sou · GNB3124 · 2000-01-01 00:00:00 — The first, major advantage of the KPI index construction outlined above is that it is largely able to avoid the so called upper level substitution bias (the Laspeyres overestimation of a true cost of living index). This advantage is due to the use of weights that are between the two periods for which prices are compared. A simple measure of the magnitude of this substitution bias is the difference between the short-term and the long-term index, although this difference is to some extent disturbed by other kinds of new information that is brought into the long-term index. Table 0 gives these differences between 1979 and 1998. We note that the difference is positive in 18 years out of 20. The order of size of the mean difference, 0.1–0.2, is similar to estimates of upper level substitution bias in the U.S. CPI, see for example Aizcorbe and Jackman (1993) who estimated this bias for the years 1982–91. , we will return to other measures of substitution bias for the KPI. Table 0: Differences between short- and long-term , Dec y-1 to Dec y. The second advantage of the present index construction is that it is possible to take in new products and up-to-date weights every year. This advantage is, however, shared with other countries that apply chain and is not due to the specific long-term index definition. However, there are also some problems. The first one is that there is a slight asymmetry in the long-term index formula. Prices are generally measured in the week in which the 15th occurs. This means that the period between the two points in time that the 1943 Commission referred to in formulae (3) and (4) above is , y-1 to , y rather than 1, y to Dec 31, y as in todays long-term index. Another way to look at this problem is to consider the following decomposition of the long- term index. å Pky ,12Qky å Pky ,12Qky å Pky - 1,12Qky å Pky - 1,12Qky Here we have divided L into two factors. The first one is a Paasche index from y-1 to the year y, a period of 6½ months on average. The second one is a Laspeyres index from the year y to y, a period of 5½ months on average. This means that the long-term index is slightly biased in the Paasche direction, which may lead to an underestimating substitution bias! This may seem as a minor point but an additional factor may tend to aggravate this bias. This is to the extent that price changes tend to be concentrated to the turn of the year, in January, or else to the first part of the year so that quantity weights would largely reflect the consumption pattern after the price increase. This problem will be looked at empirically in the next section. A second problem with the way that the long-term index is estimated is that price updating is applied below the level where there is new information on consumption values. (1992) points out that this procedure amounts to an assumption of unitary price elasticity and that an underestimating bias occurs if the price elasticity is smaller than one. (1996) shows that the type of ratio estimator used for estimating (8b) gives rise to an underestimating sampling bias. The third problem is due to seasonal products, especially those for which there is no consumption and thus no price in . It is difficult to cover them in a consistent way in an index construction that has a single month as its price reference period. The fourth problem is of a more practical nature and concerns the easy-to-analyse criterion. When the 12-month rate of inflation is now published, the effect of the long-term index is removed. The official motive for this practice is that the annual revision of the figures may refer to changes (substitutions), which are not within the 12 months to which the inflation measure refers. However, it gives rise to some confusion among the index users to have a measure of 12-month inflation, which is not the simple ratio of the index numbers. 4Basic (upper level) index construction The approach to index construction put forward by the 1943 Commission could be seen as an early attempt to estimate a true cost of living index. Today, we know that as far as the functional form is concerned, the best approach to this estimation is through the concept of a superlative index. See Bilaga 2 for a definition of this concept. Three important examples of superlative index number formulae, based on four vectors of prices and quantities (P1,P2,Q1,Q2) and referring to two periods in time, 1 and 2, are I 12;F å Pk1Qk1 å Pk1Qk2 ö(V k1 + V k2 )/ 2 for t = 1 or 2 and Walshs index: I 12;W å W k ;W where W k ;W An index formula, which numerically comes very close to being a superlative index and will be considered later, is: å Pk2 (Qk1 + Qk2 ) = åWk;E I1;E = , (13½) + Qk )Pk with Wk ;E = å Vk1 + å The fundamental difficulty of all superlative is that they require weights from both ends of the comparison. The demand for timeliness normally prevents the use of weights for the last period in a short-term indicator like the CPI. They also run into difficulties in a chain index where the price reference period is only a month, since the weights would then also refer to only a month. Besides the lack of monthly consumption statistics, it cannot be reasonable even in principle to choose a months consumption as the basis for weights, since there is a seasonal variation that would make such a choice less representative of a households total consumption. If, on the other hand, we choose a years consumption as our quantity weight in an index with a month as a price reference period we introduce an asymmetry into the index which results in a loss of its true superlativity property. , we will look at these effects empirically. 4.2Index formulae using only period 1 weights At least for the last link in a chain index it will be necessary to use a base-weighted index, i.e. an index formula where only period 1 quantities are included. Some such formulae that we will consider are (remember that W k1 å Vk1 ): = å W k The geometric index: I The of Substitution (CES) index: ö1- s ù1 /( 1- s ) The Laspeyres index is only consistent with zero price elasticity and tends to overestimate a cost of living index, where elasticity is larger than that. The geometric index is only consistent with unitary price elasticity and over-/underestimates a cost of living index where elasticity is larger/smaller than one1. The CES index is a generalisation of these two and is consistent with a price elasticity of substitution, which is equal to s between all pairs of products and assumed to be known in advance. (In practice, one would use data from before the present index link to estimate it). The CES index is a relatively novel theoretical invention. It was first proposed by (1975) and later rediscovered by Moulton (1996). At the highest aggregation level, empirical results suggest a price elasticity between zero and one. Based on U.S. CPI data for 1988–1995 and Wilcox gave a value of s=0.7 as the best estimate. Moulton and (1997) present series of both fixed weight and chained Laspeyres and geometric and compare them to and 1 We adopt the convention of using positive values for elasticity. based on U.S. CPI data for 1987–1995. Their results likewise show that the superlative fall below the Laspeyres but above the geometric index – and closer to the geometric than to the Laspeyres. In both these studies, a disaggregation of the U.S CPI data into 207 item groups and 44 geographic strata was used. 4.3The chain/link choice Four approaches to the chain/link choice will be investigated. In this alternative, we would choose one of (14)– (16) as our aggregation formula. It would be natural to have a full year as the reference period for both prices and quantities. The link could last 3–10 years, below we will look at 5-year links. The long run development of the chained index would then be like I0Y ,m = I05 I510 ...I5Yn,m Annually chained , linking by month. This is the present chaining method used in the KPI with as the link month. The long run development of the index follows the following chain formula:2 = IY - 1,12 Õ I y - 1 ,12 Annually chained , linking by year. For reasons discussed below we want to try a different form of the chain index, not used earlier, in which we use a full year as our reference period for both prices and quantities. With this option the chained index will be: = IY - 1 Õ I y - 1 Delayed chain index, linking by year. There are some practical problems with implementing (19), which will be discussed more below. Another option would be to have two-year Laspeyres link in the end of 2 The extra link from 0 to 0,12 is needed for having a full year as the index Its exact definition is: I the chain, allowing more time for weight data for earlier years to arrive. This gives the following expression for the index: = IY - 2 Õ I y - 1 We will now take a closer look at the various set-ups that are possible within these four frameworks and compare them numerically based on historic data. A number of numerical experiments were carried out for comparing various index alternatives in order to estimate the sizes of biases. For this purpose, a database of KPI subindices for 72 National Accounts (NA) purposes was used, covering the period from 1980 to 1998. Imputed rent for owner-occupied housing was excluded from the simulations and so were a few other, very small categories for which comparable data for the whole period were not available. The simulations were thus designed to measure upper level substitution bias, i.e. the bias resulting from failure to use a superlative index in the final aggregation step where subindices for these 72 purposes are being put together into an All Item Index. Substitution biases at lower levels are not covered by this simulation. The annual values thus obtained were price updated to – the link base month. Algebraically, the and weights in the data base are defined in the following way (the superscript y,m means month m in year y and only y means the whole year y): The short-term index (STIX): å Pky ,m Qky - 1 P y ,m S y - 1 ,12 y- 1 ,12 P y - 1 ,12 where the weights are W STIX y - 1 ,12 å Vky - 1 The long-term index (LTIX): å Pky ,12Qky P y ,12 y- 1 ,12 where the weights are W LTIX The data base consists of the and weights defined above. However, in the simulations we need the actual values, which are not in the database. The value shares were instead obtained by backdating the weights defined above.3 Throughout the analysis, we will use the two present index types defined in (5) and (6) as standards with which we compare our alternatives. Further, when comparing two whole years, we will use the following computational rules: y - 1 ,12 S y - 1 = S y - 2 ,12 S y- 1 ,12 / S y - 2 ,12 y - 1 ,12 = Ly- 2 ,12 S y - 1 ,12 / S y - 2 ,12 will be geometric or arithmetic averages of the monthly S y- 1 ,12 and S y - 2 ,12 We will use 5-year links in our simulations. Our base-weighted formulae will now be defined as follows: Laspeyres - FL: I yy + n ,m = å Pky + n ,m Qky , where 0<=n<=5. 3 The different levels of values and quantities over the years get lost in this exercise. This problem is not essential. However, it is a slight problem when estimating the index. The form of the Laspeyres index to be used in the simulations will be y + 5 ,12 P y + 5 ,12 P y ,12Q y I y ,12 = å W k which are identical to the short-term weights of year y+1. Note that (25b) is identical to the ratio of y+5,12 to y,12 in a run of strict Laspeyresindices with year y as the reference period. The second option is subindices as in (25): s I y + 5 ,12 CES - FC( ): y ,12 a CES index: based on the same weights and æP y + 5 ,12 We will compute (26) for s=0.1, 0.2, 0.9. For s=0 it coincides with FL and for s=1 it turns into the geometric index as the limiting case: æP y + 5 ,12 Geometric - FG: I y + 5 In Table 1 we look at simulated for rolling 5-year links, i.e. 79– 84, 80–85, 81–86 etc. We present one-year averages of 5-year changes. We notice that the fixed base Laspeyres is on average 0.27 points above the fixed base geometric index. These two variants are the extremes and the CES effectively interpolate the whole interval in between for s=0-1. Comparing with present index construction we see that the long- term index (taken as our preliminary yardstick) corresponds to a value of about s=0.7 which happens to be the same value as found by and Wilcox (1997). We also see that the difference FC(.7) – LTIX over the whole period varies from +0.25 to –0.13. This difference can be interpreted as the annual bias, if an FC index were used. Also, the chained short-term (STIX) which uses recent weights are on average smaller than the fixed base Laspeyres, indicating that bias tends to increase with the age of the weights. Table 1: Fixed base compared to chained short and long term . FC(.7) Table 1 (cont.) FC(.9) FC(.8) FC(.7) FC(.6) FC(.5) FC(.4) FC(.3) FC(.2) FC(.1) 4.4.2Chain : linking by month When linking by month, all index formulae have to be somewhat twisted, since quantity and price data will necessarily be from different periods. We choose only one superlative index in as our reference here. P y ,12 öW k ;T - MT: I y - 1,12 y - 1 ,12 with Wk ;T (Wky - 1 + Wky ). We immediately note the asymmetry in the formula with weights reflecting years and prices being for months. This problem is difficult to avoid when linking by month since consumption values with good quality exist only by year and since we want to represent the whole year in our index. This index would still have to be a long-term index, since year y values are available only early in year y+1. å Pky ,12 (Qky- 1 + Qky ) P y ,12 P y - 1 ,12 (Q y - 1 + Q y )= åk I y- 1 ,12 P y - 1,12 V y - 1 P y - 1,12 P y - 1,12 P y - 1,12 P y - 1 ,12 P y - 1 Two variants of geometric have been simulated. In both of those we change the weights in (28) so that for: Geometric - MG1: As (28) with weights =W ky- 1 Geometric - MG2: As (28) with weights= W kSTIX in (21b) above. Table 2: Chained , linking by month.Change December- . MC(.6) The CES-index now takes on the following shape: P y ,12 CES - MC(s): I y - 1 ,12 y - 1 ,12 The numerical results of this comparison are shown in Table 2. For brevity, only one CES index is shown – the one with the optimum s value. Some noteworthy features are: 1.The present long-term index on average well approximates a index. the years, LTIX-MT goes from +0.12 to – 0.11. 2.The s value in the MC index which best approximates a index is 0.6. Looking at single years shows, however that MC(.6) – MT varies from +0.20 to -0.25. This means that, if we have MC(.6) as a short-term index and a superlative long-term index, then the shift when going from a short- to a long-term index would be between these numbers. As a comparison, the present difference between STIX and LTIX varies between -0.08 and 0.27. That is, STIX is biased but with somewhat smaller variation than MC(.6). 4.4.3Chain : linking by year When linking by year it becomes necessary to establish a connection between the annual and the monthly index, since the I yy- 1 in (19) or (20) have to be built up from monthly . For each product group k, we P y ,m P y- 1,m P y ,m P y - 1 ,m For the simulations, we used the geometric time aggregation principle throughout, that is P y ,m y - 1,m ÷ (, in Section 5, we discuss the calculation method of the year-to- year index further.) The following index types were explored in the simulations: - YT : I y - 1 where the weights Wk;T are defined according to (28b) å W k ;W – YW: I y- where Wk ;W Vky - 1V ky 1 = å W k ;E - YE: I y - where Wk ;E çå Vky- 1 + å Note that the weights in (32), (33) and (34) are now symmetric with respect to the pricing periods. according to (32) and (33) are thus, by definition, true superlative . This is a clear advantage of the year-to-year index, which no other index formulation can fully meet. For the final year-to-month index, a base-weighted index is necessary and we again define the three formulae that are possible for this situation. = å W k P y ,m Laspeyres - YL: I y - 1 The year-to-year index that will be used in the simulations below will then be: = å W k I y - 1 Geometric – YG: As (28) with weights =W ky- 1 . æP y ,m CES - YC(s): I y ,m = ê çP y - 1 ÷ The year-to-year index that will be used in the simulations below will then be: Pky ö1- s ù = êå W k In Table 3, we present the simulation results on year-to-year . For reference, we have also included the monthly index. We now note that: 1.The true superlative , YT and YF are very close together as is also YE, an almost superlative index. They differ by at most 0.01and usually much less. On average, the difference is 0.001- 0.002. For all practical purposes, they could thus be considered numerically equivalent.4 2.The long-term index is below the superlative , on average by 0.04.It is larger, however, in 7 years out of 17 so the difference is not necessarily significant. It could still be interpreted as some evidence of the Paasche effect demonstrated in (10) above. 3.The Laspeyres index is about 0.05 and the short-term index about 0.08larger than the superlative . These differences are persistent over almost all years and clearly significant. 4 This statement has to be qualified for the situation where one price is zero – YT then collapses to zero or infinity. This situation does not show up in simulations at this high aggregation level but happens at lower levels from time to time. Table 3: Chained , linking by year. Change from whole year Y-1 to Y. YC(.4) 4.A CES index with s slightly smaller than 0.4 approximates a superlative index on average over the whole period. Looking at single years shows that [YC(.4) – YT] varies from 0.06 to -0.17. In most years the difference is smaller than 0.1. This means that, if we would have YC(.4) as a short-term index, then we could expect the shift when going from a short- to a long-term index to be between these numbers. 5.The geometric index with old weights significantly underestimates a superlative index, on average by about 0.08. 6.The linking-by-month index (MT) is biased downwards by 0.02 on average. The reason for this is that the weights on average tend to be biased backwards in time, moving MT to some extent towards a base-weighted geometric index. 4.5Data access end index construction in practice The present production of an index series according to (5) – (7) involves an annual reweighting process based on various data sources, which takes place in January-February each year. We will here discuss briefly the data and timing problems that the other chain/link choices will give rise to. Weight data access. The new household expenditure surveys (HES) are planned to run continuously and calendar year results will become available about six months after the reference year. However, the sample size is small (2000 households) and large non-response rates are expected. National Accounts (NA) values are available for some 90 categories (purposes). A preliminary version, used at present, is available in early January, t+1 (immediately after the end of the year). A revised version is available in , t+1 and a final version in , t+2. An ambition, not yet fully reached, is also to have a division into finer categories, finished in a preliminary version by , t+1 and revised in , t+2. In this case, it is no point in using the rather crude estimates in January, t+1. Thus a new link with base year t would start from January, t+2, (t+3 if one wants to wait for the revised values). In the case of a 5-year link continuing to , t+6 (t+7), weights would on average, over the link, be 3½ (4½) years old. Annually chained , linking by month. In this case, the presently used formulae – (5)–(7) – are one choice. Other combination of long and short-term links to be entered into (18) would also be possible such as (with the above acronyms) i) MT+MG1, ii) MT+MG2, iii) MT+MC, iv) ME + STIX, v) ME + MC and vi) only STIX. The list is not exhaustive. Annually chained , linking by year. Here one idea would be to have a superlative index (YF, YT, YW or YE, although the latter is not strictly superlative) for the year-to-year links and either a Laspeyres, a geometric or a CES index for the final year-to-month link. Another option would be to have Laspeyres throughout. 4.6An evaluation of construction alternatives We will now look at how our various alternatives perform with regard to the criteria set out in Section 2. 4.6.1Minimal long-term bias in relation to cost-of- living index/superlative index. Table 4 gives average biases over 18 years according to our simulations, for the directly comparable formulae. Some alternatives are not possible to compare directly with YT. We then used LTIX as an intermediate standard and added the bias of LTIX (–0.04) to their estimated biases according to Tables 1 and 2. Results are in Table 5. Table 4: Bias of index formulae, with YT as the standard, according to Table 3, 1981–98 Range of bias variation -0.02 to +0.01 -0.03 to +0.11 -0.38 to +0.01 YC(.4) -0.17 to +0.06 -0.15 to +0.07 -0.05 to +0.08 -0.02 to +0.19 -0.27 to +0.13 5 The small differences between the superlative are not indicative of any bias, since neither of them can be proved superior on theoretical grounds. Table 5: Average bias of various index formulae, with YT as the primary standard and LTIX as the intermediate standard, according to Table 1–2, 1981–97 FC (.7) MC (.6) Bias variation over the years is only meaningful to estimate with the YT standard. For the alternatives in Table 5, it is likely to be at least as large as for those in Table 4, however. We see that the fixed base options with FL and FG both have non- negligible biases, in different directions. The CES index (FC) appears to be unbiased on average, but this is partly illusory, since it requires perfect knowledge of s, the elasticity of substitution, which we do not have. Since s could be assumed to be rather stable FC would give small average biases, though. Linking by month will not either give us unbiased estimates in the long run, although with the MT or ME formulae, biases seem to be quite small. Again, the MC index would be nearly unbiased. Since the s would now be possible to reestimate each year we would be in a better position with MC than with FC. The LTIX or STIX will not provide unbiased – according to the simulations the present index construction adds additive biases of –0.04 each year on top of each other, whereas the alternative with only STIX leads to annual biases of +0.08. Linking by year with a superlative or index for the year- to-year index does give us an unbiased index series. This is independent of the choice of year-to-month index, since its influence is cancelled out in the long run. YL or YG are both biased in opposite directions, by half a decimal point. 4.6.2Minimal bias and disturbance in short-term changes The 12-month change of the index is a key economic statistic, which is much analysed by central banks, and economists engaged in macro- economic analyses. It is therefore desirable that this statistic in itself is not biased and that a time series of such changes is not unduly influenced by other factors than pure price changes. It is therefore instructive to look at the definition of this statistic under different chain/link constructs. Fixed base index: IYY-,m1,m I 0Y ,m I 0Y - 1 ,m 12-month change: linking by month (): I Y ,m I Y ,m Y - 1 ,m Y - 1 ,m IY - 2 ,12 Linking by year according to (19): Linking by year according to (20): I Y ,m Y - 1,m Y - 1 ,m I Y ,m Y - 1 ,m Y - 1 ,m Now, in the fixed base index the 12-month change will usually be within an index link. The numerator and the denominator use the same weights and the statistic will only depend on the fixed and the actual price changes. The long-term bias is rather stable and thus a small problem, when looking at a series of such changes.