Online Appendix“COVID-19 Infection Spread and Human Mobility” by Shibamoto,
Hayaki, and Ogisu
A.1

New positive cases of the COVID-19 infection

This subsection documents the historical time-series features of new positive COVID-19 infection
cases in Japan. It especially focuses on non-stationarity and cyclicality in infection spread.
Figure A1 shows a weekly time series of the conﬁrmed new infection cases (top panel), log of
new infection cases (middle panel), and log changes in infection cases (bottom panel) from theweek of February 16, 2020, to the week of May 9, 2021. The orange shaded areas show the
weeks coinciding with the period during the state of emergency declaration in Japan. 1
The non-stationarity in the time series of the number of new infection cases is almost acertainty. The top panel of Figure A1 shows four peaks in the number of new infection cases,
despite only three states of emergency. This shows that the expected value of the number of
new infection cases tends to increase and decrease stochastically. That is, the time series of thenumber of new infection cases follows a non-stationary process.
However, the number of new infection cases seems to be cyclical. Figure A1 shows thatthe number of new infection cases decreased, despite no declaration of a state of emergency.
Interestingly, the three peak-to-bottom periods are approximately the same (about a monthand a half).2 This similarity suggests that the decline in new infection cases tends to continue
for approximately a month and a half, regardless of whether a state of emergency has been
declared.
1
In Japan, a state of emergency was declared three times during the sampling period: from April 7th to May
25th 2020, from January 8th to March 21st 2021, and from April 25th 2021 onward. The legal basis for thepolicy responses of the governors of the prefectures subject to the emergency measures, is the “Act on Special
Measures for Pandemic Inﬂuenza and New Infectious Diseases Preparedness and Response.” Under the ﬁrst
declaration of emergency, prefectural governors were able to request people to refrain from going out of theirhomes, and to request and instruct facility administrators of schools, social welfare facilities, and entertainment
venues to restrict the use of those facilities, in accordance with the provisions of Article 45 of the Act. However,the Act does not stipulate any penalties for disobeying the instructions under Article 45, and Japan’s curfew
was extremely loose compared to the lockdowns in China, United States, and many European countries. On
February 13th 2021, during the second declaration of the state of emergency, the Act has been amended to allow
the prefectural governor to instruct the facility manager regarding restrictions. In addition, based on the newly
established Article 79, facility managers who do not comply with the order will be subject to a ﬁne of up to
300,000 yen.
2
The ﬁrst peak-to-bottom period is ﬁve weeks: from the week of April 12, 2020, to the week of May 17, 2020.
The second peak-to-bottom period is seven weeks: from the week of August 2, 2020, to the week of September
20, 2020. The third peak-to-bottom period is seven weeks: from the week of January 10, 2021, to the week ofFebruary 28, 2021.

1

Figure A1: Weekly new positive cases of COVID-19 infection in Japan
Notes: The orange shaded areas show the weeks coinciding with the period during the state of emergency
declaration in Japan. The sample period is the week of February 16, 2020, to the week of May 9, 2021.

Figure A2 shows a weekly time series of log changes in infection cases from the week ofMarch 1, 2020, to the week of May 9, 2021 in Tokyo and Osaka.3 The infection spread in eachprefecture can be diﬀerentiated into spread at the same or diﬀerent times. As seen in the ﬁgure,there appears to be a common trend in the infection cases across the two prefectures. We noticethat two prefectures have experienced a sustained decrease in the infection cases during thedeclaration of the state of emergency, and a persistent increase in cases after the declaration
was lifted. On the other hand, there are regional diﬀerences in the infection spread. The growthrate of new cases in the two prefectures can often be seen to temporarily diverge.

A.2

Human mobility index

We measure the movement of people using Google’s mobility indices. The weekly mobility index
we use matches the median of each week of the daily mobility index, to eliminate the eﬀects
3
Following Hoshi et al. (2021), we set the value of the log levels of new infection cases to −1 for observation
with zero weekly cases

2

Figure A2: Weekly log changes in new positive cases of COVID-19 infection in Tokyo and Osaka
Notes: The orange shaded areas show the weeks coinciding with the period during the state of emergency
declaration in Japan. The sample period spans from the week of March 1, 2020, to the week of May 9, 2021.
Diﬀerent colors represent diﬀerent prefectures, where the navy blue is for Tokyo and the dark orange is for Osaka.

of holidays as much as possible. These mobility indices represent the rate of change in humanmobility for six categories: retail & recreation, grocery & pharmacy, parks, transit stations,workplaces, and residential spaces.
First, we present the contemporaneous relationships among the mobility indices during thesample period. Table A1 reports the Pearson’s correlation coeﬃcients between two mobility
indices. From Table A1, the six mobility indices are highly correlated. In particular, retail &recreation, transit stations, and workplaces are strongly intercorrelated, and residential showsa strong negative correlation. Hence, there are common components that help summarize thecharacteristics of these highly correlated variables.
To capture the common components among the mobility indices, we consider a model tosummarize the time-series characteristics of these indices. Speciﬁcally, we assume that the
representative measure for human mobility as the common factor of individual mobility indices,′signiﬁcantly drives the time-series patterns of the indices. Let Mt = (m1t , · · · , m6t ) be a six-3

Table A1: Correlation between the Google mobility indices[1] Retail & recreation[2] Grocery & pharmacy[3] Parks[4] Transit stations[5] Workplaces[6] Residential

[1]

[2]

[3]

[4]

[5]

0.528
0.292
0.857
0.394
-0.7490.516
0.293-0.093-0.1480.200
0.022-0.1080.751-0.948-0.854

Notes: This table shows the Pearson’s correlation coeﬃcient between the Google mobility indices from the weekof February 16, 2020, to the week of May 9, 2021.

by-one vector of the mobility indices (retail & recreation, grocery & pharmacy, parks, transit
stations, workplaces, residential). We consider the following factor model:

Mt = λ y y t + λ 1 + v t ,(1)

where yt is a single common factor, vt is a six-by-one vector of mean-zero idiosyncratic components, and λy , λ1 are six-by-one vectors of slope and intercept coeﬃcients, respectively. We
impose the ﬁrst element of coeﬃcient vector λy to be equal to one, to endow the factor yt withinterpretable units and signs, where one unit increase in yt reﬂects an increase in the mobility
index with retail & recreation by one.
We adopt the principal component (PC) approach to measure the representative measurefor human mobility yt . First, we calculate the ﬁrst PC, pct , using the standardized (mean-zeroand unit variance) six mobility indices. Next, we regress human mobility of retail & recreation
m1t on pct and the constant term:

m1,t = δ1,pc pct + δ1,1 + ν1,t .

(2)

where ν1,t is an error term. We can then compute the representative measure for humanmobility, satisﬁed with the restriction imposed in the factor model as yt = δ̂1,pc pct where δ̂1,pc
is an ordinary least squares (OLS) estimate of the parameter δ1,pc in Equation (2). Since therepresentative measure for human mobility yt is based on PC, it is a composite index of themobility indices.
Table A2 shows the empirical results for the factor model (1) by OLS regression. From the4table, the regression coeﬃcients of four indices (retail & recreation, transit stations, workplaces,and [negative] residential) on the composite index are statistically signiﬁcant at the 1% level.
Moreover, the composite index explains most of the mobility index variations with the retail &recreation, transit stations, and residential categories, although the mobility indices with thegrocery & pharmacy and parks categories are explained to a lesser degree, given the large shareexplained by the idiosyncratic components. Thus, the composite index as the representative
measure for human mobility summarizes the common characteristics among mobility indices.
Table A2: Factor model for the Google mobility indicesDependent
variable:

Retail &recreation

Grocery &pharmacy

Parks

Transitstations

Workplaces

Residential

λy1(0.11)-0.138(0.007)0.17(0.06)
0.001(0.005)0.47(0.36)-0.045(0.023)1.34(0.05)-0.254(0.003)0.92(0.15)-0.131(0.007)-0.42(0.02)
0.066(0.002)

0.77

0.16

0.10

0.96

0.57

0.90

λ1
R-squared

Notes: This table shows the results for the ordinary least squares regression (1) of the mobility index with eachcolumn category on the composite index of mobility and constant term. We obtain the composite index by scaling
and signing the ﬁrst principal component calculated using the Google mobility indices to the index for retail &recreation. The sample period is the week of February 16, 2020, through the week of May 9, 2021. The numbers
in parentheses are Newey & West (1987) heteroskedasticity and autocorrelation robust standard errors for leastsquares with a four-week lag truncation.

Each panel in Figure A3 displays each mobility index time series and the composite index.
We graphically conﬁrm that the composite index tracks the common time-series patterns among
the mobility indices, during the sample period.
Each panel of Figures A4 and A5 displays each mobility index time series and the composite
index in Tokyo and Osaka, respectively.

A.3

Comparison of in-sample forecasting accuracy among a historical average, AR model, and VAR model for the log-changes in new infection
cases

We assess multistep-ahead forecasts, computed by iterating forward the VAR model (10) in
the main text. It examines current-, two-, four-, and eight-week forecast horizons. The HH∑
week-ahead forecast is computed by∆π̂t+h where ∆π̂t+h is the h-week-ahead predictor of
h=05

Figure A3: Weekly Google mobility index in Japan
Notes: The dotted line (left-hand scale) indicates the Google mobility index with each category. The solidline (right-hand scale) indicates the composite index of mobility. We obtain the composite index by scaling andsigning the ﬁrst principal component calculated using Google mobility indices to the index for retail & recreation.
The orange shaded areas show the weeks coinciding with the period during the state of emergency declaration in
Japan. The sample period is the week of February 16, 2020, through the week of May 9, 2021.

6

Figure A4: Weekly Google mobility index in Tokyo
Notes: The dotted line (left-hand scale) indicates the Google mobility index with each category. The solidline (right-hand scale) indicates the composite index of mobility. We obtain the composite index by scaling andsigning the ﬁrst principal component calculated using Google mobility indices to the index for retail & recreation.
The orange shaded areas show the weeks coinciding with the period during the state of emergency declaration in
Japan. The sample period spans from the week of February 16, 2020, to the week of May 9, 2021.

7

Figure A5: Weekly Google mobility index in Osaka
Notes: The dotted line (left-hand scale) indicates the Google mobility index with each category. The solidline (right-hand scale) indicates the composite index of mobility. We obtain the composite index by scaling andsigning the ﬁrst principal component calculated using Google mobility indices to the index for retail & recreation.
The orange shaded areas show the weeks coinciding with the period during the state of emergency declaration in
Japan. The sample period spans from the week of February 16, 2020, to the week of May 9, 2021.

∆πt+h conditional on the information available at time t − 1 using the forecasting models. As
a comparison, we compute forecasts for a univariate AR with three-week lags and a historical
average. Table A3 shows the mean square forecast error (MSE) and the percentage reduction
in MSE for each of the forecasting models, relative to the historical average.
Accordingly, the predictions of the VAR model improve over the AR model and the historical
average for any forecast horizon. Hence, the composite index of mobility contains valuable
information for predicting the number of new infection cases.
The AR model forecast does not work adequately, especially for longer horizons. The upper
four panels in Figure A6 shows the prediction values of the log changes in the new COVID-19
infection cases over the current (two, four, and eight) week(s) using the estimated parametersin the AR model. Although it has good predictive power in the shorter horizon, the predictive
power of the AR model in the longer horizon is relatively poor.

8

Table A3: Prediction comparison for the log changes in infection cases
Horizon (H)

M SE1

M SEAR2RAR

M SEV ARRV2 AR

0
2
4
8

0.129
0.818
1.581
2.190

0.083
0.629
1.507
2.739

35.7
23.1
4.7
-25.1

0.054
0.363
0.847
2.015

57.8
55.6
46.4
8.0

Notes: This table shows the results from predicting the log changes in infection cases from the last weekthrough the current (following two, four, and eight) week(s), conditioned on the information at the last week.
M SE1 , M SEAR , and M SEV AR show the mean square forecast error (MSE) for the historical average, autoregres2
sive (AR), and vector autoregressive model (VAR)forecasts, respectively. RAR(= 100 × (1 − M SEAR /M SE1 ))and RV2 AR (= 100 × (1 − M SEV AR /M SE1 )) measure the percentage reduction in MSE for the AR and VAR
models, respectively, relative to the historical average. We set the lag length to three weeks in the AR and VAR
estimation. Estimation samples in the AR and VAR model span the week of March 1, 2020, through the weekof May 9, 2021. The forecasts are computed over the sample period depending on data availability of the actual
H-week-ahead log changes in infection cases.

In contrast, the VAR model forecast works adequately over the sample period. The lower
four panels in Figure A6 shows the prediction values of the log changes in the new COVID-19
infection cases over the forecast horizon using the estimated parameters in the VAR model. The
predicted values explain the actual time-series pattern from March 2020 to May 2021, though
the predictions after the state (exit) of (from) emergency declaration in early April (June)
2020 are slightly poor. This result suggests that the proposed VAR model simply and thriftily
captures the dynamics of the log changes in new infection cases.

A.4

Estimation of the COVID-19 infection–mobility trade-oﬀ in Japan using
each Google mobility index

As well as the benchmark, we estimate the following speciﬁcation of the infection–mobility
trade-oﬀ:∆πt+1 = κyt−1 + ι + ϵπ,t+1 .

(3)

For human mobility yt , we use the Google mobility index with each category (retail & recreation,
grocery & pharmacy, parks, transit stations, workplaces, residential) and the composite index
of mobility. The sample period spans from the week of February 23rd, 2020, to the week of May
2nd, 2021.
Table A4 reports estimation results for the infection–mobility trade-oﬀ (3). The coeﬃcients κdescribing the response of the one-week-ahead log changes in infection cases, to the one-week lag
9Autoregressive model with three-week lags

Vector autoregressive model with three-week lags

Figure A6: Prediction comparison for the log changes in infection cases
Notes: The solid line indicates the actual log changes in the new positive COVID-19 infection cases from the last
week through the current (following two, four, or eight) week(s). The dotted line indicates the prediction valuesof the log changes, conditioned on the information at the last week; it is calculated using the parameter estimatesof the autoregressive (AR) or vector autoregressive (VAR) model. We set the lag length to three weeks in the
AR and VAR estimations. Dark and light yellow areas respectively denote ±1 and ±2 mean squared prediction
error bands, calculated using 1,000 bootstrap samples. The orange shaded areas show the weeks coinciding withthe period during the state of emergency declaration in Japan. Estimation samples in the AR and VAR models
span from the week of March 1, 2020, to the week of May 9, 2021.

10

Table A4: Estimation results for the COVID-19 infection–mobility trade-oﬀ in Japan

Dependent
variable:κι
Adj-R2Retail &recreation

Grocery &pharmacy

Parks

Transitstations

Workplaces

Residential

MobilityCI

2.67(0.60)
0.46(0.11)

4.97(1.33)
0.08(0.06)

0.76(0.60)
0.12(0.08)

2.33(0.41)
0.68(0.12)

1.76(0.90)
0.32(0.13)-6.22(1.43)
0.50(0.11)

3.28(0.54)
0.09(0.05)

0.38

0.18

0.04

0.40

0.18

0.30

0.43

Notes: This table shows the ordinary least squares regression results (3) of the log changes in infection cases on
the mobility index with each column category and constant term. Mobility CI denotes the composite index ofmobility. We obtain the composite index by scaling and signing the ﬁrst principal component, calculated using
Google mobility indices, to the index for retail & recreation. The sample period spans from the week of February
23, 2020, to the week of May 2, 2021. The numbers in parentheses are Newey & West (1987) heteroskedasticity
and autocorrelation robust standard errors for least squares with a four-week lag truncation.

of human mobility, are positive and statistically signiﬁcant for human mobility under the retail& recreation, grocery & pharmacy, transit stations, and workplaces categories. The coeﬃcientis negative and statistically signiﬁcant for human mobility under the residential category; therelationship is statistically insigniﬁcant for human mobility under the parks category.
Further, there are some diﬀerences in the explanatory power of the infection Phillips curveamong the categories of the mobility index. In particular, the speciﬁcation using the mobility
index, especially with the parks and workplaces categories, has a lower R-squared than thatusing the mobility index with the retail & recreation, transit station, and residential categories.
Thus, human mobility in parks and workplaces may be limited in explaining the time-series
pattern of COVID-19 infection cases. These results are consistent with Nagata et al. (2021),
where the mobility changes in nightlife and residential places are signiﬁcantly associated with
COVID-19 infection cases. Meanwhile, the association of mobility changes in workplaces isstatistically insigniﬁcant. Moreover, the speciﬁcation using the composite index of mobility has
a higher R-squared than that using individual mobility indices. Hence, using the composite
index improves the ﬁt of the infection–mobility trade-oﬀ, by removing possible measurement
errors for human mobility in individual mobility indices.

11

A.5

VAR-IV model

In this subsection, we describe the derivation of the VAR–IV model from a simultaneous equations system of infection-mobility trade-oﬀ and mobility demand.
We consider the following simultaneous equations system;′∆πt = ζ Wt + ξa,t + ϕξmd,t ,(4)′

yt = γ Wt + b∆πt + ξmd,t ,(5)where Wt is a vector of control variables comprising the one- to three-week lags of yt and ∆πt
and a constant. ξa,t with a mean of zero and a variance of σa2 represents an anxiety shock.
2 represents a mobility demand shock. ξξmd,t with a mean of zero and a variance of σmd
a,t andξmd,t are assumed to be serially uncorrelated and independent each other. Equation (4) is avariant of the infection–mobility trade-oﬀ (Equation (15) in main manuscript) conditioning on
the information at time t − 1. ξa,t + ϕξmd,t reﬂects the stochastic term of the new infection
production, that is, unexpected changes in new infection cases at time t. Equation (5) is themobility demand (Equation (16) in main manuscript).
We can express the structural VAR representation as follows;

A(L)Xt = a0 + Θa ξa,t + Θmd ξmd,t ,
′

′

′

(6)
′

where Θa = (Θ1a , Θ2a ) = (1, b) , Θmd = (Θ1md , Θ2md ) = (ϕ, 1 + bϕ) .
To identify the model, we use an external instrument from the VAR system that captures
surprise variation in the new infection cases that people perceive as the change in infection risk.
We can identify b under the premise that there is an instrument zt for the anxiety shock thatsatisﬁes the following conditions:E[ξa,t zt ] ̸= 0, and(7)E[ξmd,t zt ] = 0.

(8)Thus, we estimate the regression (5) by an IV estimation to obtain the estimate of b. Given
the estimate of b, we can identify the mobility demand shocks as the residuals obtained fromthe IV regression. Thus, we estimate the regression (18) in the main manuscript to obtain the12estimate of Θmd .

A.6

The information content of the log level of weekly cases for the dynamics
of human mobility in the VAR model

We consider a model referring to the speciﬁcation by Chernozhukov et al. (2021) and Hoshi et al.
(2021) that adds log change and log level of weekly cases at time t, ∆πt and πt , respectively, tothe equation for the human mobility in the reduced-form VAR model;′

yt = b1 ∆πt + b2 πt + γ Wt + ẽy,t .

(9)where Wt is a vector of control variables comprising the one- to three-week lags of yt and ∆πt
and a constant, and ẽy,t is the error term orthogonal to ∆πt and πt . We estimate the regression(9) with and without ∆πt and πt as independent variables by an OLS, and compare the ﬁt of
the models.
Table A5 shows the empirical results. In the column (1), neither log changes nor log levels
in infection cases at week t are included in the explanatory variables in Equation (9), i.e.,
the equation for the human mobility in the reduced-form VAR model. In column (2), theindependent variable in the equation is the log changes in infection cases, i.e., the speciﬁcation of
the mobility demand in Equation (16) in the main manuscript. In column (3), the independent
variable in the equation is the log levels in infection cases. In column (4), the independent
variables in the equation are both the log changes and the log levels in infection cases. As
shown in the column (1) of the Table, we ﬁnd that about 70% of the variation in the composite
index of mobility can be explained by the lag values of log changes in the new infection cases
and the mobility index. As shown in the column (2) of the Table, the coeﬃcient of the growthrate of cases at time t is statistically positive and signiﬁcant, it does not have much additional
explanatory power. As shown in the columns (3) and (4) of the Table, the coeﬃcient of the log
level of cases at time t is not statistically signiﬁcant. This implies that adding the log level ofweekly cases in our VAR model, plays a limited role in explaining the dynamics of the composite
index of mobility.

13

Table A5: OLS Estimation results for the composit index of mobility with/without the log levelof the new infection cases
Dependent variable:∆πtπtAdj-R2

Mobility CI
(1)
(2)

(3)

(4)

-

0.027
(0.013)--0.003(0.004)0.025(0.014)-0.002(0.004)

0.693

0.697

0.691

0.693

Notes: The dependent variable is the composite index of mobility. In column (1), both log changes and log levels
in infection cases at week t are not included in the explanatory variables. In column (2), the independent variable
is the log changes in infection cases. In column (3), the independent variable is the log levels in infection cases. In
column (4), the independent variables are both the log changes and the log levels in infection cases. The constantand one- to three-week lags of log changes in infection cases, and the composite index of mobility are included
as control variables in the linear regression model. We estimate the regression by ordinary least squares. The
numbers in parentheses are White (1980) heteroskedasticity-robust standard errors. Coeﬃcients and standard
errors for the control variables are not reported. The sample period spans from the week of March 1, 2020, to
the week of May 9, 2021.

A.7

The search volume of “感染者数” in Google

Figure A7 plots the weekly log changes in the search volume of the Japanese term “ 感染者数”(number of infected individuals in English) over time.4

A.8

Robustness check and sensitivity analysis

In this subsection, we re-estimate the model under several alternative settings to examine therobustness and sensitivity of our empirical analyses.5

Additional principal component for the Google mobility indices
We examine the robustness of the representative measure of human mobility. Although weassume that the time-series patterns of the Google mobility indices are signiﬁcantly driven by
their ﬁrst PC, one may suspect a misspeciﬁcation of our model for new infection cases andhuman behavior, due to the possibility that the other PCs also contain informational content
about beneﬁcial infection dynamics. Thus, we describe additional results that show that theﬁrst PC can suﬃciently summarize the common features of the mobility indices using the factor
4

We retrieved the search volume data from Google Trends (https://trends.google.co.jp/trends/?geo=JP)
on June 1, 2021.
5
Detailed results of the following exercises can be obtained from the corresponding author upon request.

14

Figure A7: Weekly log changes in the search volume of “感染者数” (number of infected individuals in English) in Google
Notes: The orange shaded areas show the weeks coinciding with the period during the state of emergency
declaration in Japan. The sample period spans from the week of March 1, 2020, to the week of May 9, 2021.

model, and the additional PC is limited in explaining the infection dynamics.
One way to justify the assumption that the number of common factors is equal to one is tocheck the information criteria for determining the number of factors. Speciﬁcally, we use theinformation criteria proposed by Ahn & Horenstein (2013), which suggests that the preferredmodel is the one that maximizes the information criteria, deﬁned as follows:

IC(k) =log(SSR(k − 1))/ log(SSR(k)),
log(SSR(k))/ log(SSR(k + 1))(10)

k = m − (λk ykwhere SSR(k) is the sum of squared residuals, vi,ti,ti,y1 1,t + · · · + λi,yk yk,t + λi,1 ) fori = 1, · · · , 6 given the number of factors, k, such that1 ∑ ∑ k2
SSR(k) =vi,t .
6TT

6

t=1 i=1

15(11)

Table A6: The number of common factors underlying the Google mobility indices
Number of factors (k)SSR(k)
IC(k)

0

1

2

3

4

64.48
n.a.

28.43
0.94

12.61
0.58

3.02
-0.68

0.72
-11.97kNotes: SSR(k) denotes the sum of squared residuals, vi,t= mi,t −(λki,y1 y1,t +· · ·+λki,yk yk,t +λi,1 ) for i = 1, · · · , 6

given the number of factors, k, deﬁned as Equation (11). IC(k) denotes the Ahn & Horenstein (2013) information
criteria given the number of factors, k, deﬁned as Equation (10). The sample period spans from the week ofFebruary 16, 2020, to the week of May 9, 2021.

Table A6 reports the calculated sum of squared residuals SSR(k) and the Ahn & Horenstein(2013) information criteria IC(k) for each number of factors k. The criteria suggest that weshould adopt one common factor in summarizing the common features of the mobility indicesusing the factor model.
Second, although the inclusion of additional PCs increases explanatory power regarding
mobility indices, it does not mean they are useful as a representative time-series measure forhuman mobility. Table A7 reports the empirical results for the factor model, including thetwo common factors estimated in the same manner as the benchmarks. In particular, fromthe table, the second PC strongly captures the characteristics of grocery & pharmacy andparks. Given that the mobility indices of these categories is limited in explaining the infection–
mobility trade-oﬀ reported in Table A4, the second PC is unlikely to signiﬁcantly capture theworkings of the human mobility associated with the spread of infection. In fact, we extend
the regression of the infection–mobility trade-oﬀ such that the second PC also aﬀects the log
changes in infection cases, conﬁrming that the second PC has little explanatory power regarding
the rate of new infection cases. Each panel in Figure A8, which displays the time series of each
mobility index and the second composite index, shows that a large part of the ﬂuctuations of
the second PC is a signiﬁcant spike, because of two long vacations: The former is the summer
vacation in the week of August 9, 2020, and the latter is the winter vacation in the week ofDecember 27, 2020. Although transitory changes in mobility during the long vacations may
have aﬀected new infection cases, such changes are not appropriate indicators to capture thetime-series characteristics of the representative measure for human mobility, when modeling theintertemporal relationships between human mobility and new infection cases.
Next, we examine the information content of the second PC for the infection dynamics.

16

Table A7: Two factors model for the Google mobility indicesDependent
variable:

Retail &recreation

Grocery &pharmacy

Parks

Transitstations

Workplaces

Residential

λy1

1(0.12)
1(0.50)-0.138(0.008)0.17(0.04)
1.32(0.14)
0.001(0.002)0.47(0.27)
4.10(0.78)-0.045(0.013)1.34(0.04)-0.58(0.18)-0.254(0.002)0.92(0.14)-2.32(0.58)-0.131(0.008)-0.42(0.01)
0.47(0.04)
0.066(0.001)

0.82

0.81

0.63

0.97

0.83

0.98

λy2
λ1
R-squared

Notes: This table shows the results for the ordinary least squares regression of the mobility index, with eachcolumn category on the composite indices of mobility constructed from the ﬁrst and second principal components
and constant term. We obtain the composite indices by scaling and signing the ﬁrst and second principal
components calculated using Google mobility indices to the index for retail & recreation. The sample periodspans from the week of February 16, 2020, to the week of May 9, 2021. The numbers in parentheses are Newey& West (1987) heteroskedasticity and autocorrelation robust standard errors for least squares with a four-week
lag truncation.

Table A8 shows the forecasting assessment of multistep-ahead forecasts, computed by iterating
forward the three-variable VAR model comprising the log changes in the infection cases andthe ﬁrst and second composite indices for mobility, conducted in the same manner as thebenchmarks. Four panels in Figure A9 show the prediction values of the log changes in the new
infection cases, using the estimated parameters in the three-variable VAR model.
While the inclusion of the second composite index has contributed to some improvementin the model—speciﬁcally, reducing the prediction error—it has not signiﬁcantly impacted thefuture movement of new infection cases. Comparing the results in Tables A3 and A8, the threevariable VAR model has a somewhat better prediction accuracy regarding in-sample prediction
than the bivariate VAR model. However, comparing Figures A6 and A9, it is not apparent thatthe three-variable VAR model predictions have changed signiﬁcantly from the bivariate VAR
model predictions. Thus, although the additional PC contributed to reducing the prediction
error of the VAR model, it does not change the dynamics of the VAR model. That is, theinclusion of the second composite index in the time-series model is not expected to be signiﬁcantin analyzing the dynamic causal relationship between the number of infection cases and humanmobility.
Finally, we conduct a sensitivity analysis, which includes the second composite index in ourmacroeconometric model. Speciﬁcally, we rerun the VAR model, including one- to three-week17

Figure A8: Weekly Google mobility index and the second composite index in Japan
Notes: The dotted line (left-hand scale) indicates the Google mobility index with each category. The solid line
(right-hand scale) indicates the second composite index of mobility. We obtain the second composite index byscaling and signing the second principal component, calculated using Google mobility indices, to the index forretail & recreation. The orange shaded areas show the weeks coinciding with the period during which the stateof emergency was declared in Japan. The sample period spans the week of February 16, 2020, to the week of
May 9, 2021.

lags of the second composite index as exogenous variables. As expected, we conﬁrm that themodel produces, quantitatively and qualitatively, results that are similar to the benchmark
results.

Assumption of epidemiological rigidity
We check the robustness for the empirical assessment of the infection–mobility trade-oﬀ basedon the assumption of epidemiological rigidity. In particular, we assume epidemiological rigidity
such that when infection increases due to increased mobility, it takes two weeks for the disease
to develop under the benchmark speciﬁcations. One may attribute the sensitivity of the results
to diﬀerent settings for the rigidity because it may not take much time for the increased mobility
to aﬀect the spread of new infections. Thus, to consider the possibility that increased mobility

18

Table A8: Prediction comparison for the log changes in infection cases
HorizonM SE3varV AR

2
R3varV
AR

0.049
0.329
0.781
1.852

61.7
59.8
50.6
15.4

0
2
4
8

Notes: This table shows the results of predicting the log changes in infection cases from the last week throughthe current and the following two, four, and eight weeks conditioned on the information at the last week.
M SE3varV AR shows the mean square forecast error (MSE) for the three-variable vector autoregressive (VAR)
2model forecast. R3varV
AR (= 100 × (1 − M SE3varV AR /M SE1 )) measures the percent reduction in the MSE forthe three-variable VAR model relative to the historical average. We set the lag length to three weeks in thethree-variable VAR estimation. Estimation samples in the three-variable VAR model span from the week ofMarch 1, 2020, to the week of May 9, 2021. The forecasts are computed over the sample period, depending ondata availability of the actual H-week-ahead log changes in infection cases.

may increase the new infection cases even after one week, we extend the regression of theinfection–mobility trade-oﬀ, such that human mobility in period t also aﬀects the log changes
in infection cases during the period t + 1:

∆πt+1 = κ1 yt + κ2 yt−1 + ι + ϵπ,t+1 .

(12)Along with the benchmark case in Equation (3), we use the Google mobility index with eachcategory (retail & recreation, grocery & pharmacy, parks, transit stations, workplaces, andresidential) and the composite index of mobility for human mobility yt . Table A9 reports theestimation results for the extended infection–mobility trade-oﬀ in Equation (12).
Table A9 shows that our empirical assessment regarding the infection–mobility trade-oﬀ ismost robust to the possibility that increased mobility may increase the new infection cases, after
just one week. In particular, we conﬁrm that the human mobility indices with transit stations
and residential categories, and the composite index of mobility in week t, do not statisticallycontribute to the increase in the new infection cases in week t + 1. This result suggests that
our assumption of epidemiological rigidity in the benchmark model is realistic. We also conﬁrm
that the mobility in parks and workplaces is limited in explaining the time-series pattern of
COVID-19 infection cases. Moreover, the speciﬁcation using the composite index of mobility
has a higher R-squared than that using individual mobility indices.
Note that there is a slight diﬀerence in the benchmarks. Regarding the retail & recreation

19

Three-variable vector autoregressive model with three-week lags

Figure A9: Prediction of the log changes in infection cases using a three-variable vector autoregressive model
Notes: The solid line indicates the actual log changes in new positive COVID-19 infection cases from the last
week to the current (following two, four, and eight) week(s). The dotted line indicates the prediction values of thelog changes, conditioned on the information at the last week. It is calculated using the parameter estimates of thethree-variable vector autoregressive (VAR) model. We set the lag length to three weeks in the VAR estimation.
Dark and light yellow areas denote ±1 and ±2 mean squared prediction error bands, calculated using 1,000
bootstrap samples, respectively. The orange shaded areas show the weeks coinciding with the period during thestate of emergency declaration in Japan. Estimation samples in the three-variable VAR model span from theweek of March 1, 2020, to the week of May 9, 2021.

and grocery & pharmacy categories, the mobility in week t contributes to a statistically significant increase in new infection cases in week t + 1. In particular, the explanatory power of themodel on the new infection cases, using the mobility index with the grocery & pharmacy category, has increased to a reasonable degree. Although we cannot explain why the explanatory
power of the new infection rate increases only for a certain human mobility category, humanmobility in the grocery & pharmacy category can explain the time-series pattern of the new
infection cases.

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Table A9: Estimation results for the extended COVID-19 infection–mobility trade-oﬀ in Japan

Dependent
variable:

Retail &recreation

Grocery &pharmacy

Parks

Transitstations

Workplaces

Residential

MobilityCI

1.36(0.55)
1.54(0.79)
0.49(0.11)

4.86(1.09)
2.40(1.19)
0.09(0.05)

0.15(0.39)
0.65(0.51)
0.12(0.08)

0.25(0.59)
2.13(0.48)
0.70(0.14)

0.63(0.72)
1.57(0.63)
0.38(0.16)-0.79(1.74)-5.64(1.15)
0.51(0.13)

0.53(0.89)
2.86(0.63)
0.09(0.05)

0.39

0.30

0.02

0.39

0.19

0.29

0.42

κ1
κ2
ι
Adj-R2

Notes: This table shows the results for the ordinary least squares regression (12) of the log changes in infection
cases on the mobility index, with each column category and constant term. Mobility CI denotes the composite
index of mobility. We obtain the composite index by scaling and signing the ﬁrst principal component, calculatedusing Google mobility indices, to the index for retail & recreation. The sample period is the week of February 23,
2020, through the week of May 2, 2021. The numbers in parentheses are Newey & West (1987) heteroskedasticity
and autocorrelation robust standard errors for least squares with a four-week lag truncation.

VAR settings
First, we check the sensitivity of the results to the lag length of the VAR model. In applications
using the VAR model with long-run restrictions, one may attribute the sensitivity of the results
to minor speciﬁcation changes that are expected if identiﬁcation is weak. In particular, whenlong-run restrictions are used to identify the VAR model, the lag length of VAR plays a rolein identifying restrictions.6 Even when we use the lag length of the VAR model from one week
to ﬁve weeks, the models generate, qualitatively and quantitatively, results that are similar to
those of a benchmark speciﬁcation.
Second, we perform exercises incorporating measures for the strength of the government’s
reaction to the pandemic into the system. Speciﬁcally, we rerun the VAR model, including thecurrent and one- to three-week lags of log changes of Oxford’s Stringency Index, as exogenous
variables. Surprisingly, none of the exercises changed the qualitative nature of the results.
Third, we perform exercises to incorporate some event dummies into the system. Speciﬁcally,
we construct two long vacation dummies. The summer vacation dummy is set to one at the weekof August 9, 2020, and zero elsewhere. The winter vacation dummy is set to one at the week ofDecember 27, 2020, and zero elsewhere. It may not be appropriate to include the winter vacation

6

See Faust & Leeper (1997) for details.

21dummy as a control variable, due to the possibility of losing useful information regarding theidentiﬁcation of structural shocks pertaining to the Japanese people, who seemed to change
their mobility patterns during the winter vacation at the end of 2020 (The following week, theJapanese government declared a second state of emergency, on January 7, 2021. ). Nevertheless,the model produced quantitatively similar results for impulse response functions. However, thehistorical decomposition due to structural shocks around January 2021, was diﬀerent from thebenchmark, because the signiﬁcant drop in the permanent shock at the week of December 27
disappeared in the alternative model.

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