Discussion Paper Series A  No.722 
Lower-Level Substitution Bias 
in the Japanese Consumer Price Index: Evidence from Government Micro Data 
Shigenori Shiratsuka 
(Keio University) 
         
June  2021 
                                                Institute of Economic Research                                       Hitotsubashi University                                       Kunitachi, Tokyo, 186-8603 Japan 
Lower-Level Substitution Bias
in the Japanese Consumer Price Index:
Evidence from Government Micro Data *
Shigenori Shiratsuka†
June 16, 2021
Abstract
This paper explores measurement errors in the Japanese Consumer Price Index (CPI) stem-ming from lower-level substitution within items. The CPI is widely used as a measure forinﬂation or the cost of living. The Japanese CPI employs the one-speciﬁcation for one-itempolicy in surveying individual prices. The policy speciﬁes a few most popular speciﬁca-tions for each item and continuously surveys their prices at speciﬁc outlets. As a result,the price homogeneity is generally maintained, limiting the impact of the differences inthe elementary aggregation formulas, which corresponds to the narrow deﬁnition of thelower-level substitution bias. In contrast, the price representativeness becomes difﬁcult tobe maintained for highly heterogeneous and differentiated products. That is another as-pect of the lower-level substitution bias particular to the Japanese CPI, encompassed by thebroad deﬁnition of the lower-level substitution bias. However, quantitative assessments onthe lower-level substitution bias in the Japanese CPI are very limited since the detailed CPIdata at individual price observations was not readily available for a long time. This paper isthe ﬁrst trial on a quantitative assessment of the lower-level substitution bias using the mi-cro data for the Retail Price Survey (RPS), which is the primary source data for the JapaneseCPI. Empirical evidence conﬁrms that the lower-level substitution bias in the Japanese CPIdiffers from that for the U.S. CPI. On the one hand, the one-speciﬁcation for one-item pol-icy in the price survey succeeds in keeping price observations homogeneous, limiting theelementary aggregation bias. On the other hand, the policy also weakens price representa-tiveness, which requires additional quantitative assessments using alternative data sources,such as scanner data and web-scraping data.
Keywords: Consumer Price Index; Measurement Errors; Substitution Effects; Elementary AggregationFormula; Price Representativeness; Price Survey Method
JEL codes: C43; E31
*This paper was prepared as a part of the FY2020 research project entitled “An Economic Analysis for Con-sumer Prices using Government Micro Data” under the Joint Usage and Research Center, Institute of EconomicResearch, Hitotsubashi University. Micro data for the Retail Price Survey used in this paper was provided by theStatistic Bureau of Japan, based on Article 33 of the Statistics Act. The author thanks Erwin Diewert, Shin’ichiFukuda, Masahiro Higo, Yukinobu Kitamura, Nao Sudo, Tsutomu Watanabe, and seminar participants at Hitot-subashi University and TCER Financial Section for their constructive comments and discussions. The author alsothanks Masayuki Tanaka for his assistance. The author acknowledges the ﬁnancial support from Joint Usage andResearch Center, Institute of Economic Research, Hitotsubashi University, and Zengin Foundation for Studies onEconomics and Finance.†Keio University. E-mail: shigenori.shiratsuka@keio.jp
1Introduction
In this paper, I explore measurement errors in the Japanese Consumer Price Index (CPI) stem-
ming from lower-level substitution within items.
The CPI is widely used as a measure of inﬂation or the cost of living by generally using
a ﬁxed-weight Laspeyres formula. This formula is used mainly because of its ease of calcu-
lation and comprehension, limiting the total cost of constructing the statistics. However, such
simplicity makes it difﬁcult for the CPI to account for the dynamic nature of the economic activ-
ity, such as changes in consumers’ behavior in response to relative price ﬂuctuations between
goods and services, the introduction of new goods, and the disappearance of old goods. As a
result, measurement errors are introduced into the CPI.
Although the measurement problems for the CPI attract considerable attention from prac-
titioners and academics, Japan has very limited research on assessing the CPI measurement
errors compared with the United States, as summarized in Table 1. My previous study, Shi-
ratsuka (1999), is the ﬁrst trial to make an overall assessment of measurement errors in the
Japanese CPI, based on the best available information. However, given such limited previ-
ous studies, the estimate needs various rather bold assumptions. Since then, the situation has
almost unchanged over 20 years, as seen in the very limited follow-up studies to assess the
overall measurement errors in the Japanese CPI. That is a very contrasting situation in the
United States as we observe many following-up studies on the so-called Boskin Report, Boskin
et al. (1996).1
Among various factors for the measurement bias of the CPI, the lower-level substitution
bias has the least previous studies with rigorous quantitative assessment in Japan, since the
data on the CPI lower than the item level has not been readily available for a long time. Without
micro data for the CPI at an individual price level, it is impossible to examine the elementary
aggregation problem from individual price observations to item index. This paper is the ﬁrst
trial on a quantitative assessment of the lower-level substitution bias using the micro data for
the Retail Price Survey (RPS), which is the source data for the Japanese CPI.
The substitution bias arises in the CPI because consumers change their purchasing behavior
in response to relative price changes. This substitution bias can be divided into the low-level
and upper-level substitution biases, corresponding to the two-step procedure in the CPI compi-
lation. In the ﬁrst step, the elementary aggregation compiles the price index for the item level at
each region from individual price observations at outlets.2 That is the lowest level of groups of
goods and services surveyed in the CPI, which is supposed to include relatively homogeneous
1 The winter issue in 1998 of The Journal of Economic Perspectives features the symposium entitled “Measuringthe CPI,” which include the summary of the so-called Boskin Report, Boskin et al. (1998), the responses from staffat the Bureau of Labor Statistics, Abraham et al. (1998), and other related issues by major economists in this ﬁeld,such as Deaton (1998), Diewert (1998), Nordhaus (1998), and Pollak (1998). In addition, there exists a huge list offollowing-up studies on Boskin et al. (1996), such as General Accounting Ofﬁce (2000) and Lebow and Rudd (2003).The Bureau of Labor Statics staff makes additional public responses to such studies in Greenlees (2006). In contrast,there are very few studies on the CPI measurement issues, including Ariga and Matsui (2003).2 For the details on the elementary aggregation index formulas, see Diewert (2021) and International MonetaryFund (2020).
Table 1: Measurement Errors in CPI
JapanUnited States
Shiratsuka(1999)Shiratsuka(2006)
Broda &Weinstein(2007)
BoskinReport(1996)
Lebow &Rudd(2003)
Broda &Weinstein(2007)
Upper-levelsubstitution0.00→0.20.150.300.0
Lower-levelsubstitution0.10→0.40.250.050.0
Quality changes/ new products0.70↓0.8 + 0.20.600.370.8
Outletsubstitution0.10Ů0.20.100.050.2
Weighting————0.10—
Total0.90↓1.81.100.871.0
goods and services with similar characteristics, content, price, and price ﬂuctuation. Second,
those elementary price indexes are aggregated into higher-level price indexes by employing
respeciﬁed index formula, basically Laspeyres price index.
The Japanese CPI was thought to be less inﬂuenced by the lower-level substitution bias
than the U.S. CPI due to their differences in price survey methods. The Japanese CPI employs
the one-speciﬁcation for one-item policy in surveying individual prices. The policy speciﬁes a
few most popular speciﬁcations for each item and continuously surveys their prices at speciﬁc
outlets. As a result, the dispersion of price observations is generally restricted at a low level,
limiting the impact of the differences in elementary aggregation formulas for aggregating in-
dividual price observations into item indexes. However, the price representativeness becomes
difﬁcult to maintain for highly differentiated products and services.
In contrast, the U.S. CPI employs random sampling of many price observations within
broadly deﬁned items. This sampling method becomes a source of the lower-level substitu-
tion bias when using an arithmetic mean of price observations as an elementary aggregation
formula at the lower level. The random sampling method effectively maintains the price rep-
resentativeness while ineffectively controls the price homogeneity.
The U.S. CPI changed its elementary aggregation formula from the arithmetic mean to the
geometric mean in about two-third of the item strata in 1999. In that sense, the lower-level sub-
stitution bias stemming from the elementary aggregation formula is unique to the United States
because of the random sampling of a large number of price observations. At least, this problem
is less relevant to Japan. Instead, Japan’s problem arises mainly from the one-speciﬁcation for
one-item policy by weakening the price representativeness. I call this issue the broadly-deﬁned
lower-level substitution bias, encompassing the elementary aggregation bias as the narrowly-
deﬁned lower-level substitution bias.
With the ﬁrst trial to employ the micro data for the RPS, I conﬁrm that the lower-level sub-
stitution bias in the Japanese CPI differs from that examined by Boskin et al. (1996) for the
U.S. CPI. On the one hand, the Japanese CPI successfully constrains the elementary aggrega-
tion bias, or the narrowly-deﬁned lower-level substitution bias, at a smaller margin than the
U.S. CPI, reﬂecting the more homogeneous price observations within the items with the one-
speciﬁcation for one-item policy. On the other hand, the Japanese CPI has problems of the
broadly-deﬁned lower-level substitution bias, which corresponds to the weakened price repre-
sentativeness due to the one-speciﬁcation for one-item policy in the price survey. Some items
with the loosely deﬁned speciﬁcation have features of highly heterogeneous and differentiated
products within the item, resulting in the intensiﬁed price representative problems, in addition
to the elementary aggregation problem.
I should emphasize two things in making a quantitative assessment on the CPI measure-
ment errors. First, the measurement of individual prices raises more difﬁcult and fundamental
problems in examining price measurement than the aggregation of those prices into the overall
price index. The problems of aggregation are well understood by economists, and workable so-
lutions are now readily available to some extent, including the introduction of the chain-weight
index formula. However, collecting individual prices with high accuracy is fairly difﬁcult than
generally thought since it is impossible to survey prices for all goods and services traded in the
economy. In addition, the nominal values of goods and services are difﬁcult to be decomposed
into quantity and price on a quality-adjusted basis. Second, the assessment of the measurement
errors in the CPI needs to consider practical differences in CPI compilation methods from coun-
try to country, even though the measurement biases are examined with a common and general
framework.3In that respect, the most signiﬁcant differences across countries are observed in
collecting individual prices for goods and services.
This paper is structured as follows. Section 2 explains RPS as the primary source data for
the CPI in Japan with focusing on its particular feature of the one-speciﬁcation for one-item
policy in the price survey. Section 3 examines the practice of price substitutions in the Japanese
CPI by estimating the implied quality index from the CPI and RPS. Section 4 summarizes the
major elementary aggregation formula and estimates the elementary aggregation bias, which
corresponds to the narrowly-deﬁned lower-level substitution bias. Section 5 carries out a case
study analysis for selected items to examine the price representativeness issues. Finally, Section
6 concludes the paper by assessing the lower-level substitution bias in the Japanese CPI and
providing some proposals to improve its precision.
3 See Shiratsuka (2007) for the counterargument against Broda and Weinstein (2007), which naively apply U.S.empirical results to Japan, without considering the differences between the two countries, such as economic andprice developments and CPI compilation methods.
2Retail Price Survey and Data Employed
2.1Overview of Retail Price Survey
The RPS surveys retail prices of goods and services selected based on the importance of house-
hold consumption expenditure.4 The RPS is a primary data source for the CPI in Japan.
The price survey is structured by three types, as summarized in Table 2. First, enumerators
survey relatively standardized products and services purchased at outlets nearby, categorized
into four groups: Item group A covers goods and services consumers usually purchase in their
residential districts, and the prices vary by district, such as food and domestic non-durables;
Item group B covers goods and services consumers usually purchase in their representative
commercial areas or large retail stores with price differentials across stores, such as clothing
and electric appliances; Item group C covers goods and services with small price differentials
across stores, including recreational goods; Item group S includes other goods and services that
are surveyed without specifying survey district, such as contact lenses, and gasoline.
Second, the prefectural governments survey goods and services with almost uniform prices
in each prefecture or city, town, and village, such as water charges, hospital charges for delivery
(item group D). Third, the Statistics Bureau surveys goods and services with uniform prices
throughout the country or within a region, such as electricity and telephone charges (item
group E).
Table 2: Price Survey Framework in the Retail Price Survey
SurveyGroupSurveyorTypes of ProductsItemsNumber ofPricesSurveyed
AenumeratorShopped nearbyFood, Dailynecessities, ...42
BenumeratorShopped at largeretailersClothing, Electricappliances, ...21
CenumeratorSmall pricedifferences atoutlets
Recreationalgoods, ...12
DPrefecturalGovernmentUtility charges atlocal levelWater charges, ...—
EStatistics BureauUtility charges atnation-wide level
Electricity,Telephone charges,...—
SenumeratorNo speciﬁc surveydistrictsGasoline, ...
Source: Statitics Bureau of Japan, “Outline of the Retail Price Survey”(https://www.stat.go.jp/english/data/kouri/doukou/pdf/outline.pdf)
4 In January 2013, “Trend Survey” part was added to the RPS to explore the price structure by region and storetype.
2.2Data employed
To examine the lower-level substitution bias in the Japanese CPI, I narrow the focus in two
respects among the all available micro data provided by the Statistic Bureau of Japan (see Table
3): First, only data for Tokyo Metropolitan Area, which is commonly available both the CPI
and the RPS at an item level; and data for the period from January 2015 to December 2019,
corresponding to the 2015-base CPI.
Table 3: Retail Price Survey as a Source Data for the Consumer Pice Index
Number of itemsWeight
Numberof itemsShareweightShare
in totalincategoryin totalincategory
CPI (Tokyo Metropolitan)58210,000
Total46279.4%79.4%4,39243.9%43.9%Goods40269.1%91.2%3,28732.9%76.1%Every 10-day survey447.6%97.8%3323.3%99.6%Services6010.3%42.6%1,10511.0%19.4%
The table shows that the data used in this paper covers 79.4 percent of the CPI in term of the
number of items, and 43.9 percent in terms of the weight, since the RPS data does not include
the CPI items with large weight, such as house rent, public transportation fares, school fees,
and medical services charges. The data also covers 91.2 percent of goods surveyed, while just
42.6 percent of services surveyed in terms of the number of items. The coverage declines to 76.1
percent for goods and 19.4 percent for services in terms of the weight. Among the goods sur-
veyed, some 44 items are surveyed every ten days since some goods with high price volatility
require a more frequent price survey than other standard goods, to trace price movements.
2.3One-speciﬁcation for one-item policy
In exploring the lower-level substitution bias in the Japanese CPI, it should be noted that
sources of the bias are not restricted for the elementary aggregation formula.
The Japanese CPI employs the one-speciﬁcation for one-item policy in surveying individual
prices. This method speciﬁes a few most popular speciﬁcations for each item and continuously
surveys its prices at some speciﬁc outlets.
The policy has an advantage in that monitoring the speciﬁcation changes for the items sur-
veyed, including product downsizing/upsizing, becomes easier. The policy, however, has a
disadvantage in that the speciﬁed item is not necessarily representative of all items in an ex-
penditure category. The tightly deﬁned speciﬁcation can sometimes introduce a signiﬁcant
noise when there are major changes in the pricing strategy of the products.
Figure 1 plots the price indexes for chocolate in the CPI and the CGPI (Corporate Goods
Price Index).5The CGPI surveys average prices for readily available some speciﬁc products for
chocolates, while the CPI makes a very detailed speciﬁcation of products by listing three prod-
uct names: “Meiji milk chocolate,” “Lotte Ghana milk chocolate,” or “Morinaga milk choco-
late.” The ﬁgure shows a signiﬁcant downward shift in 2001 when major speciﬁcation changes
occurred for these speciﬁcations. Just ignoring the sharp downward shift in the CPI caused by
these changes and linking the trend before and after the speciﬁcation changes, the CPI move-
ments are very close to those in the CGPI. The ﬁgure also shows two sharp upward shifts in
the CPI in 2008 and 2015, while the upward shifts are much mild and gradual in the CGPI.
Figure 1: CPI and CGPI for Chocolate
Sources: Statistics Bureau of Japan, “Consumer Price Index”; Bank of Japan, “Corporate Goods Price Index.”
The price survey practice in the Japanese CPI seems to constrain the lower-level substitu-
tion bias stemming from the elementary aggregation formula by keeping price observations
homogeneous. It likely induces another lower-level substitution bias due to the failure in rep-
resenting the overall price movements for the expenditure categories.
3Price Substitution Practice
As a preparation for the quantitative analysis on the lower-level substitution bias, I examine
price substitution practice in the CPI compilation procedure through simulation analysis. The
simulation clariﬁes the price linkage between the RPS and the CPI in a practical manner.
3.1Price Substitution Simulations
The CPI is computed from the RPS (or unit prices in general) by making a quality adjustment
when changing the item speciﬁcations. Quality adjustment covers various aspects of product
characteristics, such as product performances and unit size. The CPI is computed from the RPS
with taking account of quality changes at price substitution by applying the equation below.
5The CGPI covers goods traded in the corporate sector, compiled by the Bank of Japan. The prices of productsare continuously surveyed by keeping the quality and contract terms ﬁxed.
CPIi,t = RPSi,t
QALi,t,(1)
where CPIi,t, RPSi,t, and QALi,t denote the CPI, the RPS, and implied quality index of item i at
period t with normalized at base period 0. Thus implied quality index is computed from the
CPI and the RPS, although details of the quality adjustment at an individual price level are not
readily available.
QALi,t =
RPSi,tRPSi,0CPIi,t(2)
The Statistics Bureau of Japan makes an announcement on price substitutions when the item
speciﬁcation changes. The announced price substitutions are common to all price observations
in the item. In practice, however, the Statistics Bureau of Japan continues to make various
adjustments to link discontinued price observations to compile the CPI.
Using equation (2), I compute the monthly quality adjustment implied by the CPI and the
RPS as follows.QALi,tQALi,t−1=
RPSi,tRPSi,0CPIi,t×CPIi,t−1
RPSi,t−1RPSi,0=RPSi,tRPSi,t−1× CPIi,t−1
CPIi,t(3)
I compute two series of the implied quality index. The ﬁrst series is to apply equation (3)
for all periods (implicit quality index: IQI hereafter). The second series is to apply equation
(3) only for the period with the ofﬁcial announcement of the price substitution to the item
(ofﬁcial quality index: OQI hereafter). The differences in the two series indicate implicit and
undisclosed quality adjustment for discontinued individual price observations, not common to
all observations for the item. Such effort is critical in maintaining the precision of the CPI at a
high level, but, at the same time, it makes it difﬁcult to make use of the time series information
in the micro data for the RPS.
3.2Simulation Results
Figure 2 provides the results for price substitution simulation by plotting the IQI and the OQI
in December 2019 (the end of the sample period). The IQI and the OQI are log-transformed
indexes for December 2019 by normalizing with 2015 at one, thus indicating the cumulative
effects of quality adjustment at the price substitutions from January 2015 to December 2019.
The ﬁgure indicates three points: First, many observations are placed along with the 45-
degree line, suggesting that many items show almost identical quality adjustment in both of-
ﬁcial and implicit price substitutions. Second, many other observations are located along the
vertical axis, implying that implicit quality adjustments are made at individual price observa-
tions without the ofﬁcial announcement of price substitutions. Third, other observations that
deviate from the vertical axis and 45-degree line suggest that both ofﬁcial and implicit substi-
tutions are made, but quality improvement and deterioration are diverse across the items.
Figure 2: Results for Price Substitution Simulation
Figure 3 plots the data for horizontal and vertical axes in Figure 2 as frequency distributions
in separate panels. The left panel of the ﬁgure for OQI shows a spike at exactly zero with
more than 50 percent of items. The right panel for IQI also shows spikes around zero, but no
observations exist at exactly zero, and two peaks exist for the intervals from -0.1 to zero and
from zero to 0.1. That suggests that price substitutions at individual price observations are
made at all the items, even though no ofﬁcial price substitutions at an item level are made for
more than half of the items. Such adjustments in an individual price observation are critically
important in maintaining the precision of price information in practice. That, however, makes
it very difﬁcult to employ time-series information in micro data for the RPS since it produces
many discontinuities in individual price observations.
Figure 3: Frequency Distributions for Ofﬁcial and Observed Price Substitutions
Figure 4 plots the distribution of the ratio of the IQI to the OQI with mean, median, max,
min, 10-percentile, and 90-percentile values over the whole sample period from January 2015 to
December 2019. The ﬁgure shows that the intervals between 10- an 90-percentiles are relatively
tight, but the intervals become wider for the max and min values. That observation is consis-
tent with Figure 3, suggesting the existence of some outlier items with signiﬁcant differences
between ofﬁcial and implicit price substitutions.
Figure 4: Cumulative Effects of Implicit Price Substitutions
Given the price substitution simulation results with discontinuous price observations, I de-
cided to focus solely on cross-sectional information on price observations for each item, avail-
able from the micro data for the RPS.
4Elementary Aggregation Bias
4.1Elementary aggregation formula
The following three formulas are mainly used as an elementary aggregation formula, which
aggregates individual price observations at the lowest level to the item price index.
ICi,t = 1
pi,j,tpi,j,0,(4)
IDi,t =
1n ∑nj=1 pi,j,t
1n ∑nj=1 pi,j,0,(5)
IJi,t =n∏j=1
� pi,j,t
pi,j,0
n= ∏nj=1(pi,j,t)1n
∏nj=1(pi,j,0)1n,(6)
where pi,j,t denotes the price of j-th price observation in i-th item at time t, n is the number of
price observations, and t = 0 is the base period.
Equation (4) is called the Carli index formula of the arithmetic mean of price relatives at
the current period and the base period. Equation (5) is the Dutot index formula of the ratio
of the arithmetic mean of prices at the current and base periods. Equation (6) is the Jevons
index formula of the geometric mean of price relatives, comparable to equation (4), as well as
the ratio of the geometric mean of prices at the current period and base period, comparable to
equation (5).6
With very homogeneous price observations, all the elementary aggregation formulas pro-
duce very close results, and elementary aggregation bias, or narrowly deﬁned lower-level sub-
stitution bias, is not a major issue. However, once price observations become heterogeneous
and diverse, the elementary aggregation formula matters a lot.
That point becomes clearer as price dispersion becomes wide, as shown in two types of
simple mean-preserving-spread exercises. First, the left-hand side panel in Figure 5 carries out
a standard mean-preserving-spread exercise by computing arithmetic and geometric means
for three observations (1 − x, 1, 1 + x) by increasing x from zero to 0.5. In this exercise, the
arithmetic mean remains unchanged while the geometric mean continues to decline as x be-
comes larger. Second, the right-hand side panel conducts a mean-preserving-spread exercise
in terms of the geometric mean by computing both the arithmetic and geometric means for
(1/(1 + x), 1, 1, x). In this case, the geometric mean remains unchanged, while the arithmetic
mean continues to increase.
Figure 5: Difference between Arithmetic and Geometric Means
Note: Mean-preserving spreads exercise using data for (1-x, 1, 1+x) and (1/(1+x), 1, 1+x), respectively .
Two simple exercises shown above reveal that dispersion of price observations becomes
larger, the arithmetic mean exceeds the geometric mean at an accelerated pace. That is a prac-
6 International Monetary Fund (2020), which summarizes the international recommendations on the CPI com-pilation methods and practices, the Jevons index formula is preferred in most circumstances when weights arenot available. The Dutot index formula can be employed only for homogeneous elementary aggregates since theprice relatives are implicitly weighted by the price level in the reference period. However, Diewert (2021) stronglyrecommends using the Jevons index formula in any circumstances from the theoretical perspective.As a more general formulation, the CES aggregator is an alternative index formula, which encompasses boththe Dutot and Jevons index formulas, equations (5) and (6), with the elasticity of substitution σ of zero and one,respectively. Shapiro and Wilcox (1997) for the details of the CES index formula. Shiratsuka (1999) applies the CESindex formula to the upper-level aggregation for the Japanese CPI as an alternative real-time index formula.
ICESi,t=
n∑j=1
�pi,j,tpi,j,0
tical thing that the effects of elementary aggregation formula on measurement errors in the
CPI. In that sense, the Japanese CPI is robust against the elementary aggregation bias since
it employs the one-speciﬁcation for one-item policy to keep price observations for each item
homogeneous as possible.
4.2Heterogeneity in price observations
To examine the price heterogeneity at an item level, I introduce the ratio for arithmetic and
geometric means (hereafter AG-ratio) for item i at period t, AGRi,t, given by
AGRi,t =
1n ∑nj=1 pi,j,t
∏nj=1(pi,j,t)1n.(7)
If all prices surveyed are identical, then AG-ratio will be one since the arithmetic mean and
the geometric mean produce the same mean value. However, as price observations become
diverse, the arithmetic mean constantly overstates the geometric mean, making the AG-ratio
larger.
Figure 6 shows a scatter plot for the AG-ratio and the coefﬁcient of variation (CV) for all
common items in the CPI and the RPI by dividing them into two groups: mid-month survey
items and every ten-day survey items. This ﬁgure clearly shows a positive correlation between
the AG-ratio and CV. Focusing on the mid-month survey items, the relationship between the
AG-ratio and CV seems nonlinear, and the AG-ratio increases at an accelerated pace as CV
increases.
Figure 6: Ratio of Arithmetic-Geometric Means
The nonlinear relationship between the AG-ratio and CV becomes evident with panel re-
gression results in Table 4. The AG-ratio is regressed on CV and its quadratic term by applying
three panel data estimation procedures: pooled, ﬁxed-effect, and between estimations.
AGRi,t = Const. + αCVi,t + βCV2i,t + ϵi,t,
where CVi,t denotes the coefﬁcient of variation for item i at period t.
Looking at the table, pooled and between estimates are closer than ﬁxed-effect estimates,
suggesting that the between variation across items is more important than the within variation
over time.
Table 4: Panel Regression on AG-Ratio
PooledFixed-effectBetween
Coeff.S.E.Coeff.S.E.Coeff.S.E.
CV0.0300.002**0.0220.009*0.0350.010**CV20.4060.006**0.4270.023**0.3940.024**Constant0.9980.000**0.9980.001*0.9970.001**
obs.23,82523,825418groups—418—R20.9610.9600.978
4.3Measurement bias from elementary aggregation formula
Next, I will compute annual inﬂation rates based on the arithmetic and geometric mean formu-
las, thereby estimating a lower-level substitution bias from their differences. It should be noted
that this is a lower-level substitution bias in a narrow sense, examined by Boskin et al. (1996)
for the U.S. CPI.
Figure 7 plots inﬂation rates for mid-month survey items in the upper panel and every
ten-day survey items in lower panel. Differences are generally small, especially in the case of
every ten-day items. Every ten-day survey covers volatile prices, mainly perishables inﬂuenced
heavily by weather conditions but succeeds in controlling the price homogeneity.
As a result, the impact of lower-level substitution bias on the overall CPI inﬂation is also
limited over time, as shown in Figure 8, with the maximum value of 0.08 percentage point in
an absolute term. The ﬁgure also shows that the directions of the bias are unstable over time
with both negative and positive values, even though the bias exhibits mostly negative values.
The arithmetic mean constantly overstates the geometric mean, but their ratio varies over time,
thereby pushing inﬂation rates upward and downward.7
7 Very minor impacts from the elementary aggregation bias, as shown in this paper, do not necessarily implythat we do not need to reexamine the current practice of the Japanese CPI to apply the Dutot index formula to allthe items in the elementary aggregation. As International Monetary Fund (2020) explains, the preferable elemen-tary aggregation formula without weight information is the Jevons index formula, and the Dutot index formula isapplicable only to the homogeneous elementary aggregate. Thus some items with highly heterogeneous and differ-entiated price observations must consider the elementary aggregation formula to the Jevons. It should be also notedthat Diewert (2021) recommends not using arithmetic mean formula since it is not invariant to general changes inthe units of measurement.
Figure 7: Difference in Inﬂation Rates between Arithmetic and Geometric Means
(a) Mid-month survey items
(b) Every 10-day survey items
Figure 8: Elementary Aggregation Formula Bias
To sum up, lower-level substitution bias stemming from the elementary aggregation for-
mula is relatively minor and produces only negligible impacts on the measurement errors in
the Japanese CPI. However, the more fundamental problem exists regarding the lower-level
substitution bias through the price representativeness due to the one-speciﬁcation for one-item
policy in the price survey.
5Case Study on Heterogeneity of Price Observations
5.1Framework for case study analysis
I carry out a case study analysis to examine the effects of elementary aggregation formulas
using selected items based on the AG-ratio and the degree of detail in the speciﬁcation. Ta-
ble 5 summarizes selected items for case study analysis. Items from number one to three are
stipulated with the detailed speciﬁcation, while those from number four to nine are stipulated
with the slightly loose speciﬁcation. Items from ten to 12 are every ten-day survey items of
perishables with different price volatility.
Table 5: Selected Items for Case Studies
No.ItemsSpeciﬁcation / Price volatility
(Mid-month Survey Items)1WhiskyDetailedSuntory Kaku Bin
2Soy SauceKikkoman or Yamasa (1-little bottle)
3Washing MachineSpeciﬁc brand and product (undisclosed)
4Butter
200-gram box (excluding unsalted butter)
5GasolineRegular gasoline (1 little, excluding self-service station)
6CapStandard product with logo (chemical ﬁber)
7Women’s JeansStandard product with logo (chemical ﬁber)
8PanSaucepan (standard product, stainless steel, 18cm, 2.0-2.5 little)
9BatteryAA alkaline battery (packed 4 batteries)
(Every Ten-day Survey Items)10Sea Bream (Tai)—Low price volatility
11Horse Mackerel (Aji)High price volatility12Lettuce
To examine the effects of price heterogeneity in each item, I compute the arithmetic and
geometric means on a trimmed basis, given by
TRA(k)i,t =1
n − 2k
n−k∑j=1+kpi,j,t,(8)
TRG(k)i,t =n−k∏j=1+k(pi,j,t)1/(n−2k).(9)
where k denotes the number of trimmed price observations in both upper and lower tails of
price distribution at period t. k is determined based on the number of price observations in
each item as follows.
1(n = 12)
2(n = 21).
3(n = 42)
If price observations within the item become more homogeneous by trimming outlier obser-
vations, the arithmetic mean and the geometric mean on a trimmed basis show closer values,
making the AG-ratio on a trimmed basis, given below, closer to one than the simple AG-ratio.8
AGR(k)i,t = TRA(k)i,t
TRG(k)i,t.(10)
Figure 9 exhibits a scatter plot for the AG-ratios for the simple and trimmed mean bases.
The ﬁgure shows that all the observations are located below the 45-degree line, suggesting that
price observations become more homogeneous in all items, but that the effectiveness of the
trimmed mean procedure is limited in some outlier items.
Figure 9: AG-Ratio for simple and trimmed means
5.2Case study analysis for mid-month survey items
I summarize the results for case study analysis for nine mid-month survey items in Figures
10. The ﬁgures for case study analysis includes the item CPI and ﬁve statistics: the arithmetic
mean, the geometric mean, median, the trimmed arithmetic mean, and the trimmed geometric
mean, which are normalized by the arithmetic mean for 2015 average being equal to 100 with
a one standard error band around the arithmetic mean.
Whisky (Figure 10a): The CPI for whisky is one of the most typical item for the price sur-
vey based on the one-speciﬁcation for one-item policy. That speciﬁes a single brand of Suntory
Kaku Bin, one of Japan’s most popular whiskey. As a result, arithmetic and geometric means are
very close, suggesting fairly homogeneous price observations and resultant high precision of
elementary aggregation. However, signiﬁcant gaps between the simple and trimmed means ex-
ists from 2015 to 2017, indicating possible distortion from outlier price observation (high price
observations in this case). Still, it is difﬁcult to conﬁrm whether the current CPI for whisky rep-
resents the overall price movements for the diversiﬁed whisky products in Japan. For example,
8 If k = 0, equation 10 falls into equation 7 for the ratio of the simple arithmetic and geometric means.
the price hike in April 2016 reﬂects upward price revisions for Suntory’s whisky products, and
it is not clear whether it accurately reﬂects price trends for whisky products in general.
Soy Sauce (Figure 10b): The CPI for soy sauces is also the typical item for the price survey
based on the one-speciﬁcation for one-item policy. It surveys two, not one, the most popular
brands for soy sauce, Kikkoman and Yamasa. That makes price observations homogeneous,
thereby achieving high precision of the elementary aggregation. It should be noted that small
but constant deviations between the arithmetic and geometric means are observed, indicating
the possible distortion from low price observations. And such deviations become smaller when
using the the arithmetic and geometric means on a trimmed basis.
Washing Machine (Figure 10c): The CPI for washing machines is another typical item for
the price survey based on the one-speciﬁcation for one-item policy, applied to goods and ser-
vices with clear product cycles by periodic price substitutions. The ﬁgure shows strong season-
ality in price ﬂuctuations: prices jump in October and gradually decline afterward every year.
The ﬁgure also shows that the CPI for washing machines remains generally ﬂat, while other
unit value indicators gradually increase over time, suggesting constant quality improvement.
Even with such large price ﬂuctuation with quality improvement, the ﬁgure suggests relatively
high precision in the elementary aggregation. Still, it is difﬁcult to conﬁrm whether the current
CPI for washing machines represents the overall price movements of the diversiﬁed washing
machine market in Japan.
Butter (Figure 10d): The CPI for butter has the relatively loose deﬁnition of the product
speciﬁcation, without clearly stipulating the brand, but seems successful in maintaining high
precision in the elementary aggregation. However, it should be noted that some failures in
the price survey, evidence as widening of the error bands for a couple of times, induce the
distortions from outlier observations. In that respect, the arithmetic and geometric means on a
trimmed basis show smoother movements at those times by limiting such outlier impacts.
Gasoline (Figure 10e): The CPI for gasoline also successfully maintains high precision in el-
ementary aggregation even with relatively large price ﬂuctuations over time. Price differences
across gas stations seem to be the outlet-speciﬁc factor, generally constant over time.
Cap (Figure 10a): The CPI for caps achieves good precision in elementary aggregation
even with relatively loose speciﬁcation, saying “a standard product with logo (chemical ﬁber).”
However, it shows a puzzling deviation between the CPI and the RPS in April 2017, indicating
signiﬁcant quality deterioration at that time.
Women’s Jeans (Figure 10g): The CPI for women’s jeans shows a large AG-ratio and high
coefﬁcient of variation with a very wide range of price observations. It has a very loose speci-
ﬁcation, just saying “standard product for Blue denim jeans.” The CPI contains the distortion
from outlier prices through practical effort at individual price observations to some extent.
However, it needs to review the price survey practice and examine the introduction of the ge-
ometric mean formula in the elementary aggregation.
Pan (Figure 10h): The CPI for pans also shows a relatively large AG-ratio and high coefﬁ-
cient of variation with a wide range of price observations. It also shows a puzzling drop in the
RPS in June 2018, while the CPI remains unchanged, suggesting signiﬁcant quality deteriora-
tion at that time.
Battery (Figure 10i): The CPI for batteries widens the price range after 2017, showing signif-
icant deterioration in the elementary aggregation precision. Such deterioration is attributable
to the expansion of private label products in price observations, suggesting the difﬁculty in
balancing the elementary aggregation precision and the price representativeness.
5.3Case study analysis for every ten-day survey items
The case study analysis for every ten-day survey items include perishable foods with generally
volatile price ﬂuctuations due to changes in weather conditions.
I choose three items with slightly different volatility in price ﬂuctuations: sea bream (tai) in
Figure 11a, horse mackerel (aji) in Figure 11b, and lettuce in Figure 11c. Among the three items,
sea bream shows relatively stable price ﬂuctuation both in the short-term and longer-term due
to increased aquaculture. Horse mackerel shows large short-term price volatility, but the range
of price ﬂuctuations is not so large in the longer term. Lettuce shows large swings in prices
with seasonal ﬂuctuations.
However, the three items with large price volatility show generally common movements for
arithmetic and geometric means in both all observations and trimmed observations. That sug-
gests that large price volatility does not necessarily produce distortionary effects from outlier
price observations.
Figure 10: Case Study Analysis for Mid-Month Survey Items
(a) Whisky (Item-1)
(b) Soy Sauce (Item-2)
(c) Washing Machine (Item-3)
(d) Butter (Item-4)
(e) Gasoline (Item-5)
Figure 10: (Continued)
(f) Cap (Item-6)
(g) Women’s Jeans (Item-7)
(h) Pan (Item-8)
(i) Battery (Item-9)
Figure 11: Case Study Analysis for Every-Ten-Day Survey Items
(a) Sea Bream: Tai (Item-10)
(b) Horse Mackerel: Aji (Item-11)
(c) Lettuce (Item-12)
6Conclusions
In this paper, I explored the CPI measurement errors stemming from the lower-level substitu-
tion in the Japanese CPI and provided some quantitative evaluations.
With the ﬁrst trial to employ the micro data for the RPS, I made it clear that the lower-level
substitution bias in the Japanese CPI differs from that examined by Boskin et al. (1996) for the
U.S. CPI. On the one hand, the Japanese CPI successfully constrains the elementary aggrega-
tion bias, or the narrowly-deﬁned lower-level substitution bias, at a smaller margin than the
U.S. CPI, reﬂecting the more homogeneous price observations within the items with the one-
speciﬁcation for one-item policy in the price survey. On the other hand, the Japanese CPI has
problems of the broadly-deﬁned lower-level substitution bias, which corresponds to the weak-
ened price representativeness due to the one-speciﬁcation for one-item policy. Some items
with the loosely deﬁned speciﬁcation have features of highly heterogeneous and differentiated
products within the item, resulting in the intensiﬁed price representative problems.
In addition, the current price survey method does not necessarily guarantee a stable price
survey in practice. The micro data for the RPS reveals that it has many price discontinuities
for temporary price survey failures due to the disappearance of the products surveyed at the
outlets. Although the Statistics Bureau of Japan makes the best effort to link such discontinued
price observations in compiling the CPI, it is undeniable that the Japanese CPI is vulnerable to
outlier price observations stemming from the temporary price survey failure.
For further analysis on the price representativeness issues, it is necessary to employ alterna-
tive data sources, such as scanner data and web-scraping data. The Statistical Bureau of Japan
has already employed such alternative data in a limited area, such as PCs, TV sets, and travel
tour charges. It is deemed effective to widen the application of such alternative data step-by-
step since the issues on the one-speciﬁcation for one-item policy need to be examined on an
item-by-item basis.
Concerning the price representativeness issues, another important point is how to deal with
the downsizing of the products, reﬂecting the decrease in the household member size. Figure 12
plots the relative size of the items surveyed in 2019, normalized by the average size in 2015. The
ﬁgure shows that the median size declines to 0.987, about 12 percent over four years. It is often
argued that product prices typically remain unchanged with product downsizing, resulting
in hidden price hikes.9 However, it is not necessarily the case if household preference shifts
toward products with smaller unit sizes. For example, single-person households prefer small-
sized food staffs considering the risk of leftovers. In this case, higher unit prices for small-
sized products reﬂect convenience premia compact and reasonable sizes. That is another open
question that requires employing alternative data for a rigorous empirical analysis.
9 Imai and Watanabe (2014) argue that the Japanese CPI is downwardly biased since the quality downgradeassociated with product downsizing is not appropriately adjusted, based on their analysis using daily scannerdata for supermarkets from 2000 to 2012. Abe, Inakura and Tonogi (2017) also point out that product turnovereffects produce the signiﬁcant price gap between new and old goods, using weekly scanner data covering generalmerchandise stores, supermarkets, drug stores and convenience stores from 2006 to 2016.
Figure 12: Distribution of Relative Size in 2019
Despite the efforts of statistical agencies in constructing accurate statistics, measurement
errors are unavoidable to some extent in the CPI. Thus, the most important point is to ascertain
whether they are small enough to be safely ignored in practice or serious enough to mislead
users. The debate about the accuracy of the CPI should investigate the sources of measurement
errors and how they affect accuracy. Making improvements in the CPI is difﬁcult and requires
a signiﬁcant commitment of resources since each item needs to be examined separately.
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