1 L P2a L =P1 L (23:57) (23:58) 23.89 Let d1 be the period 1 depreciation rate for the structure.
Then the anticipated beginning of period 2 value for the structure and the associated land is equal to V 2a=P2a S (1d0)(1d1)Q0 S+P2a L Q0 L (23:59) 23.90 The following end of period 1 user cost or 1, for a one-period-old dwelling imputed rental cost, R1 unit is obtained using equations (23.56)–(23.59): R1 [P2a S+P1 1 V 1(1+r1)V 2a S(1d0)Q0 LQ0 =[P1 L](1+r1) S+P2a S (1d0)(1d1)Q0 L Q0 L] =[P1 L](1+r1) LQ0 S(1d0)Q0 S)(1d0)(1d1)Q0 [P1 S(1+i1 LQ0 S1(1d0)Q0 L S+P1 S+p1 =p1 S+P1 L(1+i1 L)Q0 L] (23:60) where the period 1 user costs of one-period-old structures and land, p1 L, are deﬁned as follows: S1 and p1 432 S1 [(1+r1)(1+i1 p1 S+d1(1+i1 =[r1i1 S)(1d1)]P1 S S)]P1 S L [(1+r1)(1+i1 p1 L)]P1 L=[r1i1 L]P1 L (23:61) (23:62) 23.91 Comparing the period 0 user cost of land, p0 L, deﬁned by equation (23.53) with the period 1 user cost of land, p1 L, deﬁned by equation (23.62), it can be seen that these user costs have exactly the same form and hence are comparable.
However, comparing the period 0 user cost for a new structure, p0 S, deﬁned by equation (23.52) with the period 1 user cost for a one- period-old structure, p1 S, deﬁned by equation (23.61), it can be seen that these user costs are not quite com- parable unless the period 0 depreciation rate d0 is equal to the period 1 depreciation rate d1.

If declining balance depreciation for structures is assumed, then d0=d1=d, where d is the common depreciation rate across all periods.
Under this assumption, p1 S is com- parable to the period 0 user cost for a new unit of structures p0 S. Even under the assumption of geometric depreciation, it can be seen that the period 1 imputed rent for a one-period-old dwelling unit, R1 1, deﬁned by equation (23.60) is not comparable to the corre- sponding period 0 imputed rent for a new dwelling unit, R0, deﬁned by equation (23.51).
The imputed rent R1 that would be comparable to R0 can be deﬁned as follows: R1 p1 SQ0 S+p1 LQ0 L=R1 1+p1 SdQ0 S (23:63) where the period 1 user cost of structures, p1 S, is deﬁned by the right-hand side of equation (23.61), with d1 equal to the common depreciation rate d and the period 1 user cost of land, p1 L, deﬁned by equation (23.62).
Equation (23.63) has the following implication for the quality adjustment of the price of a rented property: if R0 is the observed rent of the unit in period 0 and R1 1 is the observed rent for the same dwelling unit in period 1, then the observed rent R1 1 is too low compared to R0 and so the period 1 observed rent should be quality adjusted upwards by the period 1 rental price for structures, p1 S, times the amount of physical depreciation, dQ0 S, in the structure that occurred in the previous period.
This is the same point that was made in paragraphs 23.69 to 23.78, but in this section the complications arising from the fact that housing services are a mixture of structure and land services are taken into account.
23.92 It is evident that the main drivers for the user costs of structures and land are a price index for new dwelling construction, Pt S, and a price index for resi- dential land, Pt L. Most statistical agencies have a con- stant quality price index for new residential structures, since this index is required in the national accounts in order to deﬂate investment expenditures on residential structures.
This index could be used as an approxima- S.45 The national accounts also require an tion to Pt 45 This index may only be an approximation since it covers the con- struction of rented properties, as well as owner-occupied dwellings.
imputation for the services of owner-occupied housing and thus the constant quality price component of this imputation may be suitable for CPI purposes.46 If the national accounts division also computes quarterly real balance sheets for the economy, then a price index for residential land may be available to the prices division.
Even if this is the case, there will be problems in pro- ducing this price index for land on a timely basis and at a monthly frequency.47 A further possible source of information on land prices may be found in land title registry oﬃces and in the records of real estate ﬁrms.
The associated information on transactions involving the same property can be used in a hedonic regression framework; see for example, Malpezzi, Ozanne and Thibodeau (1987).48 23.93 There are many other diﬃculties associated with measuring the price and quantity of owner-occupied housing services.

The following section discusses some of the problems involved in modelling the costs of certain expenditures that are tied to the ownership of a home.
The treatment of costs that are tied to owner-occupied housing 23.94 There are many costs that are quite directly tied to home-ownership.

It is not always clear, however, how these costs can be decomposed into price and quantity components.

Several of these cost components are listed below, and some ways of forming their asso- ciated prices are suggested.
The treatment of mortgage interest costs 23.95 The derivation of the user cost or expected rental price that an owner of a home should charge for the use of the dwelling unit for one period implicitly assumes that the owner has no mortgage interest costs, so the interest rate r0 refers to the owner’s opportunity cost of equity capital.

In this section, the case where the owner has a mortgage on the property is considered.
46 However, the national accounts imputation for the services of owner-occupied housing will only be produced on a quarterly basis, and so some additional work will be required to produce a price deﬂator on a monthly basis.

Also, even though the SNA 1993 recommends that the imputation for the services of owner-occupied housing be based on the rental equivalent method, it may be the case that the imputation covers only the imputed depreciation on the structure part of owner-occupied housing.
As was pointed out above, there are two other important additional components that should also be included in owner-occupied housing services; namely, the imputed real interest on the structures and the land on which the structure is situated.

These latter two components of imputed expenditures are likely to be considerably larger than the depreciation component.
47 Another source of information on the value of residential land may be available from the local property tax authorities, particularly if properties are assessed at market values.

48 Many hedonic regression studies use the logarithm of a transaction price as the dependent variable.
This speciﬁcation of the hedonic model is usually not consistent with the additive nature of the structure and land components of a property, and the multiplicative nature of the depreciation adjustment, as appearing in equations (23.47) and (23.56) which deﬁned the value of a speciﬁc property in successive periods.

DURABLES AND USER COSTS 23.96 Recall the notation in the previous section where the user cost or imputed rental cost, R0, for an equity-ﬁnanced dwelling unit is obtained; see equation (23.51).
Suppose now that the property purchase is partly ﬁnanced by a mortgage of M 0 dollars at the beginning of period 0.

Let f 0 be the fraction of the beginning of period 0 market value of the property that is ﬁnanced by the mortgage so that M0=f 0V 0=f 0[P0 SQ0 S+P0 LQ0 L] (23:64) 23.97 Let the one-period nominal mortgage interest rate be r0 M. The owner’s period 0 beneﬁts of owning the dwelling unit remain the same as in the previous section and are equal to V1a, deﬁned by equation (23.50).
However, the period 0 costs are now made up of an explicit mortgage interest cost equal to M0(1+r0 M) plus an imputed equity cost equal to (1f 0)V 0(1+r0).
Thus the new imputed rent for using the property during period 0 is now M)V 1a R0 (1f 0)V 0(1+r0)+M0(1+r0 L](1+r0) SQ0 S+P0 L Q0 L =(1f 0)[P0 SQ0 +f 0[P0 S+p0* S Q0 LQ0 L](1+r0 S+P0 LQ0 =p0* M)[P1a S (1d0)Q0 S+P1a L Q0 L] (23:65) where the new mortgage interest adjusted period 0 user costs of structures and land, p0* L , are deﬁned as follows: S and p0* S [(1+r0)(1f 0)+(1+r0 p0* S )(1d0)]P0 S =[r0(1f 0)+r0 =[(r0i 0 Mf 0i 0 S )(1f 0)+(r0 S +d0(1+i 0 Mi 0 S )f 0+d0(1+i 0 S )]P0 S M)f 0(1+i 0 S )]P0 S L [(1+r0)(1f 0)+(1+r0 p0* L]P0 L L)f 0]P0 Mi 0 L =[r0(1f 0)+r0 =[(r0i 0 Mf 0i 0 L)(1f 0)+(r0 M)f 0(1+i 0 L)]P0 L (23:66) (23:67) 23.98 Comparing the new user costs for structures and land deﬁned by equations (23.66) and (23.67) with the corresponding equity-ﬁnanced user costs deﬁned by equations (23.52) and (23.53), it can be seen that the old equity opportunity cost of capital r0 is now replaced by a weighted average of this equity opportunity cost and Mf 0, where f 0 is the mortgage interest rate, r 0(1f 0)+r0 the fraction of the beginning of period 0 value of the dwelling unit that is ﬁnanced by the mortgage.
23.99 Central bankers often object to the inclusion of mortgage interest in a CPI.

Examination of the last equation in (23.66) and in (23.67) nevertheless shows that the nominal mortgage interest rate r0 M has an oﬀ- setting beneﬁt resulting from anticipated price inﬂation in the price of structures, i 0 S in equation (23.66), and in the price of land, i 0 L in equation (23.67), so, as usual, what counts in these user cost formulae are real interest costs rather than nominal ones.
433 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE The treatment of property taxes 23.100 Recall the user costs of structures and land deﬁned by equations (23.52) and (23.53).
It is now supposed that the owner of the housing unit must pay the property taxes T 0 L for the use of the structure and land, respectively, during period 0.49 Deﬁne the period 0 structures tax rate t0 L, as follows: S and land tax rate t0 S and T 0 t0 S t0 L T 0 S SQ0 P0 S T 0 L LQ0 P0 L (23:68) (23:69) 23.101 The new imputed rent for using the property during period 0, R0, including the property tax costs, is deﬁned as follows: =[P0 R0 V 0(1+r0)+T 0 LQ0 SQ0 S+P0 S(1+i 0 [P0 S )(1d0)Q0 LQ0 S+p0 SQ0 L SV 1a S+T 0 L](1+r0)+t0 S+P0 =p0 SP0 SQ0 L(1+i 0 S+t0 LP0 L)Q0 L] LQ0 L (23:70) where separate period 0 tax-adjusted user costs of struc- tures and land, p0 L, are deﬁned as follows: S and p0 S [(1+r0)(1+i 0 p0 S +d0(1+i 0 =[r0i 0 S )(1d0)+t0 S )+t0 S]P0 S S]P0 S L [(1+r0)(1+i 0 p0 L+t0 =[r0i 0 L]P0 L L)+t0 L]P0 L (23:71) (23:72) S and t0 Thus the property tax rates, t0 L, deﬁned by equations (23.68) and (23.69), enter the user costs of S and p0 structures and land, p0 L, deﬁned by equations (23.71) and (23.72), in a simple additive manner; i.e., these terms are additive to the previous depreciation and real interest rate terms.50 The treatment of property insurance 23.102 At ﬁrst glance, it would seem that property insurance could be treated in the same manner as the treatment of property taxes in the previous subsection.
Thus, let C0 S be the cost of insuring the structure at the beginning of period 0 and deﬁne the period 0 structures premium rate g0 S as follows: g0 S C0 S SQ0 P0 S (23:73) 49 If there is no breakdown of the property taxes into structure and land components, then just impute the overall tax into structure and land components based on the beginning of the period values of both components.
50 If the price statistician uses the national accounts imputation for the value of owner-occupied housing services, care should be taken to ensure that the value of property taxes is included in this imputation.
434 23.103 The new imputed rent for using the property during period 0, R0, including property tax and insur- ance costs, is deﬁned as follows: S +C 0 R0 V 0(1+r0)+T 0 SV 1a=p0 S +T 0 S+p0 SQ0 LQ0 L (23:74) where separate period 0 tax and insurance adjusted user costs of structures and land, p0 L, are deﬁned as follows: S and p0 S [(1+r0)(1+i 0 p0 S )(1d0)+t0 S+g0 S] S=[r0i 0 P0 S +d0(1+i 0 S )+t0 S+g0 S]P0 S (23:75) L [(1+r0)(1+i 0 p0 L)+t0 L]P0 L=[r0i 0 L+t0 L]P0 L (23:76) SP1 SP0 SQ0 S(1d)Q0 23.104 Thus the insurance premium rate g0 S appears in the user cost of structures, p0 S, deﬁned by equation (23.75), in an additive manner, analogous to the additive property tax rate term.51 If it is desired to have a sepa- rate CPI price component for insurance, then the cor- SP0 responding period 0 and 1 prices can be deﬁned as g0 S and g1 S, respectively, while the corresponding period 0 and 1 expenditures can be deﬁned as g0 S and S, respectively.52 Of course, if this separate g1 SP1 treatment is implemented, then these terms have to be dropped from the corresponding user costs of structures.
23.105 The above treatment of property taxation and insurance assumes that the property taxes and the premium payments are made at the end of the period under consideration; see equation (23.74) above.

While this may be an acceptable approximation for the pay- ment of property taxes, it is not acceptable for the payment of insurance premiums: the premium must be paid at the beginning of the period of protection rather than at the end.
When this complication is taken into account, the user cost of structures becomes S [(1+r0)(1+i 0 p0 S +d0(1+i 0 =[r0i 0 S )(1d0)+t0 S+g0 S )+t0 23.106 There are some additional problems asso- S+g0 S(1+r0)]P0 S S(1+r0)]P0 S (23:77) ciated with the modelling of property insurance: The above user cost derivations assume that the risk of property damage remains constant from period to period.
If the risk of damage changes, then an argu- ment can be made for quality adjustment of the pre- mium to hold constant the risk so that like can be compared with like.

 The gross premium approach to insurance is taken in the above treatment; i.e., it is assumed that dwel- ling owners pay premiums for property protection 51 This treatment of property insurance dates back to Walras (1954, pp.

268–269).
268–269).

52 Similarly, if it is desired to have a separate CPI price component for property taxes on structures, then the corresponding period 0 and 1 prices can be deﬁned as t0 S, respectively, while the corre- sponding period 0 and 1 expenditures can be deﬁned as t0 S and SP1 t1 S, respectively.

S(1d)Q0 S and t1 SQ0 SP1 SP0 SP0 services, no matter whether they have a claim or not.

In the net premium approach, payments to settle claims are subtracted from the gross premium payments.
The property protection may not be complete; i.e., the insurance policy may have various limitations on the type of claim that is allowed and there may be a deductible or damage threshold, below which no claim is allowed.

If the deductible changes from period to period, then the price statistician is faced with a rather complex quality adjustment problem.

Thus it can be seen that there are many diﬃcult prob- lems that remain to be resolved in this area.
The treatment of maintenance and renovation expenditures 23.107 Another problem associated with home- ownership is the treatment of maintenance expenditures, major repair expenditures and expenditures associated with renovations or additions.

23.108 Empirical evidence suggests that the normal decline in a structure resulting from the eﬀects of ageing and use can be oﬀset by maintenance and renovation expenditures.
How exactly should these expenditures be treated in the context of modelling the costs and beneﬁts of home-ownership?

23.109 A common approach in the national accounts literature is to treat major renovation and repair expen- ditures as capital formation, and smaller routine main- tenance and repair expenditures as current expenditures.
If this approach is followed in the CPI context, then these smaller routine maintenance expenditures can be treated in the same manner as other non-durable goods and services.

The major renovation and repair expen- ditures do not enter the CPI in the period that they are made, but are capitalized and added to expenditures on new structures for the period under consideration, so in structures in constant that period 0 investment S say,53 would include both types of expendi- dollars, I 0 tures.
Let Q0 S be the stocks (in constant quality units) of owner-occupied structures in the reference population at the beginning of period 0 and 1, respec- tively.
Then if the geometric model of depreciation is used, so that the constant period-to-period depreciation rate d is applicable, then the beginning of period 1 stock of owner-occupied structures Q1 S is related to the beginning of period 0 stock of structures Q0 S and the period 0 investment in structures I 0 S, according to the following equation: S and Q1 Q1 S=(1d)Q0 S+I 0 S (23:78) 23.110 Thus, if declining balance depreciation is assumed for structures, then the treatment of major repair and renovation expenditures does not pose major conceptual problems using a conventional capital accu- mulation model: it is only necessary to have an estimate 53 Let VI 0 S be the nominal value of investment in new owner-occupied structures in period 0, plus the value of major renovation expenditures made during period 0.
Then the constant dollar quantity of investment S=P0 could be deﬁned as I 0 S is the period 0 construction price index for new structures.
S, where P0 S VI 0 DURABLES AND USER COSTS for the monthly or quarterly depreciation rate d, a starting value for the stock of owner-occupied structures for some period, information on new purchases of resi- dential housing structures by the household sector, information on expenditures by owners on major repairs and renovations, and a construction price index for new residential structures.
With this information on a timely basis, up-to-date CPI weights for the stock of owner- occupied structures could be constructed.54 23.111 This section concludes by looking at how major repair and renovation expenditures could be treated in a repeat sales regression model that uses trans- actions data on the sale of the same housing unit in two or more periods.
In order to minimize notational com- plexities, consider a highly simpliﬁed situation where data on the sale of N houses of a relatively homogeneous type for two consecutive periods are available.

Suppose that these sale prices are V 0 n for period 1, for n=1, 2, .

.

.
, N. Suppose that a price index for structures of this type of property in period 0, P0 S, and a corresponding price index for land in period 0, L, have been constructed.55 The price statistician’s P0 problem is to use the data on the matched sales for the two periods in order to construct estimates of these two indices for period 1; i.e., the problem is to construct P1 S and P1 L. n for period 0 and V 1 23.112 The period 0 dwelling unit values for the N properties can be decomposed into the structure and land components as follows: V 0 n =V 0 Sn+V 0 Ln=anP0 SQ0 Sn+bnP0 LQ0 Ln; n=1, 2, .
, N (23:79) Sn and V 0 where V 0 Ln are the estimated period 0 values of the structure and land of property n in period 0, P0 S and P0 L are the (known) price index values for structures and land for all properties of this type in period 0, and Q0 Sn and Q0 Ln are (known) estimates of the quantity of struc- tures and land for property n. The numbers an and bn are property n quality adjustment factors that convert the property standardized values of structures and land, Sn and P0 SQ0 P0 Sn, respectively, into the period 0 actual market values, V 0 Ln, respectively; i.e., if esti- mates of the period 0 market values of the structures and land for property n are available, then an and bn can be deﬁned as follows: Sn and V 0 LQ0 an V 0 Sn SQ0 Sn P0 ; bn V 0 Ln LQ0 Ln P0 ; n=1, .
.

.

, N (23:80) 23.113 Suppose that information on the dollar amount of major repairs and renovations made to property n during period 0, VR0 n, are also available for each property n in the sample of properties.

Then the period 1 value for property n, V 1 n , should be approxi- mately equal to 54 The practical problems involved in obtaining all this information on a timely basis are, however, not trivial.

55 If these period 0 indices are not available, then set P 0 to 1.
S and P 0 L equal 435 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE V 1 n =anP1 S(1d)Q0 Sn+VR0 n+bnP1 LQ0 Ln; n=1, .

.

.

, N (23:81) S, P1 where d is the geometric depreciation rate for structures.
All the variables on the right-hand side of equation (23.81) are assumed to be known, with the exception of the period 1 price index values for structures and land, P1 S and P1 L, respectively, and the one-period geometric depreciation rate, d. If the number of observations N is greater than three, then it would appear that these three parameters, P1 L and d, could be estimated by a linear regression using the N equations in (23.81) as estimating equations.
It turns out, however, that this is not quite correct.

The problem is that the parameters P1 S and (1d) appear in equation (23.81) in a multiplicative fashion so that while the product of these two terms will be identiﬁed by the regression, the individual terms cannot be uniquely identiﬁed.
This is just a reappearance of the same problem that was discussed in paragraphs 23.69 to 23.78 in relation to unique consumer durables: the separate eﬀects of ageing of the asset (depreciation or capital consumption) and price appreciation over time cannot be separately identiﬁed using just market data.56 23.114 There are three possible solutions to this identiﬁcation problem: use an external estimate of the depreciation rate d; use an external construction price index P1 S instead of estimating it as a parameter in equations (23.81); abandon the repeat sales approach and use a hedonic regression approach instead.
23.115 What would a hedonic regression model look like, taking into account the approximate additivity of the value of the housing structure and the value of the land on which the structure is situated?
If the renova- tions problem is ignored and geometric depreciation of the structure is assumed, then the value of a housing unit n in period t that is v periods old, V t n, should be approximately equal to the depreciated value of the structure plus the value of the land plus an error term; i.e., the following relationship should hold approxi- mately: V t n=Pt S(1d)vQSn+Pt LQL+ut n (23:82) where d is the one-period geometric depreciation rate, QSn is the number of square metres of ﬂoor space of the original structure for housing unit n, QL is the number of square metres of land on which the housing structure 56 Recall equation (23.46).
This fundamental identiﬁcation problem was recognized by Bailey, Muth and Nourse (1963, p. 936) in the original repeat sales housing article, but it was ignored by them and subsequent users of the repeat sales methodology.

Another problem with the housing hedonic regression literature is that, usually, the logarithm of the purchase price is taken as the dependent variable in the regression.
While this speciﬁcation has some advantages, it does not recognize properly the additive nature of the structure and land components of the housing property.

A ﬁnal problem with the tradi- tional hedonic housing literature is that, usually, separate price indices for land and structures are not estimated.
It is important to allow for separate price indices for these two components since, usually, the price of land is more volatile and tends to increase faster than the price of structures over long periods of time.

436 S, Pt is situated, and ut n is an error term.

Pt S is the beginning of period t price level for structures of this type and Pt L is the corresponding price of land for this class of housing units.
As long as there is more than one vintage of structure in the sample (i.e., more than one v), then the parameters Pt L and d can be identiﬁed by running a non-linear regression model using equation (23.82).

Why can the price levels be identiﬁed in the present hedonic regression model, whereas they could not be identiﬁed in the repeat sales model?
The answer is that the hedonic model (23.82) does not assume property- speciﬁc quality adjustment factors for each housing unit; instead, all the housing units in the sample are assumed to be of comparable quality once prices are adjusted for the age of the unit and the quantity (in square metres) of original structure and the quantity of land.
23.116 Unfortunately, many housing structures that may have started oﬀ as identical structures do not remain the same over time, because of diﬀering stan- dards of maintenance, as well as major renovations and additions to some of the structures.

To model this phenomenon, let Rt n be real maintenance, repair and renovation expenditures on housing unit n during period t, and suppose that these real expenditures depreciate at the geometric rate dR.
It is reasonable to assume that these expenditures add to the value of the housing unit, and so equation (23.82) should be replaced by the fol- lowing equation: V t n=Pt S(1d)vQSn+Pt +(1dR)2Rt2 n+(1dR)Rt1 n + +(1dR)vRtv R[Rt n n ]+Pt LQL+ut n (23:83) n , , Rt2 n n, Rt1 n , .

.

.

, Rtv where Pt R is the period t price level for real maintenance, repair and renovation expenditures on this class of housing units.
If information on these real renovation and repair expenditures, Rt is available for each housing unit in the sample of housing units that sold in period t, then the parameters Pt S, Pt L, Pt R, d and dR can be identiﬁed by running a non-linear regression model using equation (23.83).57 A major practical problem with implementing a hedonic regres- sion model along the above lines is that, usually, accu- rate data on renovation and repair expenditures on a particular dwelling unit between the construction of the initial housing unit and the present period are not available.
Without accurate data on repairs and reno- vations, it will be impossible to obtain accurate esti- mates of the unknown parameters in the hedonic regression model.

23.117 A ﬁnal practical problem with the above hedonic regression model will be mentioned.

Theoreti- ‘‘normal’’ maintenance expenditures could be cally, included in the renovation expenditure terms, Rt n, in equation (23.83).
If this is done, then including normal maintenance expenditures in Rt n will have the eﬀect of increasing the estimated depreciation rates d and dR.

Thus diﬀerent statistical agencies, with diﬀerent criteria 57 Alternatively, if price levels are available for P t R from con- struction price indices, then these parameters do not have to be esti- mated.
S and P t for deciding where to draw the line between ‘‘normal’’ maintenance and ‘‘major’’ repair and renovations, will produce diﬀerent estimated depreciation rates.

The treatment of the transactions costs of home purchase 23.118 Another cost of home-ownership needs to be discussed.
Normally, when a family purchases a dwell- ing unit, they have to pay certain fees and costs, which can include: the commissions of real estate agents who help the family ﬁnd the ‘‘right’’ property; various transactions taxes that governments can impose on the sale of the property; various legal fees that might be associated with the transfer of title for the property.
23.119 Should the above fees be considered as expenditure in the period of purchase, or should they simply be regarded as part of the purchase price of the property and hence be depreciated over time in a man- ner analogous to the treatment of structures in the national accounts?

23.120 An argument can be made for either treat- ment.
From the viewpoint of the opportunity cost treatment of purchases of durable goods, the relevant price of the dwelling unit in the periods following the purchase of the property is the after tax and transactions fees value of the property.

This viewpoint suggests that the transactions costs of the purchaser should be counted as expenses in the period of purchase.
From the view- point of a landlord who has just purchased a dwelling unit for rental purposes, however, it would not be sen- sible to charge the tenant the full cost of these transac- tions fees in the ﬁrst month of rent.

The landlord would tend to capitalize these costs and recover them gradually over the time period that the landlord expects to own the property.
Thus, either treatment could be justiﬁed and the statistical agency will have to decide which treatment is most convenient from its particular perspective.

User costs for landlords versus owners 23.121 In the previous section, the various costs associated with home-ownership were discussed.

Both home-owners and landlords face these costs.

Thus, they will be reﬂected in market rents, and this must be kept in mind if the imputed rent approach is used to value the services of owner-occupied housing.
If some or all of these associated costs of owner-occupied housing are covered elsewhere in the CPI (e.g., home insurance could be separately covered), then the value of imputed rents for owner-occupied housing must be reduced by the amount of these expenditures covered elsewhere.

23.122 In addition to the costs of home-ownership covered in the previous section, landlords face a number of additional costs compared to the home-owner.

These additional costs will be reﬂected in market rents.
Thus, if market rents are used to impute the services provided by the ownership of a dwelling unit, then these extra costs DURABLES AND USER COSTS should also be removed from the market rents that are used for imputation purposes, since they will not be relevant for owner-occupiers.

These additional landlord- speciﬁc costs are discussed in paragraphs 23.123 to 23.133.
Damage costs 23.123 Tenants do not have the same incentive to take care of a rented property compared to an owned property, so depreciation costs for a rented property are likely to exceed depreciation rates for comparable owned properties.

Usually, landlords demand damage deposits, but often these deposits are not suﬃcient to cover the costs of the actual damages that some tenants inﬂict.
Non-payment of rent and vacancy costs 23.124 At times, tenants run into ﬁnancial diﬃcul- ties and are unable to pay landlords the rent that is owned.

Usually, eviction is a long-drawn-out process, and so landlords can lose several months of rent before a non-paying tenant ﬁnally leaves.
The landlord also incurs extra costs compared to a home-owner when a rented property remains vacant because of lack of demand.58 These extra costs will be reﬂected in market rents but should not be reﬂected in the user costs of owner-occupied housing.

Billing and maintenance costs 23.125 A landlord may have to rent oﬃce space and employ staﬀ to send out monthly bills to tenants, and employ staﬀ to respond to requests for maintenance.
A home-owner who gives his or her time in order to pro- vide maintenance services59 oﬀers this time at his or her after income tax wage rate, which may be lower than the before income tax wage rate that a landlord must pay his or her employees.

The net eﬀect of these factors leads to higher market rents compared to the corresponding owner-occupied user cost.
The opportunity cost of capital 23.126 The home-owner’s after tax opportunity cost of capital that appeared in the various user cost for- mulae considered earlier in this chapter will typically be lower than the landlord’s before tax opportunity cost of capital.

Put another way, the landlord has an extra income tax cost compared to the home-owner.

In addi- tion, the landlord may face a higher risk premium for the use of capital because of the risks of damage and non-payment of rent.
Care must be taken that these additional landlord costs are not counted twice.

58 The demand for rented properties can vary substantially over the business cycle, and this can lead to depressed rents or very high rents compared to the user costs of home-ownership.

Thus imputed rents based on market rents of similar properties can diﬀer substantially from the corresponding user costs of owner-occupied housing over the business cycle.
59 Typically, these imputed maintenance costs will not appear in the CPI, but if the user cost of an owned dwelling unit is to be comparable with the market rent of a similar property, these imputed labour costs should be included.
437 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE The supply of additional services for rented properties 23.127 Often, properties that are to let will contain some major consumer durables that home-owners have to provide themselves, such as refrigerators, stoves, washing-machines, driers and air conditioning units.

In addition, landlords may pay for electricity or fuel in some rented apartments.
Thus, to make the market rental comparable to an owner-occupied imputed rent, the market rental should be adjusted downwards to account for the above factors (which will appear else- where in the expenditures of owner-occupiers).
23.128 The factors listed above will tend to make observed market rental prices higher than the corre- sponding user cost for an owner-occupier of a property of the imputed rental approach is used to value the services of owner-occupied housing, then these market-based rents should be adjusted downward to account for the above factors.

the same quality.
the same quality.

Thus, if 23.129 Although all the above factors will tend to lead to an upward bias if unadjusted market rental rates are used to impute the services of owner-occupied housing, there is another factor not discussed thus far that could lead to a large downward bias.

That factor is rent controls.


23.130 Under normal conditions, the acquisitions approach to the treatment of owner-occupied housing will give rise to the smallest expenditures, the user cost approach will give rise to the next highest level of expenditures, and the use of imputed market rentals will give the largest level of expenditures for owner-occupied housing.

For the ﬁrst two approaches, a main driver of the price of owner-occupied housing is the price of new housing construction.
For the user cost approach, another main driver is the price of land.

For the imputed rent approach, the main driver of the price of owner- occupied housing is the rental price index.

23.131 The above discussion is far from being complete and deﬁnitive, but it does illustrate that it is not completely straightforward to impute market rental rates to owner-occupied dwelling units.

Care must be taken to ensure that the ‘‘correct’’ expenditure weights are constructed.
23.132 As can be seen from the material above, the treatment of owner-occupied housing presents special diﬃculties.

Astin (1999, p. 5) discussed some of the diﬃculties that the European Union encountered in trying to ﬁnd the ‘‘best’’ approach to use in its Har- monized Index of Consumer Prices (HICP), as follows: A special coverage problem concerns owner-occupied housing.

This has always been one of the most diﬃcult sectors to deal with in CPIs.
Strictly, the price of housing should not be included in a CPI because it is classiﬁed as capital.

On the other hand, the national accounts classiﬁes imputed rents of owner-occupiers as part of consumers’ expenditure.

This is a reasonable thing to do if the aim is to measure the volume of consumption of the capital resource of hous- ing.

But that is not what a CPI is measuring.

Some countries, following the compensation index concept, would prefer to have mortgage interest included in the HICP.
This approach could indeed be defended for 438 a compensation index, because there is no doubt that the monthly mortgage payment is an important element in the budget of many households: a rise in the interest rate acts in exactly the same way as a price increase from the point of view of the individual household.

But this is not acceptable for a wider inﬂation index.

So, after many hours of debate, the Working Party came to the conclusion that there were just two options.
The ﬁrst was to simply exclude owner-occupied housing from the HICP.

One could at least argue that this was a form of harmonization, although it is worrying that there are such large diﬀerences between Member States in the percentages of the population which own or rent their dwellings.

Exclusion also falls in line with the interna- tional guideline issued 10 years ago by the ILO.
Fur- thermore, it would be possible to supplement the HICP with a separate house price index, which could be used by analysts as part of a battery of inﬂation indicators.

The second option was to include owner-occupied housing on the basis of acquisition costs, essentially treating them like any other durable.
Most secondhand housing would be excluded: in practice the index would include new houses plus a small volume of housing new to the household sector (sales from the company or gov- ernment sectors to the household sector).

The main problem here is practical: several countries do not have new house price indices and their construc- tion could be diﬃcult and costly.

A Task Force is at present examining these matters.

Final recommendations are due at the end of 1999.


Because of the complexities involved in modelling the treatment of owner-occupied housing, ﬁnal recommen- dations have still not emerged for the HICP.

23.133 A fourth approach to the treatment of housing will be studied in the following section.
Since this approach has only been applied to owner-occupied dwellings, it is not as ‘‘universal’’ as the other three approaches.60 The payments approach 23.134 A fourth possible approach to the treatment of owner-occupied housing, the payments approach, is described by Charles Goodhart (2001, pp.

F350–F351) as follows: The second main approach is the payments approach, measuring actual cash outﬂows, on down payments, mortgage repayments and mortgage interest, or some subset of the above.
This approach always, however, includes mortgage interest payments.

This, though com- mon, is analytically unsound.

First, the procedure is not carried out consistently across purchases.

Other goods bought on the basis of credit, e.g., credit card credit, are usually not treated as more expensive on that account (though they have been in New Zealand).

Second, the treatment of interest ﬂows is not consistent across per- sons.
If a borrower is worse oﬀ in some sense when interest rates rise, then equivalently a lender owning an interest bearing asset is better oﬀ; why measure one and not the other?

If I sell an interest earning asset, say a 60 The acquisitions, user cost and rental equivalence approaches can be applied to any consumer durable, but to apply the rental equivalence approach, appropriate rental or leasing markets for the durable must exist.
money market mutual fund holding, to buy a house, why am I treated diﬀerently to someone who borrows on a (variable rate) mortgage?

Third, should not the question of the price of any purchase be assessed separately from the issue of how that might be ﬁnanced?

Imports, inventories and all business purchases tend to be pur- chased in part on credit.

Should we regard imports as more expensive, when the cost of trade credit rises?

Money, moreover, is fungible.


As we know from calcu- lations of mortgage equity withdrawal, the loan may be secured on the house but used to pay for furniture.

When interest rates rise, is the furniture thereby more expensive?

Moreover, the actual cash out-payments totally ignore changes in the ongoing value of the house whether by depreciation, or capital loss/gain, which will often dwarf the cash ﬂow.
Despite its problems, such a cash payment approach was used in the United Kingdom until 1994 and still is in Ireland.

23.135 Thus, the payments approach to owner- occupied housing is a kind of cash ﬂow approach to the costs of operating an owner-occupied dwelling.

Possible objections to this approach are that it ignores the oppor- tunity costs of holding the equity in the owner-occupied dwelling, it ignores depreciation, and it uses nominal interest rates without any oﬀset for inﬂation.
If the payments approach is adjusted for these imputed costs, however, then the result is a rather complicated user cost approach to the treatment of housing.

Nevertheless, as was mentioned in Chapter 10, under some condi- tions, the payments approach to the treatment of owner- occupied housing may be a reasonable compromise.
In general, the payments approach will tend to lead to much smaller monthly expenditures on owner-occupied housing than the other three main approaches, except during periods of high inﬂation, when the nominal mortgage rate term becomes very large without any oﬀsetting item for inﬂation.61 Alternative approaches for pricing owner-occupied housing 23.136 For consumer durables that have long useful lives, the usual acquisitions approach will not be ade- quate for CPI users who desire prices that measure the service ﬂows that consumer durables generate.
This is particularly true for owner-occupied housing.

Hence it will be useful to many users if, in addition to the acquisitions approach, the statistical agency implements a variant of either the rental equivalence approach or the user cost approach for long-lived consumer durables and for owner-occupied housing in particular.

Users can then decide which approach best suits their purposes.

Any one of the three main approaches could be chosen as the approach that would be used in the ‘‘headline’’ CPI.
The other two approaches could be made available to users as analytic tables.

61 If there is high inﬂation, then the statistical agency using the pay- ments approach may want to consider adjusting nominal mortgage interest rates for the inﬂation component, as is done in paragraphs 23.95 to 23.99.

For additional material on the payments approach, see Chapter 10.
DURABLES AND USER COSTS 23.137 We conclude this chapter by outlining some of the problems involved in implementing the three main approaches to the measurement of price change for owner-occupied housing.

23.138 The acquisitions approach to implement In order the acquisitions approach, a constant quality price index for the sales of new residential housing units will be required.
The rental equivalence approach Option 1: Using home-owners’ estimates of rents 23.139 In this option, home-owners would be sur- veyed and asked to estimate a rental price for their housing unit.

Problems with this approach are: Home-owners may not be able to provide very accu- rate estimates for the rental value of their dwelling unit.
The statistical agency should make an adjustment to these estimated rents over time in order to take into account the eﬀects of depreciation, which causes the quality of the unit to decline slowly over time (unless this eﬀect is oﬀset by renovation and repair expendi- tures).62 Care must be taken to determine exactly what extra services are included in the home-owner’s estimated rent; i.e., does the rent include insurance, electricity and fuel or the use of various consumer durables in addition to the structure?
If so, these extra services should be stripped out of the rent, since they are covered elsewhere in the CPI.63 Option 2: Using a hedonic regression model of the rental market to impute rents 23.140 In this option, the statistical agency would collect data on rented properties and their character- istics, and then use this information to construct a hedonic regression model for the housing rental mar- ket.64 This model would then be used to impute prices for owner-occupied properties.
Problems with this approach are: It is information intensive; in addition to requiring information on the rents and characteristics of rented properties, information on the characteristics of owner-occupied properties would also be required.

 The characteristics of the owner-occupied population could be quite diﬀerent from the characteristics of the rental population.

In particular, if the rental market for housing is subject to rent controls, this approach is not recommended.
62 Recall paragraphs 23.79 to 23.93.

63 It could be argued that these extra services that might be included in the rent are mainly a weighting issue; i.e., that the trend in the home- owner’s estimated rent would be a reasonably accurate estimate of the trend in the rents after adjusting for the extra services included in the rent.

64 See Hoﬀmann and Kurz (2002) for an example of such a model.
439 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Hedonic regression models suﬀer from a lack of reproducibility in that diﬀerent researchers will have diﬀerent characteristics in the model and use diﬀerent functional forms.

 From the previous discussions, it is seen that market rents can be considerably higher than the opportunity costs of home-owners.
Hence, using market rents to impute rents for owner-occupiers may lead to rents that are too high.65 On the other hand, if there are rent controls or a temporary glut of rented properties, then market rents could be too low compared to the opportunity costs of home-owners.

 There is some evidence that depreciation is somewhat diﬀerent for rental units compared to owner-occupied housing units.66 If this is so, then the imputation procedure will be somewhat incorrect.
However, all studies that estimate depreciation for owner-occupied housing suﬀer from biases resulting from the inade- quate treatment of land and the lack of information on repair, renovation and maintenance expenditures over the life of the dwelling unit.

It is not certain that depreciation for rental units is signiﬁcantly diﬀerent from that for owner-occupied units.

The user cost approach 23.141 It is ﬁrst necessary to decide whether an ex ante or ex post user cost of housing is to be calculated.
It seems that the ex ante approach is the more useful for CPI purposes; these are the prices that should appear in economic models of consumer choice.

Moreover, the ex post approach will lead to user costs that ﬂuctuate too much to suit the needs of most users.

Of course, the problem with the ex ante approach is that it will be diﬃcult to estimate anticipated inﬂation rates for house prices.
Option 3: The rent-to-value approach 23.142 In this option, the statistical agency collects information on market rents paid for a sample of rented properties, but it also collects information on the sales price of these rented properties when they are sold.

Using these two pieces of information, the statistical agency can form an estimated rent-to-value ratio for rented proper- ties of various types.
It can be seen that this rent-to-value ratio represents an estimate of all the terms that go into an ex ante user cost formula, except the asset price of the property; i.e., the rent-to-value ratio for a particular property can be regarded as an estimate of the interest rate less anticipated housing inﬂation plus the deprecia- tion rate plus the other miscellaneous rates discussed in paragraphs 23.94 to 23.120, such as insurance and 65 Again, it could argued that this is a mainly a weighting issue; i.e., that the trend in market rents would be a reasonably accurate estimate for the trend in home-owners’ opportunity costs.
66 According to Stephen Malpezzi, Larry Ozanne and Thomas G. Thibodeau (1987, p. 382): ‘‘The average depreciation rate for rental property is remarkably constant, ranging from 0.58 per cent to 0.60 per cent over the 25 year period.

Depreciation rates for owner-occupied units show more variation than the estimated rates for renter occupied units.

The average depreciation rate for owner-occupied housing ran- ges from 0.9 per cent in year 1 to 0.28 per cent in year 20.’’ 440 property tax rates.
Under the assumption that these rates remain reasonably constant over the short run, changes in user costs are equal to changes in the price of owner- occupied housing.

Thus, this approach can be imple- mented if a constant quality price index for the stock value of owner-occupied housing can be developed.

It may be decided to approximate the comprehensive price index for owner-occupied housing by a new housing price index.
If this is done, the approach essentially reduces to the acquisitions approach, except that the weights will generally be larger using this user cost approach than those obtained using the acquisitions approach.67 Problems with this approach include: It will require a considerable amount of resources to construct a constant quality price index for the stock of owner-occupied housing units.
If a hedonic regres- sion model is used, there are problems associated with the reproducibility of the results.

 Rent-to-value ratios can change considerably over time.

Hence it will be necessary to keep collecting information on rents and selling prices of rented properties.

 As was noted in paragraphs 23.121 and 23.122, the user cost structure of rented properties can be quite diﬀerent from the corresponding user cost structure of owner-occupied properties.
Hence, the use of rent-to- value ratios can give misleading results.68 Option 4: The simpliﬁed user cost approach 23.143 This approach is similar to that of Option 3 above, but instead of using the rent-to-value ratio to estimate the sum of the various rates in the user cost formula, direct estimates are made of these rates.
If the simpliﬁed Icelandic user cost approach discussed in paragraphs 23.79 to 23.93 is used, all that is required is a constant quality owner-occupied housing price index, an estimated real interest rate, and an estimated composite depreciation rate on the structure and land together.

Problems with this approach are: As was the case with Option 3 above, it will require a considerable amount of resources to construct a constant quality price index for the stock of owner- occupied housing units.
If a hedonic regression model is used, there are problems associated with the reproducibility of the results.

 It is not known with any degree of certainty what the appropriate real interest rate should be.

 Similarly, it is diﬃcult to determine what the ‘‘cor- rect’’ depreciation rate should be.69 67 Recall the discussion in paragraphs 23.34 to 23.42.
68 This is primarily a weighting issue, however, so the trend in the constant quality stock of owner-occupied housing price index should be an adequate approximation to the trend in owner-occupied user costs.
69 Because of the lack of information on repairs and renovations, estimated housing depreciation rates vary widely: One striking feature with the results of all three approaches used in these and related studies is their variability: estimates range from about a half per cent per year to two and a half per cent (Malpezzi, Ozanne and Thibodeau (1987, pp.

373–375).
373–375).

DURABLES AND USER COSTS This problem is complicated by the fact that, over time, the price of land tends to increase faster than the price of building a residential structure, so the land price com- ponent of an owner-occupied housing unit will tend to increase in importance, which in turn will tend to decrease the composite depreciation rate.
Option 5: A national accounting approach 23.144 This approach makes use of the fact that the national accounts division of the statistical agency will usually collect data on investment in residential hous- ing, as well as on repair and renovation expenditures on housing.

In addition, many statistical agencies will also construct estimates for the stock of residential dwelling units, so that estimates for the structures’ depreciation rates are available.
Finally, if the statistical agency also constructs a national balance sheet, then estimates for the value of residential land will also be available.

Thus, all the basic ingredients that are necessary to construct stocks for residential structures and the associated land stocks are available.
If, in addition, assumptions about the appropriate nominal interest rate and about expected prices for structures and land are made,70 then aggregate user costs of residential structures and resi- land can be constructed.

The proportion of dential these stocks that is rented can be deducted, and esti- mates for the user costs and corresponding values for owner-occupied residential land and structures can be made.
Of course, it would be almost impossible to do all this on a current basis, but all the above computa- tions can be done for a base period in order to obtain appropriate weights for owner-occupied structures and land.

Then, it can be seen that the main drivers for the monthly user costs are the price of a new structure timely and the price of residential monthly indicators for these two prices can be devel- oped, the entire procedure is feasible.
Problems with this approach include: As was the case with Option 4 above, it will be diﬃcult to determine what the ‘‘correct’’ depreciation rates and real interest rates are.71 land.

Hence, if It will be diﬃcult to construct a monthly price of residential land index.

 It may be diﬃcult to convert the residential housing investment price deﬂator from a quarterly to a monthly basis.
23.145 All the above ﬁve options have their advan- tages and disadvantages; there does not appear to be a clear ‘‘winning’’ option.

Thus, each statistical agency will have to decide whether they have the resources to implement any of these ﬁve options in addition to the usual acquisitions approach to the treatment of owner- occupied housing.
From the viewpoint of the cost of living approach to the CPI, any one of the ﬁve options would be an adequate approximation to the ideal treatment from the perspective of measuring the ﬂow of consumption services in each period.72 70 Alternatively, an appropriate real interest rate can be assumed.

71 As usual, however, it can be argued that errors in estimating these parameters will mainly aﬀect the weights used in the price index.
72 For consumer durables that do not change in quality over time, Option 5 will probably suﬃce.

441 A GLOSSARY OF MAIN TERMS The appendix to this glossary gives the main aggregate index number formulae used for consumer price index (CPI) purposes and also explains the interrelationships between them.
Acquisitions approach Additivity Aggregate Aggregation Axiomatic, or test approach Base period Basket Basket price index Bias Bouncing An approach to CPIs in which consumption is identiﬁed with the consumption goods and services acquired by a household in some period (as distinct from those wholly or partially used up for purposes of consumption).
Depending on the intended scope of the CPI, acquisitions may include not only goods and services purchased, but also those acquired by own-account production or as social transfers in kind from government or non-proﬁt institutions.

At current prices, the value of an aggregate is identical to the sum of the values of its components.
Additivity requires this identity to be preserved for the extrapolated values of the aggregate and its components when their current values in some period are extrapolated using a set of interrelated quantity indices; or, alternatively, when the current values of an aggregate and its components in some period are deﬂated using a set of interrelated price indices.
A set of transactions relating to a speciﬁed ﬂow of goods and services, such as the total purchases made by resident households on consumer goods and services in some period.

The term ‘‘aggre- gate’’ is also used to mean the value of the designated set of transactions.

The process of combining, or adding, diﬀerent sets of transactions to obtain larger sets of trans- actions.

The larger set is described as having a higher level of aggregation than the sets of which it is composed.
The term ‘‘aggregation’’ is also used to mean the process of adding the values of the lower-level aggregates to obtain higher-level aggregates.

In the case of price indices, it means the process by which price indices for lower-level aggregates are averaged, or otherwise combined, to obtain price indices for higher-level aggregates.

The approach to index number theory that determines the choice of index number formula on the basis of its mathematical properties.
A list of tests is drawn up, each test requiring an index to possess a certain property or satisfy a certain axiom.

An index number may then be chosen on the basis of the number of tests satisﬁed.

Not all tests may be considered to be equally important and the failure to satisfy one or two key tests may be considered suﬃcient grounds for rejecting an index.

The base period is usually understood to mean the period with which all the other periods are compared.
The term may, however, have diﬀerent meanings in diﬀerent contexts.

Three types of base period may be distinguished: the price reference period – the period that provides the prices to which the prices in other periods are compared.

The prices of the price reference period appear in the denominators of the price relatives, or price ratios, used to calculate the index.
The price reference period is typically designated as period 0; the weight reference period – the period, usually one or more years, of which the expenditures serve as weights for the index.

When the expenditures are hybrid (i.e., the quantities of one period are valued at the prices of some other period), the weight reference period is the period to which the quantities refer.
The weight reference period is typically designated as period b in this manual; the index reference period – the period for which the value of the index is set equal to 100.

It should be noted that, in practice, the duration of the weight reference period for a CPI is typically a year, or even two or more years, whereas the CPI is calculated monthly or quarterly, the duration of the price reference period being a month or quarter.
Thus, the weight and price reference periods seldom coincide in practice, at least when a CPI is ﬁrst calculated, although the price and index reference periods frequently coincide.

A speciﬁed set of quantities of goods and services.

In a CPI context, the set may comprise the actual quantities of consumption goods or services acquired or used by households in some period, or may be made up of hypothetical quantities.
A price index that measures the proportionate change between periods 0 and t in the total value of a speciﬁed basket of goods and services: that is, P ptq/P p0q, where the terms q are the speciﬁed quantities.

See Lowe index.

A systematic tendency for the calculated CPI to diverge from some ideal or preferred index, resulting from the method of data collection or processing, or the index formula used.

See Cost of living bias and Representativity bias.


A situation in which the set of prices for the second period is simply a reordering of the set of prices for the ﬁrst period, the price relatives thus being obtained by matching each price in the ﬁrst period with another price from the same set of prices.
443 A GLOSSARY OF MAIN TERMS Carli price index Carry forward Central product classiﬁcation (CPC) Chain index Characteristics Circularity (transitivity) Collective consumption Commensur- ability Commodity reversal test Component Conditional cost of living index Consistency in aggregation Consumer price index (CPI) Consumers Consumption Consumption of own production Continuity Cost of living bias Cost of living index (COLI) Coverage 444 An elementary price index deﬁned as a simple, or unweighted, arithmetic average of the sample price relatives.
A situation in which a missing price in some period is imputed as being equal to the last price observed for that item.

An internationally agreed classiﬁcation of goods and services based on the physical characteristics of goods or on the nature of the services rendered.

An index number series for a long sequence of periods obtained by linking together index numbers spanning shorter sequences of periods.

See Linking; see also equation (6) of the Appendix.
The physical and economic attributes of a good or service that serve to identify it and enable it to be classiﬁed.

An index number property such that, if jIk denotes a particular kind of price index that measures the change between periods j and k, then jIl:jIk kIl where the indices jIl and kIl are of the same type.

When an index is transitive, the index that compares periods j and l indirectly through period k is identical with the index that compares j and l directly.
One test that might be required under the axiomatic approach is that the index number should be transitive.

Goods and services that are consumed simultaneously by a group of consumers or by the com- munity as a whole; for example, defence services provided by the State.

See Invariance to changes in the units of measurement test.
A test that might be used under the axiomatic approach, which requires that, for a given set of products, the price index should remain unchanged when the ordering of the products is changed.

A subset of the goods and services that make up some deﬁned aggregate.
A conditional cost of living index measures the change in the cost of maintaining a given utility level, or standard of living, on the assumption that all the factors, except the prices covered by the index, that inﬂuence the consumer’s utility or welfare (e.g., the state of the physical environment) remain constant.

See Cost of living index.


An index is said to be consistent in aggregation when the index for some aggregate has the same value whether it is calculated directly in a single operation, without distinguishing its components, or whether it is calculated in two or more steps by ﬁrst calculating separate indices, or sub-indices, for its components, or sub-components, and then aggregating them, the same formula being used at each step.
A monthly or quarterly price index compiled and published by an oﬃcial statistical agency that measures changes in the prices of consumption goods and services acquired or used by households.

Its exact deﬁnition may vary from country to country.

Individual persons or groups of persons living together as households.
There are several types of consumption: intermediate consumption consists of the goods and services used by enterprises as inputs into their processes of production; it is excluded from CPIs; collective consumption consists mainly of the collective services provided by governments to the community as a whole; it is excluded from CPIs; ﬁnal individual consumption consists of goods and services that individual households may acquire in order to satisfy their own needs and wants.
See also Households’ consumption expenditures.

Goods or services that are consumed by the same household that produces them.

The housing services consumed by owner-occupiers fall within this category.

If goods and services produced and consumed within the same household are to be included in CPIs, prices must be imputed for them.

Their inclusion or exclusion depends on the intended scope of the CPI.

The property whereby the price index is a continuous function of its price and quantity vectors.
An alternative term used to describe Substitution bias.

An index that measures the change between two periods in the minimum expenditures that would be incurred by a utility-maximizing consumer, whose preferences or tastes remain unchanged, in order to maintain a given level of utility (or standard of living or welfare).

As consumers may be expected to change the quantities they consume in response to changes in relative prices (see Substitution eﬀect), the COLI is not a basket index.
The expenditures in one or other, or possibly both, periods cannot usually be observed.

COLIs cannot be directly calculated but may be approximated by superlative indices.

See Conditional cost of living index.

The set of goods and services of which the prices are actually included in the index.

For practical reasons, coverage may have to be less than the ideal scope of the index, that is, the set of goods and services that the compilers of the index would prefer to include if it were feasible.
A GLOSSARY OF MAIN TERMS Current period, or comparison period Current prices Current value In principle, the current period should refer to the most recent period for which the index has been compiled or is being compiled.

The term is widely used, however, to mean the comparison period; that is, the period that is compared with the base period, usually the price reference or index reference period.

It is also widely used simply to mean the later of the two periods being compared.
The exact meaning is usually clear in the context.

The actual prices prevailing in the period in question.

The actual value of some aggregate in the period in question: the quantities in the period multiplied by the prices of the same period.

Deﬂation Democratic index Discount Cut-oﬀ sampling A sampling procedure in which a predetermined threshold is established with all units in the universe at or above the threshold being included in the sample, and all units below the threshold being excluded.
The threshold is usually speciﬁed in terms of the size of some relevant variable, the largest sampling units being included and the rest given a zero chance of inclusion.

In the case of retail outlets, size may be deﬁned in terms of sales.

The division of the current value of some aggregate by a price index (described as a deﬂator) in order to revalue its quantities at the prices of the price reference period.
A form of CPI in which each household is given equal weight in the calculation of the index, irrespectively of the size of its expenditures.

A deduction from the list or advertised price of a good or a service that is available to speciﬁc customers under speciﬁc conditions.

Examples include cash discounts, prompt payment discounts, volume discounts, trade discounts and advertising discounts.

A price or quantity index that treats both prices and quantities as continuous functions of time.
By diﬀerentiating with respect to time, the rate of change in the value of the aggregate in question is partitioned into two components, one of which is the price index and the other the quantity index.

In practice, the indices cannot be calculated directly, but it may be possible to approximate them by chain indices in which indices measuring the changes between consecutive periods are linked together.

An alternative term for the scope of an index.


A chain index is said to drift if it does not return to unity when prices in the current period return to their levels in the base period.

Chain indices are liable to drift when prices ﬂuctuate over the periods they cover.

The arithmetic average of the Laspeyres price index and the Paasche price index.
Domain Drift Divisia index Drobisch price index Durable consumption good Dutot index Economic approach Edgeworth price index Editing Elementary aggregate Elementary price index Expenditure weights Explicit quality adjustment A consumption good that can be used repeatedly or continuously for purposes of consumption over a long period of time, typically several years.

An elementary price index deﬁned as the ratio of the unweighted arithmetic averages of the prices in the two periods compared.
The economic approach to index number theory assumes that the quantities are functions of the prices, the observed data being generated as solutions to various economic optimization problems.

In the CPI context, the economic approach usually requires the CPI to be some kind of cost of living index.

A basket price index in which the quantities in the basket are simple arithmetic averages of the quantities consumed in the two periods.
The process of scrutinizing and checking the prices reported by price collectors.

Some checks may be carried out by computers using statistical programs written for the purpose.

The smallest aggregate for which expenditure data are available and used for CPI purposes.

The values of the elementary aggregates are used to weight the price indices for elementary aggregates to obtain higher-level indices.
The range of goods and services covered by an elementary aggregate should be relatively narrow, and may be further narrowed by conﬁning the goods and services to those sold in particular types of outlet or in particular locations.

Elementary aggregates also serve as strata for the sampling of prices.

An elementary index is a price index for an elementary aggregate.
Expenditure weights cannot usually be assigned to the price relatives for the sampled products within an elementary aggregate, although other kinds of weighting may be explicitly or implicitly introduced into the calculation of elementary indices.

Three examples of elementary index number formulae are the Carli, the Dutot and the Jevons.

See Weights.

A direct estimate of how much of the change in the price of a product is attributable to changes in its physical or economic characteristics.
It requires an evaluation of the contributions of the dif- ferences in particular characteristics to the diﬀerences in the observed prices of two products.

It includes quality adjustments based on hedonic methods.

See also Implicit quality adjustment.


445 A GLOSSARY OF MAIN TERMS Factor reversal test Fisher price index Fixed basket indices Fixed weight indices Geometric Laspeyres index Goods Suppose the prices and quantities in a price index are interchanged to yield a quantity index of exactly the same functional form as the price index.
Under the axiomatic approach, the factor reversal test requires that the product of this quantity index and the original price index should be identical to the proportionate change in the value of the aggregate in question.

The geometric average of the Laspeyres price index and the Paasche price index.

It is a symmetric index and a superlative index.

A time series of basket indices that all use the same basket; see equation (4) of the Appendix.
In a CPI context, the ﬁxed basket usually consists of the total quantities consumed by the designated set of households over a period of a year or more.

An abbreviated description for a series of weighted arithmetic averages of price relatives that all use the same weights; see equation (13) of the Appendix.

The weights are usually either actual or hybrid expenditure shares.

A weighted geometric average of the price relatives using the expenditure shares of the price reference period as weights.
Also called Logarithmic Laspeyres index.

Physical objects for which a demand exists, over which ownership rights can be established and for which ownership can be transferred between units by engaging in transactions on the market.

Hedonic method A regression model in which the market prices of diﬀerent products are expressed as a function of their characteristics.

Non-numerical characteristics are represented by dummy variables.
Each regression coeﬃcient is treated as an estimate of the marginal contribution of that characteristic to the total price.

The estimates may be used to predict the price of a new product for which the mix of characteristics is diﬀerent from that of any product already on the market.

The hedonic method can therefore be used to estimate the eﬀects of quality changes on prices.

Higher-level index An aggregate index as distinct from an elementary index.
Household budget surveys (HBSs) Households Sample surveys of households in which the households are asked to provide data on, or estimates of, the amounts they spend on consumption goods and services, and for other purposes over a given period of time.

Households may be either individual persons living alone or groups of persons living together who make common provision for food or other essentials for living.
Most countries choose to exclude groups of persons living in large institutional households (barracks, retirement homes, etc.)

from the scope of their CPIs.

Expenditures on ﬁnal consumption goods and services incurred by individual households on their own behalf.

They exclude expenditures incurred by governments or non-proﬁt institutions on goods or services provided to households as free social transfers in kind.
Hypothetical values, or expenditures, in which the quantities are valued at a diﬀerent set of prices from those at which they were actually bought or sold: for example, when the quantities purchased in an earlier period, such as b, are valued at the prices prevailing in a later period, such as 0.

Households’ consumption expenditures Hybrid values or expenditures Hybrid weights Weights deﬁned as hybrid value, or hybrid expenditure, shares.
Identity test Implicit quality adjustment Imputed price Indexation A test under the axiomatic approach that requires that, if each and every price remains unchanged between the two periods, the price index must equal unity.

Inferring indirectly the change in the quality of a product of which the characteristics change over time by estimating, or assuming, the pure price change that has occurred.
For example, if the pure price change is assumed to be equal to the average for some other group of products, the implied change in quality is equal to the actual observed price change divided by the assumed pure price change.

If the whole of the observed price change is assumed to be pure price change, there is assumed to be no change in quality.

See also Explicit quality adjustment.

The price assigned to an item for which the price is missing in a particular period.
The term ‘‘imputed price’’ may also refer to the price assigned to an item that is not sold on the market, such as a good or service produced for own consumption, including housing services produced by owner- occupiers, or one received as payment in kind or as a free transfer from a government or non-proﬁt institution.

The periodic adjustment of the money values of some regular scheduled payments based on the movement of the CPI or some other price index.
The payments may be wages or salaries, social security or other pensions, other social security beneﬁts, rents, interest payments, etc.

The period for which the value of the index is set at 100.

Index reference period Institutional unit A national accounts concept deﬁned as an economic entity that is capable, in its own right, of owning assets, incurring liabilities and engaging in economic activities and transactions with other entities.

Households are institutional units.


Other kinds of units include enterprises and governments.

A test under the axiomatic approach that requires that the price index does not change when the units of quantity to which the prices refer are changed: for example, when the price of some drink is quoted per litre rather than per pint.

This test is also described as the commensurability test.
Invariance to changes in the units of measurement test 446 Invariance to proportional change in current or base quantities test Inverse proportionality in base year prices test Item Item or product rotation Jevons price index Laspeyres price index Linking Lowe index Lower-level index Matched products or models Mean value test for prices Non-probability sampling A test under the axiomatic approach that requires that the price index remains unchanged when all the base period quantities, or all the current period quantities, are multiplied by a positive scalar.
A GLOSSARY OF MAIN TERMS A test which may be invoked under the axiomatic approach that requires that, if all the base period prices are multiplied by the positive scalar l, the new price index is 1/l times the old price index.

An individual good or service in the sample of products selected for pricing.

The deliberate replacement of a sampled item, or product, for which prices are being collected, by another product before the replaced product has disappeared from the market or individual outlet.
It is designed to keep the sample of products up to date and reduce the need for forced replacements caused by the disappearance of products.

An elementary price index deﬁned as the unweighted geometric average of the sample price relatives.

A basket index in which the basket is composed of the actual quantities of goods and services in the earlier of the two periods compared, the price reference period; see equation (3) of the Appendix.
It can also be expressed as a weighted arithmetic average of the price relatives that uses the expen- diture shares in the earlier period as weights; see equations (7) to (10) of the Appendix.

The earlier period serves as both the weight reference period and the price reference period.

Splicing together two consecutive sequences of price observations, or price indices, that overlap in one or more periods.
When the two sequences overlap by a single period, the usual procedure is simply to rescale one or other sequence so that the value in the overlap period is the same in both sequences and the spliced sequences form one continuous series.

See equation (6) of the Appendix.

A price index that measures the proportionate change between periods 0 and t in the total value of a speciﬁed basket of goods and services; that is, P ptq/P p0q, where the terms q are the speciﬁed quantities.
The basket does not necessarily have to consist of the actual quantities in some period.

See Appendix.

This type of index is described in the manual as a Lowe index after the index number pioneer who ﬁrst proposed this general type of index.

The class of indices covered by this deﬁnition is very broad and includes, by appropriate speciﬁcation of the terms q, the Laspeyres, Paasche, Edgeworth and Walsh indices, for example.
Lowe indices are widely used for CPI purposes, the quantities in the basket typically being those of some weight reference period b, which precedes the price reference period 0.

An elementary index as distinct from an aggregate index.

The practice of pricing exactly the same product in two or more consecutive periods.

It is designed to ensure that the observed price changes are not aﬀected by quality change.

The change in price between two perfectly matched products is described as a pure price change.
A test under the axiomatic approach, which requires that the price index should lie between the smallest price relative and the largest price relative.

The deliberate, i.e.

non-random, selection of a sample of outlets and products on the basis of the knowledge or judgement of the person responsible.

Also known as purposive sampling and judge- mental sampling.
Outlier Owner-occupied housing ‘‘One hoss shay’’ A model of depreciation, in which the durable delivers the same services for each vintage: a chair is a chair, no matter what its age (until it falls to pieces and is scrapped).

Also known as the light bulb model of depreciation.

A term that is generally used to describe any extreme value in a set of survey data.
In a CPI context, it is used for an extreme value of price or price relative that requires further investigation or that has been veriﬁed as being correct.

Dwellings owned by the households that live in them.

The dwellings are ﬁxed assets that their owners use to produce housing services for their own consumption, these services being usually included within the scope of the CPI.

The rents may be imputed by the rents payable on the market for equivalent accommodation or by user costs.
See Rental equivalence and User cost.

A basket index in which the basket is composed of the actual quantities of goods and services in the later of the two periods compared.

The later period serves as the weight reference period and the earlier period as the price reference period.

The Paasche index can also be expressed as a weighted harmonic average of the price relatives that uses the actual expenditure shares in the later period as weights.

See equations (7) to (11) of the Appendix.


The period of which the prices appear in the denominators of the price relatives.

See also Base period.

The ratio of the price of an individual product in one period to the price of that same product in some other period.
Price reference period Price relative Paasche price index 447 A GLOSSARY OF MAIN TERMS Price updating Probability proportional to size sampling (PPS) Probability sampling Products A procedure whereby the quantities in an earlier period are revalued at the prices of a later period.

The resulting expenditures are hybrid.

In practice, the price-updated expenditures may be obtained by multiplying the original expenditures by price relatives or price indices.
A sampling procedure whereby each unit in the universe has a probability of selection proportional to the size of some known variable, such as the value of the sales of an outlet.

The random selection of a sample of units, such as outlets or products, in such a way that each unit in the universe has a known non-zero probability of selection.

A generic term used to mean a good or a service.

Individual sampled products selected for pricing are often described as items.
A test under the axiomatic approach that requires that, if all current period prices are multiplied by the positive scalar l, the new price index is l times the old price index.

Proportionality in current prices test Purchaser’s price The amount payable by the purchaser to acquire a good or service.

The purchaser’s price includes Pure price change Quality adjustment any charges incurred in order to take delivery at the time and place required by the purchaser.
The change in the price of a good or service of which the characteristics are unchanged; or the change in the price after adjusting for any change in quality.

An adjustment to the change in the price of a product of which the characteristics change over time that is designed to remove the contribution of the change in the characteristics to the observed price change.
In a CPI context, the adjustment is needed when the price of a replacement product has to be compared with the price of the product it replaces.

In practice, the required adjustment can only be estimated.

Diﬀerent methods of estimation, including hedonic methods, may be used in diﬀerent circumstances.

See Explicit quality adjustment and Implicit quality adjustment.

Quantity relative The ratio of the quantity of a product in one period to the quantity of that same product in some other period.
Quantity weights A term sometimes used to describe the quantities in the basket.

However, expenditures rather than Rebasing Reference population Rental equivalence Replacement product Representative product Representativity bias Reweighting Sample augmentation quantities act as weights for price relatives.

See Weights.

Rebasing may have diﬀerent meanings in diﬀerent contexts.
It may mean: changing the weights used for a series of indices; or changing the price reference period used for a series of indices; or changing the index reference period for a series of indices.

The weights, price reference period and index reference period may be changed separately or at the same time.

The set of households included within the scope of the index.
The estimation of the imputed rents payable by owner-occupiers on the basis of the rents payable on the market for accommodation of the same type.

A product chosen to replace a product for which prices have been collected previously, either because the previous product has disappeared altogether or because it accounts for a diminishing share of the sales of the outlet, or the expenditures within the elementary aggregate.
A product, or category of products, that accounts for a signiﬁcant proportion of the total expen- ditures within an elementary aggregate, and/or for which the average price change is expected to be close to the average for all products within the aggregate.

Bias in a basket index that results from the use of quantities that are not representative of the two periods compared; that is, that systematically diverge from the average quantities consumed in the two periods.
For example, representativity bias may result from the use of an old, out-of-date basket which deviates systematically from the baskets in both the periods compared.

In practice, representativity bias tends to be similar to substitution bias, as it is attributable to the same economic factors.

Replacing the weights used in an index by a new set of weights.

Maintaining and adding to the sample of outlets in the survey panel to ensure that they continue to be representative of the population of outlets.
A ﬁxed sample of outlets tends to be depleted over time, as outlets cease trading or stop responding.

Including new outlets also tends to facilitate the inclusion of new products in the CPI.

The price collected for a sampled product, sometimes described as a price quote.

Sampled price Sampled product An individual product that is included in the sample selected for pricing within an elementary Sample rotation aggregate.
Limiting the length of time that outlets and/or products are included in the price surveys by dropping a proportion of them, or possibly all of them, after a certain period of time and selecting a new sample of outlets and/or products.

Rotation is designed to keep the sample up to date.

448 Sampling frame Scanner data Scope Seasonal products Speciﬁcation Stochastic approach Substitute Substitution A GLOSSARY OF MAIN TERMS A list of the units in the universe from which a sample of units can be selected.
The list may contain information about the units, which may be used for PPS sampling.

Examples of lists that may be used for retail outlets are business registers, telephone directories (‘‘yellow pages’’), local authority records, trade directories, etc.

Such lists may not cover all the units in the designated universe and may also include units that do not form part of that universe.
Detailed data on sales of consumer goods obtained by scanning the bar codes for individual products at electronic points of sale in retail outlets.

The data can provide detailed information about quantities, characteristics and values of goods sold, as well as their prices.

Scanner data constitute a rapidly expanding source of data with considerable potential for CPI purposes.

They are increasingly used for purposes of hedonic analysis.
The set of products for which the index is intended to measure the price changes.

The scope of a CPI will generally be deﬁned in terms of a designated set of consumption goods and services purchased by a designated set of households.

In practice, certain goods and services or households may have to be excluded because it is too diﬃcult, time-consuming or costly to collect the relevant data on expenditures or prices: for example, illegal expenditures.
The coverage of an index denotes the actual set of products included, as distinct from the intended scope of the index.

Seasonal products are products that either are not available on the market during certain seasons or periods of the year, or are available throughout the year but with regular ﬂuctuations in their quantities and prices that are linked to the season or time of the year.

A description or list of the characteristics that can be used to identify an individual sampled product to be priced.
A tight speciﬁcation is a fairly precise description of an item intended to narrow the range of items from which a price collector might choose, possibly reducing it to a unique item, such as a particular brand of television set identiﬁed by a speciﬁc code number.

A loose speciﬁcation is a generic description of a range of items that allows the price collector some discretion as to which particular item or model to select for pricing, such as colour television sets of a particular size.
The approach to index number theory that treats the observed price relatives as if they were a random sample drawn from a deﬁned universe for which the mean can be interpreted as the general rate of inﬂation.

The sample mean provides an estimate of the rate of inﬂation.

A product of which the characteristics are similar to those of another product and that can be used to meet the same kinds of consumer needs or wants.
The replacement of products by substitutes, typically in response to changes in relative prices.

Rational utility-maximizing consumers, as price takers, typically react to changes in relative prices by reducing, at least marginally, their consumption of goods and services that have become relatively dearer and increasing their consumption of substitutes that have become relatively cheaper.

Substitution results in a negative correlation between the quantity and price relatives.
Substitution bias This is generally understood to be the bias that results when a basket index is used to estimate a cost of living index, because a basket index cannot take account of the eﬀects on the cost of living of the substitutions made by consumers in response to changes in relative prices.

In general, the earlier the period of which the basket is used, the greater the upward bias in the index; see also Representativity bias.

The eﬀect of substitution on the value of an index.
A type of index formula that can be expected to approximate to the cost of living index.

An index is said to be exact when it equals the true cost of living index for consumers whose preferences can be represented by a particular functional form.

A superlative index is then deﬁned as an index that is exact for a ﬂexible functional form that can provide a second-order approximation to other twice-diﬀerentiable functions around the same point.
The Fisher, the To¨ rnqvist and the Walsh price indices are examples of superlative indices.

Superlative indices are generally symmetric indices.

System of National Accounts (SNA) Symmetric index An index that treats both periods symmetrically by attaching equal importance to the price and expenditure data in both periods.

The price and expenditure data for both periods enter into the index formula in a symmetric way.
A coherent, consistent and integrated set of macroeconomic accounts, balance sheets and tables based on a set of internationally agreed concepts, deﬁnitions, classiﬁcations and accounting rules.

Household income and consumption expenditure accounts form part of the SNA.

The expenditure data are one of the sources that are used to estimate expenditure weights for CPI purposes.
An index number property such that, if jIk denotes a particular kind of price index formula that measures the change from period j to period k, then jIk:1/kIj where kIj measures the change from period k to period j.

When an index has this property, the change is the same whether it is measured forwards from the ﬁrst to the second period or backwards from the second to the ﬁrst period.

An index may be required to satisfy the time reversal test under the axiomatic approach.
A symmetric index deﬁned as the weighted geometric average of the price relatives in which the weights are simple arithmetic averages of the expenditure shares in the two periods.

It is a super- lative index.

Also known as the To¨ rnqvist–Theil price index.


To¨ rnqvist price index Time reversal 449 Substitution eﬀect Superlative index A GLOSSARY OF MAIN TERMS Transitivity Unit value or average value User cost Uses approach Value Walsh price index Weight reference period Weighted arithmetic average index Weights Young index See Circularity.

The unit value of a set of homogeneous products is the total value of the purchases/sales divided by the sum of the quantities.
It is therefore a quantity-weighted average of the diﬀerent prices at which the product is purchased/sold.

Unit values may change over time as a result of a change in the mix of the products sold at diﬀerent prices, even if the prices do not change.

The cost incurred over a period of time by the owner of a ﬁxed asset or consumer durable as a consequence of using it to provide a ﬂow of capital or consumption services.
User cost consists mainly of the depreciation of the asset or durable (measured at current prices and not at historic cost) plus the capital, or interest, cost.

An approach to CPIs in which the consumption in some period is identiﬁed with the consumption goods and services actually used up by a household to satisfy their needs and wants (as distinct from the consumption goods and services acquired).
In this approach, the consumption of con- sumer durables in a given period is measured by the values of the ﬂows of services provided by the stocks of durables owned by households.

These values may be estimated by the user costs.

Price times quantity.

The value of the expenditures on a set of homogeneous products can be factored uniquely into its price and quantity components.
Similarly, the change over time in the value of a set of homogeneous products can be factored uniquely into the change in the unit value and the change in the total quantities.

There are, however, many diﬀerent ways of factoring the change over time in the value of a set of heterogeneous products into its price and quantity components, a phenomenon that gives rise to the index number problem.
A basket index in which the quantities are geometric averages of the quantities in the two periods; see the Appendix.

It is a symmetric index and a superlative index.

The period of which the expenditure shares serve as the weights for a Young index, or of which the quantities make up the basket for a Lowe index.

There may be no weight reference period when the expenditure shares for the two periods are averaged, as in the To¨ rnqvist index, or when the quantities are averaged, as in the Walsh index.
See also Base period.

An index deﬁned as a weighted arithmetic average of the price relatives: namely, P w( pt/p0 ), where the weights w sum to unity.

A set of numbers summing to unity that are used to calculate averages.

In a CPI context, the weights are generally actual or hybrid expenditure shares that sum to unity by deﬁnition.

They are used to average price relatives, or elementary price indices; see the Appendix.

Quantities of diﬀerent kinds of products are not commensurate and not additive.
They cannot serve as weights.

The quantities that make up a basket should therefore not be described as quantity weights.

A Young index is a weighted arithmetic average of the price relatives, P w(pt/p0 ), in which the terms w refer to the actual expenditure shares of period b, the weight reference period; that is, w=sb=pbqb/P pbqb.

It is a weighted version of the Carli index.

450 Appendix to the glossary.

Some basic index number formulae and terminology 1.
Throughout this Appendix, the sums are understood to be running over all items n. A basket price index is an index of the form P pt P p0 nqn nqn (A:1) which compares the prices of period t with those of (an earlier) price reference period 0, using a certain speciﬁed quantity basket.

The basket does not have to consist of the actual quantities in any particular period.

This general type of index is called a Lowe price index after the index number pioneer who ﬁrst proposed this general type of index.
The family of Lowe indices includes some well-known indices as special cases: when qn=q0 when qn=qt when qn=(q0 n, we get the Laspeyres index; n, we get the Paasche index; n)=2, we get the Marshall–Edgeworth n+q1 index; nqt n)1=2, we get the Walsh index.

 and when qn=(q0 In practice, statistical oﬃces frequently work with a Lowe index in which qn=qb n, where b denotes some weight reference period that is typically prior to 0.

2.
2.

A useful feature of a Lowe index for period t relative to period 0 is that it can be decomposed, or factored, into the product of two or more indices of the same type: for instance, as the product of an index for period t1 relative to period 0 and an index for period t relative to period t1.
Formally, P pt P p0 nqn nqn = P pt1 n qn P p0 nqn P pt nqn P pt1 n qn In particular, when qn=q0 n, expression (2) turns into P pt P p0 nq0 n nq0 n = P pt1 n q0 n P p0 nq0 n P pt nq0 n P pt1 n q0 n (A:3) The left-hand side of expression (3) is a direct Laspeyres index.
Note that only the ﬁrst of the indices that make up the right- hand side is itself a Laspeyres index, the second being a Lowe index for period t relative to period t1 that uses the quantity basket of period 0 (not t1).

Some statistical oﬃces describe the index on the right-hand side of expression (3) as a modiﬁed Laspeyres index.
A GLOSSARY OF MAIN TERMS (5) into a series of the form P pT P p0 n qn nqn P pT+1 q0 n n P pT n q0 n , P pT P p0 n qn nqn P pT+2 q0 n n P pT n q0 n , P pT P p0 n qn nqn P pT+3 q0 n n P pT n q0 n , .

.

.

: (A:6) n ¼ qT n and q0 This could be termed a series of chain-linked ﬁxed basket price indices.

In particular, when qn=q0 n , we get a series of chain-linked Laspeyres indices.

Since the basket is changed at period T, the adjective ﬁxed applies literally only over a certain number of time intervals.
The basket is ﬁxed from period 1 to period T, and is again ﬁxed from period T+1 onwards.

When the time intervals during which the basket is kept ﬁxed are of the same length, such as one, two or ﬁve years, this feature can be made explicit by describing the index as an annual, bi-annual or ﬁve-yearly chain-linked ﬁxed basket price index.

5.
5.

A weighted arithmetic-average price index (so-called to distinguish it from a geometric or other kind of average) is an index of the form P wn( pt n=p0 n) (A:7) which compares the prices of period t with those of period 0, using a certain set of weights adding up to 1.

In particular, when the weights are the period b value shares wn=sb n pb nqb n=P pb nqb n (A:8) we obtain the Young index.
Note that any basket price index (1) can be expressed in the form (7), since P pt P p0 nqn nqn =P p0 nqn P p0 nqn pt n p0 n : (A:2) When the weights are the period 0 value shares, wn=s0 n p0 nq0 n=P p0 nq0 n expression (7) turns into the Laspeyres index.

When wn=p0 nqt n=P p0 nqt n (A:9) (A:10) (A:11) that is, hybrid period (0, t) value shares, we get the Paasche index.

One could also think of setting wn=sb n)=P sb that is, price-updated period b value shares.
n)=p0 n=pb n=pb n( p0 n( p0 nqb n=P p0 nqb n (A:12) Note that hybrid value shares, such as given by expressions 3.

In a time series context, say when t runs from 1 to T, the (11) or (12), are not observable but must be constructed.

series 6.

In a time series context, when t runs from 1 to T, the series P p1 P p0 nqn nqn , P p2 P p0 nqn nqn , .

.

.

, P pT P p0 n qn nqn (A:4) is termed a series of ﬁxed basket price indices.

In particular, when qn=q0 n, we get a series of Laspeyres indices.

4.
4.

At period T one could change to a new quantity basket q0, and calculate from this period onwards P pT+1 q0 n n P pT n q0 n , P pT+2 q0 n n P pT n q0 n , P pT+3 q0 n n P pT n q0 n ; .

.

.

: (A:5) P wn( p1 n=p0 n), P wn( p2 n=p0 n), .

.

.

::, P wn( pT n =p0 n) (A:13) is termed a series of ﬁxed weighted arithmetic-average price indices.
In particular, when the weights are equal to the period 0 expenditure shares, we get a series of Laspeyres indices, and when the weights are equal to the price-updated period b expenditure shares, we get a series of Lowe indices in which the quantities in the basket are those of period b.

7.

In period T one could change to a new set of weights w 0, and calculate from this period onwards To relate the prices of periods T+1, T+2, T+3, .

.

.
.

.

to those of period 0, chain linking can be used to transform the series P w0 n( pT+1 n =pT n ), P w0 n( pT+2 n =pT n ), P w0 n( pT+3 n =pT n ), .

.

.

:: (A:14) 451 n =pb0 ( pT Lowe indices.

n ) for some later period b0, we get a series of chain-linked 8.

Again, since the weights are changed at period T, the adjective ﬁxed applies literally only over a certain number of time intervals.

The weights are ﬁxed from period 1 to period T, and are again ﬁxed from period T+1 onwards.
When the time intervals during which the weights are kept ﬁxed are of the same length, this feature can be made explicit by adding a temporal adjective, such as annual, bi-annual or ﬁve-yearly.

A GLOSSARY OF MAIN TERMS or, using chain-linking to relate the prices of periods T+1, T+2, T+3, .

.

.

.

to those of period 0, P wn( pT P wn( pT n =p0 n =p0 n)P w0 n)P w0 n( pT+1 n n( pT+2 n =pT =pT n ), n ), .

.

.
.

.

: (A:15) This could be termed a series of chain-linked ﬁxed weighted arithmetic-average price indices.

In particular, when wn=s0 n and w0 n , we get a series of chain-linked Laspeyres indices.
n )=Psb0 When wn=sb n n) and w0 n)=Psb n=sb0 n=sT n =pb0 n ( pT n=pb n=pb n( p0 n( p0 452 Annex 1 Harmonized Indices of Consumer Prices (European Union) 1 Introduction The Harmonized Indices of Consumer Prices (HICPs) are a set of European Union consumer price indices (CPIs) calculated according to a harmonized approach and a single set of deﬁnitions.
This annex outlines the aims and history of the HICPs, summarizes the most important harmonized standards and notes some key items on the agenda for further harmonization.

The HICP development project is ongoing.

This annex describes the state of development of the HICPs at mid- 2003.

The HICPs have a legal basis in that their pro- duction, and many elements of the speciﬁc methodology to be used, are stated in and required by a series of legally binding European Union Regulations.
Refer- ences for the full set of HICP legal standards are given at the end of this annex.

1.1 The main HICPs The HICPs on which most attention is focused are: the Monetary Union Index of Consumer Prices (MUICP) – an aggregate index covering the countries within the euro-zone; the European Index of Consumer Prices (EICP) – for the euro-zone plus the other European Union coun- tries; the national HICPs – for each of the Member States of the European Union (EU).
Beyond these are also the European Economic Area Index of Consumer Prices (EEAICP) and HICPs for the individual EEA countries.

There are also interim HICPs for Candidate and, in particular, Acceding Countries.1 It is expected that once those countries accede to the EU their HICPs will be fully comparable to those of the existing Member States.

The national HICPs are produced by the national sta- tistical institutes, while the country-group aggregates are produced by Eurostat.
1 The composition of the country groups at the end of 2003 is as fol- lows: Euro-zone countries: Austria, Belgium, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, Netherlands, Portugal, Spain.

EU countries: Euro-zone countries plus Denmark, Sweden, United Kingdom.

European Economic Area countries: EU countries plus Iceland, Norway.

Acceding countries: Cyprus, Czech Republic, Estonia, Hungary, Lat- via, Lithuania, Malta, Poland, Slovakia, Slovenia.
Candidate countries: Acceding countries plus Bulgaria, Romania, Turkey.

When countries accede to the EU or the euro-zone, the composition of those aggregates is revised accordingly.

1.2 Uses of the HICPs As explained elsewhere in this manual, CPIs have a variety of potential uses, for example for indexing social beneﬁts or contracts, or as inputs to various types of economic analyses.
The drive for the harmonization project has been the use of the HICPs as convergence criteria and the main measure for monitoring price stability in the euro-zone.

The HICPs have been set up to provide the best measure for international compar- isons of consumer price inﬂation within the EU and the euro-zone, for assessing price convergence and stability in the context of monetary policy analysis.
In the early stages of the project, the most important use of the HICPs was in the assessment of the price stability and price convergence required for entry into the European Economic and Monetary Union.

More recently, the focus of interest has shifted to- wards country-group aggregates – and in particular the MUICP.

This change of emphasis reﬂects the European Central Bank’s objective of price stability and the view that the HICPs are the most appropriate price measure for assessing price stability.
The focus of the HICPs on measuring price stability and convergence, and on international comparisons, does not mean that a wider range of users should not or cannot use HICPs for other purposes.

Depending on the precise purpose the user has in mind, the HICPs may be the best available price statistics.

All users of the HICPs should note, however, that the HICPs are revisable; the indices may change after the ﬁrst results are published.
1.3 Brief history of the HICPs On 23 October 1995, the European Union’s Council of Ministers adopted a Regulation providing the legal basis for the establishment of a harmonized methodol- ogy for compiling consumer price indices in the Member States and European Economic Area Countries.
This Regulation2 (hereinafter referred to as the HICP Framework Regulation) required HICPs to be produced and published, that they use a common reference base, employ a common coverage of consumer goods and services, and share a common classiﬁcation.

In the context of the HICP Framework Regulation, a series of speciﬁc implementing measures has been adopted.
As mentioned above, early in the harmonization project the most important use of the HICPs was for the application of the criterion of price stability in pre- paration for Economic and Monetary Union in Europe.

On 1 January 1999, Economic and Monetary Union began, with 11 countries participating in the single currency – the euro.

From that date, there has been a common monetary policy, with common interest rates, operating within the euro-zone under the control of the European Central Bank (ECB).
The maintenance of price stability is the primary objective of the European System of Central Banks.

The President of the ECB announced in October 19983 that it would be operating a ﬂexible monetary policy 2 Council Regulation (EC) No.

2494/95.

3 ECB press release of 13 Oct. 1998.

453 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE strategy, ensuring price stability in the euro-zone based on a monetary reference value and a mix of other indi- cators.
In this context, the Governing Council of the ECB stated: ‘‘Price stability shall be deﬁned as a year- on-year increase in the Harmonized Index of Consumer Prices (HICP) for the euro area of below 2%.
Price stability is to be maintained over the medium term.’’ In 2003 the ECB reaﬃrmed its inﬂation target of October 1998, founded on the HICP for the euro-zone, and added that: ‘‘At the same time, the Governing Council agreed that in the pursuit of price stability it will aim to maintain inﬂation rates close to 2% over the medium term.’’4 1.4 Stepwise harmonization The HICP Framework Regulation laid down a step- wise each approach to harmonization, whereby step requiring speciﬁc implementing measures would be legislated in the form of further legally binding stan- dards.
The HICP Framework Regulation established that the EU’s Statistical Programme Committee would act as the so-called Regulatory Committee, and would therefore be responsible for adopting the further harmonization stan- dards to be given legal force.

The Statistical Programme Committee is the most senior statistical committee of the EU, comprising the directors of the national statistical institutes.
1.5 Minimum standards The approach taken to harmonization has been to build, as far as possible, on the EU Member States’ their existing data sources and methodologies national CPIs.

The legal standards typically take the form of minimum standards, whereby more than one solution to a harmonization issue may usually be allowed so long as comparability is not threatened.
for Within this framework, by mid-2003, a series of 13 legally binding standards and some additional guidelines had been drawn up and implemented in collaboration with the EU Member States.

1.6 Compliance monitoring Given the importance accorded to the accuracy, reliability and comparability of the HICPs in the EU, Eurostat operates a system of compliance monitoring to ensure that the legal framework is adhered to.
This includes, in particular, compliance assessments on the basis of questionnaires and visits by Eurostat oﬃcials to the EU national statistical institutes to study in more detail their work on their HICPs.

2 Basic concepts and deﬁnitions 2.1 Aim and scope of the HICPs The aim of the HICPs was stated to be to measure inﬂation on a comparable basis, taking into account diﬀerences in national deﬁnitions.

This, however, requires an operational deﬁnition of the term ‘‘inﬂation’’.
Given the opinion and the needs of the HICPs’ main users, it was decided to compute the HICPs as Las- peyres-type price indices, based on the prices of goods and services available for purchase in the economic territory of each EU Member State for the purpose of directly satisfying consumer needs.
Based on this concept and by reference to national accounts, speciﬁcally the European System of Accounts (ESA 95), the coverage in practice of the HICPs was taken to be household ﬁnal monetary consumption expenditure5 (HFMCE).

This deﬁnition eﬀectively pre- scribes the goods and services, the population and the geographic territory to be covered, as well as the prices and the weights to be used.
The HICP may thus be described as a Laspeyres- type ‘‘consumer inﬂation’’ or ‘‘pure price’’ index, which measures average price changes on the basis of the changed expenditure of maintaining the consumption pattern of households and the composition of the consumer population in the base or reference period.

The term ‘‘pure price index’’ indicates that it is only changes in prices that should be reﬂected in the HICP measure between the current and the base or reference period.
The HICP is therefore not a cost of living index.

That is, it is not intended to be a measure of the change in the minimum cost for achieving the same standard of living (i.e.

constant utility) from two diﬀerent con- sumption patterns realized in the two periods compared, and where factors other than pure price changes may enter the index.
2.2 Household ﬁnal monetary consumption expenditure The coverage of the HICPs is delimited by HFMCE, and so concerns that part of ﬁnal consumption expen- diture which is: by households irrespective of their nationality or residence status; in monetary transactions; on the economic territory of the EU Member State; on goods and services that are used for the direct satisfaction of individual needs or wants; in one or both of the time periods being compared.
The prices used in the HICP should be the prices paid by households to purchase individual goods and services in monetary transactions.

The purchaser’s price is the price for the products that the purchaser actually pays at the time of purchase.

The weights of the HICP are the aggregate expendi- tures by households on any set of goods and services covered by the HICP, expressed as a proportion of the total expenditure on all goods and services within the coverage of the HICP.
The HICPs are classiﬁed according to the four-digit categories and sub-categories of the COICOP/HICP (Classiﬁcation of Individual Consumption according to Purpose, adapted to the needs of HICPs).

4 ECB press release of 8 May 2003.

5 Council Regulation (EC) No.

1688/98.


454 HARMONIZED INDICES OF CONSUMER PRICES (EUROPEAN UNION) 2.3 Links to national accounts concepts The concept of HFMCE not only speciﬁed the cov- erage, the prices and the weights for the HICP, but also established a link between HICPs and ESA 95 that has proved useful to analysts and policy-makers.

HICP deﬁnitions follow ESA 95 wherever possible and when to do so is consistent with the aims and uses of the HICP.
That said, there are some diﬀerences between the coverage of the HICPs and that of household ﬁnal consumption expenditure (HFCE) as deﬁned by national accounts, in particular the treatment of owner-occupied housing.

A full list of these diﬀerences is given below.

2.4 Some basic requirements for HICPs The relative distribution of consumers’ expenditure on individual products varies from country to country.

Hence, there is no uniform basket applying to all EU Member States.
The weights used in the compilation of HICPs may relate to a reference period up to seven years prior to the current year.

In practice, this results in a complete weight and sample revision of national HICPs in at least ﬁve-yearly intervals, taking into account that a period of about two years may be needed to integrate results of a full consumer expenditure survey.
Adjust- ments must nevertheless be made each year for any especially large changes in expenditure patterns, to minimize any disparities that could arise from diﬀerent update frequencies.

To keep the HICPs broadly in step with each other and up to date, new products must be included when they achieve a signiﬁcant relative importance.

HICPs must also be shown to be based on appropriate sampling procedures, taking into account the national diversity of products and prices.
The samples must be kept up to date, in particular by banning the practice whereby missing prices are simply assumed to be equal to the last observed prices.

In order to measure pure price changes, the prices included in HICPs need to be adjusted for changes in the quality of goods and services.

Certain inappropriate quality adjustment prac- tices, such as so-called automatic linking, may not be used.
HICP aggregates for country groups are calculated as the weighted averages of the national HICPs, using the weights of the countries and sub-indices concerned.

The weight of a country is its share of HFMCE in the total.

For the MUICP the weights are all naturally expressed in euros, whereas for the EICP and the EEAICP the aggregations use purchasing power standards.
The MUICP is treated as a single entity within the EICP and EEAICP aggregates.6 6 For technical notes see: Eurostat News Release 21/97 of 5 March 1997, Harmonizing the way EU measures inﬂation.

Eurostat Memo 8/98 of 4 May 1998, New monetary union index of consumer prices (MUICP).

Eurostat Memo 02/00 of 18 February 2000, Improved EU Harmonized Index of Consumer Prices: Extended coverage and earlier release dates for the HICP.
Further details can be found in the Compendium of HICP reference documents : http://europa.eu.int/comm/eurostat/Public/datashop/print- catalogue/EN?catalogue=Eurostat&product=KS-AO-01-005-_-I-EN 3 Coverage 3.1 Goods and services The coverage of goods and services in the HICPs has been expanded over time.

The HICPs now cover vir- tually all of HMFCE.

The main diﬀerence to the ESA 95 concept of HFCE is the exclusion of imputed expendi- tures of owner-occupied housing.
The initial coverage of goods and services in the HICPs, although fairly comprehensive, reﬂected for the most part what was common to the national consumer price indices.

Since then, with considerable eﬀort and cooperation by EU Member States, coverage has been extended to virtually all consumers’ expenditure, in the sense of HFMCE.

In particular, the diﬃcult areas of health, education and social protection services are now covered, as are insurance and ﬁnancial services.
These are included in the HICPs according to agreed deﬁni- tions, thus ensuring comparability despite major insti- tutional diﬀerences.

In the initial coverage of the HICPs,7 some diﬃcult categories such as health and educational services, where there are major institutional diﬀerences between EU Member States, were not fully covered.

In 1998 a further legal standard8 amended the initial coverage of goods and services and laid down a staged procedure to extend the coverage of the HICP.
Another legal standard extending the coverage entered into eﬀect with the index for January 2000.9 The next step took place with the publication of the January 2001 index.10 3.2 Geographic and population coverage The HICP Framework Regulation required the HICPs to be based on the prices of goods and services available for purchase on the economic territory of the EU Member State for the purposes of directly satisfying consumer needs.
As regards the economic territory and the consumers concerned, a harmonized deﬁnition of the geographic and population coverage of the HICP was necessary, both to achieve comparability and to avoid gaps or double counting when aggregating national HICPs.

In 1998 a legal standard11 speciﬁed that the HICP should cover all HFMCE which takes place on the economic territory of an EU Member State.
In partic- ular, HICP coverage should include expenditure by foreign visitors and expenditure by individuals living in institutions, but should exclude the expenditure by residents whilst in a foreign country (the so-called domestic concept).

All private households should be included, irrespective of the area in which they live or their position in the income distribution.

Expenditure incurred for business purposes should be excluded.
The choice of the domestic concept reﬂected the role of the MUICP in measuring price stability in the euro-zone.

7 Commission Regulation (EC) No.

1749/96.

8 Council Regulation (EC) No.

1687/98.

9 Commission Regulation (EC) No.

1749/1999.

10 Commission Regulation (EC) No.

2166/1999.

11 Council Regulation (EC) No.

1688/98.


455 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Price changes in the euro-zone are measured by aggre- gating price changes taking place within the individual EU Member States.

Expenditure and price changes to be measured within the economic territory should include those aﬀecting foreign visitors and exclude those aﬀecting residents whilst in a foreign country.
It is an HICP requirement that HICPs should be compiled using weights which reﬂect the HFMCE of all households.

HICPs which cover only a subset of households should nevertheless be regarded as com- parable if this diﬀerence in practice accounts for less than one part per thousand of the total expenditure to be covered by the HICP.
4 Weights, index formulae and price sampling 4.1 Weights The HICP Framework Regulation requires HICP weights to be suﬃciently up to date to ensure compar- ability, whilst at the same time avoiding the cost of conducting household budget surveys more than every ﬁve years.

An HICP legal standard12 gives minimum standards for the quality of HICP weights.
It aims to guarantee the quality of weights used to construct the HICP and minimize the disparities between HICPs which might arise from diﬀerent update frequencies.

Concerning the quality of HICP weights, the weights should be suﬃciently up to date to ensure comparability whilst avoiding unnecessary costs.

Diﬀerences in the frequency of updating of the weights could, but not lead to diﬀerences in measured necessarily would, inﬂation and non-comparability.
Imposing the cost of high precision for all weights or frequent updating of weights was not considered justiﬁed.

On the other hand, it is diﬃcult to be sure that an HICP using weights up to seven years old will provide a reliable and relevant measure of current inﬂation.

is suﬃcient.
is suﬃcient.

The legal standard on the quality of HICP weights requires a minimum action of review and adjustment to ensure that the quality of weights used to construct HICPs It establishes a comparability threshold in relation to the proposed reference practice.

The review requirement involves checking each year those weights which are judged to be most critical for reliability and relevance and, hence, for the compar- ability of the overall HICP.
These are primarily the weights for index components where signiﬁcant market changes have accompanied atypical price movements.

Where a weight is identiﬁed as deﬁcient, EU Member States should make an improved estimate and introduce an appropriate adjustment, from the following January index, where this would exceed the threshold eﬀect of 0.1 percentage points (on average for one year compared with the previous year).
The aim is to ensure that the adjusted weights are the best estimates that can be made on the information available.

12 Commission Regulation (EC) No.

2454/97.

456 4.2 Index formulae The choice of the index formula to be used for the HICP is made at two levels: the level of the macro-formula; that is, the choice between a chained index with annual links and a ﬁxed base index with links up to ﬁve years.
the level of the micro-formula; within each level there is the issue of reference period, both for prices and for weights.

4.2.1 Macro index formula The HICP is required to be a Laspeyres-type index.13 Although the HICPs produced by the EU Member States diﬀer in detail, they can all be broadly described as Laspeyres-type indices.
They are all price indices in which the month-to-month movements in prices are measured as an average of price indices using expendi- ture weights which are an appropriate reﬂection of the consumption pattern of the consumer population in the weight reference period.
In practice, there are three types of base period used in the construction of HICPs: the base period to which the volumes of the current expenditure weights refer (‘‘weight reference period’’); the base period from which the current price change is measured, i.e.

the time reference of the prices used for the valuation of the volumes in the current weights (‘‘index reference period’’); the period in which the index base is set to 100 (‘‘index base period’’).
The HICP is, depending on to the macro-formula applied in practice for its computation, potentially a chained index.

It should be stressed that this is the equivalent chain form of the ﬁxed base index which simply allows chained and ﬁxed indices to be expressed by a common formula.
The chaining becomes eﬀective if and only if there are changes to the weights currently used, for instance on the grounds of the review as required by the HICP standard on the quality of HICP weightings.14 In practice, some EU Member States compile ﬁxed base HICPs, while others compute chain HICPs with annual weight updating.

In order to obtain a set of HICPs with sub-indices allowing for consistent aggre- gations, it is necessary to present the HICPs as if they were all computed with the same formula.
Hence, it was necessary to apply a common index base period and reference period.

By mid-2003, the HICP Framework Regulation deﬁned the common index base period with 1996=100.

In order to obtain also a common index reference per- iod, the weights are ‘‘price updated’’ to each December.

4.2.2 Elementary aggregates An HICP legal standard15 deﬁnes elementary aggre- gates by reference to the expenditure or consumption 13 Council Regulation (EC) No.

2494/95, Article 9.

14 Commission Regulation (EC) No.


2454/97.

15 Commission Regulation (EC) No.

1749/96.

HARMONIZED INDICES OF CONSUMER PRICES (EUROPEAN UNION) covered by the most detailed level of stratiﬁcation of the HICP, and within which reliable expenditure informa- tion is not available for weighting purposes.

An ele- mentary aggregate index is a price index for an ele- mentary aggregate comprising only price data.
For the HICPs the ratio of geometric mean prices or the ratio of arithmetic mean prices are the two formulae which should be used within elementary aggregates.

The arithmetic mean of price relatives may only be applied in exceptional cases and where it can be shown that it is comparable.
4.2.3 The level at which macro-aggregation changes into elementary aggregation The level of elementary aggregation interacts with other design features such as sampling procedures and the availability of weighting information.

Depending on the sources of the weights used, elementary aggregation may start at diﬀerent levels in diﬀerent countries in the product, geographic and outlet hierarchies.
Diﬀerences in national practices can aﬀect the resulting HICPs but this issue was, in the ﬁrst instance, not considered to be a priority for harmonization and no action has been undertaken up until now.

This issue is likely to be taken up again as the harmonization process develops.

4.3 Sampling of prices There are three important sampling dimensions to take into account: the item dimension; the outlet dimension; the regional dimension.
Each of these dimensions may in turn be divided into sampling stages.

In the product dimension, a selection or sample of representative items is sometimes ﬁrst made within the national statistical oﬃce, and then a further selection is made by price collectors in the ﬁeld.

In the outlet dimension, there is often ﬁrst a selection of geographical areas, followed by a sample of outlets being taken within each of them.
Random sampling is not easily achieved when it comes to the collection of prices for a CPI and in practice most EU Member States follow purposive sampling procedures for their HICPs.

Irrespective of the sampling method used, small eﬀective sampling sizes may lead to random errors of a size that in itself may constitute a comparability problem.
For Member States using purposive sampling, the elementary aggregates and of prices numbers of within the elementary aggregates give an indication of the degree of coverage of the universe of outlets and items.

HICPs should be constructed from target samples which take into account the weight of each Classiﬁca- tion of Individual Consumption according to Purpose (COICOP)/HICP category.
HICPs which have suﬃcient elementary aggregates to represent the diversity of items within the category and suﬃcient prices within each elementary aggregate to take account of the variation of price movements in the population are regarded as reliable and comparable.

Concerning the replacement of products and outlets, a replacement may take place because an item or an outlet has disappeared from the market or because an item or an outlet is no longer considered to be representative.
Market developments with regard to products and outlets are likely to constitute an important source of non-comparability.

However, there is an important interaction with the choice of the index formula and sampling practices here.

For example, the use of tight as opposed to loose product speciﬁcations may lead to quite diﬀerent issues in respect to quality adjustment.
5 Speciﬁc HICP standards 5.1 Timing of entering purchaser prices into the HICP Diﬀerences between Member States in the time of entering purchaser prices into the HICP may be parti- cularly important for products where there is a sig- niﬁcant time diﬀerence between the time of purchase, payment, or delivery and the time of consumption.
An HICP legal standard16 harmonized practices and made the compilation of the HICP more transparent by setting down detailed rules for the timing of entering purchaser prices into the HICP.

It uses ESA 95 as the source of its deﬁnitions and it is consistent with the ESA 95 deﬁnitions in as far as they are consistent with the purposes of the HICP.
In particular, ESA 95 states that goods and services should in general be recorded when the payables are created, that is, when the purchaser incurs a liability to the seller.

However, expenditure on services in ESA 95 is recorded when the delivery of the service is com- pleted.

In the HICP, volumes are generally valued at purchaser prices following the acquisition principle.
This implies for the purposes of the HICP that prices for goods shall be entered into the HICP for the month in which they are observed, and that prices for services shall be entered into the HICP for the month in which the consumption of the service at the observed prices can commence.

5.2 The treatment of price reductions An HICP legal standard17 lays down detailed rules for the treatment of price reductions.

The standard reﬂects what was common practice in many EU Member States.
The standard deﬁnes the general principles under which transient reductions in prices should be taken into account.
It requires that price reductions must be: (i) attributable to the purchase of an individual good or service; (ii) available to all potential consumers with no special conditions attached; (iii) known to the buyer at the time when he or she entered into the agreement to buy the product concerned; and (iv) claimable at the time of purchase or within such a time period from the actual purchase that they might be expected to have a 16 Commission Regulation (EC) No.

2601/2000.
2601/2000.

17 Commission Regulation (EC) No.

2602/2000.

457 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE signiﬁcant inﬂuence on the quantities buyers are willing to buy.

Further guidelines supplement the legal standard, giving advice on how various price reduction schemes should be treated, such as: sales prices (e.g.
stock-clearing sales and closing-down sales, seasonal sales, end of range or line sales, and damaged, shop-soiled or defective goods); credit and payment arrangements: a zero-interest loan when buying a new durable; inducements in the form of extra quantities or gifts; discounts available only to a restricted group of households; regular rebates or refunds (e.g.

bottle deposits); irregular rebates or refunds (e.g.

loyalty rebates schemes and cards).


5.3 Missing observations In order to ban practices which can lead to serious biases, EU Member States are asked to maintain and provide a statement of their target sample from month to month.

Where prices are not observed, they must be estimated by an appropriate procedure.

A legal standard18 deals with minimum standards for price observation.
Where the target sample requires monthly price observation, but observation fails because of the non-availability of the item or for any other reason, estimated prices may be used for the ﬁrst or second month but replacement prices shall be used from the third month.
5.4 Quality adjustment For the HICPs, quality change is said to occur whenever the Member State judges that a change in speciﬁcation has resulted in a signiﬁcant diﬀerence in utility (or functionality) to the consumer between a new variety or model of a good or service and the good or service previously selected for pricing.

A quality change does not arise when there is a comprehensive revision of the HICP sample.
Quality adjustment is deﬁned as the procedure of making an allowance for a quality change by increasing or reducing the observed current or reference prices by a factor or an amount equivalent to the value of that quality change.
Under the HICP legal standard19 EU Member States are required to examine their quality adjustment pro- cedures and to avoid the so-called automatic linking method, which is equivalent to the assumption that the diﬀerence in price between two successive models is wholly attributable to a diﬀerence in quality.

Where quality changes occur, EU Member States should make appropriate quality adjustments based on explicit estimates of the value of the quality change.
In the absence of national estimates, EU Member States should use Eurostat estimates where these are available and relevant.

Eurostat should assist in this process 18 Commission Regulation (EC) No.

1749/96, Article 6.

19 Commission Regulation (EC) No.

1749/96.

458 by setting up a database of quality change estimates themselves and provided both by Member States from other sources.
Where no estimates are available, price changes should be estimated as the whole diﬀer- ence between the price of the substitute and that of the item it has replaced.

Furthermore, Member Sates are required to monitor the incidence of quality changes and the adjustments made in order to demonstrate their compliance.
Despite the existing legal standards, diﬀerences between HICPs may arise because the same changes in the physical characteristics of an item are still perceived and treated in diﬀerent ways in diﬀerent countries.

This is not to say that the same quality characteristic must be valued to the same extent in diﬀerent EU Member States, only that the principles and procedures for valuation should be harmonized.
In practice, diﬀerences in quality adjustment procedures between countries may not average out across the goods and services covered by the indices.

On the contrary, they are likely to cumulate to diﬀerences well in excess of 0.1 percentage points.

Quality adjustment is one of the most, if not the most, intractable harmonization issues for the HICP.
Eurostat and the EU Member States are currently involved in both general and conceptual discussions of methods and in the examination of the results of empirical studies of quality change and quality adjustment.

5.5 Rejected price observations in general, HICP guidelines concerning the rejection of price observations specify the procedures to be followed for the validation and adjustment of price observations.

the prices The guidelines require that, reported by the price collectors should be accepted.
Rejection or adjustment of reported prices, for example the correction of an unusually high or low price change, should not be carried out by automatic procedures, but only by reference to speciﬁc information on the indi- vidual price observation, such as a repeated observa- tion.

If, following a validation procedure, the reported price has nevertheless still to be rejected, the rejected price should be treated according to the rules for missing observations.
The guidelines leave it to the EU Member States to apply methods other than the speciﬁed methods.

Where an EU Member State does not use the described meth- ods, Eurostat may request it to show that the resulting HICP does not diﬀer systematically from an HICP constructed in line with the described methods by more than 0.1 percentage points on average, taking one year against the previous year.

5.6 Newly signiﬁcant goods and services What is meant by the term ‘‘new goods’’ is not always precise.
In particular, there is no sharp dividing-line between new models and varieties of previously existing products and genuinely new innovative products which fulﬁl needs that could not be fulﬁlled before.
Neither the formulae used to calculate the index, nor the frequency of renewing the basket of goods and ser- vices, can fully address the basic problem: the risk of bias if the introduction of new models and varieties is HARMONIZED INDICES OF CONSUMER PRICES (EUROPEAN UNION) used as an occasion to implement price increases or decreases.
The HICP Framework Regulation20 contains a requirement to maintain the relevance of HICPs, meaning that steps must be taken to ensure that HICPs keep broadly in step with each other and are up to date in terms of market developments.
The HICP legal standard concerning newly signiﬁcant goods and ser- vices (NSGS) aims to ensure that new products are incorporated in the HICP as soon as they achieve a sales volume of one part per thousand of total consumers’ expenditure in the EU Member State.21 The term ‘‘newly signiﬁcant’’ can be interpreted in a wide sense, that is in the sense of being new to the index.
There are two ways whereby new products are introduced into the HICP if they have gained a sig- niﬁcant part of consumption: replacement: the new product replaces an already existing product that has lost importance, so a more up-to-date representative is brought into the sample; addition: the new product is brought into the index in addition to the products which are already covered, as representative for a purpose not yet represented in the index.
The HICP standard on NSGS relates to additions, not to replacements.

Additions are brought into the index for two main reasons: a new product (e.g.

mobile phones), which had not been represented in the index, and would not nor- mally be considered as a replacement because it was radically diﬀerent from the existing products.
It would be added as a new category within an existing category; a product had been previously available, but not explicitly represented in the index because the con- sumption of the product was too low.

Inclusion is not undertaken as a replacement within a category, but by adding a new category within an existing category.
In the case of additions, the price of the new product is collected in addition to the already observed products; and the minimum standard on NSGS oﬀers the fol- lowing treatments: either adjust the weights of the relevant category of COICOP/HICP; or adjust the weights within the relevant category of COICOP/HICP; or assign part of the weight speciﬁcally to the new pro- duct (i.e.

below the 4-digit level of COICOP/HICP).


5.7 Tariff prices Many tariﬀ prices faced by consumers relate to pro- ducts which are or have been regulated by government, or are or have been provided in a monopoly or a monopoly-like situation.

Changes are, however, taking place in many EU countries in such markets, as the 20 Council Regulation (EC) No.

2494/95.

21 Commission Regulation (EC) No.

1749/96, Article 2(b).


markets are opened up, and it is important that the impact on consumer inﬂation is appropriately captured in the HICPs since such products account for a large proportion of the total expenditure.

HICP sub-indices involving tariﬀ prices are, in prac- tice, often obtained centrally or directly from suppliers such as major retail chains, or computed by the Member States based on data on tariﬀ prices and their underlying consumption patterns provided by suppliers.
The requirement for the HICP was not only to determine what statistical standards are required to ensure that EU Member States measure the same price change in a comparable and reliable way, but also to give such legal powers as are necessary to ensure that Member States are in position to have access to the data they need.
An HICP legal standard22 lays down minimum stan- dards for the treatment of tariﬀs: It clariﬁes the obligation of suppliers to provide Member States with the necessary data.23 It deﬁnes the procedure to follow in the case of changes in the tariﬀ structure.

The HICP approach is based on the Laspeyres ﬁxed basket concept, with consumption patterns as up to date as necessary to determine the immediate impact of the tariﬀ change on the index population.
The HICP should reﬂect the price change on the basis of the changed expenditure of maintaining the consumption pattern chosen by households prior to the given change in the tariﬀ.

The aim of this principle is to avoid showing the changes in the con- sumption pattern because of a change in a tariﬀ.

5.8 Insurance The initial coverage of the HICP included home- contents insurance and motor insurance.
Since January 2000, the HICPs have also covered all insurance con- nected with the dwelling which is typically paid by the tenant, not only contents insurance, and private health, civil liability and travel insurance.24 Life insurance is excluded from the coverage of the HICP, as it is con- sidered as a household saving.

In 1997, an HICP legal standard25 stated that the weights and prices for insurance should be measured net of claims.
However, a price index for gross premiums may be used as a proxy or estimate for changes in the ‘‘prices’’ of net premiums.

As the initial legal standard still left scope for some procedural diﬀerences, a further standard26 was adopted in 1999.

Following the HFMCE concept, the standard retained the net concept for insurance, reﬂecting the use of the HICP as a measure of consumer price inﬂation in terms of prices actually charged.
EU household budget surveys cover all expenditure, including expenditure ﬁnanced out of claims.

Hence, the use of the net concept avoids the possibility of double 22 Commission Regulation (EC) No.

2464/98.

23 The legal basis is provided by Council Regulation (EC) No.

2494/95.

24 Council Regulation (EC) No.

1687/98.

25 Commission Regulation (EC) No.

2214/97.

26 Commission Regulation (EC) No.

1617/1999.

459 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE counting or gaps.
It also ensures that the overall HICP measures the change in the price for the insurance ser- vice, and that the other sub-indices, in particular those for the purchase and the repair of vehicles, major household appliances and other durables, measure the change in price for the repair and replacement of such products.

5.8.1 Weights for insurance According to the HICP legal standard,27 weights should reﬂect the so-called service charge.
This is deﬁned as follows: Gross insurance premiums (net of insurance tax) +/– Changes in the actuarial provisions other than for life-insurance risks = Actual premiums earned + Premium supplements Claims due +/– Changes in technical provisions against out- standing life insurance risks = Implicit service charge (net of insurance tax) (+ Insurance tax) = Implicit service charge The actuarial provisions in the above deﬁnition are the technical provisions for outstanding risks which exist for almost any insurance type.
The HICP legal standard also states that the weights should be based on average expenditure over three years.

This is aimed at using more stable estimates of the service charge and mini- mizing the risk of negative weights.

According to ESA 95, payments as a result of claims are treated as current transfers from the insurance companies to policy-holders and other parties to the claim and, therefore, enter into households’ disposable income.
The HICP standard28 on the treatment of insurance requires explicitly that the weights of other sub-indices (e.g.

those for the purchase or repair of vehicles, for major household appliances and other durables) should include all expenditure ﬁnanced out of claims where it is incurred by or on behalf of the household sector.

If, for example, a damaged car is repaired then the expenditure should be reﬂected in the weight for car maintenance and repairs.
The same holds true if the repair is paid directly by the insurance com- pany, since the latter is considered to be acting on the policy-holder’s behalf.

5.8.2 Prices for insurance As the net concept of insurance is not applicable at the individual consumer level for pricing in practice, and because the information is not available each month, gross premiums are followed.

In practice, gross pre- miums (or the value of the insured good) are often indexed by the CPI or by other price or cost indices.
This eﬀect should be reﬂected in the HICP; the gross insurance premiums should not be adjusted to exclude this indexation.

27 Commission Regulation (EC) No.

1617/1999.

28 Commission Regulation (EC) No.

1617/1999.

460 Supplementary guidelines on insurance are in the process of being elaborated.

They will aim at providing practical guidance on some further technically diﬃcult areas.
5.9 Health, education and social protection services Several HICP legal standards concern health, educa- tion and social protection services.29 The standards state that the purchaser prices of goods and services in the health, education and social protection sectors to be used in the HICP should, in accordance with the usual approach and with ESA 95, concern the amounts to be paid by consumers net of reimbursements.
Reimburse- ments are deﬁned as payments to households by gov- ernment units, social security administrations or non- proﬁt institutions serving households, that are made as direct consequences of purchases of individually speci- ﬁed goods and services, initially paid for by households.

Payments of claims to households by insurance com- panies do not constitute reimbursements.
The HICP sub-indices concerned should be calculated using a formula consistent with the Laspeyres type for- mula used for other sub-indices, i.e., they should reﬂect the price change on the basis of the changed expenditure of maintaining the consumption pattern of households and the composition of the consumer population in the base or reference period.
In accordance with the Las- peyres principle and the HICP standards on tariﬀ prices, changes in purchaser prices, which reﬂect changes in the rules determining them, should be shown as price changes in the HICP, as should changes in the purchaser prices resulting from changes in purchasers’ incomes.

If EU Member States choose to use a procedure which diﬀers from that described above, they are required to describe the procedure before it is used so that it may be assessed against the legal standard.
5.10 Financial services EU Member States had traditionally followed diﬀer- ent practices in the measurement of prices for ﬁnancial services in their national CPIs and applied diﬀerent methods for deﬁning the weights.

There was scope for non-comparability by excluding service charges expressed as a proportion of transaction values.

A har- monized methodology for the treatment of such charges was thus considered necessary.
The HICP legal standard concerning prices expressed as a proportion of transaction values30 provides some clariﬁcations, in particular with respect to the coverage of administrative charges of ‘‘private pension funds and the like’’ and ‘‘estate agent fees’’.

The standard says that where service charges are deﬁned as a proportion of the transaction value, the purchaser prices should be deﬁned as the propor- tion itself, multiplied by the value of a representative 29 Commission Regulation (EC) No.
1749/96, as amended by Council Regulation (EC) No.

1687/98, requires extended coverage in the health, education and social protection sectors.

Council Regulation (EC) No.

2166/1999 deﬁnes the methodological details.

30 Commission Regulation (EC) No.

1920/2001.

HARMONIZED INDICES OF CONSUMER PRICES (EUROPEAN UNION) unit transaction in the base or reference period.

The HICP should include charges expressed as a ﬂat fee or ﬂat rate but exclude interest payments and interest-like charges.
Changes in purchaser prices which reﬂect changes in the rules determining them, as well as changes in the purchaser prices resulting from changes in the values of the representative unit transactions, should be shown as price changes in the HICP.

The change in the values of the representative unit trans- actions may be estimated by the change in a price index which represents appropriately the unit transactions concerned.
Supplementary guidelines on ﬁnancial services are in the process of being elaborated.

These guidelines aim at providing practical guidance on some technically diﬃ- cult issues in this area.

5.11 Data processing equipment Prices for data processing equipment tend to diﬀer very much from the development of the all-items HICP.

At the same time, there has been a considerable increase in the relative importance of in HFMCE.
In view of these circumstances and the sig- niﬁcant diﬀerences in national practices in the treatment of data processing equipment, there was a need to establish some minimum common guidelines for the HICPs.

such equipment The HICP guidelines on data processing equipment specify that Member States should cover personal com- puters and associated items in their HICPs.

Where direct sellers are a signiﬁcant source of supply, they should be covered in addition to other outlets.
Prices may be obtained from magazines instead of, or as well as, direct observation in retail outlets.

The HICP standards concerning the annual review of the weights31 are extended to verifying the weights to the level of the major components of the sub-index for data processing equipment.

Those major compo- nents should comprise a component index for personal computers.

5.12 Owner-occupied housing Measurement of the services provided by owner- occupied housing is a notoriously diﬃcult issue for CPIs.
It is sometimes stated that these services are not an issue that should concern consumer price indices, and sometimes that they are an issue and so should be covered, but that practical solutions con- cerning how they should be measured are elusive.

In the HICPs, the imputed prices for the services provided by owner-occupied housing are currently excluded.
The exclusion of imputed services relating to owner-occupied housing has removed a potentially very important source of non-comparability between HICPs.

It should be noted that, for their national CPIs, EU Member States use a variety of methods–for example, some use an approach involving imputed rents, some include mortgage interest in their CPI, while others entirely exclude the shelter costs of owner-occupiers.
Other monetary consumption expenditures related to owner-occupied housing, for example, expenditure on minor property maintenance and repairs, are included in the HICP.

Currently a price index based on the net acquisitions of housing by the consumers is being piloted for possible inclusion in the HICP in future.

It will be compiled separately from the HICPs on an experimental basis before any decision is made to incorporate it within the HICPs.
5.13 Revisions Since the main purpose of the HICP is to inform the monetary policy for the euro-zone by the ECB, and the HICP is a revisable index, a clear and transparent policy with respect to revisions is of paramount importance.

Also, for the HICP harmonization process, a decision was necessary on how to implement improvements whilst at the same time minimizing the diﬃculties caused to users by introducing discontinuities into the pub- lished HICP series.
A legal standard32 states that the published HICP series may be revised for mistakes, new or improved information, and changes in the system of harmonized rules.

In particular: Mistakes should be corrected and any revisions that may result from such corrections should be imple- mented without unnecessary delay.
New or improved information, for example a more up-to-date weighting structure, may result in revisions which should be implemented, provided that Eurostat does not oppose the timing of the revisions to be made.

 Changes in the system of harmonized rules should not require revisions of published HICPs unless otherwise stated in the particular implementing measure.

The impact of such changes should be assessed.


Only if it is likely that the impact is signiﬁcant should the impact then be estimated for each of the 12 following months, starting with the index for January in which the change takes place.
6 Links between the HICP and National Accounts deﬂator of HFCE 6.1 Monetary expenditures There are some diﬀerences between the concept of household ﬁnal consumption expenditure (HFCE) used in national accounts and that of household ﬁnal mon- etary consumption expenditure (HFMCE) used for the HICP.

The HICP covers only the part of HFCE that is considered relevant for monetary policy purposes, that is, only the part which involves actual monetary transactions.
Imputed expen- in particular imputed services provided by ditures, owner-occupied shelter, are thus excluded from the HICP.

to inﬂation measurement 31 Commission Regulation (EC) No.

2454/97.

32 Commission Regulation (EC) No.

1921/2001.

461 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE 6.2 Domestic concept For the HICP, HFMCE is deﬁned on the domestic concept, while HFCE in national accounts is deﬁned on the national concept.
Thus HFMCE in the HICPs excludes household ﬁnal monetary consumption ex- penditure by residents abroad, outside the economic territory of the EU Member State, but includes house- hold ﬁnal monetary consumption expenditure by non- residents on the economic territory of the EU Member State.

HFMCE includes extraterritorial enclaves such as embassies and foreign military bases situated within the EU Member State but excludes territorial enclaves situated in the rest of the world.
6.3 Imputed expenditures and owner-occupied housing As noted above, HFMCE excludes imputed services from owner-occupied housing.

Also excluded are incomes in kind and own ﬁnal consumption, to the extent that they are covered within HFCE, because they also do not involve monetary transactions.

6.4 Life insurance and pension funds HFMCE excludes the service charge of life insurance and the administrative charges of private pension funds.
6.5 Commissions HFMCE excludes commissions to estate agents in connection with the sale or purchase of non-ﬁnancial assets.

It includes payments for services of housing agents in connection with rental transactions.

6.6 Games of chance, prostitution and narcotics On price measurement grounds, HFMCE is deﬁned to exclude games of chance, prostitution and narcotics.

6.7 Time of recording Both in HICPs and ESA 95, volumes are generally valued at purchaser prices, following the acquisition principle.
However, expenditure on services in ESA 95 is recorded when the delivery of the service is completed, whilst service prices are recorded in the HICP in the month for which consumption at the observed prices can commence.

6.8 Household consumption deﬂator The points listed above can result in diﬀerences between the national accounts deﬂator of HFCE and the HICP.

Furthermore, there are diﬀerences between these two price measures regarding the index formulae and the weighting schemes applied.
While the HICP is deﬁned as a Laspeyres type price index, national accounts deﬂators are Paasche type indices.

Deﬂators are used to derive volume indices that are of the Laspeyres type.

7 Release and timeliness of the HICPs 7.1 Full HICPs between 17 and 19 days after the end of the month in question.

This schedule has advanced signiﬁcantly since the HICP was ﬁrst published, as a result of a series of improvements to timeliness made in both the EU Member States and at Eurostat.
7.2 Flash estimate of the MUICP Eurostat also publishes each month a ﬂash estimate for the MUICP – the HICP for the euro-zone as a whole.

This ﬂash estimate is based on the results from the ﬁrst countries to publish their national estimates and on energy price data.

It gives an early indication of what the MUICP is likely to show when the full data set is available.
The estimation procedure combines his- torical information with partial information on price developments in the most recent months to give a total index for the euro-zone.

No detailed breakdown is available.

Over the two years up to June 2003, the ﬂash estimate exactly anticipated the full estimate 14 times, eight times diﬀered by 0.1, and twice diﬀered by 0.2 – the last time in April 2002.

The MUICP ﬂash estimate is generally released on the last working day of the month in question.
7.3 Data The HICP data which are released each month cover the price indices themselves, annual average price indices and rates of change, and monthly and annual rates of change.

None of these are seasonally adjusted.

As well as the all-items HICPs, the full range of around 100 COICOP/HICP indices for diﬀerent goods and services are made available.
The main headings are as follows: food; alcohol and tobacco; clothing; housing; household equipment; health; transport; communications; recreation and culture; education; hotels and restaurants; miscellaneous.
In addition, a series of special aggregates is released, including, for example: the MUICP, excluding energy; the MUICP, excluding energy, food, alcohol and tobacco; the MUICP, excluding unprocessed food; the MUICP, excluding energy and seasonal goods; the MUICP, excluding tobacco.

The weights for the component goods and services and the individual countries are also made available.
The full set of HICPs is published each month according to a pre-announced schedule – in general All of the HICPs, including the complete list of component indices and special aggregates, are accessible 462 HARMONIZED INDICES OF CONSUMER PRICES (EUROPEAN UNION) via the Eurostat web site33 and Euro-indicators web site.34 The Euro-indicators web site gives quick access to the latest headline ﬁgures and most important sub-indices.
The Eurostat web site also gives access to the monthly news releases, more detailed data, and contact points in many countries through which comprehensive data can be obtained.

7.4 Metadata The Eurostat web site also gives access to the com- pendium of HICP reference documents,35 which con- tains detailed reports on the functioning of the HICP, in addition to the HICP legal standards and guidelines, and some technical notes.
8 Agenda for further harmonization The progress made on the harmonization of CPIs does not mean that development is at an end.

There are several major issues where further harmonization will still be necessary.

Currently, work is in progress on: Quality adjustment and sampling: Eurostat and the EU Member States are following up an action plan concerning this subject.
The aim is to agree on some more concrete best practices for a range of speciﬁc goods and services, in particular for cars, consumer durables, books and CDs, clothing, computers and existing HICP telecommunications standard which addressed this issue in 1996 was only a ﬁrst step – it is not in itself a suﬃcient guarantee of full comparability.

services.
services.

The Owner-occupied housing: the imputed expenditures for the consumption of the service provided by owner- occupied housing are currently excluded from the HICPs.

Pilot calculations are being carried out using an approach based on the acquisition prices of housing that is new to the household sector – mainly newly constructed dwellings.

Indices will be compiled separately from the HICPs on an experimental basis before any decision is made to incorporate them within the HICPs.
Other issues currently on the agenda include: minimum sampling standards in the ﬁeld of price collection; more comprehensive systems to assess EU Member States’ compliance with the existing Regulations and other guidance.
More comprehensive quality assur- ance of the HICP compilation process in the widest sense is needed; support for those countries seeking to join the EU, the Acceding and Candidate Countries, to ensure that their HICPs are fully comparable; the consolidation of the legal framework for HICPs, and the production in due course of a methodological manual to assist both compilers and users.
33 http://europa.eu.int/comm/eurostat/ 34 http://europa.eu.int/comm/euroindicators/ 35 http://europa.eu.int/comm/eurostat/Public/datashop/print-catalogue/ EN?catalogue=Eurostat&product=KS-AO-01-005-_-I-EN 9 Regulations concerning HICPs (as at June 2003) Council Regulation (EC) No 2494/95 of 23 October 1995 concerning Harmonized Indices of Consumer Pri- ces (OJ L 257, 27.10.1995, p. 1).

Commission Regulation (EC) No.


1749/96 of 9 Sep- tember 1996 on initial implementing measures for Council Regulation (EC) No.

2494/95 concerning Harmonized Indices of Consumer Prices (OJ L 229, 10.9.1996, p. 3).

Commission Regulation (EC) No.

2214/96 of 20 November 1996 concerning Harmonized Indices of Consumer Prices: transmission and dissemination of sub-indices of the HICP (OJ L 296, 21.11.1996, p. 8).

Commission Regulation (EC) No.


2454/97 of 10 December 1997 laying down detailed rules for the implementation of Council Regulation (EC) No.

2494/ 95 as regards minimum standards for the quality of HICP weightings (OJ L 340, 11.12.1997, p. 24).

Council Regulation (EC) No.

1687/98 of 20 July 1998 amending Commission Regulation (EC) No.

1749/96 concerning the coverage of goods and services of the Harmonized Index of Consumer Prices (OJ L 214, 31.7.1998, p. 12).

Council Regulation (EC) No.


1688/98 of 20 July 1998 amending Commission Regulation (EC) No.

1749/96 concerning the geographic and population coverage of the Harmonized Index of Consumer Prices (OJ L 214, 31.7.1998, p. 23).

Commission Regulation (EC) No.

2646/98 of 9 December 1998 laying down detailed rules for the implementation of Council Regulation (EC) No.

2494/ 95 as regards minimum standards for the treatment of tariﬀs in the Harmonized Index of Consumer Prices (OJ L 335, 10.12.1998, p. 30).
Commission Regulation (EC) No.

1617/1999 of 23 July 1999 laying down detailed rules for the imple- mentation of Council Regulation (EC) No.

2494/95 as the treatment of regards minimum standards insurance in the Harmonized Index of Consumer Prices and modifying Commission Regulation (EC) No.

2214/ 96 (OJ L 192, 24.7.1999, p. 9).

for Commission Regulation (EC) No.

1749/1999 of 23 July 1999 amending Regulation (EC) No.
2214/96, concerning the sub-indices of the Harmonized Indices of Consumer Prices (OJ L 214, 13.8.1999, p. 1 – corrigenda published in OJ L 214, 13.8.1999, p. 1).

Council Regulation (EC) No.

2166/1999 of 8 October 1999 laying down detailed rules for the implementa- tion of Regulation (EC) No.

2494/95 as regards mini- mum standards for the treatment of products in the health, education and social protection sectors in the Harmonized Index of Consumer Prices (OJ L 266, 14.10.1999, p. 1).
Commission Regulation (EC) No.

2601/2000 of 17 November 2000 laying down detailed rules for the implementation of Council Regulation (EC) No.

2494/ 95 as regards the timing of entering purchaser prices into 463 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE the Harmonized Index of Consumer Prices (OJ L 300, 29.11.2000, p. 14).

Commission Regulation (EC) No.

2602/2000 of 17 November 2000 laying down detailed rules for the implementation of Council Regulation (EC) No.
2494/ 95 as regards minimum standards for the treatment of price reductions in the Harmonized Index of Consumer Prices (OJ L 300, 29.11.2000, p. 16).

Commission Regulation (EC) No.

1920/2001 of 28 September 2001 laying down detailed rules for the implementation of Council Regulation (EC) No.

2494/95 as regards minimum standards for the treatment of ser- vice charges proportional to transaction values in the Harmonized Index of Consumer Prices and amending Commission (EC) No.

2214/96.
2214/96.

(OJ L 261, 29.9.2001, p. 46 – corrigenda published in OJ L 295, 13.11.2001, p. 34).

Commission Regulation (EC) No.

1921/2001 of 28 September 2001 laying down detailed rules for the implementation of Council Regulation (EC) No.

2494/ 95 as regards minimum standards for revisions of the Harmonized Index of Consumer Prices and amend- ing Regulation (EC) No.

2602/2000 (OJ L 261, 29.9.2001, p. 49 – corrigenda published in OJ L 295, 13.11.2001, p. 34).
All these legal acts can be found on the following web site: http://europa.eu.int/celex/ 464 01 01.1 01.2 02 02.1 02.2 02.3 03 03.1 03.2 04 04.5 05 05.1 05.2 05.3 05.4 05.5 05.6 Annex 2 Classiﬁcation of Individual Consumption according to Purpose (COICOP)-Extract COICOP: BREAKDOWN OF INDIVIDUAL CON- SUMPTION EXPENDITURE OF HOUSEHOLDS BY DIVISION AND GROUP FOOD AND NON-ALCOHOLIC BEV- ERAGES FOOD NON-ALCOHOLIC BEVERAGES ALCOHOLIC BEVERAGES, TOBACCO AND NARCOTICS ALCOHOLIC BEVERAGES TOBACCO NARCOTICS CLOTHING AND FOOTWEAR CLOTHING FOOTWEAR HOUSING, WATER, ELECTRICITY, GAS AND OTHER FUELS ACTUAL RENTALS FOR HOUSING IMPUTED RENTALS FOR HOUSING 04.1 04.2 04.3 MAINTENANCE AND REPAIR OF THE DWELLING 04.4 WATER SUPPLY AND MISCELLANEOUS SERVICES RELATING TO THE DWELLING ELECTRICITY, GAS AND OTHER FUELS FURNISHINGS, HOUSEHOLD EQUIP- MENT AND ROUTINE HOUSEHOLD MAINTENANCE FURNITURE AND FURNISHINGS, CAR- PETS AND OTHER FLOOR COVERINGS HOUSEHOLD TEXTILES HOUSEHOLD APPLIANCES GLASSWARE, TABLEWARE AND HOUSEHOLD UTENSILS TOOLS AND EQUIPMENT FOR HOUSE AND GARDEN GOODS AND SERVICES FOR ROUTINE HOUSEHOLD MAINTENANCE HEALTH 06 06.1 MEDICAL PRODUCTS, APPLIANCES 06.2 06.3 07 07.1 07.2 07.3 08 08.1 08.2 08.3 AND EQUIPMENT OUTPATIENT SERVICES HOSPITAL SERVICES TRANSPORT PURCHASE OF VEHICLES OPERATION OF PERSONAL TRANS- PORT EQUIPMENT TRANSPORT SERVICES COMMUNICATION POSTAL SERVICES TELEPHONE AND TELEFAX EQUIPMENT TELEPHONE AND TELEFAX SERVICES 09 09.1 09.2 09.3 09.4 09.5 09.6 10 10.1 10.2 10.3 10.4 10.5 11 11.1 11.2 12 12.1 12.2 12.3
 12.4 12.5 12.6 12.7 RECREATION AND CULTURE AUDIO-VISUAL, PHOTOGRAPHIC AND INFORMATION PROCESSING EQUIP- MENT OTHER MAJOR DURABLES FOR RECREATION AND CULTURE OTHER RECREATIONAL ITEMS AND EQUIPMENT, GARDENS AND PETS RECREATIONAL AND CULTURAL SER- VICES NEWSPAPERS, BOOKS AND STATION- ERY PACKAGE HOLIDAYS EDUCATION PRE-PRIMARY AND PRIMARY EDUCA- TION SECONDARY EDUCATION POST-SECONDARY NON-TERTIARY EDUCATION TERTIARY EDUCATION EDUCATION NOT DEFINABLE BY LE- VEL RESTAURANTS AND HOTELS CATERING SERVICES ACCOMMODATION SERVICES MISCELLANEOUS GOODS AND SER- VICES PERSONAL CARE PROSTITUTION PERSONAL EFFECTS N.E.C.
SOCIAL PROTECTION INSURANCE FINANCIAL SERVICES N.E.C.

OTHER SERVICES N.E.C.

COICOP: DEFINITION BY CLASS 01–12 INDIVIDUAL CONSUMPTION EXPENDI- TURE OF HOUSEHOLDS 01 FOOD AND NON-ALCOHOLIC BEVERAGES 01.1 FOOD food products The food products classiﬁed here are those purchased for consumption at home.

The sold for group excludes: immediate consumption away from the home by hotels, restaurants, cafe´ s, bars, kiosks, street vendors, automatic vending machines, etc.
(11.1.1); cooked dishes prepared by restaurants for consumption oﬀ their premises (11.1.1); cooked dishes prepared by catering contractors whether collected by the customer or delivered to the customer’s home (11.1.1); and products sold speciﬁcally as pet foods (09.3.4).
01.1.1 Bread and cereals (ND) – Rice in all forms; – maize, wheat, barley, oats, rye and other cer- eals in the form of grain, ﬂour or meal; 465 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE – bread and other bakery products (crispbread, rusks, toasted bread, biscuits, gingerbread, wafers, waﬄes, crumpets, muﬃns, croissants, cakes, tarts, pies, quiches, pizzas, etc.
); – mixes and doughs for the preparation of bakery products; – pasta products in all forms; couscous; – cereal preparations (cornﬂakes, oatﬂakes, etc.)

and other cereal products (malt, malt ﬂour, malt extract, potato starch, tapioca, sago and other starches).

Includes: farinaceous-based products prepared with meat, ﬁsh, seafood, cheese, vegetables or fruit.

Excludes: meat pies (01.1.2); ﬁsh pies (01.1.3); sweetcorn (01.1.7).

Excludes: soups, broths and stocks containing ﬁsh and seafood (01.1.9).
01.1.4 Milk, cheese and eggs (ND) – Raw milk; pasteurized or sterilized milk; – condensed, evaporated or powdered milk; – yoghurt, cream, milk-based desserts, milk- based beverages and other similar milk- based products; – cheese and curd; – eggs and egg products made wholly from eggs.

Includes: milk, cream and yoghurt containing sugar, cocoa, fruit or ﬂavourings; dairy pro- ducts not based on milk such as soya milk.

Excludes: butter and butter products (01.1.5).


01.1.2 Meat (ND) – Fresh, chilled or frozen meat of: - bovine animals, swine, sheep and goat; - horse, mule, donkey, camel and the like; - poultry (chicken, duck, goose, turkey, guinea fowl); - hare, rabbit and game (antelope, deer, boar, pheasant, grouse, pigeon, quail, etc.

); – fresh, chilled or frozen edible oﬀal; – dried, salted or smoked meat and edible oﬀal (sausages, salami, bacon, ham, paˆ te´ , etc.
); – other preserved or processed meat and meat- (canned meat, meat based preparations extracts, meat juices, meat pies, etc.).

Includes: meat and edible oﬀal of marine mammals (seals, walruses, whales, etc.)

and exotic animals (kangaroo, ostrich, alligator, etc.

); animals and poultry purchased live for consumption as food.

Excludes: land and sea snails (01.1.3); lard and other edible animal fats (01.1.5); soups, broths and stocks containing meat (01.1.9).
Fish and seafood (ND) – Fresh, chilled or frozen ﬁsh; – fresh, chilled or frozen seafood (crustaceans, molluscs and other shellﬁsh, sea snails); – dried, smoked or salted ﬁsh and seafood; – other preserved or processed ﬁsh and sea- food and ﬁsh and seafood-based prepara- tions (canned ﬁsh and seafood, caviar and other hard roes, ﬁsh pies, etc.).

Includes: land crabs, land snails and frogs; ﬁsh and seafood purchased live for consumption as food.
01.1.3 01.1.5 Oils and fats (ND) – Butter and butter products (butter oil, ghee, etc.

); – margarine (including ‘‘diet’’ margarine) and other vegetable fats (including peanut but- ter); – edible oils (olive oil, corn oil, sunﬂower-seed oil, cottonseed oil, soybean oil, groundnut oil, walnut oil, etc.

); – edible animal fats (lard, etc.).

Excludes: cod or halibut liver oil (06.1.1).


01.1.6 Fruit (ND) – Fresh, chilled or frozen fruit; – dried fruit, fruit peel, fruit kernels, nuts and edible seeds; – preserved fruit and fruit-based products.

Includes: melons and water melons.

Excludes: vegetables cultivated for their fruit such as aubergines, cucumbers and tomatoes (01.1.7); jams, marmalades, compotes, jellies, fruit pure´ es and pastes (01.1.8); parts of plants preserved in sugar (01.1.8); fruit juices and syrups (01.2.2).
01.1.7 Vegetables (ND) – Fresh, chilled, frozen or dried vegetables cultivated for their leaves or stalks (aspar- agus, broccoli, cauliﬂower, endives, fennel, spinach, etc.

), for their fruit (aubergines, cucumbers, peppers, pumpkins, tomatoes, etc.

), and for their roots (beetroots, carrots, onions, parsnips, radishes, turnips, etc.

); courgettes, green – fresh or chilled potatoes and other tuber cassava, (manioc, arrowroot, vegetables sweet potatoes, etc.
); Note: ND, SD, D and S denote non-durable goods, semi-durable goods, durable goods and services, respectively.

– preserved or processed vegetables and vege- table-based products; 466 CLASSIFICATION OF INDIVIDUAL CONSUMPTION ACCORDING TO PURPOSE (COICOP) – products of tuber vegetables (ﬂours, meals, ﬂakes, pure´ es, chips and crisps) including frozen preparations such as chipped pota- toes.

Includes: olives; garlic; pulses; sweetcorn; sea fennel and other edible seaweed; mushrooms and other edible fungi.
starches Excludes: potato starch, tapioca, sago and other soups, broths and (01.1.1); stocks containing vegetables (01.1.9); culinary herbs (parsley, rosemary, thyme, etc.)

and spi- ces (pepper, pimento, ginger, etc.)

(01.1.9); vegetable juices (01.2.2).
Sugar, jam, honey, chocolate and confectionery (ND) – Cane or beet sugar, unreﬁned or reﬁned, powdered, crystallized or in lumps; – jams, marmalades, compotes, jellies, fruit pure´ es and pastes, natural and artiﬁcial honey, maple syrup, molasses and parts of plants preserved in sugar; – chocolate in bars or slabs, chewing gum, toﬀees, pastilles and other con- sweets, fectionery products; – cocoa-based foods and cocoa-based dessert preparations; – edible ice, ice cream and sorbet.
Includes: artiﬁcial sugar substitutes.

Excludes: cocoa and chocolate-based powder (01.2.1).

Food products n.e.c.

(ND) – Salt, spices (pepper, pimento, ginger, etc.

), culinary herbs (parsley, rosemary, thyme, etc.

), sauces, condiments, seasonings (mus- tard, mayonnaise, ketchup, soy sauce, etc.

), vinegar; – prepared baking powders, baker’s yeast, dessert preparations, soups, broths, stocks, culinary ingredients, etc.
; – homogenized baby food and dietary pre- parations irrespective of the composition.

01.1.8 01.1.9 Excludes: milk-based desserts (01.1.4); soya milk (01.1.4); substitutes (01.1.8); cocoa-based dessert preparations (01.1.8).

artiﬁcial sugar 01.2.1 Coﬀee, tea and cocoa (ND) – Coﬀee, whether or not decaﬀeinated, roasted or ground, including instant coﬀee; – tea, mate´ and other plant products for infusions; – cocoa, whether or not sweetened, and cho- colate-based powder.
Includes: cocoa-based beverage preparations; coﬀee and tea substitutes; extracts and essences of coﬀee and tea.

Excludes: chocolate in bars or slabs (01.1.8); cocoa-based food and cocoa-based dessert preparations (01.1.8).

01.2.2 Mineral waters, soft drinks, fruit and vegetable juices (ND) – Mineral or spring waters; all drinking water sold in containers; – soft drinks such as sodas, lemonades and colas; – fruit and vegetable juices; – syrups and concentrates for the preparation of beverages.
Excludes: non-alcoholic beverages which are generally alcoholic such as non-alcoholic beer (02.1).

02 ALCOHOLIC BEVERAGES, TOBACCO AND NARCOTICS 02.1 ALCOHOLIC BEVERAGES The alcoholic beverages classiﬁed here are those purchased for consumption at home.

The group excludes alcoholic beverages sold for immediate consumption away from the home by hotels, restaurants, cafe´ s, bars, kiosks, street vendors, automatic vending machines, etc.

(11.1.1).
(11.1.1).

The beverages classiﬁed here include low- or non-alcoholic beverages which are generally alcoholic such as non-alcoholic beer.

02.1.1 Spirits (ND) – Eaux-de-vie, liqueurs and other spirits.

Includes: mead; aperitifs other than wine-based aperitifs (02.1.2).

01.2 NON-ALCOHOLIC BEVERAGES The non-alcoholic beverages classiﬁed here are those purchased for consumption at home.
The group excludes non-alcoholic beverages sold for immediate consumption away from the home by hotels, restaurants, cafe´ s, bars, street vendors, automatic vending kiosks, machines, etc.

(11.1.1).

02.1.2 Wine (ND) – Wine, cider and perry, including sake; – wine-based aperitifs, fortiﬁed wines, cham- pagne and other sparkling wines.

02.1.3 Beer (ND) – All kinds of beer such as ale, lager and porter.
467 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Includes: low-alcoholic beer and non-alcoholic beer; shandy.

02.2 TOBACCO This group covers all purchases of tobacco by households, including purchases of tobacco in restaurants, cafe´ s, bars, service stations, etc.

02.2.0 Tobacco (ND) – Cigarettes; cigarette tobacco and cigarette papers; – cigars, pipe tobacco, chewing tobacco or snuﬀ.

Excludes: other smokers’ articles (12.3.2).


02.3 NARCOTICS 02.3.0 Narcotics (ND) – Marijuana, opium, cocaine and their deri- vatives; – other vegetable-based narcotics such as cola nuts, betel leaves and betel nuts; – other narcotics including chemicals and man-made drugs.

03 CLOTHING AND FOOTWEAR 03.1 CLOTHING 03.1.1 Clothing materials (SD) – Clothing materials of natural ﬁbres, of man- made ﬁbres and of their mixtures.

Excludes: furnishing fabrics (05.2.0).


03.1.2 Garments (SD) – Garments for men, women, children (3 to 13 years) and infants (0 to 2 years), either ready-to-wear or made-to-measure, in all materials (including leather, furs, plastics and rubber), for everyday wear, for sport or for work: - capes, overcoats, raincoats, anoraks, par- trousers, waist- jackets, kas, blousons, coats, suits, costumes, dresses, skirts, etc.
; - shirts, blouses, pullovers, sweaters, cardi- gans, shorts, swimsuits, tracksuits, jogging suits, sweatshirts, T-shirts, leotards, etc.

; - vests, underpants, socks, stockings, tights, petticoats, brassie` res, knickers, slips, gir- dles, corsets, body stockings, etc.

; - pyjamas, nightshirts, nightdresses, house- coats, dressing gowns, bathrobes, etc.

; - baby clothes and babies’ booties made of fabric.
Excludes: articles of medical hosiery such as elasticated stockings (06.1.2); babies’ napkins (12.1.3).

468 03.1.3 Other articles of clothing and clothing acces- sories (SD) – Ties, handkerchiefs, scarves, squares, gloves, mittens, muﬀs, belts, braces, aprons, smocks, bibs, sleeve protectors, hats, caps, berets, bonnets, etc.

; – sewing threads, knitting yarns and acces- sories for making clothing such as buckles, buttons, press studs, zip fasteners, ribbons, laces, trimmings, etc.
Includes: gardening gloves and working gloves; crash helmets for motorcycles and bicycles.

Excludes: gloves and other articles made of rubber (05.6.1); pins, safety pins, sewing nee- dles, knitting needles, thimbles (05.6.1); pro- tective headgear for sports (09.3.2); other pro- tective gear for sports such as life jackets, boxing gloves, body padding, belts, supports, etc.
(12.1.3); watches, jewellery, cuﬀ links, tiepins (12.3.1); walking sticks and canes, umbrellas and para- sols, fans, keyrings (12.3.2).

(09.3.2); paper handkerchiefs 03.1.4 Cleaning, repair and hire of clothing (S) – Dry-cleaning, garments; laundering and dyeing of – darning, mending, repair and altering of garments; – hire of garments.

Includes: total value of the repair service (that is, both the cost of labour and the cost of materials are covered).

Excludes: materials, threads, accessories, etc.


purchased by households with the intention of undertaking the repairs themselves (03.1.1) or (03.1.3); repair of household linen and other household textiles (05.2.0); dry-cleaning, laun- dering, dyeing and hiring of household linen and other household textiles (05.6.2).
03.2 FOOTWEAR 03.2.1 Shoes and other footwear (SD) – All footwear for men, women, children (3 to 13 years) and infants (0 to 2 years) including sports footwear suitable for everyday or lei- sure wear (shoes for jogging, cross-training, tennis, basket ball, boating, etc.).

Includes: gaiters, leggings and similar articles; shoelaces; parts of footwear, such as heels, soles, etc., purchased by households with the intention of repairing footwear themselves.
Excludes: babies’ booties made of fabric (03.1.2); shoe-trees, shoehorns and polishes, creams articles (05.6.1); orthopaedic footwear (06.1.3); game- speciﬁc footwear (ski boots, football boots, golﬁng shoes and other such footwear ﬁtted shoe-cleaning other and CLASSIFICATION OF INDIVIDUAL CONSUMPTION ACCORDING TO PURPOSE (COICOP) with ice-skates, rollers, spikes, studs, etc.)

(09.3.2); shin-guards, cricket pads and other such protective apparel for sport (09.3.2).
03.2.2 Repair and hire of footwear (S) – Repair of footwear; shoe-cleaning services; – hire of footwear.

Includes: total value of the repair service (that is, both the cost of labour and the cost of materials are covered).
Excludes: parts of footwear, such as heels, soles, etc., purchased by households with the intention of undertaking the repair themselves shoe- (03.2.1); polishes, creams and other cleaning articles (05.6.1); repair (09.3.2) or hire (09.4.1) of game-speciﬁc footwear (ski boots, football boots, golﬁng shoes and other such footwear ﬁtted with ice-skates, rollers, spikes, studs, etc.).

04.1.2 Other actual rentals (S) – Rentals actually paid for secondary resi- dences.
Excludes: accommodation services of holiday villages and holiday centres (11.2.0).

04.2 IMPUTED RENTALS FOR HOUSING For coverage see note to (04.1) above.

04.2.1 Imputed rentals of owner-occupiers (S) – Imputed rentals of owners occupying their main residence.

04.2.2 Other imputed rentals (S) – Imputed rentals for secondary residences; – imputed rentals of households paying a reduced rental or housed free.
04 HOUSING, WATER, ELECTRICITY, GAS AND OTHER FUELS 04.3 04.1 ACTUAL RENTALS FOR HOUSING Rentals normally include payment for the use of the land on which the property stands, the dwelling occupied, the ﬁxtures and ﬁttings for heating, plumbing, lighting, etc., and, in the case of a dwelling let furnished, the furniture.

Rentals also include payment for the use of a garage to provide parking in connection with the dwelling.
The garage does not have to be physically contiguous to the dwelling; nor does it have to be leased from the same landlord.

Rentals do not include payment for the use of garages or parking spaces not providing parking in connection with the dwelling (07.2.4).
Nor do they include charges for water supply (04.4.1), refuse collection (04.4.2) and sewage collection (04.4.3); co-proprietor char- ges for caretaking, gardening, stairwell clean- ing, heating and lighting, maintenance of lifts and refuse disposal chutes, etc.

in multi-occu- pied buildings (04.4.4); charges for electricity (04.5.1) and gas (04.5.2); charges for heating and hot water supplied by district heating plants (04.5.5).
04.1.1 Actual rentals paid by tenants (S) – Rentals actually paid by tenants or sub- tenants occupying unfurnished or furnished premises as their main residence.

Includes: payments by households occupying a room in a hotel or boarding house as their main residence.

Excludes: accommodation services of educa- tional establishments and hostels (11.2.0) and elderly persons of (12.4.0).
retirement homes for MAINTENANCE AND REPAIR OF THE DWELLING Maintenance and repair of dwellings are distinguished by two features: ﬁrst, they are activities that have to be undertaken regularly in order to maintain the dwelling in good working order; second, they do not change the dwelling’s performance, capacity or expected service life.
There are two types of maintenance and repair of dwellings: those which are minor, such as interior decoration and repairs to ﬁt- tings, and which are commonly carried out by both tenants and owners; and those which are major, such as replastering walls or repairing roofs, and which are carried out by owners only.

Only expenditures which tenants and owner- occupiers incur on materials and services for minor maintenance and repair are part of individual consumption expenditure of house- holds.
Expenditures which owner-occupiers incur on materials and services for major maintenance and repair are not part of indivi- dual consumption expenditure of households.

Purchases of materials made by tenants or owner-occupiers with the intention of under- taking the maintenance or repair themselves should be shown under (04.3.1).
If tenants or owner-occupiers pay an enterprise to carry out the maintenance or repair, the total value of the service, including the costs of the materials used, should be shown under (04.3.2).
04.3.1 Materials for the maintenance and repair of the dwelling (ND) – Products and materials, such as paints and varnishes, renderings, wallpapers, fabric wall coverings, window panes, plaster, cement, 469 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE putty, wallpaper pastes, etc., purchased for minor maintenance and repair of the dwelling.

Includes: small plumbing items (pipes, taps, joints, etc.

), surfacing materials (ﬂoorboards, ceramic tiles, etc.)
and brushes and scrapers for paint, varnish and wallpaper.

Excludes: ﬁtted carpets and linoleum (05.1.2); hand tools, door ﬁttings, power sockets, wiring ﬂex and lamp bulbs (05.5.2); brooms, scrub- bing brushes, dusting brushes and cleaning products (05.6.1); products, materials and ﬁx- tures used for major maintenance and repair (intermediate consumption) or for extension and conversion of the dwelling (capital for- mation).
Services for the maintenance and repair of the dwelling (S) – Services of plumbers, electricians, carpen- ters, glaziers, painters, decorators, ﬂoor polishers, etc.

engaged for minor main- tenance and repair of the dwelling.

Includes: total value of the service (that is, both the cost of labour and the cost of materials are covered).
Excludes: separate purchases of materials made by households with the intention of undertaking the maintenance or repair them- selves (04.3.1); services engaged for major maintenance and repair (intermediate con- sumption) or for extension and conversion of the dwelling (capital formation).

04.3.2 04.4 WATER SUPPLY AND MISCELLANEOUS SERVICES RELATING TO THE DWELL- ING 04.4.1 Water supply (ND) – Water supply.

Includes: associated expenditure such as hire of meters, reading of meters, standing charges, etc.
Excludes: drinking water sold in bottles or containers (01.2.2); hot water or steam pur- chased from district heating plants (04.5.5).

05 04.4.2 Refuse collection (S) – Refuse collection and disposal.

04.4.3 Sewage collection (S) – Sewage collection and disposal.

04.4.4 Other services relating to the dwelling 05.1 05.1.1 n.e.c.

(S) – Co-proprietor charges for caretaking, gar- cleaning, heating and stairwell dening, 470 lighting, maintenance of chutes, disposal buildings; etc.
lifts and refuse in multi-occupied – security services; – snow removal and chimney sweeping.

Excludes: household services such as window cleaning, disinfecting, fumigation and pest extermination (05.6.2); bodyguards (12.7.0).

04.5 ELECTRICITY, GAS AND OTHER FUELS 04.5.1 Electricity (ND) – Electricity.

Includes: associated expenditure such as hire of meters, reading of meters, standing charges, etc.

04.5.2 Gas (ND) – Town gas and natural gas; – liqueﬁed hydrocarbons (butane, propane, etc.).
Includes: associated expenditure such as hire of meters, reading of meters, storage containers, standing charges, etc.

04.5.3 Liquid fuels (ND) – Domestic heating and lighting oils.

04.5.4 Solid fuels (ND) – Coal, coke, briquettes, ﬁrewood, charcoal, peat and the like.

04.5.5 Heat energy (ND) – Hot water and steam purchased from district heating plants.

Includes: associated expenditure such as hire of meters, reading of meters, standing charges, etc.

; ice used for cooling and refrigeration purposes.
FURNISHINGS, HOUSEHOLD EQUIPMENT AND ROUTINE HOUSEHOLD MAINTENANCE FURNITURE AND FURNISHINGS, CARPETS AND OTHER FLOOR COVERINGS Furniture and furnishings (D) – Beds, sofas, couches, tables, chairs, cup- boards, chests of drawers and bookshelves; – lighting equipment such as ceiling lights, standard lamps, globe lights and bedside lamps; CLASSIFICATION OF INDIVIDUAL CONSUMPTION ACCORDING TO PURPOSE (COICOP) – pictures, sculptures, engravings, tapestries and other art objects including reproduc- tions of works of art and other ornaments; – screens, folding partitions and other furni- ture and ﬁxtures.
Includes: delivery and installation when applicable; base mattresses, mattresses, tata- mis; bathroom cabinets; baby furniture such as cradles, high chairs and playpens; blinds; camping and garden furniture; mirrors, can- dleholders and candlesticks.
Excludes: bedding and sunshades (05.2.0); safes (05.3.1); ornamental glass and ceramic articles (05.4.0); clocks (12.3.1); wall thermometers and barometers (12.3.2); carrycots and pushchairs (12.3.2); works of art and antique furniture acquired primarily as stores of value (capital formation).

05.1.2 Carpets and other ﬂoor coverings (D) – Loose carpets, ﬁtted carpets, linoleum and other such ﬂoor coverings.

Includes: laying of ﬂoor coverings.


Excludes: bathroom mats, rush mats and (05.2.0); antique ﬂoor coverings doormats acquired primarily as stores of value (capital formation).

05.1.3 Repair of furniture, furnishings and ﬂoor cover- ings (S) – Repair of furniture, furnishings and ﬂoor coverings.
Includes: total value of the service (that is, both the cost of labour and the cost of materials are covered); restoration of works of art, antique furniture and antique ﬂoor coverings other than those acquired primarily as stores of value (capital formation).

Excludes: separate purchases of materials made by households with the intention of under- (05.1.1) or taking the (05.1.2); dry-cleaning of carpets (05.6.2).
themselves repair 05.2 HOUSEHOLD TEXTILES 05.2.0 Household textiles (SD) – Furnishing fabrics, curtain material, cur- tains, double curtains, awnings, door cur- tains and fabric blinds; – bedding such as futons, pillows, bolsters and hammocks; – bedlinen such as sheets, pillowcases, blan- travelling rugs, plaids, eiderdowns, kets, counterpanes and mosquito nets; – table linen and bathroom linen such as tablecloths, table napkins, towels and face cloths; – other household textiles such as shopping bags, laundry bags, shoe bags, covers for clothes and furniture, ﬂags, sunshades, etc.
; – repair of such articles.

Includes: cloth bought by the piece; oilcloth; bathroom mats, rush mats and doormats.

Excludes: fabric wall coverings (04.3.1); tap- estries (05.1.1); ﬂoor coverings such as carpets and ﬁtted carpets (05.1.2); electric blankets (05.3.2); covers for motor cars, motorcycles, etc.

(07.2.1); air mattresses and sleeping bags (09.3.2).
05.3 HOUSEHOLD APPLIANCES 05.3.1 Major household appliances whether electric or not (D) – Refrigerators, freezers and fridge-freezers; – washing machines, dryers, drying cabinets, dishwashers, ironing and pressing machines; – cookers, spit roasters, hobs, ranges, ovens and microwave ovens; – air-conditioners, humidiﬁers, space heaters, water heaters, ventilators and extractor hoods; – vacuum cleaners, steam-cleaning machines, carpet shampooing machines and machines for scrubbing, waxing and polishing ﬂoors; – other major household appliances such as safes, sewing machines, knitting machines, water softeners, etc.
Includes: delivery and installation of appliances when applicable.

the Excludes: such appliances that are built into the structure of the building (capital forma- tion).

Small electric household appliances (SD) – Coﬀee mills, coﬀee-makers, juice extractors, can-openers, food mixers, deep fryers, meat grills, knives, toasters, ice cream makers, sorbet makers, yoghurt makers, hotplates, irons, kettles, fans, electric blankets, etc.
Excludes: small non-electric household articles and kitchen utensils (05.4.0); household scales (05.4.0); personal weighing machines and baby scales (12.1.3).

05.3.2 05.3.3 Repair of household appliances (S) – Repair of household appliances.

Includes: total value of the service (that is, both the cost of labour and the cost of materials are 471 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE covered); charges for the leasing or rental of major household appliances.
Excludes: separate purchases of materials made by households with the intention of under- taking the (05.3.1) or (05.3.2).
themselves repair 05.4 GLASSWARE, HOUSEHOLD UTENSILS TABLEWARE AND 05.4.0 Glassware, tableware and household utensils (SD) – Glassware, crystal ware, ceramic ware and china ware of the kind used for table, kitchen, bathroom, toilet, oﬃce and indoor decoration; – cutlery, ﬂatware and silverware; – non-electric kitchen utensils of all materials such as saucepans, stewpots, pressure coo- kers, frying pans, coﬀee mills, pure´ e makers, mincers, hotplates, household scales and other such mechanical devices; – non-electric household articles of all ma- terials such as containers for bread, coﬀee, spices, etc., waste bins, waste-paper baskets, laundry baskets, portable money boxes and strongboxes, towel rails, bottle racks, irons and ironing boards, letter boxes, feeding bottles, thermos ﬂasks and iceboxes; – repair of such articles.
Excludes: lighting equipment (05.1.1); electric household appliances (05.3.2); cardboard tableware (05.6.1); personal weigh- ing machines and baby scales (12.1.3); ashtrays (12.3.2).

(05.3.1) or 05.5 TOOLS AND EQUIPMENT FOR HOUSE AND GARDEN 05.5.1 Major tools and equipment (D) – Motorized tools and equipment such as electric drills, saws, sanders and hedge cut- ters, garden tractors, lawnmowers, cultiva- tors, chainsaws and water pumps; – repair of such articles.
Includes: charges for the leasing or rental of do- it-yourself machinery and equipment.
Small tools and miscellaneous accessories (SD) – Hand tools such as saws, hammers, screw- drivers, wrenches, spanners, pliers, trimming knives, rasps and ﬁles; – garden tools such as wheelbarrows, watering cans, hoses, spades, shovels, rakes, forks, scythes, sickles and secateurs; – ladders and steps; – door ﬁttings (hinges, handles and locks), ﬁttings for radiators and ﬁreplaces, other metal articles for the house (curtain rails, 05.5.2 472 carpet rods, hooks, etc.)
or for the garden (chains, grids, stakes and hoop segments for fencing and bordering); – small electric accessories such as power sockets, switches, wiring ﬂex, electric bulbs, torches, ﬂash- ﬂuorescent lights, hand lamps, electric batteries for general use, bells and alarms; lighting tubes, – repair of such articles.
05.6 GOODS AND SERVICES FOR ROUTINE HOUSEHOLD MAINTENANCE 05.6.1 Non-durable household goods (ND) – Cleaning and maintenance products such as soaps, washing powders, washing liquids, scouring powders, detergents, disinfectant bleaches, softeners, conditioners, window- cleaning products, waxes, polishes, dyes, unblocking agents, disinfectants, insecti- cides, pesticides, fungicides and distilled water; – articles for cleaning such as brooms, scrub- bing brushes, dustpans and dust brushes, dusters, tea towels, ﬂoorcloths, household sponges, scourers, steel wool and chamois leathers; – paper products such as ﬁlters, tablecloths and table napkins, kitchen paper, vacuum cleaner bags and cardboard tableware, including aluminium foil and plastic bin liners; – other non-durable household articles such as matches, candles, lamp wicks, methylated spirits, clothes-pegs, clothes hangers, pins, safety pins, sewing needles, knitting needles, thimbles, nails, screws, nuts and bolts, tacks, washers, glues and adhesive for household use, string, twine and rubber gloves.
tapes Includes: polishes, creams and other shoe- cleaning articles; ﬁre extinguishers for house- holds.
transport equipment Excludes: brushes and scrapers for paint, var- nish and wallpaper (04.3.1); ﬁre extinguishers for (07.2.1); products speciﬁcally for the cleaning and maintenance of transport equipment such as paints, chrome cleaners, sealing compounds and bodywork polishes (07.2.1); horticultural products for the upkeep of ornamental gardens (09.3.3); paper handkerchiefs, toilet paper, toilet soaps, toilet sponges and other products for personal hygiene (12.1.3); cigarette, cigar and pipe lighters and lighter fuel (12.3.2).
05.6.2 Domestic services and household services (S) – Domestic services supplied by paid staﬀ employed in private service such as butlers, CLASSIFICATION OF INDIVIDUAL CONSUMPTION ACCORDING TO PURPOSE (COICOP) cooks, maids, drivers, gardeners, govern- esses, secretaries, tutors and au pairs; – similar services, including babysitting and housework, supplied by enterprises or self- employed persons; – household services such as window cleaning, disinfecting, fumigation and pest extermi- nation; – dry-cleaning, laundering and dyeing of household linen, household textiles and carpets; – hire of carpets, household equipment and household linen.
furnishings, furniture, collection (04.4.3); Excludes: dry-cleaning, laundering and dyeing of garments (03.1.4); refuse collection (04.4.2); sewerage co-proprietor charges for caretaking, gardening, stairwell cleaning, heating and lighting, maintenance of lifts and refuse disposal chutes, etc.
in multi- occupied buildings (04.4.4); security services (04.4.4); snow removal and chimney sweeping (04.4.4); removal and storage services (07.3.6); services of wet-nurses, cre` ches, day-care centres and other child-minding facilities (12.4.0); bodyguards (12.7.0).

06 HEALTH 06.1 This division also includes health services purchased from school and university health centres.
MEDICAL PRODUCTS, APPLIANCES AND EQUIPMENT This group covers medicaments, prostheses, medical appliances and equipment and other health-related products purchased by indivi- duals or households, either with or without a prescription, usually from dispensing chemists, pharmacists or medical equipment suppliers.

They are intended for consumption or use outside a health facility or institution.
Such products supplied directly to outpatients by medical, dental and paramedical practitioners or to in-patients by hospitals and the like are included in outpatient services (06.2) or hospi- tal services (06.3).

06.1.1 Pharmaceutical products (ND) – Medicinal preparations, medicinal drugs, serums and vaccines, patent medicines, vitamins and minerals, cod liver oil and halibut liver oil, oral contraceptives.
Excludes: veterinary products (09.3.4); articles for personal hygiene such as medicinal soaps (12.1.3).

06.1.2 Other medical products (ND) – Clinical thermometers, adhesive and non- adhesive bandages, hypodermic syringes, ﬁrst-aid kits, hot-water bottles and ice bags, medical hosiery items such as elasticated stockings and knee supports, pregnancy tests, condoms and other mechanical con- traceptive devices.
06.1.3 Therapeutic appliances and equipment (D) – Corrective eyeglasses and contact lenses, hearing aids, glass eyes, artiﬁcial limbs and other prosthetic devices, orthopaedic braces and supports, orthopaedic footwear, surgical belts, trusses and supports, neck braces, medical massage equipment and health lamps, powered and unpowered wheelchairs and invalid carriages, ‘‘special’’ beds, crut- ches, electronic and other devices for mon- itoring blood pressure, etc.

; – repair of such articles.


Includes: dentures but not ﬁtting costs.

therapeutic Excludes: hire of equipment (06.2.3); protective goggles, belts and supports for sport (09.3.2); sunglasses not ﬁtted with corrective lenses (12.3.2).

06.2 OUTPATIENT SERVICES This group covers medical, dental and paramedical services delivered to outpatients by medical, dental and paramedical practi- tioners and auxiliaries.
The services may be in individual or group delivered at home, consulting facilities, dispensaries or the out- patient clinics of hospitals and the like.

Outpatient services include the medicaments, prostheses, medical appliances and equipment and other health-related products supplied directly to outpatients by medical, dental and paramedical practitioners and auxiliaries.

Medical, dental and paramedical services provided to in-patients by hospitals and the like are included in hospital services (06.3).
06.2.1 Medical services (S) – Consultations of physicians in general or specialist practice.

Includes: services of orthodontic specialists.

Excludes: services of medical analysis labora- tories and x-ray centres (06.2.3); services of practitioners of traditional medicine (06.2.3).

06.2.2 Dental services (S) – Services of dentists, oral hygienists and other dental auxiliaries.

Includes: ﬁtting costs of dentures.


Excludes: dentures (06.1.3); services of ortho- dontic specialists (06.2.1); services of medical analysis laboratories and x-ray centres (06.2.3).

473 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE 06.2.3 Paramedical services (S) – Services of medical analysis laboratories and x-ray centres; – services of freelance nurses and midwives; – services of freelance acupuncturists, chir- opractors, optometrists, physiotherapists, speech therapists, etc.
; – medically prescribed corrective-gymnastic therapy; – outpatient thermal bath or sea-water treat- ments; – ambulance services; – hire of therapeutic equipment.

Includes: services of practitioners of traditional medicine.

06.3 HOSPITAL SERVICES Hospitalization is deﬁned as occurring when a patient is accommodated in a hospital for the duration of the treatment.

Hospital day-care and home-based hospital treatment are inclu- ded, as are hospices for terminally ill persons.
This group covers the services of general and specialist hospitals, the services of medical centres, maternity centres, nursing homes and convalescent homes which chieﬂy provide in- patient health care, the services of institutions serving old people in which medical monitoring is an essential component and the services of rehabilitation centres providing in-patient health care and rehabilitative therapy where the objective is to treat the patient rather than to provide long-term support.
Hospitals are deﬁned as institutions which oﬀer in-patient care under direct supervision of qualiﬁed medical doctors.

Medical centres, maternity centres, nursing homes and con- valescent homes also provide in-patient care but their services are supervised and frequently delivered by staﬀ of lower qualiﬁcation than medical doctors.

This group does not cover the services of facilities, such as surgeries, clinics and dis- pensaries, devoted exclusively to outpatient care (06.2).
Nor does it include the services of retirement homes for elderly persons, institu- tions for disabled persons and rehabilitation centres providing primarily long-term support (12.4).
06.3.0 Hospital services (S) 07.2 – Hospital services comprise the provision of the following services to hospital in-patients: services: administration; accom- - basic modation; food and drink; supervision and care by non-specialist staﬀ (nursing auxiliaries); ﬁrst aid and resuscitation; ambulance transport; provision of medi- cines and other pharmaceutical products; 474 provision of therapeutic appliances and equipment; - medical services: services of physicians in general or specialist practice, of surgeons and of dentists; medical analyses and x- rays; paramedical services such as those of nurses, midwives, chiropractors, opto- metrists, physiotherapists, speech thera- pists, etc.
07 TRANSPORT 07.1 PURCHASE OF VEHICLES Purchases of recreational vehicles such as camper vans, caravans, trailers, aeroplanes and boats are covered by (09.2.1).

07.1.1 Motor cars (D) – Motor cars, passenger vans, station wagons, estate cars and the like with either two-wheel drive or four-wheel drive.

Excludes: vans (09.2.1); golf carts (09.2.1).

invalid carriages (06.1.3); camper 07.1.2 Motor cycles (D) – Motor cycles of all powered bicycles.

types, scooters and Includes: sidecars; snowmobiles.
Excludes: invalid carriages (06.1.3); golf carts (09.2.1).

07.1.3 Bicycles (D) – Bicycles and tricycles of all types.

Includes: rickshaws.

Excludes: toy bicycles and tricycles (09.3.1).

07.1.4 Animal-drawn vehicles (D) – Animal-drawn vehicles.

Includes: animals required to draw the vehicles and related equipment (yokes, collars, har- nesses, bridles, reins, etc.).

Excludes: horses and ponies, horse- or pony- drawn vehicles and related equipment pur- chased for recreational purposes (09.2.1).
OPERATION OF PERSONAL TRANS- PORT EQUIPMENT Purchases of spare parts, accessories or lubricants made by households with the inten- tion of undertaking the maintenance, repair or intervention themselves should be shown under (07.2.1) or (07.2.2).

If households pay an enterprise to carry out the maintenance, repair the service, the total value of or ﬁtting, CLASSIFICATION OF INDIVIDUAL CONSUMPTION ACCORDING TO PURPOSE (COICOP) 07.2.1 07.2.2 including the costs of should be shown under (07.2.3).
the materials used, Spare parts and accessories for personal trans- port equipment (SD) – Tyres (new, used or retreaded), inner tubes, spark plugs, batteries, shock absorbers, ﬁl- ters, pumps and other spare parts or acces- sories for personal transport equipment.
for Includes: ﬁre transport extinguishers equipment; products speciﬁcally for the clean- ing and maintenance of transport equipment such as paints, chrome cleaners, sealing com- pounds and bodywork polishes; covers for motor cars, motorcycles, etc.

(03.1.3); non-speciﬁc products Excludes: crash helmets for motorcycles and bicycles for cleaning and maintenance such as distilled water, household sponges, chamois leathers, detergents, etc.
(05.6.1); charges for the ﬁtting of spare parts and accessories and for the painting, washing and polishing of bodywork (07.2.3); radio-telephones (08.2.0); car radios (09.1.1); baby seats for cars (12.3.2).

Fuels and lubricants for personal equipment (ND) – Petrol and other fuels such as diesel, liquid petroleum gas, alcohol and two-stroke mix- tures; transport – lubricants, brake and transmission ﬂuids, coolants and additives.
Includes: fuel for major tools and equipment covered under (05.5.1) and recreational veh- icles covered under (09.2.1).

Excludes: charges for oil changes and greasing (07.2.3).

07.2.3 Maintenance and repair of personal transport equipment (S) – Services purchased for the maintenance and repair of personal transport equipment such as ﬁtting of parts and accessories, wheel balancing, technical inspection, breakdown services, oil changes, greasing and washing.
Includes: total value of the service (that is, both the cost of labour and the cost of materials are covered).

Excludes: separate purchases of spare parts, accessories or lubricants made by households with the intention of undertaking the main- (07.2.1) or tenance or (07.2.2); roadworthiness tests (07.2.4).
themselves repair – toll facilities (bridges, tunnels, shuttle ferries, motorways) and parking meters; – driving lessons, driving tests and driving licences; – roadworthiness tests; – hire of personal transport equipment with- out drivers.

Excludes: hire of a car with driver (07.3.2); service charges for insurance in respect of personal transport equipment (12.5.4).

07.3 TRANSPORT SERVICES Purchases of transport services are generally classiﬁed by mode of transport.
When a ticket covers two or more modes of transport – for example, intra-urban bus and underground or inter-urban train and ferry – and the expendi- ture cannot be apportioned between them, then such purchases should be classiﬁed in (07.3.5).

Costs of meals, snacks, drinks, refreshments or accommodation services have to be included if covered by the fare and not separately priced.

If separately priced, these costs have to be classiﬁed in Division 11.
School transport services are included, but ambulance services are excluded (06.2.3).

07.3.1 Passenger transport by railway (S) – Transport of individuals and groups of per- sons and luggage by train, tram and under- ground.

Includes: transport of private vehicles.

Excludes: funicular transport (07.3.6).

07.3.2 Passenger transport by road (S) – Transport of individuals and groups of per- sons and luggage by bus, coach, taxi and hired car with driver.
07.3.3 Passenger transport by air (S) – Transport of individuals and groups of per- sons and luggage by aeroplane and heli- copter.

07.3.4 Passenger transport by sea and inland waterway (S) – Transport of individuals and groups of per- ferry, sons and luggage by ship, boat, hovercraft and hydrofoil.

Includes: transport of private vehicles.


07.2.4 Other services in respect of personal transport equipment (S) – Hire of garages or parking spaces not provid- ing parking in connection with the dwelling; – Transport of individuals and groups of per- sons and luggage by two or more modes of transport when the expenditure cannot be apportioned between them.

07.3.5 Combined passenger transport (S) 475 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Includes: transport of private vehicles.
09 RECREATION AND CULTURE Excludes: package holidays (09.6.0).

07.3.6 Other purchased transport services (S) – Funicular, cable-car and chairlift transport; – removal and storage services; – services of porters and left-luggage and lug- gage-forwarding oﬃces; – travel agents’ commissions, if separately priced.

Excludes: cable-car and chairlift transport at ski resorts and holiday centres (09.4.1).
08 COMMUNICATION 08.1 POSTAL SERVICES 08.1.0 Postal services (S) – Payments for the delivery of letters, post- cards and parcels; – private mail and parcel delivery.

Includes: all purchases of new postage stamps, pre-franked postcards and aerogrammes.

Excludes: purchase of used or cancelled post- age stamps (09.3.1); ﬁnancial services of post oﬃces (12.6.2).
08.2 TELEPHONE AND TELEFAX EQUIP- MENT 08.2.0 Telephone and telefax equipment (D) – Purchases of telephones, radio-telephones, telephone-answering telefax machines, machines and telephone loudspeakers; 09.1.3 – repair of such equipment.

Excludes: facilities provided by personal (09.1.3).
telefax and telephone-answering computers 08.3 TELEPHONE AND TELEFAX SERVICES 08.3.0 Telephone and telefax services (S) 09.1 AUDIO-VISUAL, PHOTOGRAPHIC AND INFORMATION PROCESSING EQUIP- MENT 09.1.1 Equipment for the reception, recording and reproduction of sound and pictures (D) – Television sets, video cassette players and recorders, television aerials of all types; – radio sets, car radios, radio clocks, two-way radios, amateur radio receivers and trans- mitters; – gramophones, tape players and recorders, cassette players and recorders, CD-players, personal stereos, stereo systems and their constituent units (turntables, tuners, ampli- ﬁers, speakers, etc.
), microphones and ear- phones.

Excludes: video cameras, sound-recording cameras (09.1.2).

camcorders and 09.1.2 Photographic and cinematographic equipment and optical instruments (D) – Still cameras, movie cameras and sound- recording cameras, video cameras and cam- corders, ﬁlm and slide projectors, enlargers and ﬁlm processing equipment, accessories (screens, viewers, lenses, ﬂash attachments, ﬁlters, exposure meters, etc.

); – binoculars, microscopes, telescopes and compasses.
Information processing equipment (D) – Personal computers, visual display units, accessories and miscellaneous printers accompanying them; software packages such as operating systems, appli- cations, languages, etc.

; computer – calculators, including pocket calculators; – typewriters and word processors.

Includes: facilities provided by personal computers.
and telephone-answering telefax – Installation and subscription costs of per- sonal telephone equipment; – telephone calls from a private line or from a public line (public telephone box, post oﬃce telephone calls from hotels, cabin, etc.

); cafe´ s, restaurants and the like; – telegraphy, telex and telefax services; – information transmission services; Internet connection services; – hire of telephones, telefax machines, tele- phone-answering machines and telephone loudspeakers.
Includes: and radiotelex services.

radio-telephony, radio-telegraphy Excludes: pre-recorded diskettes and CD- ROMs containing books, dictionaries, ency- clopaedias, foreign language trainers, multi- media presentations, etc.

in the form of soft- ware (09.1.4); video game software (09.3.1); video game computers that plug into a televi- sion set (09.3.1); typewriter ribbons (09.5.4); toner and ink cartridges (09.5.4); slide rules (09.5.4).
09.1.4 Recording media (SD) – Records and compact discs; – pre-recorded tapes, cassettes, video cassettes, diskettes and CD-ROMs for tape recorders, 476 CLASSIFICATION OF INDIVIDUAL CONSUMPTION ACCORDING TO PURPOSE (COICOP) cassette recorders, video recorders and per- sonal computers; – unrecorded tapes, cassettes, video cassettes, diskettes and CD-ROMs for tape recorders, cassette recorders, video recorders and per- sonal computers; – unexposed ﬁlms, cartridges and disks for photographic and cinematographic use.
encyclopaedias, Includes: pre-recorded tapes and compact discs of novels, plays, poetry, etc.

; pre-recorded dis- kettes and CD-ROMs containing books, dic- tionaries, foreign language trainers, multimedia presentations, etc.

in the form of software; photographic supplies such as paper and ﬂashbulbs; unexposed ﬁlm the price of which includes the cost of processing without separately identifying it.

languages, etc.
languages, etc.

Excludes: batteries (05.5.2); computer software packages such as operating systems, applica- tions, (09.1.3); video game software, video game cassettes and video game CD-ROMs (09.3.1); development of ﬁlms and printing of photographs (09.4.2).

09.1.5 Repair of audio-visual, photographic and infor- mation processing equipment (S) – Repair of audio-visual, photographic and information processing equipment.
Includes: total value of the service (that is, both the cost of labour and the cost of materials are covered).

Excludes: separate purchases of materials made by households with the intention of under- taking the repair themselves (09.1.1), (09.1.2) or (09.1.3).
09.2 OTHER MAJOR RECREATION AND CULTURE DURABLES FOR 09.2.1 Major durables for outdoor recreation (D) – Camper vans, caravans and trailers; – aeroplanes, microlight aircraft, hang-gliders and hot-air balloons; gliders, – boats, outboard motors, sails, rigging and superstructures; – horses and ponies, horse- or pony-drawn vehicles and related equipment (harnesses, bridles, reins, saddles, etc.
); – major items for games and sport such as canoes, kayaks, windsurﬁng boards, sea- diving equipment and golf carts.

Includes: ﬁtting out of boats, camper vans, caravans, etc.

Excludes: horses and ponies, horse- or pony- drawn vehicles and related equipment pur- chased for personal transport (07.1.4); inﬂa- table boats, rafts and swimming pools for children and the beach (09.3.2).
09.2.2 Musical instruments and major durables for indoor recreation (D) – Musical instruments of all sizes, including electronic musical such as pianos, organs, violins, guitars, drums, trumpets, clarinets, ﬂutes, recorders, har- monicas, etc.

; instruments, – billiard tables, ping-pong tables, pinball machines, gaming machines, etc.

Excludes: toys (09.3.1).


09.2.3 Maintenance and repair of other major durables for recreation and culture (S) – Maintenance and repair of other major durables for recreation and culture.

Includes: total value of the service (that is, both the cost of labour and the cost of materials are covered); laying up for winter of boats, camper vans, caravans, etc.

; hangar services for private planes; marina services for boats; veterinary and other services (stabling, feeding, farriery, etc.)
for horses and ponies purchased for recreational purposes.

Excludes: fuel for recreational vehicles (07.2.2); separate purchases of materials made by households with the intention of undertaking the maintenance or repair themselves (09.2.1) or (09.2.2); veterinary and other services for pets (09.3.5).
09.3 OTHER RECREATIONAL ITEMS AND EQUIPMENT, GARDENS AND PETS 09.3.1 Games, toys and hobbies (SD) – Card games, parlour games, chess sets and the like; – toys of all kinds including dolls, soft toys, toy cars and trains, toy bicycles and tricycles, toy construction sets, puzzles, plasticine, electronic games, masks, disguises, jokes, novelties, ﬁreworks and rockets, festoons and Christmas tree decorations; – stamp-collecting requisites (used or cancelled postage stamps, stamp albums, etc.
), other items for collections (coins, medals, miner- als, zoological and botanical specimens, etc.)

and other tools and articles n.e.c.

for hob- bies.

Includes: video-game software; video-game computers that plug into a television set; video- game cassettes and video-game CD-ROMs.

Excludes: collectors’ items falling into the category of works of art or antiques (05.1.1); unused postage stamps (08.1.0); Christmas trees (09.3.3); children’s scrapbooks (09.5.1).
477 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE 09.3.2 Equipment for sport, camping and open-air recreation (SD) – Gymnastic, physical education and sport equipment such as balls, shuttlecocks, nets, rackets, bats, skis, golf clubs, foils, sabres, poles, weights, discuses, javelins, dumb-bells, chest expanders and other body-building equipment; – parachutes and other sky-diving equipment; – ﬁrearms and ammunition for hunting, sport and personal protection; – ﬁshing rods and other equipment for ﬁshing; – equipment for beach and open-air games, such as bowls, croquet, frisbee, volleyball, and inﬂatable boats, rafts and swimming pools; such as – camping equipment tents and accessories, sleeping bags, backpacks, air mattresses and inﬂating pumps, camping stoves and barbecues; – repair of such articles.
Includes: game-speciﬁc footwear (ski boots, football boots, golﬁng shoes and other such footwear ﬁtted with ice-skates, rollers, spikes, studs, etc.

); protective headgear for sports; other protective gear for sports such as life jackets, boxing gloves, body padding, shin- guards, goggles, belts, supports, etc.

Excludes: crash helmets for motor cycles and bicycles (03.1.3); camping and garden furniture (05.1.1).
09.3.3 Gardens, plants and ﬂowers (ND) – Natural or artiﬁcial ﬂowers and foliage, plants, shrubs, bulbs, tubers, seeds, fertili- zers, composts, garden peat, turf for lawns, specially treated soils for ornamental gar- dens, horticultural preparations, pots and pot holders.

Includes: natural and artiﬁcial Christmas trees; delivery charges for ﬂowers and plants.
Excludes: gardening gloves (03.1.3); gardening services (04.4.4) or (05.6.2); gardening equip- ment (05.5.1); gardening tools (05.5.2); insec- ticides and pesticides for household use (05.6.1).

09.3.4 Pets and related products (ND) – Pets, pet foods, veterinary and grooming products for pets, collars, leashes, kennels, birdcages, ﬁsh tanks, cat litter, etc.

Excludes: horses and ponies (09.2.1); veterinary services (09.3.5).
(07.1.4) or 09.3.5 Veterinary and other services for pets (S) – Veterinary and other services for pets such as grooming, boarding, tattooing and training.

478 Excludes: veterinary and other services (sta- for horses and ponies bling, purchased for recreational purposes (09.2.3).

farriery, etc.)

09.4 RECREATIONAL AND CULTURAL SER- VICES 09.4.1 Recreational and sporting services (S) – Services provided by: - sports stadiums, horse-racing courses, motor-racing circuits, velodromes, etc.
; - skating rinks, swimming pools, golf cour- ses, gymnasia, ﬁtness tennis courts, squash courts and bowling alleys; centres, - fairgrounds and amusement parks; - roundabouts, see-saws and other play- ground facilities for children; - pin-ball machines and other games for adults other than games of chance; - ski slopes, ski lifts and the like; – hire of equipment and accessories for sport and recreation, such as aeroplanes, boats, horses, skiing and camping equip- ment; – out-of-school individual or group lessons in chess, aerobics, dancing, music, bridge, skating, skiing, swimming or other pastimes; – services of mountain guides, tour guides, etc.
; – navigational aid services for boating.

Includes: hire of game-speciﬁc footwear (ski boots, football boots, golﬁng shoes and other such footwear ﬁtted with ice-skates, rollers, spikes, studs, etc.).

Excludes: cable-car and chairlift transport not at ski resorts or holiday centres (07.3.6).
09.4.2 Cultural services (S) – Services provided by: - cinemas, theatres, opera houses, concert halls, music halls, circuses, sound and light shows; - museums, libraries, art galleries, exhibi- tions; - historic monuments, national parks, zo- ological and botanical gardens, aquaria; – hire of equipment and accessories for cul- ture, such as television sets, video cassettes, etc.
; – television and radio broadcasting, in partic- ular licence fees for television equipment and subscriptions to television networks; – services of photographers developing, print processing, portrait graphy, etc.

photography, wedding such as ﬁlm enlarging, photo- CLASSIFICATION OF INDIVIDUAL CONSUMPTION ACCORDING TO PURPOSE (COICOP) Includes: services of musicians, clowns, per- formers for private entertainments.
09.4.3 Games of chance (S) – Service charges for lotteries, bookmakers, totalizators, casinos and other gambling establishments, gaming machines, bingo halls, scratch cards, sweepstakes, etc.

(Ser- vice charge is deﬁned as the diﬀerence between the amounts paid for lottery tickets or placed in bets and the amounts paid out to winners.)

09.5 NEWSPAPERS, BOOKS AND STATION- ERY 09.5.1 Books (SD) – Books, including atlases, dictionaries, ency- and guidebooks textbooks, clopaedias, musical scores.
Includes: scrapbooks and albums for children; bookbinding.

Excludes: pre-recorded tapes and compact discs of novels, plays, poetry, etc.

(09.1.4); pre- recorded diskettes and CD-ROMs containing foreign books, dictionaries, encyclopaedias, language trainers, etc.

in the form of software (09.1.4); stamp albums (09.3.1).

09.5.2 Newspapers and periodicals (ND) – Newspapers, magazines and other periodi- cals.
09.5.3 Miscellaneous printed matter (ND) – Catalogues and advertising material; – posters, plain or picture postcards, calen- dars; – greeting cards and visiting cards, announce- ment and message cards; – maps and globes.

Excludes: pre-franked postcards and aero- grammes (08.1.0); stamp albums (09.3.1).

09.5.4 Stationery and drawing materials (ND) – Writing pads, envelopes, account books, notebooks, diaries, etc.
; – pens, pencils, fountain pens, ballpoint pens, felt-tip pens, inks, erasers, pencil sharpeners, etc.

; – stencils, carbon paper, typewriter ribbons, inking pads, correcting ﬂuids, etc.

; – paper punches, paper cutters, paper scissors, oﬃce glues and adhesives, staplers and sta- ples, paper clips, drawing pins, etc.

; – drawing and painting materials such as can- vas, paper, card, paints, crayons, pastels and brushes.
Includes: toner and ink cartridges; educational materials such as exercise books, slide rules, geometry instruments, slates, chalks and pencil boxes.

Excludes: pocket calculators (09.1.3).

09.6 PACKAGE HOLIDAYS 09.6.0 Package holidays (S) – All-inclusive holidays or tours which provide for travel, food, accommodation, guides, etc.

Includes: half-day and one-day excursion tours; pilgrimages.

10 EDUCATION This division covers educational services only.
It does not include expenditures on edu- cational materials, such as books (09.5.1) and stationery (09.5.4), or education support ser- vices, such as health-care services (06), trans- port services (07.3), catering services (11.1.2) and accommodation services (11.2.0).

It includes education by radio or television broadcasting.
The breakdown of educational services is based upon the level categories of the 1997 International Standard Classiﬁcation of Edu- the United Nations cation (ISCED-97) of Educational, Scientiﬁc and Cultural Organiz- ation (UNESCO).

10.1 PRE-PRIMARY AND PRIMARY EDUCA- TION 10.1.0 Pre-primary and primary education (S) – Levels 0 and 1 of ISCED-97: pre-primary and primary education.

Includes: literacy programmes for students too old for primary school.
10.2 SECONDARY EDUCATION 10.2.0 Secondary education (S) – Levels 2 and 3 of ISCED-97: lower-second- ary and upper-secondary education.

Includes: out-of-school secondary education for adults and young people.

10.3 POST-SECONDARY EDUCATION NON-TERTIARY 10.3.0 Post-secondary non-tertiary education (S) – Level 4 of ISCED-97: post-secondary non- tertiary education.

479 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Includes: out-of-school post-secondary non- tertiary education for adults and young people.
10.4 TERTIARY EDUCATION 10.4.0 Tertiary education (S) versities and other educational establish- ments.

Includes: university refectories, military messes and wardrooms.

Excludes: food and drink provided to hospital in-patients (06.3.0).

– Levels 5 and 6 of ISCED-97: ﬁrst stage and second stage of tertiary education.
11.2 ACCOMMODATION SERVICES 10.5 EDUCATION NOT DEFINABLE LEVEL BY 11.2.0 Accommodation services (S) – Accommodation services of: 10.5.0 Education not deﬁnable by level (S) – Educational programmes, for adults, which do not require any special prior instruction, in particular vocational training and cultural development.

generally Excludes: driving lessons (07.2.4); recreational training courses such as sport or bridge lessons given by independent teachers (09.4.1).
- hotels, boarding houses, motels, inns and establishments oﬀering ‘‘bed and break- fast’’; - holiday villages and holiday centres, camping and caravan sites, youth hostels and mountain chalets; - boarding schools, universities and other educational establishments; - public transport (trains, boats, etc.)

when priced separately; - hostels for young workers or immigrants.

11 RESTAURANTS AND HOTELS Includes: tips, porters.


11.1 CATERING SERVICES 11.1.1 Restaurants, cafe´ s and the like (S) – Catering services (meals, snacks, drinks and refreshments) provided by restaurants, cafe´ s, buﬀets, bars, tearooms, etc., including those provided: - in places providing recreational, cultural, sporting or entertainment services: theatres, cinemas, sports stadiums, swimming pools, sports complexes, museums, art galleries, nightclubs, dancing establishments, etc.
; - on public transport (coaches, trains, boats, aeroplanes, etc.)
when priced separately; – also included are: - the sale of food products and beverages for immediate consumption by kiosks, street vendors and the like, including food products and beverages dispensed ready for consumption by automatic vending machines; - the sale of cooked dishes by restaurants for consumption oﬀ their premises; - the sale of cooked dishes by catering contractors whether collected by the cus- tomer or delivered to the customer’s home.

Includes: tips.

Excludes: phone calls (08.3.0).
tobacco purchases (02.2.0); tele- 11.1.2 Canteens (S) – Catering services of works canteens, oﬃce in schools, uni- canteens and canteens 480 Excludes: payments of households occupying a room in a hotel or boarding house as their rentals paid by (04.1.1); main residence households for a secondary residence for the duration of a holiday (04.1.2); telephone calls (08.3.0); catering services in such establish- ments except for breakfast or other meals included in the price of the accommodation (11.1.1); housing in orphanages, homes for disabled or maladjusted persons (12.4.0).
12 MISCELLANEOUS GOODS AND SERVICES 12.1 PERSONAL CARE 12.1.1 Hairdressing salons and personal grooming establishments (S) – Services of hairdressing salons, barbers, beauty shops, manicures, pedicures, Turkish baths, saunas, solariums, non-medical mas- sages, etc.

Includes: bodycare, depilation and the like.

Excludes: spas (06.2.3) or (06.3.0); ﬁtness cen- tres (09.4.1).
12.1.2 Electric appliances for personal care (SD) – Electric razors and hair trimmers, hand-held and hood hairdryers, curling tongs and sty- ling combs, sunlamps, vibrators, electric toothbrushes and other electric appliances for dental hygiene, etc.

; – repair of such appliances.


12.1.3 Other appliances, articles and products for per- – articles for smokers: pipes, lighters, ciga- CLASSIFICATION OF INDIVIDUAL CONSUMPTION ACCORDING TO PURPOSE (COICOP) sonal care (ND) – Non-electric appliances: razors and hair trimmers and blades therefor, scissors, nail ﬁles, combs, shaving brushes, hairbrushes, toothbrushes, nail brushes, hairpins, curlers, personal weighing machines, baby scales, etc.
; – articles for personal hygiene: toilet soap, soap, cleansing oil and milk, medicinal shaving soap, shaving cream and foam, toothpaste, etc.

; – beauty products: lipstick, nail varnish, make- up and make-up removal products (including compacts, brushes and powder powder puﬀs), hair lacquers and lotions, pre-shave and after-shave products, sunbathing pro- ducts, hair removers, perfumes and toilet waters, personal deodorants, bath products, etc.
; – other products: toilet paper, paper hand- kerchiefs, paper sanitary towels, towels, cotton wool, cotton tops, babies’ napkins, toilet sponges, etc.

Excludes: (03.1.3).

handkerchiefs made of fabric 12.2 PROSTITUTION 12.2.0 Prostitution (S) – Services provided by prostitutes and the like.

12.3 PERSONAL EFFECTS N.E.C.


12.3.1 Jewellery, clocks and watches (D) – Precious stones and metals and jewellery fashioned out of such stones and metals; – costume jewellery, cuﬀ links and tiepins; – clocks, watches, stopwatches, alarm clocks, travel clocks; – repair of such articles.

Excludes: ornaments (05.1.1) or (05.4.0); radio clocks (09.1.1); precious stones and metals and jewellery fashioned out of such stones and metals acquired primarily as stores of value (capital formation).
12.3.2 Other personal eﬀects (SD) – Travel goods and other carriers of personal eﬀects: suitcases, trunks, travel bags, attache´ cases, satchels, hand-bags, wallets, purses, etc.

; – articles for babies: baby carriages, push- chairs, carrycots, recliners, car beds and seats, back-carriers, front carriers, reins and harnesses, etc.

; rette cases, cigar cutters, ashtrays, etc.

; – miscellaneous personal articles: sunglasses, walking sticks and canes, umbrellas and parasols, fans, keyrings, etc.
; – funerary articles: coﬃns, gravestones, urns, etc.

; – repair of such articles.

Includes: lighter fuel; wall thermometers and barometers.

Excludes: baby furniture (05.1.1); shopping bags (05.2.0); feeding bottles (05.4.0).
12.4 SOCIAL PROTECTION Social protection as deﬁned here covers assistance and support services provided to persons who are: elderly, disabled, having occupational injuries and diseases, survivors, unemployed, destitute, homeless, low-income earners, indigenous people, immigrants, refu- gees, alcohol and substance abusers, etc.

It also covers assistance and support services provided to families and children.
12.4.0 Social protection (S) Such services include residential care, home help, day care and rehabilitation.
More speci- ﬁcally, this class covers payments by house- holds for: – Retirement homes for elderly persons, resi- dences for disabled persons, rehabilitation centres providing long-term support for patients rather than health care and rehabi- litative therapy, schools for disabled persons where the main aim is to help students overcome their disability; – help to maintain elderly and disabled per- sons at home (home-cleaning services, meal programmes, day-care centres, day-care ser- vices and holiday-care services); – wet-nurses, cre` ches, play schools and other child-minding facilities; – counselling, guidance, arbitration, fostering and adoption services for families.
12.5 INSURANCE Service charges for insurance are classi- ﬁed by type of insurance, namely: life insu- rance and non-life insurance (that is, insurance in connection with the dwelling, health, trans- port, etc.).

Service charges for multi-risk insurance covering several risks should be classiﬁed on the basis of the principal risk if it is not possible to allocate risks the covered.

to the various the cost of charges service 481 12.6 FINANCIAL SERVICES N.E.C.
12.6.1 FISIM (S) – Financial intermediation services indirectly measured.

12.6.2 Other ﬁnancial services n.e.c.

(S) – Actual charges for the ﬁnancial services of banks, post oﬃces, saving banks, money changers and similar ﬁnancial institutions; – fees and service charges of brokers, invest- ment counsellors, tax consultants and the like; – administrative charges of private pension funds and the like.

12.7 OTHER SERVICES N.E.C.

12.7.0 Other services n.e.c.


(S) – Fees for legal services, employment agencies, etc.
; – charges for undertaking and other funeral services; – payment for the estate agents, housing agents, auctioneers, sales- room operators and other intermediaries; – payment for photocopies and other repro- services of ductions of documents; – fees for the issue of birth, marriage and death certiﬁcates and other administrative documents; – payment for newspaper notices and adver- tisements; – payment for the services of graphologists, astrologers, private detectives, bodyguards, matrimonial agencies and marriage guidance counsellors, public writers, miscellaneous concessions (seats, toilets, cloakrooms), etc.
CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Service charge is deﬁned as the diﬀerence between claims due and premiums earned and premium supplement.

12.5.1 Life insurance (S) – Service charges for life assurance, death beneﬁt assurance, education assurance, etc.

12.5.2 Insurance connected with the dwelling (S) – Service charges paid by owner-occupiers and by tenants for the kinds of insurance typi- cally taken out by tenants against ﬁre, theft, water damage, etc.
Excludes: service charges paid by owner-occu- piers for the kinds of insurance typically taken out by landlords (intermediate consumption).

12.5.3 Insurance connected with health (S) – Service charges for private sickness and accident insurance.

12.5.4 Insurance connected with transport (S) – Service charges for insurance in respect of personal transport equipment; – service charges for travel insurance and luggage insurance.
12.5.5 Other insurance (S) – Service charges for other insurance such as civil liability for injury or damage to third parties or their property.

Excludes: civil liability or damage to third par- ties or their property arising from the operation of personal transport equipment (12.5.4).
482 Annex 3 Resolution concerning consumer price indices adopted by the Seventeenth International Conference of Labour Statisticians, 2003 Preamble* The Seventeenth International Conference of Labour Statisticians, Having been convened at Geneva by the Governing Body of the ILO and having met from 24 November to 3 December 2003, Recalling the resolution adopted by the Fourteenth International Conference of Labour Statisticians con- cerning consumer price indices and recognizing the continuing validity of the basic principles recommended therein and, in particular, the fact that the consumer price index (CPI) is designed primarily to measure the changes over time in the general level of prices of goods and services that a reference population acquires, uses or pays for, Recognizing the need to modify and broaden the existing standards in the light of recent methodological and computational developments to enhance the use- fulness of the international standards in the provision of technical guidelines to all countries, Recognizing the usefulness of in enhancing the international comparability of the statistics, Recognizing that the CPI is used for a wide variety of purposes and that governments should be encouraged to identify the (priority) purposes a CPI is to serve, to provide adequate resources for its compilation, and to guarantee the professional independence of its compilers, Recognizing that the (priority) objectives and uses of a CPI diﬀer among countries and that, therefore, a single standard could not be applied universally, such standards Recognizing that the CPI needs to be credible to observers and users, both national and international, and that better understanding of the principles and procedures used to compile the index will enhance the users’ conﬁdence in the index, Agrees that the principles and methods used in con- structing a CPI should be based on the guidelines and methods that are generally accepted as constituting good statistical practices; Adopts, this third day of December 2003, the fol- lowing resolution which replaces the previous one adopted in 1987: The nature and meaning of a consumer price index 1.
The CPI is a current social and economic indicator that is constructed to measure changes over time in the general level of prices of consumer goods and services that households acquire, use or pay for consumption.

2.

The index aims to measure the change in consumer prices over time.
This may be done by measuring the cost of purchasing a ﬁxed basket of consumer goods and services of constant quality and similar characteristics, with the products in the basket being selected to be representative of households’ expenditure during a year or other speciﬁed period.

Such an index is called a ﬁxed basket price index.

3.

The index may also aim to measure the eﬀects of price changes on the cost of achieving a constant stan- dard of living (i.e.

level of utility or welfare).


This con- cept is called a cost-of-living index (COLI).

A ﬁxed basket price index, or another appropriate design, may be employed as an approximation to a COLI.

The uses of a consumer price index 4.


The CPI is used for a wide variety of purposes, the two most common ones being: (i) to adjust wages as well as social security and other beneﬁts to compensate, partly or completely, for changes in the cost of living or in consumer prices; and (ii) to provide an average measure of price inﬂation for the household sector as a whole, for use as a macroeconomic indicator.
CPI sub- indices are also used to deﬂate components of household ﬁnal consumption expenditure in the national accounts and the value of retail sales to obtain estimates of changes in their volume.

5.
5.

CPIs are also used for other purposes, such as monitoring the overall rate of price inﬂation for all sec- tors of the economy, the adjustment of government fees and charges, the adjustment of payments in commercial contracts, and for formulating and assessing ﬁscal and monetary policies and trade and exchange rate policies.

In these types of cases, the CPI is used as more appro- priate measures do not exist at present, or because other characteristics of the CPI (e.g.
high proﬁle, wide accep- tance, predictable publication schedule, etc.)

are seen to outweigh any conceptual or technical deﬁciencies.

6.

Given that the CPI may be used for many purposes, it is unlikely that one index can perform equally satis- factorily in all applications.

It may therefore be appro- priate to construct a number of alternative price indices for speciﬁc purposes, if the requirements of the users justify the extra expense.
Each index should be properly deﬁned and named to avoid confusion and a ‘‘headline’’ CPI measure should be explicitly identiﬁed.

7.

Where only one index is compiled, it is the main use that should determine the type of index compiled, the its geographic range of goods and services covered, coverage, the households it relates to, as well as to the concept of price and the formula used.
If there are several major uses, it is likely that compromises may have to be made with regard to how the CPI is con- structed.

Users should be informed of the compromises made and of the limitations of such an index.

Scope of the index *All annexes referred to in the footnotes are the annexes to the reso- lution.

8.
8.

The scope of the index depends on the main use for which it is intended, and should be deﬁned in terms of 483 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE the type of households, geographic areas, and the cate- gories of consumer goods and services acquired, used or paid for by the reference population.

9.

If the primary use of the CPI is for adjusting money incomes, a relevant group of households, such as wage and salary earners, may be the appropriate target population.
For this use, all consumption expenditures by these households, at home and abroad, may be covered.

If the primary use of the CPI is to measure inﬂation in the domestic economy, it may be appropriate to cover consumption expenditures made within the country, rather than the expenditures of households resident within the country.

10.

In general, the reference population for a national index should be deﬁned very widely.
If any income groups, types of households or particular geographic areas are excluded, for example, for cost or practical considerations, then this should be explicitly stated.

11.

The geographic scope refers to the geographic coverage of price collection and of consumption expenditures of the reference population and both should be deﬁned as widely as possible, and preferably consistently.

If price collection is restricted to particular areas due to resource constraints, then this should be speciﬁed.
The geographic coverage of the consumption expenditure may be deﬁned either as covering con- sumption expenditure of the resident population (resi- dent consumption) or consumption expenditure within the country (domestic consumption).

12.

Signiﬁcant diﬀerences in the expenditure patterns and/or price movements between speciﬁc population groups or regions may exist, and care should be taken to ensure that they are represented in the index.
Separate indices for these population groups or regions may be computed if there is suﬃcient demand to justify the additional cost.

13.

In accordance with its main purpose, the CPI should conceptually cover all types of consumer goods and services of signiﬁcance to the reference population, without any omission of those that may not be legally available or may be considered socially undesirable.
Where appropriate, special aggregates may be con- structed to assist those users who may wish to exclude certain categories of goods or services for particular applications or for analysis.

Whenever certain goods or services have been excluded from the index, this should be clearly documented.

14.
14.

Goods and services purchased for business pur- poses, expenditures on assets such as works of art, ﬁnancial investment (as distinct from ﬁnancial services), and payments of income taxes, social security con- tributions and ﬁnes are not considered to be consumer goods or services and should be excluded from the coverage of the index.

Some countries regard expendi- tures on the purchase of houses entirely as a capital investment and, as such, exclude them from the index.

Acquisition, use or payment 15.


In determining the scope of the index, the time of recording and valuation of consumption, it is important to consider whether the purposes for which the index is 484 used are best satisﬁed by deﬁning consumption in terms of ‘‘acquisition’’, ‘‘use’’, or ‘‘payment’’.1 The ‘‘acquisi- tion’’ approach is often used when the primary purpose of the index is to serve as a macroeconomic indicator.
The ‘‘payment’’ approach is often used when the pri- mary purpose of the index is for the adjustment of compensation or income.

Where the aim of the index is to measure changes in the cost of living, the ‘‘use’’ approach may be most suitable.
The decision regarding the approach to follow for a particular group of pro- ducts should in principle be based on the purpose of the index, as well as on the costs and the acceptability of the decision to the users who should be informed of the approach followed for diﬀerent products.

Because of the practical diﬃculties in uniformly deﬁning consumption and estimating the ﬂow of services provided by other durable goods in terms of ‘‘use’’, it may be necessary to adopt a mixed approach, e.g.
‘‘use’’ for owner-occupied housing and ‘‘acquisition’’ or ‘‘payment’’ basis for other consumer durables.

16.

The diﬀerences between the three approaches are most pronounced in dealing with products for which the times of acquisition, use and payment do not coincide, such as owner-occupied housing, durable goods and products acquired on credit.

the 17.

The most complex and important of the products mentioned above is owner-occupied housing.
In most countries, a signiﬁcant proportion of households are owner-occupiers of their housing, with the housing being characterized by a long useful life and a high purchase outlay (price).

Under ‘‘acquisition’’ approach, the value of the new dwellings acquired in the weight reference period may be used for deriving the weight (and the full price of the dwelling is included in the CPI at the time of acquisition, regardless of when the consumption is taking place).
Under the ‘‘payment’’ approach, the weights reﬂect the amounts actually paid out for housing (and the prices enter the CPI in the period(s) when the prices are paid).

Under the ‘‘use’’ approach the weights are based on the value of the ﬂow of housing services consumed during the weight refer- ence period estimated using an implicit or notional cost (and prices or estimated opportunity costs enter the CPI when the consumption is taking place).

18.
18.

Own-account consumption, remuneration in kind and/or goods and services provided without charge or subsidized by governments and non-proﬁt institutions serving households may be important in some countries where the purpose of the index is best satisﬁed by deﬁning consumption in terms of ‘‘use’’ or ‘‘acquisition’’ (under the payment approach these are out of scope).

The inclusion of these products will require special valuation and pricing techniques.

Basket and weights 19.


Decisions on the composition of the basket and the weights follow directly from the scope, as well as from the choice between the ‘‘acquisition’’, ‘‘use’’ or ‘‘payment’’ approaches.

1See Annex 1.

STATISTICIANS RESOLUTION CONCERNING CONSUMER PRICE INDICES, 2003 20.

Once deﬁned, the expenditures that fall within the scope of the index should be grouped into similar cate- gories in a hierarchical classiﬁcation system, e.g.
divi- sions/groups/classes, for compilation as well as analy- tical purposes.

There should be consistency between the classiﬁcation used for index compilation and the one used for household expenditure statistics.

The CPI classiﬁcation should meet the needs of users for special sub-indices.
For the purposes of international compar- isons, the classiﬁcation should also be reconcilable with the most recent version of the UN Classiﬁcation of Individual Consumption according to Purpose (COICOP), at least at its division level.2 21.

In order to facilitate the analysis and interpreta- tion of the results of the index, it may be desirable to classify goods and services according to various sup- plementary classiﬁcations, e.g.

source of origin, dur- ability and seasonality.


Calculation of the CPI by using various classiﬁcations should generate the same overall results as the original index.

22.

The classiﬁcation should also provide a frame- work for the allocation of expenditure weights.

Expen- ditures at the lowest level of the classiﬁcation system, expressed as a proportion of the total expenditure, determine the weights to be used at this level.
When the weights are to remain ﬁxed for several years, the objective should be to adopt weights that are repre- sentative of the contemporary household behaviour.

23.

The two main sources for deriving the weights are the results from household expenditure surveys (HESs) and national accounts estimates on household con- sumption expenditure.
The results from an HES are appropriate for an index deﬁned to cover the con- sumption expenditures of reference population groups resident within the country, while national account estimates are suitable for an index deﬁned to cover consumption expenditures within the country.

The decision about what source or sources to use and how they should be used depends on the main purpose of the index and on the availability and quality of appropriate data.

24.
24.

The information from the main source (HESs or national accounts) should be supplemented with all other available information on the expenditure pattern.

Sources of such information that can be used for dis- aggregating the expenditures are surveys of sales in retail outlets, point-of-purchase surveys, surveys of produc- tion, export and import data and administrative sources.

Based on these data the weights for certain products may be further disaggregated by region and type of outlet.
Where the data obtained from diﬀerent sources relate to diﬀerent periods, it is important to ensure, before weights are allocated, that expenditures are adjusted so that they have the same reference period.

25.

Where the weight reference period diﬀers sig- niﬁcantly from the price reference period, the weights should be price updated to take account of price changes between the weight reference period and price reference period.
Where it is likely that price-updated weights are 2 See Annex 4. less representative of the consumption pattern in the price reference period this procedure may be omitted.

26.

Weights should be reviewed and if appropriate revised as often as accurate and reliable data are avail- able for this to be done, but at least once every ﬁve years.
Revisions are important to reduce the impact on the index of product substitutions and to ensure the basket of goods and services and their weights remain representative.3 For some categories, it may be neces- sary to update the weights more frequently as such weights are likely to become out of date more quickly than higher-level weights.

In periods of high inﬂation, the weights should be updated frequently.

27.
27.

When a new basket (structure or weights) replaces the old, a continuous CPI series should be created by linking4 together the index numbers based on the new basket of goods and services to those based on the earlier basket.

The particular procedure used to link index number series will depend on the particular index com- pilation technique used.

The objective is to ensure that the technique used to introduce a new basket does not, of itself, alter the level of the index.

28.
28.

Completely new types of goods and services (i.e.

goods and services that cannot be classiﬁed to any of the existing elementary aggregates) should normally be considered for inclusion only during one of the periodic review and reweighting exercises.

A new model or variety of an existing product that can be ﬁtted within an existing elementary aggregate should be included at the time it is assessed as having a signiﬁcant and sus- tainable market share.
If a quality change is detected an appropriate quality adjustment should be made.5 29.

Some products such as seasonal products, insur- ance, second-hand goods, expenditure abroad, interest, own production, expenditures on purchase and con- struction of dwellings, etc., may need special treatment when constructing their weights.

The way these products are dealt with should be determined by the main pur- pose of the index, national circumstances and the practicalities of compilation.

30.
30.

Seasonal products should be included in the bas- ket.

It is possible to use: (i) a ﬁxed weight approach which uses the same weight for the seasonal product in all months using an imputed price in the out-of-season months; or (ii) a variable weights approach where a changing weight is attached to the product in various months.

The decision on the approach should be based on national circumstances.

31.
31.

The expenditure weights for second-hand goods should be based either on the net expenditure of the reference population on such goods, or the gross expenditure, depending on the purpose of the index.

32.

When consumption from own production is within the scope of the index, the weights should be based on the value of quantities consumed from own production.

Valuation of consumption from own pro- duction should be made on the basis of prices prevailing 3See Annex 1.

4See Annex 2.

5See Annex 2.


485 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE on the market, unless there is some reason to conclude that market prices are not relevant or cannot be reliably observed, or there is no interest in using hypothetically imputed prices.

In this case the expenditures and prices for the inputs into the production of these goods and services could be used instead.

The third option is to valuate it by using quality-adjusted market prices.

Sampling for price collection 33.


A CPI is an estimate based on a sample of households to estimate weights, and a sample of zones within regions, a sample of outlets, a sample of goods and services and a sample of time periods for price observation.

34.
34.

The sample size and sample selection methods for both outlets and the goods and services for which price movements over time are to be observed should ensure that the prices collected are representative and suﬃcient to meet the requirements for the accuracy of the index, but also that the collection process is cost-eﬀective.
The sample of prices should reﬂect the importance, in terms of relative expenditures, of the goods and services available for purchase by consumers in the reference period, the number, types and geographic spread of outlets that are relevant for each good and service, and the dispersion of prices and price changes across outlets.

35.

Probability sampling techniques are the preferred methods, in principle, as they permit sound statistical inference and control over the representativity of the sample.
In addition, they permit estimation of sampling variation (errors).

However, they may be costly to implement and can result in the selection of products that are very diﬃcult to price to constant quality.

36.

In cases where appropriate sampling frames are lacking and it is too costly to obtain them, samples of outlets and products have to be obtained by non-prob- should use available ability methods.
Statisticians information and apply their best judgement to ensure that representative samples are selected.

The possibility of applying cut-oﬀ or detailed quota sampling6 strategy may be considered, especially where the sample size is small.

A mixture of probability and non-probability sampling techniques may be used.

37.

Eﬃcient and representative sampling, whether random or purposive, requires comprehensive and up- to-date sampling frames for outlets and products.
Sample selection can be done either by head oﬃce from centrally held sampling frames, or in the ﬁeld by price collectors, or by a mixture of the two.

In the ﬁrst case, price collectors should be given precise instructions on which outlets to visit and which products to price.

In the second case, price collectors should be given detailed and unambiguous guidelines on the local sampling procedures to be adopted.
Statistical business registers, business telephone directories, results from the point-of-purchase surveys or from surveys of sales in diﬀerent types of outlets, and lists of Internet sellers may be used as sampling frames for the central selection of outlets.

Catalogues or other product lists drawn up 6See Annex 1.
486 by major manufacturers, wholesalers or trade associa- tions, or lists of products that are speciﬁc to individual outlets such as large supermarkets might be used as the sampling frame for selection of products.

Data scanned by bar-code readers at the cashier’s desk (electronic databases) can be particularly helpful in the selection of goods and services.

38.

The sample of outlets and of goods and services should be reviewed periodically and updated where necessary to maintain its representativeness.
Index calculation 39.

The compilation of a CPI consists of collecting and processing price and expenditure data according to speciﬁed concepts, deﬁnitions, methods and practices.

The detailed procedures that are applied will depend on particular circumstances.

40.

CPIs are calculated in steps.

In the ﬁrst step, the elementary aggregate indices are calculated.

In the subsequent steps, higher-level indices are calculated by aggregating the elementary aggregate indices.

Elementary aggregate indices 41.


The elementary aggregate is the smallest and relatively homogeneous set of goods or services for which expenditure data are deﬁned (used) for CPI pur- poses.

It is the only aggregate for which an index number is constructed without any explicit expenditure weights, although other kinds of weights might be explicitly or implicitly introduced into the calculation.
The set of goods or services covered by an elementary aggregate should be similar in their end-uses and are expected to have similar price movements.

They may be deﬁned not only in terms of their characteristics but also in terms of the type of location and outlet in which they are sold.

The degree of homogeneity achieved in practice will depend on the availability of corresponding expen- diture data.

42.

An elementary index is a price index for an ele- mentary aggregate.
As expenditure weights usually cannot be attached to the prices or price relatives for the sampled products within the elementary aggregate, an elementary index is usually calculated as an unweighted average of the prices or price relatives.

When some information on weights is available, this should be taken into account when compiling the elementary indices.

43.

There are several ways in which the prices, or the price relatives, might be averaged.
The three most commonly used formulae are the ratio of arithmetic mean prices (RAP), the geometric mean (GM) and the arithmetic mean of price relatives (APR).

The choice of formula depends on the purpose of the index, the sample design and the mathematical properties of the formula.

It is possible to use diﬀerent formulae for diﬀerent ele- mentary aggregates within the same CPI.
It is recom- mended that the GM formula be used, particularly where there is a need to reﬂect substitution within the elementary aggregate or where the dispersion in prices or price changes within the elementary aggregate is large.

The GM has many advantages because of its mathematical properties.
The RAP may be used for STATISTICIANS RESOLUTION CONCERNING CONSUMER PRICE INDICES, 2003 elementary aggregates that are homogeneous and where consumers have only limited opportunity to substitute or where substitution is not to be reﬂected in the index.

The APR formula should be avoided in its chained form, as it is known to result in biased estimates of the elementary indices.

44.

The elementary index may be computed by using either a chained or direct form of the formula chosen.
The use of a chained form may make the estimation of missing prices and the introduction of replacement products easier.

Upper-level indices 45.

These price indices are constructed as weighted averages of elementary aggregate indices.

Several types of formulae can be used to average the elementary aggregate indices.

In order to compile a timely index, the practical option is to use a formula that relies on the weights relating to some past period.
One such formula is the Laspeyres-type index, the formula used by most national statistical agencies.

46.

For some purposes it may be appropriate to cal- culate the index retrospectively by using an index number formula that employs both base period weights and current period weights, such as the Fisher, To¨ rn- qvist or Walsh index.
Comparing the diﬀerence between the index of this type and the Laspeyres-type index can give some indication of the combined impact of income changes, preference changes and substitution eﬀects over the period in question, providing important infor- mation for producers and users of the CPI.

47.
47.

Where the change in an upper-level index between two consecutive periods such as t1 and t is calculated as the weighted average of the individual indices between t1 and t, care should be taken to ensure that the weights are updated to take account of the price changes between the price reference period 0 and the preceding period t1.

Failure to do so may result in a biased index.

Price observations 48.


The number and quality of the prices collected are critical determinants of the reliability of the index, along with the speciﬁcations of the products priced.

Standard methods for collecting and processing price information should be developed and procedures put in place for collecting them systematically and accurately at regular intervals.
Price collectors should be well trained and well supervised, and should be provided with a comprehen- sive manual explaining the procedures they have to follow.

Collection 49.

An important consideration is whether the index or parts of the index should relate to monthly (or quarterly) average prices or to prices for a speciﬁc per- iod of time (e.g.

a single day or week in a month).


This decision is related to a number of issues, which include the use of an index, the practicalities of carrying out price collection and the pattern of price movements.

When point-in-time pricing is adopted, prices should be collected over a very small number of days each month (or quarter).

The interval between price observations should be uniform for each product.

Since the length of the month (or quarter) varies, this uniformity needs to be deﬁned carefully.
When the aim is monthly (or quarterly) average prices, the prices collected should be representative of the period to which they refer.

50.

Attention should also be paid to the time of day selected for price observation.

For example, in the case of perishable goods, price observations may need to be collected at the same time on the same day of the week and not just before closing time, when stocks may be low, or sold cheaply to minimize wastage.

51.
51.

Price collection should be carried out in such a way as to be representative of all geographical areas within the scope of the index.

Special care should be taken where signiﬁcant diﬀerences in price movements between areas may be expected.

52.

Prices should be collected in all types of outlets that are important, including Internet sellers, open-air markets and informal markets, and in free markets as well as price-controlled markets.
Where more than one type of outlet is important for a particular type of product, this should be reﬂected in the initial sample design and an appropriately weighted average should be used in the calculation of the index.

53.

Speciﬁcations should be provided detailing the variety and size of the products for which price infor- mation is to be collected.
These should be precise enough to identify all the price-determining characteristics that are necessary to ensure that, as far as possible, the same goods and services are priced in successive periods in the for same outlet.

The speciﬁcations should include, example, make, model, size, terms of payment, condi- tions of delivery, type of guarantees and type of outlet.

This information could be used in the procedures used for replacement and for quality adjustment.

54.
54.

Prices to be collected are actual transaction prices, including indirect taxes and non-conditional discounts, that would be paid, agreed or costed (accepted) by the reference population.

Where prices are not displayed or have to be negotiated, where quantity units are poorly deﬁned or where actual purchase prices may deviate from listed or ﬁxed prices, it may be necessary for the price collectors to purchase products in order to determine the transaction prices.
A budget may be provided for any such purchases.

When this is not possible, consideration may be given to interviewing customers about the prices actually paid.

Tips for services, where compulsory, should be treated as part of the price paid.

55.

Exceptional prices charged for stale, shop-soiled, damaged or otherwise imperfect goods sold at clearance prices should be excluded, unless the sale of such pro- ducts is a permanent and widespread phenomenon.
Sale prices, discounts, cut prices and special oﬀers should be included when applicable to all customers without there being signiﬁcant limits to the quantities that can be purchased by each customer.

56.
56.

In periods of price control or rationing, where limited supplies are available at prices which are held at a low level by measures such as subsidies to the sellers, government procurement, price control, etc., such prices 487 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE as well as those charged on any signiﬁcant unrestricted markets should be collected.
The diﬀerent price obser- vations should be combined in a way that uses the best information available with respect to the actual prices paid and the relative importance of the diﬀerent types of sales.

57.

For each type of product, diﬀerent alternatives for collecting prices should be carefully investigated, to ensure that the price observations could be made reli- ably and eﬀectively.
Means of collection could include visits to outlets with paper forms or hand-held devices, interviews with customers, computer-assisted telephone interviews, mail-out questionnaires, brochures, price lists provided by large or monopoly suppliers of services, scanner data and prices posted on the Internet.

For each alternative, the possible cost advantages need to be balanced against an assessment of the reliability and timeliness of each of the alternatives.

58.
58.

Where centrally regulated or centrally ﬁxed prices are collected from the regulatory authorities, checks should be made to ascertain whether the goods and services in question are actually sold and whether these prices are in fact paid.

For goods and services where the prices paid are determined by combinations of sub- scription fees and piece rates (e.g.

for newspapers, journals, tele- communications) care must be taken to ensure that a representative range of price oﬀers is observed.
Care must also be taken to ensure that prices charged to diﬀerent types of consumers are observed, e.g.

those linked to the age of the purchaser or to memberships of particular associations.

transport, electricity public and 59.


The collected price information should be reviewed for comparability and consistency with pre- vious observations, the presence of replacements, unu- sual or large price changes and to ensure that price conversions of goods priced in multiple units or varying quantities are properly calculated.

Extremely large or unusual price changes should be examined to determine whether they are genuine price changes or are due to changes in quality.
Procedures should be put in place for checking the reliability of all price observations.

This could include a programme of direct pricing and/or selective repricing of some products shortly after the initial observation was made.

60.

Consistent procedures should be established for dealing with missing price observations because of, e.g.

inability to contact the seller, non-response, observation rejected as unreliable or products temporarily unavail- able.
Prices of non-seasonal products that are tem- porarily unavailable should be estimated until they reappear or are replaced, by using appropriate estima- tion procedures, e.g.

imputation on the basis of price changes of similar non-missing products.

Carrying for- ward the last observed price should be avoided, espe- cially in periods of high inﬂation.

Replacements 61.

Replacement of a product will be necessary when it disappears permanently.
Replacement should be made within the ﬁrst three months (quarter) of the product 488 becoming unavailable.

It may also be necessary when the product is no longer available or sold in signiﬁcant quantities or under normal sale conditions.

Clear and precise rules should be developed for selecting the replacement product.
Depending on the frequency of sampling and the potential for accurate quality adjust- ment, the most commonly used alternatives are to select: (i) the most similar to the replaced variety; (ii) the most popular variety among those that belong to the same elementary aggregate; and (iii) the variety most likely to be available in the future.
Precise procedures should be laid down for price adjustments with respect to the diﬀerence in characteristics when replacements are necessary, so that the impact of changes in quality is excluded from the observed price.

62.

Replacement of an outlet may be motivated if prices cannot be obtained, e.g.

because it has closed permanently, because of a decline in representativeness or because the outlet no longer cooperates.
Clear rules should be established on when to discontinue price observations from a selected outlet, on the criteria for selecting a replacement, as well as on the adjustments that may be required to price observations or weights.

Such rules should be consistent with the objectives of the index and with the way in which the outlet sample has been determined.

63.
63.

Deletion of an entire elementary aggregate will be necessary if all products in that elementary aggregate disappear from most or all outlets and it is not possible to locate a suﬃcient number of price observations to continue to compile a reliable index for this elementary aggregate.

In such situations, it is necessary to redis- tribute the weight assigned to the elementary aggregate among the other elementary aggregates included in the next level of aggregation.

Quality changes 64.
Quality changes 64.

The same product should be priced in each period as long as it is representative.

However, in practice, products that can be observed at diﬀerent time periods may diﬀer with respect to package sizes, weights, volumes, features and terms of sale, as well as other characteristics.
Thus it is necessary to monitor the characteristics of the products being priced to ensure that the impact of any diﬀerences in price-relevant or utility-relevant characteristics can be excluded from the estimated price change.

65.

Identifying changes in quality or utility is rela- tively more diﬃcult for complex durable goods and services.

It is necessary, therefore, to collect a con- siderable amount of information on the relevant char- acteristics of the products for which prices are collected.
The most important information can be obtained in the course of collecting prices.

Other sources of information on price-relevant or utility-relevant characteristics can be producers, importers or wholesalers of the goods included and the study of articles and advertisements in trade publications.

66.

When a quality change is detected, an adjustment must be made to the price, so that the index reﬂects as nearly as possible the pure price change.
If this is STATISTICIANS RESOLUTION CONCERNING CONSUMER PRICE INDICES, 2003 not done, the index will either record a price change that has not taken place or fail to record a price change that did happen.

The choice of method for such adjustments will depend on the particular goods and services involved.

Great care needs to be exercised because the accuracy of the resulting index depends on the quality of this process.
To assume automatically that all price change is a reﬂection of the change in quality should be avoided, as should the automatic assumption that products with diﬀerent qualities are essentially equivalent.

67.

The methods for estimating quality-adjusted pri- ces7 may be: (a) Explicit (or direct) quality adjustment methods that directly estimate the value of the quality diﬀerence between the old and new product and adjust one of the prices accordingly.
Pure price change is then implicitly estimated as the diﬀerence in the adjusted prices.

(b) Implicit (or indirect) quality adjustment methods which estimate the pure price change component of the price diﬀerence between the old and new products based on the price changes observed for similar products.

The diﬀerence between the estimate of pure price change and the observed price change is considered as change due to quality diﬀerence.

Some of these methods are complex, costly and diﬃcult to apply.
The methods used should as far as possible be based on objective criteria.

Accuracy 68.

As with all statistics, CPI estimates are subject to errors that may arise from a variety of sources.8 Com- pilers of CPIs need to be aware of the possible sources of error, and to take steps during the design of the index, its construction and compilation processes to minimize their impact, for which adequate resources should be allocated.

69.
The following are some well-known sources of potential error, either in pricing or in index construc- tion, that over time can lead to errors in the overall CPI: incorrect selection of products and incorrect observation and recording of their prices; incorrect selection of outlets and timing of price collection; failure to observe and adjust correctly for quality changes; appearance of new goods and outlets; failure to adjust for product and outlet substitution or loss of representativity; the use of inappropriate computing elementary for aggregate and upper level indices.
formulae 70.

To reduce the index’s potential for giving a mis- leading picture, it is in general essential to update weights and baskets regularly, to employ unbiased ele- mentary aggregate formulae, to make appropriate adjustments for quality change, to allow adequately and correctly for new products, and to take proper account of substitution issues as well as quality control of the entire compilation process.

Dissemination 71.
Dissemination 71.

The CPI estimate should be computed and pub- licly released as quickly as possible after the end of the period to which it refers, and according to a pre- announced timetable.

It should be made available to all users at the same time, in a convenient form, and should be accompanied by a short methodological explanation.

Rules relating to its release should be made publicly available and strictly observed.
In particular, they should include details of who has pre-release access to the results, why, under what conditions, and how long before the oﬃcial release time.

72.

The general CPI should be compiled and released monthly.

Where there is no strong user demand for a monthly series or countries do not have the necessary resources, the CPI may be prepared and released quar- terly.

Depending on national circumstances, sub-indices may be released with a frequency that corresponds to users’ needs.

73.
73.

When it is found that published index estimates have been seriously distorted because of errors or mis- takes made in their compilation, corrections should be made and published.

Such corrections should be made as soon as possible after detection according to publicly available policy for correction.

Where the CPI is widely used for adjustment purposes for wages and contracts, retrospective revisions should be avoided to the extent possible.

74.
74.

The publication of the CPI results should show the index level from the index reference period.

It is also useful to present derived indices, such as the one that shows changes in the major aggregates between: (i) the current month and the previous month; (ii) the current month and the same month of the previous year; and (iii) the average of the latest 12 months and the average of the previous 12 months.
The indices should be pre- sented in both seasonally adjusted and unadjusted terms, if seasonally adjusted data are available.

75.

Comments and interpretation of the index should accompany its publication to assist users.

An analysis of the contributions of various products or group of pro- ducts to the overall change and an explanation of any unusual factors aﬀecting the price changes of the major contributors to the overall change should be included.

76.
76.

Indices for the major expenditure groups should also be compiled and released.

Consideration should be given to compiling indices for the divisions and groups of the COICOP.9 Sub-indices for diﬀerent regions or population groups, and alternative indices designed for analytical purposes, may be compiled and publicly released if there is a demand from users, they are judged to be reliable and their preparation is cost eﬀective.

77.
77.

The index reference period may be chosen to coincide with the latest weight reference period or it could be established to coincide with the base period of other statistical series.

It should be changed as fre- quently as necessary to ensure that the index numbers remain easy to present and understand.

7See Annex 2.

8See Annex 3.

9See Annex 4.

489 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE 78.
Average prices and price ranges for important and reasonably homogeneous products may be estimated and published in order to support the research and analytical needs of users.

79.

Countries should report national CPI results and methodological information to the International Labour Oﬃce as soon as possible after their national release.

80.
80.

Comparing national CPI movements across countries is diﬃcult because of the diﬀerent measure- ment approaches used by countries for certain pro- ducts, particularly housing and ﬁnancial services.

The exclusion of housing (actual rents and either imputed rents or acquisition of new houses, and maintenance and repair of dwelling) and ﬁnancial services from the all-items index will make the resulting estimates of price change for the remaining products more comparable across countries.
Therefore, in addition to the all-items index, countries should, if possible, compile and pro- vide for dissemination to the international community an index that excludes housing and ﬁnancial services.

It should be emphasized, that even for the remaining products in scope, there can still be diﬃcul- ties when making international comparisons of changes in consumer prices.

though, Consultations and integrity 81.


The compiling agency should have the profes- sional independence, competence and resources neces- sary to support a high quality CPI programme.

The UN Fundamental Principles of Oﬃcial Statistics10 and the ILO Guidelines concerning dissemination practices for labour statistics11 should be respected.

82.
82.

The agency responsible for the index should con- sult representatives of users on issues of importance for the CPI, particularly during preparations for any changes to the methodology used in compiling the CPI.

One way of organizing such consultations is through the establishment of advisory committee(s) on which social partners, as well as other users and independent experts, might be represented.

83.
83.

In order to ensure public conﬁdence in the index, a full description of the data collection procedures and the index methodology should be prepared and made widely available.

Reference to this description should be made when the CPI is published.

The documentation should include an explanation of the main objectives of the index, details of the weights, the index number formulae used, and a discussion of the accuracy of the index estimates.
The precise identities of the outlets and goods and services used for price collection should not be revealed.

84.

Users should be informed in advance of any changes that are going to be made to the scope, weights or methodology used to estimate the CPI.

85.

Technical guidance on the compilation of con- sumer price indices is provided in the Consumer price index manual: Theory and practice.12 This manual 10UN Economic and Social Council, 1994.
11Sixteenth International Conference of Labour Statisticians, 1998.

12ILO/IMF/OECD/Eurostat/UNECE/World Bank, Geneva, 2004.

490 should be updated periodically in order to reﬂect current best practice.

Annex 1 Terminology and deﬁnitions (a) ‘‘Consumer goods’’ are goods or services that are used by households for the satisfaction of individual needs or wants.
(b) ‘‘Consumption expenditures’’ are expenditure on consumer goods and services and can be deﬁned in terms of ‘‘acquisition’’,13 ‘‘use’’, or ‘‘payment’’: – ‘‘acquisition’’ indicates that it is the total value of the goods and services acquired during a given period that should be taken into account, irre- spective of whether they were wholly paid for or used during the period.
This approach could be extended to include the estimated values of own- account production and social transfers in kind received from government or non-proﬁt institu- tions.
The prices enter the CPI in the period when consumers accept or agree prices, as distinct from the time payment is made; – ‘‘use’’ indicates that it is the total value of all goods and services actually consumed during a given period that should be taken into account; for durable goods this approach requires valuing the services provided by these goods during the period.
The prices (opportunity costs) enter the CPI in the period of consumption; – ‘‘payment’’ indicates that it is the total payment made for goods and services during a given period that should be taken into account, without regard to whether they were delivered or used during the period.

The prices enter the CPI in the period or periods when the payment is made.

(c) ‘‘Scope of the index’’ refers to the population groups, geographic areas, products and outlets for which the index is constructed.
(d) ‘‘Coverage’’ of the index is the set of goods and services represented in the index.

For practical rea- sons, coverage may have to be less than what cor- responds to the deﬁned scope of the index.

to that speciﬁc population group for which the index has been constructed.
(e) ‘‘Reference population’’ refers (f ) ‘‘Weights’’ are the aggregate consumption expendi- tures on any set of goods and services expressed as a proportion of the total consumption expenditures on all goods and services within the scope of the index in the weight reference period.

They are a set of numbers summing up to unity.

(g) ‘‘Price updating of weights’’ is a procedure that is used to bring the expenditure weights in line with the index or price reference period.
The price- updated weights are calculated by multiplying the 13This deﬁnition diﬀers from the one adopted by the Fourteenth ICLS (1987).

STATISTICIANS RESOLUTION CONCERNING CONSUMER PRICE INDICES, 2003 weights from the weight reference period by ele- changes mentary indices measuring the price between weight reference and price reference period and rescaling to sum to unity.

(h) ‘‘Index reference period’’ is the period for which the value of the index is set at 100.0.
(i) ‘‘Price reference period’’ is the period whose prices are compared with the prices in the current period.

The period whose prices appear in the denominators of the price relatives.

( j) The ‘‘weight reference period’’ is the period, usually a year, whose estimates of the volume of consumption and its components are used to calculate the weights.
(k) ‘‘Probability sampling’’ is the selection of a sample of units, such as outlets or products, in such a way that each unit in the universe has a known non-zero probability of selection.

(l) ‘‘Cut-oﬀ sampling’’ is a sampling procedure in which a predetermined threshold is established with all units in the relevant population at or above the threshold being eligible for inclusion in the sample and all units below the threshold being excluded.
The threshold is usually speciﬁed in terms of the size of some relevant variable (such as some percentage of total sales), the largest sampling units being included and the rest excluded.

(m) ‘‘Quota sampling’’ is a non-probability method where the population is divided into certain strata.

For each stratum, the number (‘‘quota’’) of ele- ments to be included in the sample is speciﬁed.
The price collector simply ‘‘ﬁlls the quotas’’, which in the case of outlet sampling, that the means, selection of the outlets is based on the judgement of the price collectors and the speciﬁed criteria.
the are (n) ‘‘Imputed expenditures’’ expenditures assigned to a product that has not been purchased, such as a product that has been produced by the household for its own consumption (including housing services produced by owner-occupiers), a product received as payment in kind or as a free transfer from government or non-proﬁt institutions.

(o) ‘‘Imputed price’’ refers to the estimated price of a product whose price during a particular period has not been observed and is therefore missing.
It is also the price assigned to a product for which the expenditures have been imputed, see (n).

(p) ‘‘Outlet’’ indicates a shop, market stall, service establishment, Internet seller or other place where goods and/or services are sold or provided to con- sumers for non-business use.
(q) ‘‘Linking’’ means joining together two consecutive sequences of price observations, or price indices, that overlap in one or more periods, by rescaling one of them so that the value in the overlap period is the same in both sequences, thus combining them into a single continuous series.

is deﬁned as the value of one unit of a (r) ‘‘Price’’ product, for which the quantities are perfectly homogeneous not only in a physical sense but also in respect of a number of other characteristics.
(s) ‘‘Pure price change’’ is that change in the price of a good or service which is not due to any change in its quality.

When the quality does change, the pure price change is the price change remaining after eliminating the estimated contribution of the change in quality to the observed price change.
refers (t) ‘‘Quality adjustment’’ to the process of adjusting the observed prices of a product to remove the eﬀect of any changes in the quality of that pro- duct over time so that pure price change may be identiﬁed.

(u) ‘‘Consumer substitution’’ occurs when, faced with changes in relative price, consumers buy more of the good that has become relatively cheaper and less of the good that has become relatively more expensive.
It may occur between varieties of the same product or between diﬀerent expenditure categories.

Annex 2 Quality adjustment methods Implicit quality adjustment methods 1.

The ‘‘overlap’’ method assumes that the entire price diﬀerence at a common point in time between the disappearing product and its replacement is due to a diﬀerence in quality.

2.
2.

The ‘‘overall mean imputation’’ method ﬁrst calcu- lates the average price change for an aggregate without the disappearing product and its replace- ment, and then uses that rate of price change to impute a price change for the disappearing product.

It assumes that the pure price diﬀerence between the disappearing product and its replacement is equal to the average price changes for continuing (non-miss- ing) products.

3.
3.

The ‘‘class mean imputation’’ method is a variant of the overall mean imputation method.

The only dif- ference is in the source of the imputed rate of price change to period t+1 for the disappearing product.

Rather than using the average index change for all the non-missing products in the aggregate, the imputed rate of price change is estimated using only those price changes of the products that were judged essentially equivalent or were directly quality- adjusted.
Explicit quality adjustment methods 4.

The ‘‘expert’s adjustment’’ method relies on the jud- gement of one or more industry experts, commodity specialists, price statisticians or price collectors on the value of any quality diﬀerence between the old and replacement product.

None, some, or all of the price diﬀerence may be attributed to the improved quality.

5.
5.

The ‘‘diﬀerences in production costs’’ approach relies on the information provided by the manufacturers on the production costs of new features of the replace- ments to which retail mark-ups and associated indirect taxes are then added.

This approach is most practicable in markets with a (new models), 491 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE relatively small number of producers, with infrequent and predictable model updates.
However, it should be used with caution as it is possible for new pro- duction techniques to reduce costs while simulta- neously improving quality.

6.

The ‘‘quantity adjustment’’ method is applicable to products for which the replacement product is of a diﬀerent size to the previously available one.

It should only be used if the diﬀerences in quantities do not have an impact on the quality of the good.

7.
7.

The ‘‘option cost’’ method adjusts the price of the replacements for the value of the new observable characteristics.

An example of this is the addition of a feature that earlier has been a priced option as standard to a new automobile model.

8.

A ‘‘hedonic’’ regression method estimates the price of a product as a function of the characteristics it pos- sesses.
The relationship between the prices and all relevant and observable price-determining char- acteristics is ﬁrst estimated and then results are used in the estimation of the index.

Annex 3 Types of errors – ‘‘Quality change error’’ is the error that can occur as a result of the index’s failure to make proper allowance for changes in the quality of goods and services.
– ‘‘New goods error’’ is the failure to reﬂect either price changes in new products not yet sampled, or given a COLI objective, the welfare gain to consumers when those products appear.

– ‘‘Outlet substitution error’’ can occur when con- sumers shift their purchases among outlets for the same product without proper reﬂection of this shift in the data collection for the index.

– ‘‘New outlets error’’ is conceptually identical to new goods error.
It arises because of the failure to reﬂect either price changes in new outlets not yet sampled, or the welfare gain to consumers when the new outlets appear.

– ‘‘Upper-level substitution error’’ arises when the index does not reﬂect consumer substitution among the basic categories of consumption owing to the use of an inappropriate method for aggregating elementary aggregates in the construction of the overall index value.
Only relevant to a COLI, although an equiva- lent (representativity error) may be deﬁned from the perspective of the pure price index.

– ‘‘Elementary index error’’ arises from the use of an inappropriate method for aggregating price quota- tions at the very lowest level of aggregation.

The elementary index error can take two forms: formula error and lower-level substitution error.
The index suﬀers from formula error if, as a result of the properties of the formula, the result produced is biased relative to what would have been the result if a pure price change could have been estimated.

The index suﬀers from lower-level substitution error if it 492 does not reﬂect consumer substitution among the products contained in the elementary aggregate.
– ‘‘Selection error’’ arises when the sample of price observations is not fully representative of the intended population of outlets or products.

The ﬁrst four types of errors listed above can be seen as special cases of this type of error.
Annex 4 Classiﬁcation of Individual Consumption according to Purpose (COICOP)14 (breakdown of individual consumption expenditure of households by division and group) 01 Food and non-alcoholic beverages 01.1 Food 01.2 Non-alcoholic beverages 02 Alcoholic beverages, tobacco and narcotics 02.1 Alcoholic beverages 02.2 Tobacco 02.3 Narcotics 03 Clothing and footwear 03.1 Clothing 03.2 Footwear 04 Housing, water, electricity, gas and other fuels 04.1 Actual rentals for housing 04.2 Imputed rentals for housing 04.3 Maintenance and repair of the dwelling 04.4 Water supply and miscellaneous services related to the dwelling 04.5 Electricity, gas and other fuels 05 Furnishings, household equipment and routine household maintenance 05.1 Furniture and furnishings, carpets and other ﬂoor coverings 05.2 Household textiles 05.3 Household appliances 05.4 Glassware, tableware and household utensils 05.5 Tools and equipment for house and garden 05.6 Goods and services for routine household maintenance 06 Health 06.1 Medical products, appliances and equipment 06.2 Outpatient services 06.3 Hospital services 07 Transport 07.1 Purchase of vehicles 07.2 Operation of personal transport equipment 07.3 Transport services 08 Communication 08.1 Postal services 08.2 Telephone and telefax equipment 08.3 Telephone and telefax services 14See Annex 2 on p. 465 for explanatory notes.
STATISTICIANS RESOLUTION CONCERNING CONSUMER PRICE INDICES, 2003 09 Recreation and culture 09.1 Audio-visual, photographic and information processing equipment 09.2 Other major durables for recreation and culture 09.3 Other recreational products and equipment, gardens and pets 09.4 Recreational and cultural services 09.5 Newspapers, books and stationery 09.6 Package holidays 10 Education 10.1 Pre-primary and primary education 10.2 Secondary education 10.3 Post-secondary non-tertiary education 10.4 Tertiary education 10.5 Education not deﬁnable by level 11 Restaurants and hotels 11.1 Catering services 11.2 Accommodation services 12 Miscellaneous goods and services 12.1 Personal care 12.2 Prostitution 12.3 Personal eﬀects n.e.c.
12.4 Social protection 12.5 Insurance 12.6 Financial services n.e.c.

12.7 Other services n.e.c.

493 Annex 4 Spatial comparisons of consumer prices, purchasing power parities and the International Comparison Program 1 Introduction This annex deals with the problem of comparing price levels across diﬀerent areas or regions within a country, as well as across countries.
Even though international price comparisons are required to handle diﬀerences in currencies in diﬀerent countries, the index number pro- blems involved in price comparisons across countries mirror those encountered in comparisons over time.

There is a large body of literature on cross-country comparisons of prices and real income undertaken under the auspices of the International Comparison Program (ICP).
While not providing an exhaustive account of the related problems and relevant aggrega- tion methods, this annex aims to achieve a degree of completeness in the coverage of the problem of con- sumer price comparisons in the manual by adding the spatial and international dimensions to the temporal comparisons dealt with in various chapters of the manual.

The annex also attempts to identify possible avenues for a closer integration between spatial and temporal comparisons of consumer prices.
The main objectives of the annex are: (i) to provide a brief summary of the index number problems encoun- tered in the process of international and inter-area price comparisons and to highlight the need for the develop- ment and use of specialized aggregation methods; (ii) to describe a few aggregation methods used in deriving purchasing power parities (PPPs) and spatial measures of price levels; (iii) to examine the relationship between the ICP and PPPs for cross-country comparisons with the CPI; and (iv) to explore the feasibility of integrating the ICP activities with the streamlined activities of national statistical oﬃces for the compilation of the CPI.
The annex is also designed to provide an introduction for the statisticians in various national statistical oﬃces who may currently be involved in consumer price index (CPI) compilation to the issues and methods involved in spatial comparisons of consumer prices.

The annex outlines in the approaches to spatial comparisons.
Countries embark- ing on inter-area or regional consumer price compar- isons, as well as those countries that may participate in the ICP in the near future, may ﬁnd the contents of the annex useful.

the principal diﬀerences some of 2 Differences between temporal and spatial comparisons There are several major qualitative diﬀerences in the nature of price comparisons involved in the standard CPI comparisons over time and price comparisons over space involving regions or countries.
These diﬀerences high- light the need for specialized methods for aggregating price data in deriving summary measures of price levels, as well as speciﬁc types of data requirements associated with cross-country and inter-area comparisons.

The foremost diﬀerence is the absence of a natural ordering of price and quantity observations in the con- text of cross-country or inter-area comparisons.

The CPI framework and methods are devised to measure changes over time.
Therefore, the price observations appear in a chronological order.

The presence of a natural ordering of price observations makes it possible to examine the feasibility and relative merits of the ﬁxed and chain index numbers.

For example, in the context of constructing price comparisons across countries within the OECD, or across states within the United States, it is impossible to arrive at an ordering which facilitates chained comparisons.
The multilateral nature of spatial comparisons is a distinguishing feature of price comparisons across regions and countries.

When price levels of goods and services across diﬀerent countries are compared, it is essential that such comparisons are undertaken for every pair of regions being considered.

If the World Bank is interested in comparisons of real income in diﬀerent countries, it is necessary for the Bank to be able to make comparisons between all pairs of countries involved.
This multilateral nature of comparisons creates several problems.

First, the number of comparisons (one for each pair) can be quite large, and presentation and use of such results may be quite unwieldy.

For example, if a particular comparison exercise involves 20 countries, then it requires 190 (20 19/2) separate binary com- parisons involving distinct pairs of countries.

Second, results from such a large tableau of binary comparisons require a degree of consistency.
This requirement trans- lates into the ‘‘transitivity’’ condition described below.

The uses and applications of inter-area price compar- isons may diﬀer signiﬁcantly from general consumer price indices.

The CPI is probably the most signiﬁcant economic statistic produced in any country.

It is not only used as a general measure of price changes over time, but often in assessing and calibrating monetary policy.
Despite the conceptual similarities in price comparisons over time and across space, spatial price comparisons are useful in comparisons of standards of living and well- being in diﬀerent regions of a country or across coun- tries.

Such comparisons are essential in assessing devel- opment and in ensuring more balanced growth in dif- ferent regions.

There is considerable demand for measures of CPI across diﬀerent cities and diﬀerent states and regions (rural versus urban) within countries.
How- ever, there are very few countries where inter-area price level comparisons are readily available.

Kokoski et al.

(1999) demonstrate the feasibility of deriving meaningful inter-area price comparisons within the United States using the price data collected across diﬀerent states.
International comparisons of prices, in the form of PPPs from the ICP, are used by international organi- zations and individual researchers in assessing growth and productivity performance of countries, and also in 495 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE making meaningful comparisons of various national income aggregates (including government expenditure) across diﬀerent countries.
Currently, consensus is emerging among researchers and practitioners that price comparisons and PPPs are necessary in assessing the nature and extent of global poverty and its distribution across countries and regions of the world.

Several recent research papers by Ward (2001), Prennushi (2001), Astin (2001) and Dwyer et al.
(2001), presented at the joint World Bank–OECD Seminar on Purchasing Power Parities, 30 January to 2 February 2001, highlight a number of important applications of PPPs derived from international comparisons of prices under the ICP.

Eurostat has recently embarked on a programme to extend the current ICP for its EU Member States to cover regional comparisons within diﬀerent countries.
In recognition of the major analytical diﬀerences between the standard CPI comparisons over time and spatial and cross-country comparisons of consumer prices and PPPs, considerable research eﬀorts have focused on the development of the data and methods necessary for spatial comparisons of prices.

A brief summary of the results of this research is given below.
3 Data requirements for spatial comparisons The basic data requirements for spatial comparisons are very similar to the data required for standard CPI calculation.

The main components are the data on prices of a large range of products representative of the con- sumption baskets of households and information on weights associated with various product categories reﬂecting the importance attached to diﬀerent products.
Within the CPI, it is common practice to collect price quotations from diﬀerent outlets scattered throughout the country.

The selection of the outlets and areas from which prices are collected is based on complex multi- stage sampling designs.

The expenditure weights are based on a classiﬁcation of goods and services using a standard system such as the Classiﬁcation of Individual Consumption according to Purpose (COICOP) or a similar national classiﬁcation.
The lowest level of pro- duct classiﬁcation at which expenditure weights are available is used in identifying the elementary indices and higher-level indices at progressively higher levels of aggregation, leading ultimately to the total household expenditure level.

Spatial comparisons pose several problems in terms of identifying products that are to be priced from diﬀerent areas, regions or countries involved in a comparison exercise.
This problem is less severe when fairly similar or homogeneous areas are being compared.

In cases where comparisons involve areas that are fairly hetero- geneous, two problems arise.

The ﬁrst arises from major diﬀerences in the consumption baskets.

For example, when comparisons are made between two states, say Minnesota and Florida, there may be major diﬀerences in the consumption baskets at the detailed level, even though the major expenditure categories may be iden- tical.
This problem is somewhat similar to the treatment of disappearing and new goods in the context of the 496 CPI, but is more serious when cross-country compar- isons are being attempted.

The second problem arises from major diﬀerences in the quality of items.

The quality diﬀerences may be measured through several product characteristics, with allowances and adjust- ments being made at the appropriate stage of index number calculation.

Kokoski et al.
Kokoski et al.

(1999) demonstrate the feasibility of making inter-area price comparisons for heterogeneous goods.

Changes in quality are likely to be more gradual in the case of temporal comparisons, but can be a serious problem when comparisons across countries are attempted.

The ICP follows the principle of identity in dealing with the problem of quality diﬀerences across countries.
A comprehensive list of products with detailed product speciﬁcations is developed at the planning stages of any cross-country comparison exer- cise.

These items are priced in diﬀerent countries from various outlets distributed across the country, a pro- cedure very similar to that used in the CPI.

Develop- ment of the product listing is, however, a diﬃcult step, with the degree of diﬃculty depending upon the size and heterogeneity of the group of countries involved.
The use of a product listing, based on the identity principle, can have serious implications for the representativeness of the product list of the consumption baskets in dif- ferent countries.

There are several operational proce- dures used by international organizations in handling these problems relating to the compilation of price data.
A more detailed account of the problems and recom- mended solutions can be found in ICP Handbook (United Nations, 1992) and the recent publication by OECD (1999) on its international comparison work.

Once the price data are compiled, the next stage in the CPI compilation is the aggregation of item-speciﬁc price changes to measure price movements for various cate- gories of consumption expenditure.

At this stage, it is necessary to have information on consumption patterns.
This information is usually drawn from household expenditure surveys.

These surveys are regularly con- ducted in most countries by the respective national statistical organizations.

For purposes of making inter- area comparisons of consumer prices, the corresponding requirement is the availability of household expenditure survey data speciﬁc to each area included in the com- parisons.
In many cases, for reasons relating to sampling and statistical reliability, detailed expenditure pattern data may not be available for all the regions.

Spatial comparisons of consumer prices pose speciﬁc problems because of the non-overlapping nature of the consumption baskets, major diﬀerences in the quality of items priced in diﬀerent regions and countries, and the crucial data on region-speciﬁc non-availability of expenditure patterns.
These problems the development of new analytical techniques that can handle major diﬀerences in quality.

National statistical oﬃces may need additional ﬁnancial resources in order to provide reliable and meaningful price comparisons between diﬀerent cities, areas and regions within coun- tries, and to compile reliable data for the more diﬃcult task of inter-country comparisons of prices and real consumption.
require 4 Aggregation methods for spatial comparisons This section brieﬂy describes the types of aggregation methods that are commonly used in cross-country comparisons of prices.

Since most of these methods have been developed in the context of the ICP, and are equally valid for inter-area or regional comparisons, the discussion below uses countries as spatial entities.

This section is further divided into three parts.


The ﬁrst deals with the notation and conceptual framework necessary to deal with multilateral spatial comparisons.

The sec- ond describes the construction of elementary indices for aggregation of prices when no quantity or expenditure information is available.

Finally, a small selection of index number methods used in spatial price comparisons are presented.
4.1 Notation and conceptual framework Consider the case involving comparisons across M countries, and price and quantity data on N commod- ities.

These commodities refer to goods and services that are priced in all the countries.

If the commodities refer to items below the elementary level at which no quantity or expenditure share data are available, we make use of only the price data.
At this stage, all the problems relating to non-overlapping commodity lists and exis- tence of quality diﬀerences are set aside so that the main focus is just on the aggregation issues.

Let p j=[p j N] represent the price and quantity vectors from country j ( j=1, 2, .

.

.

, M).

In the case of international comparisons, all the prices are expressed in the respective national currency units.
As in the case of the CPI computation, the problem is one of decomposing the diﬀerences in the value aggregates N] and q j=[q j 1, .

.

.

q j 1, .

.

.

p j N V j=P i=1 i q j p j i (A4:1) into measures of price and real expenditure components.

Since there are M sets of price and quantity vectors and, therefore, M(M1)/2 binary comparisons between all distinct pairs of countries, a simpler notation is used in this annex in the place of the notation generally used in the manual.
Let Ijk denote the (consumer) price index number for country k with country j as the base.

If j and k are, respectively, the United States and India, and if Ijk=22.50, then the index is interpreted to mean that 22.50 Indian rupees have the same purchasing power as one US dollar for the goods and services covered in computing the index.

Thus the index can also be inter- preted as the PPP between the currencies of j and k. This interpretation is consistent with the meaning accorded to the CPI.
Since currency denominations are involved here, a proper measure of relative price level diﬀerences can be obtained if the PPP is compared to the exchange rate prevailing at the time when comparisons are made.

Because of the multilateral nature of spatial compar- isons, when M countries are involved, it is necessary to provide comparisons between all pairs of countries.
Thus, it becomes necessary to compute each and every entry in the following matrix of binary comparisons: SPATIAL COMPARISONS OF CONSUMER PRICES I= 2 6 6 4 I11 I21 Ij1 IM1 I12 I22 Ij2 IM2 I1k I2k Ijk IMk 3 7 7 5 I1M I2M IjM IMM (A4:2) Several points concerning the matrix, I, are worth noting.

First, the matrix can be large if the number of countries (or regions) involved is large.

Second, the results recorded in the matrix need to be internally consistent.
All the index number issues and various approaches discussed in the manual apply directly to each binary comparison involving two countries.

Die- wert (1986, 1999b) provides a summary of the micro- economic theoretical and test approaches to cross- it is possible to apply country comparisons.

Thus, Fisher, To¨ rnqvist, Walsh or other index number for- mulae described in the manual.
In order to ensure meaningful interpretation of the results from multilateral cross-country comparisons, the index number methods applied need to satisfy a number of basic requirements, only the most important of which are discussed below.

Kravis et al.

(1982), OECD (1999) and United Nations (1992) provide a complete list of these requirements.

Transitivity.
Transitivity.

An index number formula Ijk is said to satisfy the transitivity property if and only if for all choices of j, k and ‘ ( j, k, ‘=1, 2,..., M), the index satisﬁes Ijk=Ijl Ilk (A4:3) Equation (A4.3) requires that the application of a formula to make a direct comparison, Ijk, should result in the same numerical measure as an indirect compar- ison between j and k through a link country ‘.
Note that the transitivity property ensures internal consistency of index numbers in the matrix given in equation (A4.2).

It guarantees that the PPP for two currencies, say A and B, is the same whether it is derived through a direct com- parison of A and B or through an indirect comparison that compares A with C and C with B, which are then combined to provide an indirect PPP for A and B.
This requirement arises mainly from the spatial nature of the comparisons where no natural ordering of the countries involved could be imposed without a value judgement.

Most of the commonly used index number procedures do not satisfy this requirement.

The following result is useful in constructing transitive index numbers.

An index number formula Ijk satisﬁes the transitivity property in (A4.3) if and only if there exist M positive real numbers l1, l2, .

.

.
.

.

, lM, such that Ijk= lk lj (A4:4) for all j and k. The proof of this result is straightforward (Rao and Banerjee, 1984).

The result is important since it shows that, when the transitivity property is satisﬁed, all that is necessary is to measure M real numbers l1, l2, .

.

.

, lM, and then all the necessary indices in (A4.2) can be cal- culated using these M numbers, thus reducing the dimension of the problem involved.
Two important 497 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE points may be noted.

First, the numbers lj in equation (A4.4) are not unique, since any scalar multiplication of a vector of lj can also lead to the same matrix of index numbers as that derived from the original lj.

Therefore, these lj need to be determined (in any empirical exercise) up to a factor of proportionality.

Second, these lj can be interpreted as the PPP of currencies involved.
This particular result formed the basis of the work of statis- ticians such as Geary (1958) and Khamis (1970), who proposed aggregation methods designed to compute PPPs directly from the price and quantity data without invoking the index number literature.

Base invariance.

An index number formula is said to be base invariant if a comparison between a given pair of countries (j, k) is invariant to the order in which the countries are listed.
This implies that multilateral com- parisons should be invariant to all possible permutations of the data set.

For example, consider a set of transitive comparisons derived using a particular country (say the United States) as a star country.

Under this scheme, price comparison between any pair of countries, say A and B, is eﬀected through the United States which serves as a link country.
Therefore, PPPA, B=PPPA, USA PPPUSA, B This scheme is inadmissible under the base invariance criterion since the choice of the star country clearly aﬀects the PPP of currencies of countries A and B.

Further, the United States is accorded a special status, in the form of a link country, in deriving transitive multilateral comparisons.

Characteristicity.

This is a requirement outlined in Drechsler (1973).
This property requires that any set of multilateral comparisons satisfying the transitivity property should retain the essential features of the binary comparisons constructed without the transitivity implies that a requirement.
Since condition (A4.3) transitive comparison between a pair of countries j and k is necessarily inﬂuenced by the price and quan- tity data for all the other countries, the characteris- ticity property requires resulting from adherence to the transitivity property should be kept at a minimum.
Balk (2001) shows that a complete adherence to the characteristicity principle in its extreme (complete preservation of all binary compar- isons) would imply that price indices, and hence PPPs, cannot depend upon any quantity or expendi- ture share weights.

This is an extreme result, which is to be avoided in all index number comparisons.

The Elteto–Koves–Szulc (EKS) method for multilateral comparisons, discussed below, has its origins in the characteristicity property.
that distortions 4.2 Index number methods for spatial comparisons Spatial price comparisons in general, and interna- tional comparisons in particular, use index number methods for aggregating price and quantity data at two diﬀerent levels.

The ﬁrst is the basic heading level.

This is normally the lowest level of aggregation at which expenditure data and weights are available.
These basic 498 headings usually consist of a fairly homogeneous group of items that are priced in diﬀerent outlets in the countries.

The subsequent levels of aggregation lead to indices for broad expenditure categories, and ﬁnally to the whole consumption basket.

4.2.1 Aggregation below the basic heading level Two commonly used index number methods are described below.
These procedures explicitly allow for the possibility that price data may not be available for items in the product list constructed for a given all international comparison exercise.

Such a situation is possible in the case of temporal comparisons, but is usually limited to a small number of commodities that are either disappearing or new goods.

The Elteto–Koves–Szulc (EKS) Method.


A variant of the original method proposed in Elteto and Koves (1964) and Szulc (1964) is generally used in aggregating price data below the basic heading level.

The EKS method involves two stages.

In the ﬁrst stage, binary comparisons are constructed using price relatives of those commodities for which prices are available in both countries.
If njk is the number of commodities that are priced in both countries, then the current practice within ICP constructs a binary elementary index using the following formula: njk Ijk= Q i=1 " #1=njk pk i p j i (A4:5) Obviously, these indices are not transitive, since each index is based on prices of a diﬀerent set of commod- ities.

The EKS procedure is then used in deriving a transitive set of indices.
The resulting formula for the construction of elementary indices for spatial compar- isons is given by M jk = Q I EKS ‘=1 Ij‘I‘k 1=M (A4:6) The elementary index number formula in (A4.5) is similar to the formula used in the construction of the CPI.

The principal diﬀerence results from the fact that not all commodities are priced in all countries and that there is a need for transitivity at all stages of aggrega- tion.
The properties of these indices are discussed in Chapter 20 of the manual.

The OECD (1999) uses a slightly modiﬁed variant of the binary indices shown in equation (A4.5).

A formula that mimics the standard Fisher index is used, but without the use of any expenditure shares since the aggregation is below the basic heading level.

This procedure tries to account for the fact that not all commodities for which prices are collected are really characteristic or important in one or both of the countries.
The procedure takes explicit account of those commodities which are starred, indicating that the item is important in a given country.
The modiﬁed EKS method uses the same formula as above, but the binary index on the right-hand side is replaced by: 8 < Ijs= Q : i 2 MðsÞ " # 1 nðsÞ ps i p j i " # 1 nðjÞ ps i p j i 1 2 9 = ; (A4:7) Q i 2 Mð jÞ where n(s) and n( j) are, respectively, the number of starred items in countries s and j; M(s) and M( j) are, respectively, the sets of commodities that are starred (considered representative) in diﬀerent countries.
Use of equations (A4.5) and (A4.6) for the construc- tion of spatial CPI numbers at the basic heading level has its problems.

The most important problem is that these formulae do not take into consideration whether or not the commodities priced in diﬀerent countries are ‘‘representative’’ of the consumption in diﬀerent coun- tries within the basic heading.

A related problem is whether or not the coverage of the commodities, priced with respect to the basic heading to which they belong, is adequate.
These issues are currently being researched, and Rao (2001b) oﬀers a modiﬁed approach that atta- ches weights proportional to coverage and representa- tiveness.

While there has been much research on the properties of index number formulae for the construction of ele- mentary indices within the CPI framework (Diewert (1995a), Dale´ n (1992) and Turvey (1996)), there has been very little research on the properties of elementary indices within the context of international comparisons.
The Country–Product–Dummy (CPD) Method.

The CPD method was originally proposed by Summers (1973) as a tool to deal with missing price observations.

The method is a simple statistical device that can be used in deriving the PPPs for a particular basic heading by simply regressing the logarithm of observed prices against a set of dummy variables, deﬁned with respect to commodities and countries.

Thus the procedure involves the model: ln p j i =1D1+2D2+ .

.

.

+nDn+p1D* + .

.

.


+pMDM+u j i 1+p2D* 2 (A4:8) where Di (I=1,2, .

.

.

, n) and D* j (j=1, 2, .

.

.

, M) are, respectively, dummy variables for the N commodities in the basic heading and M countries involved in the comparisons.

Once this regression equation is estimated, the PPP for currency of country k with country j as base can be obtained by PPPj= exp(^ppj) (A4:9) where ^ppj is the estimator of pj in equation (A4.8).
Then the desired index at the basic heading level is given by Ijk= PPPk PPPj : (A4:10) The exponential of the diﬀerence in the estimates of pj and pk obtained from the regression equation provides the necessary index number.

The CPD model oﬀers a number of generalizations that can explicitly account for a number of data-related problems.

The model can be easily generalized to account for diﬀerences in quality measured through a SPATIAL COMPARISONS OF CONSUMER PRICES set of product characteristics.
The feasibility of this approach to inter-area comparisons of consumer prices within the United States was demonstrated in Kokoski et al.

(1999).

Rao and Timmer (2000) examined the feasibility of using a generalized CPD model to incor- porate various measures of reliability in the context of aggregating unit value ratios to provide comparisons at the manufacturing branch level.
Rao (2001b) discusses a few model that are appropriate for aggregation below the basic heading level within the context of the ICP.

speciﬁcations The EKS method, described in equations (A4.6)– (A4.8), is the procedure that is currently used by all the international organizations for purposes of aggregation below the basic heading level.

The CPD and EKS methods yield identical basic heading parities when all the commodities are priced in all the countries.

Ferrari and Riani (1998) and Ferrari et al.


(1996) present a number of analytical results relating to these methods.

While the sampling issues relating to the selection and distribution of outlets and the frequency of price quotations are considered important for the construc- tion of the elementary indices within the CPI, issues that are crucial for the construction of PPPs below the basic heading for inter-country comparisons are quite diﬀerent.
Issues of quality diﬀerences and non-avail- ability of goods and services in all the comparison countries are far more important in spatial compar- isons of consumer prices.

4.2.2 Aggregation above the basic heading level This section presents a small selection of the range of aggregation methods used in the context of spatial comparisons.

A more comprehensive analysis of the spatial aggregation methods developed over the past three decades is presented by Balk (2001).
This level of aggregation is similar to the stage where elementary indices are aggregated to derive the overall CPI.

In the case of temporal comparisons involving two time periods, all the methods and approaches described in the manual are appropriate, and in most cases the national statistical oﬃces use the Laspeyres, Fisher or some variants of these formulae for CPI construction.

However, the multilateral nature of spatial comparisons necessitates slightly diﬀerent approaches to their con- struction.
A number of index number methods for aggregation above the basic heading level have been developed over the past three decades but, in the interest of brevity, only the principal methods are discussed below.

These are the Geary–Khamis and EKS methods for international comparisons, the principal aggregation methods used in various international comparison exercises by the ICP, the OECD, Eurostat and FAO.
Several approaches to the construction of multilateral index numbers that satisfy transitivity and base invar- iance properties are discussed below.

Four distinct approaches emerged during ICP work carried out between 1970 and the early 2000s.

The ﬁrst and most straightforward is the EKS approach, which uses binary results as building-blocks for multilateral comparisons.
The second is the Geary–Khamis approach, which 499 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE provides a methodology for computing PPPs of cur- rencies and international average prices of commodities, using the price–quantity data at the basic heading level.

The third is the stochastic approach based on the CPD method and its generalizations that can be used in econometrically estimating the PPPs using a regression framework.
The fourth and the last approach discussed here is the linking approach to the construction of chained comparisons based on the concept of the minimum spanning tree.

This is generating considerable interest and is explored further in the last section of this annex.

These four approaches are by no means exhaustive, but they represent major strands of research and development in this area.

The EKS Method.
The EKS Method.

The EKS system is a simple method of generating transitive multilateral index numbers from a system of binary index numbers, with the property that the resulting multilateral indices deviate the least (according to a speciﬁc criterion) from the binary indi- ces.

Since the seminal paper by Drechsler (1973), it has been well recognized that (transitive) multilateral sys- tems necessarily deviate from their binary counterparts and therefore result in a loss of ‘‘characteristicity’’.
The EKS system is designed to minimize such loss of char- acteristicity.

The original EKS system uses the Fisher binary indices, but the work of Caves, Christensen and Diewert (1982b) and Rao and Banerjee (1984) recog- nizes that other binary indices could be used in con- junction with the EKS technique.

For any pair of countries j and k, if Fjk represents the Fisher binary index, then M EKSjk= Q Fj‘:F‘k 1=M ‘=1 (A4:11) provides the EKS index.
There are several notable features of the EKS tech- nique.

First, it is based on the premise that direct binary comparisons, derived using any chosen formula, provide the best comparison between pairs of countries.

Second, even though the EKS index in equation (A4.11) is deﬁned using the Fisher index, this approach can be applied in conjunction with any other index number formula.
For example, the Fisher index in equation (A4.11) may be replaced by another superlative index, such as the To¨ rnqvist index.

Caves, Christensen and Diewert (1982b) suggest the use of a To¨ rnqvist-based EKS formula for spatial comparisons.

Third, the EKS index in equation (A4.11) is the multilateral index that deviates the least from the matrix of non-transitive binary indices, when the deviations are measured using a logarithmic distance function.
Finally, the EKS index can be interpreted as a simple geometric mean of all indirect comparisons between j and k through all pos- sible link countries.

The simple unweighted nature of the EKS index has attracted attention in recent years.

Since diﬀerent binary comparisons have diﬀerent levels of reliability, measured using various criteria, it is necessary to reﬂect these diﬀerences in deﬁning weighted EKS index numbers.

Rao and Timmer (2000), Rao et al.


(2000) and Rao (2001b) provide illustrations of how weighted EKS 500 indices can be generated in order to account for various data-related problems.

The Geary–Khamis (GK) Method.

The GK method was originally proposed by Geary (1958) and subse- quently developed by Khamis (1970, 1972 and 1984).

The GK method has been the principal aggregation method in most of the ICP phases to date.
Since 1996, the OECD has produced and published international comparisons based on both the EKS and GK methods.

The GK method provides a way of calculating PPPs of currencies of diﬀerent countries from the observed price and quantity data (applied at the basic heading level).

The concept of PPP is applicable even when the currency unit is the same in several areas of a country.

The GK method simultaneously determines interna- tional average prices of diﬀerent countries.
Let Pi denote the international average price of ith commodity.

The GK method is deﬁned through the following system of interrelated equations, deﬁned for each country j and each commodity i, Pi= M P j=1 i q j p j i =PPPj M P j=1 q j i and PPPj= N P i=1 N P i=1 i q j p j i Piq j i (A4:12) These simultaneous equations are then solved to yield numerical values of PPPs and Ps, after selecting one of the currencies as a numeraire.
Once the PPPs are solved, the spatial price index numbers are simply deﬁned as Ijk= PPPk PPPj (A4:13) One of the main reasons for the continued use of the GK method is ‘‘additivity’’.

Additivity requires that aggregates, such as real domestic product, derived by converting national aggregates using PPPs, should be to aggregates derived through valuation of equal quantities at international prices.
Thus additivity requires n P i=1 i q j p j i =PPPj=P Piq j i (A4:14) n i=1 This requirement is satisﬁed automatically by the PPPs and Ps derived from the GK system deﬁned in equation (A4.12).

The GK system is also useful in analysing the structure of real GDP and shares of dif- ferent components across diﬀerent countries.

This sys- tem provides a framework within which internationally comparable national accounts could be constructed.
However, the GK system is not rooted in standard economic theory and fails several test properties (Die- wert, 1986).

There has been considerable debate among practitioners concerning the average prices resulting from the GK system.

The system has the potential to reﬂect the price structure of the richer countries, and therefore has the tendency to overstate the real income of the poorer countries.

Weighted Country–Product–Dummy (CPD) Method.


It is possible to generalize the CPD method discussed in the context of aggregation below the basic heading level.

Rao (1995) has generalized the CPD method by incor- porating quantity and value data directly into the CPD method described in equation (A4.8).

The basic idea behind this generalization comes from the fact that the standard CPD regression model attempts to track the logarithm of the observed prices using an unweighted residual sum of squares.
In the spirit of the standard index number approach, where price index numbers are required to track price changes of more important commodities more closely, a more appropriate pro- cedure would be to ﬁnd estimates of the parameters that are likely to track important commodities more closely.

This is achieved by minimizing a weighted residual sum of squares, with each observation weighted according to the expenditure share of the commodity in a given country.
Thus the generalized CPD method suggests that estimation of the equation ln pij=p1D1+p2D2+ .

.

.

+pMDM+1D* n+uij + .

.

.

+nD* 1+2D* 2 (A4:15) be conducted after weighting each observation accord- ing to its value share.

This is equivalent to the applica- tion of ordinary least squares to the following trans- formed equation obtained by premultiplying equation (A4.15) by p .

The resulting equation is: ﬃﬃﬃﬃﬃﬃwij p p p ﬃﬃﬃﬃﬃﬃwij ln pij=p1 D1+p2 ﬃﬃﬃﬃﬃﬃwij p +pM ﬃﬃﬃﬃﬃﬃwij + .

.

.
.

.

+n ﬃﬃﬃﬃﬃﬃwij D2+ .

.

.

p ﬃﬃﬃﬃﬃﬃwij D* DM+1 1 p ﬃﬃﬃﬃﬃﬃwij D* n+vij (A4:16) where wij=pijqij=PN basic heading in the jth country.

i=1pijqij is the value share of the ith Rao (1995) has shown that the international prices and PPPs resulting from the estimates of parameters in equation (A4.13) are identical to those derived using the Rao method for international comparisons, described in Rao (1990).
Thus the weighted CPD method may be considered as a bridge between the GK approach to international comparisons and the standard stochastic approach to index numbers.

Kokoski et al.

(1999) outline a procedure which makes use of the CPD method to adjust for diﬀerences in quality characteristics of various consumer items for the construction of inter-area CPIs using United States data.
Estimates derived from the CPD method are subsequently used in the application of the multilateral To¨ rnqvist index derived after imposing the transitivity condition.

Hence the approach used in Kokoski et al.

(1999) may be described as a mixed stochastic and index number approach to multilateral spatial comparisons of CPI numbers.

Spatial linking and chaining approach.

In recent years, a new approach to inter-area and inter-country com- parisons of prices has been given serious consideration.
This approach advocates spatial chaining of binary comparisons where links are identiﬁed using a procedure based on a measure of distance or reliability of binary SPATIAL COMPARISONS OF CONSUMER PRICES comparisons involved.

This approach is in sharp con- trast to the general approach to multilateral compar- isons, where either all the binary comparisons are used, as in the case of the EKS method, or all the price and quantity data are simultaneously used, as in the case of the GK and CPD approaches.
Using the graphical theoretical concept of minimum spanning trees, Hill (1999c, 1999d) proposed a method of deriving a system of transitive multilateral compar- isons from a matrix of binary comparisons.

The Hill approach is based on the fact that direct binary com- parisons may not always be the best.
For any pair of countries j and k, Hill suggests a measure of distance (indicating the reliability of the binary comparison) using the Laspeyres–Paasche spread deﬁned as D(j, k)= ln L(j, k P(j, k) (A4:17) where L(j, k) and P(j, k) are, respectively, binary Las- peyres and Paasche price index numbers.

Note that the same distance function emerges if price index numbers are replaced by quantity index numbers.
D(j, k) is equal to zero if the price structures or quantity structures are identical in countries, j and k. Thus, this distance func- tion serves as an indicator of similarity of price and quantity structures in these countries.

Using a matrix of distances calculated for all pairs of countries, Hill (1999c, 1999d) suggests that a minimum spanning tree (MST) be extracted and used in con- structing chained links between all pairs of countries.
The MST has the property that a chained comparison between any pair of countries has the least distance, and therefore can be considered as the most reliable.

It also has the property that the sum of the distances between all the links, in the MST, is the least when compared to all possible tree-conﬁgurations.

For purposes of illus- trating the concepts involved, Figure A4.1.1 shows the MST for Europe constructed using ICP data for Europe for the 1985 benchmark year.
It is evident from the MST presented here that a comparison between Germany and Portugal is through a chain involving Ireland, Luxembourg and Spain.

This chained comparison is deemed to be better than a direct comparison between these two countries.

There are a number of issues yet to be resolved regarding the use of the MST.

Nonetheless, an MST provides a formaliza- tion of a somewhat intuitive notion of linking dissimilar countries using a chain of similar countries.
Once the MST is identiﬁed, a transitive comparison between a given pair of countries in a particular exercise is constructed using binary indices calculated using a chosen formula, such as the Fisher or To¨ rnqvist index, and the links indicated in the MST.

Thus, if a compar- ison between Sweden and Denmark is needed, then the MST approach suggests the following index for this comparison: I MST Sweden, Denmark=FSweden, Finland FFinland, Denmark where F denotes the Fisher index.
501 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE Figure A4.1 A minimum spanning tree: Europe Austria Germany Luxembourg Greece Italy Netherlands Ireland France Denmark Belgium United Kingdom Hungary Finland Poland Spain Yugoslavia Portugal Sweden Norway Since the MST provides a unique chain of links between any two countries, comparisons are uniquely deﬁned.
The spanning trees are, however, sensitive to the countries included, and the types of measures used in assessing the degree of reliability or comparability of any two countries.

Aten et al.

(2001) examine the sensitivity of the spanning trees and the resulting comparisons based on a range of measures including some similarity indices.

Rao et al.
Rao et al.

(2000) applied the spanning tree approach to the construction of multi- lateral, agricultural input, output and productivity indices, using United States state-level agricultural production data.

5 Integration of the CPI and inter-area and international comparisons The best available inter-area and international price comparisons for consumer goods and services use data from the national statistical oﬃces compiling CPIs.
In recent critical reviews of the ICP by Ryten (1998), and of the OECD–Eurostat PPPs by Castles (1997), the accu- racy and reliability of international comparisons have been assessed as being less than adequate.

Castles notes the diﬃculties encountered in comparing like goods from country to country, and Ryten argues for the need to secure greater support from national statistical oﬃces in compiling price data for the ICP.
Both reviews recommended examination of the feasibility of inte- grating ICP work with that of the normal CPI work undertaken by the national statistical oﬃces.

Since this annex deals with spatial and international comparisons, it is appropriate now to examine and identify possible steps involved in a more integrated approach to the CPI and ICP activities.
This section ﬁrst provides an assessment of potential beneﬁts that could ﬂow from the integration of CPI and ICP activities with statistical systems in general, and price statistics in particular, at both global and national levels.

It is useful to note here that the ICP pro- vides cross-country comparisons of prices of goods and services that enter private consumption, government consumption and investment.

Thus ICP encompasses all components of gross domestic product (GDP).
In con- trast, the CPI focuses mainly on movements in prices of consumer goods and services.

The nature and scope of the CPI and ICP activities and the limits on the extent of an integrated approach are discussed brieﬂy.

The ﬁnal subsection identiﬁes a number of useful initiatives that could provide a framework for a practical, more inte- grated approach to these important activities involving inter-area and international compar- inter-temporal, isons of consumer prices and the ICP.
5.1 Beneﬁts from CPI and ICP integration Globalization and the resulting expansion in inter- national trade and ﬁnancial ﬂows have led to an ever- internationally comparable increasing demand for 502 statistics that can be used in assessing the economic performance of nations.

The ICP plays a major role in meeting this demand by providing internationally com- parable national income aggregates, such as private and government consumption and capital formation.
The ICP has also provided valuable information on relative international prices of goods and services, at a reason- ably disaggregated level, which is used by researchers around the world.

The most popular by-products of the international comparisons are the Penn World Tables and the World Development Indicators, which are considered invaluable sources of data for research on global inequality, poverty and econometric analysis of productivity growth, and the study of catch-up and convergence among nations.
The potential beneﬁts of the ICP are somewhat diminished by the long lags in making international comparisons available to potential users.

The coverage of the ICP is not extensive and the number of countries covered varies across regions.

The limited coverage of ICP in some regions reﬂects the resource needs asso- ciated with the compilation of price data speciﬁcally for the purpose of the ICP.
It is in this area where signiﬁcant beneﬁts can be derived if the ICP activities can be integrated with the CPI work of the national statistical oﬃces.

At the global level, the potential beneﬁts are many and varied.
A few are listed below: increased country coverage, framework for extrapolations; leading to a better improved quality of the estimates resulting from the use of extensive price data collected for CPI purposes, rather than basing comparisons on products with speciﬁcations that may not be representative of the consumption baskets of the countries involved; from research on methods for quality adjustment.
Such methods are necessary to make adjustments for diﬀerences in product quality across countries; beneﬁts the development of regional PPPs, which are likely to make ICP results more consistent with domestic price movements and more acceptable to national govern- ments; construction of internationally comparable national accounts, in a common currency unit, complementing the existing national accounts in national currency units.
Such accounts will be a useful addition to the international statistical data bases that will enable global-level research on country and regional eco- nomic performance, and long-term catch-up, and convergence among countries; reliable estimates of PPPs along with domestic rates of inﬂation, providing a complete matrix of temporal– spatial price diﬀerences that can be used to better understand the factors inﬂuencing national price levels and exchange rate movements.
Several beneﬁts may also be derived by the national statistical oﬃces from an integrated approach to the compilation of CPI and PPPs.

The nature of beneﬁts derived will, however, vary depending on the stage of SPATIAL COMPARISONS OF CONSUMER PRICES the countries development of involved.

For more developed countries with well-established statistical agencies and programmes, the beneﬁts derived are through the synergies arising out of a joint approach to spatial and temporal comparisons.
These are as follows: Recent work on CPI and ICP manuals addresses the important issue of quality change over time and quality diﬀerences across countries.

Eﬀorts are being channelled into ﬁnding suitable statistical methods that can be used in the measurement of price-level changes leading to reductions in potential biases.
The treatment of new and disappearing products within the CPI is also an important problem asso- ciated with increased globalization, the expansion of free trade and the removal of tariﬀ barriers.

Thus national statistical oﬃces, especially in developed countries, are often confronted with the problem of accounting for new goods, and goods subject to rapid quality changes.
The regular compilation of the national accounts statistics and the measurement of price change through the CPI, and extending to inter-area and international comparisons, can provide information needed to assess the levels of, and movements in, real income.

In the case of developing countries with inadequate statistical infrastructure, beneﬁts from an integrated approach could be signiﬁcant and aid statistical capacity building in such countries.
These beneﬁts include: strengthening statistical infrastructure and institution building.

Eﬀorts to implement the CPI manual and its recommendations, along with participation in the ICP activity, are likely to identify deﬁciencies in a coun- try’s infrastructure and the lack of institutional capacity.
In some countries, it may be necessary to strengthen management and planning functions, recruit new staﬀ, and provide training to conduct household expenditure and other general price sur- veys; strengthening of data collection, processing systems and dissemination.

Designing and conducting sample surveys to international standards is a lengthy and demanding exercise.
It may be necessary to develop a systematic plan to improve survey designs to coordi- nate surveys of economic and business entities, and to conduct periodic censuses of economic activity.

Computerization of data-processing activities may also be required; improvement in the measurement of income inequal- ity and poverty.

Improved price and expenditure data can be used to improve national and international estimates of poverty; improved regional comparisons.
PPPs compiled for regions and for neighbouring countries can provide useful insights into the dynamics of regional develop- ment, and help identify the regions with special needs and required assistance.

Comparisons with neigh- bouring countries, in terms of relative price diﬀerences and real income changes, can provide powerful incen- tives to pursue policies for growth and low inﬂation.

503 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE There are many potential applications of the PPPs from the ICP.
Several papers (by Astin (2001), Ward (2001) and Prennushi the recent OECD–World Bank (2001) seminar dealt with some important applications of PPPs at Eurostat and the OECD, and at the global level in the assessment of poverty and inequality.
in particular) at (2001), 5.2 Salient features of CPI and ICP integration In order to identify strategies for closer integration of temporal CPI compilation with wider inter-area and cross-country comparisons of consumer prices and PPPs at the level of GDP, it is necessary to examine the main features and the context in which such an integration will occur.
The important issues are the scope and coverage of these two endeavours from the perspective of national statistical oﬃces, and a general framework of price comparisons within which both these activities are placed.

This section deals with these two aspects.

The scope and coverage of the CPI and ICP are vastly diﬀerent.

The CPI is a measure of changes, over time, in prices of goods and services that belong to the consumption baskets of households in a given country.
In contrast, the ICP provides a measure of price-level diﬀerences across countries, covering all components of the expenditure side of the national accounts.

The main components of GDP used in ICP comparisons are household consumption, government consumption, capital formation and net exports.

In line with the SNA 1993, the ICP merges the portion of government expenditure to households with private consumption to form house- hold consumption.
Thus, the scope and coverage of goods and services in the ICP is much wider than that of the CPI.

The household consumption concept used to that used in con- identical is, however, almost structing the CPI.

Any integration of CPI and ICP work will necessarily be conﬁned to the household consumption aggregate of the national accounts.
that provides goods and services Within the ICP, price quotations from diﬀerent countries are obtained for a large number of goods and services with very well-deﬁned product speciﬁcations.

This approach, described as the ‘‘tight speciﬁcation’’ approach, is used in the ICP.
The tight speciﬁcation approach requires a suﬃcient description of the product so that it is uniquely deﬁned in the ‘‘law of one price’’ sense, and so that it can be recognized in a range of localities and time periods wherever and whenever it is available.

The product listing is determined on the basis of the group of countries included in an international comparison exercise.
While this approach provides a solution to the problem of quality variations across countries, the commodities priced for ICP purposes may not be representative of the consumption baskets in respective countries.

Thus, commodities priced may not be representative of the items consumed in the countries, which usually make up the bundle of goods and services for the construction of the CPI.
The degree of success of integrating ICP activity with the CPI compilation depends upon the extent to which 504 these two activities can draw on a common pool of data level.

The and information available at the national intersection of data sets for the CPI and the ICP is represented in ﬁgure A4.2.

The following marked areas are of particular sig- niﬁcance: (1) Common set of goods and services between the CPI and ICP lists.
These price data can be used directly in an integrated approach to these two activities.

(4) (5) (2) and (3) Subset of ICP goods and services for which prices can be derived after making quality adjustments to products listed in the CPI basket.

These are the goods and services that identical, but close enough for are not quality adjustments to be made based on the characteristics of the goods and services.

Set of goods and services in the CPI basket that has no direct component in the ICP basket.
ICP basket of goods and services under headings of the national accounts that has no direct rele- vance or correspondence with CPI.

Figure A4.2 shows that if the integration between the ICP and the CPI is to be successful, the ICP compar- isons should necessarily be restricted to country group- ings where the basket of goods and services representing household consumption within the ICP has a signiﬁcant overlap with the country-speciﬁc CPI baskets.
Such an overlap can be achieved only when country groupings within the ICP comparisons exhibit similarities in their CPI consumption baskets.

This has implications for the ICP and its regionalization programme.

the expenditure side of In examining the CPI and ICP activities, it is also necessary to consider a range of temporal and spatial to national statistical interest price comparisons of oﬃces in the course of providing a comprehensive set of economic statistics for policy-makers and other analysts.
It is possible to consider these activities in a sequence indicating the progression involved in these price com- parison activities (ﬁgure A4.3).

The schematic diagram uses the standard CPI activity of estimating annual or quarterly price changes for the nation as a whole.

In most countries, national CPI ﬁg- ures are supplemented by area-speciﬁc CPI estimates for either capital cities or regions within the country.

A natural progression, where data permit, is to undertake spatial comparisons of prices.
At present, very few Figure A4.2 Price data for CPI and ICP activities CPI basket ICP basket 4 3 1 5 2 SPATIAL COMPARISONS OF CONSUMER PRICES Figure A4.3 A sequence of price comparisons CPI Price comparison at national level (quarterly and annual) CPI Price changes for capital cities and regions (quarterly and annual) PPP Spatial price comparison across regions within the country (annual) PPP Spatial price comparison across neighbouring countries within the region (annual) PPP Cross-country price comparison on a global level (annual or benchmark year) countries appear to produce such indices on a regular basis.
A fairly signiﬁcant jump from this level is to undertake price comparisons, on a bilateral or multi- lateral basis, with geographically contiguous countries or countries within a political or economic grouping.

The last element in this chain is the participation of the national statistical oﬃces in a global price comparison exercise such as the ICP.

At present, national statistical oﬃces are mostly involved at the two extreme ends of this spectrum.
The general level of involvement of national statistical oﬃces in the ICP is likely to be more enthusiastic, however, when they gain experience from their participation in inter-area comparisons within the country and intra-regional involving countries in geographical proximity.
comparisons 5.3 Two core strategies for CPI and ICP integration Based on the brief discussion of the CPI and ICP activities undertaken by participating national statis- tical oﬃces, it is possible to identify two major strate- gies that will result in a level of integration between these two activities that can beneﬁt both programmes and the systems of economic statistics in the partici- pating countries.
Both of these strategies emerge from the need to maximize the ﬂow of data from the CPI and ICP, and at the same time provide a framework for improving temporal and inter-area consumer price comparisons within a country.

Use of characteristics approach.

This approach was proposed in Zieschang et al.

(2001) and alluded to in Rao (2001a).

The characteristics approach begins with a market study by a national statistical oﬃce analyst to determine a set of price-determining characteristics.
These product characteristics, such as size, features, nature of the sale transaction, type of outlet, and so on, are determined according to the available information about the impact of the detailed characteristic on price at a point in time or over a speciﬁed reference period, such as a year.

Under this approach, product prices are collected and at the same time the product character- istics are also recorded.
the product In the standard CPI context, listing remains fairly constant, except in cases where an old variety or product is replaced by a new one.

When spa- tial comparisons are undertaken, however, the overlap in products may be limited.

In such cases, the character- istics approach becomes useful.

This approach is in direct contrast with pricing very speciﬁc products in all the countries, or in areas within a country, thus limiting the overlap and the usefulness of the resulting CPIs.
The characteristics approach requires price and characteristics data for a suﬃciently wide variety of detailed commodities or speciﬁcations in the item group to estimate a regression model of price on character- istics.

Such models are known as hedonic regression models, where the logarithm of price is regressed on various characteristics.

For spatial comparisons, this approach is very similar to the country–product–dummy (CPD) method discussed above in this annex.

Kokoski et al.
Kokoski et al.

(1999) describe a method of producing ‘‘exact’’, characteristics-adjusted, economic index number com- parisons between areas that allow for hedonic equation parameters to diﬀer from one area to another.

This method can be classiﬁed as a variant of the weighted CPD method along with a multilateral To¨ rnqvist index number.
The main advantages of the characteristics approach is that it is not critically based on the overlap in com- modity bundles for diﬀerent areas or countries, but on the sample size and number of commodities for which prices and quality characteristics are collected.

The sample size needs to be large enough to enable eﬃcient estimation of the parameters involved.
505 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE In order to facilitate cooperation with the ICP, in addition to having a well-structured database for the CPI that is extended to cover characteristics, the inter- ests of national statistical oﬃces are ultimately best served by implementing product and characteristic classiﬁcation schemes consistent with internationally agreed standards.
In order to make this approach operational, is necessary that such classiﬁcation schemes are established by making use of some of the existing classiﬁcations, for example the Central Product Classiﬁcation (CPC) or the Classiﬁcation of Individual Consumption according to Purpose (COICOP).

These product classiﬁcations need to be extended to include a core set of standard characteristics for each category within the classiﬁcation.
it Notwithstanding the beneﬁts of the characteristics approach, implementing a scheme of product cha- racteristics classiﬁcation requires all the countries or parties involved to agree to a speciﬁc standard and to allocate the necessary resources for such an endeavour.
Until comprehensive data sets with price and char- acteristics of products become available, it may be necessary to explore other possible approaches that can be used in conjunction with the present ‘‘tight speciﬁca- tion’’ approach used in ICP exercises.

Linking approach to international comparisons.


If maximizing the overlap with the CPI is one of the principal objectives of the ICP, then multilateral cross- country comparisons need to be built up from bilateral comparisons, where pairs of countries are identiﬁed on the basis of the maximum overlap in their national CPI baskets.

Once such pairs of countries are identiﬁed, then multilateral comparisons can be built using chains constructed on the basis of links.
This approach is somewhat similar to the MST approach proposed by Hill.

While the basic criterion in Hill’s approach uses variability in price relatives, measured using the Las- peyres–Paasche spread, the approach suggested here requires measures of overlap of price data as the prin- cipal criterion.

The linking approach needs a multi-stage framework.

In the ﬁrst stage, it is necessary to identify groups of countries to form regions or clusters.
The principal cri- teria that should be used are, ﬁrst, the extent of overlap that can be achieved between pairs of countries within the group and some measures of similarity in the expenditure patterns.

In the second stage, an MST approach should be used to identify the exact links within a regional cluster of countries.
Once multilateral regional PPPs are constructed, and PPPs for the GDP and its major components are derived, then the next stage will involve linking various regional comparisons to derive a set of global comparisons and PPPs.

Application of the chaining procedure represents a major shift from the present approach to ICP work.
Currently, comparisons within the ICP are based essentially on a top-down approach, where a com- modity listing based on the ‘‘tight speciﬁcation’’ approach is determined in the ﬁrst instance, and then price data from diﬀerent countries are collected.

Where the ICP work is regionalized, regions are essentially determined on geographical considerations and not on any data-based considerations.
If cross-country com- parisons are to be based on nationally available CPI data, it is necessary to use a bottom-up approach, where all the operational procedures, including the determi- nation of clusters and links, need to be built with the data available from the national sources provided by the national statistical oﬃces.

The application of spatial linking procedures will minimize the need for quality adjustments of the type described under the character- istics approach to price data collection.
The integration of spatial and temporal comparisons has the potential to provide a consistent set of temporal, inter-area and inter-country comparisons, and at the same time improve the quality of the underlying com- parisons.

It is an exciting prospect, but several chal- lenges need to be met before a truly integrated approach to spatial and temporal consumer price comparisons can be achieved.

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(t*)) indicates that the table, etc.

is located immediately after - but not relevant to - that paragraph (the paragraph number is merely a locator).
accounts see financial services; SNA acquisitions approach (g) CPIs based on 1.159–161, 3.26–29, 14.6 durable goods 1.151–164, 3.18–29, 23.3, 23.9–15 expenditures versus acquisitions 1.162 net acquisitions approach 23.9–12 owner-occupied housing 10.39–50, 23.136–138 alterations and additions 10.40–41, 10.47, 10.50 new constructions 10.40–41, 10.46–50 purchase of dwelling 10.40–45 relationship with user cost approach 23.34–42 services 1.157, 3.7–8, 3.18–21, 3.24–25, 10.166 actual final consumption 14.36 additivity test 1.70, 16.62–73 advisory committees 13.62–63 aggregates (g) for major treatments see elementary aggregates; value aggregates aggregation (g) 1.109–113 consistency (g) 9.45–46 one-stage aggregation see elementary aggregates; value aggregates; single-stage indices problems see elementary aggregates structure 4.4–15(f), 9.9, 9.16(f *) two-stage aggregation 17.55–60, 19.23–27, 19.28(t*) agriculture, production for own consumption 1.180–181, 3.79–80, 14.32–33 algorithms 9.160, 12.52–55 annual indices see seasonal products annual weights and chaining 9.135–136 and monthly price indices 15.33–64 Lowe index and mid-year indices 15.49–53 Lowe index with monthly prices and annual base year quantities 15.33–48 Young index 15.54–64 assets see financial assets auditors (price collection) 12.27–40 backchecking 12.33–38 monitoring 12.29–32 other functions 12.39–40 axiomatic approach (g) 1.53–84, 16 elementary indices 1.133–136, 9.25–30, 20.58–70 first approach (bilateral price indices) 1.54–79, 16.30–73
 additivity test 1.70, 16.62–73 early tests 1.55, 16.30–36 homogeneity tests 1.55, 16.37–41 invariance and symmetry tests 1.55–61, 1.68–69, 16.40–46 mean value tests 1.55, 16.47–49 monotonicity tests 1.55, 16.50–52 ranking of indices 1.65–67, 16.57–61 first approach (unilateral price indices) 16.11–21 overview 16.1–10 second approach (bilateral price indices) 1.80–84, 16.94–129 basic framework and preliminary tests 16.94–98 geometric Paasche and Laspeyres bounding test 16.123 homogeneity tests 16.99–105, 16.132
 invariance and symmetry tests 16.101–103, 16.106–111 mean value test for prices (g) 16.112 monotonicity tests 16.113 price relative reciprocal test 16.125–126 test of determinateness as to prices 16.127 weighting tests 16.114–119 second approach (unilateral price indices) 16.22–29 backchecking (price collection) 12.33–38 bargaining (prices) 6.92–102, 6.103(t*) bonus items 6.99–102 defined 6.92 survey by purchase of products 6.95, 6.97 survey of purchasers 6.96–97 survey of wholesale prices 6.98 barter transactions 1.163, 3.7, 3.13, 6.33 base invariance (price comparisons) see Annex 4 (4.1) base period (g) 5.31, 7.41, 15.8 see also index reference period; price reference period; weight reference period basket indices (g and g appendix) 1.16–52 see also fixed basket indices; individual indices benchmarking 12.80–82 bias (g) 11.30–66 aggregation bias/error 20.20 estimator, of the 5.62–64, 5.72 first-order approximation to the bias of Lowe index 17.74–75 fixed-weight indices 9.131–138 non-probability sampling 5.29 second-order approximation to the substitution bias of Lowe index 17.76–83 types elementary aggregate bias 11.35, 11.37, 11.43–50 new goods bias 11.35–36, 11.56–59 new outlet (outlet substitution) bias 3.118, 11.35–36, 11.60–63 omitted variable bias 21.1(appendix) outlet rotation bias 3.120 quality change bias 11.35–36, 11.51–55 representativity bias (g) 1.102–105 upper-level substitution bias 11.35, 11.37–42 see also errors bilateral price indices,
 axiomatic approach see axiomatic approach bonus offers 3.142, 6.84, 6.99–102 borrowing and credit 1.172, 3.56–71 credit card payments 6.91 debt price indices 10.33, 10.38(t*) hire purchase 3.65–66 interest see interest 521 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE mortgages 10.20–21, 10.23–38(t), 23.95–99 see also financial services bouncing (g) price bouncing test 20.60–61 business activities (household production) 3.73 business registers 5.14 capital account 14.18, 14.46–48(t) see also SNA capital gains
 tax 2.19 Carli index (g) 19.5–8(t) axiomatic approach 9.25–26, 9.30, 20.44, 20.70 chained 9.40, 19.7–8(t) economic approach 9.32, 9.34–37, 20.78, 20.80 elementary index 9.17–24(t), 20.40 estimators of stratum index 5.67 fixed base 19.5–6(t) missing price observations 9.53(t*), 9.58(t*), 9.64(t*) relationships with other elementary indices 1.128–129, 20.46–49, 20.53–57 sampling properties 9.38 Carruthers-Sellwood-Ward-Dale´ n index 20.43 relationships with other elementary indices 20.53–57 test performance 20.44, 20.69 carry forward of unavailable prices (g) 7.71, 7.124, 9.48, 9.50, 10.81–82, 10.83(t*), 10.85(t*) annual indices 22.78–84(f)(t), 22.88–89(t), 22.91–93(f)(t) cars 4.68–70, 6.36, 6.86, 7.20, 7.84 see also vehicles cash flow approach see payment approach cash register data 4.32 catalogues, price collection from 6.43 central and head office price collection 1.212, 6.69–79 documentation 6.124 electronic see electronic price collection quality checks 12.41–55 algorithms 12.52–55 reports 12.47–51 retail chains 6.43, 6.66, 6.69–70, 12.13 services 6.71, 12.14 techniques 6.43–49, 12.13–14 catalogues 6.43 data provided in the form of a price index 6.74 Internet 6.43, 6.44–45, 6.47, 6.109 letters 6.43, 6.72 telephone 6.43, 6.46, 6.109 Central Product Classification (CPC) (g) 3.147, 3.159 chain indices (g and g appendix) and annual weights 9.135–136 and reweighting 1.274, 9.105–126 partial reweighting 9.120–122 asymmetrically weighted indices 19.11(t*), 19.13–16(t)
 calculation of chain index 9.112–114(t) elementary aggregates 9.19(t), 9.39–44 hedonic functions with dummy variables for time 7.134–135, 21.41 frequency 9.108–111 long-term and short-term links 9.123–126 new and disappearing items 9.43, 9.58(t), 9.60(t), 9.62 new elementary aggregates 9.115–117 new higher-level indices 9.118–119 quality adjustment 7.153–158, 8.21 seasonal products month-to-month indices 22.64–77(t) rolling year annual 22.49–53(f)(t), 22.57(f *) year-over-year annual 22.40(t*), 22.41–43(t)
 year-over-year monthly 22.30–34(t) 522 symmetrically weighted indices 19.20–22(t) temporarily (non-seasonal) missing prices 9.53(t) versus fixed base indices 1.46–52 and Divisia approach 1.52, 15.76–97 see also individual indices change of units test 16.43 characteristics approach see Annex 4 (5.3) circularity (transitivity) test (g) 9.25, 15.88–96, 16.110, 16.132, 20.63 see also Annex 4 (4.1) classification 1.187–189, 3.144–168 major factors 3.144 product detail capital account 14.47(t), 14.54(t*) external goods and services account 14.54(t*) production account 14.28, 14.31(t*) use of income account 14.40–41, 14.45(t) by product purpose 3.147, 3.150–159 mixed purpose goods and services 3.155–157 multi-purpose goods and services 3.151–154 by product type 3.147, 3.148–149, 3.158–159, 3.166–168 publication level 3.160–161 weighting structure 4.4–15(f), 4.56–61 Classification of Individual Consumption according to Purpose (COICOP) 1.187–189, 3.23, 3.162–168, 4.57–58, 20.31–34(t), 20.35(t*) see also Annex 2 class mean imputation 7.67–68, 7.121 clothing 10.51–89 brands 10.61–63, 10.68 market 10.53–57 non-seasonal 10.58–65, 10.87 quality adjustment 10.66–72 seasonal 10.73–88(t) Cobb-Douglas preferences 9.33, 20.81–85 codes 6.1(appendix), 6.55–56, 6.60–61 coefficients (hedonic quality adjustment) 7.101–103 COICOP see Classification of Individual Consumption according to Purpose see also Annex 2 COLI see cost of living index collection (prices) 1.200–225, 1.276–279, 6,
 12.1–55(f) central and head office see central and head office price collection electronic see electronic price collection frequency and timing 6.5–22 collection dates 6.14–20 frequency 6.11–12, 6.76 HICPs see Annex 1 (5.1, 6.7) hyperinflation 6.23–24 influence of major price setters 6.22 less often than monthly 6.42 period versus point in time 6.6–11 publication of indices 6.13 tariffs, demand-dependent and limited availability goods and services 6.21 time of day 6.21, 6.34–35 geographical coverage see
 coverage procedures 6.32–107(f) quality checks (head office) 12.41–55 quality checks (local collection) 12.27–40 backchecking 12.33–38 monitoring 12.29–32 other auditor functions 12.39–40 quality in the field 6.67–68, 6.123, 12.15–26 continuity 1.218–221, 6.57, 12.18–20 data entry queries 12.21–25 descriptions 12.16–17 feedback 12.26 questionnaire design 6.50–66 codes 6.1(appendix), 6.55–56, 6.60–61 forms 6.1(appendix), 6.53, 6.99(t*) sampling see sampling collective consumption (g) collectors (prices) auditors see auditors geographical distribution 5.32, 5.55 training and development 12.93–99 documentation 12.100–102 reviews 12.103 commensurability (invariance to changes in the units of measurement) test (g) 9.25, 9.27-28, 16.132, 20.64–65 first axiomatic approach 1.55, 1.56–58, 16.43 second axiomatic approach 16.107–108 commission fees see fees commodity reversal (invariance to changes in the ordering of commodities) test (g) 16.42, 16.106, 16.132 comparable items 6.39–40, 7.69, 7.118 see also substitution comparison (or current) period (g) 5.31, 7.41, 15.8 comparisons (prices) inflation 2.38 long-run and short-run comparisons 7.42–43, 7.159–173(t) spatial comparisons see Annex 4 (4) compensation indices see wages component (g) composite transactions 1.173–174 computers HICPs see Annex 1 (5.11) new products, introduction 8.39 quality adjustment chaining 7.157 data 7.1(appendix) hedonic approach 7.1(appendix), 7.90–107(f)(t) high turnover of models 7.128 option
 costs 7.86, 7.89 confidentiality (price data) 13.53–54 constant prices (or identity) test (g) 15.94, 16.132, 20.59 first axiomatic approach 1.55, 16.35 second axiomatic approach 16.98 constant quantities (or fixed basket) test 16.35–36 consumer price indices (CPIs) (g) among major price indices 14.57–68(t), 14.75(t*) concerns about current methods see Preface independence and integrity 2.42–43 and national accounts price deflators 3.1(appendix) origins and uses 1.8–12 popularity as economic statistics
 2.39–41 range of 2.2–4, 2.11–12 consumers (g) 1.148–149, 3.1–3 consumption (g) 1.148–149, 2.20–30, 3.1–2 see also household consumption expenditures; individual types of consumption consumption CPI 14.6–7 consumption goods and services see goods; services consumption of own production (g) 1.175–182, 3.15, 3.74–89 agriculture 1.180–181, 3.79–80, 14.32–33 housing services 1.180–182, 3.81–89 in production account 14.32–33 continuity (price collection) 1.218–221, 6.57, 12.18–20 continuity test 16.132, 20.59 first axiomatic approach 16.34 second axiomatic approach 16.98 INDEX core (underlying) inflation 13.24–26 cost of living bias see substitution bias cost of living index (COLI) (g) 1.85–113, 1.165–166, 3.32–38 aggregation 1.109–113, 9.31–38, 17.55–60 alternative index 13.31 conditional (g) 1.165–166, 3.36–38, 7.31–32 data requirements and calculation issues 1.106–107 democratic COLIs (g) 18.23–35 Laspeyres 18.25–26, 18.30–31, 18.33 Paasche 18.25, 18.27, 18.30–31, 18.33 estimating by superlative indices 1.97–101 Konu¨ s COLI 17.9–17 Lloyd–Moulton index 1.108, 17.61–64 Lowe index as approximation to true COLI 17.66–73 first-order approximation to the bias of Lowe index 17.74–75 second-order approximation to substitution bias of Lowe index 17.76–83 overview 1.85–90, 3.30 plutocratic COLIs 18.3–13 Fisher 18.14–22 Laspeyres 18.6–9, 18.33 Paasche 18.6, 18.10–12, 18.33 versus democratic COLIs 18.23–35 representativity bias (g) 1.102–105 seasonal products 17.84–89 special
 cases (measurement) 1.94–96 substitution, allowing for 1.108 superlative indices 1.98–99, 17.4, 17.27–49, 17.55–60 true COLI 17.11–26(f) when preferences are homothetic 17.18–26 unconditional 1.165–166, 3.35, 3.37 upper and lower bounds 1.91–93 see also superlative indices Country–Product–Dummy (CPD) method (aggregation) see Annex 4 (4.2.1, 4.2.2) coverage (g) 3.90–107, 14.58, 14.60(t*) financial services 10.117–124 geographical 3.97–104 foreign purchases of residents and domestic purchases of
 non-residents 3.99–100, 3.128–129, 4.73–75 regional indices 3.101–104 sampling 5.3–6, 5.32, 5.55 urban and rural 3.97–98 HICPs see Annex 1 (3) households 1.183–184 definition of household 3.91–92 geographical coverage 3.97–104 national accounts price deflators 3.1(appendix) types 3.93–96 luxury goods and services 3.125–126 minor products 4.36–38 outlets 3.105–107 over/undercoverage 11.6, 11.9 social security benefits 2.9–13 undesirable and illegal goods and services 1.169, 3.123–124 wages 2.8 CPC see Central Product Classification CPD method see Country–Product–Dummy method CPIs see consumer price indices credit see borrowing and credit cross-border shopping see household consumption expenditures currency exchange 10.119, 10.125–129 currency, foreign 3.55 current (or comparison) period (g) 5.31, 7.41, 15.8 current prices (g) 523 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE current value (g) cut-off sampling (g) 5.36–41, 11.19 data dissemination see production and publication errors see data editing (below) HICPs see Annex 1 (5.11, 7.3–4) quality and accuracy 6.123, 13.46–48 sources 4.16–33 electronic see electronic price collection in the form of a price index 6.74 household expenditure surveys see household expenditure surveys national accounts 1.194–196, 4.25–28, 4.41 point-of-purchase surveys 4.30–31 population censuses 4.33 retail sales data 4.29 data editing 9.139–177, 11.27 detection of errors and outliers 9.139–140, 9.142–165 non-statistical checking
 of input data 9.147–153 output checking 9.147, 9.163–165 statistical checking of input data 9.147, 9.154–162 verification and correction 9.139, 9.141–145, 9.166–177, 12.21–25 missing price observations 9.172–177 outliers 9.171 debt see borrowing and credit declining balance (or geometric) depreciation 23.55–58 decomposition Fisher price index 19.31(t*), 19.32–36(e)(t) of index changes 9.127–130(t) Laspeyres price index 1.24–25, 15.11–17 Paaschee price index 1.24–25, 15.11–17 deflation (g) 3.1(appendix)
 Delphi method (quality adjustment) 7.76 democratic COLIs (g) 18.23–35 Laspeyres 18.25–26, 18.30–31, 18.33 Paasche 18.25, 18.27, 18.30–31, 18.33 depreciation 23.43–68 general model (unchanging durables) 23.43–54 geometric or declining balance model 23.55–58 one hoss shay (light bulb) model 23.62–68 straight line model 23.59–61 determinateness as to prices, test of 16.127 directed replacements 8.49–51, 8.57 direct indices 9.39–44 chain versus direct indices 9.40–41 new and disappearing items 9.41–43, 9.57–61(t), 9.63 temporarily (non-seasonal) missing prices 9.53(t) direct quality adjustment see explicit quality adjustment disappeared items see missing observations discontinued items see quality adjustment discounted prices (g) 3.138–143, 6.81–82 dissemination (CPIs) see production and publication Divisia approach (g) 1.52, 15.65–97 discrete approximations to continuous time Divisia index 15.72–75 fixed base versus chain indices 1.52, 15.76–97 price and quantity indices 15.65–71 relationship with economic approach 15.4(appendix) see also chain indices domain (g) double-logarithmic (log-log) form (hedonic function) 21.1(appendix) for major treatment see acquisitions approach; classification; depreciation; household consumption expenditures; housing; uses approach in SNA 14.40(b*), 23.1 Dutot index (g) 9.17–24(t) axiomatic approach 1.56–57, 9.25–30, 20.44, 20.68 chain versus direct indices 9.40–41 economic approach 9.32, 20.74–77, 20.80 elementary index (defined) 9.17,
 20.39 estimators of stratum index 5.67 missing price observations 9.53(t), 9.58(t*), 9.64(t*) relationships with other elementary indices 1.130–131, 20.48–52, 20.57 dwellings see housing economic approach (g) 17, 18 elementary indices 1.137–146, 9.31–38, 20.71–86 many-household case 1.109–113, 18 democratic versus plutocratic COLIs 18.23–35 errors in measurement 18.33 Fisher plutocratic price index 18.14–22 overview 1.109–113, 18.1–2 plutocratic COLIs and observable bounds 18.3–13 single-household case 17
 first-order approximation to the bias of Lowe index 17.74–75 Konu¨ s cost of living index 17.9–17 Lloyd–Moulton index 1.108, 17.61–64 Lowe index as an approximation to a true COLI 17.66–73 seasonal products 17.84–89 second-order approximation to the substitution bias of Lowe index 17.76–83 superlative indices see superlative indices true COLI when preferences are homothetic 17.18–26 zero price increasing to a positive price 17.90–94 Edgeworth (and Marshall) index (g and g appendix) 15.27, 15.30, 19.18–22(t) editing (g) see data editing education services see Annex 1 (5.9) see also social transfers in kind EKS method see Elteto–Koves–Szulc method electronic dissemination (CPI data) 13.55–58 electronic price collection 6.108–118 electronic forms 6.53, 6.65 electronic point of sale (EPOS) 6.117 hand-held computers 6.65, 6.110–116 scanner data 1.213–214, 4.32, 5.60, 6.117–118, 9.72–73, 20.88–99 elementary aggregates (g) 1.265–270, 4.4–8(f), 9.6–73 aggregation and classification problems 20.23–37 elementary aggregate bias 11.35, 11.37, 11.43–50 product dimension 20.23, 20.27, 20.31–37(t) sectoral dimension 20.23, 20.28–30 spatial dimension 20.23, 20.26, 20.31–37(t) time dimension 5.3, 20.15–17, 20.20, 20.23, 20.25 aggregation structure 9.9, 9.16(f *) consistency in aggregation 9.45–46 HICPs see Annex 1 (4.2.2–3) item selection 9.8 missing price observations 9.47–63(t) permanent 9.55–63(t) temporary (non-seasonal) 9.48–54(t) Drobisch price index (g) 19.18–22(t) dummy variables
 for time 7.134–136, 21.40–42, 21.58–60 durable goods (g) 1.154–158, 3.166–168, 23 sampling 9.38, 20.87 weights 1.124–126, 4.14–15(t), 9.11–14, 9.115–117 see also value aggregates 524 INDEX elementary indices (g) 1.120–123, 9.16–27(t), 20 aggregation and classification problems see elementary insurance services see insurance real estate agency services 3.122, 10.20, 10.40, 10.149–155, aggregates axiomatic approach 1.133–136, 9.25–30, 20.58–70 chain indices 9.19(t), 9.39–44, 9.115–117 economic approach
 1.137–146, 9.31–38, 20.71–86 harmonic mean formulae see harmonic elementary indices ideal elementary indices 20.11–22 individual indices 20.38–45 see also individual headings other formulae 9.64–69 relationships between elementary indices 1.127–132, 20.46–57 stochastic approach 20.100–111 unit value indices 9.70–71 Elteto–Koves–Szulc (EKS) method see Annex 4 (4.2.1, 4.2.2) environmental factors, and quality change 7.31–32 EPOS (electronic point of sale) data 6.117 equilibrium supply and demand 21.22–23 errors 11.1–29(t), 20.19–20 data see data editing measurement 11.12–16, 18.33 minimizing 11.17–29 non-sampling 11.5–11, 11.16 sampling 5.38, 11.3–4(t), 11.21, 20.20 weighting 4.76–77 see also bias estimation 5.61–99 bias 5.62–64, 5.72, 11.64–66 COLIs 1.97–101 errors 11.4, 11.12–15, 11.17 implementation for CPIs 5.65–72 variance estimation see variance estimation weighted least squares (WLS) estimator 21.1(appendix) see also sampling evolutionary goods 8.35, 21.62–63 directed replacements 8.49–51, 8.57 sample augmentation 8.56 exclusions see coverage expenditures see household consumption expenditures expenditure weights see weights expert judgement quality adjustment 1.251, 7.73–76 sampling 11.19 explicit quality adjustment methods (g) 1.249–255, 7.35, 7.38, 7.72–115 choice of 7.119–120(f) expert judgement 1.251, 7.73–76 hedonic approach see hedonic approach production or option costs, differences in 1.250, 7.81–89, 7.119 quantity adjustment 1.249, 7.77–80(f),
 7.81(t*) export price index (XPI) 14.4, 14.51, 14.57–58, 14.60(t*), 14.75(t*) coverage 14.58, 14.60(t*) external goods and services account 14.18, 14.49–52, 14.54(t*) see also SNA extra charges 6.91 extra quantity offers 6.84 see also bonus offers factoring and price updated weights 9.92–94 factor reversal (functional form symmetry) test (g) 1.68–69, 16.56 fees 23.118–120 see also Annex 1 (6.5) see also financial services filtering (data) see data editing final consumption 14.34–45(b)(t) final uses, price indices
 for 14.72 financial assets 3.4, 3.52–55 borrowing and credit see borrowing and credit fees 3.54, 10.125–136 foreign currency 3.55 financial services 10.117–148 currency exchange 10.119, 10.125–129 definition and coverage 10.117–124 deposit and loan facilities 10.119, 10.137–148 example calculation 10.1(appendix) mortgages 10.20–21, 10.23–38(t), 23.95–99 HICPs see Annex 1 (5.10) payments approach 10.120 SNA 14.42–43 stockbroking services 10.119, 10.130–136 see also borrowing and credit; fees; insurance financial transactions 1.170–172 Fisher price index (g) 1.42 aggregation and classification problems 20.33–34, 20.36(t) as average of Paasche and Laspeyres indices 15.18–23 axiomatic approach 16.53–56 chained and Lloyd–Moulton indices 19.28–31(t) symmetrically weighted 19.20–22(t) two-stage aggregation 19.26–27, 19.28(t*) decomposition 19.31(t*), 19.32–36(t) democratic price index 18.29 economic approach 17.27–32 fixed base symmetrically weighted 19.17–19(t) two-stage aggregation 19.24–25, 19.28(t*) hedonic indices 21.51 month-to-month indices 22.65–77(t) plutocratic price index 18.14–22 relationship with Lowe index 1.32–33 rolling year annual index 22.47, 22.57(f *) chained 22.49–53(f)(t), 22.57(f *) fixed base 22.48–53(f)(t) superlative index 1.96–97, 1.99–101, 17.4, 17.27–32, 17.50–54 two-stage aggregation 17.57–58 year-over-year annual index 22.36–43(t) chained 22.40(t*), 22.41–43(t) fixed base 22.38–40(t) year-over-year monthly index 22.19–34(t) chained 22.30,
 22.31(t*), 22.32–34(t) fixed base 22.26, 22.27(t*), 22.28, 22.31(t*) fixed assets 3.23 fixed base indices asymmetrically weighted indices 19.9–12(t), 19.16(t*) hedonic functions with dummy variables for time 7.134–135, 21.41 rolling year annual indices 22.48–53(f)(t), 22.56–59(f)(t), 22.60(f *) superlative indices 19.17–19(t) symmetrically weighted 19.17–19(t) versus chain indices 1.46–52 deposit and loan facilities 10.1(appendix), 10.140, 10.143–147 financial assets 3.54, 10.125–136 and Divisia approach 1.52,
 15.76–97 year-over-year annual 22.38–40(t) 525 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE year-over-year monthly 22.26–29(t), 22.31(t*) see also individual indices fixed basket indices (g and g appendix) 2.14–15, 15.24 see also basket indices; individual indices fixed basket (or constant quantities) test 16.35–36 fixed weight indices (g and g appendix) bias 9.131–138 fixed weights for seasonal products 4.63–66 foreign purchases of residents see household consumption expenditures formula approximation error 20.19 free goods and services 3.7, 3.142, 6.84 functional form (hedonic function) 21.1(appendix) functional form symmetry (factor reversal) test (g) 1.68–69, 16.56 gambling 3.51 garments see clothing GDDS see General Data Dissemination System GDP see gross domestic product Geary–Khamis (GK) method see Annex 4 (4.2.2) Geary–Khamis quantity index 1.70 General Data Dissemination System (GDDS) 13.43–44 geographical coverage see coverage geometric averages, weighted 9.137 geometric indices 1.38–40 see also individual indices geometric (or declining balance) depreciation 23.55–58 geometric Paasche and Laspeyres bounding test 16.123 gifts 3.7, 3.45 goods (g) 3.3 for major treatment see under acquisitions approach; classification; household consumption expenditures; uses approach see also durable goods; non-durable goods gratuities 3.46, 6.88 gross domestic product (GDP) 14.12, 14.73–74, 14.75(t*) groups (aggregation/weighting structure) 4.6(f), 4.9, 4.58, 5.58, 9.9,
 9.16(f *) harmonic elementary indices 9.67–68, 20.42 economic approach 20.79–80 relationships with other elementary indices 1.128–130, 20.46–47, 20.53–57 test performance 20.44, 20.70 harmonic Laspeyres index 16.88, 19.9–16(t) Harmonized Indices of Consumer Prices (HICPs) 13.30, 23.132, Annex 1 note: locators are Annex 1 paragraph numbers basic concepts and definitions 2 coverage 3 harmonization, agenda for further 8 index formulae 4.2 links with National Accounts deflator of HFCE 6 overview and history 1
 price sampling 4.3 regulations 9 release and timeliness 7 specific standards 5 weights 4.1 health services 3.24–25 HICPs see Annex 1 (5.9) see also social transfers in kind hedonic approach (g) 7.132–149, 21 equilibrium supply and demand 21.22–23 explicit quality adjustment 1.252–254, 7.90–115(f)(t) adjustments 7.108–109 choice of methods 7.120 coefficients 7.101–103 data illustrating 7.1(appendix) imputation 7.103–107 limitations 7.110–115 functional form 21.1(appendix) functions with dummy variables for time 7.134–136, 21.40–42, 21.58–60 hedonic prices, meaning of 21.24–28 and matched models method 7.150–152, 21.59–60 multicollinearity 21.1(appendix) omitted variable bias 21.1(appendix) period-to-period indices 7.137–141 producer or supply side 21.1(appendix), 21.18–21 regression model of rental market 23.140 superlative and exact hedonic indices (SEHI) 7.142–149, 21.48–58 tastes and technologies, changing 21.1(appendix) theoretical characteristics price indices 21.38–39 tied bundles of characteristics 21.12 unweighted indices 21.59–60 weighting 21.1(appendix), 21.58 see also quality adjustment HES see household expenditure surveys HFCE see household final consumption expenditure HFMCE see household final monetary consumption expenditure HICPs see Harmonized Indices of Consumer Prices higher-level indices (g) 1.271–274, 4.8, 9.74–138 choice of 9.74–76 weighted averages of elementary indices 1.273, 9.77–138 alternatives to fixed-weight indices 9.131–138 calculation
 9.77–86(t) decomposition of index changes 9.127–130(t) price-updating from weight reference period 9.95–104(t) reweighting and chain linking 1.274, 9.105–126 Young and Lowe indices 9.87–91 see also individual indices high-technology goods 7.125–131, 19.4(t) see also computers high-technology services 19.4(t) hire purchase 3.65–66 homogeneity tests first axiomatic approach 1.55, 16.37–41 second axiomatic approach 16.99–105, 16.132 homothetic preferences 17.18–26 household consumption expenditures 1.162,
 3.9–17, 14.34–45(b)(t), 14.75(t*) defined (g) 14.36 foreign purchases of residents 3.99–100, 3.128–129, 4.73–75, 6.36, 14.66–68 hierarchy 14.42–44 imputed 3.130–132, 14.38–39, 14.42, 14.44 monetary 1.162, 3.7, 3.12, 3.16–17, 14.38, 14.42 non-monetary 1.163, 3.12–15 barter transactions 1.163, 3.7, 3.13, 6.33 free gifts or transfers 3.7 imputed values 3.130–132 remuneration in kind 1.163, 3.7, 3.14 self-produced goods and services see household production consumer or demand side 21.13–17(f) alternative formulation 21.29–36 identification and appropriate estimators 21.1(appendix) outside the scope of CPIs 3.4, 3.39–54, 4.35 financial assets see financial assets gambling 3.51 526 INDEX insurance 3.47–50 transfers 3.7, 3.41–46 product detail 14.40–41, 14.45(t), 14.47(t) SNA 14.34–45(b)(t), 14.75(t*) social transfers in kind 3.9–11 time incurred/used 3.18–21, 3.56–57 see also individual goods and services household expenditure surveys (HES) (g) 1.192–197 weighting data 4.17–28, 4.40–41 adjustment 4.42–43 classification 4.56–57 reliability 4.21 hedonic approach 21.43–47 linked to show no price change 7.70, 7.124 non-overlapping qualities 1.241–248 overall mean or targeted mean imputation 7.53–66(t), 7.121 overlap 1.236–240, 7.35, 7.45–52(t), 7.123 short-run comparisons 7.165–170 implicit service charge 3.70 import price index (MPI) 14.57–58, 14.60(t*) SNA 14.4, 14.51–52, 14.54(t*), 14.57–58, 14.60(t*), 14.75(t*) imputation (g) 7.35 household final consumption expenditure
 (HFCE) 14.36 see also Annex 1 (6) class mean imputation 7.67–68, 7.121 household consumption expenditures 3.130–132, 14.38–39, household final monetary consumption expenditure 14.42, 14.44 (HFMCE) see Annex 1 (2.2) household production 1.175–182, 3.72–89 business activities 3.73 consumption of own production 1.175–182, 3.15, 3.74–89 agriculture 1.180–181, 3.79–80, 14.32–33 housing services 1.180–182, 3.81–89 in production account 14.32–33 households (g) institutional units 14.14, 14.17(b) housing
 (owner-occupied) (g) 10.4–50 acquisitions approach 10.39–50, 23.136–138 alterations and additions 10.40–41, 10.47, 10.50 new constructions 10.40–41, 10.46–50 purchase of dwelling 10.40–45 HICPs see Annex 1 (5.12, 6.3, 8) housing services for own consumption 1.180–182, 3.81–89 payments approach 10.20–38, 10.120, 10.159–163, 23.134–135 SNA 14.40(b*), 23.144 user cost approach 10.7–19, 23.69–93 alternative 23.141–144 relationship with acquisitions approach 23.39 rental equivalence approach 10.14–19, 23.3, 23.16–21, 23.139–140 insurance 23.102–106 maintenance and renovation 10.20, 10.40, 23.107–117 mortgage interest costs 10.23–38(t), 23.95–99 property taxes 23.100–101 transaction costs (home purchase) 23.118–120 user costs 23.94–120 additional services 23.127 billing and maintenance 23.123, 23.125 landlord-specific 23.121–131 non-payment of rent and vacancy costs 23.124 opportunity cost of capital 23.126 weights 4.3 hybrid values or expenditures (g) hybrid weights (g) identity (or constant prices) test (g) 15.94, 16.132, 20.59 first axiomatic approach 1.55, 16.35 second axiomatic approach 16.98 IDI see implicit deflator index illegal goods and services 1.169, 3.123–124 implicit deflator index (IDI) 14.70, 14.75(t*) implicit quality adjustment methods (g) 1.236–248, 7.38, 7.44–71 carry forward see carry forward of unavailable prices choice of 7.121–124 class mean imputation 7.67–68, 7.121 comparable replacement 7.69, 7.118 overall mean imputation 7.53–66(t), 7.121
 permanently missing prices 9.58–63(t) seasonal products 10.79–87(t), 22.85–86(t), 22.88(f *), 22.93–95(t), 22.96(f *) temporarily (non-seasonal) missing prices 9.48, 9.51–54(t) income generation of income account 14.71(t*), 14.75(t) household subsectors by 14.17(b) real income 2.20–23 use of income account see use of income account see also wages income tax 2.19, 3.42, 13.27, 23.126 indexation (g) 1.9, 2.5–19 index number theory 15 aggregation see elementary aggregates; value aggregates annual weights and
 monthly price indices 15.33–64 axiomatic approach see axiomatic approach choice of index number 1.13–15 Divisia approach 1.52, 15.65–97 economic approach see economic approach formulae and terminology (g appendix) national accounts price deflators 1.190, 3.1(appendix) overview 15.1–6 stochastic approach see stochastic approach symmetric averages 1.41–45, 15.18–32 index reference period (g) 9.81–83 indirect quality adjustment see implicit quality adjustment inflation 2.34–38 accounting 2.31–33 core (underlying) 13.24–26 CPIs as measure of/proxy for 1.11, 1.12, 14.63–65 hyperinflation 6.23–24 international comparisons 2.38 targets 2.36–37 inflation (transactions) CPI 14.6–7 institutional units (g) 14.14–17(b) for major accounts see capital account; external goods and services account; production account; use of income account insurance, life 3.47–50 HICPs see Annex 1 (6.4) insurance, property 3.47–50, 10.156–173 acquisitions approach 10.40, 10.166 defined 10.157 HICPs see Annex 1 (5.8) owner-occupied housing 23.102–106 payments approach 10.20, 10.159–163 premiums 10.159–163, 10.167–173(t) taxes 10.171–173(t) uses approach 10.164–165 weights 4.67 see also financial services 527 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE interest 1.174, 3.63, 3.67–71, 6.91, 10.1(appendix), 10.139–141 indexation 1.11, 2.16–17 mortgages 10.23–38(t), 23.95–99 see also borrowing and credit intermediate consumption price indices (IPIs) 14.71, 14.75(t*) International Standard
 Industrial Classification (ISIC) 3.147, 14.30, 14.31(t*) international standards 13.42–44, 14.30, 14.41, Annex 3 see also Preface Internet 3.100, 3.106, 10.101–102 CPI data dissemination by 13.55–58 price collection by 6.43, 6.44–45, 6.47, 6.109 see also telecommunication services invariance and symmetry tests first axiomatic approach 1.55–61, 1.68–69, 16.40–46 second axiomatic approach 16.101–103, 16.106–111 invariance to changes in the ordering of commodities (commodity reversal) test (g) 16.42, 16.106, 16.132
 invariance to changes in the units of measurement (commensurability) test (g) 9.25, 9.27–28, 16.132, 20.64–65 first axiomatic approach 1.55, 1.56–58, 16.43 second axiomatic approach 16.107–108 invariance to proportional changes in base quantities test (g) 16.40–41 invariance to proportional changes in base values test 16.101–103 invariance to proportional changes in current quantities test (g) 16.40–41 invariance to proportional changes in current values test 1.71, 16.101–103 inverse proportionality in base period prices test (g) 20.59 first axiomatic approach 16.39 second axiomatic approach 16.100 IPIs see intermediate consumption price indices irrelevance of price change with tiny value weights test 16.117–118 ISIC see International Standard Industrial Classification) item (g) item or product rotation (g) item specification (g) 6.25–31 broad/loose item specification 6.27–29, 6.31 narrow/tight item specification 6.27–30 Jevons price index (g) 1.76, 9.17–24(t), 19.5–8(t) axiomatic approach 9.25–26, 9.30, 20.44, 20.67 chained 9.40–41, 19.7–8(t) economic approach 9.33–37, 20.83–84, 20.86 elementary index (defined) 9.17, 20.41 estimators of stratum index 5.67 fixed base 19.5–6(t) missing price observations 9.53(t), 9.58(t*), 9.64(t*) relationships with other elementary indices 1.128–130, 20.46–47, 20.50–57 sampling properties 9.38 judgemental (expert choice) sampling 11.19 kind, remuneration in 1.163, 3.7, 3.14 Konu¨ s cost of living index 17.9–17 Laspeyres–Konu¨ s true
 cost of living index 17.12–17(f) Paasche–Konu¨ s true cost of living index 17.13–17(f) true COLI when preferences are homothetic 17.18–26 see also cost of living index Laspeyres–Konu¨ s true cost of living index 17.12–17(f) Laspeyres price index (g and g appendix) 2.14–15 aggregation and classification problems 20.33–36(t) chained 19.7–8(t) geometric/harmonic 19.11(t*), 19.13–16(t) month-to-month indices 22.65–77(t) rolling year 22.49–53(f)(t), 22.57(f *) year-over-year 22.30–34(t), 22.40(t*), 22.41–43(t)
 decomposition 1.24–25, 15.11–170 defined 1.21–22, 9.65–66, 15.12–13 economic approach 20.73–75 fixed base 19.5–6(t) geometric/harmonic 19.9–12(t), 19.16(t*) rolling year 22.48–53(f)(t), 22.56–59(f)(t), 22.60(f *), 22.80(t*), 22.81(f *), 22.82–83 year-over-year 22.26–28(t), 22.38–40(t) geometric (g) 1.38–40, 9.66 annual basket indices with carry forward of unavailable prices 22.80(t*), 22.81–84(f), 22.91–92(t), 22.93(f *) annual basket indices with imputation of unavailable prices 22.85–86(t), 22.88(f *), 22.93–95(t), 22.96(f *) chained 19.11(t*), 19.13–16(t) fixed base 19.9–12(t), 19.16(t*) year-over-year 22.40(t) harmonic 16.88, 19.9–16(t) chained 19.11(t*), 19.13–16(t) fixed base 19.9–12(t), 19.16(t*) hedonic 21.48–49 Paasche and Laspeyres bounding test 1.55, 16.49, 16.123 plutocratic 18.6–9, 18.16–17, 18.20–21, 18.33 relationship with Lowe index 1.32, 15.2(appendix) relationship with Paasche index 1.30–34, 15.1(appendix) relationship with Young index 1.36, 15.56–58 rolling year annual index 22.47 chained 22.49–53(f)(t), 22.57(f *) fixed base 22.48–53(f)(t), 22.56–59(f)(t), 22.60(f *), 22.80(t*), 22.81(f *), 22.82–83 symmetric averages 15.18–23, 15.32 test performance 1.60–61, 1.69, 16.58, 16.61 two-stage aggregation 17.55–57 year-over-year annual index (seasonal products) 22.36–43(t) chained 22.40(t*), 22.41–43(t) fixed base 22.38–40(t) geometric 22.40(t) year-over-year monthly index (seasonal products) 22.19–34(t) chained 22.30–34(t) fixed base 22.26–28(t) legal fees
 23.118 Leontief preferences 9.32 licences 3.43–44 light bulb (one hoss shay) depreciation (g) 23.62–68 linear form (hedonic function) 21.1(appendix) linking see chain indices lists of products/prices 5.15, 6.43 Lloyd–Moulton index 1.108, 9.69, 9.137, 17.61–64, 19.28–31(t) loans see borrowing and credit log–log (double-logarithmic) form (hedonic function) 21.1(appendix) long-run comparisons and short-run comparisons 7.42–43, 7.159–173(t) implicit short-run comparisons 7.165–170 quality adjustment methods
 7.160–164 single-stage and two-stage indices 7.171–173 labour services, price indices for 14.71(t*), 14.75(t) land (property) 10.44 Lowe index (g and g appendix) 1.17–20, 2.14, 3.30–31, 15.11, 15.24, 19.34(t*), 19.37–38 528 INDEX annual indices 22.78–86(f)(t), 22.88–89(f)(t), 22.91–95(f)(t) bias 17.74–83 chain indices see chain indices (versus fixed base indices) and COLI 1.91–93, 17.66–83 conditional 3.38 economic approach 17.66–83 higher-level indices 9.87–91 mid-year indices 15.49–53, 19.41–44, 19.46(t*) monthly indices 15.33–53 relationship with Fisher index 1.32–33 relationship with Laspeyres index 1.32, 15.2(appendix) relationship with Paasche index 1.32 relationship with Young index 9.87–91 test performance 1.61, 1.83, 16.130–134 updating 1.28–29 lower-level index (g) loyalty schemes 3.138, 3.142, 3.143 luxury goods and services 3.125–126 mail order catalogues 6.43 maintenance expenditures (owner-occupied housing) 10.20, 10.40, 23.107–117 landlord-specific 23.123, 23.125 annual weights 15.33–64 Lowe index and mid-year indices 15.49–53 Lowe index with monthly prices and annual base year quantities 15.33–48 Young index 15.54–64 year-over-year 22.16–34(t) chained 22.30–34(t) fixed base 22.26–29(t), 22.31(t*) used in predicting rolling year index 22.55–62(f)(t) month-to-month indices 9.19(t), 9.86(t) Rothwell or Bean and Stine Type C indices 22.87–89(t), 22.91(f *), 22.93–94(t), 22.96(f *) maximum overlap 22.63–77(t) used in predicting rolling year indices
 22.91–96(f)(t) mortgages 10.20–21, 10.23–38(t) interest 10.23–38(t), 23.95–99 motor cars 4.68–70, 6.36, 6.86, 7.20, 7.84 see also vehicles moving-weights for seasonal products 4.62–66 multicollinearity (hedonic quality adjustment) 21.1(appendix) multiperiod identity test 15.94 multiplicative quality adjustment 7.39–40 management process see production and publication; quality management manuals (training and development) 12.100–102 many-household case (economic approach) see economic net price indices 3.137
 new outlet (outlet substitution) bias 3.118, 11.35–36, 11.60–63 new outlets 8.58 new products 8.32–62, 21.61–68 approach Marshall-Edgeworth index (g and g appendix) 15.27, 15.30, 19.18–22(t) matched models method (g) 7.2, 7.5, 7.6–18 dynamic universe 8.1(appendix), 8.7 and hedonic indices 7.150–152, 21.59–60 missing items 7.7–13, 8.2 new products 7.18, 8.4 sampling concerns 7.14–17, 8.3, 8.6–8 telecommunication services 10.97–104(t) mean square error (sampling) 5.38, 5.63 mean value tests 16.47–49, 16.132, 20.59 mean value test for prices (g) 1.55, 16.47, 16.112 mean value test for quantities 16.48 Paasche and Laspeyres bounding test 1.55, 16.49, 16.123 medical services see health services metadata system (quality adjustment) 8.23–31 methodology, presentation 13.41, 13.44(b*), 13.60–61 mid-year indices (Lowe index) 15.49–53, 19.41–44, 19.46(t*) minimum spanning tree (MST) see Annex 4 (4.2.2(f)) missing observations 9.47–63(t) chain indices 9.43, 9.53(t) deleting items that disappear 8.1(appendix) direct indices 9.41–43, 9.53(t) HICPs see Annex 1 (5.3) matched models method 7.7–13, 8.2 permanent 9.55–63(t), 9.64(t*) temporary (non-seasonal) 9.48–54(t) verification and correction 9.172–177 see also quality adjustment; seasonal products mobile telephones 8.37, 10.110, 10.112(t*) see also telecommunication services money-off vouchers 3.138, 3.141, 3.143 monitoring (price collection) 5.30, 12.29–32 monotonicity tests 16.50–52, 16.132 base period prices 16.50, 16.113,
 20.59 base period quantities 16.50–51 current period prices 1.55, 16.50, 16.113, 20.59 current period quantities 16.50–51 monthly indices chain indices 9.43, 9.58(t*), 9.60(t), 9.62 defining/terminology 8.32–35, 21.61–62 directed replacements 8.49–51, 8.57 direct indices 9.41–43, 9.57–61(t), 9.63 evolutionary goods 8.35, 8.49–51, 8.56–57, 21.62–63 HICPs see Annex 1 (5.6) matched models method 7.18, 8.4 new goods bias 11.35–36, 11.56–59 reservation prices 8.59–60 revolutionary goods 8.35, 8.52–55(t), 8.57–58,
 21.62–64 sample augmentation 8.52–58(t), 21.64 sample rebasing and rotation 1.222–225, 8.1(appendix), 8.18–21, 8.44–48 substitution 1.259–262, 8.2(appendix) virtual prices 21.65–67 see also replacement products non-durable goods 1.154–158, 3.22, 3.166–168 non-market output (production account) 14.25(t*), 14.26–27, 14.75(t*) non-observation errors 11.9–11 non-probability sampling (g) 5.6, 5.27–50 bias 5.29 cut-off sampling 5.36–41, 11.19 quota sampling 5.42–45, 11.19 reasons for using 5.28–35 representative item method 5.6, 5.46–48 sampling in time 5.49–50 see also sampling non-profit institutions serving households (NPISHs) 14.14, 14.17(b*) non-residents, expenditures 3.99, 4.73 non-response errors 11.10–11, 11.28–29 non-sampling errors 11.5–11, 11.16 NPISHs see non-profit institutions serving households observations errors 11.5–8 omitting 9.48–49, 9.53(t*), 9.58(t) off-cycle items see missing observations offer functions (quality adjustment) 21.21 529 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE omitted variable bias 21.1(appendix) one hoss shay (light bulb) depreciation (g) 23.62–68 opportunity cost of capital (landlords) 23.126 option costs (quality adjustment) 7.83–89, 7.119 order PPS sampling 5.19–23(t) order r, quadratic mean of 1.98–99, 17.33–43, 17.59 outlet rotation bias 3.120 outlets 3.105–107, 6.35, 8.17, 8.58 price variation between 3.116–120 retail chains 6.43, 6.66, 6.69–70, 12.13 rotation 3.119–120 sampling 5.3–6, 5.15, 5.55 symmetric treatment
 of outlets test 20.59 weights 4.13, 4.30–31 see also sectoral dimension (elementary aggregates) outlet substitution (new outlet) bias 3.118, 11.35–36, 11.60–63 outliers (g) 9.139–141, 9.171 output price index (YPI) 14.70, 14.75(t*) out of stock items 6.60 overall mean or targeted mean imputation (quality adjustment) 7.53–66(t), 7.121 overlap method (quality adjustment) 1.236–240, 7.35, 7.45–52(t), 7.123, 22.63–77(t) overseas purchases see household consumption expenditures owner-occupied housing see housing
 (owner-occupied) own production consumption see consumption of own production own share price weighting test 16.115–116 Paasche–Konu¨ s true cost of living index 17.13–17(f) Paasche and Laspeyres bounding test 1.55, 16.49, 16.123 Paasche price index (g and g appendix) 1.21–23, 2.14, 19.5–8(t) chained 19.7–8(t) month-to-month 22.65–77(t) rolling year 22.49–53(f)(t), 22.57(f *) year-over-year 22.30–34(t), 22.40(t*), 22.41–43(t) COLIs democratic 18.25, 18.27, 18.30–31, 18.33 Paasche-Konu¨ s true cost of living index 17.13–17(f) plutocratic 18.6, 18.10–12, 18.33 decomposition 1.24–25, 15.11–17 economic approach 20.77 fixed base 19.5–6(t) rolling year 22.48–53(f)(t), 22.57(f *) year-over-year 22.26–29(t), 22.38–40(t) geometric 1.38–39 chained 19.11(t*), 19.13–16(t) fixed base 19.9–12(t), 19.16(t*) hedonic 21.50–51 national accounts price deflators 3.1(appendix) plutocratic 18.6, 18.10–12, 18.16, 18.18–21, 18.33 relationship with Laspeyres index 1.30–34, 15.1(appendix) relationship with Lowe index 1.32 rolling year annual index 22.47 chained 22.49–53(f)(t), 22.57(f *) fixed base 22.48–53(f)(t), 22.57(f *) symmetric averages 15.18–23, 15.32 test performance 1.60, 1.69, 16.58, 16.61 year-over-year annual index 22.36–43(t) chained 22.40(t*), 22.41–43(t) fixed base 22.38–40(t) year-over-year monthly index 22.19–34(t) chained 22.30–34(t) fixed base 22.26–29(t) Palgrave index 19.9–16(t) Pareto PPS sampling 5.19–21(t), 5.23 530 payment in kind 1.163, 3.7, 3.14 payments
 approach 10.20–38, 10.120, 10.159–163, 23.134–135 pensions 1.11, 3.54 HICPs see Annex 1 (6.4) period-to-period hedonic indices 7.137–141 plutocratic COLIs 18.3–13 Fisher 18.14–22 Laspeyres 18.6–9, 18.33 Paasche 18.6, 18.10–12, 18.33 versus democratic COLIs 18.23–35 point-of-purchase surveys 4.30–31 positivity test 16.132 first axiomatic approach 1.55, 16.34 second axiomatic approach 16.98 PPI see producer price index PPP see purchasing power parity PPS sampling see probability proportional to size sampling
 premiums (property insurance services) 10.159–163, 10.167–173(t) gross premiums, gross expenditures approach 10.163 gross premiums, net expenditures approach 10.160 net premiums, gross expenditures approach 10.161–162 as a proxy for the net insurance service 10.171–173(t) see also fees press releases 13.38–41(b) price bouncing test 20.60–61 price collection see collection price collectors see collectors price comparisons see comparisons price deflators, national accounts 1.190, 3.1(appendix) price discrimination 3.112–115, 6.82 price indices, collection data in the form of 6.74 price quote reports 12.51 price reductions see reductions price reference period (g) 1.20, 9.81–82 price-updating from weight reference period 9.95–104(t) price relative (g) price relative reciprocal test 16.125–126 price reversal (price weights symmetry) test 16.46 price updated weights (g) 1.28–29, 9.95–104(t) price variation 1.185–186, 3.108–120 probability proportional to size (PPS) sampling (g) 5.11, 5.16–23, 11.18 order sampling 5.19–23(t) Pareto sampling 5.19–21(t), 5.23 sequential sampling 5.19, 5.21 systematic sampling 5.17–18(t) probability sampling (g) 5.6, 5.8–26, 5.108 frames 5.13–15, 5.59 PPS sampling see above simple random sampling 1.203–210, 5.10, 11.18 stratification 5.12, 11.18 systematic sampling 5.10 see also sampling processing errors 11.8, 11.27 producer price index (PPI) 14.4, 14.57–58, 14.60(t*), 14.75(t*) coverage 14.31, 14.32(b*), 14.58, 14.60(t*), 14.64 output
 14.31(t), 14.32(b*), 14.75(t*) see also production account production account 14.18, 14.25–33(b)(t), 14.75(t*) see also producer price index; SNA production costs 7.81–82, 7.119 production for own consumption see consumption of own production production and publication (indices) 1.280–284, 12.56–70, replacement/substitution of products 6.40, 6.103–107, INDEX 13.1–58 access to data 13.49–52 confidentiality 13.53–54 contributions to price change, analysis 13.19–21 disaster recovery 12.66–70 electronic 13.55–58
 HICPs see Annex 1 (7) international standards 13.42–44 interpretation 13.22–23 monthly compilation 12.58–60 press release, bulletin and methodological statement 13.38–41(b) publication level 3.160–161 related/alternative measures, presentation 13.24–37 seasonal adjustment and smoothing 13.13–18 spreadsheets 12.61–64 sub-aggregate indices 13.32–37 time series presentation of level and change 13.4–12, 13.38(b*) timing 6.13, 13.45–48 see also quality management; user consultation product lists 5.15 products (g) product test 15.7–10 property taxes see taxation proportionality in current prices test (g) 9.25, 20.59 first axiomatic approach 1.55, 16.37–38 second axiomatic approach 16.99 publication of CPIs see production and publication purchase of products, during price collection 6.95, 6.97 purchaser’s price (g) purchases from abroad see household consumption expenditures purchasing power parities (PPPs) 2.28–30, 6.119–122, 14.77–79 see also Annex 4 pure price change (g) pure price index 15.24–32 see also fixed basket indices pure quantity index 16.63–67 quadratic mean of order r indices 1.98–99, 17.33–43, 17.59 quality adjustment (g) 1.226–255, 7, 21 HICPs see Annex 1 (5.4) hedonic approach see hedonic approach high turnover industries 7.125–158 chaining 7.153–158, 8.21 difference between hedonic and matched models indices 7.150–152 examples 7.128–131 hedonic indices 7.132–149 information requirements 8.22–31 statistical metadata system 8.23–31 long-run and short-run
 comparisons 7.42–43, 7.159–173(t) implicit short-run comparisons 7.165–170 quality adjustment methods 7.160–164 single-stage and two-stage indices 7.171–173 matched models 7.2, 7.5, 7.6–18 dynamic universe 8.1(appendix), 8.7 and hedonic indices 7.150–152, 21.59–60 missing items 7.7–13, 8.2 new products 7.18, 8.4 sampling concerns 7.14–17, 8.3, 8.6–8 telecommunication services 10.97–104(t) overview 1.226–229, 21.1–11 quality change see quality change (below) 7.69, 7.118 seasonal products 7.8–9 user value
 7.29–30 when matched models are unavailable 7.33–43 additive versus multiplicative adjustment 7.39–40 base versus current period adjustment 7.41 long-run versus short-run comparisons 7.42–43 see also missing observations quality adjustment methods 1.226–255, 7, 21 choice of methods 7.116–124(f) explicit methods 1.249–255, 7.35, 7.38, 7.72–115 choice of 7.119–120(f) expert judgement 1.251, 7.73–76 hedonic approach see hedonic approach production or option costs, differences in 1.250, 7.81–89, 7.119 quantity adjustment 1.249, 7.77–80(f), 7.81(t*) implicit methods 1.236–248, 7.38, 7.44–71 carry-forward see carry forward of unavailable prices choice of 7.121–124 class mean imputation 7.67–68, 7.121 comparable replacement 7.69, 7.118 hedonic approach 21.43–47 linked to show no price change 7.70, 7.124 non-overlapping qualities 1.241–248 overall mean or targeted mean imputation 7.53–66(t), 7.121 overlap 1.236–240, 7.35, 7.45–52(t), 7.123 short-run comparisons 7.165–170 quality change 7.19–32 conditional indices 7.31–32 effect on price 1.230–235 utility-based approach 7.24–30 quality change bias 11.35–36, 11.51–55 quality checks (price collection) central and head office 12.41–55 algorithms 12.52–55 reports 12.47–51 local 12.27–40 backchecking 12.33–38 monitoring 12.29–32 other auditor functions 12.39–40 quality in the field (price collection) 6.67–68, 6.123, 12.15–26 continuity 1.218–221, 6.57, 12.18–20 data entry queries 12.21–25 descriptions 12.16–17 feedback 12.26 see
 also collectors quality management (user needs) 12.71–91 performance management 12.92 systems 12.78–89 training and development (staff) 12.93–99 documentation 12.100–102 reviews 12.103 see also production and publication quantities mean value test for 16.48 price collection by 6.63–64 target quantity (sampling) 5.2 quantity adjustment explicit quality adjustment (g) 1.249, 7.77–80(f), 7.81(t*) Geary–Khamis quantity index 1.70 quantity indices Divisia approach 15.65–71 pure quantity index 16.63–67 quantity
 relative (g) 531 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE quantity reversal (quantity weights symmetry) test 16.45, 16.111 quantity weights (g) questionnaires (price collection) design 6.50–66 codes 6.1(appendix), 6.55–56, 6.60–61 forms 6.1(appendix), 6.53, 6.99(t*) quota sampling 5.42–45, 11.19 ranking variables (sampling) 5.19 real estate agency fees 3.122, 10.20, 10.40, 10.149–155, 23.118–120 see also Annex 1 (6.5) rebasing (g), 8.44–48 rebates 3.138, 3.141–143, 6.89–90 recording transactions 14.19–24 timing 14.20 valuation 14.21 reductions (prices) 6.80–91 reservation prices 8.59–60 response errors 11.7, 11.25–27 retailers see outlets revisions (indices) see Annex 1 (5.13) revolutionary goods 8.35, 21.62–64 directed replacements 8.57 sample augmentation 8.52–55(t), 8.57–58, 21.64 reweighting see weights rolling year annual indices 22.45–54(f)(t), 22.57(f *) chained 22.49–53(f)(t), 22.57(f *) fixed base 22.48–53(f)(t), 22.56–59(f)(t), 22.60(f *), 22.80(t*), 22.81(f *), 22.82–83 by using month-to-month annual basket indices 22.91–96(f)(t) by using year-over-year monthly indices 22.55–62(f)(t) rotation item or product (g) outlets 3.119–120 sample (g) 8.1(appendix), 8.18–21, 8.44–48 bonus offers, extras and free gifts 3.142, 6.84, Rothwell or Bean and Stine Type C indices 22.87–89(t), 6.99–102 discounted prices 3.138–143, 6.81–82 HICPs see Annex 1 (5.2) payments involving interest, service charges or extra charges 6.91 price discrimination 3.112–115, 6.82
 rebates or refunds 3.138, 3.141–143, 6.89–90 sale or special offer prices 6.1(appendix), 6.60, 6.83, 6.84 sales taxes 6.87 stamps 6.85 tips for services 3.46, 6.88 trade-ins 6.86 reference periods 1.20, 4.44–46, 9.81–84, 9.95–104(t) reference population (g) refunds 6.89–90 regional indices 3.101–104 see also subsidiary indices regional weights 4.10–12, 4.14–15(t) regulatory authorities, prices obtained from 6.43 reimbursements, by governments or social security schemes 3.134 rejected price observations
 (validation) see Annex 1 (5.5) remuneration in kind 1.163, 3.7, 3.14 renovation expenditures (owner-occupied housing) 10.20, 10.40, 23.107–117 rental equivalence (leasing equivalence) approach (g) 10.14–19, 23.3, 23.16–21, 23.139–140 rents 23.127–131 durables 3.28 home-owner’s estimates 23.139 indexation 1.11, 2.16 non-payment 23.124 22.91(f *), 22.93–94(t), 22.96(f *) rural areas, coverage 3.97–98 sales (reductions) 6.83 sales taxes 6.87, 13.28–29 sampled product (g) 4.7(f), 9.9, 9.16(f *) sampling (g) 1.203–210, 5 allocation of resources 5.100–105 augmentation (g) 8.52–58(t), 21.64 see also directed replacements choice of method 5.51–60 elementary indices 9.38, 20.87 errors 5.38, 11.3–4(t), 11.21, 20.20 estimation see estimation frame (g) 5.13–15, 5.28, 5.59 HICPs see Annex 1 (4.3, 8) and item replacement/substitution 8.9–17 matched models method 7.14–17, 8.3, 8.6–8 non-probability see non–probability sampling outlets 5.3–6, 5.15, 5.55 probability see probability sampling rotation/rebasing 1.222–225, 8.1(appendix), 8.18–21, 8.44–48 size of samples 5.33, 5.52–54 specific recommendations 5.106–110 in time 5.49–50 universe 5.2–6 see also collection; estimation; surveys scanner data (g) 1.213–214, 4.32, 5.60, 6.117–118, 9.72–73, 20.88–99 scope (g) seasonal adjustment (index publication) 13.13–18 seasonal products (g) 1.116–119, 6.41, 22 carry forward of unavailable prices 22.78–84(f)(t), repair expenditures (owner-occupied housing) 10.20, 10.40, 22.88–89(t),
 22.91–93(f)(t) 23.107–117 replacement products (g) 8.9–17, 9.55–63(t) clothing 10.66–72 directed 8.49–51, 8.57 quality adjustment 6.40, 6.103–107, 7.69, 7.118 replacement universe 8.1(appendix), 8.7 see also new products; substitution representative products (g) 4.7(f), 9.9, 9.16(f *) representative item method (sampling) 5.6, 5.46–48 telecommunication services 10.97–107(t) representativity 5.109 representativity bias (g) 1.102–105 re-sampling 1.222–225, 8.1(appendix), 8.18–21, 8.44–48 532 clothing 10.73–88(t) data set
 22.14–15(t) defined 22.1 economic approach 17.84–89 imputation 10.79–87(t), 22.85–86(t), 22.88(f *), 22.93–95(t), 22.96(f *) month-to-month indices Rothwell or Bean and Stine Type C indices 22.87–89(t), 22.91(f *), 22.93–94(t), 22.96(f *) maximum overlap 22.63–77(t) used in predicting rolling year indices 22.91–96(f)(t) quality adjustment 7.8–9 rolling year indices 22.45–54(f)(t), 22.57(f *) chained 22.49–53(f)(t), 22.57(f *) fixed base 22.48–53(f)(t), 22.56–59(f)(t), 22.60(f *) by using year-over-year monthly indices 22.55–62(f)(t) weights (fixed and variable) 4.62–66, 22.74 year-over-year annual indices 22.35–44(t) chained 22.40(t*), 22.41–43(t) fixed base 22.38–40(t) year-over-year monthly indices 22.16–34(t) chained 22.30–34(t) fixed base 22.26–29(t), 22.31(t*) used in predicting rolling year index 22.55–62(f)(t) see also missing observations second-hand goods 3.127–129, 4.68–72 selection errors 11.4 semi-durables 3.166, 3.167, 3.168 semi-logarithmic form 21.1(appendix) sequential PPS sampling 5.19, 5.21 services 3.3 acquisitions approach 1.157, 3.7–8, 3.18–21, 3.24–25, 10.166 central price collection 6.71, 12.14 durable services 1.157, 3.24–25 flow of provided by durable goods 3.27–29 tips for 3.46, 6.88 uses approach 1.151, 3.18–21, 3.24–25 see also individual services shops see outlets shortages see missing observations short-run comparisons (quality adjustment) 7.42–43, 7.159–173(t) implicit short-run comparisons using imputations 7.165–170 quality adjustment
 methods 7.160–164 single-stage and two-stage indices 7.171–173 simple random sampling 1.203–210, 5.10, 11.18 single-household case (economic approach) see economic approach single-stage indices (quality adjustment) 7.171–173 smoothing (index publication) 13.13–18 SNA see System of National Accounts social security 1.11, 2.9–13, 3.42, 3.134 HICPs see Annex 1 (5.9) social transfers in kind 3.9–11, 14.35, 14.37, 14.42 see also education services; health services spatial price comparisons aggregation methods see
 Annex 4 (4) data requirements see Annex 4 (3) differences between temporal and, see Annex 4 (2) overview see Annex 4 (1) Special Data Dissemination Standard (SDDS) 13.43 special offer prices 6.1(appendix), 6.60, 6.83, 6.84 stamp duty (stockbroking services) 10.130–132 stamps, given with purchases 6.85 stochastic approach (g) 1.73–79, 16.74–93 elementary indices 20.100–111 unweighted 1.75–76, 16.74–78 weighted 1.77–79, 16.79–93 stockbroking services 10.119, 10.130–136 straight line depreciation 23.59–61 stratification (sampling) 5.12, 11.18 sub-aggregate indices 13.32–37 subscriptions 3.45 subsidiary indices 3.95–96, 3.103–104 see also regional indices subsidies 3.135–137 substitute (g) substitution (g) 1.108, 1.256–262 and new products 1.259–262, 8.2(appendix) quality adjustment (g) 6.40, 6.103–107 INDEX sampling (g) 8.9–17 see also comparable items; missing observations; replacement products substitution bias (g) Lowe index 17.76–83 outlet substitution (new outlet) bias 3.118, 11.35–36, 11.60–63 upper-level substitution bias 11.35, 11.37–42 substitution effect (g) 1.31–32 superlative and exact hedonic indices (SEHI) 7.142–149, 21.48–58 superlative indices (g) chained 19.20–22(t) choice of 1.13–14, 1.100–101, 17.50–54 COLIs 1.98–99, 17.4, 17.27–49, 17.55–60 estimating COLIs by 1.97–101 Fisher index 15.18–23, 17.27–32 fixed base 19.17–19(t) quadratic mean of order r 1.98–99, 17.33–43, 17.59 retrospective 9.138 symmetrically weighted 19.17–22(t) To¨ rnqvist index
 15.93, 17.4, 17.44–49 two-stage aggregation 17.55–60, 19.23–27, 19.28(t*) Walsh index 15.24–32(e) see also cost of living index; symmetric indices supply price index (SPI) 14.70, 14.75(t*) supply and use table (SUT) 14.9, 14.13, 14.18, 14.53–56(t) surveys 4.30–31 household expenditure surveys see household expenditure surveys see also sampling symmetric indices (g) 1.41–45, 15.18–32 Fisher index as average of Paasche and Laspeyres indices 15.18–23 Walsh index and theory of pure price index 15.24–32 see
 also superlative indices symmetry tests see invariance and symmetry tests systematic sampling 5.10, 5.17–18(t) System of National Accounts (SNA) (g) 14 data sources 1.194–196, 4.25–28, 4.41 durables 14.40(b*), 23.1, 23.16 household consumption expenditures 14.34–45(b)(t), 14.75(t*) institutional units and establishments 14.14–17(b) for major accounts see capital account; external goods and services account; production account; use of income account owner-occupied housing 14.40(b*), 23.144 price deflators 1.190, 3.1(appendix) links with HICPs see Annex 1 (6) price indices export price index 14.4, 14.51, 14.57–58, 14.60(t*), 14.75(t*) for final uses 14.72 for gross domestic product 14.73–74, 14.75(t*) import price index 14.4, 14.51–52, 14.54(t*), 14.57–58, 14.60(t*), 14.75(t*) for intermediate consumption 14.71, 14.75(t*) for labour services 14.71(t*), 14.75(t) producer price index see producer price index for total supply 14.69–70, 14.75(t*) recording transactions 14.19–24 supply and use of goods and services 14.11–13 supply and use table 14.9, 14.13, 14.18, 14.53–56(t) terminology 14.6 weighting data 4.25–28, 4.41 533 CONSUMER PRICE INDEX MANUAL: THEORY AND PRACTICE system of price statistics framework 14.75(t*) national accounts see SNA see also data targeted mean or overall mean imputation 7.53–66(t), 7.121 target quantity (sampling) 5.2 tariffs 6.21, 9.14 HICPs see Annex 1 (5.7) taxation 13.27–29 capital gains 2.19 coverage 3.135–137 financial services
 10.1(appendix) income and wealth 2.19, 3.42, 13.27, 23.126 indexation 2.19 on insurance 10.171–173(t) on products 3.135–137 property 10.20, 10.40, 10.171–173(t), 23.100–101, 23.118–120 sales taxes 6.87, 13.28–29 in SNA 14.21–24 tax and prices index 13.27, 13.29 technology see high-technology telecommunication services 10.90–116 bills, sample of 10.113–116 contracts 10.92 customer profiles 10.108–112(t) matched samples 10.97–104(t) unit values 10.105–107 see also Internet; mobile telephones telephone services see
 telecommunication services tenants (rented accommodation) 23.123–124 test approach see axiomatic approach test of determinateness as to prices 16.127 theoretical indices in characteristic space (hedonic approach) 21.38–39 tied bundles of characteristics (hedonic quality adjustment) 21.12 time aggregation problem (elementary indices) 20.15–17, 20.20 time reversal tests (g) 9.25, 16.132, 20.44, 20.62 first axiomatic approach 1.55, 1.59–61, 16.44 second axiomatic approach 16.109 time, sampling in 5.49–50 timing price collection see collection publication of indices 6.13, 13.45–48 HICPs see Annex 1 (7) tips and gratuities 3.46, 6.88 To¨ rnqvist index (g) 1.44, 15.93, 17.4, 17.44–49 chained 19.20–22(t), 19.26–27, 19.28(t*) and COLI 1.99, 17.44–49 fixed base 19.17–19(t), 19.24–25, 19.28(t*) stochastic approach 1.78–79 superlative index 1.99–101, 17.4, 17.44–49, 17.50–54 test performance 1.66–67, 1.81–82, 1.84, 16.1(appendix), 16.60–61, 16.120–129 and two-stage aggregation 17.57–58, 19.24–27, 19.28(t*) To¨ rnqvist–Theil index 1.78, 1.81 trade-ins 6.86 training and development (staff) 12.93–99 documentation 12.100–102 reviews 12.103 transactions (inflation) CPI 14.6–7 transfers 1.168, 3.41–46 definition 3.41 gifts and subscriptions 3.45 534 licences 3.43 social security contributions 3.42 taxes, income and wealth 3.42 tips and gratuities 3.46 see also financial assets; gambling; insurance transitivity (circularity) test (g) 9.25, 15.88–96, 16.110, 16.132, 20.63 see also Annex 4
 (4.1) transport services see social transfers in kind Turkey algorithm 9.160, 12.53–54 turnover of models and quality adjustment see quality adjustment two-stage aggregation 17.55–60, 19.23–27, 19.28(t*) two-stage indices (quality adjustment) 7.171–173 unavailable items see quality adjustment unavailable prices see missing observations underlying (core) inflation 13.24–26 undesirable goods and services 1.169, 3.123–124 unemployment benefits see social security unequivocal indices 15.24, 16.63–67 unilateral
 price indices first axiomatic approach 16.11–21 second axiomatic approach 16.22–29 unit values (g) 9.70–71, 10.105–107 universe, sampling 5.2–6 unweighted hedonic indices 21.59–60 unweighted stochastic approach 1.75–76, 16.74–78 upper-level substitution bias 11.35, 11.37–42 urban areas, coverage 3.97–98 use of income account 14.18, 14.34–45(b)(t) see also SNA user consultation 13.59–65 advisory committees 13.62–63 index quality, explaining 13.64–65 methodology, presentation 13.41, 13.44(b*), 13.60–61 uses of CPIs, explaining 13.59 see also production and publication user cost approach (g) 23.3, 23.22–33 owner-occupied housing 10.7–13, 23.69–93 alternative 23.141–144 relationship with acquisitions approach 23.34–42 unique durable goods 23.69–78 user costs (owner-occupied housing) 23.94–120 insurance 23.102–106 landlord-specific 23.121–131 additional services 23.127 billing and maintenance 23.123, 23.125 non-payment of rent and vacancy costs 23.124 opportunity cost of capital 23.126 maintenance and renovation 10.20, 10.40, 23.107–117 mortgage interest costs 10.23–38(t), 23.95–99 property taxes 23.100–101 transaction costs (home purchase) 23.118–120 user value (quality adjustment) 7.29–30 uses approach (g) 1.151–158, 3.18–29 CPIs based on 3.26–29 insurance 10.164–165 owner-occupied housing 10.7–19, 23.3, 23.16–21, 23.139–140 services 3.24–25 SNA 14.6, 14.54 utility-based approach (quality change) 7.24–30 vacancy costs (rented accommodation) 23.124 validation checks
 (price collection) 12.21–25, 12.45–46 see also Annex 1 (5.5) valuation, in recording transactions 14.21 value added tax (VAT) 3.137 value aggregates 14.1–7, 15.7–17 Laspeyres and Paasche indices 15.11–17 product test 15.7–10 see also elementary aggregates value (g) variable weights for seasonal products 4.63–66 variance estimation 5.62, 5.73–99 elementary index formulae 5.74–75 errors 5.110, 11.12–15 see also estimation vehicles 3.129, 7.81 see also cars virtual prices (new products) 21.65–67 vouchers,
 money-off 3.138, 3.141, 3.143 wages 1.9, 2.8 see also income Walsh price index (g and g appendix) 1.43 chained symmetrically weighted 19.20–22(t) two-stage aggregation 19.26–27, 19.28(t*) fixed base symmetrically weighted 19.17–19(t) two-stage aggregation 19.24–25, 19.28(t*) representativity bias 1.102–105 superlative index 1.99, 1.100–101, 17.4 symmetric averages 15.24–32 test performance 1.66–67, 1.71, 16.59, 16.61 see also quadratic mean of order r indices; symmetric indices wealthy households, coverage 3.94 weighted arithmetic average index (g and g appendix) weighted price indices asymmetrically weighted 19.9–16(t) chained 19.11(t*), 19.13–16(t) fixed base 19.9–12(t), 19.16(t*) symmetrically weighted 19.17–22(t) chained 19.20–22(t) fixed base 19.17–19(t) weighted least squares (WLS) estimator 21.1(appendix) weight, price collection by 6.63–64 weight reference period (g) 1.20, 4.44–46, 9.81–82 price-updating from weight reference period 9.95–104(t) weights (g) 1.191–199, 4, 14.58, 14.60(t*) adjustment 4.42–43 annual 9.135–136, 15.33–64 chaining 9.135–136, 19.11(t*), 19.13–16(t), 19.20–22(t) data sources 4.16–33, 4.40–41 elementary aggregates 1.124–126, 4.14–15(t), 9.11–14, 9.115–117 errors 4.76–77 fixed base indices 19.9–12(t), 19.17–19(t) INDEX fixed-weight indices 9.131–138 geometric averages 9.137 HICPs see Annex 1 (4.1) hedonic approach 21.1(appendix), 21.58 housing (owner-occupied) 4.3 insurance 4.67 monthly indices 15.33–64 outlets 4.13, 4.30–31 price
 updated weights (g) 1.28–29, 9.95–104(t) regional 4.10–12, 4.14–15(t) revision 4.47–55 reweighting 1.274, 9.105–126 calculation of chain index 9.112–114(t) frequency 9.108–111 long-term and short-term links 9.123–126 new elementary aggregates 9.115–117 new higher-level indices 9.118–119 partial reweighting 9.120–122 seasonal products (fixed and variable weights) 4.63–66 second-hand goods 4.68–72 stochastic approach 1.75–79, 16.79–93 structure 4.4–15(f)(t), 4.56–61 superlative indices 19.17–22(t) see also
 higher–level indices tests 16.45–46, 16.114–119 see also household consumption expenditures WLS estimator see weighted least squares estimator yearly indices see seasonal products year-over-year annual indices (seasonal products) 22.35–44(t) chained 22.40(t*), 22.41–43(t) fixed base 22.38–40(t) year-over-year monthly indices (seasonal products) 22.16–34(t) chained 22.30–34(t) fixed base 22.26–29(t), 22.31(t*) used in predicting rolling year index 22.55–62(f)(t) Young index (g and g appendix) 1.35–37, 19.34(t*), 19.37–40 annual indices 22.80(t), 22.81(f *), 22.83–86(t), 22.91–95(f)(t) and COLI 1.95 factoring and price updated weights 9.92–94 geometric 1.38, 1.40 higher-level indices 9.78–80, 9.87–94 monthly indices 15.54–64 relationship with its time antithesis 15.3(appendix) relationship with Laspeyres index 1.36, 15.56–58 relationship with Lowe index 9.87–91 test performance 1.59, 1.83, 16.130–134 Young type indices 19.45–48(t) zero price, increasing to a positive price 17.90–94 535