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HOME ARCHIVES FOR AUTHORS ABOUT SUBSCRIBE Search MLR GO 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. 1 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 2 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 3 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 4 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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.) 5 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 6 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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.) 7 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 8 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 10 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. 11 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 12 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 13 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. Related Subjects Labor dynamics Statistical programs and methods Pay 17 Job creation Employment Layoffs Recession 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] 2 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 4 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 6 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 7 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 8 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 9 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 10 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 11 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW Chart 5 displays the results when the Törnqvist formula is used. 12 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 13 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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). 14 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 15 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 16 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 17 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 18 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 19 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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- 20 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 21 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 22 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 23 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 24 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 25 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 29 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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. 30 U.S. BUREAU OF LABOR STATISTICS MONTHLY LABOR REVIEW 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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