Institute for Economic Studies, Keio University
Keio-IES Discussion Paper Series

On the Source of Seasonality in Price Changes:
The Role of Seasonality in Menu Costs
宗像晃、篠原武史、白塚重典、須藤直、渡辺努
2023 年 11 月 21 日

DP2023-016
https://ies.keio.ac.jp/publications/23228/

Institute for Economic Studies, Keio University
2-15-45 Mita, Minato-ku, Tokyo 108-8345, Japan
ies-office@adst.keio.ac.jp
21 November, 2023

On the Source of Seasonality in Price Changes: The Role of Seasonality in Menu Costs
宗像晃、篠原武史、白塚重典、須藤直、渡辺努
IES Keio DP2023-016
2023 年 11 月 21 日
JEL Classification: E31, E32, E37
キーワード: POS データ、価格変動の季節性、ニューケインジアンモデル、メニューコスト
【要旨】
季節性は物価変動の最も顕著な特徴の一つであるが、数量の季節性や物価変動の景気循環的要素に比べると、その分析は著しく少ない。このギャップを埋めるため、日本の199カテゴリーの商品のスキャナーデータを使用し、1990年から2021年までの価格変動の季節性を実証的に研究する。分析結果からは、以下の４つの特徴がほとんどの品目分類で確認できる。(1)物価上昇・下落の頻度は3月と9月に上昇する、(2)物価変動の頻度に関する季節成分は物価変動の大きさの季節成分と負の相関がある、(3)インフレ率の季節成分はネットでみた物価変動の頻度の季節成分と一致する、(4)物価変動の頻度の季節パターンは物価変動の大きさの季節パターンに比べて安定している。しかし、このパターンは、その年の品目分類レベルの年間インフレ率の変化に反応する。次に、簡便な状態依存価格モデルを用いてシミュレーション分析を行い、メニュー
コストの季節的サイクルが、データで観察される価格変動の季節性を生み出す上で本質的な役割を果たしていることを示す。最後に、メニューコストの季節的サイクルの特性とマクロ経済
変動への影響について議論する。
宗像晃
日本銀行金融機構局
kou.munakata@boj.or.jp
篠原武史
日本銀行金融研究所
takeshi.shinohara@boj.or.jp
白塚重典
慶應義塾大学経済学部
shigenori.shiratsuka@keio.jp
須藤直
日本銀行金融研究所
nao.sudou@boj.or.jp
渡辺努
東京大学大学院経済学研究科
watanabe@e.u-tokyo.ac.jp
謝辞：本論文の作成にあたっては、青木浩介、藤本淳一、M. Hsu、伊藤隆敏、松本英彦、M.
McMahon、P. Porapakkarm、R. Schoenle、新谷元嗣、戸村肇、上田晃三、M. Wynneの各氏、ならびにGRIPsのセミナー参加者、2023年BOJ-IMESコンファランス「金融政策の古くて新しい課
題」およびSWETへの参加者、日本銀行スタッフから貴重なコメントを頂戴した。記して感謝の意を表したい。ただし、本論文で示されている意見は、筆者たち自身に属し、日本銀行の公式見解を示すものではない。本論文は以前「日本における季節的サイクルと物価変動の同期化」
として配布されている。

1

Introduction
It is widely known among both scholars and policymakers that the time series of prices

have a sizable degree of seasonality, similar to quantity series. The top and middle
panels of Figure 1 show the month-to-month change in the consumer price index (CPI)in Japan for all items and for goods less fresh food and energy, respectively, for selected
years during the 1990s and 2000s. Generally speaking, prices tend to rise from theprevious month around March and September and decline around June, though thepattern is less visible in the data for all items. The bottom panels of the same figure show
the month-to-month change in the CPI in the U.S. for commodities less food and energy
commodities. Indeed, the seasonal patterns are similar in the two countries. Figure 2
shows the decomposition of the yearly growth rate of the CPI, again for all items andfor goods less fresh food and energy, into twelve month-to-month changes within thesame year in Japan. It can be seen that there are months in which prices generally
increase, such as March and April, and months in which prices generally decrease, such
as January and February. Such seasonal patterns have been stable from the 1990s to
2020s.
Not surprisingly, such seasonal patterns of price changes have attracted the attentionof macroeconomists and the presence of seasonal patterns itself has been documented
in a good number of existing studies in the literature of macroeconomics. For example,
a seminal paper by Nakamura and Steinsson (2008), pointing out that the frequency of
price changes tends to be the highest in the first quarter and declines monotonicallythrough the fourth quarter for consumer prices, states “seasonality of the frequency of
price change” as one of the five noteworthy facts regarding price dynamics in the U.S.
In the case of Europe, Álvarez et al. (2006), using granular data on consumer prices andproducer prices, document that price changes tend to occur in the first quarter, inparticular in January, and in September. Despite drawing attention to these facts, thesestudies do no more than report the existence of seasonal patterns and do not makethese facts the central focus of their analysis. 1 Consequently, there remain issues that

1This contrasts with seasonality of quantity, which has been taken as the main theme and central

subject of analysis in macroeconomics. See, for example, the pioneering work by Barsky and Miron(1989).
1are not fully analyzed or not well understood regarding the seasonality of price changes.
We aim to fill this gap by answering three questions: What are the key features of theseasonality of price changes? What types of economic structures are consistent with theobserved features? What are the macroeconomic implications of seasonality of pricechanges? To this end, we conduct empirical and theoretical analyses. For the empiricalanalysis, we study the seasonality of price changes from January 1990 to December 2021
using scanner data in Japan. The data contain about 11 billion observations of food anddaily commodities, except for fresh food. We first decompose the inflation rate intofrequency and size for upward price changes and downward price changes following
Klenow and Kryvtsov (2008). We then extract the seasonal components of these series
following Geremew and Gourio (2018) and study their characteristics including theirinterrelationships across categories and their relationship with the annual inflation rate.
In extracting seasonal components, we use a regression with monthly dummy variables
in our baseline. In addition to the baseline, as a robustness check, we also extract theseasonal components by two other methodologies, X12-ARIMA and dividing theoriginal series by the annual average of the series, and redo the exercises.
For the theoretical analysis, we build a simple menu cost model and examine whatmodel features are needed to bring the model close to the observed features of thedata. Regarding the causes of the seasonality of price changes, Olivei and Tenreyro(2010) stress the importance of differences in wage flexibility across different months ofthe year. Nakamura and Steinsson (2008) point out the role of time-dependency in pricesetting on top of the seasonal changes in wage flexibility. We address these two viewsin our exercise. We simulate two models, one with seasonal changes in real wages andthe other with seasonal changes in menu costs and compare which of the two models
can successfully generate seasonal patterns of price changes seen in the data.
Our findings can be summarized as follows. Regarding the empirical analysis, weobserve four key features. First, the frequency of price increases and decreases tends torise in March and September for most categories. In other words, the timing of pricechanges coincides within a category and across categories as well as for both upwardand downward price changes. Second, for the majority of categories, seasonal
components of the frequency of price changes are negatively correlated with those of
the size of price changes, though seasonal components of the size are less pronounced2and less synchronized across categories and across the direction of price change relative
to seasonal components of the frequency. Third, there is also seasonality in the inflation
rate. The seasonal patterns of the inflation track the seasonal patterns of net frequency,
i.e. the difference between the frequency of price increases and price decreases, andare moderately synchronized across categories. Fourth, the seasonal pattern of thefrequency of price changes has been stable over the sample period, in contrast to thatof the size of price changes. The pattern is, however, responsive to changes in the annual
category-level inflation rate for the year. That is, the seasonal component of thefrequency of price increases (decreases) of a category becomes more volatile (lessvolatile) in the sense that the standard deviation of monthly changes in the frequency
becomes larger (smaller) when the category-level annual inflation rate for the year ishigh (low). In other words, a rise in the frequency of price increase in March and
September of a category becomes more pronounced when the inflation rate of thecategory for the year is high. Such responses are not seen in the seasonal componentof the size of price changes.
Regarding our theoretical analysis, we find, based on our simulation exercises, thatseasonal cycles in menu costs, i.e., declines in March and September, are needed tobring the model close to the observations. Indeed, the model with seasonal cycles inmenu costs is able to produce the four features qualitatively. First, under the assumptionthat menu costs are low in March and September, firms face an added incentive tochange their prices for both upward and downward price changes. Consequently, thefrequency of both price increases and decreases becomes high in these two months
compared with other months. Second, because a larger portion of firms, including those
whose prices are not very far from the target price, changes their prices in the twomonths, the average size of price changes falls, yielding a negative correlation between
the frequency and the size of price changes. Third, as seasonal cycles of menu costs
generate larger seasonal variations in frequency rather than those in size, the seasonalpatterns of net frequency trace those of the inflation rate. Forth, the seasonal pattern of
the frequency of price increases (decreases) becomes more (less) pronounced as thesteady-state inflation rate rises, since a larger (smaller) portion of firms finds it necessary
to change their prices. In contrast, a model with seasonal increases in real marginal costs
does not generate the first two features, which starkly contrasts with the data.
Our paper contributes to the literature on price dynamics in three aspects. First, it
3

empirically uncovers characteristics of seasonality of price changes that have been
unexplored in detail in existing studies. For example, to the best of our knowledge, therelationship between the seasonal components of the size and frequency of pricechanges and the responsiveness of seasonal patterns of price changes to the annual
inflation rate for the year have not been studied. 2 Second, our paper offers a theoreticalexplanation as to why there are such seasonal patterns by using a state-dependent
pricing model. While existing studies such as Álvarez et al. (2006) and Nakamura andSteinsson (2008) document the presence of seasonality and discuss the potential
sources of the seasonality, they do not explicitly construct a model that accounts for theseasonal patterns. Third, our result that there are seasonal cycles in menu costs
underscore the importance of the “month” in macroeconomic dynamics. For example,
because there should be months in which price changes are likely to occur and months
they are not, the transmission of shocks to output and prices is considered as being
affected by the months in which the original shocks occur. This point is, in fact, consistent
with the argument made by Olivei and Tenreyro (2010) that monetary policy
transmission to goods and prices is affected by the month in which monetary policy
shocks occur.
In addition, our finding regarding the seasonal cycles of menu costs provides insights
into the nature of menu costs themselves. The feature that menu costs fall in a specificmonth in a synchronized fashion across different categories agrees with the arguments
made in early works by Zbaracki et al. (2004) and Blinder et al. (1998). Using data froma large U.S. industrial manufacturer and its customers, Zbaracki et al. (2004) study thenature of menu costs and document that in addition to physical menu costs there arethree types of managerial costs—information gathering, decision making andcommunication costs, and two types of customer costs—communication, andnegotiation costs. Blinder et al. (1998), based on a survey of firms in the U.S., documents
2

The characteristics of the frequency and size of price changes, including their relationships withthe annual inflation rate, have already been investigated in existing studies, such as Klenow andKryvtsov (2008) and Blanco et al. (2022), using original data and seasonally adjusted data. Blanco etal. (2022), for example, documents that the frequency of price changes increases from 10% to 14%
when the sectoral inflation rate increases from around zero to 5% in the original data from theUnited Kingdom Office for National Statistics. However, these studies do not analyze the seasonal
components themselves.
4that an important portion of firms indicate coordination failure as a potential theory forprice stickiness. One potential interpretation of the seasonal cycles of menu costs is that,
due to commonly held expectations by firms, there is implicit coordination among firmsin specific months, so that the non-physical components of menu costs, such as those
associated with communication and negotiation, decline in these months.
The structure of this paper is as follows. Section 2 overviews the literature. Section 3
explains the scanner data. Section 4 provides stylized facts on the seasonality of pricechanges. Section 5 develops a menu cost model that provides explanations for theseasonality of the changes. Section 6 concludes.

2 Literature Review
Broadly speaking, our study is related to three strands of literature. The first strand ofstudies includes works that specifically focus on seasonality of macroeconomic variables.
Almost all of these works focus on quantity variables. They document characteristics of
seasonality and explore the interaction between seasonal components and the businesscycle component of a variable or derive macroeconomic implications of seasonality.
Seminal works by Barsky and Miron (1989) and Miron (1996) show that, for example, for
GDP and its components, seasonal variations are sizable and that seasonal cyclicality ofthese variables resembles business cycle variations in various dimensions, including comovement of the variables. They also show that variations in the seasonal componentof prices are small compared with those of quantity variables. From a slightly different
angle, Beaulieu et al. (1992) show there is a strong positive correlation between thestandard deviation of seasonal component and that of non-seasonal component ofvariables across countries and industries. Cecchetti and Kashyap (1996) and Matas-Mir
and Osborn (2004) study interactions between seasonal cycles and business cycles using
the data of advanced countries and show that summer shut-downs are shorter during
boom years, which they interpret as the result of reallocation of production inputs fromhigh output months to low output months. 3 Olivei and Tenreyro (2010) compare the3

Cecchetti et al. (1997) also show that, in several U.S. industries, economic booms are associatedwith a reduction in the seasonal variations of production and either no change or an increase in theinventories during the high-production season. Based on this findings, they conclude that firms in5

estimated response of industrial goods production to a monetary policy shock across
selected developed countries, including Japan, and show that in Japan monetary policy
shocks that occur in the first quarter yield a muted impact on output compared withshocks that occur in the third quarter of a year. They argue that this is because the renegotiation of wages tends to take place in a good number of firms during the annual
wage negotiations, known as Shunto, that occur in the first and second quarters in Japan.
Our paper is related to Cecchetti and Kashyap (1996) and Matas-Mir and Osborn (2004)
in the sense that it provides a potential channel that generates interactions between
seasonal and business cycles in price dynamics. As in the data, with seasonal cycles inmenu costs, our model predicts a larger increase in the frequency of price increases
when the inflation rate of the category is high. Our paper is related to Olivei andTenreyro (2010) in the sense that it offers an alternative explanation for their empiricalfinding. Namely, both their paper and our paper emphasize that a monetary policy
shock is affected by the month in which it occurs, though they stress the importance ofchanges in wage flexibility and our paper stresses the role of changes in the size ofmenu costs across months.
The second strand of studies includes works on macroeconomic price dynamics that
exploit granular data, including scanner data. These studies particularly focus on the
distinction between the intensive and extensive margins of price changes behind pricestickiness, i.e., the size and frequency of price changes, and explore whether or not thedata agree with the implications of models used in the literature of macroeconomics,
such as the Taylor model, Calvo model, and state-dependent pricing model. These
works include, for example, Levy et al. (1997), Bils and Klenow (2004), Klenow andKryvtsov (2008), and Nakamura and Steinsson (2008). A good survey is conducted by
Mackowiak and Smets (2009), Klenow and Malin (2010), and Nakamura and Steinsson(2013). There are also studies that exploit granular data and study Japan's micro pricedynamics. Such works include the Bank of Japan (2000), Higo and Saita (2007), Ikeda
these industries face upward-sloping and convex marginal-production-cost curves. Krane andWascher (1999) investigate interactions between seasonal and cyclical movements in U.S. payroll
employment using a multivariate unobserved components model, and find statistically significant
interactions in a number of industries. Geremew and Gourio (2018) present some stylized facts about
seasonality in U.S. employment and argue that seasonality has rarely associated with cyclicality
contrasting to earlier studies.
6and Nishioka (2007), Mizuno et al. (2010), Abe and Tonogi (2010), Watanabe and
Watanabe (2014), Sudo et al. (2018), and Ueda et al. (2019) among others. In terms ofthe data, our data is the same as that used in Abe and Tonogi (2010), Sudo et al. (2018),
and Ueda et al. (2019), though our data set is longer than theirs. Our paper is similar tosome of these studies in the sense that it empirically studies the intensive and extensive
margins of price changes and theoretically examines if the data findings are consistent
with the implications of a state-dependent pricing model. However, our paper differs
starkly from these studies in focusing on the seasonal components of price changes.
As discussed in the introduction, seasonality itself is already documented in some ofthese studies. For example, Nakamura and Steinsson (2008) and Álvarez et al. (2006)
document that the frequency of price changes increases in January for both the U.S.
and the euro area, respectively, and Bunn and Ellis (2012) document a rise in non-saleprices in April for the U.K. In contrast to our study, however, the seasonality is not thecentral focus of their study. Consequently, they do not formally extract the seasonal
component of price changes nor study causes of seasonality using a theoretical model.

3 Data and Definition of Variables
3.1 Data
We employ Point-of-Sale (POS) scanner data collected by Nikkei Digital Media fromretail shops located in Japan. The data have been widely used by existing studies on
micro-level price dynamics in Japan, including Abe and Tonogi (2010), Sudo et al. (2014),
Sudo et al. (2018), and Ueda et al. (2019). The data are daily and the sample period
covers the period from March 1, 1988 to February 10, 2022, excluding November andDecember of 2003. The data are taken from 575 stores and the sampled stores are
spread across Japan. According to Abe and Tonogi (2010), among the sampled stores,
even small stores have 2,000 customers a day. The data consist of 11 billion records andeach record contains the number of units sold and sales in yen for product 𝑖𝑖, identified
by 13-digit Japanese Article Number (JAN) code, at shop 𝑠𝑠 on date d. The cumulative
number of products appearing during the sample period is 1.8 million.

The data include processed food and domestic articles and, unlike the CPI, do not7include fresh food, recreational durable goods, such as TVs and PCs, and services, suchas rent and utilities. The coverage of the POS scanner data in the CPI is 201 out of 582
items, which constitutes 20.5% of households’ expenditure covered by the official CPI
with the base year of 2020.
For the purpose of the analysis, we aggregate the 13-digit JAN product level data to
a 3-digit level, such as “tofu,” “yogurt,” “beer,” “tobacco,” and “laundry detergent,” asdefined by Nikkei and hereafter refer to the data aggregated at this level as a “category.”
Table 1 shows the list of categories studied in this paper.
We exclude the data of years 1988, 1989, 2003, 2004 and 2022 from our analysis,
either because the data of some months are missing (1988, 2003, and 2022), the dataneeded for computing the regular price of some months are missing (2004), or becauseof a complication associated with the introduction of the consumption tax (1989).
Consequently, the data period for our analysis is reduced to that from the beginning ofJanuary 1990 to the end of December 2021, excluding the years 2003 and 2004.
We also make adjustments to the sampled products and categories. First, we exclude
products whose first and last days of sale are separated by less than 365+90 days. Thisis because the “monthly changes in the seasonal pattern” of the “regular price” of suchproducts cannot be well measured due to the constraint associated with our filtering
methodologies of regular prices and seasonal components of the prices. Second, foreach category, we identify the years that contain months with fewer than 15 regular
price increases or decreases per month and exclude these years from the sample toaccurately measure the size of price changes. After that, we drop 18 categories thathave short sample periods (less than 15 years). Consequently, we use 199 categories outof 217 categories for our analysis.

3.2 Regular Prices
We focus on regular prices, excluding sales, following existing studies such as
Nakamura and Steinsson (2008). We use a mode filter to obtain the regular prices, as in
Abe and Tonogi (2010) and Eichenbaum et al. (2011). Namely, the regular price of a
product at a particular date is defined as the mode of the daily prices of the productwithin a window of 89 days i.e., between 44 days before and 44 days after the day of8measurement. Due to this definition of the regular price, we exclude the first and last
45 days of the sample for each product.

3.3 Definition of Variables
For the purpose of analyzing price dynamics in detail, following existing studies, suchas Klenow and Kryvtsov (2008), we look at the intensive and extensive margins of pricechanges. To this end, we decompose the inflation rate into the frequency of pricechanges and the size of price changes. In addition, for both the frequency and size of
price changes, we separate upward and downward changes, again following Klenow
and Kryvtsov (2008). More precisely, the regular price inflation of category 𝐽𝐽 in month𝑡𝑡, which is expressed as 𝜋𝜋𝐽𝐽𝐽𝐽 , can be decomposed into the following four elements.
−−
𝜋𝜋𝐽𝐽𝐽𝐽 = 𝐹𝐹𝐹𝐹𝐹𝐹𝑄𝑄+𝑆𝑆𝑆𝑆𝑆𝑆𝐸𝐸+𝐽𝐽𝐽𝐽 − 𝐹𝐹𝐹𝐹𝐹𝐹𝑄𝑄𝐽𝐽𝐽𝐽 𝑆𝑆𝑆𝑆𝑆𝑆𝐸𝐸𝐽𝐽𝐽𝐽
𝐽𝐽𝐽𝐽

( 1)

Here, the frequency of upward (downward) price adjustment of category 𝐽𝐽 in month+−𝑡𝑡, which is expressed as 𝐹𝐹𝐹𝐹𝐹𝐹𝑄𝑄𝐽𝐽𝐽𝐽(𝐹𝐹𝐹𝐹𝐹𝐹𝑄𝑄𝐽𝐽𝐽𝐽), is given as the number of products 𝑖𝑖 thatbelong to category 𝐽𝐽 and have changed their price upwards (downwards) on any day

𝑑𝑑 in the month divided by the total number of products 𝑖𝑖 that belong to category 𝐽𝐽,

as described in the equation below.
𝐹𝐹𝐹𝐹𝐹𝐹𝑄𝑄±=
𝐽𝐽𝐽𝐽∑𝑖𝑖∈𝐽𝐽, 𝑑𝑑∈𝑡𝑡 1�𝑝𝑝𝑖𝑖𝑖𝑖 ≷ 𝑝𝑝𝑖𝑖𝑖𝑖−1 �∑𝑖𝑖∈𝐽𝐽,𝑑𝑑∈𝑡𝑡�1�𝑝𝑝𝑖𝑖𝑖𝑖 > 𝑝𝑝𝑖𝑖𝑖𝑖−1 � + 1�𝑝𝑝𝑖𝑖𝑖𝑖 = 𝑝𝑝𝑖𝑖𝑖𝑖−1 � + 1�𝑝𝑝𝑖𝑖𝑖𝑖 < 𝑝𝑝𝑖𝑖𝑖𝑖−1 ��( 2)Note that when the regular price of a product on a day does not differ by more than
3 yen from the previous observation, we regard the regular price as having remained
unchanged, following Sudo et al. (2014). 4
Similarly, the size of the upward (downward) price adjustment of category 𝐽𝐽 in month𝑡𝑡, which is expressed as 𝑆𝑆𝑆𝑆𝑆𝑆𝐸𝐸𝐽𝐽𝐽𝐽+ (𝑆𝑆𝑆𝑆𝑆𝑆𝐸𝐸𝐽𝐽𝐽𝐽− ) is given as the size of the price change of product4

In equation (2), we compute the frequency of price changes using changes from the previous day.

In the analysis below, because we focus on the monthly difference, we construct the monthly
frequency of price changes from the daily frequency of price changes using the formula30±±(𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚ℎ𝑙𝑙𝑙𝑙) = 1 − �1 − 𝐹𝐹𝐹𝐹𝐹𝐹𝑄𝑄𝐽𝐽𝐽𝐽𝐹𝐹𝐹𝐹𝐹𝐹𝑄𝑄𝐽𝐽𝐽𝐽�for all months. We use 30 instead of the actual number ofdays for each month, such as 29 for February and 31 for January, in order to control the effects of
the length of the month. The results are little changed, however, if the actual number of days is usedinstead of 30 when constructing the monthly frequency of price changes.
9

𝑖𝑖 whose price is changed upward (downward) on any day 𝑑𝑑 in the month divided by

the total number of products 𝑖𝑖 that belong to category 𝐽𝐽 whose price has changed

upward (downward) as described in the equation below.
𝑆𝑆𝑆𝑆𝑆𝑆𝐸𝐸±𝐽𝐽𝐽𝐽 =∑𝑖𝑖∈𝐽𝐽, 𝑑𝑑∈𝑡𝑡�𝑙𝑙𝑙𝑙𝑙𝑙�𝑝𝑝𝑖𝑖𝑖𝑖 /𝑝𝑝𝑖𝑖𝑖𝑖−1 ��1�𝑝𝑝𝑖𝑖𝑖𝑖 ≷ 𝑝𝑝𝑖𝑖𝑖𝑖−1 �∑𝑖𝑖∈𝐽𝐽, 𝑑𝑑∈𝑡𝑡 1�𝑝𝑝𝑖𝑖𝑖𝑖 ≷ 𝑝𝑝𝑖𝑖𝑖𝑖−1 �( 3)Figure 3 shows the time path of the aggregate POS inflation rate obtained by thescanner data together with the CPI inflation rate. Note that the latter represents the CPIof goods less fresh food and energy so that the coverage becomes close to that of thescanner data. There are some commonalities in the way that the two series havedeveloped. 5 For example, both series exhibit a high inflation rate during the early 1990s
followed by a decline in the inflation rate lasting until the early 2000s. They also both
exhibit a sharp decline after the global financial crisis.

3.4 Seasonality
To measure the seasonality for each category-level variable of interest 𝑦𝑦𝐽𝐽𝐽𝐽 , i.e., thefrequency and size of price adjustments, we estimate the following equation for eachof these variables, following Geremew and Gourio (2018).
12

𝑦𝑦𝐽𝐽𝐽𝐽 = �𝑚𝑚=1�𝑎𝑎𝐽𝐽𝐽𝐽 𝑑𝑑𝑑𝑑𝑑𝑑𝑚𝑚,𝑡𝑡 � + 𝛽𝛽𝐽𝐽0 + 𝛽𝛽𝐽𝐽1 × 𝑡𝑡 + 𝛽𝛽𝐽𝐽2 × 𝑡𝑡 2 + 𝜖𝜖𝐽𝐽𝐽𝐽
5(4),Clearly, there are differences in terms of how the two series have developed during the sample

period. In terms of the compiling methodology, the official CPI and our POS inflation differ mainlyin the following three aspects. First, the scope of the sample is different. While our POS inflation uses
data on all products sold in the sampled stores to calculate an index for each of the categories, the
CPI first selects specific products that are considered representative and then makes calculations
based on the prices of those products only. Second the definition of regular price differs. The regular
price in the CPI is defined as the price on any one day from Wednesday through Friday of the week
containing the 12th of each month and that lasts for at least eight days. On the other hand, theregular price in our POS inflation calculation is, as discussed below, calculated by taking the modein a rolling window. Third, when a firm changes the quantity of a product without changing its price,
the CPI and our POS data treat this change differently: in the CPI, an increase (decrease) in thequantity of some items at the same price is measured as a price reduction (price increase). In our
POS inflation calculation, such cases are considered as the exit of an old product and the entry of anew product since the barcode has changed.
10

12

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 �𝑚𝑚=1 𝑎𝑎𝐽𝐽𝐽𝐽 = 0,

where 𝑑𝑑𝑑𝑑𝑑𝑑𝑚𝑚,𝑡𝑡 is a dummy variable that takes a value of unity when time 𝑡𝑡 occurs inmonth 𝑚𝑚 and zero otherwise. The coefficient 𝑎𝑎𝐽𝐽𝐽𝐽 captures the effect of a particularmonth 𝑚𝑚. The coefficients 𝛽𝛽𝐽𝐽1 and 𝛽𝛽𝐽𝐽2 capture the effects of a linear trend and aquadratic trend, respectively. 6 Because, as we argue in Section 4, the degree ofseasonality is time-varying, we estimate the above equation using rolling regressions

with a three-year window. The degree of seasonality for a specific year 𝑦𝑦 is obtainedby taking the average of the estimates of the rolling regressions whose sample period
includes year 𝑦𝑦.

We use as the baseline the seasonal components of variables extracted from equation(4) throughout our analysis as they are simple and easily interpretable. For the purposeof robustness checking, however, we study seasonal components obtained by two othermethodologies as well. In Appendix B, we show the results when we use X12 forextracting seasonal components from the original series to analyze seasonal patterns.
In Appendix C, we show the results when we normalize the original series with theaverage of the value of twelve months in the same year. In both of the methodologies,
most of the results are barely changed for the four key observations that are describedin the main text.
How important are seasonal components quantitatively relative to variations of thevariable in other frequencies? To see this, in Figure 4 we show three measures, all ofwhich are constructed from the estimation results of equation (4). The upper panel
shows variations of seasonal components normalized by the average of the original
series for the frequency and size of upward and downward price changes. The heightof the bars represents the median of the categories and the error bands indicate the25th and 75th percentiles of all categories. The middle panel shows variations of
seasonal components relative to variations of the detrended series. Two observations
can be made. The first is that the size of seasonal variations is importantly large and thesecond is that the seasonal variations are more pronounced in the frequency of pricechanges than in the size of price changes. The bottom panel shows the statistical
6

While Geremew and Gourio (2018) use the Hodrick-Prescott filter to remove the trend, we use thelinear and quadratic trend dummies because there are missing observations in our data and the
Hodrick-Prescott filter cannot be applied.
11

significance of monthly dummies in equation (4), tested using F-tests. Because we
execute three-year rolling estimates for each of the series and our sample period ranges
from 1990 to 2021, excluding 2003 and 2004, we have at most 36 estimation results foreach variable 𝑦𝑦𝐽𝐽𝐽𝐽 . For each of the regression results of each of the categories, we test

the null hypothesis that “all the coefficients of the monthly dummies, namely
𝑎𝑎𝐽𝐽𝐽𝐽 for m = 1, … 12, are zero,” count the share of regression results in which the null
hypothesis is rejected at the 5% for each category, and show the median and the 25thand 75th percentiles across the categories. For the frequency of price changes, the null

hypothesis is rejected at the 5% level for more than half of the categories for both
upward and downward price changes. In contrast, for the size of price changes, the null
hypothesis is rejected in around 20% of the regressions, indicating seasonality mattersless for the size of price changes.

4 Observations
This section documents characteristics of the seasonal component of price changes
based on our scanner data. In summary, there are four key observations.
[1] For most categories, the frequency of both price increases and decreases tends
to rise in March and September, exhibiting a two-humped pattern, with theformer more pronounced than the latter.
[2] For the majority of categories and for both price increases and decreases, theseasonal components of the frequency of price changes are negatively correlatedwith those of the size of price changes, though the seasonal components of thesize of price changes are less pronounced and less synchronized across
categories and across the direction of price change than those of the frequency
of price changes.
[3] For most categories, the seasonal component of the overall inflation rate tracks
the seasonal component of net frequency, i.e., the difference between thefrequency of price increases and that of price decreases and is moderately
synchronized across categories.
[4] The seasonal pattern of the frequency of price changes has been stable over our
12sample period relative to that of the size of price changes. The pattern is, however,
responsive to changes in the category-level annual inflation rate for the year. Thatis, the seasonal component of the frequency of price increases (decreases)
becomes more (less) volatile when the category-level inflation rate is high. Such
responses are not seen for the seasonal component of the size of price changes.
We focus our analysis on the seasonal components of the series extracted by themethodology described in equation (4). Unless otherwise noted, all of thecharacteristics documented below are those of the seasonal components rather thanthose of the original series.

4.1 Seasonality of the Frequency of Price Changes
Figure 5 shows the characteristics of the seasonal component of the frequency of
price changes for the category “tofu” at the top and all categories at the bottom. The
top left panel shows the time series of the frequency for upward and downward pricechanges over the sample period and the top right panel shows their monthly seasonal
components averaged over time. The bottom panels show the seasonal components
averaged over time for all categories. Specifically, the solid lines represent the medianand the shaded area represents the 25th-75th percentile bands across categories.
There are two points worth noting. First, for “tofu,” there are noticeable seasonalpatterns for both upward and downward price changes. That is, the frequency is high inMarch and September and low in other months. In other words, the seasonal patternsof the frequency of price changes are synchronized for both upward and downward
price changes. Second, similar seasonal patterns are observed for all categories, which
implies that the synchronization is present across categories. .
Figure 6 studies the degree of synchronization of the seasonal component of thefrequency of price changes both across categories and across directions (i.e. upward or
downward price changes). The top panels show the degree of synchronization across
categories, represented by the distribution of pairwise correlation coefficients. In
particular, we compute the correlation of the seasonal components of the frequency of
price increases (left) and decreases (right) over the sample period for every pairwise
combination of categories. Note that there are 19,701 pairs (199 times 198/2). It can beseen that the seasonal components of the frequency of price changes are positively
13correlated for most of the pairs for price increases and decreases. For price increases,
the peak lies around 0.1-0.2 and the median is 0.19. Furthermore, 12,233 pairs, which isabout 62% of the total number of pairs, are significantly positively correlated at the 5%
level. For price decreases, the peak lies around 0.2-0.3 and the median is 0.21.
Furthermore, 12,975 pairs, which is about 66% of the total number of pairs, aresignificantly positively correlated at the 5% level.
The bottom panels show the synchronization of price increases and decreases withina category. We compute the correlation of the two time series, i.e., the seasonal
components of the frequency of upward price changes and downward price changes,
for each of the 199 categories. The correlations are computed using Pearson’s
correlation shown at the left and Spearman’s rank correlation shown at the right. The
two measures agree that the frequency of upward and downward price changes of agiven category are positively correlated. For Pearson’s correlation, the median of thecorrelations is 0.33 and the peak lies around 0.3-0.4. Furthermore, a positive correlation
is observed at a statistical level of 5% for 150 of the 199 categories. On the other hand,a negative correlation is observed at a statistical level of 5% for only 11 categories. For
Spearman’s rank correlation, the median of the correlation is 0.44 and the peak lies
around 0.6-0.7.

4.2 Seasonality of the Size of Price Changes
Figure 7 shows the characteristics of the seasonal component of the size of pricechanges for the category “tofu” at the top and all categories at the bottom. The top left
panel shows the time series of the size of upward and downward price changes overthe sample period and the top right panel shows the monthly seasonal components ofthe size of price changes averaged over time for the same category. The bottom panels
show the seasonal components of the size of upward and downward price changes
averaged over time for all categories. Specifically, the solid lines represent the medianand the shaded area represents the 25th-75th percentile bands across categories. While
there are some seasonal variations for both price increases and decreases, it is lessobvious that these seasonal variations are synchronized across categories and/or acrossdirections than is the case with the frequency of price changes. For the category “tofu,”
the size of price changes tends to be large in January, November, and December andtends to be small in March and around September for both upward and downward
14

price changes, exhibiting a negative correlation with the frequency of price changes.
This observation, however, does not hold clearly for all categories shown at the bottom.
Indeed, for all categories, differences in the size of price changes across months are lessvisible compared with the case of the frequency of price changes.
How are the seasonal components of the size of price changes related to those of
the frequency of price changes? To see this, we compute the correlation between theseasonal components of the frequency and size of price changes for each of thecategories. The top left panel in Figure 8 shows the distribution across categories of thecorrelation between the frequency and the size of price increases. For Pearson’s
correlation, the median of the distribution is -0.11 and the peak lies around -0.2 to -0.3.
For 97 out of the 199 categories, the correlation is negative at the 5% level. For
Spearman’s rank correlation, the median of the correlation is -0.01 and the peak lies
around -0.1-0.0. These observations indicate that the seasonal components of thefrequency and size of price increases are negatively, though modestly, correlated. The
bottom panels of the same figure show the case for price decreases. Similar
observations can be made as in the case of price increases. Namely, for Pearson’s
correlation, the median of the distribution is -0.17 and the peak lies around -0.2 to -0.3.
Furthermore, for 114 out of the 199 categories the correlation is negative at the 5% level.
For Spearman’s rank correlation, the median of the correlation is -0.14 and the peak lies
around -0.1-0.0.
Similar to the case of the frequency of price changes, we study the degree ofsynchronization for the size of price changes. In Figure 9, the top panels show thedegree of synchronization across categories, represented by the distribution of pairwise
correlation coefficients. We compute the correlation of the seasonal components of thesize of price changes for every pairwise combination of categories over the sample
period for upward (left) and downward (right) price changes separately. It can be seenthat seasonal components of the size of price changes are positively correlated in mostof the pairs, but the proportion of pairs exhibiting a positive relationship is limited
compared with the case of the frequency of price changes. For upward price changes,
the median is 0.07 and the peak lies around 0.0-0.1. Furthermore, 7,634 pairs,
representing about 39% of the total number of pairs, are positively correlated at the 5%
level and 2,694 pairs, representing about 14% of the total number of pairs, arenegatively correlated at the 5% level. For downward price changes, the median is 0.08
15

and the peak again lies around 0.0-0.1. Furthermore, 8,082 pairs, representing about
41% of the total number of pairs, are positively correlated at the 5% level and 2,729
pairs, representing about 14% of the total number of pairs, are negatively correlated at
the 5% level.
The bottom panels of the figure capture the degree of the synchronization between
upward and downward price changes within a category. Using Pearson's correlation,
we see that the sizes of price increases and decreases are positively correlated at a
statistical level of 5% for 96 out of the 199 goods. The median of the distribution is 0.10
and the peak lies around 0.0 to 0.1. Using Spearman’s rank correlation, we find that themedian of the correlation is 0.10 and the peak lies around 0.1-0.3. Again, these numbers
are less positive than in the case of the frequency of price changes where, for example,
the median of the two correlation measures are 0.33 and 0.44.
To summarize, while synchronization of the seasonal component of the size of pricechanges is generally positive both across categories and across the direction of pricechange, the correlation coefficients are less positive than in the case of the seasonal
component of the frequency of price changes, indicating that the degree ofsynchronization is more moderate for the former than the latter. In addition, for the
majority of categories and for both price increases and decreases, seasonal componentof the frequency of price changes are negatively correlated with those of the size of
price changes.

4.3 Seasonal Patterns of Inflation
Now we turn our attention to the seasonal component of the category-level monthly
inflation rate. Figure 10 shows the seasonal component of the POS inflation rate for the
category “tofu” at the top and all categories at the bottom. The top left panel shows themonthly inflation rate over the sample period for “tofu,” and the top right panel shows
the seasonal component of the inflation rate for “tofu” averaged over time. The bottom
panel shows the seasonal component of the monthly inflation rate averaged over time
for all categories. Specifically, the solid line represents the median and the shaded area
represents the 25th-75th percentile band across categories. For the category “tofu,” theinflation rate increases in March and September, as is seen in the frequency of pricechanges. For all categories, however, the seasonal pattern differs slightly from that of16

the frequency of price changes. It tends to be high in January, February, March, April,and September and low in other months.
To see how the frequency and size of price changes affect the seasonal pattern of
the inflation rate, we construct two series which we refer to as net frequency and net
size hereafter. For the former, we subtract the seasonal component of the frequency of
price decreases from that of price increases and for the latter we subtract the seasonal
component of the size of price decreases from that of price increases. Figure 11 shows
the net frequency and the net size of price changes for all categories, with the median
depicted in blue, together with the median of seasonal component of the POS inflation
rate across categories. Similar to the seasonal component of the POS inflation rate, thenet frequency tends to be high in January, February, March, April, and September andlow in other months. This pattern arises from the asymmetry between the frequency of
price increases and that of price decreases. While both series tend to be high in Marchand September, a rise in the frequency of price changes in the two months and a fall in
the frequency of price changes in the months from May to August are more
pronounced for price increases and a fall in the frequency of price changes in January
and February is more pronounced for price decreases. Consequently, the seasonalpattern of the POS inflation rate generally tracks that of the net frequency. In contrast,
the net size of price changes tends to be low in January and February and high in the
fourth quarter of the year. This seasonal pattern does not match well with the seasonalpattern of the POS inflation rate.
Figure 12 studies the degree of synchronization in the POS inflation rate across
categories. Similar to the exercises conducted above, we compute the correlation of theseasonal component of the POS inflation rate over the sample period for each of the
19,701 pairs (199 times 198/2). There is synchronization across pairs, though only
modestly. The median of the distribution is 0.10 and the peak lies around 0.0-0.1.
Furthermore, while 9,160 pairs, representing about 46% of the total number of pairs, arepositively correlated at the 5% level, 3,343 pairs, representing about 17% of the total
number of pairs, are negatively correlated at the 5% level.

174.4 Changes in Seasonal Patterns over Time
Lastly, we examine how the seasonal components of the frequency and size of pricechanges have changed over time.

Variations over Time
In order to check the stability of the seasonal patterns of the frequency and size of
price changes, we first split the sample period mechanically in half and study how theseasonal patterns in the early and latter halves differ from each other. Figure 13 shows
the seasonal patterns of the frequency of price increases (top left panel), that of pricedecreases (top right panel), the size of price increases (bottom left panel) and that ofprice decreases (bottom right panel) for the two subsamples for all categories. It can beseen that the general pattern of seasonal components are little changed for thefrequency of price changes. The frequency tends to be high in March and September
and low in other months. In contrast, the seasonal pattern of the size of price changes
is less stable.
Figure 14 studies the degree to which the seasonal patterns are stable. For each of
the categories, we compute the correlation of the seasonal component in the first halfof the sample period and that in the second half of the sample period. The top panels
show the correlation for the frequency of price changes. For the frequency of priceincreases, the median of the distribution across categories is 0.75 and the peak lies
around 0.8-0.9. For the frequency of price decreases, the median of the distribution is0.80 and the peak lies again around 0.8-0.9. The bottom panels show the correlation forthe size of price changes. The seasonal pattern is clearly less stable compared with thatof the frequency of price changes. For the size of price increases, the median of the
distribution is 0.33 and the peak lies around 0.2-0.3, while for the size of price decreases,
the median of the distribution is 0.46 and the peak lies around 0.6-0.7 and around 0.80.9.

Seasonal Patterns and Economic Conditions
Though the seasonal patterns of price changes are stable at least for the frequency
18

of price changes, it does not mean that they are unresponsive to changes in themacroeconomic environment. Figure 15 shows the yearly time-series of the POS
inflation rate and the standard deviation of the seasonal components of the frequency
of price increases and decreases for the category “tofu” at the top and for all categories
at the bottom. For the standard deviation, we first estimate the coefficients in equation(4) and compute the standard deviation of the twelve coefficients for each year. The
greater the variation across months is within the year, the higher the value is. There ismodest asymmetry between price increases and decreases. For price increases, thestandard deviation roughly tracks the time path of the POS inflation rate for both “tofu”
and all categories. The standard deviation is large during the high inflation period in thefirst two years of the 1990s, falls to a low level in 1993, and remains at a low level until it
starts to increase in the mid-2000s, mirroring the increase in the POS inflation rate. In
contrast, for price decreases, the standard deviation does not move together with the
POS inflation rate. For example, it stays at a low level in the early 1990s and during the
few years before the global financial crisis when the POS inflation rate is high.
Figure 16 shows the yearly time-series of the POS inflation rate and the standard
deviation of the seasonal components of the size of price changes of "tofu" and all
categories for both upward and downward price changes. Roughly speaking, for “tofu,”
the standard deviation tracks the time path of the annual POS inflation rate. It is notable
that, compared with the frequency of price changes, not only the standard deviation of
the size of price increases but also that of price decreases is high during the early 1990s.
For all categories, the relationship is less clear.
Figure 17 looks at the relationship between the seasonal pattern of price changes andthe inflation rate from a different angle. For each of the categories, we split the sample
period into two sub-samples, a high inflation period and a low inflation period,
depending on the yearly POS inflation rate and see how the seasonal patterns differ
across the two subsample periods. It can be seen from the figure that ups and downs
across months are slightly more volatile for the frequency of price increases in the highinflation period compared with the low inflation period. Regarding the frequency of
price decreases, ups and downs across months are more volatile in the low inflation
period than the high inflation period, in particular when looking at changes in the 25th
to 75th percentiles. Such a difference is not seen for the size of price changes.

19Indeed, it can be shown more formally that the seasonal patterns of the frequency of
price changes are responsive to annual changes in the category-level inflation rate forthe year. In Figure 18, the top panels show the distribution across categories of thecorrelations between the standard deviation of seasonal component of the frequency
of price changes and its annual POS inflation rate. For price increases, the median ofthe distribution is 0.38 and the peak lies around 0.4 to 0.5. Furthermore, for 97 out of
199 categories, the correlation is positive at the 5% level. For price decreases, the medianof the distribution is -0.12 and the peak lies around 0.0 to 0.1. For 30 out of 199
categories, the correlation is negative at the 5% level. In other words, the seasonal
component of the frequency of price increases (decreases) becomes more volatile (lessvolatile) when the category-level annual inflation rate for the year is high. The
relationship with annual category-level inflation is less clear for the size of price changes.
The bottom panels show the distribution across categories of the correlations between
the standard deviation of seasonal component of size of price changes and thatcategory’s annual POS inflation rate for the year. For price increases, the median of the
distribution is 0.10 and the peak lies around 0.2 to 0.3. For 23 out of 199 categories, thecorrelation is positive at the 5% level. For price decreases, the median of the distributionis also 0.10 and the peak lies around 0.2 to 0.3. For 24 out of 199 categories, thecorrelation is positive at the 5% level.

5 Simulation Using a Menu Cost Model
What features of the economic structure are responsible for the observations
obtained above? To see this, in this section we construct a simple menu cost model andsimulate the time path of the frequency and size of price changes as well as that of
inflation under various assumptions on the source of seasonality.
Our model is built upon the partial equilibrium model used by Nakamura andSteinsson (2008) and is extended with seasonal variations to either of the two key model
ingredients: the size of menu costs and the size of real marginal costs. This setting aims
to address two views in the existing studies regarding what generates seasonality ofprice changes. In particular, Nakamura and Steinsson (2008) underscore the importanceof a time-dependency element in the observed seasonal patterns of price dynamics and20

Olivei and Tenreyro (2010) underscore the importance of changes in the flexibility ofwages during a specific quarter of the year as a source of seasonality for Japan’s pricedynamics. See also Appendix A for the details of our model.
In addition, we do not explicitly consider models that extend purely time-dependent
models, such as that of Taylor (1980) and Calvo (1983), by introducing exogenous
seasonal variations in frequency of price changes. Admittedly, among the four key
findings listed above, the first finding, i.e., an increase in frequency of both upward anddownward price changes in a specific month of the year, is considered consistent withthe prediction of such time-dependent models. However, the second and fourthfindings, i.e., the (weak and) negative correlation between the frequency and size of
price changes and the responsiveness of the frequency of price changes to changes in
category-level inflation, do not accord well with the prediction of these pure timedependent models. 7

5.1 Seasonal Cycles in Marginal Cost and Menu Cost
As in the model of Nakamura and Steinsson (2008), we consider a monopolisticallycompetitive market in which firms set their prices so as to maximize their profits subject
to costs associated with price changes. We assume that firms take changes in real wages,
denoted as 𝜔𝜔𝑚𝑚(𝑡𝑡) , and idiosyncratic technology, denoted as 𝐴𝐴𝑡𝑡 (𝑧𝑧), as given and set
their price, denoted as 𝑝𝑝𝑡𝑡 (𝑧𝑧), so as to maximize the present value of the profits fromnow and beyond. Firms are allowed to set prices only if they pay the menu cost, denotedas 𝜔𝜔𝑚𝑚(𝑡𝑡) 𝐾𝐾𝑚𝑚(𝑡𝑡) . Note that the menu cost is driven by the real wage 𝜔𝜔𝑚𝑚(𝑡𝑡) and the menu
cost specific component 𝐾𝐾𝑚𝑚(𝑡𝑡) . Due to the menu cost, firms’ current price 𝑝𝑝𝑡𝑡 (𝑧𝑧) can

deviate from the optimal price 𝑝𝑝t∗ (𝑧𝑧) that would prevail in a hypothetical economy

where the menu cost is absent. Firms change their price when the absolute value gap
between the two prices |𝑝𝑝𝑡𝑡 (𝑧𝑧) − 𝑝𝑝𝑡𝑡∗ (𝑧𝑧)| is sufficiently large so that it is profitable forthem to change the price even after paying the menu cost. The value of a firm is
described by the equation below.
7

It is also notable that existing studies, such as Golosov and Lucas (2007) and Nakamura andSteinsson (2008), already argue that the standard Taylor and Calvo models do not agree with key
facts observed in micro price data other than seasonality.

21

𝑉𝑉𝑚𝑚(𝑡𝑡) �𝑝𝑝𝑡𝑡−1 (𝑧𝑧), 𝐴𝐴𝑡𝑡 (𝑧𝑧)�
𝑃𝑃𝑡𝑡

−𝜃𝜃

𝑝𝑝𝑡𝑡 (𝑧𝑧)= max �𝐶𝐶 ��𝑝𝑝𝑡𝑡 (𝑧𝑧)𝑃𝑃𝑡𝑡+ 𝛽𝛽E𝑡𝑡 𝑉𝑉𝑚𝑚(𝑡𝑡+1) ��𝑝𝑝𝑡𝑡 (𝑧𝑧) 𝜔𝜔𝑚𝑚(𝑡𝑡)−� − 𝜔𝜔𝑚𝑚(𝑡𝑡) 𝐾𝐾𝑚𝑚(𝑡𝑡) 𝟏𝟏�𝑝𝑝𝑡𝑡 (𝑧𝑧) ≠ 𝑝𝑝𝑡𝑡−1 (𝑧𝑧)�
𝑃𝑃𝑡𝑡
𝐴𝐴𝑡𝑡 (𝑧𝑧)𝑝𝑝𝑡𝑡 (𝑧𝑧), 𝐴𝐴𝑡𝑡+1 (𝑧𝑧)�� (5)𝑃𝑃𝑡𝑡+1

Note that the state variables consist of the relative price 𝑃𝑃𝑡𝑡−1 𝑝𝑝𝑡𝑡−1 (𝑧𝑧) and thetechnology level 𝐴𝐴𝑡𝑡 (𝑧𝑧). Clearly the gap |𝑝𝑝𝑡𝑡 (𝑧𝑧) − 𝑝𝑝𝑡𝑡∗ (𝑧𝑧)| changes with these variables.

We consider three versions of the model, which we call models A, B, and C hereafter.

These models are identical except for the setting regarding either the menu cost
component 𝐾𝐾𝑚𝑚(𝑡𝑡) or the real wage 𝜔𝜔𝑚𝑚(𝑡𝑡) . In model A, we assume that the menu cost

temporarily declines twice a year, in March and September, due to a decline in 𝐾𝐾𝑚𝑚(𝑡𝑡) ,and stays at a constant value in other months, while 𝜔𝜔𝑚𝑚(𝑡𝑡) is constant. In models B andC, we assume that real wages temporarily increase in March and September, while 𝐾𝐾𝑚𝑚(𝑡𝑡)is constant. The two models differ in terms of the pace at which once-increased real
wages revert back to their original level. Note that we assume that firms are informedof these seasonal changes in menu costs or real marginal costs.
The settings are shown in Figure 19. The top panels show the settings regarding themenu cost component 𝐾𝐾𝑚𝑚(𝑡𝑡) and the real wage 𝜔𝜔𝑚𝑚(𝑡𝑡) . As for the menu cost

component 𝐾𝐾𝑚𝑚(𝑡𝑡) , it declines in March and September in model A, whereas it stays
constant throughout the year in models B and C. As for the real wage 𝜔𝜔𝑚𝑚(𝑡𝑡) , it stays

constant throughout the year in model A, whereas it increases in these two months inmodels B and C. 8 The middle table shows other model parameters. The values are8

For models B and C, we choose these settings only for the purpose of showing how seasonal cycles

of real marginal costs affect seasonality of price changes. Admittedly, there are other ways tocalibrate seasonal cycles of real marginal costs. One way is to use the actual seasonal patterns ofreal wages themselves instead of the series shown in the upper panels of Figure 19 that are
generated in an ad hoc manner. Based on the data of the Monthly Labour Survey from 1990 to 2019,the seasonal component extracted by the X12 filter has two peaks, similar to model B, but in Juneand December for “Total Cash Earnings,” reflecting the bonuses typically paid during the summer
22

almost the same as those in Nakamura and Steinsson (2008). The bottom panels show
the seasonal patterns of the implied nominal wage that are computed from the annual
inflation rate, exogenously set to be 2%, and variations in the real wage in the threemodels. In model A, the nominal wage increases one-for-one with the inflation rate. Inmodels B and C, the nominal wage exhibits seasonal variations.
In our simulation, for each of the models, we focus on what we refer to as the cyclicalsteady state. In this steady state, while each firm faces uncertainty due to the presenceof idiosyncratic shocks, there is no uncertainty at the aggregate level. Because of
seasonal variations in parameters, i.e., 𝐾𝐾𝑚𝑚(𝑡𝑡) for model A and 𝜔𝜔𝑚𝑚(𝑡𝑡) for models B andC, there are monthly changes in the endogenous variables, such as the frequency andsize of price changes. However, these variables return to the same value after a periodof one cycle. Note that the sum of monthly inflation rates over 12 months equals thepredetermined value of 2% unless noted otherwise.

5.2 Model-Generated Seasonal Patterns
Figure 20 shows the time path of (a) the frequency of price changes and (b) the sizeof price changes under the three models from January to December. In model A, thefrequency increases in March and September for both price increases and decreases.
Other things being equal, in this model, firms are incentivized to change prices in Marchand September even when the absolute value of the gap between the desired andactual price |𝑝𝑝𝑡𝑡 (𝑧𝑧) − 𝑝𝑝𝑡𝑡∗ (𝑧𝑧)| is not so large. Consequently, the average of the size of

price changes made by firms changing prices in these months tends to be smaller thanin the other months. For the twelve samples here, the correlation between the frequency
of price increases and decreases is 0.96, that is, positive. The correlation between thefrequency and the size of price changes is negative, specifically -0.92 and -0.86 for priceincreases and decreases, respectively. Admittedly, there are quantitative differences

and winter. The seasonal component of the “Scheduled Cash Earnings” data, that is considered lessaffected by bonuses show no pattern of seasonal peaks except that it declines in January and slightly
increases in April and June. Based on our simple state-dependent pricing model, both seasonalpatterns of real wages induce a negative correlation between the frequency of upward price changes
and the frequency of downward price changes and a positive correlation between the frequency
and size of price changes, contrasting with the observations based on our scanner data.
23

from the actual data, but model A at least agrees with the data qualitatively in terms of
the sign of the correlation of these variables.
The frequency of price increases rises in March and September as well in models B
and C. In these models, the desired price for firms 𝑝𝑝𝑡𝑡∗ (𝑧𝑧) rises in March and September
due to the rise in real marginal costs. Therefore, for firms whose current price is lower

than the desired price, the price gap 𝑝𝑝𝑡𝑡∗ (𝑧𝑧) − 𝑝𝑝𝑡𝑡 (𝑧𝑧) becomes larger, which motivatesthese firms to pay the menu costs and set a higher price. In contrast to model A,

however, the frequency of price decreases falls in March and September. This is because
a rise in the real marginal cost reduces the price gap 𝑝𝑝𝑡𝑡 (𝑧𝑧) − 𝑝𝑝𝑡𝑡∗ (𝑧𝑧) for firms whose

current price is higher than the desired price, which in turn disincentivizes these firms
from reducing their prices.

The dynamics of the size of price changes in models B and C is more complicated.
On the one hand, if the problem were static and firms independently maximized theircurrent profits, the rise in the desired price in March and September would contributeto larger price increases and to smaller price decreases in these months than in othermonths. On the other hand, the dynamic and collective nature of equation (5)introduces additional mechanisms. For example, higher frequency of price increases inMarch and September leaves a smaller number of firms whose prices are not adjusted
to the desired prices, shrinking the range of the distribution of firms’ prices. Thismechanism contributes to the reduction of the average size of price increases in thefollowing months (April and October). The mixture of these different mechanisms leadto complicated behavior in the evolution of the size of price changes.
The key differences between model A and the other two models are two-fold. First,
whereas in model A the frequency of price increases and that of price decreases are
positively correlated, in the latter two models the correlation is negative. Second,
whereas the frequency and size of price changes are negatively correlated in model A,
the same does not hold in models B and C. The results in model B and C do not agreewith the empirical observations even qualitatively. For model B, the correlation between
the frequency of price increases and decreases is -0.99, which is negative, and thecorrelation between the frequency and the size of price changes is modestly positive,
specifically 0.57 and 0.06 for price increases and decreases, respectively. For model C,
the correlation between the frequency of price increases and decreases is -0.96, which24

is again negative, and the correlation between the frequency and the size of pricechanges is again modestly positive, specifically 0.61 and 0.44 for price increases anddecreases, respectively.
Figure 21 shows the time path of inflation rates at the top and that of net frequency
of price changes at the bottom under the three different models. In all of the models,the inflation rate increases in March and September and, as in the data, the time pathof net frequency traces that of the inflation rate. This is because, as shown in Figure 20,
the seasonal variations in the frequency of price changes are more volatile than thosein the size of price changes in all three of the models, which is consistent with ourscanner data shown in Figure 4 as well.
Figure 22 shows the time path of the frequency and size of price changes undervarious settings of the annual inflation rate in model A. It can be seen that as the annual
inflation rate increases, the seasonal patterns of the frequency of price increases
become more volatile. In other words, the difference between the level of the frequency
in the two months, March and September, and the other ten months becomes larger
when the annual inflation rate is higher. In contrast, as the annual inflation rate increases,
the seasonal patterns of the frequency of price decreases become less volatile. In otherwords, the difference between the two months and other ten months shrinks. This isbecause when the annual inflation rate is high, each month a larger portion of firms see
a widening of the negative gap, pt (z) − p∗t (𝑧𝑧) < 0, and these firms face an additional

incentive to increase prices in particular during months in which the menu costs are low.
By contrast, firms are less likely to face a positive gap, pt (z) − p∗t (𝑧𝑧) > 0 under a positive

annual inflation. These firms also reduce their prices in these months but the share ofsuch firms is limited. The volatility of seasonality in the size of price changes, on theother hand, does not seem to depend on the annual inflation rate in a simple andmonotonic way. All in all, the model-generated time path is qualitatively consistent withthe data shown in Figure 18. 9

9Figure 23 shows the results of a similar simulation based on model C. It can be seen that as theannual inflation rate rises from 1% to 8%, the volatility of the seasonal cycle of the frequency of
upward price changes increases and that of downward price changes decreases, similar to the resultsof model A. The seasonal pattern of the frequency of price changes, however, differs between priceincreases and price decreases in the sense that the frequency declines in March and September for25

5.3 Discussion
Our simulation exercise underscores the importance of seasonal variations in menu
costs in bringing the state-dependent model closer to the data. In the model, because
the menu cost declines in a particular month, increases in the frequency of priceincreases and decreases become synchronized and the frequency and size of pricechanges become negatively correlated. In contrast, seasonal changes in real marginal
costs alone are unable to reproduce these seasonal patterns observed in the data. In
this sense, our results accord with the argument made by Nakamura and Steinsson(2008) stressing the importance of time-dependent elements in price dynamics. 10, 11
Admittedly, however, while the model with seasonal cycles in menu costs does accountfor the key moments qualitatively, there is still a gap to be filled between the model andthe data quantitatively. In particular, the model predicts a strong negative correlation
between the frequency and size of price changes, contrasting with a weak correlation
seen in the data. 12 Our simulation results should therefore be interpreted as stressing
the role played by seasonal cycles in menu costs but should not be interpreted asindicating that other factors, including variations in real marginal costs, do not play a
price decreases, whereas it rises in March and September for prices increases.
10

It is also important to note that our model is not a pure time-dependent model such as that of

Taylor (1980) or Calvo (1983), but instead consists of both time-dependent elements and statedependent elements in one model. Indeed, the observation [4], i.e., the responsiveness of theseasonality of the frequency of price changes to changes in the category-level inflation rate for theyear, suggests the presence of a state-dependent element in the seasonal component of pricechanges in the data.
11

Our argument that there are seasonal cycles in menu costs is related to the discussion of the

CalvoPlus model analyzed in Nakamura and Steinsson (2010). They construct a menu cost model inwhich a certain fixed portion of firms receives an opportunity to change their prices at a low menu
cost and the presence of these low-repricing opportunities that are largely orthogonal to the firms’
desire to change the price mutes the selection effect.
12

While it is true that the frequency and size of price changes are negatively correlated for a certain

set of categories regardless of how one computes the seasonal components, the degree of thenegative correlation differs depending on the methodology used for extracting the seasonal
component. The negative correlation becomes salient when the seasonal component is computed
by normalizing the original data with the annual data, as shown in Figure C1.
26

role.
The time-varying menu costs are to some extent consistent with the explanations of
why prices are sticky as presented in early works, including Zbaracki et al. (2004) andBlinder et al. (1998). Using data from a large U.S. industrial manufacturer and its
customers, Zbaracki et al. (2004) study the nature of menu costs and argue thatmanagerial costs that consist of information gathering, decision making andcommunication costs, and customer costs that consist of communication andnegotiation costs are quantitatively larger than the physical menu cost. These arguments
imply that the size of menu costs can decline if communication and negotiation aresmooth even when the physical menu costs do not change. Blinder et al. (1998), basedon a survey of firms in the U.S., states that “firms might like to raise or lower prices, but
hesitate to do so unless and until other firms move first. Once other firms move, they
follow quickly.” Put differently, firms are less reluctant to raise or lower prices today ifthey know that other firms will move today. If there is a common expectation amongfirms that other firms will change their prices in March and September, for example,
non-physical menu costs may fall, leading to an increase in the frequency of pricechanges.
These characteristics of menu costs have several broad implications for theunderstanding of price dynamics. While the standard model does not pay attention tochanges in menu costs, assuming they are constant over time, if changes in menu costs
play an important role in shaping the seasonality of price dynamics, the samemechanism may also be at play in price dynamics beyond the seasonal variations. 13,14
In other words, the inflation dynamics can vary in response to a change in menu costs
even without changes in marginal costs or other economic conditions. In addition, withthe size of menu costs differing across months, other things being equal, it is optimal
13

Admittedly, there are works that endogenize the degree of price flexibility in a New Keynesianframework, for example, Romer (1990) and Kimura and Kurozumi (2010). Changes in the degree of
flexibility of prices in our model are, however, different from theirs in the sense that they depend ontime rather than state.
14

Indeed, Sudo et al. (2014), studying scanner data, as in this paper, from 1988 to 2013, documentthat the size of price changes has been declining and the frequency of price changes has been
increasing over the sample period and argue that these secular changes may reflect a long-term
decline in menu costs.
27

for firms to reflect a change in the economic environment, including monetary policy
shocks, in their prices in months in which menu costs are low. Consequently, thetransmission of shocks to prices and then output may be altered depending on themonth in which these shocks occur. This prediction accords well with the argument
made by Olivei and Tenreyro (2010) that the month matters to the transmission ofmonetary policy shocks.

6 Conclusion
Seasonality of price dynamics has been identified in a good number of existing studies,but a comprehensive picture, including the difference between upward price changes
and downward price changes or the mechanism behind the seasonality, has yet to be
fully drawn so far.
To fill the gap, we first study point-of-sale (POS) scanner data covering the period
from 1990 to 2021 in Japan to draw an overall picture of seasonality of goods prices.
Our empirical findings are as follows. First, the frequency of price increases anddecreases tend to be high in March and September and low in other months for mostcategories. Second, for the majority of categories, seasonal cycles of the frequency of
price changes are negatively correlated with those of the size of price changes, though
the latter is less pronounced and less synchronized across categories and between the
direction of price change. Third, the seasonal patterns of overall inflation track theseasonal pattern of net frequency, i.e., the difference between the frequency of priceincreases and that of price decreases, and are moderately synchronized across
categories. Fourth, the pattern of seasonal cycles of the frequency of price changes has
been stable relative to that of the size of price changes. The pattern is, however,
responsive to changes in the annual inflation rate of the category for the year. That is,
the seasonality of the frequency of price increases (decreases) becomes more
pronounced (less pronounced) when the category-level inflation rate for the year is high.
Such responsiveness is not observed for the size of price changes. These observations
are generally made regardless of the method used to extract seasonal components. The
same general pattern is observed whether one uses a rolling regression with monthly
dummy variables (our baseline), X12-ARIMA (shown in Appendix B), or the original series

28divided by the average of the same year (shown in Appendix C).
Next, we conduct simulation exercises using a menu cost model and explore the causesof seasonality of price changes. Our exercise underscores the importance of seasonalchanges in menu costs in generating seasonal patterns of price changes that areconsistent with the data. Theoretically, when menu costs fall in March and September,
firms are incentivized to change their prices in both upward and downward directionsin the two months because it is less costly for firms to adjust prices in these months. This
yields a positive correlation between frequency of price increases and that of pricedecreases and a negative correlation between the frequency and size of price changes.
The key contributions of the paper lies in documenting seasonal patterns of pricedynamics in detail and offering an explanation for these seasonal patterns. While
existing studies document the presence of seasonal patterns, they are not necessarily
focused on the seasonality itself, nor do they study the causes and implications of theseasonality. In addition to these points, this paper contributes to a better understandingof price dynamics in a broader context. For example, synchronized ups and downs in
the frequency of price changes indicate the importance of coordination rather thanphysical menu costs in explaining staggered price setting of firms. In addition, thepresence of seasonal cycles in menu costs indicates that the speed of transmission may
change depending on the month in which the exogenous shock occurs, similar to theargument made by Olivei and Tenreyro (2010). Also, if months matter in this sense, thenthe central bank may also need to take into account the role of months in monetary
policy implementation and communication.
There are two caveats regarding this study. First, the current analysis only indicatesthat the observed seasonal patterns of price dynamics are consistent with the presenceof seasonal cycles in menu costs and that such seasonal cycles in menu costs can beinterpreted as representing the non-physical component of “menu costs” discussed in
Zbaracki et al. (2004) and possibly representing changes in implicit coordination amongfirms, as discussed in Blinder et al. (1998). The current paper is therefore silent about
what factors have shaped seasonal patterns in menu costs in the way suggested by thedata (i.e., a fall in March and September and a rise in other months). The timing of
seasonal cycles in menu costs may be related to the fact that the fiscal year begins in
April and changes in institutional settings, including various tax rates, are often made in29

April. 15 It may also be related to Shunto, as stressed in Olivei and Tenreyro (2010),
though the changes in wages themselves do not seem to play the dominant role inshaping the seasonal patterns of price dynamics. Second, the nature of seasonalpatterns of price dynamics may be different from those documented in this study if
seasonal patterns of service prices or those of products in other jurisdictions are
examined. Regarding the distinction between goods and services, the monthly increasein the official CPI tends to be high in April and October for service prices, contrastingwith goods prices that tend to be high in March and September, as shown in Figure 1,
which may in turn indicate that the seasonal patterns of menu costs in the service sector
may be slightly different from those in the goods sector. Regarding the difference acrossborders, while there are some similarities in terms of the seasonal patterns of inflationin the U.S. and Japan, as shown in Figure 1, the seasonal pattern of the frequency of
price changes documented in Nakamura and Steinsson (2008) differs from ours.
Exploring the deeper sources of the seasonal patterns and comparing the seasonalpatterns of price changes across sectors and across countries may help improve ourunderstanding of price dynamics, including the causes of price stickiness. These are left
as the agenda for future research.

15

See Bunn and Ellis (2012) for a related discussion for U.K. consumer prices. They document that alarger portion of prices increases in April than in other months and argue that "Excluding all saleprices, consumer prices are most likely to change in April. That could reflect changes in duties and/or
firms changing prices to coincide with the start of a new fiscal year."30

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33

A. Model
We extend the simple partial equilibrium menu cost model for firms by Nakamura
and Steinsson (2008) to include seasonally varying parameters. Each month, firms, which
are subject to idiosyncratic shocks, decide whether to adjust their price while paying amenu cost or not. The firm’s policy function is influenced by seasonal variation inparameters, but is assumed to return to its original form after one cycle, corresponding
to 6 months, under the cyclical steady state equilibrium that we focus on in this paper.
In this setting, the total inflation rate over 1 cycle is exogenously given, whereas themonthly inflation rate as well as the frequency and size of price increases/decreases foreach month are endogenously determined by firms’ responses to seasonally varying
parameters.
In our model, firm 𝑧𝑧 with idiosyncratic productivity 𝐴𝐴𝑡𝑡 (𝑧𝑧) has linear production

function

𝑦𝑦𝑡𝑡 (𝑧𝑧) = 𝐴𝐴𝑡𝑡 (𝑧𝑧)𝐿𝐿𝑡𝑡 (𝑧𝑧),where the logarithm of idiosyncratic productivity follows an AR(1) process
𝜀𝜀𝑡𝑡 (𝑧𝑧)~𝑁𝑁(0, 𝜎𝜎𝜀𝜀2 ).

log�𝐴𝐴𝑡𝑡 (𝑧𝑧)� = 𝜌𝜌log�𝐴𝐴𝑡𝑡−1 (𝑧𝑧)� + 𝜀𝜀𝑡𝑡 (𝑧𝑧),Firm 𝑧𝑧 faces demand 𝑦𝑦𝑡𝑡 (𝑧𝑧) that depends both on its own price level 𝑝𝑝𝑡𝑡 (𝑧𝑧) and on theaggregate price level 𝑃𝑃𝑡𝑡 according to−𝜃𝜃𝑝𝑝𝑡𝑡 (𝑧𝑧)𝑦𝑦𝑡𝑡 (𝑧𝑧) = 𝐶𝐶 ��
𝑃𝑃𝑡𝑡Firms face a common real marginal cost, given by𝜔𝜔𝑚𝑚(𝑡𝑡) =.

𝑊𝑊𝑚𝑚(𝑡𝑡)
𝑃𝑃𝑡𝑡where 𝑚𝑚(𝑡𝑡) denotes the month of time 𝑡𝑡 satisfying 𝑚𝑚(𝑡𝑡) = mod(𝑡𝑡 − 1,6) + 1 and𝑊𝑊𝑚𝑚(𝑡𝑡) is the nominal wage rate. As in the model by Nakamura and Steinsson (2010), in

order to adjust their prices, firms have to hire additional workers and pay a menu cost
given by𝑊𝑊𝑚𝑚(𝑡𝑡) 𝐾𝐾𝑚𝑚(𝑡𝑡) .

Firms’ real profit can be written in the following form:
34𝛱𝛱𝑡𝑡 (𝑧𝑧) =𝑝𝑝𝑡𝑡 (𝑧𝑧)𝑦𝑦𝑡𝑡 (𝑧𝑧) − 𝑊𝑊𝑚𝑚(𝑡𝑡) 𝐿𝐿𝑡𝑡 (𝑧𝑧) 𝑊𝑊𝑚𝑚(𝑡𝑡) 𝐾𝐾𝑚𝑚(𝑡𝑡)−1�𝑝𝑝𝑡𝑡 (𝑧𝑧) ≠ 𝑝𝑝𝑡𝑡−1 (𝑧𝑧)�,
𝑃𝑃𝑡𝑡
𝑃𝑃𝑡𝑡

which can be rewritten as−𝜃𝜃𝑝𝑝𝑡𝑡 (𝑧𝑧)𝛱𝛱𝑡𝑡 (𝑧𝑧) = 𝐶𝐶 ��
𝑃𝑃𝑡𝑡𝑝𝑝𝑡𝑡 (𝑧𝑧) 𝜔𝜔𝑚𝑚(𝑡𝑡)�−� − 𝜔𝜔𝑚𝑚(𝑡𝑡) 𝐾𝐾𝑚𝑚(𝑡𝑡) 1�𝑝𝑝𝑡𝑡 (𝑧𝑧) ≠ 𝑝𝑝𝑡𝑡−1 (𝑧𝑧)�.
𝑃𝑃𝑡𝑡
𝐴𝐴𝑡𝑡 (𝑧𝑧)The optimization problem that each firm faces in this model is therefore written as
𝑉𝑉𝑚𝑚(𝑡𝑡) �𝑝𝑝𝑡𝑡−1 (𝑧𝑧)𝑝𝑝𝑡𝑡 (𝑧𝑧), 𝐴𝐴𝑡𝑡 (𝑧𝑧)� = max �𝛱𝛱𝑡𝑡 (𝑧𝑧) + 𝛽𝛽E𝑡𝑡 𝑉𝑉𝑚𝑚(𝑡𝑡+1) �, 𝐴𝐴 (𝑧𝑧)��.
𝑝𝑝𝑡𝑡 (𝑧𝑧)𝑃𝑃𝑡𝑡
𝑃𝑃𝑡𝑡+1 𝑡𝑡+1

In model A described in the main text, 𝐾𝐾3 > 𝐾𝐾1 = 𝐾𝐾2 = 𝐾𝐾4 ⋯ = 𝐾𝐾6 , while 𝜔𝜔𝑚𝑚(𝑡𝑡) isconstant. In contrast, in models B and C, 𝜔𝜔𝑚𝑚(𝑡𝑡) varies with the month, while 𝐾𝐾𝑚𝑚(𝑡𝑡) isconstant. In each model, under the seasonally varying parameter values, we seek thecyclical steady state satisfying 𝑉𝑉𝑚𝑚(𝑡𝑡) = 𝑉𝑉𝑚𝑚(𝑡𝑡+6) and ln(𝑃𝑃𝑡𝑡+6 ⁄𝑃𝑃𝑡𝑡 ) = 𝜋𝜋 ∗ = 0.01. The valuesof the other parameters mostly follow those used in Nakamura and Steinsson (2008)
and are shown in Figure 19.

We use the following iterative algorithm to obtain the solution:A) Specify finite grid points for the state variables 𝑝𝑝𝑡𝑡−1 (𝑧𝑧)⁄𝑃𝑃𝑡𝑡 and 𝐴𝐴𝑡𝑡 (𝑧𝑧).

B) Assume a particular path of monthly inflation rate 𝜋𝜋𝑚𝑚(𝑡𝑡) = 𝑙𝑙𝑙𝑙(𝑃𝑃𝑡𝑡 ⁄𝑃𝑃𝑡𝑡−1 ).

C) Given the monthly inflation rate in (B), solve the firms’ optimization problem
described above by value function iteration to obtain monthly policy functions.
D) From the monthly policy functions and the idiosyncratic productivity process,calculate the density of firms on the grid for each month.
E) Using the monthly density and monthly policy function, calculate the endogenous
path of the monthly inflation rate.
F) Repeat steps (C) to (E) until convergence.

35

B. Characteristics of Seasonal Components Extracted by the X12
FilterIn the main text, we extract the time series of seasonal components of price dynamics
by conducting a rolling estimation of equation (4). In this appendix, as a sensitivity
analysis, we use an alternative filter, the X12, to extract the time series of the seasonal
components. We then use the X12 filtered seasonal components for the analysis ofseasonality of price changes. Compared with the regression-based estimates ofseasonal components in our baseline, the seasonal components estimated by the X12
filter tend to vary less from year to year.
Figure B1 shows the seasonal components of the frequency and size of price changes
for all categories extracted by the X12 filter. Figure B2 shows the correlation between
the frequency of price increases and decreases, the correlation between the size of priceincreases and decreases, and the correlation between the frequency and size of pricechanges. Figure B3 shows the seasonal component of the POS inflation rate, together
with the net frequency and the net size of price changes. Figures B4 and B5 show theseasonal pattern of frequency and size for the high and low inflation periods and thecorrelation between the standard deviation of the seasonal component of the frequency
or the size of price changes and the annual inflation rate of the same year.
It can be seen that for the four key observations, three of them hold true for the series
extracted by the X12 filter. First, frequencies of both price increases and decreases tend
to rise in March and September for most categories, exhibiting a two-humped pattern,with the former more pronounced than the latter. Second, for most categories, seasonalpatterns of overall inflation track seasonal patterns of net frequency, i.e., the difference
between the frequency of price increases and that of price decreases, and aremoderately synchronized across categories. Third, the seasonal patterns of thefrequency of price changes of a category have been stable over our sample period but
are responsive to annual changes in the inflation rate of the same category for the year.
The only exception is the correlation between the frequency and size of price changes.
Though the correlation is almost zero for most of the categories, the median of thecorrelation across categories is positive, contrasting with the results obtained from ourbaseline series extracted by equation (4). The reason for the difference lies in the factthat the seasonal components of a year are computed using a longer window in the X1236

filter compared to our baseline method. For example, while the first several years of thesample from 1990 and 1992 correspond to the period in which the negative correlation
between the frequency and size of price changes is the most pronounced, the estimatedseasonal components of these years are affected by the realizations in the following
years when the negative correlation became smaller.

37

C. Characteristicsof

Seasonal

Components

Extracted

byNormalizing with the Annual ValueIn the main text, we extract the time series of seasonal components by conducting arolling estimate of equation (4) and study the characteristics of these series. In Appendix
B, we show that the key observations generally hold for seasonal components extracted
by the X12 filter. In this Appendix C, we use an alternative filter to extract the seasonal
components. For a variable 𝑦𝑦𝐽𝐽𝐽𝐽 , we divide it by the yearly value, i.e., the average of the
12 monthly values of the same year as the time t, which we denote as 𝑦𝑦�𝐽𝐽𝐽𝐽 . The series

𝑦𝑦𝐽𝐽𝐽𝐽 /𝑦𝑦�𝐽𝐽𝐽𝐽 is considered as having had the year effect removed. We then repeat the same

empirical exercises as the main text for the series. Compared with our baseline method
and X12, the method applied in this Appendix C allows for flexible changes in seasonal

components from year to year, which in turn implies that it could regard any
idiosyncratic shocks as seasonality, overestimating the variation of seasonality.
Figure C1 shows the correlation between price increases and decreases for thefrequency and size of price changes at the top and the correlation between thefrequency and size of price changes at the bottom. It can be seen that, as in the analysis
in the main text, the seasonal components of the frequency of price increases are
positively correlated with those of price decreases and those of frequency and size arenegatively correlated for both price increases and decreases, indicating thatobservations [1] and [2] hold for this methodology as well.
Figure C2 shows the correlation between the standard deviation of the seasonal
component within a year of a category and the yearly POS inflation rate of the category
for the year. It is seen that the seasonal component of the frequency of price changes
is responsive to changes in the annual category-level inflation rate for the year in themanner documented in the main text. When the yearly POS inflation is high (low),
variation in seasonality becomes high (low), indicating that observation [4] holds.
Figure C3 shows the seasonal component of the POS inflation rate, together with the
net frequency and the net size of price changes. It is seen that observation [3] also holds
true for the series extracted by normalizing with the annual value. 16

16

Because the yearly value 𝑦𝑦�𝐽𝐽𝐽𝐽 of the POS inflation rate and the net frequency could be negative,38

D. Degree of Synchronization across Foods and Others
In the main test, we report that the timing of the frequency of price changes is
synchronized across different categories. Under the premise that such synchronizationarises from implicit coordination among firms, it is likely that similar goods tend to besynchronized more. Our sample category includes 199 categories of goods and there
are 145 processed food products and 54 other non-food products, as shown in Table 1.
In order to see the relation between the degree of synchronization of seasonality ofprice changes and the degree of similarity among different goods, in this appendix we
divide the sampled 199 categories into two groups - “processed foods” and “others” and compute the correlation of categories within each of the groups and across the twogroups.
Figure D1 shows the degree of synchronization (a) across all categories, (b) within“processed foods,” (c) within “others,” and (d) between “processed foods” and “others,”
for the frequency of price increases. For (b) and (c), we compute the correlation for pairs
within the group, and for (d), we compute the correlation for pairs between a category
in “processed foods” group and a category in “others” group. It is notable that theseasonal components of the frequency of price increases are positively correlated themost for the pairs within “others” and less so for the pairs within “processed foods” interms of the median. The positive correlation also is seen for pairs between the twogroups. The median is 0.19 and 12,233 pairs, which is about 62% of the total number ofpairs, are significantly positively correlated at the 5% level. The observation that apositive relationship is more pronounced for pairs within the same group rather thanfor pairs between the groups indicates a possibility that the degree of synchronization
may be to some extent affected by the similarity of goods. Figure D2 shows the degreeof synchronization for the frequency of price decreases. Again, a positive correlation is
obtained for all of the four cases. It is also seen that the correlation is higher for pairs
within groups and less so for pairs between groups.
Figures D3 and D4 show the case for the size of price changes. Compared with thefrequency of price changes, the correlation is muted for both within and between
groups.

we subtract it from variable 𝑦𝑦𝐽𝐽𝐽𝐽 instead of dividing for these series.
39

E. Number of Statistically Significant Monthly Dummies
In the main text, we measure the size of seasonality with the size of the coefficientsof monthly dummies in equation (4). Based on the measure, we show that the frequency
of price changes has two peaks, rising in March and September and falling in June andthat the size of price changes does not have salient seasonal patterns. In this appendix,
we focus on the statistical significance of the coefficients of monthly dummies inequation (4) and show that similar observations can be made.
Figure E1 shows the number of statistically significant dummies in the estimation ofequation (4) for the frequency of price changes at the top and for the size of pricechanges at the bottom. The x-axis is month and the y-axis is the number of dummy
variables that are different from zero at 95% statistical significance. Note that because
the dummies are estimated with a three-year rolling estimation for each of 199
categories, there are 57,362 estimates. It is seen that the two months with the highest
number of statistically significant positive dummy variables are March and September
and the two months with the highest number of statistically significant negative dummy
variables are July and December for the frequency of upward price changes. For thefrequency of downward price changes, the two months with the highest number ofstatistically significant positive dummy variables are March and November and the twomonths with the highest number of statistically significant negative dummy variables
are January and August. For the size of price changes, the number of dummies that are
statistically significantly different from zero is far more moderate and the difference ofthe number of such dummies across months is less pronounced relative to thefrequency of price changes.
Figure E2 shows the case when the dummies are estimated based on the full-sample
period instead of a three-year rolling estimation. Similar observations can be made tothose shown in Figure E1.

40

Figure 1: Monthly Changes in CPI(a) Japan (CPI, all)2.0

change from previous month, %

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

1990
1992
1994

-1.0
-1.5

Jan

Mar May

Sep

2000
2002
2004

2001
2003

-0.5

1991
1993
Jul

change from previous month, %

-1.0
-1.5

Nov

Jan

Mar May

Jul

Sep

Nov

(b) Japan (CPI, goods less fresh food and energy)
2.0change from previous month, %

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

-0.5

1990
1992
1994

-1.0
-1.5

Jan

Mar May

Sep

2000
2002
2004

2001
2003

-0.5

1991
1993
Jul

change from previous month, %

-1.0
-1.5

Nov

Jan

Mar May

Jul

Sep

Nov

(c) US (CPI, commodities less food and energy)
2.0change from previous month, %
1990
1992
1994

1.5

1.0

2.0

1991
1993

1.0
0.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0
Jan

Mar May

2000
2002
2004

1.5

0.5

-1.5

change from previous month, %

Jul

Sep

-1.5

Nov

41

Jan

Mar May

2001
2003

Jul

Sep

Nov

Figure 2: Decomposition of the Annual CPI Change in Japan(a) CPI (goods less fresh food and energy)
5change from previous month, %

4
3
2
1
0
-1
-2
-3
Jan
Jun
Nov

-4
-5
CY1991

93

95

Feb
Jul
Dec
97

99

01

Mar
Aug
yoy change
03

05

07

09

Apr
Sep
11

May
Oct
13

15

17

19

(b) CPI (all)5change from previous month, %
Jan
Jun
Nov

4

Feb
Jul
Dec

Mar
Aug
yoy change

Apr
Sep

May
Oct

3
2
1
0
-1
-2
-3
CY1991

93

95

97

99

01

03

05

07

09

11

Note: The figures are adjusted for changes in the consumption tax rate.

42

13

15

17

19

Figure 3: Aggregate POS Inflation
4

12-month backward moving average, %yoy, %

4

aggregate POS inflation
CPI (goods less fresh food and energy, right scale)

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3
CY1990

1995

2000

2005

2010

2015

2020

-3

Notes: The CPI figures exclude the effects of the consumption tax hikes. Aggregate POS inflation is
calculated as the weighted average of the inflation rate for each category weighted by sales.

43Figure 4: Importance of Seasonality(a) Variations of seasonality relative to the original series
0.2
0.15
0.1
0.05
0

Freq
(+)

Freq
(-)

Size
(+)

Size
(-)

(b) Variations of seasonality relative to those of the detrended series
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Freq
(+)

Freq
(-)

Size
(+)

Size
(-)

(c) Statistical significance of seasonality
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Freq
(+)

Freq
(-)

Size
(+)

Size
(-)

Note: The upper and middle panels show the across-category distribution of the standard deviation of theseasonal component relative to the average of the original series and to the standard deviation of thedetrended series, respectively. The bottom panel shows the ratio of estimation equations for which
the F-test rejects the null hypothesis that the coefficients of the seasonal components are zero at the
95% level. All panels show the median of all categories and the error bands indicate the 25th and75th percentiles.
44Figure 5: Frequency of Price Changes(a) Category "soybean curd and its products, Tofu"%

25

4
frequency of price increases
frequency of price decreases

20

3
2

15

1
10

0

5

-1

0
CY1990

1995

2000

2005

2010

2015

-2

2020

Jan

Apr

Jul

Oct

Notes: The left panel plots the frequency of price changes of the category "soybean curd and its products."
The right panel shows the average seasonal component of the frequency of price increases and pricedecreases.

(b) Seasonal component across all categories<Frequency of price increases><Frequency of price decreases>

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

Jan

Mar

May

Jul

Sep

-3

Nov

Jan

Mar

May

Jul

Sep

Nov

Notes: The panels plot the median of the average seasonal component of frequency of price changes acrossall categories. The shaded area indicates the 25th and 75th percentile bands.
45

Figure 6: Correlation of Frequency of Price Changes

Number of combinations
p-value >= 0.05
p-value < 0.05

Median: 0.19

p-value < 0.01

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

4,500
4,000
3,500
3,000
2,500
2,000
1,500
1,000
500
0

(b) Correlation of frequency of(b) price decreases
4,500
4,000
3,500
3,000
2,500
2,000
1,500
1,000
500
0

Correlation coefficients

Number of combinations
p-value >= 0.05
p-value < 0.05

Median: 0.21

p-value < 0.01

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9(a) Correlation of frequency of(a) price increases

Correlation coefficients

Notes: The panels show histograms of the correlations of the seasonal components of (a)frequency of
price increases and (b)frequency of price decreases across pairs of 199 categories, respectively.
The data coverage is 1990 to 2021, excluding 2003 and 2004.

(c) Correlation between frequency of price increases and frequency of price decreases

40
30

Number of categories
p-value >= 0.05
Median: 0.33
p-value < 0.05
p-value < 0.01

<Spearman's rank correlation>
40
30
20

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

20Correlation coefficients

Number of categories
Median: 0.44

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9<Pearson's correlation>

Correlation coefficients

Notes: The panels show histograms of the correlations between the seasonal components of the frequency
of price increases and price decreases within the same category. The data coverage is 1990 to 2021,
excluding 2003 and 2004. The left panel is based on Pearson's correlation coefficient. The rightpanel is based on Spearman's rank coefficient, calculated using the average seasonal component.
46

Figure 7: Size of Price Changes(a) Category "soybean curd and its products, Tofu"
%

30

1.5
size of price increases
size of price decreases

28
26

1.0

24

0.5

22
20

0.0

18

-0.5

16
14

-1.0

12
10
CY1990

1995

2000

2005

2010

2015

2020

-1.5

Jan

Apr

Jul

Oct

Notes: The left panel plots the size of price changes of the category "soybean curd and its products."
The right panel shows the average seasonal component of the size of price increases and pricedecreases.

(b) Seasonal component across all categories<Size of price increases><Size of price decreases>

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1.0

Jan

Mar May

Jul

Sep

-1.0

Nov

Jan

Mar May

Jul

Sep

Nov

Notes: The panels plot the median of the average seasonal component of the size of price changes acrossNotes: all categories. The shaded area indicates the 25th and 75th percentile bands.
47

Figure 8: Correlation between Frequency and Size of Price Changes(a) Correlation between frequency of price increases and size of price increases<Pearson's correlation>
50
40
30

Number of categories
Median: -0.11

p-value >= 0.05
p-value < 0.05
p-value < 0.01

20

<Spearman's rank correlation>
30

Number of categories
Median: -0.01

20

10

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

0-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

10Correlation coefficients

Correlation coefficients(b) Correlation between frequency of price decreases and size of price decreases<Pearson's correlation>
50
40
30

Number of categories
p-value >= 0.05
p-value < 0.05
p-value < 0.01

<Spearman's rank correlation>
30

Number of categories
Median: -0.14

20Median: -0.17

20

10Correlation coefficients0-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

0-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

10Correlation coefficients

Notes: The panels show the histogram of the correlations between the seasonal component of the frequency
of price increases (decreases) and the size of price increases (decreases) for all 199 categories. The
data coverage is 1990 to 2021, excluding 2003 and 2004. The left panel is based on Pearson's
correlation coefficient. The right panel is based on Spearman's rank correlation, calculated using the
average seasonal component.
48

Figure 9: Correlation of Size of Price Changes

6,000
5,000
4,000

Number of combinations
p-value >= 0.05
p-value < 0.05

Median: 0.07

p-value < 0.01(b) Correlation of size of(b) price decreases
6,000
5,000
4,000

3,000

2,000

2,000

1,000

1,000

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

3,000Correlation coefficients

Number of combinations
p-value >= 0.05
p-value < 0.05

Median: 0.08

p-value < 0.01

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9(a) Correlation of size of(a) price increases

Correlation coefficients

Notes: The panels show histograms of the correlations of the seasonal component of (a)size of priceincreases and (b)size of price decreases across pairs of 199 categories, respectively. The data
coverage is 1990 to 2021, excluding 2003 and 2004.

(c) Correlation between size of price increases and size of price decreases

50
40

Number of categories
p-value >= 0.05
Median: 0.10
p-value < 0.05
p-value < 0.01

<Spearman's rank correlation>
50
40
30

20

20

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.930Correlation coefficients

Number of categories
Median: 0.10

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9<Pearson's correlation>

Correlation coefficients

Notes: The panels show histograms of the correlations between the seasonal component of the size of
price increases and price decreases within the same category. The data coverage is 1990 to 2021,
excluding 2003 and 2004. The left panel is based on Pearson's correlation coefficients. The rightpanel is based on Spearman's rank correlation, calculated using the average seasonal component.
49

Figure 10: POS Inflation Rate(a) Category "soybean curd and its products, Tofu"
12

%

1.5

10
1.0

8
6

0.5

4
0.0

2
0

-0.5

-2
-4
CY1990

1995

2000

2005

2010

2015

2020

-1.0

Jan

Apr

Jul

Oct

Notes: The left panel plots the POS inflation rate of the category "soybean curd and its products."
The right panel shows the average seasonal component of the POS inflation rate.

(b) Seasonal component across all categories
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5

Jan

Feb

Mar

Apr May

Jun

Jul

Aug

Sep

Oct

Nov Dec

Notes: The panel plots the median of the average seasonal component of the POS inflation rate acrossall categories. The shaded area indicates the 25th and 75th percentile bands.

50

Figure 11: Seasonality of Net Frequency and
Net Size of Price Changes(a) Net frequency of price changes
4

net frequency of pricechanges
inflation rate (right sclae)

3

1.6

0.8

1

0.4

0

0

-1

-0.4

-2

-0.8

-3

-1.2
Jan

Mar May

Jul

Sep

Nov

1.5

1.2

2

-4

(b) Net size of price changes
net size of price changes
inflation rate (right sclae)

1.0

1.6
1.2
0.8

0.5

0.4

0.0

0
-0.4

-0.5

-0.8
-1.0

-1.6

-1.5

-1.2
Jan

Mar May

Jul

Sep

Nov

-1.6

Notes: The panels plot the median of the average seasonal component of net frequency and net size of
price changes across all categories. The shaded area indicates the 25th and 75th percentile bands.

51

Figure 12: Correlation of POS Inflation Rate
across Categories

4,000

Number of combinations
p-value larger than 0.05

3,500Median: 0.10

p-value smaller than 0.05

3,000

p-value smaller than 0.01

2,500
2,000
1,500
1,000

0.8 to 0.9

0.6 to 0.7

0.4 to 0.5

0.2 to 0.3

0 to 0.1-0.2 to -0.1

-0.4 to -0.3

-0.6 to -0.5

-0.8 to -0.7

0

-1 to -0.9

500

Correlation coefficients
Notes: The plot shows the histogram for the correlations of the seasonal component of the POS inflation
rate across pairs of 199 categories. The data coverage is 1990 to 2021, excluding 2003 and 2004.

52

Figure 13: Seasonality in the Two Subsample Periods(a) Frequency of price increases
4(b) Frequency of price decreases
4

early half period
latter half period

3
2

2

1

1

0

0

-1

-1

-2

-2

-3

-3

-4

Jan

Mar May

Jul

Sep

-4

Nov

(c) Size of price increases
1.2
0.8

1.2
0.8
0.6

0.4

0.4

0.2

0.2

0.0

0.0

-0.2

-0.2

-0.4

-0.4

-0.6

-0.6

-0.8

-0.8

-1.0

-1.0

-1.2

-1.2

Mar MayJul

Sep

Mar May

Jul

Sep

Nov

early half period
latter half period

1.0

0.6

Jan

Jan

(d) Size of price decreases

early half period
latter half period

1.0

early half period
latter half period

3

Nov

Jan

Mar May

Jul

Sep

Nov

Notes: The panels plot the average seasonal component in the early and the latter halves of the sample
period for each category. The solid line and circle marker indicate the median of all categories.
The shaded area and dotted line indicate the 25th and 75th percentile bands.

53

Figure 14: Correlation of Seasonality between
the Two Subsample Periods(a) Frequency of price increases
5040(b) Frequency of price decreases

Number of categories

60

Median: 0.75

50

Number of categories
Median: 0.80

40

30

30
20

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

10Correlation coefficients

Correlation coefficients(c) Size of price increases(d) Size of price decreases

Number of categories

30

Number of categories
Median: 0.46

Median: 0.33
20

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

20Correlation coefficients-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.930-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

20Correlation coefficients

Notes: These panels show histograms of 199 categories for the correlation of the average seasonal
component between the early and latter half of the sample for each series. The data coverage
is 1990 to 2021, excluding 2003 and 2004.

54

Figure 15: Annual Inflation and Seasonality ofFrequency of Price Changes(a) Category "soybean curd and its products, Tofu"
4.0

annual average, %annual POS inflation rateseasonality of frequency of price increases (right scale)seasonality of frequency of price decreases (right scale)

3.5
3.0
2.5

2.7
2.4
2.1

2.0

1.8

1.5

1.5

1.0

1.2

0.5

0.9

0.0
-0.5

0.6

-1.0
CY1990

0.3

1995

2000

2005

2010

2015

2020

Notes: Annual POS inflation rate is the annual average of the monthly inflation rate. The dotted line andthe solid line with circle marker indicate the standard deviation of the seasonal component of thefrequency of price increases and price decreases during the year, respectively.

(b) Median of all categories
3.0

annual average, %

2.5
2.0

annual POS inflation rateseasonality of frequency of price increases (right scale)seasonality of frequency of price decreases (right scale)

1.5

3.5
3.0
2.5

1.0
0.5

2.0

0.0
-0.5

1.5

-1.0
-1.5
CY1990

1995

2000

2005

2010

2015

2020

1.0

Notes: Annual POS inflation rate is the annual average of the monthly inflation rate. The dotted line andthe solid line with circle marker indicate the standard deviation of the seasonal component of thefrequency of price increases and price decreases during the year, respectively. For all categories,
the median of all 199 categories in each year is shown.
55

Figure 16: Annual Inflation and Seasonality ofSize of Price Changes(a) Category "soybean curd and its products, Tofu"
4.0

annual average, %

3.5

2.5

annual POS inflation rateseasonality of size of price increases (right scale)seasonality of size of price decreases (right scale)

3.0
2.5

2.0

2.0

1.5

1.5
1.0

1.0

0.5
0.0

0.5

-0.5
-1.0
CY1990

1995

2000

2005

2010

2015

2020

0.0

Notes: Annual POS inflation rate is the annual average of the monthly inflation rate. The dotted line andthe solid line with circle marker indicate the standard deviation of the seasonal component of thesize of price increases and price decreases during the year, respectively.

(b) Median of all categories
3.0

annual average, %2.5

annual POS inflation rateseasonality of size of price increases (right scale)seasonality of size of price decreases (right scale)

2.5
2.0

2.0

1.5

1.5

1.0
0.5

1.0

0.0
-0.5

0.5

-1.0
-1.5
CY1990

1995

2000

2005

2010

2015

2020

0.0

Notes: Annual POS inflation rate is the annual average of the monthly inflation rate. The dotted line andthe solid line with circle marker indicate the standard deviation of the seasonal component of thesize of price increases and price decreases during the year, respectively. For all categories,
the median of all 199 categories in each year is shown.
56

Figure 17: Seasonality in High and Low Inflation Periods(a) Frequency of price increases(b) Frequency of price decreases

4

4

3

3

2

2

1

1

0

0

-1

-1

-2

-2

high inflation period
low inflation period

-3
-4

Jan

Mar May

Jul

Sep

-4

Nov

(c) Size of price increases
1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.5

high inflation period
low inflation period
Jan

Mar May

Jul

Sep

Jan

Mar May

Jul

Sep

Nov

(d) Size of price decreases

1.0

-1.0

high inflation period
low inflation period-3high inflation period
low inflation period

-1.0
-1.5

Nov

Jan

Mar May

Jul

Sep

Nov

Notes: The panels plot the average seasonal component in the high and the low inflaton periods for eachof the categories. The solid line and that with circle marker indicate the median of all categories.
The high inflation period is defined as a year in the top 10% of annual inflation during the wholesample period, and the low inflation period is defined as a year in the bottom 10% for each of thecategories. The shaded area and dotted line indicate the 25 and 75 percentile bands.
57

Figure 18: Correlation between Seasonality of Price Changes
and Annual POS Inflation

30

Number of categories
p-value >= 0.05
p-value < 0.05

40Median: 0.3830

p-value < 0.01

20

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

20Correlation coefficients

40
30

p-value < 0.05

Median: -0.12(d) Size of price decreases

Number of categories
p-value >= 0.05

p-value >= 0.05
p-value < 0.05
p-value < 0.01

Correlation coefficients(c) Size of price increases
50

Number of categories

50

Median: 0.1040p-value < 0.01

30
20

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

20Correlation coefficients

Number of categories
p-value >= 0.05

p-value < 0.05

Median: 0.10

p-value < 0.01

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.940(b) Frequency of price decreases-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9(a) Frequency of price increases

Correlation coefficients

Notes: These panels show histograms of the correlations between the annual POS inflation rate and thestandard deviation of the seasonal components for (a)the frequency of price increases, (b)thefrequency of price decreases, (c)the size of price increases, and (d)the size of price decreases
during the year, respectively. The data coverage is 1990 to 2021, excluding 2003 and 2004.
58

Figure 19: Model Parameters(a) Parameters with seasonality< K (size of menu cost) >
0.030

0.77

model Amodel B or C

0.028

0.024

0.77

model Amodel B

0.76

0.026

model C
0.76

0.75

0.022
0.020

< ω (real marginal cost) >0.74

Jan MarMay Jul Sep Nov

0.75
0.74

Jan Mar May Jul Sep Nov

Jan Mar May Jul Sep Nov(b) Other parameters
Parameters

Value

β

Subjective discount factor (monthly)

0.961/12θDemand elasticity

4

C

Demand1

ρStickiness of idiosyncratic shocks

0.66

σ

Size of idiosyncratic shocks

0.0428(c) Implied nominal wage
< Model A >< Model B >< Model C >103 Jan. = 100

103 Jan. = 100

103 Jan. = 100

102

102

102

101

101

101

100

100

100

99

Jan Mar May Jul Sep Nov

99

Jan Mar May Jul Sep Nov
59

99

Jan Mar May Jul Sep Nov

Figure 20: Seasonality of Price Changes in the Models(a) Seasonality in the frequency of price changes< Model A >< Model B >< Model C >

2.0 %

1.5 %

2.0 %

1.5

1.0

1.5

1.0
0.5

0.5

0.0

0.0

0.0

-0.5

-0.5
-1.0

1.0

0.5

Jan Mar May Jul Sep Nov-1.0

price increases
price decreases-0.5
Jan Mar May Jul Sep Nov-1.0

price increases
price decreases

Jan Mar May Jul Sep Nov

price increases
price decreases

Notes: Each panel indicates the seasonality in the frequency of price increases and decreases in eachmodel under 2% annual inflation. The seasonality is obtained by subtracting the annual average
from the corresponding series.

(b) Seasonality in the size of price changes< Model A >< Model B >< Model C >

0.6 %

0.3 %

0.6 %

0.4

0.2

0.4

0.2

0.1

0.2

0.0

0.0

0.0

-0.2

-0.1

-0.2

-0.4

-0.2

-0.4

-0.6

Jan Mar May Jul Sep Nov

price increases
price decreases-0.3

Jan Mar May Jul Sep Nov

price increases
price decreases-0.6

Jan Mar May Jul Sep Nov

price increases
price decreases

Notes: Each panel indicates the seasonality in the size of price increases and decreases in each model
under 2% annual inflation. The seasonality is obtained by subtracting the annual average fromthe corresponding series.

60

Figure 21: Seasonality of Inflation Rate in the Models(a) Seasonality in the inflation rate
< Model A >0.24 %< Model B >0.35 %
0.30
0.25

0.20

0.20
0.15

0.16

0.10
0.05

0.12

Jan Mar May Jul Sep Nov

0.00

Jan Mar May Jul Sep Nov< Model C >0.45 %
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
Jan Mar May Jul Sep Nov

Notes: Each panel indicates the monthly inflation rate in each model when the annual inflation rate is
set to 2%. The seasonality is obtained by subtracting the annual average from the corresponding
series.

(b) Seasonality in the net frequency of price changes< Model A >1.0 %< Model B >

2.0 %
1.5
1.0

0.5

0.5
0.0

0.0

-0.5
-1.0

-0.5

Jan Mar May Jul Sep Nov-1.5

Jan Mar May Jul Sep Nov< Model C >

3.0 %
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
Jan Mar May Jul Sep Nov

Notes: Each panel indicates the monthly net frequency of price changes in each model when the annual
inflation rate is set to 2%. The seasonality is obtained by subtracting the annual average fromthe corresponding series.

61

Figure 22: Seasonality and Inflation Rate in the Model A(a) Frequency of price changes<Price increases>
3.0 %
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
Jan

<Price decreases>
1.0 %

1%
2%
4%
8%

1%
2%
4%
8%

0.8
0.6
0.4
0.2
0.0
-0.2
-0.4

Mar

May

Jul

Sep

Nov

-0.6

Jan

Mar

May

Jul

Sep

Nov

Notes: Each panel indicates the seasonality in the frequency of price increases and price decreases inmodel A for different annual inflation rates. The seasonality is obtained by subtracting the annual
average from the corresponding series.

(b) Size of price changes<Price increases><Price decreases>

0.3 %

0.6 %

0.2

0.4

0.1

0.2

0.0

0.0

-0.1
-0.2
-0.3
-0.4

-0.2

1%
2%
4%
8%
Jan

Mar

May

Jul

1%
2%
4%
8%

-0.4
-0.6
Sep

Nov

-0.8

Jan

Mar

May

Jul

Sep

Notes: Each panel indicates the seasonality in the size of price increases and price decreases inmodel A for different annual inflation rates. The seasonality is obtained by subtracting theannual average from the corresponding series.

62

Nov

Figure 23: Seasonality and Inflation Rate in the Model C(a) Frequency of price changes<Price increases>
3.0 %
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
Jan

<Price decreases>
0.6 %

1%
2%
4%
8%

0.4
0.2
0.0
-0.2

1%
2%
4%
8%

-0.4
-0.6
-0.8
Mar

May

Jul

Sep

Nov

-1.0

Jan

Mar

May

Jul

Sep

Nov

Notes: Each panel indicates the seasonality in the frequency of price increases and price decreases inmodel C for different annual inflation rates. The seasonality is obtained by subtracting the annual
average from the corresponding series.

(b) Size of price changes<Price increases>
0.25 %
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
Jan

<Price decreases>
0.4 %

1%
2%
4%
8%

1%
2%
4%
8%

0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3

Mar

May

Jul

Sep

Nov

-0.4

Jan

Mar

May

Jul

Sep

Notes: Each panel indicates the seasonality in the size of price increases and price decreases inmodel C for different annual inflation rates. The seasonality is obtained by subtracting theannual average from the corresponding series.

63

Nov

Table 1: Categories in the POS Data
Tofu and tofu products
Nabe-soup
Alcohol-related beverages
Natto (fermented soybeans)
Curry
Baby and maternity food
Konnyaku
Stew and hayashi
Nutritional supplements
Pickles
Instant soups
Food gift sets and gift certificates
Cooked soybeans and kinton
Instant miso soup and Japanese soupGrainsTsukudani
Pasta sauce
Fresh eggs
Side dishes and bento
Instant noodles
Nursing and sick food
Kamaboko (fish paste)
Instant cup noodles
Frozen ingredients
Chikuwa
Instant foods
Frozen side dishes
Fish paste products
Furikake and chazuke
Regular ice cream
Deep-fried fish paste products
Rice-related instant seasonings
Premium ice cream
Processed marine products
Instant seasonings for cooking
Ice
Egg products
Fish paste products in casing
Vegetables for heating
Chilled semi-finished products
Instant soups and juices in cups
Mushrooms
Chilled seasonings
Raw instant noodles
Hair wash
Fresh and boiled noodles
Raw instant cup noodles
Soap
Ham and baconCanned agricultural products
Bath salts
SausageCanned fruits
Toothpaste
Meat Products
Canned desserts
Toothbrushes
ButterCanned seafood
Mouth fresheners
Margarine and fat spreads
Canned meat
Portable sanitary sets
Natural cheese
Canned vegetables
Sanitary products
Processed cheese
Bottled agricultural products
Contraceptives
YogurtBottled seafood
Daily paper products
Cow's milk
Bottled meat
Diapers
Dairy beverages
BreadLaundry detergent
Lactic acid beverages
Table bread
Kitchenware detergent
Fresh cream
Sweet and steamed BreadHousehold cleaners
Soy milk
Cooked bread
Deodorizers, air fresheners and sanitizers
Chilled cool desserts
Cereals
Dehumidifiers
Chilled cakes
Mochi
Insecticides and rat poison
Coffee drinks
Jams
Insect repellents
Cocoa and chocolate beverages
Spreads
Nursing and hygiene products
Tea beverages
Honey and syrups
Denture-related products
Green tea beverages
Dessert mixes
Women's basic cosmetics
Barley tea beverages
Premixes
Women's makeup cosmetics
Oolong tea beverages
Cake and bread ingredients
Women's hair cosmetics
Health tea beverages
Regular coffee
Fragrances
Carbonated soft drinks
Instant coffee
Men's cosmetics
Soft drinks
Drink mixes for cocoa and milk
Cosmetics
Fruit juice 100% beverage
Black tea
Men's hair cosmetics
Vegetable juice
Green tea
Etiquette productsSports drinks
Barley tea
RazorsDiluted beverages
Oolong tea and health tea
Household medical supplies
Nutritional support drinks
Skimmed milk powder and creaming powder Baby food supplies
Water
Chocolate
Tobacco and smoking-related products
Nori
Chewing gum
Washroom and bathroom goods
Dried marine products
Candy and candy confections
Laundry and clothes-drying goods
Powders
Snack foods
Cleaning and maintenance supplies
SesameWestern baked goods
Miscellaneous goods
Dried beans
Dessert cakeToilet cleaning supplies
Dried agricultural products
Rice crackers
Cooking and kitchenware
Dried noodles
Japanese confectionery
Sink ware
Dried pasta
Japanese cheap candies
Food containers
Sugar and sweeteners
Confectionery with toys
Mops
Salt
Bean confections
Eating utensils
Miso
Fisheries delicacies
Leisure eating supplies
Koji
Livestock delicacies
Durable sink ware
Soy sauce
Nuts
Batteries
Edible vinegar and vinegar-related seasonings Dried fruits
Stationery and paper products
Mirin and cooking sake
Assorted confectionery
Daily stationery
Edible oil
SakeWriting supplies
Table sauces
Beer
Painting supplies
Tomato seasoning
Whiskey and brandy
OA supplies
Mayonnaise
ShochuDocumentation supplies
Dressings
Wine
Hooks
Umami seasonings
Liqueurs
Pet sanitary supplies
Instant bouillon
Spirits
Dog food
SpicesChinese liquor
Cat food
Spices and mixed seasonings
Cocktail drinks
Pet food (excluding dog and cat food)
SaucesMiscellaneous liquors
Consumable houseware gift sets
Japanese seasonings and sauces
Sparkling wine
Seasoning sauces
Low alcoholic beverages

Note: The shaded area represents the categories of foods.

64Figure B1: Seasonal Components of Price Changes
Extracted by X12(a) Frequency of price increases(b) Frequency of price decreases

3

3

2

2

1

1

0

0

-1

-1

-2

-2

-3

Jan

Mar May

Jul

Sep

-3

Nov

(c) Size of price increases

Mar May

Jul

Sep

Nov

(d) Size of price decreases

0.8

0.8

0.6

0.6
0.4

0.4

0.2

0.2

0.0

0.0

-0.2

-0.2

-0.4
-0.6

-0.4

-0.8

-0.6
-0.8

Jan

-1.0
Jan

Mar May

Jul

Sep

-1.2

Nov

Jan

Mar May

Jul

Sep

Nov

Notes: The panels plot the median of the average seasonal component of frequency and size of priceNotes: changes across all categories. The shaded area indicates the 25th and 75th percentile bands. The
Notes: seasonal components are extracted by X12.
65

Figure B2: Correlation of Seasonality of Frequency andSize of Price Changes Extracted by X12(a) Correlation between frequency of
price increases and price decreases
40(b) Correlation between size of
price increases and price decreases

Number of categories

40

20

Median: 0.27

10

10

0

0

Correlation coefficients-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

20-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.930p-value >= 0.05
p-value < 0.05
p-value < 0.0130

p-value >= 0.05
p-value < 0.05 Median: 0.43
p-value < 0.01

Number of categories

Correlation coefficients(c) Correlation between frequency of(d) Correlation between frequency of
price increases and size of price increases
price decreases and size of price decreases

40

Number of categories
p-value >= 0.05
p-value < 0.05
p-value < 0.01

50Median: 0.05

40
30

20

20

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.930Correlation coefficients

Number of categories
p-value >= 0.05
Median: 0.03
p-value < 0.05
p-value < 0.01

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

50Correlation coefficients

Notes: The panels show histograms of the correlation between the original series of (a)frequency of priceincreases and price decreases, (b)size of price increases and price decreases, (c)frequency of
price increases and size of price increases and (d)frequency of price decreases and size of pricedecreases within the same category. The data coverage is 1990 to 2021, excluding 2003 and 2004.
The seasonal components are extracted by X12.
66

Figure B3: Seasonality of Net Frequency and Net Size ofPrice Changes Extracted by X12(a) Net frequency of price changes
3

net frequency of price changes
2

inflation rate (right sclae)(b) Net size of price changes

1.6

1.0

1.6

1.2

0.8

1.2

0.6

0.8
1
0
-1

0.4

0.2

0.4

0

0.0

0

-0.4

-0.2

-0.4

-0.4

-0.8
-2

-3

Jan

Mar May

Jul

Sep

Nov

0.8

0.4

-0.8

-0.6

-1.2

-0.8

-1.6

-1.0

net size of price changes
inflation rate (right sclae)
Jan

Mar May

Jul

Sep

Nov

-1.2

-1.6

Notes: The panels plot the median of the average seasonal component of net frequency and net size ofNotes: price changes across all categories. The shaded area indicates the 25th and 75th percentile bands.
Notes: The seasonal components are extracted by X12.

67

Figure B4: Seasonality in High and Low Inflation Periods
Extracted by X12(a) Frequency of price increases(b) Frequency of price decreases

3

4
3

2

2
1

1

0

0
-1

-1

-2
-2
-3

high inflation period
low inflation period
Jan

Mar May

Jul

Sep

-4

Nov

(c) Size of price increases
1.0

0.5

0.5

0.0

0.0

-0.5

-0.5

-1.5

high inflation period
low inflation period
Jan

Mar May

Jul

Sep

Jan

Mar May

Jul

Sep

Nov

(d) Size of price decreases

1.0

-1.0

high inflation period
low inflation period

-3

-1.0
-1.5

Nov

high inflation period
low inflation period
Jan

Mar May

Jul

Sep

Nov

Notes: The panels plot the average seasonal component in the high and the low inflaton period for eachof the categories. The solid line and that with circle marker indicate the median of all categories.
The high inflation period is defined as all years in the top 10% of annual inflation during thewhole sample period, and the low inflation period is defined as all years in the bottom 10% foreach of the categories. The shaded area and dotted line indicate the 25th and 75th percentile
bands. The seasonal components are extracted by X12.
68

Figure B5: Correlation between Annual POS Inflation andSeasonality of Price Changes Extracted by X12(a) Frequency of price increases

30

p-value >= 0.05

p-value < 0.05

50Median: 0.1840p-value < 0.01

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

20Correlation coefficients

Correlation coefficients(c) Size of price increases

4030(d) Size of price decreases

Number of categories
p-value >= 0.05

p-value < 0.05

p-value >= 0.05
p-value < 0.05 Median: -0.10
p-value < 0.01

30

20

50

Number of categories-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.940

Number of categories

50

Median: 0.02

40

p-value < 0.01

30
20

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

20Correlation coefficients

Number of categories
p-value >= 0.05

p-value < 0.05

Median: 0.07

p-value < 0.01

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.950(b) Frequency of price decreases

Correlation coefficients

Notes: These panels show histograms of the correlation between the annual POS inflation rate and thestandard deviation of the seasonal component of (a)the frequency of price increases, (b)thefrequency of price decreases, (c)the size of price increases, and (d)the size of price decreases
during the year, respectively, for all 199 categories. The data coverage is 1990 to 2021, excluding
2003 and 2004. The seasonal components are extracted by X12.
69

Figure C1: Correlation of Frequency and Size of Price Changes
Extracted by Normalizing with the Annual Value

70
60
50

(b) Correlation between size of
price increases and price decreases

Number of categories

70

p-value >= 0.05
p-value < 0.05 Median: 0.27
p-value < 0.01

60
50
40

30

30

20

20

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.940Correlation coefficients

Number of categories
p-value >= 0.05
p-value < 0.05
p-value < 0.01
Median: 0.16

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9(a) Correlation between frequency of
price increases and price decreases

Correlation coefficients(c) Correlation between frequency of(d) Correlation between frequency of
price increases and size of price increases
price decreases and size of price decreases

60
50

Number of categories
Median: -0.16

70p-value >= 0.05
p-value < 0.05
p-value < 0.01

60
50
40

30

30

20

20

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.940Correlation coefficients

Number of categories
Median: -0.17

p-value >= 0.05
p-value < 0.05
p-value < 0.01

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.970Correlation coefficients

Notes: The panels show histograms of the correlation between the seasonal component of (a)frequency of
price increases and price decreases, (b)size of price increases and price decreases, (c)frequency of
price increases and size of price increases and (d)frequency of price decreases and size of pricedecreases within the same category. The data coverage is 1990 to 2021, excluding 2003 and 2004.
The seasonal components are extracted by normalizing with the annual value.
70Figure C2: Correlation between Annual POS Inflation and Seasonalityof Price Changes Extracted by Normalizing with the Annual Value(a) Frequency of price increases
4030(b) Frequency of price decreases

Number of categories

50

p-value >= 0.05
p-value < 0.05

Median: 0.57

Number of categories
p-value >= 0.05
p-value < 0.05

40

p-value < 0.01

p-value < 0.01

30Median: -0.19

20
20
10

Correlation coefficients

Correlation coefficients

30(d) Size of price decreases

Number of categories
p-value >= 0.05

p-value < 0.05

40Median: 0.1730p-value < 0.01

20

10

10

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

20Correlation coefficients

Number of categories
p-value >= 0.05

p-value < 0.05

Median: 0.07

p-value < 0.01

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9(c) Size of price increases
40-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

0-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

010Correlation coefficients

Notes: These panels show histograms of the correlation between the annual POS inflation rate and thestandard deviation of the seasonal component of (a)the frequency of price increases, (b)thefrequency of price decreases, (c)the size of price increases, and (d)the size of price decreases
during the year, respectively, for all 199 categories. The data coverage is 1990 to 2021, excluding
2003 and 2004. The seasonal components are extracted by normalizing with the annual value.
71

Figure C3: Seasonality of Net Frequency and Net Size of Price Changes
Extracted by Normalizing with the Annual Value(a) Net frequency of price changes
3(b) Net size of price changes

1.6
net frequency of price changes
inflation rate (right sclae)
1.2

2

1.2

net size of price changes
inflation rate (right sclae)

0.8

0.8
1

0.4

0

0.4

0.0

-0.4

-1

0
-0.4

-0.4

-0.8
-2

-3

-0.8
-0.8

-1.2
Jan

Mar May

Jul

Sep

Nov

1.2
0.8

0.4

0

1.6

-1.6

-1.2

-1.2
Jan

Mar May

Jul

Sep

Nov

-1.6

Notes: The panels plot the median of the average seasonal component of net frequency and net size of
price changes across all categories. The shaded area indicates the 25th and 75th percentile bands.
For extracting the seasonal components of net frequency, net size, and inflation rate, we subtract
the yearly value from the original series.

72

Figure D1: Correlation of Frequency of Price Increases
across Foods and Others(a) All pairs

3,500
3,000

p-value >= 0.05
p-value < 0.05

Median: 0.19

p-value < 0.01

2,000
1,500

2,500
2,000
1,000

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

0Correlation coefficients

p-value < 0.05

Median: 0.21

p-value < 0.01

Correlation coefficients(c) Others(d) Processed foods and others

Number of combinations
p-value >= 0.05

250

p-value < 0.05

200

p-value < 0.01

Median: 0.30

1,800
1,600
1,400
1,200

Number of combinations
p-value >= 0.05
p-value < 0.05

Median: 0.14

p-value < 0.01

1,000

150

800

100

600
400

50

200
-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.90p-value >= 0.05

500

500

300

Number of combinations

1,000

1,500

0

2,500

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

4,000

Number of combinations

Correlation coefficients0-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.94,500(b) Processed foods

Correlation coefficients

Notes: The panels show histograms of the correlations of the seasonal components of frequency of priceincreases across pairs of (a)all categories, (b)categories of processed foods, and (c)categories ofothers, respectively. Panel (d) indicates the correlation between categories of processed foods andthose of others. Processed foods include 145 categries and others include 54 categories. The data
coverage is 1990 to 2021, excluding 2003 and 2004.
73

Figure D2: Correlation of Frequency of Price Decreases
across Foods and Others(a) All pairs

3,500
3,000

p-value >= 0.05
p-value < 0.05

Median: 0.21

p-value < 0.01

2,000
1,500

2,500
2,000
1,000

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

0Correlation coefficients

p-value < 0.05

Median: 0.22

p-value < 0.01

Correlation coefficients(c) Others(d) Processed foods and others

Number of combinations
p-value >= 0.05

250

p-value < 0.05

200

p-value < 0.01

Median: 0.38

1,800
1,600
1,400
1,200

Number of combinations
p-value >= 0.05
p-value < 0.05

Median: 0.19

p-value < 0.01

1,000

150

800

100

600
400

50

200
-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.90p-value >= 0.05

500

500

300

Number of combinations

1,000

1,500

0

2,500

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

4,000

Number of combinations

Correlation coefficients0-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.94,500(b) Processed foods

Correlation coefficients

Notes: The panels show histograms of the correlations of the seasonal components of frequency of pricedecreases across pairs of (a)all categories, (b)categories of processed foods, and (c)categories ofothers, respectively. Panel (d) indicates the correlation between categories of processed foods andthose of others. Processed foods include 145 categries and others include 54 categories. The data
coverage is 1990 to 2021, excluding 2003 and 2004.
74

Figure D3: Correlation of Size of Price Increases
across Foods and Others(a) All pairs
p-value >= 0.05

5,000

p-value < 0.05

4,000

p-value < 0.01

Median: 0.07

3,000

p-value < 0.05

2,000

p-value < 0.01

2,000

1,000

1,000

500

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

1,500

Correlation coefficients(d) Processed foods and others

Number of combinations
p-value < 0.05

Median: 0.09

Correlation coefficients(c) Others
p-value >= 0.05

p-value >= 0.05

2,500

3,000

Median: 0.09

p-value < 0.01

2,500
2,000
1,500

Number of combinations
p-value >= 0.05
p-value < 0.05

Median: 0.04

p-value < 0.01

1,000
500

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

500
450
400
350
300
250
200
150
100
50
0

Number of combinations-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

Number of combinations

Correlation coefficients0-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.96,000(b) Processed foods

Correlation coefficients

Notes: The panels show histograms of the correlations of the seasonal components of size of priceincreases across pairs of (a)all categories, (b)categories of processed foods, and (c)categories ofothers, respectively. Panel (d) indicates the correlation between categories of processed foods andthose of others. Processed foods include 145 categries and others include 54 categories. The data
coverage is 1990 to 2021, excluding 2003 and 2004.
75

Figure D4: Correlation of Size of Price Decreases
across Foods and Others(a) All pairs
p-value >= 0.05

5,000

p-value < 0.05

4,000

p-value < 0.01

Median: 0.08

3,000

p-value < 0.05

2,000

p-value < 0.01

2,000

1,000

1,000

500

0

0

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

1,500

Correlation coefficients

350
300

(d) Processed foods and others

Number of combinations
p-value >= 0.05
p-value < 0.05

Median: 0.13

p-value < 0.01

250
200

2,500
2,000
1,500

Number of combinations
p-value >= 0.05
p-value < 0.05

Median: 0.05

p-value < 0.01

1,000

150
100

500

-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

500Median: 0.09

Correlation coefficients(c) Others

400

p-value >= 0.05

2,500

3,000

450

Number of combinations-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.9

Number of combinations

Correlation coefficients0-1 to -0.9
-0.8 to -0.7
-0.6 to -0.5
-0.4 to -0.3
-0.2 to -0.1
0 to 0.1
0.2 to 0.3
0.4 to 0.5
0.6 to 0.7
0.8 to 0.96,000(b) Processed foods

Correlation coefficients

Notes: The panels show histograms of the correlations of the seasonal components of size of pricedecreases across pairs of (a)all categories, (b)categories of processed foods, and (c)categories ofothers, respectively. Panel (d) indicates the correlation between categories of processed foods andthose of others. Processed foods include 145 categries and others include 54 categories. The data
coverage is 1990 to 2021, excluding 2003 and 2004.
76

Figure E1: Number of Statistically Significant Monthly Dummies (1)(a) Frequency of price increases
20,000(b) Frequency of price decreases

Number of dummies

20,00015,00015,00010,00010,000

5,000

5,000

0

0

5,000

5,000

10,00010,00015,00015,00020,00025,000significantly negative
significantly positive
Jan Mar May Jul

20,00025,000

Sep Nov(c) Size of price increases
20,000

Number of dummies

20,00015,00010,00010,000

5,000

5,000

0

0

5,000

5,000

10,00010,00015,000significantly negative

15,00020,000significantly positive

20,000

Jan Mar May Jul

significantly negative
significantly positive
Jan Mar May Jul

Sep Nov(d) Size of price decreases

15,00025,000

Number of dummies

25,000

Sep Nov

Number of dummies

significantly negative
significantly positive
Jan Mar May Jul

Sep Nov

Notes: These panels show the number of statistically significant dummies in the estimation of equation (4).
Dummies are estimated using a three-year rolling estimation for each of the 199 categories. For eachcategory, there are about 300 coefficients on the dummies and overall there are 57,362. The bars
represent the number of dummies that are statistically significant at the 95% level.
77

Figure E2: Number of Statistically Significant Monthly Dummies (2)(a) Frequency of price increases
100(b) Frequency of price decreases

Number of dummies

100

50

50

0

0

50

50

100

100

150

significantly negative
significantly positive
Jan Mar May Jul

150

Sep Nov(c) Size of price increases
100

Number of dummies

100
50

0

0

50

50

150

100

significantly negative
significantly positive
Jan Mar May Jul

significantly negative
significantly positive
Jan Mar May Jul

Sep Nov(d) Size of price decreases

50

100

Number of dummies

150

Sep Nov

Number of dummies

significantly negative
significantly positive
Jan Mar May Jul

Sep Nov

Notes: These panels show the number of statistically significant dummies in the estimation of equation (4).
The dummies are estimated based on the full-sample period instead of a three-year rolling estimationand there are 12 dummy variables for each of 199 categories. The number of dummies that are
statistically significant at the 95% level is reported.
78