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BEYOND BLS
DECEMBER 2021

The valuation effects of antitrust legislation
Yavor Ivanchev

In recent years, rising market concentrations have compelled more countries to consider adopting stringent antitrust
laws in the hope of countering monopolistic behaviors by large firms. The goal of such initiatives has been to create
competitive marketplaces that would spur innovation, benefit consumers, reduce inequality, and boost economic
growth. What remains an open question, however, particularly for shareholders, is how competition laws can affect
the value of the firms subjected to them. In a recent article titled “Competition laws, governance, and firm value”
(National Bureau of Economic Research, Working Paper 28908, June 2021), economists Ross Levine, Chen Lin,
and Wensi Xie offer new empirical answers to this question.
The authors observe that, in theory, the relationship between antitrust laws and firm value is ambiguous. One
perspective, known as the “agency view,” suggests that firms hit by antitrust laws would be forced to improve their
governance structures and practices, eventually increasing their market value. The idea here is that, by exposing
organizational inefficiencies such as informational opaqueness and management rent-seeking practices, a healthy
level of competition would encourage reform that brings value to shareholders. Just as intuitive, however, is the
opposite expectation, namely, that firms exposed to greater competition would see their valuations drop. As
explained in the article, the burdens imposed by competition may reduce a firm’s market share, monopolistic rents,
investment in research and development, and overall revenue and profits.
To assess these rival hypotheses, the authors rely on a large panel dataset—in their estimation, the largest ever
compiled in the literature on antitrust laws—containing two decades (1990–2010) of information on 90 countries
and 22,000 firms. The key independent and dependent measures in the data are, respectively, the Competition Law
Index, which measures the prevalence of antitrust laws that limit monopolistic practices, and Tobin’s Q, a firm
valuation measure reflecting the type and size of firm assets. To account for potentially confounding causal
mechanisms, the authors’ regression models also include various control variables, as well as firm and industry
fixed effects.

The baseline regressions show a statistically and economically significant positive relationship between
competition laws and firm value. Although this result provides some preliminary support for the agency view, it
does not corroborate the specific causal mechanisms proposed by that view. To see whether these mechanisms
operate in practice, the authors conduct additional statistical analyses, each based on a separate corollary hypothesis
derived from the agency view. These analyses provide further support for the valuation-boosting effects of antitrust
legislation, showing that these effects are (1) stronger among firms with more severe preexisting governance
problems and (2) weaker in industries with a high level of preexisting firm competition. In addition, the authors
find that, also consistent with a mechanism implied by the agency view, competition laws improve firm operational
efficiency, especially in countries with weak investor protections.
Download PDF »

December 2021

Business employment dynamics by wage class
Currently available data on gross job gains and losses do
not capture the wage component of employment dynamics.
This article partly fills this gap by examining the distribution
of job gains and losses among establishments that pay
high, medium, and low wages.
Statistics on gross job gains and losses, which show the
dynamics of job creation and destruction, are now often
used by economists and policymakers in understanding the
labor market. Available longitudinal microrecord data on
employers and employees have allowed researchers to
observe, in detail, how employment growth is generated by
a continuous stream of job gains and losses across all
industries, geographies, and firms of different sizes and
ages. Since its first publication in 2003, the Business
Employment Dynamics (BED) program at the U.S. Bureau
of Labor Statistics (BLS) has expanded greatly, publishing
valuable data series in response to policy and research
needs for data on employment growth and labor turnover.
Currently available data series on employment dynamics

Akbar Sadeghi
sadeghi.akbar@bls.gov
Akbar Sadeghi is an economist in the Office of
Employment and Unemployment Statistics, U.S.
Bureau of Labor Statistics.
Kevin Cooksey
cooksey.kevin@bls.gov

and some entrepreneurship indicators allow economists
and policymakers to gain a better understanding of the
overall labor market and the specific nature and magnitude
of job creation and destruction.1

Kevin Cooksey is a supervisory economist in the
Office of Employment and Unemployment
Statistics, U.S. Bureau of Labor Statistics.

Missing from current series on job flows are data on the
wage component of employment dynamics. Efforts to fill
this gap are important because of a growing desire among
economists to measure job creation and employment growth in terms of wage quality. In a working paper on the
effect of minimum-wage increases on employment in Seattle, WA, Ekaterina Jardim et al. used state employment
and wage data to examine the impact of minimum-wage increases across categories of low-wage employment.
The authors reached a “markedly different conclusion” from that reported in previous studies, stating that
“employment losses associated with Seattle’s mandated wage increases are in fact large enough to have resulted
in net reductions in payroll expenses—and total employee earnings—in the low-wage job market.”2 The paper
attributed this contrasting finding to the data sources used in the analysis.

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Jardim et al. focused on low-wage employment by using data compiled by Washington State’s Employment
Security Department in the administration of unemployment insurance (UI) law. The BLS Quarterly Census of
Employment and Wages (QCEW) program gathers such data from every state, compiling them into a single
longitudinal database used in developing the BED data series. Expanding the BED scope by adding wage data
elements and tabulating job gains and losses by a quality-of-wage measure will increase the value and utility of the
BED series. In this article, we analyze BED data by wage class, showing gross job gains and losses, as well as
employment growth, by metrics that distinguish among establishments that pay high, medium, and low wages.

What are BED data?
The BED program has published quarterly statistics on gross job gains and losses since 2003.3 These statistics
are derived from QCEW establishment-level microrecords. The data gathered by the QCEW program provide a
virtual census of employees on nonfarm payrolls, covering 98 percent of such employees. In the first quarter of
2020, the QCEW program reported 147.0 million employees working across 10.4 million establishments.4
The QCEW estimates are based on mandatory quarterly reports on employment and wages submitted by all
employers subject to UI laws. The process of reviewing and editing these reports turns raw, unedited,
administrative data into high-quality, reliable, and consistent economic statistics. These QCEW statistics are the
most accurate, timely, comprehensive, and frequent employment and wage census data in the federal statistical
system at the local level. In addition to being a high-quality source of employment statistics, the QCEW serves as
the sampling frame for many BLS surveys, as a benchmark for the BLS Current Employment Statistics and
Occupational Employment and Wage Statistics surveys, and as an input to the U.S. Bureau of Economic Analysis
National Income and Product Accounts.
The QCEW records are matched across quarters to create a longitudinal history for each establishment. Records
are linked by their unique identifiers, including state codes, UI numbers, and reporting-unit numbers. This method
creates a history for continuous records and identifies establishment entries and exits, while avoiding spurious
business births and deaths that could be reported in the event of ownership changes, mergers, acquisitions,
spinoffs, or other corporate restructuring. The longitudinal database created from the linked records is used to
construct the BED data.5
The BED program tracks employment levels and establishment counts by the direction of employment changes.
The employment reported in the third month of each consecutive quarter is used to measure the over-the-quarter
employment change. Depending on the nature of this change, records correspond to four distinct groups of
establishments: opening, expanding, closing, and contracting. Expanding and contracting establishments have
continuous records with, respectively, increasing and decreasing employment over the quarter. Opening
establishments are those whose employment shifts from zero to positive, whereas closing establishments are
those whose employment shifts from positive to zero.6
The sum of employment at opening establishments and the change in employment at expanding establishments
represents gross job gains. Similarly, the sum of the prior-quarter employment at establishments that closed in the
current quarter and the change in employment of contracting establishments represents gross job losses. The net
employment growth for all firms can be measured in one of two ways: as the difference between total employment
in the current and previous quarters or as the difference between gross job gains and losses in the current quarter.

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Differentiating among high-, medium-, and low-paying establishments
In this article, we measure, by level of wages paid, gross job gains at opening or expanding establishments and
gross job losses at contracting or closing establishments. We show how much net employment growth comes from
establishments that pay high, medium, and low wages. Because there are no commonly accepted definitions of
these wage categories, we use the distribution of average weekly wages paid across all private sector
establishments and select certain wage ranges as high, medium, and low. We use the lower and upper quartiles
as boundaries for differentiating among high, medium, and low wages. Establishments paying average weekly
wages lower than the bottom quartile are defined as low-paying establishments, and those paying average weekly
wages higher than the top quartile are defined as high-paying establishments. Establishments paying average
weekly wages in the interquartile range are defined as medium-paying establishments.
We apply this wage classification to every establishment in the QCEW database, for every quarter. The wage class
into which a record falls is based on the distribution of average wages paid across all active records in the
reference quarter. The establishment wage group is reevaluated every quarter and may change from quarter to
quarter. For closing establishments, we use the wages paid in the last quarter in which the establishments were
active. The values for the upper and lower quartile wages vary across quarters as the ranks of establishments on
the wage scale move over time because of business growth and the phases of business cycles.
To show the dynamics of employment by wage class, we follow a methodology for measuring gross job gains and
losses (and their components) that is the same as that used routinely in measuring other data elements of the BED
series. We compare employment at the establishment level in two consecutive quarters and calculate net changes
in employment. On the basis of the direction of employment change, records are labeled as opening, closing,
expanding, or contracting. The gains and losses at the establishment level are then aggregated by wage class. We
seasonally adjust the four components of employment dynamics and use them to calculate the seasonally
adjusted values for gross job gains and losses, as well as the rate of gains and losses and the net change in
employment.7
In the following sections, we summarize findings from the tabulation of BED data by establishment wage class.
The latest available data at the time of writing this article were for the quarter ended in March 2020. The full impact
of the coronavirus disease 2019 (COVID-19) pandemic will not be reflected in the BED data elements until the
pandemic subsides. However, data for the first quarter of 2020 partially reflect the impact of the pandemic-induced
recession that started in February 2020.8 For that reason, in discussing a trend for a given data element, we
mainly compare data up to the fourth quarter of 2019 and, if warranted, refer to data for the first quarter of 2020.

Rising wage gap between high- and low-paying establishments
In the first quarter of 2020, average weekly wages in the U.S. private sector were, on a seasonally adjusted basis,
$369 or less at low-paying establishments and $1,193 or more at high-paying establishments. (See chart 1.) The
mean and median wages of all establishments in that quarter were $1,133 and $673, respectively. The difference
between mean and median wages indicates the skewness of the wage dispersion across payrolls of all private
sector establishments compiled in the QCEW database. The mean and median wages, as well as the upper and
lower quartile wage levels, increased over the quarter and from the same period in 2019. Average private sector

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U.S. BUREAU OF LABOR STATISTICS

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wages, which were partly affected by the COVID-19 economic slowdown, did not change considerably in the first
quarter of 2020. (See appendix table A-1.)

Low-paying establishments across the nation paid an average weekly wage of $175 or less in the third quarter of
1992, and this figure rose to $369 in the first quarter of 2020, a steady, 2.1-fold increase over the period. In highpaying establishments, the average weekly wage was $502 in the third quarter of 1992 and $1,193 in the first
quarter of 2020, a 2.4-fold increase that was faster than that at low-paying establishments. The gap between upper
and lower quartiles widened over this period, with the average weekly wage differential between top and bottom
quartiles rising from $328 in the third quarter of 1992 to $824 in the first quarter of 2020. The gap between the
three wage classes described previously is based on nominal wages, and the real gap, deflated for inflation, will be
smaller. However, the finding that wages rose more in establishments with higher wages stands.

Recent fall in wage dispersion
Chart 2 shows that wage dispersion, measured by the ratio of upper to lower quartile wages, has been declining in
recent years, despite previously trending up since the beginning of the series. In 1992, establishments in the top
quartile of the wage distribution paid average wages that were 2.89 times higher than those paid by
establishments in the lower quartile. In the fourth quarter of 2014, this ratio, which indicates a widening wage
inequality, reached 3.38, and since then, it has been on a declining trend, falling to 3.23 in the first quarter of 2020.

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U.S. BUREAU OF LABOR STATISTICS

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Low-wage establishments account for most job gains
In the first quarter of 2019, the U.S. private sector created 547,000 jobs, on net. Of these jobs, 294,000 (53.7
percent) were added by low-paying establishments (those paying an average weekly wage of $346 or less),
127,000 (23.2 percent) were added by high-paying establishments (those paying an average weekly wage of
$1,138 or more), and 126,000 (23.0 percent) were added by medium-paying establishments (those paying
between $346 and $1,138 per week, on average). (See chart 3 and appendix table A-2.)

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U.S. BUREAU OF LABOR STATISTICS

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In the first quarter of 2020, the private sector lost a net total of 663,000 jobs,9 a drop partly reflecting the effect of
the COVID-19 recession. Low-paying establishments lost a net of 40,000 jobs and high-paying establishments lost
a net of 47,000 jobs. Most net job losses occurred at medium-paying establishments, which lost 587,000 jobs
during the quarter.
The historical values of gross job gains and losses by wage class since 1992 reveal that, except for a slight decline
in a single quarter during the 2007–09 Great Recession and in the early days of the COVID-19 recession, gross
job gains at low-paying establishments have consistently exceeded gross job losses. For this reason, low-paying
establishments have had the highest share of employment growth in almost every quarter, compensating for the
frequent employment losses in other wage groups, especially high-paying establishments. (See chart 4.) For
example, between the 2001 and 2007–09 recessions—a period of recovery from the big job losses of the 2001
recession—high-paying establishments had only a few quarters of positive employment growth, whereas lowpaying establishments contributed substantially to employment growth and had the largest share of net job gains.

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U.S. BUREAU OF LABOR STATISTICS

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The cumulative effect of net job gains by wage class shows that, since the third quarter of 1992, the U.S. private
sector has created a net total of 37 million jobs. Over this period, low-paying establishments gained 36 million jobs,
medium-paying establishments gained 10 million jobs, and high-paying establishments lost 9 million jobs.

High-paying establishments have been affected most by the last two
recessions
During the 2001 and 2007–09 recessions, the share of high-paying establishments in total gross job gains fell,
whereas the shares of low- and medium-paying establishments increased. (See chart 5.) At the same time, the
share of job losses rose in high-paying establishments and fell in low- and medium-paying establishments. (See
chart 6.) In the immediate aftermath of the 2007–09 recession, establishments of all three wage classes saw their
shares in gross job gains recover and return to prerecession levels. However, in subsequent years, the job-gains
share changed little in high-paying establishments, increased in medium-paying establishments, and slightly
decreased in low-paying establishments. (See appendix table A-2.)

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8

U.S. BUREAU OF LABOR STATISTICS

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In the 2001 recession, more jobs were lost in high-paying establishments than in establishments of other wage
classes. Net job losses in low-paying establishments started to decline earlier than did job losses in high- and
medium-paying establishments, but they remained positive throughout the recession. The 2001 recession
occurred after the burst of the dot-com bubble of the late 1990s, with many high-paying technology companies
seeing large job losses and pay declines. In the 2007–09 recession, both high- and medium-paying establishments
experienced fast and similar rates of net job losses, but the net number of jobs lost was the largest in mediumpaying establishments. In low-paying establishments, net job losses began to decline before the onset of the
recession, but they only fell to negligible negative values in the first quarter of 2009 and reversed course
immediately after that quarter. Because of the breadth of recession impacts across all sectors and establishment
size classes, medium-paying establishments suffered the largest number of net job losses in that recession. (See
charts 3 and 4.)

Continuing establishments have much higher wages than opening
and closing establishments
Wages paid at the establishment level correlate with the type of job flow. Median wages are considerably higher in
expanding and contracting establishments than in opening and closing establishments. (See chart 7.) For opening
and closing establishments, the BED data by wage class reveal that, until 2011, wages at opening establishments
were consistently higher than wages at closing establishments. In most years since 2011, wages at closing
establishments have been either equal to or slightly higher than wages at opening establishments. Median wages
at expanding establishments have been slightly lower than wages at contracting establishments, but the difference

9

U.S. BUREAU OF LABOR STATISTICS

MONTHLY LABOR REVIEW

is not significant. However, the data show that this difference has widened during economic downturns. (See
appendix table A-1.)

Chart 8 compares the ratio of median wages at opening establishments to median wages at closing
establishments (hereafter referred to as “opening-to-closing wage ratio”) with the ratio of median wages at
expanding establishments to median wages at contracting establishments (hereafter referred to as “expanding-tocontracting wage ratio”). These ratios hover around 1, a value indicating wage parity for the data elements
compared. In the chart, this value is represented by a dashed line, allowing readers to visually determine at which
points and to what extent wage ratios are higher or lower than 1 for a pair of data elements. The data show that
the expanding-to-contracting wage ratios are mostly below the dashed line, suggesting that, on average,
expanding firms may pay less to their new hires, which would bring the median wage down, and/or that contracting
establishments may lay off more of their low-wage workers, which would cause the median wage to rise. In
contrast, the data show that the opening-to-closing wage ratios are mostly above the dashed line, indicating that
workers hired by opening establishments are generally paid higher wages than workers who lost their jobs in
closing establishments.

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U.S. BUREAU OF LABOR STATISTICS

MONTHLY LABOR REVIEW

As shown in chart 8, the pattern of wage ratios among the components of gross job gains and losses exhibits
some business cycle properties. This is especially evident for the expanding-to-contracting wage ratio, which
dipped more during economic downturns than during other phases of the business cycle. This ratio, which was
close to 1 in most years, fell to 0.93 at the height of the 2001 recession and to 0.88 at the height of the 2007–09
recession. The fall of wage ratios during economic downturns may result from a lower median wage paid at
expanding establishments, a higher median wage paid at contracting establishments, or a combination of both
factors.

Employment is shifting from high-paying to medium-paying
establishments
Charts 9a to 9c show employment for each establishment wage class as a percentage of total employment from
March 1993 to March 2019. Over this period, the employment share of low-paying establishments fluctuated in a
narrow range of 1 percent, from a high of 14.3 percent in 1999 to a low of 13.3 percent in 2012. In 2019, this share
was the same as it was in 1993. In contrast, the employment share of high-paying establishments has been
trending down, falling from 37.9 percent in 1993 to 33.8 percent in 2019. Over the same period, the distribution of
employment by wage class has shifted from high-paying to medium-paying establishments, with the 4.1-percent
drop in the employment share of high-paying establishments leading to a rise of a similar magnitude (from 48.3 to
52.5 percent) in the employment share of medium-paying establishments.

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U.S. BUREAU OF LABOR STATISTICS

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Summary
In this article, we presented research data on gross job gains and losses by wage class, quantifying the magnitude
of job creation and destruction in terms of three classes of wages. Using the distribution of average weekly wages
paid across all private sector establishments in the longitudinal QCEW, we distinguished among high-paying
establishments, whose wages are in the top quartile; low-paying establishments, whose wages are in the lower
quartile; and medium-paying establishments, whose wages are in the interquartile range.
For each of these establishment categories, we measured the BED data elements over time and during the
phases of business cycles, finding that the wage ratio—that is, wages paid at high-paying establishments to wages
paid at low-paying establishments—climbed over the two decades following the start of the series. Although this
trend indicates a rising wage inequality, more recent data show a sign of improvement, with wage dispersion
declining since 2014.
We also showed that a substantial share of employment growth (reflected in the gap between gross job gains and
losses) came from low-paying establishments. These establishments experienced employment growth even during
economic downturns.
Further, we analyzed wages by the direction of employment changes, comparing the opening-to-closing and
expanding-to-contracting wage ratios. We found that, during the period for which BED data are available, the
opening-to-closing wage ratios were consistently higher than 1, whereas the expanding-to-contracting wage ratios
were mostly less than 1. This result suggests that, on average, wages at opening establishments are higher than

14

U.S. BUREAU OF LABOR STATISTICS

MONTHLY LABOR REVIEW

wages at closing establishments, and wages at contracting establishments are higher than wages at expanding
establishments.
BED data by wage class show labor and wage dynamics, adding a tremendous value to the utility of the BED data
series. We highly recommend publishing these data periodically as a research or production series. The three
wage classes we introduced here are arbitrary and can be substituted or complemented either by more wage
classes or by specific wage levels or income targets. Whatever the wage grouping, the BED series by wage class
will provide a means to monitor whether employment growth is consistent with defined wage and income policy
objectives.

Appendix: Data
Table A-1. First-quarter average weekly wages, by direction of employment changes, seasonally adjusted,
1993–2000
All establishments

Opening

Expanding

Closing

Contracting

establishments

establishments

establishments

establishments

Median

Median

Median

Median

Year
Mean
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020

$454.1
466.8
493.6
519.1
543.1
581.7
594.4
648.3
677.3
754.7
694.3
712.0
743.7
797.2
834.0
853.2
837.8
835.6
872.9
899.2
911.8
934.9
969.7
982.6
1,034.3
1,079.8
1,097.8
1,132.8

Lower
quartile

Median

$173.6 $297.6
178.1 307.1
188.0 325.2
192.6 336.1
199.3 348.4
206.8 359.7
212.8 372.3
229.6 403.2
236.5 421.7
242.1 432.1
244.7 439.1
249.0 446.2
251.3 454.3
267.0 483.3
275.8 500.1
281.1 512.7
276.4 506.8
274.2 500.3
277.9
511.4
290.2 537.6
291.3 543.1
296.3 554.6
305.8 568.1
314.4 582.3
333.0 614.1
342.2 632.0
354.7 650.9
369.0 673.4

Upper
quartile
$493.8
512.1
543.3
563.1
585.7
608.1
630.3
686.2
716.2
731.5
745.4
756.7
773.9
829.8
864.4
890.0
886.3
879.7
907.8
955.3
970.4
993.7
1,019.1
1,035.2
1,097.7
1,132.7
1,164.9
1,193.2

$242.0
252.0
258.2
267.3
285.2
297.0
311.0
364.1
367.6
361.0
355.2
371.7
372.9
405.9
416.7
422.8
393.0
405.9
413.3
380.6
475.7
474.8
493.6
507.4
538.1
555.8
573.0
572.9

See footnotes at end of table.

15

$297.3
307.2
325.3
335.7
347.1
361.8
369.1
402.9
417.6
419.8
428.7
442.1
452.0
488.0
501.7
515.0
479.7
494.5
512.3
534.9
540.3
547.7
560.2
568.8
607.7
630.7
649.1
683.5

$207.6
215.8
224.5
232.5
242.8
259.7
277.0
296.0
308.3
323.5
320.7
329.0
343.9
362.5
374.2
379.9
385.9
382.7
385.6
436.0
462.5
466.4
498.0
545.5
535.6
555.0
594.0
507.9

$303.1
310.5
329.6
341.2
352.0
362.1
378.4
410.0
432.2
446.5
447.4
455.3
458.7
493.8
510.3
522.8
539.4
509.1
512.9
547.5
550.2
555.6
569.4
581.9
613.0
630.7
651.8
667.4

U.S. BUREAU OF LABOR STATISTICS

MONTHLY LABOR REVIEW

Source: U.S. Bureau of Labor Statistics.

Table A-2. Private sector gross job gains and losses by wage class, seasonally adjusted, first quarter 2019
to first quarter 2020 (in thousands)
Three
months
ended
Mar 2019

Jun 2019

Sep 2019

Dec 2019

Mar 2020

Gross job gains
Wage
class
Low
Medium
High
All
Low
Medium
High
All
Low
Medium
High
All
Low
Medium
High
All
Low
Medium
High
All

Gross job losses

Net
change Total
294
126
127
547
196
-118
32
110
158
-96
-27
35
483
271
72
826
-40
-587
-37
-663

1,864
3,898
1,690
7,452
1,907
3,985
1,750
7,642
1,852
3,893
1,667
7,412
2,009
4,120
1,734
7,863
1,726
3,610
1,607
6,943

Expanding

Opening

establishments

establishments

1,339
3,312
1,448
6,099
1,399
3,382
1,489
6,271
1,347
3,296
1,395
6,038
1,479
3,478
1,405
6,362
1,215
3,061
1,379
5,655

525
586
242
1,353
508
602
261
1,371
505
597
272
1,374
530
642
329
1,501
511
549
228
1,288

Total

Contracting

Closing

establishments

establishments

1,570
3,772
1,563
6,905
1,711
4,103
1,718
7,532
1,694
3,989
1,694
7,377
1,526
3,849
1,662
7,037
1,766
4,196
1,644
7,606

1,153
3,235
1,295
5,683
1,237
3,490
1,442
6,170
1,259
3,417
1,425
6,101
1,094
3,250
1,376
5,720
1,276
3,546
1,367
6,189

417
536
268
1,222
474
612
276
1,362
434
572
269
1,276
432
599
286
1,317
490
650
277
1,417

Source: U.S. Bureau of Labor Statistics.

SUGGESTED CITATION

Akbar Sadeghi and Kevin Cooksey, "Business employment dynamics by wage class," Monthly Labor Review, U.S.
Bureau of Labor Statistics, December 2021, https://doi.org/10.21916/mlr.2021.25.
NOTES
1 For discussions on the importance of job-flow analyses and employment dynamics, see Steven J. Davis and John Haltiwanger,
“Measuring gross worker and job flows,” Working Paper 5133 (Cambridge, MA: National Bureau of Economic Research, May 1995);
and Steven J. Davis, John C. Haltiwanger, and Scott Schuh, Job creation and destruction (Cambridge, MA: The MIT Press, 1996).
2 Ekaterina Jardim, Mark C. Long, Robert Plotnick, Emma van Inwegen, Jacob Vigdor, and Hilary Wething, “Minimum wage
increases, wages, and low-wage employment: evidence from Seattle,” Working Paper 23532 (Cambridge, MA: National Bureau of
Economic Research, June 2017).
3 For the Business Employment Dynamics (BED) program’s first release, see New quarterly data on business employment dynamics
from BLS, USDL 03-521 (U.S. Department of Labor, September 30, 2003), https://www.bls.gov/news.release/archives/
cewbd_09302003.pdf.

16

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4 For data and technical notes concerning the Quarterly Census of Employment and Wages, see County employment and wages—
first quarter 2021, USDL-21-1514 (U.S. Department of Labor, August 18, 2001), https://www.bls.gov/news.release/pdf/cewqtr.pdf.
5 James R. Spletzer, R. Jason Faberman, Akbar Sadeghi, David M. Talan, and Richard L. Clayton, “Business employment dynamics:
new data on gross job gains and losses,” Monthly Labor Review, April 2014, https://www.bls.gov/opub/mlr/2004/04/art3full.pdf.
6 For information on the definitions of the BED data elements, see “Business employment dynamics technical note” (U.S. Bureau of
Labor Statistics, last modified October 21, 2021), https://www.bls.gov/news.release/cewbd.tn.htm.
7 The seasonally adjusted data for total private employment in this article may not exactly equal official published BED data, mainly
because of our independent seasonal adjustment.
8 According to the National Bureau of Economic Research, the official arbiter of recessions in the United States, the recession lasted
from February to April 2020. See “U.S. business cycle expansions and contractions” (Cambridge, MA: National Bureau of Economic
Research, last updated July 19, 2021), https://www.nber.org/research/data/us-business-cycle-expansions-and-contractions.
9 The seasonally adjusted net change in employment for the BED series by wage class slightly differs from published BED data. This
difference is the net result of seasonal adjustment processes applied to the main BED data elements by wage class. Data that are not
seasonally adjusted match exactly the published values.

RELATED CONTENT

Related Articles
Establishment, firm, or enterprise: does the unit of analysis matter? Monthly Labor Review, November 2016.
High-employment-growth firms: defining and counting them, Monthly Labor Review, June 2013.
Business employment dynamics: annual tabulations, Monthly Labor Review, May 2009.
The births and deaths of business establishments in the United States, Monthly Labor Review, December 2008.
Employment dynamics: small and large firms over the business cycle, Monthly Labor Review, March 2007.
Business employment dynamics: new data on gross job gains and losses, Monthly Labor Review, April 2004.

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17

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December 2021

“Chain drift” in the Chained Consumer Price
Index: 1999–2017
This article employs circularity and unity tests, as well as
multilateral index comparisons, to measure the existence
and extent of “drift” in the Chained Consumer Price Index
for All Urban Consumers (C-CPI-U). It applies various
formulas to real data that were used to calculate the
Consumer Price Index during the period from December
1999 to December 2017. Overall, the findings show only
small amounts of chain drift in the C-CPI-U over the study
period.
This article presents the most comprehensive study to date
from the U.S. Bureau of Labor Statistics (BLS) to document
the extent of chain drift in the Chained Consumer Price
Index for All Urban Consumers (C-CPI-U). We apply
various formulas to real data collected from December
1999 through December 2017 that BLS used to calculate
the “official” Consumer Price Index (CPI) during that

Robert Cage
Cage.Rob@bls.gov
Robert Cage is the Assistant Commissioner of the
Division of Consumer Prices and Price Indexes,
Office of Prices and Living Conditions, U.S.
Bureau of Labor Statistics.

period.[1] Building on earlier work conducted by Joshua
Klick, we employ unity and circularity tests as well as
multilateral index comparisons to empirically assess the
impact of chain drift in the top-level C-CPI-U, U.S. city

Brendan Williams
Williams.Brendan@bls.gov

average, all-items index.[2] We also conduct circularity
tests of the C-CPI-U, U.S. city average, indexes at the

Brendan Williams is a senior economist in the
Office of Prices and Living Conditions, U.S.
Bureau of Labor Statistics.

subaggregate expenditure-class level. After comparing
multilateral and bilateral versions of this index, we found
that chain drift adds an estimated 0.11 percent to price
change in the national, all-items, chained Törnqvist index.
Estimates of chain drift that are based on unity and
circularity tests show similar levels of drift, but we find those
test results depend on the choice of base month and exhibit
strong seasonal patterns. We find more substantial drift in
certain indexes at the subaggregate level, but the majority
of expenditure-class indexes satisfy the circularity test

1

Jonathan D. Church
Church.Jonathan@bls.gov
Jonathan D. Church is an economist in the Office
of Prices and Living Conditions, U.S. Bureau of
Labor Statistics.

U.S. BUREAU OF LABOR STATISTICS

MONTHLY LABOR REVIEW

approximately. We also find that a monthly chained, Laspeyres index shows substantial drift.

Background
In August 2002, the CPI program began publishing the C-CPI-U. The C-CPI-U was implemented to address
concerns that the official Consumer Price Index for All Urban Consumers (CPI-U) suffered from upper-level
substitution bias, given that it is calculated using a Laspeyres (technically, a Lowe) formula. As described by
Robert Cage, John Greenlees, and Patrick Jackman in a 2003 conference paper, the C-CPI-U “employs a
superlative Törnqvist 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.”[3]
The Boskin Commission report submitted to the U.S. Senate Finance Committee in December 1996 estimated that
the official CPI-U was biased upward by 0.15 percentage points because of consumer substitution across item
categories (upper-level substitution) and that it was biased upward by 0.25 percentage points because of
consumer substitution within item categories (lower-level substitution).[4] Upper-level substitution refers to
substitution between these categories. Lower-level substitution refers to substitution within any of 243 item
categories at the lowest level of aggregation in the CPI classification scheme.
According to the Boskin Commission, “Substitution bias occurs because a fixed market basket fails to reflect the
fact that consumers substitute relatively less for more expensive goods when relative prices change.”[5] For the
CPI-U, U.S. city average, all-items index, an example of upper-level substitution bias is when consumers substitute
chicken for steak or beer for wine. An example of lower-level substitution bias is when consumers substitute low-fat
milk for whole milk. “Levels” refer to the placement of categories of goods and services within the CPI aggregation
structure.
The CPI-U is based on prices collected from monthly surveys—as well as alternative data sources for some item
categories—and on expenditures collected from the Consumer Expenditure Surveys administered monthly during
24-month intervals between biennial expenditure-weight updates. The CPI-U thus measures the changes in prices
for a market basket of goods and services that remains unchanged between biennial updates (specific items may
change because of item replacement and sample rotation), with quantity data captured implicitly via lagged
expenditures. Biennial updates also mean that price and quantity data are “chained” at each biennial “rebase”
using a Lowe formula. A geometric mean formula is used in the official CPI-U to address lower-level substitution
bias.[6]
In the C-CPI-U, price and quantity data are chained each month using a Törnqvist formula that calculates an index
incorporating both monthly price changes and monthly expenditure changes across item categories. Because of
the amount of time that is necessary to process expenditure data, C-CPI-U data are released first in preliminary
form. Three months later, they are released again in their first interim form. Six months later, they are released
again in their second interim form. Nine months later, they are released again in their third interim form. In these
three quarterly intervals following release in preliminary form, BLS uses a constant elasticity-of-substitution formula
for the initial estimates. Twelve months later, exactly a year after the preliminary release, a Törnqvist formula is
used to produce the final estimates.[7]

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By addressing potential substitution bias, the C-CPI-U is designed to be a closer approximation to a “true” cost-ofliving index (COLI). Given that the implicitly derived quantities that consumers purchase are taken to be the
optimal quantities on the basis of their income or wealth, COLIs, originally conceived by A. A. Konüs, are based on
the economic theory of consumer demand.[8] Under the standard framework, consumers maximize utility and
minimize cost, given a level of wealth or income. More specifically, under “Hicksian” demand, consumers minimize
the expenditures necessary to attain a given standard of living—that is, they seek to maximize their utility at the
minimal cost.[9] When relative prices change, the relative affordability of different goods and services changes,
which affects consumers’ standard of living. The COLI is designed to measure the compensation necessary to
afford the original standard of living. Consumer inflation is thus calculated as the ratio of minimum expenditures in
two periods necessary to achieve the same standard of living. In the following equation, which illustrates this ratio,
iIX(0,t)

is the index that represents consumer inflation over the set of i goods from period 0 to t, with the basket of

goods, and thus the standard of living, remaining constant; iP0 is the price of good i in period 0, iQ0 is the quantity
of good i in period 0, iPt is the price of good i in period t, and iQt is the quantity of good i in period t:

However, the choice between a chained or fixed-base index presents a tradeoff between representativeness and
transitivity. Indexes are “representative” when they accurately represent not only the price trend of a “market
basket” of goods and services but also the composition of the “market basket” as it changes over time. As the 2004
Consumer Price Index Manual explains,
The main problem with the use of fixed base Laspeyres indices is that the period 0 fixed basket of
commodities that is being priced out in period t can often be quite different from the period t basket. Thus, if
there are systematic trends in at least some of the prices and quantities in the index basket, the fixed base
Laspeyres price index PL(p0, pt, q0, qt) can be quite different from the corresponding fixed base Paasche price
index, PP(p0, pt, q0, qt). This means that both indices are likely to be an inadequate representation of the
movement in average prices over the time period under consideration.[10]
Christian G. Ehemann defines an index number formula as transitive if chaining from t = 1 to t = 2 and then from t
= 2 to t = 3 yields the same index value for the index at t = 3 as the direct index from t = 1 to t = 3: I(p0, p1, q0, q1) *
I(p1, p2, q1, q2) * I(p2, p3, q2, q3) = I(p0, p3, q0, q3). If this condition is not met, the index is nontransitive. The
monthly chained Törnqvist formula used for the C-CPI-U is nontransitive and, as a result, is subject to drift.[11]
Frequent weight updates may lead to increased drift in nontransitive indexes. The official index—the Consumer
Price Index for All Urban Consumers (CPI-U), U.S. city average, all items—is a chained index in the sense that
weights are updated biennially, but it also is a fixed-base index when it is calculated between expenditure-weight
updates. The C-CPI-U uses monthly chaining with monthly weight updates (when finalized) and, in principle, may
be at greater risk of measurable chain drift.
We now clarify some of the terminology associated with price-index methodology. A price index can be either fixed
base or chained, and it can be either bilateral or multilateral. (In this article, we use the terms “fixed base” and
“direct” interchangeably.) We rely on the Consumer Price Index Manual for precise definitions of these terms.
According to the Manual, the word “bilateral” refers to the assumption that a function P, or a price index number
formula, P(p0, p1, q0, q1), depends only on the data pertaining to the two situations or periods being compared. For
example, P is regarded as a function of the two sets of price and quantity vectors, p0, p1, q0, q1, “that are to be
3

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aggregated into a single number that summarizes the overall change in the n price ratios p1(1)/p0(1), . . . , p1(n)/
p0(n).”[12] “Multilateral index number theory,” however, “refers to the case where there are more than two
situations whose prices and quantities need to be aggregated.”[13]
On the distinction between a fixed-base index and a chained index, the Manual explains that the “chain system
measures the change in prices going from one period to a subsequent period using a bilateral index number
formula involving the prices and quantities pertaining to the two adjacent periods. These one-period rates of
change (the links in the chain) are then cumulated to yield the relative levels of prices over the entire period under
consideration.” Consider, for example, a bilateral price index P. The index is “chained” if index calculation
generates the following sequence:[14]
1, P(p0, p1, q0, q1), P(p1, p2, q1, q2).
Under a fixed-base system, however, the bilateral index number formula P “simply computes the level of prices in
period t relative to the base period 0 as P(p0, pt, q0, qt).”[15] As such, the fixed-base system generates the
following sequence of price levels for periods 0, 1, and 2:
1, P(p0, p1, q0, q1), P(p0, p2, q0, q2).
Empirically, chain drift is often defined as the difference between the chained and fixed-base versions of a price
index. Chain drift can occur when expenditure-share weight updates lag short-term price oscillations, distorting
trend reversion and leading to nontransitivity in a price index. However, several factors can lead to divergence,
including the representativity of the market basket of goods and services in the base period. Our analysis also
indicates that choice of base month and seasonal patterns may affect the amount of drift. Gregory Kurtzon refers
to divergence resulting from consumer substitution as “good” drift. Divergence resulting from nontransitivity, which
can be analyzed with the unity test, is, unambiguously, “bad” drift.[16]

Chaining: theory and practice
In addition to the choice of index number formula, chaining an index addresses substitution bias by providing an
approach for incorporating changes in quantity over time. According to F. G. Forsythe and R. F. Fowler, Francois
Divisia “put forward the new concept of an index of prices based specifically on the assumption that an index of
price changes over a period of time 0 to t should depend not only on prices and quantities at 0 and t but also on
the movement of prices and quantities throughout the interval 0 to t. In other words, the index should depend on
the path and take account of all the data relating to prices and quantities in the interval.”[17] This idea has its
earliest roots in the 19th-century work of Julius Lehr and Alfred Marshall, the latter introducing the principle of
chaining as a way of making index formulas more representative of ongoing changes in economic activity.[18] As
Forsythe and Fowler write, “Marshall was concerned only with the practical problem of allowing for the introduction
of new commodities into an index of prices which he thought would be greatly facilitated if the weights were
changed every year and the successive yearly indices linked or chained together by simple multiplication.”[19]
In principle, a Divisia index perfectly captures changes in price and quantity as they occur. Because a continuous
time index is not feasible, numerous discrete time-index formulas have been developed, such as the Lowe and
Törnqvist formulas used for the official CPI-U and the C-CPI-U, respectively. These formulas are weighted versions

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of arithmetic and geometric means. The economic theory of consumer demand provides the theoretical connection
between a Divisia index and a cost-of-living index.[20]
Forsyth and Fowler noted a tradeoff between maintaining base-period weights and using chained indexes.
Chained indexes have the potential for drift but represent recent expenditure patterns. Fixed-base indexes have no
drift, but the base-period consumption basket becomes less representative as consumption patterns change.
Forsyth and Fowler argued that the benefits of representativity generally outweigh the relatively small drift in a
Fisher index.[21]
The C-CPI-U combines a Törnqvist formula with monthly chaining to provide a measure of consumer inflation that
reflects monthly purchases in response to monthly changes in price. The C-CPI-U is not only a measure of
consumer inflation but a measure of revealed preference. One major problem that can emerge with chaining,
however, is a violation of transitivity, which is one of several axiomatic properties that are desirable for indexes to
have.[22] An index is transitive if long-term price change calculated with updated quantity data in each incremental
period is equal to long-term price change without frequent updating from a reference period to a comparison
period.
More precisely, Ehemann defines an index number formula as “transitive if chaining from t = 1 to t = 2 and then
from t = 2 to t = 3 gives the same index value for the index at t = 3 as the direct index from t = 1 to t = 3”:[23]
I(p1, p2, q1, q2) * I(p2, p3, q2, q3) = I(p1, p3, q1, q3).
A violation of the transitivity axiom could indicate that chain drift is a problem with the index choice. If indexes are
transitive, they do not exhibit chain drift. Conversely, indexes that are not transitive exhibit chain drift. Indexes that
exhibit chain drift diverge from their “true” long-term trend. As a result, a chained index makes an index more
representative of market-basket composition (what consumers are purchasing), but often at the expense of
providing an inaccurate measure of long-term inflation. Chaining can thus involve a tradeoff between
representativity and transitivity.[24] For any price index, the central question is whether, in any given case, the
benefits of representativity are outweighed by the cost of nontransitivity (i.e., chain drift).
Bohdan J. Szulc identified “bounce” behavior in prices resulting from such factors as seasonality or price wars as a
major source of drift. “Bounce” is more likely to cause drift if a “peak” or “trough” diverges from the long-term
trend.[25] Thus, following Kurtzon, we can imagine the prices of two goods with equal expenditure shares
“bouncing” between two periods, with the price of each good either $1 or $2 in every period, generating a price
relative of 2 or 1/2. In this situation, there is no long-term inflation, but, as Kurtzon states, “this index relative would
give an inflation rate of 1/2(2 + 1/2) = 1.25, or 25 [percent] inflation every period.”[26] Price oscillation may reflect
trend reversion as competition prevents sellers from charging prices that deviate from their long-term trends as
consumer substitution “puts downward pressure on prices that are comparatively high, and it may also put upward
pressure on prices that are unusually low by making them attract high sales.”[27] Lorraine Ivancic, Kevin J. Fox,
and W. Erwin Diewert provide a similar explanation of drift:
As a result, it is not necessarily the case that prices and quantities in adjacent periods are more similar than
those in periods which are not adjacent when subannual data is used. In particular, when an item goes off sale
and prices return to their “regular” price, we would expect that the use of a chained superlative index would
simply (more or less exactly) reverse the previous downward movement in the index and take us back to the
“regular” price level. However, in practice this may not happen because when an item comes off sale,
5

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consumers are likely to purchase less than the “average” quantity of that item for some period of time until
their inventories of the item have been depleted. It is only over time that the quantities sold will gradually
recover to their pre-sale levels [emphasis added]. If prices do not change over the post-sale period, all
reasonable indexes will show no price change over these “regular” price periods. Thus, under these conditions
(i.e., where sales are apparent), chained superlative indexes will tend to have a downward drift when
compared to their fixed base counterparts.[28]

Methodology: test for chain drift
Because chaining has the potential to make a price index more representative of consumption behavior and
marketplace activity, it is of considerable interest to determine if the benefits of a more representative index are
outweighed by any “drift” that causes the index to diverge from long-term price trends. In this study, we used
several tests to determine the extent to which the C-CPI-U exhibits undesirable drift. We first address the second
stage of index aggregation in which component indexes—which are constructed from aggregations of price
observations pertaining to specific geographic areas and item categories—are combined. We then focus on
subaggregate indexes at the expenditure-class level. In the CPI aggregation structure, there are 70 expenditureclass indexes, which are one level of aggregation above the “lower level” 243 item-level indexes, which, combined
with 32 index areas, form 7,776 item-area, lower-level, index-area “cells,” the building blocks of CPI index
construction. Item-area component indexes use fixed-weight formulas, but, like all bilateral indexes, they are
effectively chained at each weight update. Every time the relative weights of price quotes change within a cell,
which can occur because of subsampling and partial cell sample rotation, drift can occur.

Unity test
C. M. Walsh introduced the unity test as one method for detecting drift. As the Consumer Price Index Manual
explains, this test uses “the bilateral index formula P(p0, p1, q0, q1) to calculate the change in prices going from
period 0 to 1.” It then uses “the same formula evaluated at the data corresponding to periods 1 and 2, P(p1, p2, q1,
q2), to calculate the change in prices going from period 1 to 2” and then uses “P(pT−1, pT, qT−1, qT) to calculate the
change in prices going from period T – 1 to T.” The unity test then “introduce[s] an artificial period T + 1 that has
exactly the price and quantity of the initial period 0 and use[s] P(pT, p0, qT, q0) to calculate the change in prices
going from period T to 0.” In the last step, “multiply all of these indices together.”[29]
The unity test thus chains index formulas from point 0 to point T and sets the price and quantity in period T + 1
equal to the price and quantity observed in period 0. In the next step, I(p0, pT, q0, qT) is used to calculate the
change in prices from period T to period 0. In the final step, multiply all of the chained indexes together. According
to the Consumer Price Index Manual, if the result is an index value equal to the index value in period 0, “we end up
where we started, [and] the product of all of these indices should ideally be one.”[30] Mathematically, we have the
following:
DriftUnity,t = I(p0, p1, q0, q1) * I(p1, p2, q1, q2) *…* I(pt, p0, qt, q0).
This procedure computes the chained price index until month t and then appends the initial set of prices and
quantities (p0, q0) to the end of the series. Because the final period is the same as the first, a fully transitive index
should show no change. We produced estimates of drift over the entire series and, iteratively, sequentially
conducted this test for each period t after the base period, setting the terminal month indexes and weights equal to

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the exact values from the starting month iteratively from December 1999 to November 2017. Thus, in the first run
of the test, we conducted an iterative unity test with December 1999 as the starting point, and each successive
month up to November 2017, with December 2017 as the terminal month. Then, we ran another iterative unity test,
with January 2000 as the starting point, and each successive month up to November 2017, with December 2017
as the terminal month. We repeated this process through November 2017. We also produced unity tests on
indexes that are based on bounded price relatives. Monthly price changes in these indexes are capped at 95percent declines and 2,000-percent increases.

Circularity test
The unity test, or what Diewert called the “multiperiod identity test,” is a special case of the circularity test.[31] In
both cases, the aim is to test whether transitivity holds. If transitivity holds, index formulas generate the same
measurement of long-term price change. If it does not, fixed-base formulas and chained-index formulas diverge.
Or, stated differently, chained indexes drift. Note that the Consumer Price Index Manual recommends the
circularity and unity tests not as measures for deciding whether to use fixed-base or chained indexes, but as
measures of “how ‘good’ a particular index number formula is.”[32] Drift is also a reason why the Manual
recommends chaining for series that have smooth trends. As the Manual notes, Alterman, Diewert, and Feenstra
“show that if the logarithmic price ratios ln(
) trend linearly with time t and the expenditures shares
trend linearly with time, then the Törnqvist index PT will satisfy the circularity test exactly.”[33]

also

The circularity test originates in the work of Harald Westergaard and Irving Fisher and helps “to determine if there
are index number formulae that give the same answer when either the fixed base or chain system is used.”[34] If
an index formula yields the same calculation of long-term price change regardless of whether a fixed base or
chaining is used, it passes the circularity test. That is, an index number formula that passes the circularity test is
transitive. No drift will be detected in the amount of price change calculated by the chained index formula. As
explained by Ehemann, “The testing of an index number formula for transitivity by determining whether the chained
and direct calculation of the index value are equal is known as a circularity test.”[35] Mathematically, the test is
represented as follows:
I(p0, p1, q0, q1) * I(p1, p2, q1, q2) = I(p0, p2, q0, q2).[36]
Here, we express drift as the ratio of the chained index relative to the fixed-base index relative in period t:
.
For the Törnqvist price index, DriftCircularity,t and DriftUnity,t are equivalent. Multiplying the Törnqvist chained index
by an additional term returning to the base period is equivalent to dividing the chained Törnqvist index by the fixedbase Törnqvist index, because Torn(pt, p0, qt, q0) = Torn( p0, pt, q0, qt)–1.

Multilateral index comparisons
Ivancic et al. developed a rolling-window time version of the Gini, Eltetö, Köves, and Szulc (GEKS) formula
originally used for interarea price comparisons.[37] The full GEKS index is transitive. However, the fully transitive
GEKS index must be reestimated every month, in every period, and it produces revisions in previous period
estimates. The rolling-window GEKS formula produces a current-period estimate without revising prior months.

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Although not fully transitive, the rolling-window GEKS formula produces a chained index with attenuated drift.
Once the initial GEKS index is estimated, it is updated through a splicing method. We focus on the results of a
mean splice. Ivancic et al. originally used a Fisher formula to make the bilateral comparisons in each element of
the GEKS index. Here, as recommended by Diewert and Fox, we use a Törnqvist index in place of the Fisher
index to produce GEKS-Törnqvist indexes, also referred to as Caves-Christensen-Diewert-Inklaar (CCDI)
indexes.[38] This also allows us to make a more direct comparison with the C-CPI-U because that index is based
on a Törnqvist formula:
.
For these tests, a value equal to 1 indicates no chain drift, a value less than 1 indicates downward drift, and a
value greater than 1 indicates upward drift.

Data
The data used for this article consist of monthly item-area CPI indexes and cost weights from December 1999 to
December 2017. This period corresponds to the interval between the 1998 and 2018 geographic area sample
revisions. In January 2018, the CPI program implemented a geographic area sample revision. As part of this
revision, the number of primary sampling units (PSUs) declined from 87 to 75 and the number of index areas for
purposes of index construction declined from 38 to 32. Meanwhile, there are now 243 item categories. The present
study avoids the complications associated with reconciling the new geographic design with the geographic area
sample that prevailed during the 1999–2017 period. Although changes in item structure occurred during this
period, the geographic area sample remained stable in 38 index areas and 87 PSUs, avoiding complications that
arise from changes in the item-area index aggregation structure as a result of a new geographic area sample.
Although we briefly considered conducting an analysis across the geographic revision implemented in January
2018, we only had 1.5 years of data beyond 2018 at the time of this analysis; the change from 211 to 243 lowerlevel item categories in the CPI aggregation structure would have added complications that are unrelated to chain
drift.
The published C-CPI-U accounts for these structural changes, while the chained Törnqvist and other indexes
discussed in this article use a dataset that is based on a harmonized structure, so there are some differences
between the published C-CPI-U indexes and the research results presented here. Nevertheless, we view the
chained Törnqvist index as analogous to the C-CPI-U; the chained Törnqvist index displays a close relationship to
the C-CPI-U. The C-CPI-U shows a 39.5-percent increase from December 1999 to December 2017, while the
chained Törnqvist index shows a 39.9-percent increase (an annualized difference of just 0.02 percent).

Results
On the basis of the CCDI test, we find that the all-items monthly chained Törnqvist index displays a small amount
of drift: 0.11 percent annually, for a total of 2.1 percent over the 18-year period analyzed in this study. Results from
unity and circularity tests show similar levels of drift but vary substantially, depending on the choice of base and
end month. Iterative comparisons of different subperiods show that the drift, as measured by the circularity and
unity tests, is partly related to seasonality. Multilateral indexes also show small amounts of drift in the monthly
chained Törnqvist index. A monthly chained Laspeyres index shows large amounts of upward drift. We show that

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drift declines with more infrequent weight updates for both Törnqvist and Laspeyres indexes. At the expenditureclass level, some categories show substantial drift. Unity and circularity tests have nearly equivalent results, with
some differences because of rounding. The CCDI test helps mitigate drift in those indexes with large amounts of
drift.
Chart 1 summarizes the estimated chain drift in the chained Törnqvist index based on the CCDI and circularity
tests.

The CCDI test shows upward chain drift across the entire time span of 1 to 2 percent, depending on the month.
The circularity test is quite sensitive to the choice of base month. Of the first 12 months in the series, using
December 1999 as a base month implies the most drift, nearly 5.0 percent, while using January 2000 implies the
least drift, at 0.2 percent.

Unity test

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We relied on a dataset of continuous elementary item-area monthly indexes and weights from December 1999 to
December 2017, and we ran iterative calculations of the unity test over the same 1999–2017 period. We
systematically varied all base periods, starting with December 1999, and all end periods, ending with December
2017, until we obtained drift estimates based on the unity test for all chronological combinations of base and end
months.
Charts 2–5 show the results of this iterative, sequentially run unity test when we used bounded Laspeyres and
bounded Törnqvist formulas. Charts 2 and 3 show the per annum percent change in the monthly chained index
value from unity (the beginning month in which the index equals 100) to the ending month in each iteration, up to
the terminal month of December 2017. Chart 2 displays the results when the Laspeyres formula is used.

Chart 3 displays the results when the Törnqvist formula is used.

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Charts 4 and 5 show the cumulative percent change in the monthly chained index value from unity (the beginning
month in which the index equals 100) to the ending month in each iteration, up to the terminal month of December
2017. Chart 4 displays the results when the Laspeyres formula is used.

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Chart 5 displays the results when the Törnqvist formula is used.

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Chart 6 shows the accumulated amount of drift in the Laspeyres index between each pair of base and end months
in the form of a heatmap.

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Chart 7 shows the accumulated amount of drift in the chained Törnqvist index.

The chained Laspeyres index shows substantial upward drift in almost all periods, while the chained Törnqvist
index shows a seasonal pattern with strong upward drift when December is the base month and slight downward
drift for most of the rest of the year. Several months also stand out as having stronger effects on drift, especially
when December 1999 is the base month and the end months are in late 2008.
In these charts, we observe chain drift when the Laspeyres formula is used (charts 2 and 4) but not when the
Törnqvist formula is used (charts 3 and 5). Table 1 shows summary statistics for the per annum percent changes
from unity for the Laspeyres and Törnqvist formulas.
Table 1. Summary statistics for the per annum percent changes from unity for the Laspeyres and
Törnqvist formulas, 1999–2017
Statistic

Laspeyres

Mean
Median
Standard deviation

Törnqvist
1.00
1.00
0.43

0.05
0.00
0.37

Source: U.S. Bureau of Labor Statistics.

Table 2 shows the percentage of months in the time series in which we observe upward or downward annual drift
(on a cumulative basis).

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Table 2. Percentage of months in which upward or downward per annum drift was observed, 1999–2017
Drift

Laspeyres

Upward
Downward

Törnqvist
99.0
1.0

50.4
49.6

Source: U.S. Bureau of Labor Statistics.

From the preceding charts and tables, we conclude that chain drift in a monthly chained Törnqvist index at the U.S.
city average, all-items level, is minimal, compared with the drift in a monthly chained Laspeyres index at the same
level, which is sizeable.

Circularity test
We conducted a circularity test by comparing chained and direct versions of the CPI-U, U.S. city average, all-items
index, using both a Laspeyres formula and a Törnqvist formula. The relevant comparisons are between the
chained and direct Laspeyres index and between the chained and direct Törnqvist index. In both cases, we
observed visible drift. The CPI-U, U.S. city average, all-items index calculated using a Laspeyres formula with
monthly chaining exhibits upward drift relative to the same index directly calculated using a Laspeyres formula
(without chaining). When the CPI-U, U.S. city average, all-items index is calculated using a Törnqvist formula with
monthly chaining, it also exhibits upward drift relative to the same index directly calculated using a Törnqvist
formula. The Törnqvist finding conflicts with a previous empirical analysis by Klick that showed lower drift for the
chained CPI-U, U.S. city average, all-items index using a Törnqvist formula.[39]
The preceding analysis suggests that the existence and extent of chain drift may depend on the choice of base
period. We investigated this issue by changing the base period to 2000 and conducting additional circularity tests.
We set calendar year 2000 as the base period in the direct Törnqvist calculations. Chart 8 shows the results when
we compared the direct Törnqvist index with the monthly chained version, the former rebased so that the average
annual index in 2000 was equal to 100, and we observed no significant drift.

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Törnqvist indexes with quarterly and annual chaining behave similarly to a chained index with a 12-month base.
(See chart 9.) This implies that most of the drift we see in the monthly chained Törnqvist index is due to the base
period rather than the effects of chaining in intermediate periods.

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Multilateral index comparisons
The CCDI index provides an alternative measure of price change. In general, a multilateral chained index is more
representative than a direct index while maintaining transitivity and thereby avoiding chain drift. The CCDI index
also avoids the base-period sensitivity issue. We used the IndexNumR package in R to test various versions of the
CCDI index by varying the method of splicing and changing window length.[40]
Chart 10 displays the chained and direct Törnqvist indexes discussed previously compared with a full-period CCDI
index and the results of applying various extension methods to extend a CCDI index.

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Previous research has explored the relative merits of various extension methods and then the length of a rolling
window needed to mitigate drift from multilateral indexes. We find that three common extension methods (mean,
movement, and window splices) and three window lengths (13, 24, and 36 months) all produce indexes similar to
the monthly chained Törnqvist index. Although the direct Törnqvist and CCDI indexes increase at a slightly lower
rate, the remaining indexes lie on top of each other. The results show that methodological choices for extension
methods have little bearing for the index at the top level of aggregation.[41]
The CCDI analysis produces similar results except that the direct Törnqvist index rises slightly more slowly than
the monthly chained Törnqvist index. This difference might be driven more by sensitivity to the base period
(December 1999) than by chain drift. In chart 11, we compare these indexes with a Törnqvist direct index, iterating
over the first 12 months and taking each in turn as a fixed base. The index that uses December 1999 as the base
period stands out as rising more slowly than indexes that use other months as the base period. The rolling CCDI
index also generally falls within the range of the other months.

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Circularity test at the expenditure-class level
Although chain drift at the highest aggregate level appears to be minor and mostly explained by base-period
sensitivity and seasonality, we find that certain subaggregate indexes display relatively large amounts of drift.
Overall, our analysis shows that a minority of the 70 expenditure-class CPIs satisfy the circularity test exactly, but
most expenditure-class indexes satisfy the circularity test approximately.
During the period from December 1999 to December 2017, the CPI aggregation structure consisted of 8 major
groups, 70 expenditure classes, 211 item categories, and various intermediate-level indexes, such as the index for
all items less food and energy. We conducted a circularity test for each of the 70 expenditure classes. Using
December 1999 as the base month, we used a Törnqvist formula to calculate index values for each month from
December 1999 to December 2017. First, we performed the calculation with monthly chaining. Second, we
performed the calculation directly with December 1999 as the base month. To conduct the circularity test, we

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calculated a relative that divides the December 2017 indexes calculated with a chained Törnqvist formula by the
December 2017 indexes calculated with a direct Törnqvist formula.
If the relative between the chained and direct Törnqvist indexes is equal to 1.00, the monthly indexes are the same
—that is, the chained Törnqvist indexes are perfectly transitive and our circularity tests indicate no chain drift. If the
relative is less than 1.00, then there is downward drift; if the relative is greater than 1.00, then there is upward
drift. Our results show that 27 percent (or 19 of 70) of the expenditure-class indexes had a relative of
approximately 1.00. Chart 12 shows the results of the circularity tests.

The results of our circularity tests at the expenditure-class level were essentially equivalent to those of our unity
tests. The CCDI tests implied levels of drift similar to those of the other tests. (Appendix table A-1 shows the test
results for each of the 70 expenditure classes.)
According to the Consumer Price Index Manual, “it is not useful to ask that the price index P satisfy the circularity
test exactly.” It is, however, “of some interest to find index number formulae that satisfy the circularity test to some
degree of approximation, since the use of such an index number formula will lead to measures of aggregate price
change that are more or less the same no matter whether we use the chain or fixed base systems.”[42] As a result,
we calculated the percentage of expenditure-class indexes whose chained-to-direct relatives fell within a band of
0.95 and 1.05, as well as within a band of 0.90 and 1.10. We found that 74 percent of the expenditure-

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class indexes had relatives between 0.95 and 1.05, while 89 percent had relatives between 0.90 and 1.10.
Tables 3 and 4 summarize these results.
Table 3. Percentage of expenditure-class indexes that fall within the 0.95-to-1.05 relative bands, 1999–2017
Characteristic

Percent

Relative between 0.95 and 1.05
Relative outside the 0.95-to-1.05 band

74.0
24.0

Source: U.S. Bureau of Labor Statistics.

Table 4. Percentage of expenditure-class indexes that fall within the 0.90-to-1.10 relative bands, 1999–2017
Characteristic

Percent

Relative between 0.90 and 1.10
Relative outside the 0.90-to-1.10 band

89.0
11.0

Source: U.S. Bureau of Labor Statistics.

The results indicate that 11 percent of the expenditure-class indexes exhibited chain drift greater than 10 percent
over the 18-year period of the study. Extreme outliers are shown in table 5.
Table 5. Expenditure-class indexes that exhibited chain drift greater than 10 percent, 1999–2017
Expenditure class

Chained-to-direct relative in December 2017

Video and audio
Information technology, hardware, and services
Jewelry and watches
Photography
Other recreational goods
Fresh fruits

1.633
1.249
1.203
1.183
1.171
0.556

Note: Other recreational goods include toys; sewing machines, fabric, and supplies; music instruments and accessories; and unsampled recreation
commodities.
Source: U.S. Bureau of Labor Statistics.

The practical implication of these relatives is that chained and direct Törnqvist calculations of long-term price
change show significant divergence. In the most extreme case, for fresh fruits, the chained Törnqvist formula
shows long-term deflation of 24 percent, while the direct Törnqvist formula shows long-term inflation of 37 percent.
Table 6 summarizes the extent of divergence for the categories shown in table 5, with the rate of inflation (or
deflation) calculated from December 1999 to December 2017.

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Table 6. Extent of divergence for expenditure-class categories that exhibited drift of greater than 10
percent
Chained Törnqvist formula

Direct Törnqvist formula

Expenditure class
Percent divergenceInflation or deflation?Percent divergenceInflation or deflation?
Information technology, hardware, and
services
Other recreational goods
Photography
Fresh fruits
Jewelry and watches
Video and audio

79.0

Deflation

83.0

Deflation

61.0
29.0
24.0
21.0
19.0

Deflation
Deflation
Deflation
Inflation
Deflation

67.0
40.0
37.0
1.0
51.0

Deflation
Deflation
Inflation
Inflation
Deflation

Note: Other recreational goods include toys; sewing machines, fabric, and supplies; music instruments and accessories; and unsampled recreation
commodities.
Source: U.S. Bureau of Labor Statistics.

According to the analysis in this section, most expenditure-class indexes satisfy the circularity text to a reasonable
degree. However, outliers exist, which raises doubts about using these indexes as measures of inflation for items
in those index categories. The most glaring outlier was the index for fresh fruits, which shows a 24-percent
deflation rate with a chained Törnqvist index but a 37-percent inflation rate with a direct Törnqvist index. As we
suggested earlier in this article, drift often depends on the timing of price change in relation to the weight of an
index. As the Consumer Price Index Manual states, “the more prices and quantities are subject to large
fluctuations (rather than smooth trends), the less the correspondence” between a fixed-base and a chained
index.[43]
To investigate the issue further, we conducted a multilateral analysis on the fresh fruits aggregate index. The
multilateral indexes helped address chain drift in this index. Chart 13 shows the difference between the chained
and direct Törnqvist indexes, and the rolling multilateral indexes are close to the direct index. We explored three
“splicing” options to update the CCDI index on a monthly basis. Of these, the mean splice was closest to the direct
index, while the movement and window splices were almost identical to each other.

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Table 7 shows the correlation matrix between the different CCDI indexes produced at the expenditure-class level.
Table 7. Correlation matrix between the different CCDI indexes produced at the expenditure-class level,
1999–2017
Item
Mean, 13 months
Window splice, 13 months
Movement, 13 months
Full

Mean,

Window splice,

Movement,

13 months

13 months

13 months

1.0000

Full

[1]

0.9999
1.0000

[1]

[1]

0.9999
1.0000
1.0000

[1]

[1]

[1]

0.9988
0.9986
0.9987
1.0000

[1] Not applicable.

Note: CCDI = Caves-Christensen-Diewert-Inklaar, the names of the people for whom the CCDI index is named: Douglas W. Caves, Laurits R. Christensen, W.
Erwin Diewert, and Robert Inklaar.
Source: U.S. Bureau of Labor Statistics.

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All of the variations of the CCDI indexes produced similar results. The mean splice was closest to the full-period
CCDI index, while the movement and window splice had a slightly closer relationship to each other.

Geographic analysis
In this section, we discuss the results of the circularity tests and the CCDI tests that we conducted on the CPI-U,
all-items index, by geographic area. During the period from December 1999 to December 2017, the CPI program
produced indexes for 4 census regions, 3 population size classes, and 27 metropolitan areas, in addition to the
top-level index for the United States as a whole. Charts 14 and 15 show the results of the circularity tests and the
CCDI tests, respectively.

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Most area indexes exhibit cumulative drift of less than 10 percent over the 18-year period. In the circularity tests,
only Baltimore, Portland-Salem, San Diego, and the regional population size class consisting of small (size D)
Western cities exhibit upward cumulative drift of 10 percent or more (with Washington, DC, showing about 9.5
percent). The index for St. Louis exhibits downward cumulative drift of about 3 percent in the circularity tests. (See
chart 14.). On the other hand, the CCDI tests show less drift, with an average of about 1 percent over the
period. (See chart 15.) Arguably, the CCDI method keeps more of the “good” drift of representing consumption
pattern changes while still eliminating the “bad” drift resulting from nontransitivity.
We can expect to see some level of drift for geographic areas because their sample sizes are smaller. Drift results
from short-term price oscillations that distort long-term trends because expenditure-share weights are often
inversely proportional to price levels and thus underweight price declines and overweight price increases. Because
price oscillations at the microdata level are more likely to affect elementary indexes in areas with smaller samples,
chain drift is also more likely.

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The circularity tests indicate upward drift at the area level. Other conditions may lead to chain drift in addition to
pendular quantities in response to sales. Ludwig von Auer showed that delayed quantity responses to price
change, which he calls “sticky quantities,” lead to upward drift.[44] Recent research on explanations for chain drift
have focused on lower-level indexes and consumer behavior. It is unclear whether we are seeing drift at the area
level because small sample sizes allow microlevel effects to impact area-level aggregates or if drift is the result of
other factors at the aggregate level, such as stochastic variation, seasonality, or the process of weight
construction. Table 8 shows the results of the CCDI and circularity tests, by geographic area.
Table 8. CCDI and circularity test results, by Consumer Price Index area, 1999–2017
Area

CCDI test rank

Portland-Salem, OR-WA
West–size class D
Tampa-St. Petersburg-Clearwater, FL
South–size class D
Kansas City, MO-KS
San Diego, CA
Boston-Brockton-Nashua, MA-NH-ME-CT
Honolulu, HI
Midwest–size class D
Minneapolis-St. Paul, MN-WI
Washington, DC-MD-VA-WV
Baltimore, MD
Northeast urban: size class B/C
Denver-Boulder-Greeley, CO
New York-Connecticut Suburbs
New Jersey Suburbs
South–size class B/C
Los Angeles suburbs, CA
Anchorage, AK
Midwest–size class B/C
Milwaukee-Racine, WI
Houston-Galveston-Brazoria, TX
Dallas-Fort Worth, TX
West–size class B/C
Chicago-Gary-Kenosha, IL-IN-WI
Phoenix-Mesa, AZ
Philadelphia-Wilmington-Atlantic City, PA-NJ-DE-MD
Los Angeles-Orange, CA
Cincinnati-Hamilton, OH-KY-IN
Cleveland-Akron, OH
Miami-Fort Lauderdale, FL
Detroit-Ann Arbor-Flint, MI
Pittsburgh, PA
San Francisco-Oakland-San Jose, CA
Seattle-Tacoma-Bremerton, WA
Atlanta, GA

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36

See footnotes at end of table.

26

CCDI test
(mean 13)
1.0395
1.0331
1.0251
1.0190
1.0162
1.0153
1.0150
1.0142
1.0139
1.0116
1.0115
1.0096
1.0077
1.0070
1.0066
1.0059
0.9999
0.9986
0.9985
0.9982
0.9964
0.9956
0.9955
0.9954
0.9951
0.9940
0.9914
0.9884
0.9873
0.9852
0.9849
0.9817
0.9806
0.9798
0.9762
0.9754

Circularity test
1.1084
1.1188
1.0375
1.0473
1.0507
1.1002
1.0562
1.0180
1.0504
1.0632
1.0948
1.1127
1.0491
1.0666
1.0445
1.0578
1.0521
1.0280
1.0745
1.0428
1.0253
1.0630
1.0693
1.0301
1.0415
1.0424
1.0579
1.0647
1.0083
1.0290
1.0398
1.0552
1.0480
1.0343
1.0222
1.0241

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Table 8. CCDI and circularity test results, by Consumer Price Index area, 1999–2017
Area

CCDI test rank

New York, NY
St. Louis, MO-IL

37
38

CCDI test
(mean 13)

Circularity test

0.9615
0.9503

1.0107
0.9693

Note: CCDI = Caves-Christensen-Diewert-Inklaar, the names of the people for whom the CCDI index is named: Douglas W. Caves, Laurits R. Christensen, W.
Erwin Diewert, and Robert Inklaar. In 2017, there were 38 Consumer Price Index geographic areas, whereas in 1998, there were 34. In the 1998 geographic
sample, the New York-Northern New Jersey-Long Island, NY-NJ-CT-PA, area index was a consolidation of three A-size sampling units (geographic areas),
while the Los Angeles-Riverside-Orange County, CA, index was a consolidation of two A-size sampling units (geographic areas). The remaining data point
comes from the division of the Washington-Baltimore, DC-MD-VA-WV, index into Washington-Arlington-Alexandria, DC-VA-MD-WV, and Baltimore-ColumbiaTowson, MD.
Source: U.S. Bureau of Labor Statistics.

Conclusion
The Chained Consumer Price Index for All Urban Consumers (C-CPI-U) does not exhibit much drift. To the extent
that we see drift, much of it appears to be connected to seasonality and the use of December 1999 as the base
year. Moreover, the chained Törnqvist formula used for the C-CPI-U is less susceptible to drift than the chained
Laspeyres formula would be for the less frequently updated Consumer Price Index for All Urban Consumers (CPIU). The advantages from better representing consumer substitution and a timelier market basket appear to
outweigh the disadvantages of drift. Although the effects of these advantages and disadvantages should be
studied further, there is an implication that chain drift should not be a major concern, and thus that the chained
Törnqvist formula is a better approximation of a cost-of-living index than the direct Laspeyres formula currently
used for the CPI-U.
Our study demonstrates that some subaggregate indexes do show a certain amount of drift. We can, fortunately,
suggest that advances in multilateral indexes promise to provide methods that take advantage of timely weighting
while avoiding the problem of drift. We hope that further work at BLS and elsewhere will continue in order to
determine the optimal implementation of certain aspects of multilateral index methods, particularly splicing and
changing window length. Our results show little difference among the splicing methods. Once consensus is
reached on these issues, estimation of multilateral indexes could be used to address drift issues at the lower level.

Appendix: Estimated drift, by expenditure class
Table A-1. Estimated drift, by expenditure class, 1999–2017
Expenditure class

Unity test

Men’s apparel
Boys’ apparel
Women’s apparel
Girls’ apparel
Footwear
Infants’ and toddlers’ apparel

0.941652
0.965863
1.029028
0.994046
0.957624
0.955572

See footnotes at end of table.

27

Circularity test
0.941828
0.967153
1.031830
0.990251
0.959558
0.950252

CCDI test
0.937353
0.982250
1.037496
0.985837
0.959986
0.938945

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Table A-1. Estimated drift, by expenditure class, 1999–2017
Expenditure class

Unity test

Jewelry and watches
Educational books and supplies
Tuition, other school fees, and childcare
Postage and delivery services
Telephone services
Information technology, hardware, and services
Cereals and cereal products
Bakery products
Beef and veal
Pork
Other meats
Poultry
Fish and seafood
Eggs
Dairy and related products
Fresh fruits
Fresh vegetables
Processes fruits and vegetables
Juices and alcoholic drinks
Beverage materials including coffee and tea
Sugar and sweets
Fats and oils
Other foods
Food away from home
Alcoholic beverages at home
Alcoholic beverages away from home
Tobacco and smoking products
Personal care products
Personal care services
Miscellaneous personal services
Miscellaneous personal goods
Rent of primary residence
Lodging away from home
Owners' equivalent rent of residences
Tenants' and household insurance
Fuel oil and other fuels
Energy services
Water and sewer and trash collection services
Window and floor coverings and other linens
Furniture and bedding
Appliances
Other household equipment and furnishings
Tools, hardware, outdoor equipment and supplies
Housekeeping supplies
Household operations
Professional services
Hospital and related services

1.204530
0.988330
1.017108
1.010163
0.920595
1.255826
0.963709
0.990638
1.003275
0.910979
0.993535
0.939363
1.005897
0.996227
0.985726
0.556596
0.939375
0.986802
0.998312
0.994295
1.004349
0.971247
0.984199
1.001481
1.020370
1.009530
1.002171
0.998877
0.996371
1.102182
0.979213
1.002715
0.994458
1.002457
0.997705
1.030241
0.929984
1.003097
0.979303
0.994004
1.054744
1.104847
0.966439
1.010245
0.957911
1.008573
1.029366

See footnotes at end of table.

28

Circularity test
1.202724
0.999926
1.017554
1.010069
0.939250
1.249000
0.966526
0.990164
1.003969
0.913383
0.993299
0.937153
1.004070
0.996692
0.986577
0.555873
0.941835
0.987155
0.998188
0.997309
1.001851
0.974486
0.984025
1.001670
1.021433
1.010137
1.001898
0.997457
0.995939
1.100590
0.980455
1.002400
1.002775
1.002322
0.998390
1.030476
0.930839
1.002795
0.976859
0.992963
1.057353
1.102994
0.968313
1.011556
0.956760
1.008217
1.028234

CCDI test
1.150962
0.996042
1.029939
0.999716
0.964706
1.119879
0.976675
0.987380
0.998663
0.907360
0.994247
0.933386
1.004816
0.994645
0.985187
0.561130
0.942837
0.988594
0.999799
0.995180
1.002016
0.983493
0.987586
1.002081
1.010587
1.009256
1.001276
0.998657
0.995902
1.110237
0.975419
1.001123
1.001720
1.002382
0.996370
1.034032
0.928660
1.004454
0.995162
0.990361
1.029233
1.055055
0.973173
1.002490
0.974772
1.008779
1.015461

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Table A-1. Estimated drift, by expenditure class, 1999–2017
Expenditure class

Unity test

Health insurance
Medicinal drugs
Medical equipment and supplies
Video and audio
Pets, pet products and services
Sporting goods
Photography
Other recreational goods
Other recreational services
Recreational reading materials
New and used motor vehicles
Motor fuel
Motor vehicle parts and equipment
Motor vehicle maintenance and repair
Motor vehicle insurance
Motor vehicle fees
Public transportation

0.985036
1.013699
1.019574
1.632082
0.963472
1.069327
1.182544
1.170802
1.007153
1.024860
1.004985
1.002221
1.000997
1.003613
0.990936
0.997528
0.955231

Circularity test

CCDI test

0.985283
1.012711
1.018104
1.632832
0.961186
1.068525
1.183394
1.171422
1.008762
1.024699
1.005244
1.002224
1.002042
1.005058
0.990766
0.997395
0.954407

0.982734
1.000114
1.017068
1.260736
0.982019
1.039060
1.094163
1.077137
0.991975
1.009488
1.005059
1.002762
1.000716
1.002593
0.997521
1.004337
0.958536

Note: CCDI = Caves-Christensen-Diewert-Inklaar, the names of the people for whom the CCDI index is named: Douglas W. Caves, Laurits R. Christensen, W.
Erwin Diewert, and Robert Inklaar.
Source: U.S. Bureau of Labor Statistics.

SUGGESTED CITATION

Robert Cage, Brendan Williams, and Jonathan D. Church, "“Chain drift” in the Chained Consumer Price Index:
1999–2017," Monthly Labor Review, U.S. Bureau of Labor Statistics, December 2021, https://doi.org/10.21916/
DOI-xxx.
NOTES
1 The broadest and most comprehensive Consumer Price Index (CPI), or what is sometimes called the “official CPI,” is the Consumer
Price Index for All Urban Consumers (CPI-U), U.S. city average, all items (1982–84 = 100). See “Consumer Price Index Frequently
Asked Questions,” question 16, “Which index is the ‘official CPI’ reported in the media?,” https://www.bls.gov/cpi/questions-andanswers.htm#Question_16.
2 See Joshua Klick, “Measurement of chain drift in the Chained CPI-U,” Office of Survey Methods and Research Statistical survey
paper (U.S. Bureau of Labor Statistics, November 2017), https://www.bls.gov/osmr/research-papers/2017/pdf/st170100.pdf.
3 Robert Cage, John Greenlees, and Patrick Jackman, “Introducing the Chained Consumer Price Index” (paper presented at the
Seventh Meeting of the International Working Group on Price Indices, Paris, France, May 2003), p. ii, https://www.bls.gov/cpi/
additional-resources/chained-cpi-introduction.pdf.
4 Michael J. Boskin, Ellen R. Dulberger, Robert J. Gordon, Zvi Griliches, and Dale Jorgenson, Final Report of the Advisory
Commission to Study the Consumer Price Index (U.S. Government Printing Office, December 1996), table 3, p. 44, https://
www.finance.senate.gov/imo/media/doc/Prt104-72.pdf. The report is commonly referred to as “The Boskin Commission Report,”
named for Michael J. Boskin, the chairman of the Advisory Commission to Study the Consumer Price Index.
5 Ibid., p. 1.

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6 For more on using a geometric mean formula to calculate the Consumer Price Index, see Kenneth V. Dalton, John S. Greenlees,
and Kenneth J. Stewart, “Incorporating a geometric mean formula into the CPI,” Monthly Labor Review, October 1998, https://
www.bls.gov/opub/mlr/1998/10/art1full.pdf.
7 For more information on this process, see Joshua Klick, “Improving initial estimates of the Chained Consumer Price Index,” Monthly
Labor Review, February 2018, https://www.bls.gov/opub/mlr/2018/article/improving-initial-estimates-of-the-chained-consumer-priceindex.htm.
8 Peter Hill, ed., Consumer Price Index Manual: Theory and Practice (Geneva: International Labour Office, 2004), sec. 17.9, p. 314,
https://www.ilo.org/wcmsp5/groups/public/---dgreports/---stat/documents/presentation/wcms_331153.pdf. See also A. A. Konüs, “The
problem of the true index of the cost of living,” Econometrica, vol. 7, no. 1, January 1939, pp. 10–29, https://www.jstor.org/stable/pdf/
1906997.pdf.
9 The Hicksian demand function (or compensated demand function) is named after noted British Economist John Hicks.
10 Hill, ed., Consumer Price Index Manual, sec. 15.81, p. 281.
11 Christian G. Ehemann, “Chain drift in leading superlative indexes,” BEA Working Paper 2005-09 (U.S. Bureau of Economic
Analysis, December 6, 2005), p. 3, https://www.bea.gov/system/files/papers/WP2005-9.pdf.
12 Hill, ed., Consumer Price Index Manual, sec. 16.30, pp. 292–93.
13 Ibid., sec. 16.30, p. 292, fn. 14.
14 Ibid., sec. 15.77, pp. 264–65.
15 Ibid., sec. 15.78, p. 280.
16 See Gregory Kurtzon, “How much does formula vs. chaining matter for a cost-of-living index? The CPI-U vs. the C-CPI-U,” BLS
Working Paper 498 (U.S. Bureau of Labor Statistics, September 2018), p. 12, https://www.bls.gov/osmr/research-papers/2017/pdf/
ec170060.pdf.
17 F. G. Forsyth and R. F. Fowler, “The theory and practice of chain price index numbers,” Journal of the Royal Statistical Society:
Series A (General), vol. 144, no. 2, Spring 1981, pp. 224–46, https://doi.org/10.2307/2981921; quotation, p. 224.
18 J. Lehr, Beiträge zur Statistik der Preise (Frankfurt: J.D. Sauerlander, 1885) and A. Marshall, “Remedies for fluctuations of general
prices,” Contemporary Review 51, March 1887, pp. 355–75, as cited in Hill, ed., Consumer Price Index Manual, sec. 15.77, p. 280, fn.
58.
19 Forsyth and Fowler, “Theory and practice of chain price index numbers,” p. 224.
20 See “Appendix 15.4: The relationship between the Divisia and economic approaches,” in Hill, ed., Consumer Price Index Manual,
pp. 287–88.
21 Forsyth and Fowler, “Theory and practice of chain price index numbers,” pp. 237–39.
22 Other properties include identity, monotonicity, proportionality in current prices, invariance to proportional changes in quantities,
invariance to changes in units, time reversal, and the mean value test. See “Chapter 16: “The axiomatic and stochastic approaches to
index number theory,” in Hill, ed., Consumer Price Index Manual, pp. 289–310.
23 Ehemann, “Chain drift in leading superlative indexes,” p. 3.
24 Forsyth and Fowler, “Theory and practice of chain price index numbers,” pp. 237–39.
25 Bohdan J. Szulc, “Linking price index numbers,” in W. E. Diewert and C. Montmarquette, eds., Price Level Measurement:
Proceeding of a Conference Sponsored by Statistics Canada (Ottawa: Minister of Supply and Services Canada, 1983), p. 548. See
also, Hill, ed., Consumer Price Index Manual, sec. 15.84, p. 281, fn. 61.
26 Kurtzon, “How much does formula vs. chaining matter for a cost-of-living index?” p. 12.

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27 Forsyth and Fowler, “Theory and practice of chain price index numbers,” pp. 224–46; and Marshall B. Reinsdorf and Brent R.
Moulton, “The construction of basic components of cost-of-living indexes,” in Timothy F. Bresnahan and Robert J. Gordon, ed., The
Economics of New Goods (Chicago: University of Chicago Press, 1996), p. 405.
28 Lorraine Ivancic, Kevin J. Fox, and W. Erwin Diewert, “Scanner data, time aggregation and the construction of price
indexes” (paper presented at the Eleventh Meeting of the Ottawa Group, International Working Group on Price Indices, Neuchatel,
Switzerland, May 2009), p. 4, https://www.ottawagroup.org/Ottawa/ottawagroup.nsf/51c9a3d36edfd0dfca256acb00118404/
a49bc2a164b232c4ca2576a100773522?OpenDocument.
29 Correa Moylan Walsh, The Measurement of General Exchange Value (New York: Macmillan, 1901), as cited in Consumer Price
Index Manual, sec. 15.94, p. 283.
30 Hill, ed., Consumer Price Index Manual, sec. 15.94, p. 283.
31 W. E. Diewert, ‘‘The early history of price index research,’’ as cited in Consumer Price Index Manual, sec. 15.94, p. 283.
32 Hill, ed., Consumer Price Index Manual, sec. 15.96, pp. 283–84.
33 Ibid., sec. 15.93, p. 283.
34 Ibid., sec. 15.88, p. 282. As the Manual states (fn. 68), “The test name is attributable to Fisher” (in 1922) and “the concept
originated from Westergaard” (in 1890). See the Manual for specific source information for Fisher and Westergaard.
35 Ehemann, “Chain drift in leading superlative indexes,” p. 3.
36 Hill, ed., Consumer Price Index Manual, sec. 15.88, p. 282.
37 Ivancic et al., “Scanner data, time aggregation and the construction of price indexes,” pp. 33–34.
38 W. Erwin Diewert and Kevin J. Fox, “Substitution bias in multilateral methods for CPI construction using scanner data,” UNSW
Business School Research Paper No. 2018-13 (University of New South Wales, October 2018), pp. 12–14, 39, http://dx.doi.org/
10.2139/ssrn.3276457. See also Douglas W. Caves, Laurits R. Christensen, and W. Erwin Diewert, “The economic theory of index
numbers and the measurement of input, output, and productivity, Econometrica, vol. 50, no. 6, February 1982, pp. 1393–1414.
39 Klick, “Measurement of chain drift in the Chained CPI-U”; see graph 4.
40 For more information on the “IndexNumR” software package, see Graham White, “IndexNumR: A package for index number
calculation,” September 26, 2021, https://cran.r-project.org/web/packages/IndexNumR/vignettes/indexnumr.html.
41 For a helpful resource on rolling-window indexes and extension methods, see W. Erwin Diewert and Kevin J. Fox, “Substitution
bias in multilateral methods for CPI construction using scanner data” (presentation to Statistics Norway, Oslo, Norway, May 14, 2019),
http://research.economics.unsw.edu.au/kfox/assets/diewertfox_scanner_statsnorway_14may2019.pdf.
42 Hill, ed., Consumer Price Index Manual, sec. 15.92, pp. 282–83.
43 Ibid., p. 283.
44 Ludwid von Auer, “The nature of chain drift” (presentation at the Sixteenth Meeting of the Ottawa Group, International Working
Group on Price Indices, Rio de Janeiro, Brazil, May 2019), p. 19, https://www.ottawagroup.org/Ottawa/ottawagroup.nsf/home/
Meeting+16/$FILE/The%20Nature%20of%20Chain%20Drift%20pres.pdf.

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