BLS WORKING PAPERS 

 

U.S. Department of Labor 

U.S. Bureau of Labor Statistics 

Office of Prices and Living Conditions 

Reconsideration of Weighting and Updating Procedures in the US CPI 

 

John S. Greenlees, U.S. Bureau of Labor Statistics

Elliot Williams, U.S. Bureau of Labor Statistics

Working Paper 431

October 2009

All views expressed in this paper are those of the authors and do not necessarily reflect the views or policies of the 

U.S. Bureau of Labor Statistics. 
Reconsideration of Weighting and Updating Procedures in the US CPI 

by 

John S. Greenlees 

Elliot Williams 

U. S. Bureau of Labor Statistics

June 5, 2009 

Presented at the Eleventh Meeting of the Ottawa Group in Neuchâtel, Switzerland, May 

27, 2009. We are grateful to Rob Cage and Patrick Falwell for creating and interpreting 

CPI data sets and to Rob Cage, Rob McClelland, and the Ottawa Group meeting 

participants for helpful discussions. All views expressed in this paper are those of the 

authors and do not necessarily reflect the views or policies of the US Bureau of Labor 

Statistics. 
- 2 - 

I. Introduction 

In 2002, the US Bureau of Labor Statistics (BLS) made two important enhancements to 

the Consumer Price Index (CPI) program. First, it accelerated the frequency of 

expenditure weight updates in the headline CPI. Second, it introduced a new 

supplemental CPI employing a superlative formula. This paper provides a retrospective 

look at those changes and suggests conclusions that can be drawn from the experience 

thus far. 

The headline Consumer Price Index for All Urban Consumers (CPI-U) employs a form of 

the Lowe, or Modified Laspeyres, index structure. The aggregate US City Average All 

Items CPI-U is computed as an arithmetic average of lower-level indexes, with weights 

derived from consumer expenditures during a base period. Prior to 2002, the expenditure 

base period was updated approximately every 10 years. In December of 1998, however, 

the BLS announced that future updates would take place every two years, with each 

update introducing a new two-year base period.1 Thus, effective with the release of 

January 2002 data, month-to-month changes in the CPI-U were based on weights drawn 

from 1999 and 2000 calendar year expenditures. Three subsequent biennial updates have 

now taken place, and expenditure weights in effect since January 2008 are based on 

2005-2006 expenditures.2

The published CPI-U remains subject to the “upper level” consumer substitution bias 

inherent in the Lowe structure. To address this concern, in 2002 the BLS introduced a 

new, supplemental index, the Chained CPI for All Urban Consumers or C-CPI-U. The 

Törnqvist formula used in the aggregation of the final C-CPI-U is designed to be a closer 

approximation to a cost-of-living index (COLI) than the Lowe formula used in the 

CPI-U. It uses actual consumer expenditure estimates from both the current and previous 

months to weight the basic indexes as a means of accounting for consumer substitution 

between item categories.3

Monthly values of the C-CPI-U are published beginning with data for January 2000 

(December 1999=100) and continuing through the current month. Because of 

unavoidable lags in the collection and processing of expenditure data, however, the 

C-CPI-U is subject to two annual revisions. The most recent final monthly values, for 

calendar year 2007, became available in February 2009. It is only in this final version of 

the index that the superlative Törnqvist formula is used in the aggregation of basic 

indexes. Preliminary monthly index values have been computed using a weighted 

geometric mean formula with the weights corresponding to the same base period used in 

the CPI-U. 

The availability of eight calendar years of final C-CPI-U data and seven years of 

preliminary-to-final revisions, along with expenditure data from four consecutive twoyear CPI-U base periods, give us an opportunity to analyze both these BLS 

methodological changes. Among the issues that could be addressed are: how similar the 

 

1

 See Bureau of Labor Statistics (1999). 

2

 The statements in this paragraph apply also to the CPI for Urban Wage Earners and Clerical Workers 

(CPI-W). The CPI-U and CPI-W differ only in the sampled consumer populations used to derive their 

expenditure weights. 

3

 For more details on the C-CPI-U, see Cage, Greenlees, and Jackman (2003). 
- 3 - 

long-run movements in the monthly-chained C-CPI-U are to those of an annually-chained 

Törnqvist index; whether the evidence would support a modification of the geometric 

mean formula used in the preliminary C-CPI-U; and whether the BLS should consider 

changing or eliminating the current process of price-updating CPI-U expenditure weights 

between the base period and the so-called link month in which they are first used in the 

index.4

The issue on which we focus in this paper is whether the BLS can improve upon the 

current biennial weight update process it uses in the CPI-U. That index remains the 

headline CPI reported by the media. The CPI-U, and its fellow Lowe index the CPI-W, 

are also the BLS indexes widely used for government tax and benefit programs and in 

private-sector contracts, largely because unlike the C-CPI-U they are not subject to 

revision. Therefore, although the BLS and many economists believe that the C-CPI-U is 

a closer approximation to a COLI, improvements to the CPI-U are of the utmost 

importance. 

The primary potential alternatives to the current weighting process are updating at an 

annual or other increased frequency and/or reducing the time span of the expenditure 

base periods to one year or some other length. The question is whether such changes, by 

enhancing the timeliness of the CPI-U’s expenditure weights, would have the effect of 

bringing the index’s movements closer to those of the C-CPI-U. 

We begin with a background discussion in Section II, and follow it in Section III by 

estimating superlative and cost-of-living indexes for each annual and biennial period 

from 1999 through 2007, and comparing those index changes to those in the official BLS 

indexes. We also estimate indexes based on the constant-elasticity-of-substitution (CES) 

consumer demand model, obtaining in this way a series of simple summary statistics for 

the extent of consumer substitution behavior implied by our expenditure and price data 

for those years. Of special interest is whether that substitution behavior is statistically 

significantly different from the behavioral models implicitly underlying the Lowe and 

geometric mean formulas. 

In Section IV of the paper, we simulate various operationally feasible weight update 

formulas and compare the resulting indexes to the C-CPI-U. We examine whether the 

Lowe index moves closer to the superlative if it employs more timely base expenditure 

weights. We also ask whether using a shorter base period causes any instability or drift 

in the Lowe index, or whether employing sequential but overlapping base periods has any 

apparent ill effects. Section V presents our conclusions and recommendations for further 

research. 

II. Background and previous research 

The central empirical issue of our paper is the degree to which the Lowe-type weighting 

processes in the CPI-U index can be enhanced so as to bring the CPI-U’s movements 

closer to those of the superlative-type C-CPI-U, without undesirable sacrifices in 

volatility or infeasible changes in operational procedures. 

 

4

 As discussed further below, eliminating the weight-updating process would yield a Young index. 
- 4 - 

As discussed in the international CPI manual,5 a Lowe price index is distinguished from 

the familiar conceptual Laspeyres index by the separation of the weight reference (or 

expenditure base) period and price reference (or link) period.6

 That is, with q0k and s0k

indicating the period-0 quantity and expenditure share, respectively, of the k-th item, and 

ptk denoting its price in period t, the Laspeyres index between period 0 and period t can 

be written as 

 



  k

ktkk

k

okk

k

tkk

t pps

pq

pq

IX )/( 0 0

0

0

0,

Construction of the Lowe index recognizes the operational lag in collecting and 

compiling expenditure shares, which then necessitates a lag between the expenditure base 

period b and the price reference period 0 in which those weights are introduced into the 

index. Writing the Lowe index in share form requires that those shares be “priceupdated” to the link period. The price-updated share for the k-th item is given by 





j

bjjbj

Lo bkkbk

bk

pps

pps

s

0

0

,0

These price-updated shares can be thought of as the shares that would be observed in 

period 0 if there were no changes in relative quantities purchased between periods b and 

0. Using the price-updated shares, the Lowe index between periods 0 and t is 











k

okbk

tkbk

k

ktk

Lo

bk

Lo

bt

pq

pq

IX pps ,0, ,0 0

Note that the Lowe index can be written as the ratio of two Laspeyres indexes, each with 

prices and quantities from period b in the denominator. In the same way, the period-toperiod changes in a Laspeyres index will have the Lowe form. 

As in most countries, the CPI-U is constructed at two levels. Basic indexes for most 

individual item/area categories (for example, Apples in Chicago) are computed using a 

weighted geometric mean formula. The basic indexes are then aggregated to form 

higher-level indexes using a Lowe formula.7

 As noted earlier, at present the CPI employs 

the period 2005-2006 as its expenditure base period b and the period December 2007 as 

its link month 0. 

It is well known that in the presence of consumer price-taking and utility-maximizing 

behavior the Laspeyres index provides an upper bound to the true cost-of-living index. 

That bounding result does not apply to the Lowe index. Research suggests, however, that 

 

5

 International Labour Office (2004) paragraphs 1.17-1.23. 

6

 The weight reference and price reference periods should not be confused with the index reference period, 

at which the index is set to 100. 

7

 The same basic indexes are used in the CPI-U and the C-CPI-U. 
- 5 - 

under many reasonable conditions a Lowe index will tend to have an upward bias relative 

to the Laspeyres index and hence also to a target superlative or cost-of-living index.8

 

Consistent with that research, we will show in Section III below that increases in the 

CPI-U have exceeded those of the superlative C-CPI-U in recent years. 

The BLS uses a Continuing Consumer Expenditure (CE) Survey to provide expenditure 

weights for the CPI. The CE survey is carried out for the BLS by the US Census Bureau. 

Collection and processing of the annual data consumes the greater part of a year, meaning 

that expenditure data introduced into the CPI at the beginning of year t can pertain to year 

t-2 at the latest. On the other hand, the quarterly rotating panel design of the CE makes it 

possible, in principle, to use the final quarter of year t-2 as the expenditure base period. 

Using such a timely but short base period could, of course, have unattractive 

consequences, especially when the weights are drawn from a household survey rather 

than national accounts data. As noted in the international CPI manual,9

 

“Some countries prefer to use expenditure weights that are the average rates of 

expenditure over periods of two or three years in order to reduce ‘‘noise’’ caused 

by errors of estimation (the expenditure surveys are only samples) or erratic 

consumer behaviour over short periods of time resulting from events such as 

booms or recessions, stock market fluctuations, oil shocks, or natural or other 

disasters.” 

Recognized international best practice calls only for revising expenditure weights at least 

every five years, and more frequently if there is high inflation or evidence of rapid 

changes in consumption patterns.10 The 1998 BLS decision to move from a roughly 10-

year revision cycle to a two-year cycle reflected caution given the lack of evidence on the 

potential impacts of employing an even shorter expenditure base period or even more 

frequent revisions.11 At that time there was also little empirical evidence to demonstrate 

an impact of the frequency of weight revisions on the rate of growth of the CPI.12 These 

are the issues we examine in Section III and IV below. 

The importance of the price-updating of expenditure weights increases with the distance 

between the center of a Lowe index’s weight reference period and its price reference 

period. This distance is currently two years in the CPI-U; it was 3 1⁄2 years prior to 

processing enhancements that took effect in 2002. Even at two years, the impact of 

price-updating can be very noticeable for products with relatively strong price trends. 

Television prices, for example, have been decreasing steadily in the United States, in 

absolute as well as relative terms. During the 2001-2002 expenditure base period, 

televisions accounted for 0.21 percent of total expenditures. After price updating to 

December 2003 for the 2004 weight revision, however, their share fell to 0.16 percent. 

This pattern has been consistent over time: at the last CPI-U weight revision in January 

2008, the new CPI relative importance for televisions was 0.17 percent, although their 

share in the 2005-2006 weight reference period was 0.28 percent. 

 

8

 See, for example, ILO (2004), paragraphs 15.43-15.45, and Balk and Diewert (2003). 

9

 ILO (2004) paragraph 1.197. 

10 See International Conference of Labor Statisticians (2003), paragraph 26. 11 Bureau of Labor Statistics (1999). 

12 See Greenlees (1998). 
- 6 - 

Such effects might argue for an alternative, such as the Young index, that does not 

involve price-updating of weights. The Young formula can be expressed in share form as 

 

k

ktkbk

Y

bt IX pps ,0, 0

The Young index thus seems to combine two contradictory assumptions. The shares 

applied to the price reference period are the same as those in the weight reference period, 

consistent with an assumption of a unitary consumption elasticity and a geometric mean 

index formula. Like the Laspeyres and Lowe indexes, on the other hand, the Young 

index can be written as an arithmetic mean of long-term price relatives, which is 

consistent with a zero consumption elasticity assumption. Although the Young formula 

also has been criticized for violating desirable axioms for price indexes, it has a potential 

advantage in that it may yield a closer approximation to a superlative index. It is used in 

some national CPIs, and two recent studies present empirical comparisons of Young 

indexes to Lowe, superlative, and other alternative indexes.13 It will be one of the 

indexes we consider in Section IV. The 2003 resolution of the International Conference 

of Labor Statisticians allows for the option of not price-updating, saying14

: 

“Where the weight reference period differs significantly from the price reference 

period, the weights should be price updated to take account of price changes 

between the weights reference period and price reference period. Where it is 

likely that price updated weights are less representative of the consumption 

pattern in the price reference period this procedure may be omitted.” 

Hansen (2006, p. 8) sets forth this position clearly. He argues that the preferred choice 

between a Lowe and Young index is the one that most closely approximates an ideal 

index, by employing weights that are the best estimates of the expenditure shares between 

the current and price-reference periods—that is, between periods 0 and t in the equations 

above: 

“Whether the original or the price-updated expenditure weights are the best 

estimates of the average expenditure shares depends on the households’ response 

to change in the relative prices. ... If the households are most likely to hold fixed 

expenditure shares, i.e. the price elasticity of demand is around one, the Young 

index is the best estimate. If the households hold fixed quantities, indicating zero 

price elasticity of demand, the Lowe index is the best estimate.” 

The final formulas that we will discuss here are based on the constant-elasticity-ofsubstitution or CES formula. In the consumer price index context, if preferences take the 

CES form the resulting cost-of-living index is often referred to as the Lloyd-Moulton 

index,15 and in share form it is given by: 

)1/(11

0, 0 0 ])/([    

k

ktkk

LM

t IX pps

 

13 See Hansen (2007) and Pike (2007), which use CPI data from Denmark and New Zealand, respectively. 14 ICLS (2003), paragraph 25. 

15 See ILO (2004), paragraphs 17.61-17.64. If =0 the Lloyd-Moulton index reduces to the Laspeyres 

form, and it takes the geometric mean (Cobb-Douglas) form if =1. 
- 7 - 

Because of its economy of parameters the CES or Lloyd-Moulton form has been used 

frequently in price index studies, such as Feenstra (1994), Shapiro and Wilcox (1997), 

Balk (1999), and Broda and Weinstein (2007). Cage et al. (2007) fitted Lloyd-Moulton 

indexes to the C-CPI-U and demonstrated that for different time periods and levels of 

aggregation the optimal fit was consistently obtained by using CES substitution 

parameters between 0 and 1, that is, between corresponding Lowe and geometric mean 

indexes. For this paper we take a different approach; we make use of the fact that the 

Sato-Vartia index is exact for the CES preference system.16 That is, under the (strong) 

assumption that preferences do take the CES form, and given the availability of both 

current and base-period expenditure shares, the cost-of-living index can be calculated 

without knowing the substitution parameter  by computing the Sato-Vartia formula: 

   

k

ktkkt

SV

t IX 0, 0, ppw 0 ln /ln

The above formula is similar to the Törnqvist except that the weights are the log-means 

of the reference and comparison period shares, defined as (stk - s0k)/(ln stk – ln s0k), 

normalized to sum to unity over all cells k. We also employ a result by Feenstra and 

Reinsdorf (2003) that shows how the substitution parameter can be conveniently 

estimated consistent with the Sato-Vartia index. 

In this paper we use movements in superlative indexes as the standard against which we 

will judge alternative operational index formulas. The theory and advantages of 

superlative indexes were developed by Diewert (1976) and are discussed at length in the 

international CPI manual.17 Sweden produces an approximation to a superlative CPI,18

and other countries have examined superlative CPI series computed retrospectively. 

Empirical analysis of simulated US superlative CPI series, and comparison of these to the 

CPI-U, goes back to Aizcorbe and Jackman (1993). Design of the C-CPI-U as a monthly 

chained, Törnqvist index is explained in Cage et al. (2003). Other BLS studies of the 

final C-CPI-U include Shoemaker (2005) and Zadrozny (2008), who examines 

alternatives to the geometric mean formula for the preliminary C-CPI-U index values. 

III. Analysis of Substitution Behavior in BLS Expenditure Data 

In this section we estimate annual and biennial price indexes using BLS data and several 

different price index formulas. The expenditure data used for our analyses are taken from 

the CE Survey, which as noted above provides all weights for the CPI-U and C-CPI-U. 

Our data are drawn from the CPI expenditure weight database and thus are computed and 

classified in the same way as for the official indexes. In each period we have expenditure 

totals for 211 item categories and 38 areas, for a total of 8,018 cells. 

It is worth noting that there are very wide cross-sectional variations in the cell weights. 

Summed over all areas, the largest of the 211 item categories, Owners’ Equivalent Rent 

of Primary Residence, had a relative importance in the CPI-U of about 24.4 percent in 

December 2008.19 Other examples with large weights are Rent of Primary Residence 

 

16 See, for example, Sato (1976). 

17 ILO (2004), for example in paragraphs 1.97-1.101. 

18 See Ribe (2005). 

19 The percentages here are updated from the weight reference period. 
- 8 - 

(6.0 percent), New Vehicles (4.5 percent), and Gasoline (3.0 percent). At the other end 

of the scale, there are numerous item categories with weights under 0.05 percent, such as 

Flour and Prepared Flour Mixes; Frozen Noncarbonated Juices and Drinks; Watches; and 

Musical Instruments and Accessories. The 38 area weights also differ greatly.20

We computed five different indexes between each of the adjacent years from 1999 

through 2007 and between each adjacent biennial CPI-U base period from 1999-2000 

through 2005-2006. Note that we present these indexes for analysis of demand shifts, not 

as operational alternatives. They are only feasible retrospectively, not in “real time,” 

because in each case there is no lag between the price reference period and the end of the 

weight reference period. Our results are shown in Table 1, along with the log-changes in 

the published annual-average values of the CPI-U and the C-CPI-U. 

These results reveal several relationships, most of which are consistent with expectations. 

First, the two annual superlative series, using the well-known Fisher Ideal and Törnqvist 

index formulas, are quite close together in every year but one, 2000, although the year-toyear changes in the Törnqvist are always slightly higher. The apparent divergence in the 

first year is entirely due to the greater sensitivity of the Fisher index to extreme data 

points. One item-area index fell by more than 99 percent between 1999 and 2000, while 

its associated annual expenditures increased slightly. If we recode that index change to 

zero, the log-changes for the Fisher and Törnqvist indexes in 2000 increase to a nearly 

identical 0.0287 and 0.0288, respectively. (Similarly, recoding of that extreme 

observation would reduce the Fisher-Törnqvist difference in the first line of the biennial 

panel in Table 1 from 0.0026 to 0.0006.) 

Second, the table shows that the Sato-Vartia changes based on the CES assumption are 

slightly higher than the Törnqvist changes except in one year, 2005, when the latter’s logchange exceeds the former’s by approximately 0.00001. This closeness of the SatoVartia to the superlatives provides support for our use of the CES form to obtain a 

summary consumer substitution statistic for BLS data. 

Not surprisingly, in each year the Fisher, Törnqvist, and Sato-Vartia index changes are 

well within the Paasche and Laspeyres bounds. (The Fisher changes are identically the 

average of the Paasche and Laspeyres changes.) In the last row of the table we show the 

results of aggregating the eight annual log-changes to construct annually-chained index 

series. Over the whole period the chain Laspeyres index rises about 2.9 percentage points 

more than the chain Paasche. Meanwhile, the eight-year growth of our chain Törnqvist 

index is about 0.1 percentage point less than that of the Sato-Vartia index and 0.5 

percentage point more (0.4 in 2000 alone) than that of the Fisher. On a per-year basis, 

the Laspeyres index rises about 0.18 percentage point per year more rapidly, and the 

Paasche about 0.18 percentage point more slowly, than the Fisher. 

Turning to a comparison of our estimated indexes with the official BLS series, we first 

note that the published CPI-U has risen much more rapidly—about 0.2 percentage point 

per year—than the chain Laspeyres index. The differences are especially wide in 2000 

and 2001, when the CPI-U was still employing relatively old expenditure weights from 

 

20 CPI-U relative importances for 2008 are on the BLS website at http://www.bls.gov/cpi/cpiri2008.pdf. 
- 9 - 

the 1993-1995 period.21 Even in the later years, however, the log-change in the CPI-U 

always exceeds that of our chain Laspeyres series, which is computed identically except 

for the expenditure base period. These results imply that using more recent expenditure 

weights would typically have a downward effect on the index. For example, in 2007 the 

0.0281 log-change in the CPI-U was based on 2003-2004 weights, whereas our 0.0267 

Laspeyres index change is based on 2006 weights, which were not available for use at the 

BLS until 2008. 

The final comparison in Table 1 is between the C-CPI-U published annual-average 

changes and those of our simulated superlative series. These differences are perhaps 

wider than one would expect, given that the Chain CPI uses a Törnqvist superlative 

formula. In two years, 2005 and 2006, the C-CPI-U log-change is lower even than our 

Paasche change. The differences from the published and simulated Törnqvist indexes 

likely result from the fact that the former is a monthly- rather than annually-chained 

series. Moreover, due to volatile fuel prices the years 2005 and 2006 were both 

characterized by extremely sharp upward movements in the CPI during the summer 

months, followed by equally dramatic decreases in autumn. Such intra-year volatility 

would presumably increase the impact of the C-CPI-U’s monthly chaining process. 

Nevertheless, over the seven years 2001-2007, the aggregate increase in the C-CPI-U is 

much closer to our annual Törnqvist than to either the Laspeyres or the Paasche.22

We next turn to examining the results of estimating the CES  parameter using the 

Feenstra/Reinsdorf regression approach. For example, we used price and share data from 

1999 and 2000, weighted in proportion to the log-mean of the shares, to estimate an 

equation of the form 

dlog s = -0.00780 + 0.27294 dlog p + e 

Feenstra and Reinsdorf show that the coefficient on dlog p is an estimate of 1-, while 

the constant term equals -1 times the Sato-Vartia log-change between 1999 and 2000 

(here, -.27294 x .02857 = -.00780). Table 2 presents the estimates of  along with the 

standard error of the 1- coefficient estimate for each annual and biennial comparison. 

The annual results show a remarkable similarity from year to year, except for 2007. The 

values of  vary only within a range of about 0.52 to 0.73 from 2000 to 2006. This range 

narrows even further if we recode the extreme price change in 2000 mentioned above. 

When that change is recoded to zero, the estimated  for that year falls from 0.727 to 

0.637, and the 2000-2006 range is only from 0.52 to 0.66. In 2007, however, the 

regression coefficient falls to only 0.065 and is insignificantly different from zero (i.e., 

 insignificantly different from unity). Using a pooled regression we constructed an F 

test that rejected the null hypothesis of equality of the  parameters across all years. 

 

21 The CPI-U and C-CPI-U log-changes in Table 1 are computed from published index values, which until 

January 2007 were rounded to one decimal place. Consequently, the comparisons of the log-changes to 

our simulated indexes, which are derived from full-precision calculations, will be affected by rounding; see 

Williams (2006). 

22 Because publication of the C-CPI-U began in January 2000, no annual-average index data are available 

for 1999. The C-CPI-U data for 2008 are not yet final. 
- 10 - 

We examined the 2007 results but found no evidence of a higher variance of price or 

share changes, of a predominant role of one item category, or of a greater frequency of 

influential observations, compared to other years. Pending further analysis, therefore, we 

tentatively conclude only that the average consumer substitution elasticity varies from 

year to year and that occasional outlier years will sometimes occur. 

The three biennial estimates of  are about as widely spaced as the annual estimates 

covering the corresponding time periods, but have smaller standard errors. (The first 

biennial estimate falls from 0.776 to 0.740 when the outlier is recoded.) Both the annual 

and biennial estimates strongly reject the possibility of a zero substitution elasticity. 

None of our 11 regressions yields an elasticity less than 0.5. Zero elasticity is, of course, 

an implicit assumption justifying the Lowe index’s formula both for updating expenditure 

weights and for calculating period-to-period price change. 

Our primary overall conclusion from the results of these index simulations and 

substitution parameter estimates is that consumers vary their purchase quantities 

significantly, albeit inelastically on average, in response to relative price changes across 

the basic indexes of the CPI. Although this has long been an argument used against 

Laspeyres or Lowe indexes like the CPI-U, it also argues for the consideration of other 

BLS methodological changes that would stop short of abandoning the fixed-basket nature 

of the CPI-U. In particular, the evidence of substitution behavior supports research on 

accelerated expenditure weight updates in the CPI-U. It also suggests some potential 

value in alternative methods of updating expenditure weights between the base period 

and link month. Finally, the evidence that substitution is non-zero but inelastic would 

provide support for analysis of whether an alternative to the geometric mean formula 

would improve the accuracy of the preliminary C-CPI-U values. 

IV. Comparisons of Update Simulations 

In this section we present some exploratory results of simulating alternative monthly 

fixed-weight indexes using CPI data for 2002-2007. We consider different weightupdating rules (Lowe and Young) and different re-weighting frequencies (one-year and 

two-year). Each simulated index is compared to the official Lowe CPI-U and to the final 

values of the chained Törnqvist C-CPI-U. 

We begin, however, with a very simple examination of the correlations among successive 

annual price changes at the elementary index level. Several authors, including Balk and 

Diewert (2003), Hansen (2006), and the authors of ILO (2004), have discussed the factors 

determining the relative rates of index growth of Lowe, Young, and superlative indexes. 

For example, the international CPI manual points out that a Lowe index will exceed a 

Laspeyres index whenever (1) there are persistent and divergent trends in the underlying 

individual component indexes, and (2) consumers tend to substitute away from 

components with increasing relative prices. In that case the weight updating process in 

the Lowe index will give a higher weight than the Laspeyres to the components with 

persistently increasing relative prices.23 A Young index will tend to be higher than the 

Laspeyres when there are persistent trends in prices and consumer substitution is elastic. 

When consumer responses are inelastic, the Young index is likely to fall below the 

 

23 For example, see ILO (2004), paragraph 15.45 and Appendix 15.2. 
- 11 - 

Laspeyres.24 Finally, we know that the Laspeyres will exceed a superlative index 

whenever consumers substitute away from goods and services with increasing relative 

prices after the price reference period. 

We have established in Section III that basic index changes are negatively correlated with 

quantity changes in our CPI data, and positively correlated with expenditure shares, 

implying inelastic demand. We also found that the elasticities are closer to unity than to 

zero in absolute value. A question remains, however, whether there are systematic and 

divergent trends in relative prices in our data. 

In order to answer that question at a simple but illustrative level, we first regressed each 

year’s annual basic index relatives on the prior year’s relatives. The regressions were 

weighted by the prior-year expenditures. For this exercise we confined our attention to 

index years beginning with 2002, because that was the first year in which the biennial 

revision process was applied to the CPI-U. Thus, for example, we regressed the 

December 2001-to-December 2002 12-month price relatives for the 8,018 CPI basic 

indexes against the corresponding 2000-2001 relatives, weighted by the 2001 expenditure 

shares. The results of this exercise are shown in the first column of Table 3. Of the six 

years shown, four had a positive and highly significant correlation between the index 

relatives in the current and prior year. One of the other two years had no significant 

correlation. The years 2001 and 2002, however, exhibit a significantly negative 

correlation, with the 2002 relative decreasing by almost 25 percent of any increase in the 

2001 relative. 

The second column of Table 3 presents somewhat more specific evidence on the role of 

weight updating. It measures the correlations between the basic index changes during 

three CPI weight-update periods and the corresponding changes during the years during 

which those weights are used in the CPI-U. For example, the last lines of the table show 

that the index relatives between the 2003-2004 average (expenditure reference period) 

index levels and the December 2005 (price reference period) index levels were strongly 

and positively correlated with the December-to-December index relatives during 2007 

but insignificantly correlated with the 2006 December-to-December relatives. That 

would imply that the CPI-U’s Lowe index price updating, which increases the relative 

weights of the items that grew in relative price during the updating period, would have 

little effect on overall index growth in 2006 but a comparatively large upward effect in 

2007, as compared to a Laspeyres or Young index. 

For the other years examined, we find large positive correlations for index year 2003 and 

2005, with a smaller positive correlation for 2004 and a significant but relatively small 

negative correlation in 2002. Thus, on balance we would predict that a Young index 

would grow more slowly than the CPI-U in our data, except perhaps in 2002 and 2006. 

Algebraically deriving comparisons among Lowe indexes with different lag lengths 

between the expenditure and price reference periods, among Lowe indexes with different 

expenditure reference period lengths, and between those indexes and their corresponding 

Young indexes, would be more tedious than instructive, we believe. Instead, we now 

turn to simulations of feasible fixed-weight indexes. 

 

24 ILO (2004), paragraphs 15.56-15.57. 
- 12 - 

Five indexes are compared in Table 4. The first is a biennially chained Lowe index with 

two-year weight reference periods—that is, the CPI-U.25 The second index also uses 2-

year weight reference periods, but the index is revised each year by introducing an 

overlapping two-year set of weights. For example, the “rolling” index introduces 2002-

2003 weights in January 2005, whereas the CPI-U continued to use 2001-2002 weights. 

In January 2006, both series introduce 2003-2004 weights. The third Lowe index is 

annually chained with one-year weight reference periods—e.g., 2004 weights are 

introduced in January 2006. 

Table 4 also presents the C-CPI-U along with a Young index for comparison to the Lowe 

indexes. The Young index uses the same expenditure base periods as the CPI-U, and 

thus differs from the first index in the table only by not updating weights to the price 

reference month. Note that the use of more timely base periods would be expected to 

lead to higher index changes, the opposite of the case for Lowe indexes. This occurs 

because under inelastic consumer demand behavior the inter-temporal growth in 

expenditure shares for goods with rising prices will have an effect similar, although 

weaker, than the Lowe weight-updating process. 

The relationships in the table can be summed up as follows: 

 Relative to each other, the Lowe indexes display slightly lower index movements 

as their base periods become more timely. By construction, the first two indexes 

change by identical amounts in even-numbered years, but in the odd-numbered 

years the CPI-U change is always smaller than the change in the index that uses 

overlapping two-year base periods. The Lowe index revised annually with oneyear base periods increases by less than the rolling two-year index in four years 

out of six. 

 Examination of monthly and annual index changes in the data underlying Table 4 

gives no indication that the use of overlapping or one-year weight reference 

periods leads to more volatile indexes. This conclusion is tempered, however, by 

the fact that we have only examined all-items indexes. Lower-level subaggregates could be more sensitive. 

 The Young index increases by less than any of the three Lowe indexes in four of 

the six years. The exceptions are 2002 and 2006, which is not surprising given 

the results about price correlations that we presented earlier in this section. The 

differences between the changes in the biennial Lowe index in the first column 

and the changes in the biennial Young are always greater in the odd-numbered 

years, when the weight reference periods are less timely. 

 None of the differences between the fixed-base indexes are nearly as large as the 

gap between them and the C-CPI-U. Overall, after seven years, the difference 

between the CPI-U and Young index levels is 0.64 percentage point, while the 

Young index exceeds the C-CPI-U index level by 1.46 percentage point. Figure 1 

graphically demonstrates the clustering of the fixed-base indexes and the distance 

between them and the superlative. 

 

25 Index levels and percent changes will differ from the published CPI-U and C-CPI-U due to the different 

index reference periods and different rounding procedures. 
- 13 - 

We can contrast the last conclusion above with the results of simulations in which we 

relax our one-year lower limit on the processing lag between the end of the weight 

reference period and the link month or price reference period. Figure 2 is similar to 

Figure 1 except that it includes two additional Lowe indexes: one is an index revised 

semi-annually by introducing weights drawn from a six-month weight reference period 

ending six months prior to the price reference month. The other is a corresponding series 

revised each quarter, using a three-month weight reference period ending three months 

prior to the price reference month. These two new series approach the C-CPI-U much 

more closely than the series in Figure 1. Perhaps surprisingly, they also are not 

noticeably more volatile than the other series despite their short weight reference periods 

and Lowe index formula. 

Neither of the added series in Figure 2 is operationally feasible at the BLS at present. 

The purpose of the figure is only to suggest that reducing the processing lag could be as 

or more effective than increasing the frequency of weight revisions if one is attempting to 

approximate a superlative index with a fixed-base index. 

Finally, in Figure 3 we display the CPI-U, C-CPI-U, and biennial Young indexes from 

Figure 1 and Table 4 along with two other series. The first addition is a Lowe index with 

weights updated on a hypothetical five-year cycle: in January 2002 (with 1999-2000 

weights) and in January 2007 (with 2004-2005 weights). It is identical to the CPI-U until 

January 2004, when the latter underwent a weight revision, but subsequently the 

unrevised Lowe exhibits a higher growth rate, ending at a level of 119.50 compared to 

118.88 for the CPI-U. This provides additional evidence that less timely weights yield 

increased growth rates in Lowe indexes. 

The last series in Figure 3 is a so-called “Geometric Young” index, which like the usual 

Young index employs no price-updating of weights but which also uses a geometric 

mean formula for index calculation. It lies below the other indexes throughout the 

simulation period, ending at a level of 115.84, compared to 116.78 for the C-CPI-U. This 

is to be expected given the fact that the Geometric Young implicitly assumes a higher 

degree of consumer substitution than our superlative indexes indicate. 

V. Conclusions and Further Issues 

The analyses in this paper have confirmed, once again, that the consumer expenditure 

data underlying the US CPI imply consumer substitution away from goods and services 

with rising relative prices. This provides further evidence that Lowe index formulas like 

the CPI-U yield higher inflation estimates than would a true cost of living index. Using 

the Feenstra-Reinsdorf approach and data for 1999-2007, we can also conclude that the 

average CES elasticity is between zero and unity, but consistently closer to the latter. 

In addition, we provide evidence on the impact of using more timely weight reference 

periods. Notably, using simulated monthly indexes for 2002-2007, we find that a Lowe 

index with “rolling” two-year weight reference periods but annual revision rises by an 

average of 0.03 percentage point less per year than the CPI-U. Another 0.01 percentage 

point, on average, is subtracted by imposing annual revision with one-year expenditure 

base periods. On average, using a Young index formula with the same base periods as 

the CPI-U subtracts another 0.04 percentage point annually; however, the ordering of the 
- 14 - 

movements of the Young and Lowe index varies from year to year, due in part to 

variation in the inter-temporal correlation of basic index changes. 

None of these alternative fixed-weight index formulas comes close to eliminating the gap 

between the CPI-U and the Törnqvist C-CPI-U. This does not mean that those indexes 

are not worthy of further study. Thus far, we have seen no evidence that the advantages 

of using more timely weight data are offset by any increase in index volatility or other 

type of instability. Thus, the indexes simulated here may well offer better representation 

of current price change (because of the more timely weighting), as well as closer 

approximation to a cost-of-living index, while remaining operationally feasible (albeit 

with a potential need for some additional resources because of more frequent revisions). 

Increased flexibility in weight revisions, such as the use of quarterly expenditure data, 

may be possible at the BLS in the near future due to improvements in CE survey 

processing. Further examination of the weighting issue therefore appears to be a 

potentially fruitful avenue of research. 
- 15 - 

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- 17 - 

Base Current Paasche Fisher Tornqvist Sato-Vartia Laspeyres C-CPI-U CPI-U

1999-2000 2001-2002 0.0349 0.0407 0.0433 0.0435 0.0465 ... 0.0523

2001-2002 2003-2004 0.0357 0.0383 0.0384 0.0386 0.0408 0.0392 0.0436

2003-2004 2005-2006 0.0534 0.0565 0.0572 0.0571 0.0596 0.0552 0.0624

0.1240 0.1355 0.1388 0.1391 0.1469 ... 0.1583

1999 2000 0.0188 0.0245 0.0285 0.0286 0.0302 ... 0.0331

2000 2001 0.0216 0.0229 0.0231 0.0232 0.0241 0.0223 0.0281

2001 2002 0.0115 0.0126 0.0127 0.0128 0.0138 0.0124 0.0157

2002 2003 0.0190 0.0201 0.0201 0.0203 0.0212 0.0206 0.0225

2003 2004 0.0225 0.0238 0.0238 0.0240 0.0250 0.0247 0.0263

2004 2005 0.0288 0.0300 0.0302 0.0302 0.0313 0.0285 0.0333

2005 2006 0.0294 0.0304 0.0304 0.0305 0.0313 0.0286 0.0317

2006 2007 0.0239 0.0253 0.0253 0.0255 0.0266 0.0250 0.0281

0.1754 0.1895 0.1942 0.1950 0.2036 ... 0.2188

Period

Total 1999-2007

Total

Table 1. Estimated Index Log-changes

Index Formula

Base Current Estimate Standard Error

1999-2000 2001-2002 0.776 0.027

2001-2002 2003-2004 0.602 0.030

2003-2004 2005-2006 0.544 0.031

1999 2000 0.727 0.043

2000 2001 0.521 0.051

2001 2002 0.631 0.053

2002 2003 0.583 0.052

2003 2004 0.655 0.054

2004 2005 0.553 0.059

2005 2006 0.650 0.060

2006 2007 0.935 0.062

Table 2. Estimated Substitution Parameters

Period CES substitution parameter
- 18 - 

Biennial Revisions 

(CPI-U)

Rolling 2-Year 

Revisions Annual Revisions Biennial Young C-CPI-U

December Index Level

2001 100.00 100.00 100.00 100.00 100.00

2002 102.40 102.40 102.36 102.44 102.02

2003 104.31 104.25 104.22 104.24 103.75

2004 107.72 107.66 107.64 107.59 107.03

2005 111.37 111.21 111.12 111.07 110.11

2006 114.23 114.07 113.97 113.92 112.61

2007 118.88 118.65 118.52 118.24 116.78

Percent Change

2002 2.40 2.40 2.36 2.44 2.02

2003 1.87 1.81 1.82 1.76 1.70

2004 3.27 3.27 3.28 3.22 3.15

2005 3.39 3.29 3.24 3.23 2.88

2006 2.57 2.57 2.57 2.57 2.27

2007 4.07 4.02 3.99 3.79 3.70

Annual Average 2.50 2.47 2.46 2.42 2.24

Lowe Indexes

Table 4. Alternative Simulated Index Values

Index Year

On Prior Year Index 

Relative

On Biennial Update 

Period Relative

2002 -0.246 -0.035

2003 0.111 0.124

2004 0.054 0.057

2005 0.187 0.167

2006 -0.005 0.004

2007 0.163 0.271

Note: Coefficients in bold are not statistically significant.

Table 3. Correlations of Index Relatives

Regression Coefficient of Current Year Relative:
- 19 - 

Figure 1. Simulated Indexes With Fixed Lag Lengths

100102104106108110112114116118120

2001 12

2002 02

2002 04

2002 06

2002 08

2002 10

2002 12

2003 02

2003 04

2003 06

2003 08

2003 10

2003 12

2004 02

2004 04

2004 06

2004 08

2004 10

2004 12

2005 02

2005 04

2005 06

2005 08

2005 10

2005 12

2006 02

2006 04

2006 06

2006 08

2006 10

2006 12

Year and Month

Index Level, Dec. 2001=100

CPI-U Annual Lowe Rolling 2-Year Lowe Biennial Young C-C
- 20 - 
- 21 - 

100102104106108110112114116118120

Index Level, Dec. 2001=100

Year and Month

Figure 3. Selected Alternative Simulated Indexes

CPI-U Biennial Young C-CPI-U Geometric Young 5-Yea