INTRODUCING THE CHAINED CONSUMER PRICE INDEX 

Robert Cage, John Greenlees, and Patrick Jackman 

U.S. Bureau of Labor Statistics 

To be presented at the Seventh Meeting of the 

International Working Group on Price Indices 

Paris, France, May 2003 

The authors would like to thank Ralph Bradley, David Scott Johnson, Joshua 

Klick, and Cassandra Wirth for helpful comments and suggestions on this paper. 

They also thank the organizers of and participants in seminars on the Chained CPI at 

the Federal Reserve Banks of New York, Philadelphia, Dallas, Atlanta, San Francisco, 

Chicago, St. Louis, and Richmond. Particular thanks are also due to Erwin Diewert 

for his insights on superlative index design problems, without assigning him any 

responsibility for the choices ultimately made for the C-CPI-U. Finally, credit for 

many of the analyses reported in the paper goes to the Superlative Design Team of 

the Bureau of Labor Statistics: Ralph Bradley, Robert Cage, Alan Dorfman, Dennis 

Fixler, Richard Kerr, Janice Lent, Robert McClelland, Gary McMullin, and Janet 

Williams. 
- ii - 

Abstract 

Introducing the Chained Consumer Price Index 

Robert Cage, John Greenlees, and Patrick Jackman 

In August 2002, the U.S. Bureau of Labor Statistics began publishing a 

consumer price index (CPI) called the Chained Consumer Price Index for All Urban 

Consumers. Designated the C-CPI-U, the index employs a superlative Tornqvist 

formula and utilizes expenditure data in adjacent time periods in order to reflect the 

effect of any substitution that consumers make across item categories in response to 

changes in relative prices. The new measure is designed to be a closer approximation 

to a "cost-of- living" index than the existing BLS measures. 

Expenditure data required for the calculation of the C-CPI-U are available only 

with a time lag. Thus, monthly values of the C-CPI-U are issued first in preliminary 

form using the latest available expenditure data and are subject to two subsequent 

revisions. Accordingly, at the time of its introduction in August 2002, "Final" values 

of the C-CPI-U were issued for the 12 months of 2000, "Interim" values were issued 

for the 12 months of 2001, and "Initial" values were issued for January-July of 2002. 

In February 2003, with release of the January 2003 index, revised Interim indexes 

for the 12 months of 2002 were published, and the index values for 2001 were 

revised and became Final. 

This paper details the calculation of the C-CPI-U and discusses the issues that 

were addressed in its design. The paper also describes the differences between the 

new index and the existing Laspeyres-formula CPI-U, and the February 2003 data 

revisions. 
- iii - 

TABLE OF CONTENTS PAGE

I. INTRODUCTION ......................................................................... 1

II. CONCEPTUAL FRAMEWORK ........................................................... 1

III. ESTIMATION METHODOLOGY ....................................................... 6

 A. Construction of the Consumer Price Index ................................ 8 

 B. Availability of requisite expenditure data .................................. 10 

 C. Estimation of upper-level price change ..................................... 11 

 Input elementary price indexes ............................................ 11 

 Input elementary expenditure weights .................................. 13 

 a. CPI-U ................................................................. 14 

 b. Final C-CPI-U ...................................................... 17 

 c. Initial and Interim CPI-U ....................................... 18 

 Aggregation formula ............................................................ 19 

 a. CPI-U ................................................................. 19 

 b. Final C-CPI-U ...................................................... 20 

 c. Initial and Interim CPI-U ....................................... 21 

IV. COMPARING AND CONTRASTING C-CPI-U VERSIONS ................... 28

 A. Difference in estimation ......................................................... 28 

V. COMPARING AND CONTRASTING THE C-CPI-U AND THE CPI-U..... 30

 A. Difference in estimation ......................................................... 30 

 B. Index simulations .................................................................. 30 

VI. INAUGURAL PUBLISHED INDEXES ............................................... 33 

VII. 2003 REVISIONS AND MOST CURRENT PUBLISHED INDEXES ...... 33 

 Comparisons with Personal Consumption Expenditure Indexes ......... 35 

VIII SUMMARY AND DIRECTIONS OF FURTHER RESEARCH .................. 35 

TABLES AND FIGURES ......................................................................... 39 
 

I. INTRODUCTION 

In August 2002, the U.S. Bureau of Labor Statistics (BLS) began publishing a 

new index of consumer price change called the Chained Consumer Price Index for All 

Urban Consumers. Designated the C-CPI-U, the index supplements the existing 

indexes already produced by the BLS: the CPI for All Urban Consumers (CPI-U) and 

the CPI for Urban Wage Earners and Clerical Workers (CPI-W). The BLS is producing 

the C-CPI-U in order to address a perceived weakness of the CPI-U: upper-level 

substitution bias. By utilizing a superlative price index aggregator across items, the 

C-CPI-U is designed to be a closer approximation to a “cost-of-living” index (COLI) 

than the CPI-U and the CPI-W. 

This paper provides comprehensive information on concepts, definitions, 

statistical procedures, and estimation methods used by BLS to compile the C-CPI-U. 

Because the primary motivation behind publishing the C-CPI-U is to provide data 

users an index that more closely approximates a COLI, the paper begins with a brief 

summary of cost-of-living and consumer substitution theory. A cursory review of 

prior research on the substitution bias inherent in the CPI-U is also provided. Next, a 

detailed explanation of how BLS computes the C-CPI-U is outlined in a step-by-step 

fashion. Estimation methodologies for the C-CPI-U and CPI-U are then compared and 

contrasted, and differences between the two indexes are evaluated. Finally, because 

price indexes can be used for many purposes and indexes well-suited for one 

purpose may be ill-suited for another, the limitations of the C-CPI-U are identified so 

that users may evaluate the suitability of the index for their needs. 

II. CONCEPTUAL FRAMEWORK 

The Consumer Price Index is a measure of the average price change of a fixed 

market basket of goods and services purchased by the average urban household in 
- 2 - 

the United States. The market basket consists of a sample of items – food, clothing, 

shelter, fuels, and other goods and services –that consumers buy for day-to-day 

living. Price change is measured by repricing essentially the same market basket of 

goods and services at regular intervals and comparing aggregate costs with the costs 

of the same market basket in an arbitrarily selected base period. 

BLS calculates a CPI for two population groups: (1) urban wage-earners and 

clerical workers and (2) all urban consumers. The Consumer Price Index for Urban 

Wage Earners and Clerical Workers (CPI-W) is a continuation of the historical index 

introduced by BLS in the early 1900’s for use in wage negotiations.1

 The Consumer 

Price Index for All Urban Consumers (CPI-U) was introduced in 1978 as a broader 

and more representative index of the urban, non-institutional population of the 

United States. Because the same basic methodology is used for calculating both the 

CPI-W and CPI—U, the discussion in this paper focuses on the relationship between 

the CPI-U and the new superlative index. 

Over the past 80 years, the methodology for producing both CPI’s has been 

refined by way of improvements in price data collection techniques, adjustments in 

estimation methods, and continuous surveys of consumer spending behavior. 

Comprehensive revisions to the CPI have occurred in 1940, 1953, 1964, 1978, 1987, 

and 1998.2

 Other improvements have been made over the years that reflect not 

only BLS’s own experience and research, but also the criticisms and 

recommendations of outsiders.3

 The goal of each improvement has been to 

effectuate in the index a more accurate representation of contemporaneous buying 

habits and consumption costs. A unifying framework for dealing with practical 

 

1

 See U.S. Department of Labor, BLS Handbook of Methods, Bulletin 2490, 1997, Chapter 17: The Consumer Price 

Index, p.167. 

2

 For an overview of historical CPI revisions, see John S. Greenlees and Charles C. Mason, “Overview of the 1998 

revision of the Consumer Price Index.” Monthly Labor Review, December 1996, pp. 3-9. 3

 For an early example, see, Report of The President’s Committee on the Cost of Living (Washington, Office of Economic 

Stabilization, 1945). 
- 3 - 

questions that arise in the construction of the CPI, in assessment of the CPI’s quality, 

and in guiding improvements made to the index, has been the economic approach to 

index numbers and, specifically, the concept of the cost-of-living index (COLI). 

While the use of index numbers to measure price change dates back to 1707, 

the economic theory of index numbers is of much more recent vintage.4

 The theory 

underlying the COLI was developed by A. A. Konus in 1924.5

 Under the assumption 

of utility maximizing behavior, a COLI is defined as the ratio of the minimum 

expenditure required to attain a particular level of satisfaction in two price situations, 

a comparison period and a base period. The CPI-U and the CPI-W are modified 

Laspeyres indexes that hold the standard of living constant in the span between 

major revisions by keeping quantities fixed at the level consumed in the base period, 

but allowing prices to vary. The restriction imposed on these CPI’s - holding the 

quantities fixed and not allowing substitution among goods in response to relative 

price change - results in a divergence between the CPI (or any other index with fixed 

quantity weights) and the COLI. In the case of a Laspeyres index, the effect is such 

that it is greater than or equal to the true cost of living. Indeed, it is well known that 

a Laspeyres index is an upper bound to the true COLI.6

 In analyses of the CPI by 

outside reviewers, it often has been argued that the BLS should establish a cost-ofliving index as the objective in measuring consumer prices.7

 The BLS has long said 

that it operates within a COLI framework in producing the CPI.8

 There are, of 

 

4

 The first quantitative study of price levels was apparently made by Bishop Fleetwood in Chronicon-Preciosum in 1707. 

For a historical survey of price measurement, see W. Erwin Diewert, “The Early History of Price Index Research,” W. 

Erwin Diewert and A.O. Nakamura (eds.), Essays in Index Number Theory, Volume I, 1993. 5

 “The Problem of the True Index of the Cost of Living,” English translation of the original in Russian published in 1939 in 

Econometrica 7, pp. 10-29. 6

 In general, a Laspeyres index is only an upper bound to a COLI when comparing the “comparison” period with the “base” 

period, not with an intermediate period. 

7

 See, for example, The Price Statistics of the Federal Government, prepared by the Price Statistics Review Committee of 

the National Bureau of Economic Research, 1961, the Final Report of the Advisory Commission to Study the Consumer 

Price Index, submitted to the Committee on Finance, U.S. Senate, 1996, and Charles Schultze and Chris Mackie (eds.), 

At What Price? Conceptualizing and Measuring Cost-of-Living and Price Indexes, 2002. 8

 See, Robert F. Gillingham, “A Conceptual Framework for the Consumer Price Index,” Proceedings, Business and 

Economics Statistics Section, American Statistical Association, 1974, pp. 246-52, and John S. Greenlees, “The U.S. CPI 
- 4 - 

course, a number of differences between the CPI and a complete COLI other than the 

ability to reflect how consumers adjust their consumption to changes in relative 

prices. For example, the CPI ignores the fact that household preferences extend to 

choices between labor and leisure and among different types of leisure. It also 

ignores time by assuming that all consumption takes place in a single period. It 

generally ignores the impact of both the environment and government goods on 

household welfare. Reflection of substitution in response to changes in relative 

prices is, therefore, only part of what is required for a complete COLI. It is, 

however, that aspect of a COLI that has been the focus of the technical criticism of 

the CPI. The recent study by the Committee on National Statistics stated that: 

“Within the general conceptual framework of cost-of-living indexes, the appropriate 

theoretical concept for the CPI is a conditional cost-of-living index that is restricted 

to private goods and services and in which environmental background factors are 

held constant.”9

 

 The theory underlying the C-CPI-U was based largely on the work of W. Erwin 

Diewert, who demonstrated that a family of indexes, termed superlative, could be 

calculated that provided a close approximation to a COLI using only the observable 

price and quantity data.10 That is, it would not be necessary to econometrically 

estimate the elasticities of substitution of all of the items with each other. The most 

widely known index number formulas that belong to the superlative class identified 

by Diewert are the Fisher Ideal index and the Tornqvist index. The Fisher Ideal 

Index is a geometric average of a base-period-weighted Laspeyres index and a 

current-period-weighted Paasche index. The Tornqvist index utilizes expenditure 

 and the Cost-of-Living Objective,” presented at the Joint ECE/ILO Meeting on Consumer Price Indices in Geneva, 

Switzerland, November 2, 2001. 

9

 Schultze and Mackie, At What Price?, p. 73. 10 See, W. Erwin Diewert, “Exact and Superlative Index Numbers,” Journal of Econometrics 4, pp. 114-45, 1976. By a 

close approximation, Diewert states that the functional form for the price index is “superlative” if it is exact for a function

that can provide a second order approximation to an arbitrary twice differentiable linearly homogenous function. 
- 5 - 

data in both the current and base time periods in order to reflect the effect of any 

substitution that consumers may make across item categories in response to 

changes in relative prices. Hence, both indexes arrive at an estimate of average 

price change over two time periods by utilizing in some fashion the expenditure 

experience in both periods as weights. 

Although funds were initially appropriated in 1998, the development of the CCPI-U has its roots back at least as far as 1961, with the recommendation from the 

Price Statistics Review Committee, known as the Stigler Committee, that a constant 

utility index is the appropriate index for the main purposes of the CPI.11 At the same 

time, the Stigler Committee recognized the need to fund research organizations 

within the price collection agencies to deal with price and index number issues 

outside of a production framework. Following the formation of BLS’s Division of Price 

and Index Number Research in 1966, research on developing a COLI was instituted. 

Much of the theoretical work on the COLI was done by Robert Pollak, first working at 

BLS on sabbatical from the University of Pennsylvania and later as a consultant.12 

On an empirical side, several studies have examined the extent of the substitution 

effect between a Laspeyres measure and the COLI. On an aggregate level, these 

include studies by Steven D. Braithwait,13 Marilyn E. Manser and Richard J. 

McDonald,14 and Ana A. Aizcorbe and Patrick C. Jackman.15 Braithwait utilized 

annual price and quantity data on Personal Consumption Expenditures from the 

National Income and Product Accounts for 53 commodities and found that for the 

fifteen year period 1958-73, the Laspeyres index overstated the COLI by 1.5 

percent, or about 0.1 percent a year. The work by Manser and McDonald covered 

 

11 The Price Statistics of the Federal Government, p. 52. 12 See Robert A. Pollak, The Theory of the Cost-of-Living Index. New York: Oxford University Press, 1989. 13 Steven D. Braithwait, “Substitution Bias of the Laspeyres Price Index: An Analysis Using Estimated Cost-of Living 

Indexes,” American Economic Review, March 1980, pp.64-77. 14 Marilyn E. Manser and Richard J. McDonald, “An Analysis of Substitution Bias in Measuring Inflation, 1959-85,” 

Econometrica, July 1988, pp.909-30. 
- 6 - 

the period from 1959-85 and utilized data on Personal Consumption Expenditures for 

101 commodities. On an annual basis this resulted in an estimate of the substitution 

effect of .19 percent per year. In addition to the numerical estimate of substitution 

effect, the Manser-McDonald study established the upper and lower bounds for the 

amount of substitution effect in the Laspeyres index - between .14 and .22 percent 

per year in the 1959-1985 period. The Aizcorbe-Jackman study examined the 

substitution effect issue using detailed expenditure data from the Consumer 

Expenditure (CE) Surveys, the official source used for the CPI. Consumer 

Expenditure data include a finer level of disaggregation in the commodity classes as 

well as geographic detail than do data from the Bureau of Economic Analysis. At that 

time, there were 207 item categories and 44 areas, which allowed the different 

formulas to be applied at a lower level of aggregation than the earlier studies. 

Aizcorbe-Jackman’s estimate of the substitution effect for the period from 1982 

through 1991 was .20 or .27 percent per annum, depending on whether a fixed base 

or chained index was used. 

III. ESTIMATION METHODOLOGY 

Specification and development of an official superlative index presented many 

challenges for the BLS. Foremost among these were the issues surrounding the 

sampling errors in CPI price and expenditure data. In most theoretical discussions of 

superlative indexes, price indexes and expenditure shares are taken as known with 

certainty, as if the data reflected a single representative consumer who allocates his 

or her spending in response to the observed price indexes. The CPI pricing surveys 

and the Consumer Expenditure Survey, however, are based on finite and distinct 

 15 Ana A. Aizcorbe and Patrick C. Jackman, “The Commodity Substitution Effect in CPI Data, 1982-91,” Monthly Labor 

Review, December 1993, pps 25-33.
- 7 - 

samples of outlets and consumers, respectively. Sample sizes are especially small at 

the level of basic item-area indexes and weights, such as were used in the AizcorbeJackman work discussed above. The existence of sampling variation in the 

underlying CPI data, and the independence of the sampling errors in price indexes 

and expenditure shares, can affect the expectation of a superlative index computed 

from those data.16

The impact of sampling variance could have been mitigated by constructing 

the C-CPI-U using only national-level data, but this would have tended to 

underestimate the true substitution effect.17 Alternatively, the BLS could have 

chosen to produce only an annual C-CPI-U, using price and expenditure data 

averaged over the year. 18 This option was also rejected, on the basis that the 

usefulness of the index would have been sharply reduced if it were not made 

available on a monthly basis. 

It was also determined that, rather than publishing the C-CPI-U only with a 

lag, the BLS would publish the index in preliminary form and revise it when more 

timely expenditure data were received and processed. As with the decision to 

produce a monthly C-CPI-U, this decision was motivated in part by an expectation 

that some users would desire a monthly, timely superlative index and would 

estimate such indexes themselves if the BLS did not publish the C-CPI-U in that 

form. 

The BLS was thus faced with the problem of choosing how to employ volatile 

price and expenditure data from small samples to compute a monthly superlative 

series. After much discussion, it was further decided to chain the monthly index 

 

16 For a discussion of these issues see John S. Greenlees, "Random Errors and Superlative Indexes," Bureau of Labor 

Statistics Working Paper 343, March 2001, or A.H. Dorfman, S.G. Leaver, and J. Lent, “Some Observations on Price 

Index Estimators,” Proceedings of the Federal Committee on Statistical Methodology Research Conference, Monday B 

Sessions, pp. 56-65, 1999. 

17 See Greenlees, op. cit. 
- 8 - 

values, despite the volatile and potentially seasonal nature of the underlying data, 

rather than attempting to impose chaining on an annual or other frequency. Many of 

the specific design features described below reflect difficult choices made among 

several imperfect alternatives, without the benefit of guidance from index number 

theory. Fortunately, analyses of historical data indicated that simulated C-CPI-U 

series were surprisingly robust to their specific design features, particularly at the 

aggregate level. 

A second significant issue was determining how to compute the preliminary CCPI-U indexes, which could not use a true superlative formula. The objective was to 

develop a specification that would yield accurate forecasts of the Final C-CPI-U. 

Again, a variety of alternative formulas were evaluated, but simulation studies 

showed that the differences in the resulting preliminary index values were relatively 

small. 

A. Construction of the Consumer Price Index 

The CPI is built in two stages. In the first stage, price changes for roughly 

80,000 specific items per month are averaged to yield 8,018 estimates of aggregate 

price change. This stage is often referred to as “lower-level aggregation” or 

“elementary-level aggregation” as it involves averaging the most fundamental 

component of the index - observed price change for specifically defined consumer 

goods, services, and products.19 For example, the prices of approximately ten 

different brands and styles of watches at various locations in Chicago are observed 

each month, compared to the prices observed in the previous month, and averaged 

together to produce an index of price change for watches in Chicago. Watches 

(ITEM=AG01) is one of 211 elementary items, and Chicago (AREA=A207) is one of 

 18 Erwin Diewert, “Notes on Producing an Annual Superlative Index Using Monthly Price Data,” University of British 

Columbia Economics Department Discussion Paper No. 00-08 (July 2000) discusses issues in constructing an annual 

superlative index. 
- 9 - 

38 elementary areas in the current CPI market basket structure. The Chicago-watch 

index is one of the 8,018 (211 items x 38 areas) elementary indexes produced in the 

first stage of CPI construction. 

In the second stage, the elementary indexes are averaged together to yield 

various aggregate indexes and ultimately the All-Items, U.S. City Average index of 

price change. See Figure 3.1. 

Within the two-tiered scheme of calculating the CPI, consumer substitution 

can and does occur at both levels. Ideally, a superlative formula would be employed 

at both stages of CPI index construction to account for consumer substitution that 

might occur intra-item, that is among specific products within an elementary item 

(e.g., a leather band watch versus a stainless steel band watch or whole wheat bread 

versus white bread) and inter-item, that is across elementary items (e.g., theater 

admission versus video rental or beer versus wine). However, the BLS is currently 

precluded from using a superlative formula at the first stage because reliable 

monthly expenditure data for each of the 80,000 lower-level quotes are not readily 

ascertainable.20 As an alternative to a superlative index as a means of addressing 

intra-item substitution, the BLS began using a hybrid combination of Geometric Mean 

indexes and Laspeyres indexes for lower-level aggregation in 1999.21 Zero elasticity 

of substitution within item categories is assumed for the small number of item 

categories in which the Laspeyres formula is used.22 Unitary elasticity of substitution 

intra-item is assumed for items using the Geometric Mean formula. 

 19 In BLS terminology, the specific goods and services at the lower-level are called price quotes. 20 Collecting monthly information from CPI outlets on the sales of individual items would be extremely costly and would 

impose an unacceptable respondent burden. 

21 For more information on the use of the Geometric Mean index in lower-level CPI aggregation CPI, see Kenneth V. 

Dalton, John S. Greenlees, and Kenneth J. Stewart: “Incorporating a geometric mean formula into the CPI,” Monthly 

Labor Review, October, 3-7, 1998. 22 Laspeyres items are Local telephone service, Rent, Housing at school, Owners’ equivalent rent, Electricity, Natural gas, 

Water and sewerage service, Physicians’ services, Dental services, Eyeglasses and eye care, Other medical professional 

service, Hospital services, Nursing homes and adult daycare, Cable television, and State and local vehicle registration, 

license, and motor vehicle property tax. 
- 10 - 

The use of a superlative formula for upper-level aggregation is designed to 

address inter-item substitution. In order to use a superlative index formula at the 

upper-level, monthly expenditure estimates for each of the 8,018 elementary itemarea combinations are required. Expenditure weights for CPI upper-level 

aggregation are derived from the Consumer Expenditure Surveys.23 Effective with 

data collected in 1999, the CE sample size was increased by 50 percent, in part to 

accommodate production of the C-CPI-U. 

B. Availability of Requisite Expenditure Data 

Consumer Expenditure Survey data are processed annually and made 

available for CPI use with a substantial lag. For example, data for calendar year 

2001 were not available until the fourth quarter of 2002. Data for calendar year 2002 

will not be available until the fourth quarter of 2003. This lag in data availability 

prevents BLS from calculating and publishing a superlative index in real time, i.e., on 

the same contemporaneous publication schedule as the CPI-U. BLS could have 

opted to simply calculate and publish the C-CPI-U on a two-year lag schedule. For 

example, superlative indexes for all months of calendar year 2002 could have been 

released in early 2004. However, as noted above, demand for the superlative 

indexes concomitant with the release of the CPI-U was anticipated to be high. 

Therefore, BLS developed the following general estimation and publication schedule 

for the C-CPI-U index for a given month (t) in year (y): 

Following Month (t+1): Calculate and publish an Initial estimate of the 

month (t) superlative index using lagged expenditure data 

February of Following Year (y+1): Revise and publish each Initial monthly 

superlative index from year (y) with an Interim estimate based upon 

updated, but still lagged expenditure data 

 

23 The Consumer Expenditure Surveys have been conducted continuously since 1980 by the U.S. Bureau of the Census 

under contract with the BLS. 
- 11 - 

February of Following Year (y+2): Calculate and publish a Final

superlative index for each month of year (y) using the now-available 

monthly expenditure data from year (y) 

Hence, the C-CPI-U index for any given month will be published 

simultaneously with the CPI-U as an initial estimate. This initial estimate will be 

revised and republished the following February, in the news release containing CPI 

data for January. Then, a final estimate will be published in the February of the 

following year. Accordingly, there will be three versions of each monthly C-CPI-U 

index: an Initial, Interim, and Final version. At any point in time, the historical CCPI-U index series will be comprised of (a) Initial values for all months in the current 

year, (b) Interim values for all months in the previous year, and (c) Final values for 

all months prior to the previous year. See Figure 3.2 for an illustration of the 

publication and revision schedule. 

C. Estimation of Upper-Level Price Change 

Aggregation of elementary CPI data into published indexes requires three 

ingredients: input elementary indexes, input elementary expenditures to use as 

aggregation weights, and a price index number formula that employs the 

expenditures to aggregate the sample of elementary indexes into a published index. 

Input Elementary Price Indexes. All three versions of the C-CPI-U index 

utilize the exact same hybrid Laspeyres and Geometric Mean lower-level indexes as 

are currently used in the construction of the CPI-U. The estimation of price change 

in the CPI at the lower-level is defined by Equation 3.1.a and Equation 3.1.b below.24

There is one exception to this general rule, having to do with interpolation of 

index values. The Final version of the C-CPI-U uses a slightly different method of 

imputing off-cycle elementary index values for elementary item-area combinations 

 

24 Equations 2.1.a and 2.1.b provide the general estimator formulas for most commodity and services items. Rent and 

owners’ equivalent rent are estimated using a slightly different approach. See Frank Ptacek and Robert M. Baskin, 

“Revision of the CPI housing sample and estimators,” Monthly Labor Review, December 1996. 
- 12 - 

priced on a bimonthly schedule.25 In the CPI-U, off-cycle bimonthly elementary 

indexes are set equal to the previous-month index value.26 

Estimation of Lower-Level Price Change in the CPI

 

Equation 3.1.a. Laspeyres items: 

Equation 3.1.b. Geometric Mean items: 

where,

 a = CPI elementary area 

 i = CPI elementary item 

 k = unique good or service (CPI price quote) 

 t = year and month 

 0 = base-period reference month 

kpt = price in year (y) in month (t) for quote (k) 

kp0 = price in base-period reference month (0) for quote (k) 

ks0 = expenditure weight for quote (k) in base-period reference month (0), divided by 

expenditure weight for all (k) quotes in elementary item (i), area (a) 

 i,a IXL

[0;t] = Laspeyres index of price change for elementary item (i) in area (a) from baseperiod reference month (0) to month (t) in year (y) 

 i,a IXG

[0;t] = Geometric Mean index of price change for elementary item (i) in area (a) from 

base-period reference month (0) to month (t) in year (y) 

Publishing the Final C-CPI-U with a substantial lag presents the opportunity to 

adopt an alternative approach for imputing off-cycle indexes and to eliminate any 

potential bias associated with using previous-month imputation. Specifically, offcycle indexes used in Final C-CPI-U aggregation are set equal to the geometric mean 

of the immediately previous and subsequent on-cycle month indexes. See Equation 

3.2.27 

 

25 BLS field economists do not price the samples of quotes in all 8,018 elementary item-area cells on a monthly basis. 

Approximately 40 percent of the elementary item-area combinations are priced on a bi-monthly schedule, with half of 

these priced every even month and the other half priced every odd month. Months for which prices are not collected are 

called “off-cycle” months. 

26 For example, the index for “Sports vehicles” (ITEM=RC01) in “Seattle” (AREA=A423) in December 2001 was 116.4, 

relative to June 1985. This elementary index is off-cycle in odd months. Hence, prices are not collected in January, and 

the January 2002 index was set equal to the December 2002 index value of 116.4. The February 2002 index is computed 

using prices observed in February. The index value was 116.8 relative to June 1985. 

27 For example, the January 2002 index for “Sports vehicles” in Seattle was set equal to 116.6, the geometric average of 

the December 2001 index (116.4) and the February 2002 index (116.8). 

∈ 













 =

k i a k

k t

k

L

i a t

p

p IX s , 0

, [0; ] 0

∏∈ 













 =

k i a

k

k

G k t

i a t

s

p

p IX , 0

, [0; ]

0
- 13 - 

( )1/ 2

i,a [0;t] = i,a [0;(t)−1]

×i,a [0;(t)+1] IX IX IX

Equation 3.2. Imputation of Off-Cycle Indexes 

 in Final C-CPI-U estimation 

where,

 a = CPI elementary area 

 i = CPI elementary item 

 t = year and month 

 0 = base-period reference month 

i,a IX[0;t] = index of price change for elementary item (i) in area (a) from base-period (0) to 

month (t) 

Imputing off-cycle elementary indexes with geometric averaging of bounding 

month indexes, rather than using previous-month index values, is not expected to 

have a dramatic impact on published Final C-CPI-U indexes. To measure the impact, 

Final C-CPI-U indexes were estimated using both methods of off-cycle bimonthly 

elementary index imputation for the 1987 to 2000 time period.28 Use of geometric 

averaging of off-cycle elementary indexes produced, on average, a 0.001 percent 

increase in the All-Items, U.S. City Average Final C-CPI-U index per annum over this 

time frame. See Table 3.1. The differences ranged from –0.033 percent in 1992 to 

0.025 percent in 1996. The fact that geometric averaging yielded a larger index 

value for some years (in 1990, 1993-1999) and a smaller index value in other years 

(in 1988, 1989, 1991, 1992, and 2000) is evidence that previous-month imputation 

does not result in any systematic effect on inflation measurement. 

Input Elementary Expenditure Weights. In order to aggregate 

elementary indexes into published indexes, an aggregation weight for each 

elementary item-area combination is required. The function of the aggregation 

 

28 For simulation analysis, a laboratory of input elementary price indexes and elementary expenditures was created using 

official CPI lower-level indexes and Consumer Expenditure Survey data, from December 1986 to December 2000, in 

order to evaluate features of C-CPI-U index construction. The CPI item structure changed significantly in January 1998, 

with the number of item categories increasing from 207 to 211 and the number of areas decreasing from 46 to 38. See 

Walter Lane, “Changing the Item Structure of the Consumer Price Index,” Monthly Labor Review, vol. 119, no. 12, 

December 1996, pp. 18-25. In order to achieve a continuous time series of data from 1986 to 2000 for each of the 8,018 

elementary item-area combinations currently in the CPI sample, item-area index and expenditure levels were roughly 

approximated for all months in the 1986 to 1997 time span, using available indexes and expenditures based on the 1987 

CPI market basket structure. The monthly expenditure data were adjusted according to official methodologies adopted for 

Final C-CPI-U construction. However, the monthly expenditure data for years 1986 through 1999 in the simulation 

laboratory are based on underlying CE sample sizes significantly smaller than that achieved in 2000, when the CE sample 

was increased by 50 percent. This affects the variance of the simulated indexes discussed herein. The sampling error 
- 14 - 

weight is to assign each elementary index a relative importance or contribution in 

the resulting aggregate index. The aggregation weight corresponds to consumer 

tastes and preferences and resulting expenditure choices among the 211 items in the 

38 areas comprising the CPI sample, for a specified time period, by the population 

the index is designed to represent. This section compares the estimation and use of 

aggregation weights in the Laspeyres CPI-U, preliminary C-CPI-U series, and the 

Final C-CPI-U. 

a. CPI-U. In the CPI-U, aggregation weights are defined as: 

Equation 3.3 

100

, , , ,

, ,

α β

β

P Q AW i a p i a p

i a p

  × =

where i a p Pα



, , is the estimated price of item (i) faced by population (p) in area (a) in 

time period (α), and i a p Qβ



, , is the estimated quantity of item (i) purchased by 

population (p) in area (a) in time period (β). Time period (α) is the base period of 

the corresponding elementary item-area index—i.e., the period at which the index 

equals 100.29 

Time period (β) corresponds to the reference period of the expenditures used 

to derive the implicit quantity weights needed for Laspeyres aggregation. Currently, 

the CPI-U has an expenditure reference period of β=1999-2000. Historically, the CPI 

expenditure reference period has been updated approximately every ten years (see 

Table 3.2). In 1998, BLS announced that it would institute a biennial rotation 

schedule for updating the expenditure reference period. Effective with the January 

2004 index, the expenditure reference period will change from β=1999-2000 to 

β=2001-2002; effective with the January 2006 index it will be updated again to 

 associated with monthly expenditure data from 2000 and beyond is expected to be lower than that observed in the 

experimental lab. 

29 For example, the “Sports equipment” (ITEM=RC02) in Seattle (AREA=A423) index has a base period of =June 1985. 

CPI elementary indexes have varying base periods that are not updated on a regular basis. Most published indexes have 

an index base period of =1982-1984. 
- 15 - 

2003-2004; and so forth. Note that a change in the expenditure reference period 

results in a change in the implicit quantity (Q) assigned to each elementary index, 

but not the implicit price component (P) of the aggregation weight (AW). 

Aggregation weights for the CPI-U are derived from estimates of household 

expenditures collected in the Consumer Expenditure Survey data. Despite an 

increase in the CE sample size in 1999, expenditure estimates at the elementary 

item-area level would be unreliable due to sampling error without the use of 

statistical smoothing procedures. BLS uses two basic techniques to minimize the 

variance associated with each elementary item-area base-period expenditure 

estimate. First, data are pooled over an extended time period in order to build the 

expenditure estimates upon an adequate sample size. The current reference period 

(β) uses 24 months of data. Second, elementary item-area expenditures are 

averaged, or composite-estimated, with item-regional expenditures.30 This has the 

effect of lowering the variance of each elementary item-area expenditure at the 

expense of biasing it toward the expenditure patterns observed in the larger 

geographical area.31 

The CPI-U aggregation weight for item (i) in area (a) in reference period (β) is 

computed by first calculating an aggregate annual expenditure estimate, i a PQ βn ( ) , , 

for each year (βn) in reference period (β). This estimate is derived directly from 

Consumer Expenditure Survey data.32 Next, the share of total area expenditure is 

computed for each item in each area, for each year. Similarly, the share of total 

expenditure in each major-area (m) is computed for each item for each year. A 

composite-estimated share of total expenditures is computed for each item for each 

 

30 Elementary areas area grouped into region x city-size classifications for the purpose of composite-estimation. There are 

four regions and two city-size classifications for a total of eight region-city-size classifications. 

31 Aggregation weights for the CPI-U and CPI-W are each derived separately according to the steps outlined in the text. 32 For a detailed explanation of how aggregate expenditure estimates are computed from CE data, see BLS Handbook of 

Methods, Bulletin 2490, 1997, Chapter 17. 
- 16 - 

year by taking a weighted average of its area share and corresponding major-area 

share. The weight (δ) assigned to the major-area (m) and the weight (1−δ) assigned 

to the elementary area (a) is a function of the variance and covariance of each 

measure.33 The resulting average share ( n s i a β ˆ , ) is then multiplied by the sum of all 

expenditures in the elementary area in the corresponding year, to obtain a 

composite-estimated item expenditure in year (βn). This estimate is in turn multiplied 

by a “raking factor” which is equivalent to the ratio of unadjusted expenditures 

( i a PQ βn ( ) , ) summed to the expenditure-class, major-area level, to the compositeestimated expenditures ( i a PQ βn ) ~ ~ ( , ) summed to the expenditure-class, major-area 

level. The raking factor is designed to limit the degree to which composite-estimation 

can change relative expenditures among item-area cells. Next, the compositeestimated-and-raked expenditures for each year (βn) are averaged to obtain the final 

estimate of annual aggregate expenditures in reference period (β). 

The CPI-U aggregation weight for each item-area combination is then derived 

from the composite-estimated-and-raked expenditure estimate by first multiplying it 

by the index of price change from reference period (β) to pivot-month (v).34 The 

resulting product is a cost weight: an estimate of item-area expenditures in pivotmonth (v), based upon quantities purchased in reference period (β). Finally, the cost 

weight is divided by the corresponding pivot-month index to obtain the aggregation 

weight: an estimate of item-area expenditure based upon quantities purchased in 

reference period (β) and prices of time-period (α). 

 

33 For more information on composite-estimation, see Michael P. Cohn and John P. Sommers, “Evaluation of the Methods 

of Composite Estimation of Cost Weights for the CPI,” Proceedings of the Business and Economic Statistics Section, 

American Statistical Association, 1984. pp. 466-7. 

34 The pivot-month is the first month in which expenditures from reference period (β) are used in the CPI. 
- 17 - 

Estimation of Monthly Expenditures at the Elementary Level in the C-CPI-U 

Equation 3.4. Estimated monthly expenditures 

 





























= ×







∈ ∈

∈

∈

a

i a A

T

t T

i a t

T

t T

a i a t

i a i A

i a t i a t

PQ

PQ

PQ PQ

,

,

,

( , ) ( , )

, ,

( )

( )

) ( ) ˆ ˆ (

where, 

 a = CPI elementary area 

 i = CPI elementary item 

 A = all CPI elementary areas; “U.S. City Average” 

 P = price 

 Q = quantity 

 t = month 

 T = time period covering month (t) and 11 months prior to month (t) 

 

b. Final C-CPI-U. For the Final C-CPI-U, which uses the Tornqvist index for 

upper-level aggregation in a monthly-chained construct, monthly expenditure 

estimates for each elementary item-area combination are required as aggregation 

weights. Like the biennial data used for CPI-U aggregation, adequacy of the 

underlying CE sample size from which the expenditure weights are estimated is an 

issue for C-CPI-U aggregation. In order to minimize the variance of the elementary 

item-area-monthly expenditures, a ratio-allocation procedure is used to estimate 

each item-area-monthly expenditure from item-U.S.-monthly expenditures. See 

Equation 3.4. 

Moreover, the CPI-U estimate is refined by averaging area data with majorarea data, and the C-CPI-U estimate is refined by allocating U.S. expenditures to 

each area based upon previous-year expenditure patterns among the areas. Hence, 

both methods “borrow” information across time and geography in order to estimate 

item-area-reference period expenditures. Once summed, the estimated item-areamonthly expenditure data are equal to the composite-estimated-and-raked 

expenditure data at the item-U.S.-year level. See Figure 3.3 for an illustrative 
- 18 - 

example of monthly weight estimation of an elementary item-area cell in the 

C-CPI-U. 

c. Initial and Interim C-CPI-U. Lacking a satisfactory method to forecast the 

requisite monthly expenditure data, BLS opted to select an aggregation methodology 

for the Initial and Interim versions of the C-CPI-U that would best predict the Final 

C-CPI-U Tornqvist version – constrained by the use of the most contemporaneous 

expenditure data available at the time of index publication, i.e., expenditures from 

the CPI-U expenditure reference period (β). An adjusted Geometric Mean index 

formula was ultimately adopted. See the discussion of Aggregation Formula for 

Initial and Interim Indexes below. 

Since the Initial version of the C-CPI-U is published simultaneously with the 

CPI-U, it uses expenditure data from the same expenditure reference period (β) as 

the CPI-U as aggregation weights. In contrast to the CPI-U, however, it is not 

necessary to adjust the expenditures forward to a December “pivot” month and 

rebase them such that the implicit price corresponds to the corresponding item-area 

index base period (α). Rather, the estimated expenditure weights with implicit prices 

of time period (β) and implicit quantities of time period (β) are used as aggregation 

weights. This is consistent with the underlying assumption behind a geometric mean 

price index aggregator: consumers respond to changing relative prices by holding 

their expenditure shares constant over time. Hence, it is implicitly assumed that 

each item-area expenditure share derived from reference period (β) will be equal to 

each monthly item-area expenditure share over the time period in which aggregation 

weights based on reference period (β) are used to construct the Initial and Interim 

C-CPI-U indexes. In other words, i,a i,a t1 i,a t 2 i,a t 24 s = s = s = = s β , where t1 to t24 are 

the 24 months for which aggregation weights derived from (β) are used to construct 

the Initial and Interim C-CPI-U indexes. 
- 19 - 

The Interim version of each monthly C-CPI-U index will be published in 

February of the ensuing year. If the ensuing year is a weight update year, then the 

Interim version of each monthly C-CPI-U will be based upon more contemporaneous 

expenditures than its Initial version. For example, 2002 Initial indexes produced in 

2002 will use β=1999,2000. Interim indexes for 2002 will be produced in 2003 and 

will likewise use β=1999,2000. Initial indexes for 2003 will also use β=1999,2000. 

However, 2003 Interim indexes will be produced in 2004, a weight update year. 

Hence they will be constructed using β=2001,2002. 

Aggregation Formula. 

a. CPI-U. The Laspeyres price index is used to aggregate elementary indexes 

into published CPI-U indexes. The Laspeyres index uses quantities from the 

predetermined expenditure reference period (β) in order to weight each elementary 

item-area index. These quantity weights remain fixed for a two-year period and are 

then replaced each January in each even year when the aggregation weights are 

updated. Zero elasticity of substitution within item categories is assumed. An 

aggregate index for any given month is computed as a quantity-weighted average of 

the current month index divided by the index value in the index base period. See 

Equation 3.6.a. Month-to-month price change is then calculated as a ratio of the 

long-term monthly indexes. See Equation 3.6.b. 
- 20 - 

CPI-U Upper-level Aggregation Formula 

Equation 3.6.a. Long-term Price Change 

Equation 3.6.b. Month-to-Month Price Change 

where, 

 a = CPI elementary area 

 A = all elementary areas; “U.S. City Average” 

 i = CPI elementary item 

 I = all elementary items; “All-items” 

 t = month 

 z = base period of the aggregate index (NOTE: the All-Items, U.S. City Average 

CPI-U index has a base-period of z=1982-84) 

 α = base period of the elementary index (i) in area (a) 

 v = year and month, usually December, prior to the month expenditure weights 

from reference period (β) are first used in the CPI 

 i,a IX[α;t] = lower-level index of price change from period (α) to month (t) for item (i) 

in area (a) 

 i,a IX[α;v] = lower-level index of price change from period (α) to pivot-month (v) for 

item (i) in area (a) 

 i,a AWβ = aggregation weight from reference period (β) for item (i) in area (a) 

 I,A IXL

[z;v] = aggregate-level CPI-U index of price change from period (z) to pivot-month 

(v) for aggregate item (I) in aggregate area (A) 

 

b. Final C-CPI-U. In contrast, the C-CPI-U is built by chaining together 

indexes of one-month price change. For the Final C-CPI-U index, each monthly index 

is computed using the Tornqvist formula and monthly weights from the current 

month and previous month. Consumer substitution behavior is not assumed by the 

Tornqvist formula, but rather implicitly accounted for by use of current and basemonth expenditures. An index of one-month price change is calculated and then 

multiplied by the previous month index value to obtain the current month index 

value. See Equations 3.7.a and 3.7.b. 





∈

∈

×

×

= ×

( , ) ( , )

, , [ ; ]

( , ) ( , )

, , [ ; ]

, [ ; ] , [ ; ]

i a I A

i a i a

L G

v

i a I A

i a i a

L G

t

L

I A z v

L

I A z t

AW IX

AW IX

IX IX or

or

β α

β α

L

I A z t

L

L I A z t

I A t t IX

IX

IX

, [ ; 1]

, [ ; ]

, [ 1; ]

−

− =
- 21 - 

Final C-CPI-U Upper-level Aggregation Formula 

Equation 3.7.a. Long-term Price Change 

Equation 3.7.b. Month-to-Month Price Change 

 

 where, 

 a = CPI elementary area 

 A = all elementary areas; “U.S. City Average” 

 i = CPI elementary item 

 I = all elementary items; “All-items” 

 z = base period of the aggregate index (NOTE: the All-Items, U.S. City Average 

C-CPI-U index has a base-period of z=December 1999) 

 α = base period of the elementary index (i) in area (a) 

 i,a IX[α;t] = lower-level index of price change from period (α) to month (t) for item (i) 

in area (a) 

 i,a IX[α;t-1] = lower-level index of price change from period (α) to month (t-1) for item 

(i) in area (a) 

 i,a st = expenditure in month (t) for item (i) in area (a) as percent of total 

expenditures in month (t) for aggregate item (I) in aggregate area (A) 

 i,a st-1 = expenditure in month (t-1) for item (i) in area (a) as percent of total 

expenditures in month (t-1) for aggregate item (I) in aggregate area (A) 

 I,A IXT

[z;t] = aggregate-level C-CPI-U index of price change from period (z) to month (t) 

for aggregate item (I) in aggregate area (A) 

c. Initial and Interim C-CPI-U. Selecting an aggregation formula for the Initial 

and Interim versions of the C-CPI-U was more complex. As discussed above, 

construction of the Initial and Interim versions of the C-CPI-U is constrained by the 

use of aggregation weights from a lagged expenditure reference period (β). The 

Geometric Mean index and the Constant-Elasticity-of-Substitution (CES) index were 

identified as possible aggregation formula alternatives, since both of these indexes 

can be produced in real-time using expenditures from the lagged base-period (β). 

The CES long-term index is defined by Equation 3.8. The CES function was 

developed by Arrow, Chenery, Minhas, and Solow in 1961.35 Use of the CES 

functional form as a price index number formula was derived independently by Lloyd 

T

I A t t

T

I A z t

T

I A z t , IX[ ; ]= , IX[ ; −1]

× , IX[ −1; ]

∏∈

+

−

−

−

















=

( , ) ( , )

2

, [ ; 1]

, [ ; ]

, [ 1; ]

, 1 ,

i a I A i a

L G

t

i a

L G

T t

I A t t

i a t i a t

or

or

s s

IX

IX

IX

α

α
- 22 - 

(1975) and Moulton (1996).36 In 1996, Shapiro and Wilcox advocated the use of the 

CES as a feasible, “real-time” preliminary index for the Tornqvist, arguing it allows 

for substitution effects while not requiring current expenditure data.37 

Constant Elasticity of Substitution Index 

Equation 3.8. Long-term Price Change. 

σ σ

α β

α

β β

− −























	









 = × 

1

1

,

1

, [ ; ]

, [ ; ]

, [ ; ] , [ ; ] ,

i a

LorG

i a

LorG

i a t

i a

C

I A z

C

I A z t IX

IX

IX IX s

where, 

 a = CPI elementary area 

 A = all CPI areas (U.S. City Average) 

 i = CPI elementary item 

 I = all CPI items (All-items) 

 z = base period of the aggregate index 

 t = month 

 α = base period of the elementary index (i) in area (a) 

 β = expenditure reference period 

 i,a sβ = expenditure in period (β) for item (i) in area (a) as percent of total expenditures in 

period (β) for aggregate item (I) in aggregate area (A) 

 i,a IXα,β = lower-level index of price change from index base-period (α) to period (β) for item 

(i) in area (a); NOTE: because period β encompasses 24 months, this index is 

computed as an unweighted arithmetic average of the 24 monthly indexes 

contained in (β) 

 i,a IXα,τ = lower-level index of price change from index base-period (α) to period (t) for item 

(i) in area (a) 

 σ = elasticity of substitution parameter 

 I,A IXC

[z;t] = aggregate-level CES index of price change from period (z) to month (t) for 

aggregate item (I) in aggregate area (A) 

The distinguishing feature of the CES is the elasticity of substitution 

parameter (σ). Roughly speaking, substitution elasticity measures the proportionate 

change in the relative quantity demanded of a commodity over the proportionate 

change in its relative price. The CES is a pliable functional form, collapsing into 

various other price index formula in the limiting values of (σ). When σ=1 (unitary 

elasticity of substitution), Equation 3.8 reduces to the Geometric Mean price index 

 35 K.J. Arrow, H.B. Chenery, B.S. Minhas, and R.M. Solow, “Capital-Labor Substitution and Economic Efficiency,” Review 

of Economics and Statistics 63, pp. 225-250. 36 See P.J. Lloyd, “Substitution Effects and Bias in Non True Price Indexes.” American Economic Review, 1975, pp. 301-

313 and Brent Moulton, “Constant Elasticity Cost of Living Index in Share Relative Form.” BLS manuscript, 1996. 
- 23 - 

(Cobb-Douglas preferences). When σ=0 (zero elasticity of substitution), Equation 3.8 

reduces to the Laspeyres price index (Leontief preferences). 

The CES was analyzed in detail by BLS as a potential Initial and Interim CCPI-U aggregator. For a variety of reasons, however, the CES was judged ill-suited 

as an approximation of the Final C-CPI-U. First, estimation of (σ) is problematic 

from a theoretical perspective. In principle, to aggregate the 211 elementary CPI 

indexes into aggregate CPI indexes the elasticity of substitution among the 211 

items must be estimated in each of the 38 elementary CPI areas. This requires 

estimation of a complete demand system, i.e., estimation of σ for all 22,155 possible 

pairs of elementary items, in each of the 38 areas (i.e., 841,890 possible 

combinations). A representative price index aggregator could have been selected 

from a class of variable-elasticity-of-substitution functions to accomplish this task.38

However, a variable-elasticity aggregator suffers from several undesirable index 

qualities, most notably inconsistency in aggregation, and is infeasible to produce with 

the data currently available to BLS. 

Alternatively, following Shapiro and Wilcox, the CES functional form with a 

single value of (σ) could be assumed to hold across all possible pairs of item strata. 

The CES is much more feasible to produce than a variable-elasticity aggregator. One 

difficulty with using the CES, however, resides in the selection of the optimal value of 

(σ). Using the laboratory of elementary price indexes and expenditure weights from 

December 1986 to December 2000, monthly CES indexes were simulated in order to 

find solutions to the fitting parameter, i.e., the value of (σ) yielding a CES index 

best-approximating a Tornqvist index. The simulations revealed that the optimal 

elasticity of substitution parameter was unstable across time (ranging from a low of 

 37 See, Matthew D. Shapiro and David W. Wilcox (1997): “Alternative Strategies for Aggregating Prices in the CPI,” 

Federal Reserve Bank of St. Louis Review, (May/June), 113-125. 38 See, for example, Nagesh S. Revankar, “A Class of Variable Elasticity of Substitution Production Functions,” 

Econometrica, 1971, vol. 39, issue 1, pages 61-71. 
- 24 - 

σ=0.06 to a high of σ=2.78). See Figure 3.4. Accordingly, the assumption of a 

constant value of (σ) across month could result in specification error. This was 

judged a major weakness of the CES. 

Moreover, the conceptual underpinnings of the CES was judged ill-suited for 

use in building a monthly-chained time series. A CES constructed with expenditure 

shares derived from the same base-period month as used in the Tornqvist, i.e. 

i,a t−1 s , could be used to approximate the Final C-CPI-U month-to-month index for 

month (t). These CES month-to-month indexes could then be chained together to 

produce the long-term Initial and Interim C-CPI-U index series. There are two 

problems with this approach: lags in data availability preclude producing a (t-1) 

weighted CES in real-time, and a monthly-chained CES could be impacted by chain 

drift. Instead, expenditure shares derived from available data, i.e., β s i,a , must be 

used in a biennially-chained construct. 

The CES functional form further requires the expenditure shares (s) to be 

measured over the same time period as the denominator of the price relative.39 If 

the only available expenditure shares are those derived from expenditure reference 

period (β), which encompasses 24 months, it follows that the denominator price 

index in the CES relative must be some average index that is representative of the 

24 discrete indexes available in time (β). The choices (weighted versus unweighted 

average, arithmetic versus geometric average, etc.) introduce additional estimation 

complexity and potential specification error. 

Due to these impediments surrounding use of the CES, the Geometric Mean 

index was selected as a plausible, and simpler, approximation of the Tornqvist in 

 .

39 See Moulton, op. cit. 
- 25 - 

real-time. The general functional form of the Geometric Mean price index is given by 

Equation 3.9. 

Geometric Mean Index 

Equation 3.9. Long-term Price Change. 

 

where, 

 a = CPI elementary area 

 A = all CPI areas (U.S. City Average) 

 i = CPI elementary item 

 I = all CPI items (All-items) 

 0 = base period of the aggregate index 

 t = current month 

 α = base period of the elementary index (i) in area (a) 

 i,a s0 = expenditure in period (0) for item (i) in area (a) as percent of total expenditures in 

period (0) for aggregate item (I) in aggregate area (A) 

 i,a IXα,0 = lower-level index of price change from index base-period (α) to period (0) for item 

(i) in area (a); 

 i,a IXα,τ = lower-level index of price change from index base-period (α) to period (t) for item 

(i) in area (a) 

If consumers exhibit Cobb-Douglas utility preferences by holding expenditure 

shares constant over time, i.e. st1=st2=...=stn, then the Geometric Mean is an exact 

approximation of the Tornqvist. Empirical evidence from CE expenditure data 

suggests that expenditure shares do not change radically over time, at high levels of 

aggregation at least. See Table 3.5. 

The optimal value of the elasticity parameter (σ) that yields the minimum 

difference between the CES and the Tornqvist averages approximately 0.9 in CPI 

data.40 Hence, the assumption of unitary elasticity of substitution over short time 

intervals may well be a satisfactory approximation. Second, under the assumption of 

Cobb-Douglas preferences, sβ=st-1=st=sn. Because it is implicitly assumed sβ=st-1, 

the denominator price index in the Geometric Mean relative need not be an average 

of β period indexes. Month-to-month indexes can be computed directly using base-

∏∈ 















=

( , ) ( , )

,

, [ ;0]

, [ ; ]

, [0; ]

0

i a I A

i a

i a

G i a t

I A t

s

IX

IX

IX

α

α
- 26 - 

period price indexes (t-1) in the denominator, thus evading any potential 

specification error caused by use of an average price relative in the denominator. 

For any given month (t), the Geometric Mean month-to-month index will 

differ from the corresponding Tornqvist index to the extent that the i,a t−1 s differ from 

i a t s , and to the extent that the β s i,a poorly predict i,a t−1 s . Empirical evidence of 

upper-level aggregation in the CPI suggests that a Geometric Mean index is biased 

slightly below a Tornqvist index, when computed using expenditure shares derived 

from the same base-period as the Tornqvist.41 Therefore, it is anticipated that use of 

i,a t−1 s in a Geometric Mean index would tend to produce a lower measure of price 

change than would result from use of ( i,a t−1 s + i a t s , )/2 in a Tornqvist index on a 

consistent basis (e.g., the Geometric Mean month-to-month index averaged 0.006 

percent below the corresponding Tornqvist month-to-month index in CPI data over 

the 1990 to 2000 time period.) A notable exception to this general rule appears to 

occur in December, where the Geometric Mean index tends to produces a higher 

measure of price change than the Tornqvist. See Figure 3.5. Moreover, evidence 

from CPI data further suggests that use of lagged expenditure shares β s i,a in a 

Geometric Mean index consistently over predicts use of i,a t−1 s , albeit by a small 

amount. See Figure 3.5. 

In order to mitigate any specification error associated with use of a Geometric 

Mean index built with lagged expenditure shares, the BLS decided to adopt an 

“Adjusted Geometric Mean” approach for the Initial and Interim C-CPI-U. That is, 

elementary indexes are first aggregated using the Geometric Mean index. Then, the 

 40 .93 from 1987 to 2000. 41 See Ana M. Aizcorbe, Robert A. Cage, and Patrick C. Jackman, “Commodity Substitution Bias in Laspeyres Indexes: 

Analysis Using CPI Source Data for 1982-1994,” paper presented at the Western Economic Association International 

Conference in San Francisco, July 1996 (Washington, DC, Bureau of Labor Statistics). 
- 27 - 

resulting measure of price change is multiplied by an adjustment factor (λ) that 

represents the historically observed difference between Tornqvist and Geometric 

Mean upper-level aggregation of CPI elementary indexes.42 See Equation 3.10.c and 

3.10.d. The function of the adjustment factor is to more closely align the Geometric 

Mean month-to-month index, computed with lagged base-period expenditure weights 

(β), to a Tornqvist month-to-month index, computed with contemporaneous monthly 

expenditures (t-1 and t). 

Finally, the adjusted Geometric Mean month-to-month index is multiplied by 

the previous-month C-CPI-U index value to obtain the current month C-CPI-U index 

value. See Equations 3.10.a and 3.10.b. Note that each Interim month-to-month 

index is chained onto an Interim long-term index value, with the exception of the 

January index which is chained onto the previous year December index, which is in 

Final C-CPI-U form. Each Initial month-to-month index is chained onto an Initial 

long-term index value, with the exception of the January index which is chained onto 

the previous year December index, which is in Interim C-CPI-U form. 

For all months of 2002 and 2003, the adjustment factor has been set equal to 

unity. BLS plans to use Initial and Interim indexes calculated for 2002 and 2003, in 

conjunction with Initial and Interim versions of 2000 and 2001, to evaluate further 

how the Geometric Mean behaves relative to the Tornqvist. A permanent 

methodology for calculating the adjustment factor will be implemented at a future 

date. 

 

42 The set of data available to compute the adjustment factor is limited to all time periods for which the Final C-CPI-U has 

been computed. In 2002, for example, Final C-CPI-U indexes area available only for the 12 months of 2000. In 2003, an 

additional 12 months of data will become available. Lacking a sufficient time-series of historical data, the adjustment factor 

for 2002 and 2003 Initial and Interim C-CPI-U indexes will be set equal to λ=1. Official methodology for calculating the 

adjustment factor will be implemented with the calculation of January 2004 Initial indexes, when 36 months of historical 

data will be available. 
- 28 - 

Initial and Interim C-CPI-U Upper-level Aggregation Formula 

Equation 3.10.a. Initial C-CPI-U 

Long-term Price Change 

Equation 3.10.b. Interim C-CPI-U 

Long-term Price Change 

Equation 3.10.c. Initial C-CPI-U 

 Month-to-Month Price Change 

Equation 3.10.d. Interim C-CPI-U 

Month-to-Month Price Change 

 where, 

 a = CPI elementary area 

 A = all elementary areas; “U.S. City Average” 

 i = CPI elementary item 

 I = all elementary items; “All-items” 

 z = base period of the aggregate index (NOTE: the All-Items, U.S. City Average 

C-CPI-U index has a base-period of z=December 1999) 

 α = base period of the elementary index (i) in area (a) 

 i,a IX[α;t] = lower-level index of price change from period (α) to month (t) for item (i) 

in area (a) 

 i,a IX[α;t-1] = lower-level index of price change from period (α) to month (t-1) for item 

(i) in area (a) 

 i,a sβ = expenditure in reference period (β) for item (i) in area (a) as percent of 

total expenditures in reference period (β) for aggregate item (I) in 

aggregate area (A) 

 I,A IXT

[z;t] = aggregate-level C-CPI-U index of price change from period (z) to month (t) 

for aggregate item (I) in aggregate area (A) 

IV. COMPARING AND CONTRASTING C-CPI-U VERSIONS 

Difference in Estimation. Initial versus Interim Indexes. The long-term 

Interim C-CPI-U index for a given year and month can differ from the corresponding 

and previously released Initial version for three reasons. First, the historical time 

∏=

= − × −

t

n

G

I A y n y n

G

I A z y

G

I A z y t

i r i

IX IX IX

1

, [ ; , ] , [ ; 1,12] , [ , 1; , ]

∏=

= − × −

t

n

G

I A y n y n

T

I A z y

G

I A z y t

r r

IX IX IX

1

, [ ; , ] , [ ; 1,12] , [ , 1; , ]

∏∈ −

− 















= i a I A

i a

i a

L G

t

i a

L G

G t

I A t t

s

IX

IX

IX or

or

i

, ,

,

, [ ; 1]

, [ ; ]

, [ 1;, ]

β

α

α λ

∏∈ −− 















= i a I A i a

L G

t

i a

L G

G t

I A t t

s

IX

IX

IX

i a

or

or

r

, , , [ ; 1]

, [ ; ]

, [ 1; ]

, β

α

α λ
- 29 - 

series to which the Interim index is chained will contain an additional year of Final CCPI-U indexes. Consequently, the terminal December index value to which the 

January Interim index is chained may be different from the terminal December index 

to which the Initial index is chained. Second, the relative expenditure weight 

patterns used for aggregation may be different. This is highly probable in oddnumbered years – when the expenditure reference period (β) used for the calculation 

of Initial and Interim indexes will be different. For example, 2003 Initial indexes will 

use β=1999,2000, but 2003 Interim indexes (released in 2004) will use 

β=2001,2002. The aggregation weights for even-year Initial and Interim indexes will 

be identical. Third, the adjustment factor (λ) applied to the Initial Geometric Mean 

aggregation and the Interim Geometric Mean aggregation may be different. 

Final versus Interim Indexes. Similarly, the long-term Final C-CPI-U index for 

a given year and month can differ from the corresponding and previously released 

Initial and Interim versions for three reasons. First, the historical time series to 

which the Final index is chained will contain additional years of Final C-CPI-U 

indexes. Consequently, the terminal December index value to which the January 

Final index is chained may be different from the terminal December indexes to which 

the Initial and Interim indexes were chained. Second, the relative expenditure weight 

patterns used for aggregation most likely will be different. The difference in 

aggregation weights is the primary distinction in functional form between the Final 

and two preliminary versions of the C-CPI-U. Variation in the relative monthly 

expenditures used for the Final from the lagged constant-within-year relative 

expenditures used for the Initial and Interim may result in differing estimates of 

aggregate price change. Third, off-cycle elementary index values may be slightly 

different between the Final and preliminary versions, as the Final version will use 

geometric averaging of bounding month indexes. 
- 30 - 

V. COMPARING AND CONTRASTING THE C-CPI-U AND THE CPI-U 

Difference in Estimation. Because the CPI-U and all three versions of the 

C-CPI-U differ in the set of input elementary price indexes and expenditures used for 

aggregation, as well as in aggregation formula, each index may yield a different 

measure of aggregate price change for a given year and month. Table 5.1 

summarizes the differences between CPI-U and C-CPI-U index construction. 

The Initial C-CPI-U index is published in the same news release as the CPI-U. 

The long-term index values are not directly comparable, as the CPI-U will be on a 

1982-84=100 base and the C-CPI-U will be on a December 1999=100 base. The 

Initial C-CPI-U month-to-month index for a given year and month will differ from the 

corresponding CPI-U index by upper-level aggregation method only. The input prices 

and expenditures will be the same. Similarly, the Interim C-CPI-U month-to-month 

indexes will differ from the CPI-U month-to-month indexes in aggregation method. 

In addition, odd-year Interim indexes will differ in input expenditures used for 

aggregation. Final C-CPI-U month-to-month indexes will differ from the CPI-U 

month-to-month indexes in all aspects of index construction: (a) the input 

elementary price indexes will be the same, with the exception of off-cycle bimonthly 

indexes; (b) input elementary expenditures will be different, and (c) aggregation 

method will differ. 

Index Simulations. Simulating official estimation methodology, a biennially 

weight-updated CPI-U index series was calculated and compared to a Final C-CPI-U 

index series over the 1990 to 2000 period in order to measure the anticipated 

difference between the two series. See Figure 5.1. The average difference between 

the weight-updated CPI-U and the C-CPI-U was 0.32 percent per year over this time 

period. See Table 5.2. This estimate is at the upper end of the 0.1 percent to 0.4 

percent range of upper-level substitution bias estimated in prior BLS research, in 
- 31 - 

which the average annual percent difference was closer to 0.2 percent. Simulations 

for the 1990 to 2000 period define the range at 0.1 percent to 0.5 percent. 

There are several factors contributing to this result. First, the prior average of 

0.2 percent was based on data from 1991 to 1995 using the 1987 CPI market basket 

structure of 207 elementary items and 46 elementary areas (i.e., 9,522 elementary 

cells). The estimated average in Table 5.2 of 0.3 percent is based on the 1998 CPI 

market basket structure of 8,018 elementary cells from 1990 to 2000. In order to 

obtain data on the 1998 structure for years prior to 1998, price indexes and 

expenditures were approximated using a rough concordance between the old and 

new structures. Moreover, the current estimate is based on composite-estimatedand-raked biennial expenditures whereas the prior estimates were not. These 

differences function to produce an average annual substitution effect estimate for the 

overlap years of 1991 to 1995 that differs by 0.05 percent, i.e. 0.24 percent versus 

0.19 percent. 

Second, the gap between the weight-updated CPI-U and C-CPI-U appears to 

have widened in the later part of the decade. The average annual percent difference 

between the two indexes rose to 0.40 percent in 1996 to 2000, almost double that 

observed from 1991-1995. Analogously, the percent difference in simulated weightupdated CPI-U and C-CPI-U 12-month indexes steadily increased over the decade. 

See Figure 5.2. 

A likely contributor to the growing gap is increased dispersion in relative 

elementary index changes. In general, the CPI-U and the C-CPI-U will diverge to the 

extent that (a) component elementary indexes have rates of inflation that differ from 

each other, and (b) expenditure shares reflect a shift in consumer purchases toward 

those item categories that have fallen in relative price. Consequently, when there is 
- 32 - 

more variation in price movement among elementary indexes, there is more room 

for the Laspeyres-based CPI-U and the superlative-based C-CPI-U to diverge. 

Price change in CPI elementary indexes varied more widely during the later 

part of the 1990s. See Figure 5.3. Two examples of indexes with unusual index 

movements in 1999 and 2000 are computers and natural gas. The series for 

ITEM=EE01 “Personal computers and peripheral equipment” decreased by 22.7 

percent from December 1999 to December 2000. In contrast, the series for 

ITEM=HF02 “Utility natural gas service” increased by 36.7 percent over the same 

interval. The median elementary index change over this time period was 2.2 

percent. 

The significance of outlier elementary index series can by quantified by 

excluding them from CPI-U and C-CPI-U calculations and measuring the gap between 

the resulting “trimmed” indexes. This exercise was performed using the December 

1999 to December 2000 12-month simulated index for the C-CPI-U and weightupdated CPI-U, at varying outlier thresholds. See Table 5.3. When computing the 

indexes using the set of elementary indexes in the middle quartile, i.e. trimming the 

lower and upper quartiles from the calculations, the percent difference between the 

CPI-U and C-CPI-U is diminutive, 0.07 percent. The gap increases to 0.2 percent 

when the set of elementary items used in the calculations is limited to those that 

increased or decreased in price by 10 percent or less (roughly 75 percent of all 

elementary items over the December 1999 to December 2000 period). When the set 

is expanded to include all items exhibiting 20 percent price change (90 percent of all 

elementary items) the gap increases to 0.3 percent. The difference between the 

CPI-U and C-CPI-U is greatest when restricting aggregation over the lowest quartile 

of elementary index price change (1.5 percent) and highest quartile (0.7 percent). 

These trimmed indexes demonstrate that extreme changes in elementary price 
- 33 - 

indexes cause the CPI-U and C-CPI-U to diverge, and suggest that deflationary 

outliers contribute heavily to the gap. 

VI. INAUGURAL PUBLISHED INDEXES 

 C-CPI-U indexes were published for the first time in August 2002. Indexes are 

published for the urban population only. There are no plans at this time to calculate 

and publish a C-CPI-W. Published C-CPI-U indexes are available for the U.S. City 

Average only. No regional or local area indexes are published. Moreover, a limited 

set of indexes are available. See Table 6.1. 

Table 6.2, and Figure 6.1, display the inaugural published values of the 

C-CPI-U in relation to the CPI-U.43 The most surprising result was the 0.8 

percentage point gap between the two estimates of 12-month change during 

calendar year 2000, which was the only year for which the C-CPI-U index values 

were published in Final form. The reasons for this were discussed in section V 

above; additionally, rounding played a role in exaggerating the differences. The 

Interim and Initial values for 2001 and 2002 indicated some significant narrowing of 

the gap in those years. 

VII. 2003 REVISIONS AND MOST CURRENT PUBLISHED INDEXES 

In accordance with its previously-announced schedule, the BLS issued the 

first set of revised values of the C-CPI-U effective with the release of January 2003 

CPI data. While the magnitude of these revisions is expected to be small in the 

aggregate, in theory each monthly index is subject to two revisions until monthly 

expenditure data have been generated for that period and introduced two years 

later. With the availability of consumer expenditure data for 2001, the monthly 

C-CPI-U index values for 2001 became Final, and the values for 2002 moved from 

 

43 It should be noted that the published values for 2000 differ from the simulated values reported in section V for a variety 

of minor reasons. For example, the CPI-U values in Table 5.2 were simulated using a biennial weight update process, 
- 34 - 

Initial to Interim status. The revised index values and 12-month changes are shown 

in Table 7.1. 

In most cases, the revisions were upward and had the effect of narrowing the 

differences between the C-CPI-U and the CPI-U index series. At the All Items level, 

the Final C-CPI-U index for December 2001 increased from 103.6 to 103.9 and the 

12-month change rose from 1.0 to 1.3 percent, compared to a 1.6 percent increase 

in the CPI-U. It should be noted that rounding exaggerates the impact of revisions 

in this case: the unrounded change in the December 2001 index level is only 0.22 

(103.65 to 103.87). 

The bulk of the 2001 revisions can be attributed to a small number of item 

categories. The three largest contributors—owner’s equivalent rent, natural gas, and 

gasoline—account for about 92 percent of the revision to the All Items C-CPI-U for 

the period December 2000 to December 2001. In each case the item’s expenditure 

share moved in the same direction as its relative price change, indicating less than 

unitary elasticity of substitution. Owners’ equivalent rent rose somewhat more than 

the average CPI during 2001. Its overall impact results from its very large weight. 

Natural gas and gasoline prices both rose sharply early in 2001, then fell sharply. 

Their impacts result from their extreme price movements. 

C-CPI-U index levels for 2002 change only because they are spliced onto the 

revised 2001 Final values. Rates of change in months after December 2001 are 

unaffected by the revisions, except for rounding. Percent index changes during 2002 

and 2003 are still based on the geometric mean formula and weights from 1999-

2000, as they were before the revisions. At the All Items level, the CPI-U increased 

2.4 percent from December 2001 to December 2002 and the C-CPI-U increased 2.1 

percent. 

 whereas the published CPI-U in that year used 1993-95 base period weights. The values in Table 5.2 also are annual-
- 35 - 

Comparisons with Personal Consumption Expenditure Index. In 1996, 

the Bureau of Economic Analysis (BEA) at the Department of Commerce introduced 

chain-type, annual-weighted indexes as its featured measures of real output and 

prices of the National Income and Product Accounts (NIPA’s). The chain-type price 

index for personal consumption expenditures (PCE) is similar in concept to the 

C-CPI-U in several aspects. Both employ a superlative aggregator—BEA’s PCE price 

index is constructed using the Fisher Ideal index formula. Both are chained, albeit 

the C-CPI-U at monthly intervals with monthly weights, while the PCE index is 

chained quarterly, with annual adjustments. On the other hand, there are significant 

distinctions between the two series. Notably, the PCE index is broader in scope and 

includes spending on behalf of consumers by employers and government health 

agencies. The PCE index also uses NIPA weights rather than Consumer Expenditure 

Survey values, and relies in part on price series other than those used in the CPI. 

Table 7.2 compares December-to-December changes in the CPI-U, the 

C-CPI-U, and the PCE index for the three years for which the C-CPI-U has been 

published. As the table shows, the C-CPI-U and PCE series are remarkably close 

given the differences in their scope and construction. This lends some indirect 

support to the C-CPI-U. Note that the 2002 C-CPI-U estimate is not a Final number. 

VIII. SUMMARY AND DIRECTIONS OF FURTHER RESEARCH 

The release of the C-CPI-U marks a significant step forward in the U.S. CPI 

program. A superlative index formula, by employing expenditure data from two 

separate points in time rather than from a single base period, is designed to reflect 

substitution among items in response to changes in relative prices and to provide a 

closer approximation to a cost-of-living index than is possible using the Laspeyres 

 average figures, rather than December-to-December changes. 
- 36 - 

formula. The unavoidable delays in receiving expenditure data result in the inability 

to produce the final C-CPI-U on the same schedule as the fixed-base CPI-U and 

CPI-W indexes. The BLS therefore designed the C-CPI-U as its first CPI series to be 

produced in preliminary form and subject to revision. In August 2002 the inaugural 

C-CPI-U series were released, followed in February 2003 by the first set of annual 

revisions. 

One notable difference between the C-CPI-U and the existing fixed-base 

indexes is the interpretation of the sub-indexes--e.g. food, energy, all items less 

food and energy, etc. In the fixed-base CPIs, the subindexes are separable and 

additive. In the C-CPI-U the subindexes are conditional upon the behavior of the 

expenditures and prices of items not included within the subindex, and are therefore 

not additive. For example, in the C-CPI-U, the behavior of the food index is a 

function not only of the expenditure and price movements of the food items included 

within the food aggregate, but also of the expenditure and price movements of all 

other components within the All Items C-CPI-U. In addition, the All Items C-CPI-U in 

general is not the weighted average of the subindexes that comprise it, either for all 

211 item strata or for sub-aggregates such as food, energy, and all items less food 

and energy. That is, the C-CPI-U is not consistent in aggregation. 

As discussed above in section III, many challenges faced the BLS in the 

course of designing the C-CPI-U. Corresponding to each of these challenges are 

issues that can be addressed in the future with the aid of growing time series of 

published data. Further analysis may, in some cases, lead to modifications in the 

way the C-CPI-U is constructed. Among the issues that the BLS is planning to study 

are: 

(i) Analyze the gap between the CPI-U and the C-CPI-U. Sections V and VI 

described how the difference between the two estimates of annual inflation widened 
- 37 - 

in the late 1990s. In the first year of published C-CPI-U data, the gap of 0.8 

percentage point greatly exceeded the earlier BLS prediction range. The gap was 

smaller in 2001 than in 2002, and it is reasonable to expect that during the next few 

years CPI-U inflation estimates will exceed those of the C-CPI-U by more than 0.2 

percentage point but less than 0.8 percentage point. BLS staff are now working to 

develop alternative means of decomposing the inflation differences, in order to more 

easily identify the contributions of individual index series.44

 (ii) Determine the best estimator for the Initial and Interim indexes. The 

BLS has no experience yet in the C-CPI-U upon which to base estimates of the future 

size of revisions between Initial and Final index values. The revisions to 2001 

Interim indexes were not unduly large, but the fact that they were upward revisions 

was relatively unusual compared to most years in historical simulations. Thus far, 

the simulation studies have been insufficient to estimate an adjustment factor 

reliably, and thus far the BLS has employed a factor of one—i.e., using an 

unadjusted geometric mean formula—to produce the preliminary index values. 

Future work will address how best to estimate the adjustment factor when longer 

time series are available for that purpose. Other potential topics include the use of 

different factors for different months of the year and, more generally, the analysis of 

alternative formulas and approaches to predicting the final superlative values. 

 (iii) Evaluate the monthly weights. The issues of seasonality, volatility, 

sampling error, and other special features of the Consumer Expenditure Survey data 

comprise a rich and important area of research. The BLS plans to study how best to 

 

44 On the topic of superlative index decomposition, see Marshall B. Reinsdorf, W. Erwin Diewert and Christian Ehemann, 

“Additive decompositions for Fisher, Tornqvist and geometric mean indexes,” Journal of Economic and Social 

Measurement 28, pps.51–61.

. 
- 38 - 

deal with seasonal and durable goods, and whether alternative methods of composite 

weight estimation could improve index performance. 

 (iv) Evaluate the elementary price indexes. Parallel to the analysis of weight 

data, research on elementary price indexes would include outlier mitigation, 

collapsing of areas, composite estimation, and other means of reducing volatility and 

making the price series more appropriate to their use in a monthly superlative 

formula. 

In the future, with the production and publication of a longer time series, the 

BLS also will revisit the issue of generating seasonally adjusted C-CPI-U indexes. It 

should be noted that the method employed in the CPI-U, dependent seasonal 

adjustment (i.e., aggregating a selected set of seasonally adjusted components to 

obtain seasonally adjusted aggregates), is inappropriate for the C-CPI-U, since it is 

not consistent in aggregation. The Bureau also intends to produce and publish 

statistical measures of reliability for the C-CPI-U such as are presently published for 

the fixed-base CPIs. 
- 39 - 

Figure 3.1. 2-Stage Construction of the CPI 

Figure 3.2. Publication Schedule of C-CPI-U Index Versions

Aggregate

Indexes

Lower-Level Aggregation Lower-Level Aggregation

Upper-Level Aggregation Upper-Level Aggregation

Price

Quotes

80,000

Elementary

Indexes

8,018

20 03 200 4 2 005

20 03

20 04

20 05

Y E A R O F P U B L I C A T I O N

YEAR OF INDEX

IN ITIAL

IN ITIAL

IN TER IM FIN AL

IN TER IM

IN ITIA L
- 40 - 

Table 3.1. Percent Difference in Final C-CPI-U annual average indexes: 

Geometric averaging versus previous-month imputation of off-cycle elementary indexes.

NOTES: 

1. Indexes are for U. S. City Average, All-items and are not seasonally adjusted. 

2. Each index is computed as the average annual index divided by the previous-year 

average annual index. 

Year

G eom etric 

M ean o f 

B ou nd ing 

M onths

P rev io u s 

M onth 

V a lu e

Percent 

D ifferen ce

1 9 8 8 1 .0 3 7 1 9 1 .0 3 7 3 0 -0 .0 1 1

1 9 8 9 1 .0 4 4 8 0 1 .0 4 4 9 7 -0 .0 1 7

19 90 1 .048 90 1 .04 879 0 .010

1 9 9 1 1 .0 3 9 2 7 1 .0 3 9 5 2 -0 .0 2 4

1 9 9 2 1 .0 2 6 8 2 1 .0 2 7 1 6 -0 .0 3 3

19 93 1 .027 79 1 .02 769 0 .010

19 94 1 .024 42 1 .02 438 0 .005

19 95 1 .024 09 1 .02 395 0 .014

19 96 1 .027 27 1 .02 701 0 .025

19 97 1 .018 19 1 .01 800 0 .019

19 98 1 .009 12 1 .00 903 0 .009

19 99 1 .017 14 1 .01 705 0 .009

2 0 0 0 1 .0 2 7 8 2 1 .0 2 7 8 3 -0 .0 0 1

A v erag e A n n u a l D iffe re n ce : 0 .0 0 1
- 41 - 

Table 3.2. Expenditure Reference Periods in the CPI.

Expenditure 

Reference Period

Month 

Introduced Terminal Month

1917-1919 1919 1939

1934-1936 Jan 1940 Dec 1952

1950 Jan 1953 Dec 1963

1960 -1961 Jan 1964 Dec 1977

1972 - 1973 Jan 1978 Dec 1986

1982 - 1984 Jan 1987 Dec 1997

1993 - 1995 Jan 1998 Dec 2001

1999 - 2000 Jan 2002 Dec 2003

2001 - 2002 Jan 2004 Dec 2005
- 42 - 

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12

Figure 3.3. Monthly expenditures for ITEM=HF02 (Fuel Oil) as percent of All-item 

expenditures: Washington, DC and U.S. City Average, 1999 and 2000 

U.S. City average 

Washington, DC adjusted 

Washington, DC unadjusted 
- 43 - 

Figure 3.4. Value of σ minimizing (Tornqvist – CES)2 

Table 3.5. Percent distribution of CPI market basket expenditures by major group 

0.00

0.50

1.00

1.50

2.00

2.50

3.00

1987.11988.11989.11990.11991.11992.11993.11994.11995.11996.11997.11998.11999.12000.1
- 44 - 

Figure 3.5. Average percent difference in Tornqvist and Geometric mean 

month-to-month indexes, 1990 to 2000, by month 

Geometric Mean (st-1 weighted) 

minus Tornqvist 

Geometric Mean (sβ weighted) minus 

Geometric Mean (st-1 weighted) 
- 45 - 

 Table 5.1. Components of CPI-U and C-CPI-U Index Construction 
- 46 - 

Figure 5.1. Simulated Weight-updated CPI-U and C-CPI-U indexes

100.0

105.0

110.0

115.0

120.0

125.0

130.0

135.0

140.0

145.0

1990.1

1990.5

1990.9

1991.1

1991.5

1991.9

1992.1

1992.5

1992.9

1993.1

1993.5

1993.9

1994.1

1994.5

1994.9

1995.1

1995.5

1995.9

1996.1

1996.5

1996.9

1997.1

1997.5

1997.9

1998.1

1998.5

1998.9

1999.1

1999.5

1999.9

2000.1

2000.5

2000.9

Final C-CPI-U Fixed-weighted CPI-U Biennially Weight-updated CPI-U

New Market Basket Introduced

Use of Hybrid

Elementary Indexes

begins

New PSUs introduced

into CE sample

Direct mapping of CE expenditures

into New Market Basket structure

begins
- 47 - 

Table 5.2. Simulated Weight-updated CPI-U and C-CPI-U average annual 

indexes, relative to previous year average-annual index, 1991 to 2000 
- 48 - 

Figure 5.2. Percent difference between CPI-U (biennially weight updated) 

and Final C-CPI-U Simulated 12-month Indexes 

Figure 5.3. Elementary Price Index Extremes: Number of Elementary Monthto-Month Price Indexes less than 0.95 or greater than 1.05, by month
- 49 - 

Table 5.3. Percent difference between simulated CPI-U and C-CPI-U trimmed 

December 1999 to December 2000 12-month indexes

Table 6.1. List of Published C-CPI-U Indexes 
- 50 - 

Table 6.2. Inaugural C-CPI-U published indexes for All-items, U.S. City Average 

compared to CPI-U 
- 51 - 

Figure 6.1. Percent difference in 12-month CPI-U and C-CPI-U indexes
- 52 - 

Table 7.1 C-CPI-U ALL-ITEMS, U.S. CITY AVERAGE INDEXES: Differences in 

Original and Revised Values 
- 53 - 

Table 7.2. December-to-December percent changes in CPI-U, C-CPI-U 

and PCE Chain Price Indexes

 Index Series 

Year CPI-U C-CPI-U PCE 

2000 3.4 2.6 2.5 

2001 1.6 1.3 1.2 

2002 2.4 2.0 2.0