DP2021-16

COVID-19 Infection Spread andHuman Mobility
Masahiko SHIBAMOTO
Shoka HAYAKI
Yoshitaka OGISU

Revised February 3, 2022

COVID-19 Infection Spread and Human Mobility*
Masahiko Shibamoto

Shoka Hayaki

Yoshitaka Ogisu§

January 31, 2022

Abstract
Given that real-world infection-spread scenarios pose many uncertainties, and predictions
and simulations may diﬀer from reality, this study explores factors essential for more realistically describing an infection situation. It furnishes three approaches to the argumentthat human mobility can create an acceleration of the spread of COVID-19 infection andits cyclicality under the simultaneous relationship. First, the study presents a dynamic
model comprising the infection–mobility trade-oﬀ and mobility demand, where an increasein human mobility can cause infection explosion and where, conversely, an increase in new
infections can be made temporary by suppressing mobility. Second, using time-series data
for Japan, it presents empirical evidence for a stochastic trend and cycle in new infection
cases. Third, it employs macroeconometrics to ascertain the feasibility of our model’s predictions. Accordingly, from March 2020 to May 2021, the sources of COVID-19 infection
spread in Japan varied signiﬁcantly over time, and each change in the trend and cycle of
new infection cases explained approximately half the respective variation.
JEL Classiﬁcation: C32; E31; E32; I10.
Keywords: COVID-19; new infection cases; infection–mobility trade-oﬀ; mobility demand;
stochastic trend and cycle; macroeconometrics.
*

We extend our special thanks to Takashi Kamihigashi, Ryuzo Miyao, and Masato Shizume for their extensive
help and encouragement in the writing of this paper. We are also grateful to Kenya Fujiwara, Nobuaki Hamaguchi,
Ken-ichi Hashimoto, Tomoko Hashino, Takahiro Sato, the seminar participants at Kobe University, and two
anonymous referees for helpful comments and discussions. This study received ﬁnancial support from the Japan
Society for the Promotion of Science (JSPS KAKENHI Grant Number 20H05633 and 21K01579) and JST-Mirai
Program, Japan (Grant Number JPMJMI20B4).

Corresponding author: Masahiko Shibamoto, Research Institute for Economics and Business Administration(RIEB) and Center for Computational Social Science (CCSS), Kobe University, 2-1, Rokkodai, Nada, Kobe,
657-8501, Japan. E-mail: shibamoto@rieb.kobe-u.ac.jp.
Shoka Hayaki, Graduate School of Business Administration, Kobe University, 2-1, Rokkodai, Nada, Kobe,
657-8501, Japan. E-mail: 179b011b@stu.kobe-u.ac.jp.
§
Yoshitaka Ogisu, Graduate School of Economics, Kobe University, 2-1, Rokkodai, Nada, Kobe, 657-8501,
Japan. E-mail: 197e109e@stu.kobe-u.ac.jp.

1

Introduction

Many epidemiologists have sounded alerts regarding the COVID-19 infection spread, ever since
the initial outbreak in early 2020. They have conducted detailed analyses of the pandemic situation from the primary stages. For instance, the results of the Susceptible-Infected-Recovered(SIR) model, a benchmark for modeling epidemics in the ﬁeld, hinted at the likelihood of apandemic. Moreover, in February and early March 2020, the models predicted exponential
growth, massive infections, hospitalizations, and deaths.1 The models further suggested thateither eliminating physical contact between people, to reduce the probability of being infected,
or inducing herd immunity, where infected persons come into contact with suﬃcient people whohave recovered from or have become immune to the infection, can slow the infection spread at aﬁxed reproduction rate. Thus, relevant bodies opted for the former approach, by imposing restrictions on mobility and in-person economic activities to alleviate a possible COVID-19-related
tragedy.2
Unsurprisingly, predictions regarding COVID-19 infection spread garnered much attention
because anxieties and fears were strong amid extensive reporting of the outbreak by the media
in early 2020. Epidemiological experts, researchers, and commentators have since disseminatedknowledge and opinions in various media, closely following the evolution of the spread into thecurrent long-running pandemic.3
However, epidemiological model predictions and simulations are not necessarily consistent
with reality. For example, in Japan, the number of new infection cases did not increase as
expected during the ﬁrst wave of the pandemic. Even though infection cases may have decreased
because of the mobility restrictions and the subsequent declaration of the ﬁrst state of emergency
by the government, the number of new infection cases did not decrease easily during the thirdand fourth waves, despite the further restrictions of the second and third state of emergency
declarations. To be sure, real-world scenarios pose many uncertainties, and predictions and1

In Japan, on February 24, 2020, members of the Expert Committee on Countermeasures to Combat Infectious
Diseases of Novel Coronaviruses posited that the infection in Japan could spread rapidly. https://www.mhlw.
go.jp/stf/seisakunitsuite/newpage_00006.html (Accessed June 12, 2021)
2
As per the Expert Committee on Countermeasures against Novel Coronavirus Infections report on February
24, 2020, the committee sought a policy that reduces the speed of the rise in infection cases and lowers the peakof the epidemic wave, while strengthening the medical response system. https://www.kantei.go.jp/jp/singi/
novel_coronavirus/senmonkakaigi/sidai_r020224.pdf (Accessed June 12, 2021)
3News items on COVID-19 on Nippon Hoso Kyokai (Japan Broadcasting Corporation), Japan’s only publicbroadcaster, were 289 in January 2020, increased to 1,638 in February, and reached more than 4,000 in April.
Monthly news stories have since been always approximately 2,000, as of June 2021.

1

simulations may diﬀer from reality if assumptions change. However, to inform relevant policies
and contribute to the literature, it is worthwhile to explore factors essential for a more realistic
description of the evolution of an infection situation.
Arguably, COVID-19 infection cases are closely linked to human mobility. Accordingly, therise in new infection cases is positively associated with human mobility because the probabilityof being infected via interaction with other infected people increases as mobility increases.
However, people curb their mobility in response to the infection situation. Thus, whenever
the number of new infection cases increases, people will restrict their mobility. Therefore, asimultaneous relationship between human mobility and new infection cases likely underlies the
dynamics of an infection spread.
Hence, this study explores the dynamics of the COVID-19 infection and human mobility,
to explain the reality of the COVID-19 spread via the underlying simultaneous relationship.
In particular, the supply side of the COVID-19 infection spread and demand side of humanmobility, are likely to determine the number of new infection cases endogenously. Thus, humanmobility can create a dire situation. That is, given a rising number of new infection cases,
unregulated economic activities can induce an explosive increase in such cases. However, systemic mechanisms can converge the increase or decrease in the number of new infection cases.
When people respond to the current infection situation and refrain from activities, the eﬀective
reproduction rate changes systematically according to their behavior and stabilizes, relative tothe standard SIR model prediction.
Under the presence of the simultaneous relationship, an unexpected observed increase in
new infection cases can produce diﬀerent predictions for infection spread, depending on theunderlying factors. As the infectivity of the virus changes over time with the emergence ofvariant strains, the infection risk changes as well; hence, the number of new infections cansuddenly increase. If people are suﬃciently aware of and sensitive to changes in infection
risk and change their behavior to avoid infection, the infection spread can be expected to betransient. In contrast, if people’s patience with the infection situation is limited and their
behavioral preferences change so that they are less sensitive to changes in infection risk, theinfection spread is expected to be prolonged. Therefore, the key challenge in predicting infection
spread is to monitor not only changes in the infection risk itself, but also the changes in people’s
perception of the risk.

2

The 2020–2021 outbreak situation in Japan is a valuable example to explore the role ofhuman mobility in COVID-19 infection cases empirically. Unlike China, the US, and European countries that imposed mandatory lockdowns, the Japanese government has not legally
restricted social activities owing to the spread of COVID-19, but has simply asked people torefrain from going out and closed restaurants during the declared emergency. 4 Nevertheless,
COVID-19-infection status in Japan has been and remains lower than in most other countries.
As of May 21, 2021, the total number of conﬁrmed cases in Japan was approximately 700,000,
less than 1% of the total population. However, the spread of the infection has profoundly impacted the economy; Japan’s real GDP in 2020 was down by 4.8% from the previous year, thesecond-largest drop on record, after the recession in 2009 due to the global ﬁnancial crisis. Thissituation suggests that despite the absence of legal or behavioral restrictions, people in Japantend to voluntarily restrict their behavior in response to infection spread, at the expense of
economic gain.
This study contributes to the literature in three ways. First, it conducts a model analysis of
the dynamics of new infection cases. In particular, it emphasizes the role of mobility demand.
Changes in human behavior preferences, such as familiarity with and ignorance of the pandemic,
can accelerate infection spread. However, a systematic response of mobility demand to suddenchanges in new infection cases can induce cyclicality of such cases. Second, the study presents
empirical evidence of a stochastic trend and cycle in new infection cases. Third, this study
employs macroeconometrics to present ﬁndings supporting the argument in the model analysis.
Further, sources of the COVID-19 infection spread vary signiﬁcantly over time, and the changes
in the trend and cycle of new infection cases explain approximately half the respective variation,from February 2020 to May 2021 in Japan.
The rest of this paper is organized as follows. Section 2 reviews the literature relevant tothis study. Section 3 presents a model analysis of new COVID-19 infection dynamics. Section
4 introduces the time-series data and shows the time-series features on the number of new
4
The stringency index, calculated by the Oxford Coronavirus Government Response Tracker project, is ameasure of the strictness of government policies and provides evidence that policy impact in Japan is extremely
low. On average, Japan’s stringency index during the sampling period was 39.9, the seventeenth lowest amongthe 181 countries for which data can be obtained. Moreover, it is the second lowest among Organisation for
Economic Co-operation and Development member countries after New Zealand (36.9) and the second lowest
among countries with a population of 50 million or more after Tanzania (21.8). The global average of thestringency index is 59.1, and the population-weighted global average is 66.1. The index in major countries suchas Brazil (68.6), Canada (68.6), China (71.9), France (65.0), Germany (66.4), India (74.6), Italy (71.8), Russia(53.9), the UK (70.2), and the US (66.1) is generally high.

3

COVID-19 infection cases. Section 5 presents empirical results via macroeconometric analysis.
Section 6 discusses the ﬁndings and concludes the study. The Appendix provides data sources
and weekly data construction. The Online Appendix provides additional analyses, and reports
the robustness check and sensitivity analysis of the empirical results in the benchmark model.

2

Related work

Model prediction for equilibrium reproduction number equal to one
Several studies (e.g., Gans, 2020) present theoretical models that can generate the tendency
toward an equilibrium point, where the reproduction number is equal to one. Such works
are inspired by the regularity in the reproduction number pattern, showing a sudden rise andsubsequent fall to approximately one or just below one (not zero), that has been documented
across all states in the US and many countries worldwide, documented by Atkeson et al. (2020).
The key mechanism underlying the equilibrium reproduction number equal to one in the model,
is that individuals can base their behavior on prevalence rather than the falling set of those whoare susceptible, to slow an infection using the standard SIR model with a ﬁxed reproduction
rate. In this paper, we present a dynamic model by retaining its implication. We abbreviate
the SIR part of the model for simplicity and specify the supply side of new infection cases,
to capture a trade-oﬀ between the new cases as something undesirable and human mobility.
Further, we emphasize the role of mobility demand to generate the stochastic trend and cyclein the new infection cases.

Time-series analysis for the COVID-19 infections
Time-series modeling considers the model that captures the non-stationary nature of the infection spread in the dynamics of new infection cases. Jiang et al. (forthcoming) develop the
self-normalization technique to capture the phase transitions of an epidemic growth rate via
multiple change-points and apply it to the log of the cumulative COVID-19 conﬁrmed casesand deaths. They argue that the forecasts using time-series modeling can be a meaningful
addition to the other forecasting models, including complex mechanistic models for tracking
the COVID-19 pandemic. While their model characterizes the non-stationary nature of theinfection spread as the piecewise linear trend, we consider time-series modeling that does the4same as a stochastic trend. Speciﬁcally, we provide empirical evidence of a stochastic trend andcycle in the log of the new infection cases and apply the vector autoregressive (VAR) modelcomprising the log changes in such cases and the measure of human mobility to investigate therole of the trend and cycle.5

Forecasting and simulating COVID-19 infections using complex mechanistic
modelsMany studies have generated COVID-19 infection forecasts and simulations. In particular,
epidemiological models, such as the SIR model, are commonly used for estimations and predictions to quantify knowledge (or the absence of it) on the current infection status, and generate
simulations to explore suggestions for policymakers (Kissler et al., 2020; Atkeson, 2020). Ariket al. (2020) propose complex mechanistic models modifying epidemiological models, such as
the Susceptible-Exposed-Infectious-Removed model, by means of information-bearing covariates using machine learning and artiﬁcial intelligence. This study does not aim to explore orprovide a better forecasting model than other proposed forecasting methods. Rather, it focuseson investigating the role of human mobility in COVID-19 dynamics, using a model that cancapture the dynamics of the number of new infection cases and human mobility.

Epidemiological models with human behavior
Following the seminal work by Eichenbaum et al. (2021), many studies are increasingly incorporating epidemiological SIR models into economic analysis; these are often referred to as the
SIR–Macro models. Regarding Japan, Kubota (2021) and Hosono (2021) present the models
with diﬀerent ingredients per their interests. They mainly focus on the eﬀect of the spread of
COVID-19 infection on economic activities, using the dynamic stochastic general equilibrium
models. In contrast, this study primarily examines the role of human mobility in the dynamics
of the new infection cases using macroeconometric models.

5
The VAR model yields consistent estimates even if each variable in the model is non-stationary. See Hamilton(1994) pp.651-653.

5

Empirical evidence on the association between the COVID-19 infection andhuman mobility
Many empirical studies have demonstrated a signiﬁcant relationship between COVID-19 infection status and human mobility using various data sources. They conduct an empirical analysis
for each of the two causal relationships between human mobility and the number of new infection
cases. Some studies report empirical evidence supporting a trade-oﬀ between human mobility
and COVID-19 infection. For example, Kraemer et al. (2020) use rich data on COVID-19 infection cases (including the dates when people ﬁrst reported symptoms) and real-time travel
data in China, and ﬁnd that mobility measures oﬀer a precise prediction of COVID-19 spreadin Chinese cities at the start of 2020. In Japan, Nagata et al. (2021) estimate the impact ofmobility changes on the new conﬁrmed cases using mobile device data and argue that mobilitychanges, especially in areas active at night, were positively and signiﬁcantly associated with
COVID-19 spread. Moreover, Fujii & Nakata (2021) and Fukao & Shioji (2022) use daily dataon the Google mobility index and the number of new infection cases to analyze the determinants of reproduction rate and mobility.6 Others report the response of human mobility to new
infection cases. For example, Goolsbee & Syverson (2021) use mobile phone data in the US between March and May 2020, to compare the reductions in visits to businesses between counties
that were in government-mandated lockdown states with those in non-lockdown states. Theyﬁnd that the reductions in visits primarily stemmed from peoples’ choices; those attributable
to a government-mandated lockdown were few. Further, Watanabe & Yabu (2021) use a daily
prefecture-level mobile device data to examine the degree of voluntary response to infection
cases in Japan and ﬁnd results quantitatively and qualitatively similar to Goolsbee & Syverson(2021).7 This study conducts empirical analyses using the macroeconometric model comprisinga system of simultaneous equations that considers the two causal relationships between humanmobility and the number of new infection cases.8

6

Fujii & Nakata (2021) connect SIR dynamics with economic activity, though they do not incorporate optimization of economic agents in their analysis, unlike the studies by (for example) Eichenbaum et al. (2021).
7
Watanabe & Yabu (2021) refer to the voluntary lockdown as refraining from going out by choice, which is
diﬀerent from the mandatory lockdowns by governments in China, the US, and European countries.
8
Although this study has diﬀerent objectives relative to prior studies, the closest study to this work is theone by Fukao & Shioji (2022) in that they regress each of the system comprising the infection–mobility trade-oﬀ(i.e., the pandemic Phillips curve) and mobility demand (i.e., the pandemic IS curve and pandemic Taylor rule).

6

3

Rationale for the existence of both a stochastic trend andcycle in the dynamics of the COVID-19 infection

This section explains the logic behind our argument that while an increase in human mobility
can cause infection explosion, such an increase in new infections can be made temporary bysuppressing mobility.9 Speciﬁcally, we present the analysis using a straightforward model forthe dynamics of new COVID-19 infection cases, comprising a system of simultaneous equations
that considers the two causal relationships (i.e., the infection–mobility trade-oﬀ and mobility
demand). Our model can produce a stochastic trend and cycle of new infection cases undercertain conditions.

3.1

The dynamic model of the COVID-19 infection and human mobility

We consider a model that can describe COVID-19 infection dynamics and human mobility. The
variable πt represents the new infection cases at time t, deﬁned as the log of new conﬁrmed
COVID-19 infection cases. The variable yt represents the mobility level chosen by people attime t, deﬁned as the log deviation from the steady state.10
First, we model the causal relationship representing an infection–mobility trade-oﬀ. Specifically, we consider the new infection production function as follows:

πt+1 = πt + κyt−1 + ϵπ,t+1 ,(1)where κ > 0 is the parameter, and ϵπ,t is the stochastic term of the new infection production
function. Equation (1) can easily express an infection–mobility trade-oﬀ, where new infection
cases, as something undesirable, are positively associated with human activities. Further, in theformulation of Equation (1), a new infection case at time t + 1, πt+1 , is regarded as a functionof human mobility two terms prior, yt−1 . This development reﬂects “epidemiological rigidity,”
which is the time lag between physical contact with infected people and COVID-19 infection(including worsening symptoms after an incubation period, diagnosis, testing, and reporting).
We assume the time lag to be two weeks in the modeling.11
9The purpose of this section is not to propose a model which is suitable for real-world applications, such aspredicting or simulating the infection spread.
10
In the model economy, we assume weekly frequency for the duration of time t.
11
Lauer et al. (2020) report that the median incubation period is 5.1 days and that 97.5% of people develop
symptoms within 11.5 days. Thus, it takes approximately a week from the time of infection to the time one

7Now consider that there are two types of shocks related to new infection case at time t,
denoting ϵp,t and ϵc,t respectively. We then decompose the stochastic term of the new infection
production function as ϵπ,t = ϵp,t + ϵc,t . Regarding each shock process, we assume that

ϵi,t = ρi ϵi,t−1 + ξi,t ,ρi ∈ [0, 1], i ∈ {p, c},(2)where each ξi,t is independently and identically distributed with mean-zero, and is independentof each other. We will consider that people’s responses to new infection cases vary across shocks,
as shown below.
Next, we model the systematic response of human mobility demand, as people decide theirmobility level by observing the latest number of new infection cases. Given the premise of theavailability of information on the new infection cases at time t, we specify the mobility demand
function as follows:
yt = b(πt − πt∗ ),(3)where b < 0 denotes the parameter representing the sensitivity of mobility demand to the infection situation. Equation (3) expresses the demand of people’s mobility where people negatively
change their mobility demand in response to the outbreak situation. In particular, mobility
demand systematically responds to the deviation in new infection cases from its reference level.
πt∗ is the reference level of crowds. From (3), the reference level works as a threshold of theactivity level to react; if the number of new infection cases increases (decreases) above (below)
the reference level, people refrain from their activities (become active).
We relate the degree of people’s patience about the COVID-19 situation to the dynamics ofthe reference level. In practice, as the constrained lifestyle owing to the COVID-19 pandemic
is prolonged, infection anxiety weakens: this phenomenon is often called Corona habituation.
As reports of high numbers of new infection cases lead people to assume an air of permanence,people begin to entertain the thought that they “deserve it,” even as the infection situation
worsens.12 Thus, to incorporate the degree of people’s patience toward the infection situation
develops symptoms; further, it takes a certain number of days after the actual symptoms appear for those
infected to go to the hospital and undergo tests, after which the test results are reported. In practice, as per theannouncement by the Japanese government of the basic policy on countermeasures against COVID-19 infections,the average time between the date of the onset to the date of the patients’ diagnosis (reported until the earlystage of the COVID-19 pandemic at the end of March 2020) was 9.0 days, though this seemed rather short at thetime of preparing this paper. https://www.kantei.go.jp/jp/singi/novel_coronavirus/th_siryou/kihon_h_0525.pdf (Accessed June 14, 2021)
12
Kuga (2021) reports the results of an empirical analysis of the status of Corona habituation in Japan.

8

into the model, we parameterize the reference level in Equation (3) of the mobility demand as
follows:
πt∗ = dπ̄t ,(4)where d > 0 denotes the parameter representing the degree of people’s patience and π̄t is thepotential and trend level of the number of new infection cases, which are time-varying given
ϵp,t , following π̄t = π̄t−1 + ϵp,t . In d = 1, the reference level of new infection cases is neutral
to their potential and trend level. In d > 1, people are not patient toward infection cases. As
the long-term level of new infection cases increases, the reference level rises above the potential
level, as described by Corona habituation. In d < 1, people are conservative about infection
cases. Even if the long-term level of new infection cases rises even higher, people do not change
their reference level drastically.
In our model setting, the change in new infection cases given ϵp,t is associated with a shock in
mobility demand. We incorporate the degree of people’s patience about the infection situation
into the mobility demand in the modelyt = bπt − bdπ̄t−1 − bdϵp,t ,(5)

by substituting Equation (4) into Equation (3). Regarding d > 1, the increase in new infection
cases given ϵp,t results in an increase in mobility demand, separate from its systematic response.
Despite the increasing number of new infection cases, people aggressively continue their economic activities. However, regarding 0 < d < 1, the increase in new infection cases given ϵp,tresults in a decrease in mobility demand. Although they increase their reference level, people
partially refrain from activities as per the infection situation. Moreover, regarding d = 1, people’s behavior remains unchanged despite the rise in new infection cases given ϵp,t , since theirreference level also rises by the same amount.
Here, we analytically investigate the dynamic properties of the system. Replacing yt−1 inSpeciﬁcally, the researcher quantitatively analyzes the state of people’s COVID-19 anxiety using the “Survey on
Changes in Daily Life Due to the New Coronavirus” (『新型コロナウイルスによる暮らしの変化に関する調査』),
conducted by the Nissay Research Institute every three months since June 2020 on approximately 2,000 men andwomen aged 20 to 69 nationwide. The author notes that, relative to September 2020, when the second wave had
passed, infection fears had weakened in late March 2021, despite a signiﬁcant increase in the size of the infected
population.

9

Equation (1) by Equation (3) generates∑∗∆πt+1 = κb(πt−1 − πt−1)+

ϵi,t+1 .

(6)i∈{p,c}Equation (6) means that when the number of new infection cases two weeks ago is above (below)
the reference value, the current number of new infection cases is decreasing (increasing).
The model suggests that the change in new infection cases due to the infection shock ϵc is
t−1
∑
∗
∗
temporary. Replacing πt−1 in (6) by πt−1 = dπ̄t−1 = dϵp,τ under the assumption of theτ =1initial value of π̄0 = 0 generates∆πt+1 = κbπt−1 − κbdt−1∑ϵp,τ +

τ =1∑

ϵi,t+1 .

(7)i∈{p,c}This relationship can also be described as a downward-sloping line given that ∆πt+1 is a functionof πt−1 , as in the left panel of Figure 1. The steady state of the new infection cases is wheret−1∑
s∆πt+1 = 0, which occurs at π = dϵp,τ .13 Since ϵc does not aﬀect the steady state, the changeτ =1

in infection cases given ϵc can be shown as movements along the horizontal axis: leftwards whenπ > π s and rightwards when π < π s . Hence, the ﬂuctuation due to infection shocks ϵc creates
cyclicality in new infection cases in the model economy.
However, this study’s model suggests that the infection shock ϵp can permanently impact
new infection cases. The right panel of Figure 1 illustrates this impact diagrammatically. Thechange in infection cases given ϵp generates a rightward shift in the line, representing a change′

in the stochastic trend of the new infection cases. π stabilizes at the new steady state π s in
the diagram. Thus, the ﬂuctuation given ϵp is the source of the non-stationary nature of new
infection cases in the model economy.
The model can also be used to understand how the explosive spread of new infection cases
occurs. For simplicity, we assume that new infection cases are in a steady state at time t = 0,
without losing generality. Consider that an infection shock ξp hits the system at time 1. From
Equation (7) with Equation (2); we can then easily compute ∆π1 = ξp,1 and ∆π2 = ρp ξp,1 <∆π1 . Meanwhile, the rise in new cases at time 3 is given by ∆π3 = {ρ2p + κb(1 − d)}ξp,1 . The
rise in new infection cases can then re-accelerate (i.e., ∆π3 > ∆π2 ) if the parameter values in

13

We can derive the steady state of human mobility y s = 0.

10∆πt+1

−κbd

∆πt+1

Pt−1

τ = 1 ²p,τ

−κbd

Pt−1

τ = 1 ²p,τ

−→

πs

πs

> > 0> > > < < < < π<t−1

0

π s0

> > >

πt−1

Figure 1: Dynamics of the new COVID-19 infection cases in the model, the eﬀect of ϵc and ϵp .
Notes: This ﬁgure shows a phase diagram describing the dynamics of new COVID-19 infection cases in ourmodel. The left panel displays the eﬀect of an increase in new infection cases due to the shock ϵc . The rightpanel displays the eﬀect of an increase in new infection cases due to the shock ϵp .

the system are satisﬁed with the following condition;−b(d − 1) >ρp (1 − ρp )
.
κ

(8)

Intuitively, Equation (8) means that the new infection cases could form a hump-shaped curvewith some delay if people’s reaction to such cases is too sensitive (|b| is suﬃciently large) or
people have a fairly impatient preference for new infection cases (d > 1 is large enough).

3.2

Numerical example

We illustrate the dynamic relationship between the new infection cases and human mobility
using the dynamic model with given parameter sets. The system in our dynamic model can besummarized by the VAR form with vector St = (πt π̄t yt−1 ϵp,t ϵc,t ), as follows:



1
1
0
κ ρp ρc






1
0
1
0 ρp 0 








St+1 =  b −bd 0 0 0  St + 
0






1
0
0
0 ρp 0 
00
0 0 ρc
0

11


1




0

 ξp,t+1 .
0 ξc,t+1

0

1

(9)

Table 1: Parameters for the numerical example
κ

b

d

ρp

ρc

4

−0.05

5

0.3

0.3

Notes: The entries show the values of the parameters in the dynamic model, summarized by the vector autoregressive form (9) in calculating the calibrated impulse responses of the variables in the model to infection shocks,
ξp , ξ c .

We can then calculate impulse responses of the variables in the system under a given set ofparameters to examine the dynamic causal eﬀect of the infection shocks, ξp,t and ξc,t .
Table 1 presents the selected parameters. As a benchmark, we set the parameter value ofκ = 4, selected as per the empirical ﬁndings of Nagata et al. (2021). Moreover, we also set theparameter value of b = −0.05, selected based on the empirical ﬁndings of Watanabe & Yabu
(2021). We set the value of d to be equal to 5 as a calibration benchmark by assuming a Corona
habituation where people are impatient about the outbreak situation. We set the values of theautoregressive parameters, ρp and ρc , to be equal to 0.3, reﬂecting the persistence of infection
shocks.
The upper and lower four panels of Figure 2 show the calibrated dynamic eﬀects of the two
infection shocks, ξp , and ξc , respectively, on the variables in the model economy. The solid linewith circles in each panel represents the calibrated impulse responses of up to 20 times to one
unit of each infection shock at time 0.
The dynamic model with given parameter sets predicts that the infection shock ξp can induce
the explosion of new infection cases, accompanied by increases in human mobility. From theupper-left panel of Figure 2, new infection cases π increase exponentially, as suggested by the
SIR model. Further, the log change in the number of new infection cases ∆π shows a humpshaped response with a second peak after some time, while leading to a sustained increase in
human mobility y. Hence, in our model economy, the variation induced by the infection shockξp results in an explosion of new infection cases that produces a stochastic trend, accompanied
by a positive correlation between the new infection cases and human mobility.
In our model, an increase in human mobility, as per people’s response to an outbreak situation, induces an infection explosion. Figure 3 shows the comparison of the dynamic responses
to the shock ξp with diﬀerent values of parameter d representing the degree of people’s patience
about the infection situation. The dashed (dotted) line in each panel represents the responses
12Figure 2: Calibrated responses of variables in the model economy to the infection shocks, ξpand ξc .
Notes: The solid line with circles in each of the upper and lower four panels represents the calibrated impulse
responses to one unit of infection shocks, ξp and ξc , respectively. π: the log of new infection cases, π̄: the potential
and trend level of new infection cases, ∆π: the log change in new infection cases, y: the mobility level, ξp , ξc :the infection shocks described in the text.

13

Figure 3: Calibrated responses of variables in the model economy to the infection shock ξp with
diﬀerent sensitivities of reference levels
Notes: The solid line with circles, dashed line, and dotted line in each panel represent the calibrated impulse
responses to one unit of infection shock ξp with diﬀerent parameter values d, 5, 1 and 0.5, respectively. π: thelog of new infection cases, π̄: the potential and trend level of new infection cases, ∆π: the log change in new
infection cases, y: the mobility level, ξp : the infection shocks described in the text.

with d as equal to one (d equal to 0.5). From the ﬁgure, the absence of mobility responses
regarding d = 1 results in a limited rise in new infection cases relative to the benchmark, because the increase in reference levels π ∗ is comparable to the increase in the number of new
infection cases π, π̄ due to an infection shock ξp . Further, people’s behavior remains unchanged.
Additionally, regarding d = 0.5, as human mobility decreases, π converges to a level lower
than 1, which is the impact immediately after the shock. Therefore, the model predicts thatnew infection cases can exponentially increase with a stochastic trend, due to increased humanmobility associated with patience about the infection situation.
However, the increases in new infection cases induced by the shock ξc are temporary. Fromthe lower four panels of Figure 2, the new infection cases π eventually converge to zero after
its transitory rise, while leading to a persistent decrease in human mobility y. Therefore, theﬂuctuation due to infection shocks ξc creates cyclicality in the new infection cases in the model
economy.

14

Figure 4: Calibrated responses of variables in the model economy to the infection shock ξc withor without systematic responses of mobility demand
Notes: The solid line with circles and dashed line in each panel represent the calibrated impulse responses to
one unit of infection shock ξc with the diﬀerent parameter values b, −0.05, and 0, respectively. π: the log of new
infection cases, π̄: the potential and trend level of new infection cases, ∆π: the log change in new infection cases,
y: the mobility level, ξc : the infection shocks described in the text.

The key mechanism underlying the cyclicality in new infection cases in our model is themobility demand based on which people react to the increase in new cases by suppressing
mobility. Figure 4 shows the comparison of the dynamic responses to the shock ξc with diﬀerent
values of parameter b representing the systematic response of a mobility demand to the new
infection cases. The red-dashed line in each panel represents the responses with b equal to zero.
From the ﬁgure, the absence of mobility responses regarding b = 0 does not result in a steadydecline in ∆π for the convergence of π to zero. Thus, a systematic response of mobility demand
to changes in new infection cases is critical in the cyclicality of such cases in our model economy.
From our model, we can propose the following hypotheses to be examined empirically. The
ﬁrst is that there are, in reality, both a stochastic trend and cycle in new infection cases. The
second is that there is a mixture of infection shocks that cause an infection explosion withincreased mobility, and infection shocks that cause a temporary increase in new infection caseswith decreased mobility. The third is that we can explain these diﬀerences in the impact of15

infection shocks in relation to the role of mobility demand; systematic changes of mobility
demand in response to the infection spread, result in only a temporary increase in new infection
cases, while exogenous changes in mobility demand have a persistent impact on new infection
cases.

4

Evidence on the stochastic trend and cycle in new positive
cases of the COVID-19 infection

This section presents evidence on the stochastic trend and cycle in new infection cases. We
document the statistical time-series features of new positive COVID-19 infection cases in Japan.
We especially focus on non-stationarity and cyclicality in infection spread.
We note the time-series data on COVID-19 spread in Japan during the sample period. Basedon data availability, the sampling period is set from the week of February 16, 2020, to the weekof May 9, 2021. The frequency of all data is weekly, and the sampling period spans up to 65
weeks. We constructed the weekly time series of conﬁrmed new infection cases as the total cases
over the week from Sunday to Saturday.14
First, we statistically examine the time-series characteristics of new infection cases. Table 2
shows the mean and standard deviation of the log changes in such cases. It reports the sum of
the univariate autoregressive (AR) model coeﬃcients with three-week lags for the log changes
in new infection cases, as a measure of their persistence. Table 3 shows the standardized longrun variance of the log changes as a measure of persistent ﬂuctuations in the log level of new
infection cases and its standard error, as proposed by Cochrane (1988).15
The summary statistics document the non-stationary nature of the log scale of the numberof new infection cases. From Table 2, a historical average weekly rate of new infection cases
is about 10%, but its deviation from the average is quite large over the sample period. More
importantly, Table 2 shows that the persistence of log changes in new cases is 0.585; thus, thesudden increase in the new infection cases tends to persistently raise the reproduction rate.
14

The analysis focuses on the weekly time-series variation, as it does not include diﬀerences by day of the week,
as in a daily series of the number of new infection cases.
15
See Cochrane (1988) for more details. The researcher proposes the ratio of far-future long-run variance of theﬁrst diﬀerence of the time-series variable of interest to its variance (we call the ratio the standardized long-run
variance) as a measure of the persistence of the series level. This measure is zero for a stationary time series, one
for a pure random walk, a number greater than one for a series that continues to diverge following a shock, and anumber between zero and one for a series that returns to a stochastic trend in the future. The author estimates
the long-run variance by 1/l times the variance of l-diﬀerences of the series.

16

Table 2: Summary statistics for log change in new cases

Log change in new cases

meanstd. dev.

persistence

0.098

0.364

0.585

Notes: This table shows the mean, standard deviation, and persistence measure of log change in new cases
from the week of February 16, 2020, to the week of May 9, 2021. It reports the sum of the coeﬃcients of
the univariate autoregressive model with three-week lags for the log changes in the new infection cases as thepersistence measure.

Table 3: Standardized long-run variance of log change in new cases
Lag (k)Standardized long-run variance

1

4

8

12

16

20

1.622
(0.331)2.667(0.861)

1.978(0.857)0.820(0.427)0.344(0.205)0.412(0.273)

Notes: This table shows the standardized long-run variance of log change in new cases from the week of February
16, 2020, to the week of May 9, 2021. The standardized long-run variance with k lag(s) is calculated as 1/(k + 1)
times the variance of k + 1-diﬀerences of the log of the new infection cases, divided by the variance of its ﬁrst
diﬀerence. The numbers in parentheses are the Bartlett standard errors, calculated as (4(k + 1)/3T ).5 times
standardized long-run variance.

Hence, the number of infection cases in Japan can increase exponentially, as suggested by the
SIR model.
However, the persistence measure for the log level of new cases suggests cyclicality as well
as non-stationarity in new infection cases. Table 3 shows the standardized long-run variances oflog change in new cases from lag 1 to lag 20; the values in parentheses are asymptotic standard
errors. The standardized long-run variance is above one for a shorter horizon; for example,
standardized long-run variance with lag k = 1 is above one and is statistically signiﬁcant.
Nonetheless, it is signiﬁcantly below one for far longer horizons, such as lag k = 20. Thus, theﬂuctuation in the log level of new cases should have a cyclical component, and the deviation
following the component should return to a stochastic trend in the future. 16
Next, we graph the dynamic properties of the log of COVID-19 new infection cases in reality.
Figure 5 shows the scatter plot of the one-week-ahead log changes in the number of COVID-19

16

The standardized long-run variance reaches its peak with lag k = 5 and decreases from there. Since thestandardized long-run variance reﬂects the cumulative autocorrelation, the autocorrelation begins a course toward
negativity from around lag k = 5. This observation corresponds to the period from the peak-to-bottom of thenumber of new infection cases.

17

infection cases and the one-week lag of its log levels, which is a sample analogous to Figure 1. 17
Although there does not seem to be much of a relationship between log changes and levels in
the new infection cases over the entire sample period, we can identify some informative patterns
by separating the periods. In particular, the scatter plot appears to depict a downward-sloping
line for each subsample, which implies a cyclicality in the new infection cases in our dynamic
model in Section 3. This suggests that the number of new infections sometimes tended to return
to a certain level. On the other hand, looking across the subsamples, the downward-sloping
line appears to have shifted to the right, especially around mid-2020, which implies a change inthe stochastic trend of the new infection cases in the dynamic model. This suggests that theinfection spread was long-lasting, depending on the time of year.

5

Macroeconometric analysis

In this section, we employ macroeconometrics to examine whether the data support the prediction in our model analysis of Section 3. Speciﬁcally, we use the VAR model to specify thejoint dynamics of COVID-19 infection cases and human mobility in reality. 18 Thus, this section presents empirical evidence on the (1) the information content of human mobility for the
COVID-19 infection dynamics, (2) dynamic eﬀect of the changes in the stochastic trend andcycle of the new infection cases, (3) role of human mobility in the dynamics of the new infection cases, (4) simultaneous equations system comprising the infection–mobility trade-oﬀ andmobility demand, and (5) application to the infection situation in Japanese prefectures.

5.1

Information content of human mobility for the dynamics of the COVID19 infection

First, we examine the information content of human mobility for the COVID-19 infection
dynamics. We measure the movement of people using six Google mobility indices (retail &recreation, grocery & pharmacy, parks, transit stations, workplaces, and residential), and thenconstruct the composite index of mobility as the ﬁrst principal component (PC) using six standardized (mean-zero and unit variance) mobility indices, of which the unit is normalized to the17

We thank an anonymous referee for suggesting the following analysis.
A possible alternative approach is to estimate the state space model derived from the dynamic model in
Section 3. However, the model is very simple, so we have concerns about whether the model is valid with theactual data. The plausibility of the dynamic model with respect to actual data is a subject for future research.
18

18

1.5

1.0

0.5

∆πt+1

0.0

−0.5

−1.0

−1.5

−2.0

2020-02-23 ≤ t ≤ 2020-06-20
2020-06-21 ≤ t ≤ 2020-08-15
2020-08-16 < t ≤ 2021-01-02
2021-01-03 < t ≤ 2021-03-27
2021-03-28 < t ≤ 2021-05-08
0

2

4

6

8

10

12

π t −1Figure 5: Scatter plot of the log changes and levels in infection cases
Notes: The sample period spans from the week of February 23, 2020, to the week of May 2, 2021. Blue circles
indicate the scatter plot of the one-week-ahead log changes in the number of COVID-19 infection cases and theone-week lag of its log levels, from the week of February 23, 2020 to the week of June 14, 2020. Orange stars
indicate the plot from the week of June 21, 2020 to the week of August 9, 2020. Green triangles indicate theplot from the week of August 16, 2020 to the week of December 27, 2020. Red squares indicate the plot fromthe week of January 3, 2021 to the week of March 21, 2021. Purple pentagons indicate the plot from the week ofMarch 28, 2021 to the week of May 2, 2021.

19

unit of retail & recreation mobility index.19 Although using more data and complicated models
could generate a more accurate forecast of new infection cases, we expect the composite index
of mobility as a representative measure for human mobility to contain signiﬁcant information
for the COVID-19 infection dynamics.
We consider the time-series model to capture the dynamics of new positive infection casesand human mobility. Speciﬁcally, we construct the following bivariate reduced-form VAR model:
Xt = a0 + A1 Xt−1 + · · · + Ap Xt−p + et ,(10)

A(L)Xt = a0 + et ,(11)

expressed as′where Xt = (∆πt , yt ) is a two-by-one vector comprising time-series variables of the log change
in new positive cases ∆πt and mobility measure yt at week t, a0 is a two-by-one constant vector,
A(L) = I − A1 L − · · · − Ap Lp is a pth order lag polynomial of a two-by-two coeﬃcient matrix
Aj (j = 1, · · · , p) and et is a two-by-one vector of serially uncorrelated innovation with a meanof zero and a covariance matrix of Σe .
The reduced-form weekly VAR model is estimated from the week of March 1, 2020, to theweek of May 9, 2021. The lag length p in the reduced-form VAR estimation is set to three
weeks. We conﬁrm that taking three-week lags is suﬃcient to capture the system dynamics. 20
We statistically conﬁrm that human mobility plays an important role in constructing a timeseries model of the dynamics of the number of new infection cases. Table 4 shows the results
calculating Akaike and Bayesian information criteria (AIC and BIC, respectively) for the VAR
and other candidate models. As other candidate models, we consider a historical average anda univariate AR with three-week lags.
From the perspective of model selection, there is statistical support for adding information
on human mobility to the time-series model. We ﬁnd that the VAR model has the smallest
19
The weekly mobility index we use matches the median of each week of the daily mobility index, to eliminate
the eﬀects of holidays as much as possible. See the online appendix for more details on the Google mobility
indices and the data construction for the composite index of mobility.
20
The Bayesian information criterion selects one lag, and the Akaike information criterion selects two lags. We
perform a modiﬁed likelihood ratio test, proposed by Sims (1980), to check whether taking one or two lags is
suﬃcient. The chi-squared statistics indicate that the null hypothesis of one or two lags is rejected at the 5%
signiﬁcance level against the alternative of three lags. They also indicate that conventional signiﬁcance levels
do not reject the null of three lags, as against the alternative of four lags. Moreover, the estimated results areinsensitive when four and ﬁve lags are used.

20

Table 4: Information criteria for constant vs AR vs VAR models and F statistics for humanmobility in equation for new infection cases
Information criteria for the model:
VAR (3)
Hist. ave.
AR (3)

H0 : y Fails
to Cause ∆π

AIC

BIC

AIC

BIC

AIC

BIC

Granger-F

-2.68

-2.44

-2.02

-1.98

-2.36

-2.22

4.31 [0.04]

Notes: Information criteria indicate the log of the mean square forecast error for one-step model forecasts plus
the penalty term for model complexity. AIC and BIC indicate the Akaike information criterion and Bayesian
information criterion, respectively. For AIC, the penalty is 2/T times the number of parameters in the model
equation. For BIC, the penalty is log(T )/T times the number of parameters in the model equation. Granger-F
indicates the White (1980) heteroskedasticity-robust F -statistic for the Granger causality test under the null
hypothesis that the coeﬃcients on the one- to three-week lags of the composite index of mobility from theregression of log-changes in new infection cases in the reduced-form VARs are all equal to zero. The numbers inbrackets are p-values for the Granger causality test. We set the lag length to three weeks in the AR and VAR
estimation. Estimation samples span from the week of March 1, 2020, to the week of May 9, 2021.

information criterion computed, whether AIC or BIC, of candidate models, which indicatesthe best model ﬁt, including a penalty for complexity. This result suggests that the proposed
VAR model with the lags of the composite index of mobility simply and thriftily captures the
dynamics of the log changes in new infection cases.
Empirical evidence points to the signiﬁcant information content of human mobility for new
infection dynamics. The last column in Table 4 provides the F statistics for the Granger
causality test under the null hypothesis that the coeﬃcients on the one- to three-week lagsof the composite index of mobility from the regression of log-changes in infection cases in the
reduced-form VARs are all equal to zero. The F statistics show that the composite index ofmobility contains valuable information for one-step model forecasts for new infection cases,
which is statistically signiﬁcant at the 5% level.21

5.2

Dynamic eﬀect of the changes in stochastic trend and cycle of COVID-19
infection

In this subsection, we examine the role of “structural” shocks on COVID-19 infection casesand human mobility. In particular, we interpret ﬂuctuations in COVID-19 infection cases and21
We conﬁrm that the predictions of the VAR model improve over the AR model and the historical averagefor multistep-ahead model forecasts. In particular, although it has good predictive power in the shorter horizon,
the predictive power of the AR model in the longer horizon is relatively poor. See the online appendix in moredetails for the comparison of in-sample forecasting accuracy among a historical average, AR model, and VAR
model for the log-changes in new infection cases.

21

human mobility, as being attributable to changes in the stochastic trend and cycle underlying
the time-series pattern of the infection cases. Accordingly, we consider two types of structural
shocks: a shock with a permanent eﬀect on the log level of new infection cases and a shock thatcauses temporary changes in new infection cases.22 We identify two types of infection shocks,
which are in reality mixed together. Thus, we examine whether it is consistent with the model
prediction in section 3 that the former shock type increases mobility and the latter decreases it.
Let ξp,t and ξc,t be a permanent and non-permanent shock, respectively. In a structural
VAR, the innovations et in (11) are assumed to be linear combinations of the structural shocks:

et = Θξt ,

(12)

′

where ξt = (ξp,t , ξc,t ) , the elements of which are assumed to be independent of each other andhave a unit variance, and Θ = (Θp , Θc ) represents the impact matrix for the responses of theVAR variables Xt to the structural shocks ξt . Equations (11) and (12) yield a moving average
representation regarding the structural shocks as follows:

Xt = b0 + B(L)Θξt ,(13)

where b0 = A(L)−1 a0 and B(L) = A(L)−1 .
To identify Θ, we impose the long-run restriction on the VAR developed by Blanchard& Quah (1989) and King et al. (1991).23 In particular, we assume that the non-permanent
shocks do not aﬀect the log level of new positive infection cases in the long run. This long-run
restriction implies that the cumulative response of the log changes in new positive cases to thenon-permanent shock, is constrained to zero:[B(1)Θ]12 = 0,
22

(14)

We label a shock to the trend component of the new infection cases as a permanent shock and a shock totheir cyclical component as a non-permanent shock. In our bivariate VAR model, the innovations in the reducedform VARs are, by construction, decomposed into two structural shocks independent of each other. However,in reality, there can be other shocks than the ones we are interested in, such as idiosyncratic shocks that aﬀect
human mobility, unrelated to the number of new infection cases. In particular, the non-permanent shocks would
be contaminated by such shocks, under the assumption that only a permanent shock aﬀects the log level of newcases in the long run. They cannot be identiﬁed in our bivariate VAR model. We address this issue by using thecomposite index of mobility as a representative measure of mobility, so that the VAR model excludes as many
idiosyncratic factors of mobility as possible. Nevertheless, it should be noted that our VAR model has such alimitation.
23
Blanchard & Quah (1989) and King et al. (1991) develop the VAR model to identify structural shocks, by
imposing restrictions on the long-run eﬀect of a given shock on a given variable.

22

Figure 6: Estimated responses of the log changes in infection cases and human mobility topermanent and non-permanent shocks
Notes: The solid line with circles in the upper and lower panels represent the point estimates of the impulse
responses to one standard deviation permanent and non-permanent shock, respectively. The shaded areas denote
one-standard-error bands, calculated using 1,000 bootstrap samples. Mobility CI denotes the composite index ofmobility. We set the lag length to three weeks in the reduced-form vector autoregressive estimation. Estimation
samples span from the week of March 1, 2020, to the week of May 9, 2021.

where [B(1)Θ]12 is the ﬁrst row and second column of the matrix B(1)Θ. The use of the′restriction (14) with the long-run covariance matrix B(1)Σe B(1) identiﬁes Θ. To determine
the sign of the structural shocks, we assume that positive structural shocks have a positive
impact on the log changes in infection cases, Θ1,p , Θ1,c > 0.
The estimated impulse response functions, B̂(L)Θ̂, are summarized in Figure 6. As describedabove, structural shocks are normalized to have unit variance and signed to positively aﬀect thelog changes in infection cases on impact. The solid line with the circles indicates the estimatedresponse for the log changes and the composite index of mobility for up to 20 weeks. The shaded
areas denote one-standard-error bands, calculated using 1,000 bootstrap samples.
Overall, this ﬁgure shows that structural shocks have a plausible eﬀect, consistent with the
dynamic model analysis discussed in Section 3. We ﬁnd that the log changes in infection cases

23

have a hump-shaped response to a permanent shock, with the second peak eﬀect emerging
after two weeks. Quantitatively, permanent shocks increase the level of new infections overthe following eight weeks, by more than ﬁve times the impact, immediately after the shock.
Moreover, human mobility rises in response to a permanent shock. However, a non-permanent
shock, which raises the log changes in infection cases for two weeks and declines steadily for about
eight weeks, induces a persistent decrease in human mobility to its lowest values, approximately
four weeks after the shock occurs.
Next, we report the time series of structural shocks identiﬁed using the VAR model. In
particular, it is useful to decompose the unexpected changes in COVID-19 infection cases intochanges in the stochastic trend and in the cycle. Based on our model in section 3, we expectthat permanent shocks would capture exogenous changes in people’s demand for human mobility that deviates from systematic responses to infection risk, such as changes in behavioral
preferences like Corona habituation and emergency restrictions on mobility due to certain prevalence measures by the government. On the other hand, we expect that non-permanent shocks
would capture sudden changes in the number of new infection cases, which people perceive as
the rise in infection risk, which is reﬂected in changes in the degree of fear and anxiety about
the COVID-19. The identiﬁed structural shocks would have historically captured changes inpeople’s perception of infection risk, over the sample period.
Figure 7 displays the time series of permanent and non-permanent shocks. The bars in
the upper and lower panels indicate permanent and non-permanent shocks, respectively, as
identiﬁed using the estimated VAR model. The orange shaded areas show the weeks coinciding
with the period during the declaration of a state of emergency in Japan. 24
The upper panel of Figure 7 shows the persistent troughs representing permanent shocks forthe period during the ﬁrst state of emergency declaration on April 7, 2020, followed by upward
swings. Thus, the ﬁrst state of emergency declaration can contribute to inducing a downward
24

In Japan, a state of emergency was declared three times during the sampling period: from April 7 to May 25,
2020; from January 8, to March 21, 2021; and from April 25, 2021 onward. The legal basis for policy responses
by governors of 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 that people refrain from going out of their homes, and to request andinstruct 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 comparedto the lockdowns in China, the United States, and many European countries, for example. On February 13, 2021,
during the second declaration of a state of emergency, the Act was amended to allow prefectural governors toorder facility managers. Based on the newly established Article 79, facility managers who do not comply withthe order will be subject to a ﬁne of up to 300,000 yen.

24

Figure 7: Identiﬁed permanent and non-permanent shocks
Notes: The bars in the upper and lower panels indicate the permanent and non-permanent shocks, respectively,
identiﬁed using the estimated vector autoregressive (VAR) model (11) with the restriction (14). We set the lag
length to three weeks in the reduced-form VAR estimation. 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 ofMarch 1, 2020, to the week of May 9, 2021.

trend in the number of new infection cases in response to the ﬁrst wave of the COVID-19
pandemic. Moreover, there are upward swings in permanent shocks for several weeks from theweek of May 24, 2020, to the second half of August 2020. Hence, the termination of the stateof emergency on May 25 may contribute to an upward trend in the number of new infection
cases. However, there are no large troughs for the period during the second and third states of
emergency, except for a signiﬁcant drop at the week of the end of 2020. Therefore, the second
and third states of emergency have not contributed to a decline in the number of new infection
cases. Regarding non-permanent shocks, as in the lower panel of Figure 7, we see persistenttroughs from the ﬁrst half of August 2020 to the second half of October 2020, and from the endof November 2020 to the end of December 2020.
We investigate the extent to which past movements in the new positive cases of COVID-19 in-25

Figure 8: Historical decomposition of changes in COVID-19 infection cases and human mobility
due to permanent and non-permanent shocks
Notes: All the series are displayed as deviations from the deterministic component. Mobility CI denotes thecomposite index of mobility. The purple bar shows the decomposed series explained by the permanent shocks.
The light blue bar shows the decomposed series explained by the non-permanent shocks. The solid line indicatesthe estimated stochastic component before decomposition. The orange shaded areas show the weeks coinciding
with the period during the state of emergency declaration in Japan. We set the lag length to three weeks in the
reduced-form vector autoregressive estimation. Estimation samples span from the week of March 1, 2020, to theweek of May 9, 2021.

fection and human mobility resulted from permanent or non-permanent shocks. Speciﬁcally, we
use the historical decomposition technique to decompose the historical value of the log changes
in new positive infection cases and the mobility index into the accumulated eﬀects of current andpast permanent or non-permanent shocks. The upper panel of Figure 8 shows the time seriesof the log changes in new positive infection cases explained by permanent and non-permanent
shocks, whereas the lower panel of the ﬁgure shows the time series of the composite index ofmobility explained by the shocks. The purple bar shows the decomposed series explained by
the permanent shocks, ξp,t , and the light blue bar shows the decomposed series explained by
non-permanent shocks, ξc,t . The solid line indicates the estimated stochastic components before
decomposition.
26

Table 5: A fraction of the forecast error variance explained by permanent or non-permanent
shocks for COVID-19 infection cases and human mobility

Horizon
0
4
8
20

Log changes in infection cases

Composite index of mobility

Permanent

Non-permanent

Permanent

Non-permanent

18.6
51.4
48.3
48.1

81.4
48.6
51.7
51.9

92.8
71.1
62.1
61.9

7.2
28.9
37.9
38.1

Notes: The entries show the percentage variance of the forecast error made in the column variable, as explainedby the permanent or non-permanent shock at a given horizon. The results are computed from the vector autoregressive (VAR) model (11) with the restriction (14) over the sample period from the week of March 1, 2020, to
the week of May 9, 2021. We set the lag length to three weeks in the reduced-form VAR estimation.

The results reveal that sources of the epidemics of the COVID-19 infection vary greatly fromtime to time. From the upper panel of Figure 8, much of the ﬂuctuation in the log changes in
infection cases, during and several weeks after the ﬁrst state of emergency declaration, occur due
to permanent shocks. However, much of this ﬂuctuation also stems from non-permanent shocks
beginning in the ﬁrst half of August. Even during the second and third states of emergency, adecline in the log changes in infection cases stems from non-permanent shocks.
Finally, we assess the relative contributions of permanent and cyclical components for
COVID-19 infection cases and human mobility dynamics. Speciﬁcally, given the permanentand non-permanent shocks in the VAR model, we decompose the forecast error variances ofthe log changes in infection cases and the composite index of mobility into the variances ofpermanent and non-permanent shocks. Next, we estimate the percentage contribution of eachshock to the forecast errors of the log changes in infection cases and the composite index ofmobility.
Table 5 presents the results of the forecast error variance decomposition. The relative
contribution of non-permanent shocks to the ﬂuctuations of the log changes in infection cases
at the current week is about 80%. Although this contribution falls to about 50% at a longer
horizon, the results suggest an important role of the permanent and non-permanent shocks inthe dynamics of COVID-19 infection cases.

27

5.3

Role of human mobility in the dynamics of COVID-19 infection

In this subsection, we explore the role of human mobility in the dynamics of COVID-19 infection
cases. We predict that the systematic response of human mobility generates the hump-shaped
response of the changes in the new infection cases and the cyclicality therein. Thus, to investigate the plausibility of our prediction, we conduct counterfactual simulations using the VAR
model developed by Bernanke et al. (1997) and Sims & Zha (2006) to measure the role of thesystematic response of mobility in response to structural shocks.
First, we measure the dynamic eﬀect of exogenous changes in human mobility. Speciﬁcally,
we assume that the mobility changes, ξy,t , t = 1, · · · , T , produce a unit increase in the composite
index of mobility, with no impact on the log changes in infection cases at the time a shock occurs,due to epidemiological rigidity. Accordingly, we can calculate the impulse response functions to′an exogenous mobility change as B(L)Θy , where Θy = (0, 1) .
Figure 9 shows the estimated impulse response function to an exogenous mobility change,
B̂(L)Θy . We ﬁnd that, in response to an increase in human mobility, the log changes in infection
cases remain roughly zero for two weeks, due to epidemiological rigidity, and then rapidly
increase for eight weeks. The maximum impact is about four. Thus, a percentage increasein the composite index of mobility, which reﬂects a percentage increase in the mobility index
with retail & recreation, raises the rate of new infection cases by approximately 4%. Hence,exogenous changes in people’s behavior induce an increase in the number of new infection cases,
along with an empirical trade-oﬀ relationship, reported in the next subsection.
Next, we investigate the role of human mobility in the dynamic response to COVID-19
infection cases given permanent and non-permanent shocks. Following Bernanke et al. (1997)
and Sims & Zha (2006), we use the VAR model to conduct counterfactual simulations to assess
how the number of new infection cases would have performed, without the systematic responseof mobility to the structural shocks. Speciﬁcally, we measure the counterfactual values of thevariables in the VAR model by taking the following steps. Given the response function to permanent and non-permanent shocks B(L)Θp , B(L)Θc , we calculate each sequence of exogenousjjjmobility changes {ξˇy,0, ξˇy,+1, · · · , ξˇy,+H} for j = p and c that would reset y̌0 , y̌+1 , · · · , y̌+H tozero in response to permanent and non-permanent shocks, respectively. We then add each sequence before calculating the impulse response function of ∆π to permanent and non-permanent
shocks.
28

Figure 9: Dynamic impacts of the exogenous mobility change on COVID-19 infection cases
Notes: The solid line with circles represents the point estimates of the impulse responses to an exogenous mobility′change, B(L)Θy , where Θy = (0, 1) . Mobility CI denotes the composite index of mobility. The mobility change
increases the mobility CI by one unit. Estimation samples span from the week of March 1, 2020, to the week of
May 9, 2021. The shaded areas denote one-standard-error bands, calculated using 1,000 bootstrap samples. We
set the lag length to three weeks in the reduced-form vector autoregressive estimation.

Figure 10 summarizes the simulation results. The red dashed lines represent the simulated
responses to each structural shock and sequence of exogenous mobility changes, to eliminate thenormal response of mobility to the structural shocks. The diﬀerence between the baseline result
represented by the solid lines and the simulated result indicates a magnitude of the dynamic
eﬀect of the structural shocks through human mobility.
Our simulation results in the estimated VAR model are consistent with our model prediction
in Section 3. From the upper-left panel of Figure 10, the absence of mobility responses results inlimited rise in the log changes in infection cases relative to the benchmark. Moreover, as in thelower-left panel of Figure 10, the absence of mobility responses does not result in a continuous
decline to converge to a steady-state level. These results of the macroeconometric analysis
support the argument that human mobility creates an accelerated increase in new infection
cases underlying the changes in its trend. Further, the systematic response of mobility demand29

Figure 10: Responses to permanent and non-permanent shocks, no mobility response
Notes: The solid line with circles in the upper and lower panels represent the point estimates of the impulse
responses to one standard deviation permanent and non-permanent shock, respectively. The shaded areas denote
one-standard-error bands, calculated using 1,000 bootstrap samples. The red dashed lines represent the simulated
responses to each structural shock and sequence of mobility shocks to eliminate the normal response of mobility
to the structural shocks. Mobility CI denotes the composite index of mobility. We set the lag length to three
weeks in the reduced-form vector autoregressive estimation. Estimation samples span from the week of March 1,
2020, to the week of May 9, 2021.

automatically reduces the number of new infection cases underlying changes in its cycle.

5.4

Simultaneous equations system of infection-mobility trade-oﬀ and mobility demand

In this subsection, we examine the feasibility of our argument from another perspective. In
Sections 5.2 and 5.3, we conducted a macroeconometric analysis on the premise of a stochastic
trend and cycle in the new infection cases. We examined the dynamics of COVID-19 infection
cases and human mobility to permanent and non-permanent shocks in the number of new
infection cases, using the VAR model with a long-run restriction. While our dynamic modelin Section 3 explains that the system comprising the infection–mobility trade-oﬀ and mobility

30demand can create the stochastic trend and cycle in the new infection cases, it is not obvious
whether, in reality, the dynamics generated by permanent shocks and by non-permanent shocks
employ the same system. Thus, we conduct another macroeconometric analysis using the VAR–
IV model on the explicit premise of simultaneous equations comprising the infection–mobility
trade-oﬀ and mobility demand.25 We then show that we can explain the diﬀerences in theimpact of permanent and non-permanent infection shocks in relation to the role of mobility
demand, in that the dynamics suggested by the empirical results employ the same mechanism
as those described in the previous subsections.26
First, we empirically investigate the existence of the infection–mobility trade-oﬀ. Speciﬁcally, we estimate the following speciﬁcation of the infection–mobility trade-oﬀ (1):∆πt+1 = κyt−1 + ι + ϵπ,t+1 .

(15)

The sample period spans from the week of February 23, 2020, to the week of May 2, 2021.
Under the premise that the human mobility should be predetermined in the supply side of theinfection cases due to the epidemiological rigidity, we can estimate the regression (15) by OLS.
Empirical evidence supports the existence of the trade-oﬀ between COVID-19 infection andhuman mobility. Table 6 reports estimation results for the infection–mobility trade-oﬀ (15). The
coeﬃcient κ describing the response of the one-week-ahead log changes in infection cases to theone-week lag of human mobility is positive and statistically signiﬁcant for human mobility. The
OLS estimate of κ implies that a percentage increase in the composite index of mobility raises
the rate of new infection cases by about 3% after two weeks. It is quantitatively comparable withthe ﬁnding using other mobility data and statistical models by Nagata et al. (2021).27 Figure 11
shows the scatter plot of the one-week-ahead log changes in the number of COVID-19 infection
cases and the one-week lag of the composite index of mobility, which graphically conﬁrms thestability of the positive relationship of COVID-19 infection cases and human mobility during
the sample period.
25

See the online appendix for the derivation of the VAR–IV model from a simultaneous equations system of
infection-mobility trade-oﬀ and mobility demand.
26
Note that the VAR model with a long-run restriction and the VAR–IV model should be considered to be
inherently diﬀerent from each other, as they have diﬀerent restrictions. Nevertheless, as shown in the following
analysis, we ﬁnd that the dynamics estimated by using the VAR-IV model, are almost the same as those estimatedby using the VAR with a long-run restriction.
27
Nagata et al. (2021) report an impact of mobility changes in nightlife places on new conﬁrmed cases of about
1% to 4%, despite large regional diﬀerences in sensitivity intensity.

31

Table 6: Estimation results for the COVID-19 infection–mobility trade-oﬀ in Japan
Dependent variable:

log changes in infection cases

κ

3.28(0.54)
0.09(0.05)

ι
Adj-R20.43

Notes: This table shows the ordinary least squares regression results (15) of the log changes in infection cases on
the composite index of mobility and constant term. We obtain the composite index by scaling and signing theﬁrst principal component calculated using six 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 afour-week lag truncation.

Next, we empirically investigate the existence of the mobility demand. Speciﬁcally, we
estimate the following speciﬁcation of the mobility demand (5):′yt = b∆πt + γ Wt + ξmd,t ,(16)

where Wt is a vector of control variables comprising the one- to three-week lags of yt and ∆πt
2 represents the nonsystemand a constant.28 ξmd,t with a mean of zero and a variance of σmd

atic component of the reaction of mobility demand, which we refer to as a “mobility demand
shock.” In interpreting the mobility demand shocks, ξmd,t may reﬂect transitory changes in thepreferences of human behavior and emergent changes in human mobility patterns due to someprevalence measures, irrespective of whether they are compulsory or not, by the government.
We estimate the regression (16) by an instrumental variable (IV) estimation, because thenumber of infection cases can be contemporaneously positively correlated with mobility demand
shock. For example, changing preferences for mobility, such as getting used to the pandemic
28
In specifying the dynamics of human mobility, one might argue that log level should also be added to the
VAR model. In particular, Chernozhukov et al. (2021) and Hoshi et al. (2021) ﬁnd evidence that not only the log
changes, but also the log levels of cases are an important determinant of mobility, using panel data for the US andJapan. Nevertheless, even with the addition of log level of cases, there should be a less signiﬁcant diﬀerence in themobility dynamics that can be captured by our VAR model. In the online appendix, we provide empirical results
showing that adding log level of cases in our VAR model plays a limited role in explaining the dynamics of thecomposite index of mobility. It is important to note that we do not claim that our results are inconsistent withthe ﬁndings of Chernozhukov et al. (2021) and Hoshi et al. (2021). We interpret that the detailed information
contained in the log level of weekly cases in explaining the dynamics of human mobility is already contained in
the lagged values of log changes of cases and human mobility in our VAR model. In addition, the results of ourIV estimation of mobility demand, taking into account the presence of endogenous bias, are consistent with theclaims of Chernozhukov et al. (2021) and Hoshi et al. (2021) that the higher number of cases reduced people’s
mobility.

32

Figure 11: Scatter plot of the log changes in infection cases and human mobility
Notes: Blue circles indicate the scatter plot of the one-week-ahead log changes in the number of COVID-19
infection cases and the one-week lag of the composite index of mobility. The dotted line indicates the ﬁtted value
for the regression of the one-week-ahead log changes in the number of infection cases on the one-week lag of thecomposite index of mobility based on the ordinary least squares estimation. Mobility CI denotes the composite
index of mobility. The sample period spans from the week of February 23, 2020, to the week of May 2, 2021.

lifestyle and experiencing complacency toward the COVID-19 outbreak, can make people more
active, despite the worsening infection situation. Thus, the OLS estimate of b suﬀers from an
upward endogeneity bias. However, estimation using an IV, such as surprise increases in new
infection cases given the emergence of variant strains (which satisﬁes the relevance of the changes
in infection cases and the exogeneity of the mobility demand shocks), allows for a consistentestimate of b.
The instrument in the analysis is the weekly log changes in the search volume of the Japaneseterm “感染者数” (number of infected individuals in English).29 We expect this measure to reﬂect
the degree of anxiety regarding the surprise changes in the spread of the COVID-19 infection
among Japanese people, which correlates with the log changes in infection cases, but less so with29
We retrieved the search volume data from Google Trends (https://trends.google.co.jp/trends/?geo=JP)
on June 1, 2021.

33

familiarity/ignorance toward such cases. To verify the robustness of the proposed instrument, we
compute the F -statistic under the null hypothesis that the coeﬃcient on the instrument fromthe ﬁrst-stage regression of the log changes in infection cases on the instrument and control
variables is equal to zero. We also conduct the Hausman (1978) test for endogeneity.
Table 7 reports the estimation results for the mobility demand (16). The coeﬃcient b
describing the systematic response of human mobility to the log changes in infection cases isnegative (positive) in the IV (OLS) estimation. This result is consistent with our predictionthat the OLS estimate of b suﬀers from an upward endogeneity bias. Moreover, the IV estimateof b shows that a percentage increase in new infection cases reduces the composite index ofmobility by about 0.062, which is quantitatively comparable with the ﬁnding obtained by using
other mobility data and the statistical model by Watanabe & Yabu (2021). 30 The instrument
is robust, with a heteroskedasticity-robust ﬁrst-stage F -statistic of 21.488.31 Furthermore,
the Hausman test result detects endogeneity of the log changes in infection cases, although
at a 10% statistical level, possibly because of eﬃciency loss from the IV estimation. This
result statistically supports the implication of our dynamic model, where the infection–mobility
trade-oﬀ and mobility demand endogenously and simultaneously determine the number of new
infection cases.
Next, we examine the dynamic eﬀect of changes in anxiety related to surprise changes in the
spread of COVID-19 infection. Let ξa,t be an anxiety shock and Θa , the impact vector for the
responses of the VAR variables Xt to the anxiety shock. We can express the impulse response
functions of Xt to an anxiety shock ξa,t as B(L)Θa .
′To identify Θa = (Θ1a , Θ2a ) , we adopt the VAR–IV model proposed by Stock & Watson(2012, 2018). Assume that a unit increase in ϵa,t increases ∆πt by one unit Θ1a = 1. We use
the IV regression to yield a consistent estimate of Θ2ae2,t = Θ2a e1,t + ξmd,t ,(17)using the instrument, where e1,t and e2,t are the reduced-form VAR innovations of the log
30

Given the result of the factor model in the online appendix, a percentage increase in new infection cases
increases mobility in residence by 0.026(≃ −0.062 × −0.42). Watanabe & Yabu (2021) report a reduction inpeople’s outings by 0.026% due to a 1% increase in new infection cases.
31
To ensure that a weak instrument problem is not present, Stock et al. (2002) and Stock & Yogo (2005)
recommend the rule of thumb that requires the F -statistic from the ﬁrst-stage regression of the two-stage leastsquares to exceed 10.

34

Table 7: Estimation results for mobility demand in Japan
Dependent variable:

Mobility CI

OLS estimate of b

0.027
(0.013)-0.062(0.055)

IV estimate of b
Wald F
Hausman

21.4882.764[0.096]

Notes: The dependent variable is the composite index of mobility. The independent variable is the log changes
in infection cases. The constant and 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. The ordinary least squares(OLS) estimate of b indicates the estimate of b by the OLS regression. The instrumental variable (IV) estimateof b indicates the estimate of b by the IV regression. We use the weekly log changes in the search volume of “ 感染者数” (number of infected individuals in English) in Google as an external instrument in the IV regression. The

numbers in parentheses are White (1980) heteroskedasticity-robust standard errors. Coeﬃcients and standard
errors for the control variables are not reported. Wald F indicates White (1980) heteroskedasticity-robust F statistic under the null hypothesis that the coeﬃcient on the instrument from the ﬁrst-stage regression of thelog changes in infection cases on the instrument and control variables is equal to zero. Hausman indicates thestatistic on the Hausman (1978) test under the null hypothesis that the OLS and IV estimators are consistent,
but the OLS estimate is eﬃcient. The numbers in brackets are p-values for the Hausman test. The sample periodspans from the week of March 1, 2020, to the week of May 9, 2021.

changes in infection cases and the composite index of mobility, respectively. Given that the
regression in Equation (17) is mathematically equivalent to the regression in Equation (16), we
can set Θ2a as the IV estimate of b.
Figure 12 summarizes the estimated impulse response functions B̂(L)Θ̂a . Accordingly, aunit increase in infection cases due to an anxiety shock leads to a persistent decrease in humanmobility to its lowest values approximately four weeks after the shock occurs. Moreover, after
the transitory rise in the log changes in infection cases for two weeks, they decline steadily forabout eight weeks. These responses are qualitatively similar to those regarding a non-permanent
shock reported in Figure 6.
We can conﬁrm that the anxiety shocks identiﬁed by the VAR–IV model are the same as
the non-permanent shocks identiﬁed by the VAR model with long-run restriction. Figure 13
displays the time series of anxiety shocks indicated by the solid line (left-hand scale). Relative
to the time series of non-permanent shocks (the dotted line, right-hand scale) identiﬁed with thelong-run restriction (14), the two series are observationally equivalent (the Pearson’s correlation

35

Figure 12: Dynamic impacts of an anxiety shock on COVID-19 infection cases and humanmobility
Notes: The solid lines with circles represent the point estimates of the impulse responses to an anxiety shock
identiﬁed using the estimated vector autoregressive (VAR) model (11) with the weekly log changes in the searchvolume of “感染者数” (number of infected individuals in English) in Google as an external instrument. The
anxiety shock increases the log changes in infection cases by one unit. The shaded areas denote one-standarderror bands, calculated using 1,000 bootstrap samples. Mobility CI denotes the composite index of mobility. We
set the lag length to three weeks in the reduced-form VAR estimation. Estimation samples span from the weekof March 1, 2020, to the week of May 9, 2021.

coeﬃcient between them is 0.997), except for the size of variances. This evidence implies thatthe space covered by the anxiety shocks should cover the space of the non-permanent shocks
identiﬁed by imposing the long-run restriction. Therefore, it supports our view that a systematic
response of mobility demand to a surprise change in the new infection cases generates thecyclicality of COVID-19 infection cases.
Finally, we examine the dynamic impact of a mobility demand shock on COVID-19 infection
cases and human mobility. The straightforward way to analyze it within our framework is toinclude the shocks obtained from the IV regression (16) in the VAR as an exogenous variable:
A(L)Xt = a0 + Θmd ξmd,t + e⊥
t ,
36

(18)

Figure 13: Identiﬁed anxiety shocks
Notes: The solid line (left-hand scale) indicates the anxiety shocks identiﬁed using the estimated vector autoregressive (VAR) model (11) with the weekly log changes in the search volume of “ 感染者数” (number of infectedindividuals in English) in Google as an external instrument. The dotted line (right-hand scale) indicates thenon-permanent shocks identiﬁed using the estimated VAR model (11) with the restriction (14). We set the lag
length to three weeks in the reduced-form VAR estimation. The sample period spans from the week of March 1,
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.

where Θmd is a two-by-one vector capturing the contemporaneous response of the elements of
Xt to a mobility demand shock. The innovation term e⊥t is, by construction, orthogonal tothe mobility shock. We can obtain the consistent estimates of Θmd by regressing the VAR
innovations et on the mobility demand shocks ξmd,t . We can express impulse response functionsof Xt to a mobility demand shock ξmb,t as B(L)Θmd .
In addition to the anxiety and non-permanent shocks, the mobility demand shocks are thesame as the permanent shocks identiﬁed in the VAR model with the long-run restriction. From
Figure 14, the estimated responses to a mobility demand shock are qualitatively similar to thoseto a permanent shock. In response to the rise in human mobility due to a mobility demand
shock, the log changes in infection cases have a hump-shaped response, with the second peakeﬀect occurring after two weeks. Moreover, from Figure 15, the mobility demand shocks track37

Figure 14: Dynamic impacts of a mobility demand shock on COVID-19 infection cases andhuman mobility
Notes: The solid line with circles represents the point estimates of the impulse responses to one unit of a mobility
demand shock obtained from the residual of the instrumental variable regression (16) with the weekly log changes
in the search volume of “感染者数” (number of infected individuals in English) in Google as an instrument. The
shaded areas denote one-standard-error bands, calculated using 1,000 bootstrap samples. Mobility CI denotes
the composite index of mobility. We set the lag length to three weeks in the reduced-form vector autoregressive
estimation. Estimation samples spans from the week of March 1, 2020, to the week of May 9, 2021.

the movement of the permanent shocks precisely, which changes the stochastic trend in the new
infection cases. Thus, the changes in mobility demand, such as the changes in human behavioral
preferences and emergent changes in human mobility patterns, can accelerate changes in thestochastic trend in new infection cases.
The results can reconcile highly controversial evidence on the association between COVID19 infection cases and human mobility. Studies such as Kraemer et al. (2020) and Nagata et al.
(2021) report empirical evidence on the positive causal relationship between human mobility andinfection cases. Others, such as Goolsbee & Syverson (2021) and Watanabe & Yabu (2021),
report empirical evidence on the negative causal relationship between human mobility andinfection cases. Our proposed simultaneous equations system, which takes into account the two38

Figure 15: Identiﬁed mobility demand shocks
Notes: The solid line (left-hand scale) indicates the mobility demand shocks obtained from the residual of the
instrumental variable regression (16) with the weekly log changes in the search volume of “ 感染者数” (numberof infected individuals in English) in Google as an instrument. The dotted line (right-hand scale) indicates thepermanent shocks identiﬁed using the estimated vector autoregressive (VAR) model (11) with the restriction(14). We set the lag length to three weeks in the reduced-form VAR estimation. The orange shaded areas show
the weeks coinciding with the period during the state of emergency declaration in Japan. The sample periodspans from the week of March 1, 2020, to the week of May 9, 2021.

causal relationships between human mobility and new infections, can interpret these conﬂicting
empirical results without contradiction, and is useful for understanding infection dynamics.
Therefore, analyses using macroeconometrics provide a more realistic picture of the COVID-19
infection cases and human mobility to capture the role of the stochastic trend and cycle in theirdynamics.

5.5

Application to the infection situation in Japanese prefectures

In this subsection, we present the empirical results of applying our empirical framework totime-series data of new infection cases in Japanese prefectures.32 In particular, we investigate
whether there is a dynamic causal relationship as obtained in section 5.2 for these regional
32

We thank an anonymous referee for suggesting the following analysis.

39

time-series data. We further illustrate how regional diﬀerences in the evolution of an infection
situation can be interpreted in our framework.
We limit our analysis to the infection situation in the two prefectures of Tokyo and Osaka,
where the number of the new infection cases is relatively high, compared to elsewhere in Japan. 33
Both of these prefectures experienced a sustained decrease in the infection cases during thestate of emergency declaration, and a persistent increase in cases after the state of emergency
declaration was lifted. Nevertheless, there are regional diﬀerences in the infection spread. In
particular, when the ﬁrst emergency declaration was lifted, the infection situation in Osaka
improved steadily, and the emergency declaration was lifted earlier on May 21, but the infection
situation in Tokyo was not reassuring.34 Also, after the second state of emergency declaration
was lifted, the infection situation in Osaka deteriorated rapidly. 35
We apply our empirical framework to the infection situation in each prefecture. Speciﬁcally,
we estimate the VAR model with a long-run restriction separately, using the time-series data of
the log changes in infection cases and human mobility in each prefecture. As well as carryingout an analysis with the aggregated data, we construct the composite index of mobility for eachprefecture, using six of Google’s mobility indices and employ it as an endogenous variable inthe VAR model.36 The one- to three-week lags of the endogenous variables are included in the
VAR model for each prefecture.
Figure 16 shows the estimated impulse response of the log changes in infection cases and thecomposite index of mobility to permanent and non-permanent shocks in Tokyo and Osaka. We
ﬁnd that, although there are quantitative regional diﬀerences, the estimated dynamic causal
relationship between the infection cases and human mobility is qualitatively similar to theresults for Japan as a whole in section 5.2; a permanent infection shock causes a hump-shaped
increase in infection cases with increased human mobility and a non-permanent shock causes a
temporary increase in cases with decreased human mobility.
33

In the online appendix, we take a look at the evolution of an infection situation in Tokyo and Osaka toillustrate a common trend and regional diﬀerences in infection spread.
34
On June 2, 2020, immediately after the ﬁrst state of emergency declaration was lifted, the Tokyo Metropolitan
Government issued “Tokyo Alert,” its own standard for providing warning signals regarding the infection spreadsituation of COVID-19, for the ﬁrst time, to warn Tokyo residents about signs of a resurgence of the infection.
35
In Osaka, the second state of emergency declaration was lifted ahead of schedule, on February 28, 2021.
However, in response to the reemergence of the infection, on April 7, the Osaka Prefectural Government called
for the prefectural residents to be vigilant, stating that the infection situation was at the most serious levelbased on the “Osaka Model,” its own standard for a warning signal regarding the infection spread situation of
COVID-19.
36
See the online appendix for the time-series data on the composite index of mobility in Tokyo and Osaka that
we use in the analysis.

40

Figure 16: Estimated responses of the log changes in infection cases and human mobility topermanent and non-permanent shocks in Tokyo and Osaka
Notes: The solid lines in the upper and lower panels represent the point estimates of the impulse responses to
one standard deviation permanent and non-permanent shock, respectively, by prefecture. Mobility CI denotes
the composite index of mobility. We set the lag length to three weeks in the reduced-form vector autoregressive
estimation. Estimation samples span 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.

41

Figure 17 displays the time series of identiﬁed permanent and non-permanent shocks inTokyo and Osaka. As seen in the top panel of the ﬁgure, we ﬁnd that the permanent shocks
tend to occur simultaneously across the two prefectures.37 On the other hand, as seen in
the bottom panel of the ﬁgure, we ﬁnd that the time series of non-permanent shocks varyconsiderably between Tokyo and Osaka.38 In particular, there is a large discrepancy between
the non-permanent shocks in Tokyo and Osaka at the end of the ﬁrst and second states of
emergency declarations. This implies that the source of the diﬀerences in the evolution of an
infection situation between the two prefectures, can be interpreted as occurring primarily due
to non-permanent shocks in our framework.

Figure 17: Identiﬁed permanent and non-permanent shocks in Tokyo and Osaka
Notes: The solid lines in the upper and lower panels indicate the permanent and non-permanent shocks, respectively, identiﬁed using the estimated vector autoregressive (VAR) model (11) with the restriction (14) byprefecture. We set the lag length to three weeks in the reduced-form VAR estimation. 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.
37

We conﬁrm that there is a statistically positive correlation of permanent shocks between Tokyo and Osaka(correlation coeﬃcient is 0.81).
38
The correlation of the non-permanent shocks between Tokyo and Osaka is small, with a correlation coeﬃcientof 0.11.

42

Figure 18 gives the historical decomposition of the log changes in infection cases in Tokyo andOsaka. We ﬁnd that the decomposed series explained by permanent shocks is similar to the onesin the upper panel of Figure 8. In particular, there is a common tendency in the two prefectures
for the decomposed series to drop signiﬁcantly during the ﬁrst state of emergency declaration,and to rise signiﬁcantly in the weeks following the lifting of the declaration. Nevertheless,
we ﬁnd that the ﬂuctuation in the decomposed series explained by non-permanent shocks is
considerably larger than that explained by permanent shocks. This implies that when analyzing
the relationship between infection spread and human mobility using prefecture-speciﬁc data,
variations in the growth rate of new cases by prefecture is likely to be dominated by the inﬂuenceof non-permanent shocks.

Figure 18: Historical decomposition of log changes in COVID-19 infection cases due to thepermanent and non-permanent shocks in Tokyo and Osaka
Notes: All the series are displayed as deviations from the deterministic component. The purple bar shows thedecomposed series explained by the permanent shocks. The light blue bar shows the decomposed series explainedby the non-permanent shocks. The solid line indicates the estimated stochastic component before decomposition.
The orange shaded areas show the weeks coinciding with the period during the state of emergency declarationin Japan. We set the lag length to three weeks in the reduced-form vector autoregressive estimation in eachprefecture. Estimation samples span from the week of March 1, 2020, to the week of May 9, 2021.

43

The application in this subsection suggests the importance of monitoring the changes inpeople’s perception of infection risk, in controlling infection spread throughout Japan. The
infection spread can vary from prefecture to prefecture, but much of it is temporary, thanks topeople in each prefecture curbing their mobility. On the other hand, permanent shocks, possibly
due to exogenous changes in mobility demand, may create a common tendency of infection
spread across prefectures. Policymakers may need to intervene to limit people’s behavior if theinfection spread is expected to be long-lasting and widespread, as people’s perception of theinfection risk changes.

6

Summary and discussion

Human mobility is pivotal in considering the dynamics of the number of COVID-19 infection
cases. Arguably, there is a trade-oﬀ between the increase in new infection cases and humanmobility. The number of new infection cases increases, because the probability of susceptible
people being infected by interacting with other infected people becomes higher as more people
move around. However, there is a systematic response of mobility demand regarding the new
infection cases; that is, when the number of new infection cases increases, people restrain theirmobility. Hence, there is a stochastic trend and cycle in the infection dynamics. As describedin our analysis using the dynamic model of COVID-19 infection and human mobility, humanmobility can create an acceleration of the spread of COVID-19 infection and its cyclicality under
the simultaneous relationship.
This study has provided empirical evidence in support of the argument. Using time-series
data from February 2020 to May 2021 in Japan, we demonstrated the stochastic trend and cyclein the new infection cases. Using macroeconometric analysis applied to the time series of log
changes in the new infection cases and human mobility, we demonstrated the feasibility of ourpredictions. Moreover, sources of the COVID-19 infection spread vary signiﬁcantly from time
to time, and the changes in the trend and cycle of the new infection cases explain approximately
half of its variation, respectively, from March 2020 to May 2021 in Japan.
The ﬁndings of this study oﬀer suggestions on some of the current concerns regarding
COVID-19. For instance, there is concern that the infection rate may vary from time to time,
because the severe acute respiratory syndrome coronavirus 2, an RNA virus, is prone to muta-

44

tion. The emergence of a variant strain would cause surprise changes in the new infection cases,
reﬂecting a transitory change in the parameter κ in our model. Nevertheless, we predict that
if we respond appropriately to the situation and change our behavior, we can keep the numberof new infection cases at a certain level, under the premise of system stability.
However, the primary concern of this study is that, as the COVID-19 infection spread
continues, people may ignore the situation, and partly or fully continue (or resume) economic
activities because they have become accustomed to living amid the reality of the pandemic and alife of restrictions. Through this study, we hope to curb the speed of infection spread by helping
people realize that how they regulate their behavioral preferences and freedom of mobility can
reduce the probability of infections. Such decisions are reﬂected in practical measures suchas wearing masks, avoiding close encounters, social distancing, and refraining from going out
if one has COVID-19-related symptoms, to stall the spread as much as possible. Admittedly,
despite the fact that the declaration of a state of emergency due to the spread of the COVID-19
infection and the request to refrain from needlessly going out have greatly restricted economic
activities, Japan has not experienced a major medical collapse, and the probability of people inthe country contracting the virus has been kept extremely low. However, people are likely to bewearied by the repeated declarations of a state of emergency and calls for self-restraint. If this
scenario becomes unbearable and people begin ignoring the situation, the number of infection
cases could increase exponentially.
Hence, what actions can be taken to prevent an escalation of the infection rate? Assumingthat economic activities continue, it would be challenging to completely control the spreadof infections, and a certain level of new infection cases will be inevitable. People must act
to avoid increasing the number of new infection cases, due to the changes in their preferences
regarding systematic behavior. Even without relying on strict countermeasures against COVID19 infections, avoiding a sustained increase in the number of new infection cases via people’s
behavioral responses, is possible by employing the practical measures highlighted above.
Although this study assumes a stable system, structural changes are expected. 39 In particular, we expect the widespread administration of vaccines to cause a structural change in the39
It would be interesting to analyze the fourth wave and the subsequent decline of COVID-19 in Japan. At
this time, vaccination in Japan progressed rapidly. We speculate that the widespread use of vaccines may haveplayed a major role in the decrease in the number of cases after the fourth wave. However, because there is little
data after the vaccine has been distributed throughout Japan, it is diﬃcult to verify the structural changes inthe VAR model. Therefore, we leave the analysis of possible structural changes in the system as a future issue.

45

parameter κ, in that it becomes ﬂat in our model, thereby inducing a weak trade-oﬀ between the
new infection cases and human mobility. Nevertheless, if the widespread availability of vaccineschanges people’s preferences in terms of ignoring the situation, which pushes the parameterin the mobility demand, b, toward zero in our dynamic model, then the spread of COVID-19
infections may emulate the seasonal ﬂu epidemics in winter. As of June 2021, when this paper
was being prepared, this speculation is consistent with events in countries where vaccines arewidely available.
There are many possibilities for an extension of the analysis in this study, leaving roomfor future research to proceed in several directions. First, investigating the system stability inthe dynamics of the number of new infection cases is an important theoretical and empiricalchallenge. Although our analysis is premised on cyclicality and the stochastic trends in new
infection cases, the model predictions and empirical results do not necessarily hold for every
economy. The dynamics in our model strongly depend on the set of its parameters. In particular, as the parameter representing the systematic response of mobility demand to the new
infection cases, b, tends toward zero, the model predicts less cyclicality in such cases. Thus,
the irresponsibility of human mobility in the economy is associated with the probability of an
infection explosion, which can vary signiﬁcantly by country and region, depending on behavioral factors such as testing and quarantine procedures for positive cases, lifestyle, and culture.
Hence, future research can theoretically examine conditions that are satisﬁed with cyclicality
in the new infection cases and quantitatively assess its degree, using time-series data in othercountries and regions.
Second, although this study considers the formation of the reference level of new infection
cases brieﬂy in the models, future studies can provide empirical evidence on the formation ofpeople’s opinions and beliefs regarding the reality of epidemics. The study’s ﬁndings present
important policy implications. An important task for policymakers in predicting and controlling
infections is to monitor changes in people’s perception of infection risk. If people continue
to ignore the risk of infection and engage in economic activities, policymakers should takeinterventions that will change the way people view the risk of infections. To do so, we needto consult quantitative assessments on how people form their opinions and beliefs regarding
the reality of epidemics. It is also important to extend the model to include the policy sector,
in order to analyze the role of policy interventions, such as a systematic control of human46

mobility in response to increases or decreases in the number of new infections. 40 Moreover,signiﬁcant heterogeneity in the level of concern people have about the situation is expected.
It is conceivable that those who are at high risk of serious illness may respond to government
measures and requests to refrain from economic activities, while those at low risk may ignore
them.
Finally, the study ﬁndings can encourage important developments in building epidemiological models incorporating human behavior. From the theoretical analysis in Gans (2020), in
environments where the systematic responses of mobility demand to changes in the number of
new infection cases fully work, an equilibrium level would be reached long before herd immunity is achieved. However, even in the SIR–Macro model, as in Eichenbaum et al. (2021), where
the reproduction rate systematically changes depending on economic activities, this mechanism
seems to be unattractive. Better predictions and simulations of new infection cases may beprovided by modifying the SIR models, implying an economically convenient reduction in the
reproduction rate.

Appendix: Data sources and weekly data construction
Data on the new conﬁrmed cases of COVID-19 and the Oxford Stringency Index are obtained
from Our World in Data.41 The Oxford Stringency Index is calculated by the Oxford Coronavirus Government Response Tracker project.42 This index averages nine indices: school
closures, workplace closures, cancellation of public events, restrictions on public gatherings, closures of public transport, stay-at-home requirements, public information campaigns, restrictions
on internal movements, and international travel controls. Each index takes a value from 0 to100. The larger the index, the stricter the government’s response.
Data on the new conﬁrmed cases in Tokyo and Osaka are obtained from the Japan Broadcasting Corporation (NHK) special website for new coronaviruses43 because the data on COVID-19
Data Repository by CSSE at Johns Hopkins University, which is the source for Our World in
Data, have missing values for the sample period from February 16, 2020, to May 9, 2021. We
40

Fukao & Shioji (2022) use the system comprising the infection–economic activity trade-oﬀ, people’s choiceof economic activity levels, as well as a policy reaction, and point out the importance of a trade-oﬀ between
economic activities and COVID-19 infection cases in understanding the role of public policies.
41
https://ourworldindata.org/
42
https://covidtracker.bsg.ox.ac.uk/
43
https://www3.nhk.or.jp/news/special/coronavirus/

47

conﬁrm that both data are consistent except for the missing values.
The mobility indices are from the COVID-19 Community Mobility Reports by Google. 44
Each mobility variable is calculated as the rate of deviation from the reference value, which isset for each day of the week. The reference value for each day of the week, is the median value
for each day of the week for ﬁve weeks, from January 3 to February 6, 2020.
The frequency of all data is daily. We convert to weekly data from Sunday to Saturday,
in order to eliminate the transitory factor. The number of new weekly infection cases and theweekly mobility indices, are calculated as the cumulative number of new daily infection cases
and the median of the daily mobility indices each week, respectively.

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