### 3.1 Simple model

Our key determinants of online purchases are two measures of pandemic shocks, i.e., the number of COVID-19 cases and the dummy (indicator) variable for a state of emergency. Conceptually, pandemic shocks affect online purchasing behavior for two major reasons, one psychological and the other physical, as summarized in the Introduction. First, in facing the pandemic, people feel anxiety and fear about future supplies and thus purchase more essential products. Second, because of government restrictions, consumers should stay home more than they did before and thus naturally rely more on online shopping. The combination of these two aspects results in an increase in online purchases, particularly of essential products. Among the two aspects, the number of cases is more related to the psychological factors of the pandemic effect because an increase in these numbers generates fear among consumers. A state of emergency is more related to the physical restrictions that affect online purchases, i.e., staying home and closing businesses. However, the two factors cannot be clearly distinguished because an increase in the number of cases may raise fear and thus lead consumers to stay home more. Additionally, a state of emergency restricts consumers physically but promotes fear among them psychologically.

The timing of the two treatments (i.e., the waves of COVID-19 cases and the stages of states of emergency) varies across time and prefectures. Therefore, to estimate the effect of the two measures on online purchases, we employ the following simple two-way fixed-effects (TWFE) estimation that incorporates individual and time fixed effects, assuming no persistent treatment effect [39]:

$$ \ln Y_{cpt} = \beta _{1} \ln \mathrm{CASE}_{pt} + \beta _{2} \mathrm{SoE}_{pt} + \delta _{c} + \delta _{t} + \delta _{d} + \epsilon _{cpt}, $$

(1)

where \(Y_{cpt}\) is either the total purchase amount, the number of buyers, or the purchase amount per buyer in city *c* in prefecture *p* on date *t*; \(\mathrm{CASE}_{pt}\) is the number of COVID-19 cases per 1000 persons in *p* on *t*; and \(\mathrm{SoE}_{pt}\) is a dummy variable that takes a value of one if *p* is in a state of emergency on *t* and zero otherwise. Although the *Y* values are given at the city level, we lack data on the number of cases at the city level. However, our use of prefecture-level cases is justified because consumers are more concerned about the number of cases in their prefecture as reported every day by mass media [17], while information at the city level is rarely reported.

We take a natural log of the purchase amount, the number of buyers, and purchases per buyer, and we take a log of the number of COVID-19 cases per 1000 after adding 0.001 because, otherwise, the result may be zero. Therefore, \(\beta _{1}\) can be interpreted as the quasi-elasticity of online purchases with respect to COVID-19 cases, i.e., the percentage change in the online shopping variables due to the percentage change in the number of cases. Similarly, \(\beta _{2}\) is the percentage change in online shopping due to the declaration of a state of emergency. We incorporate fixed effects at the city (\(\delta _{c}\)), date (\(\delta _{t}\)), and day-of-the-week (\(\delta _{d}\)) levels. The city fixed effects control for any unobservable time-invariant factor, such as geographic and cultural factors, that affects online purchasing behaviors, whereas the date fixed effects represent the time trend common to all cities. The day-of-the-week fixed effects are included to capture fluctuations across days in a week.

In Equation (1), \(\beta _{1}\) and \(\beta _{2}\) indicate the effect of the number of COVID-19 cases and a state of emergency on the online purchases. In this simple model, we assume that once the number of COVID-19 cases becomes zero or a state of emergency is lifted, the effect on online purchases disappears; i.e., online purchases return to the trend before any COVID-19 case or state of emergency. This assumption is different from that of the standard impact evaluation literature, where the effect of a treatment persists after the treatment. We extend the model and examine the possible persistence of the effect later.

We estimate Equation (1) by ordinary least squares (OLS) regression. This estimation strategy is justified because changes in the two measures of pandemic shocks can be considered exogenous shocks to purchasing behaviors. Standard errors are clustered at the city level, incorporating possible correlations between the error terms within the city across time.

### 3.2 Benchmark model

Our benchmark model used throughout this study extends Equation (1) in three ways. First, we examine heterogeneity in the effect of the COVID-19 cases and states of emergency across time, following the recent literature on impact evaluation [39–41]. Because the pandemic lasted for one and a half years in the sample period, consumers may have coped with the pandemic and thus may have responded less to the pandemic shocks as time progressed [24]. In the presence of heterogeneity in treatment effects across time, the coefficient of the treatment variables in Equation (1) is the weighted average of the time-varying effects and can be biased [39, 40, 42]. Therefore, we incorporate time heterogeneity in the effect of pandemic shocks by distinguishing among the number of COVID-19 cases across the five waves and among the states of emergency in different months.

Second, we include dummy variables for before and after a state of emergency. Specifically, we incorporate a dummy variable that takes a value of one if prefecture *p* declared a state of emergency on date \(t+1\) (i.e., *p* is one week before a state of emergency is declared) and zero otherwise. Similarly, three other dummy variables for 2-4 weeks before a state of emergency are included. We include pre-state-of-emergency dummies to check whether a state of emergency has any effect prior to its declaration through the anticipation effect. Usually, a few days before the central government declares a state of emergency in a prefecture, its governor requests a declaration from the central government in response to increasing cases in the prefecture. Therefore, a state of emergency is generally predictable a few days before its declaration. In addition, we incorporate dummy variables for 1-4 weeks after any state of emergency. Doing so enables us to test whether the possible effect of a state of emergency persisted even after the declaration was lifted, a major hypothesis in this study, as described in the Introduction. The absence of the post-state of emergency effect is a major assumption of static TWFE estimations using equations such as (1) and thus should be tested.

Finally, we examine spillover effects, i.e., whether consumer behavior in prefecture *p* is affected by any COVID-19 case or state of emergency in neighboring prefectures because of fear of the spread of COVID-19 and the resulting state of emergency in *p* in the future. If spillover effects exist, then \(\beta _{1}\) and \(\beta _{2}\) in Equation (1) may be biased [43]. For this purpose, we include in Equation (1) the total number of cases in prefectures that share any border with prefecture *p* and the number of neighboring prefectures in a state of emergency. As in the previously defined variables, we distinguish among the number of cases in neighboring prefectures by wave and among the number of neighboring prefectures in a state of emergency by month.

Accordingly, our city-level analysis is based on the following equation:

$$ \begin{aligned}[b] \ln Y_{cpt} ={} &\sum _{w} \beta _{1w} \ln \mathrm{CASE}_{ptw} + \sum_{m} \beta _{2m} \mathrm{SoE}_{ptm} \\ &{}+ \sum_{y=2020}^{2021} \sum _{k=1}^{4} \lambda _{1yk} \mathrm{PreSoE}_{ptyk} + \sum_{y=2020}^{2021} \sum_{k=1}^{4} \lambda _{2yk} \mathrm{PostSoE}_{ptyk} \\ &{}+ \sum_{w} \mu _{w} \ln \biggl( \sum_{q \in Q_{p}} \mathrm{CASE}_{qtw} \biggr) + \sum_{m} \mu _{m} \sum _{q \in Q_{p}} \mathrm{SoE}_{qtm} + \delta _{c} + \delta _{t} + \delta _{d} + \epsilon _{cpt}, \end{aligned} $$

(2)

where \(\mathrm{CASE}_{ptw}\) is the number of COVID-19 cases per 1000 persons in prefecture *p* on date *t* in wave *w* (*w* is from one to five), and \(\mathrm{SoE}_{ptm}\) is the dummy variable that takes a value of one if *p* is in a state of emergency in *t* in month *m*. \(\mathrm{PreSoE}_{ptyk}\) and \(\mathrm{PostSoE}_{ptyk}\) are one if prefecture *p* is *k* weeks before and after a state of emergency, respectively, in *t* in year *y* and zero otherwise. For example, \(\mathrm{PreSoE}_{pt,2020,2}\) is one if prefecture *p* is two weeks before a state of emergency on date *t* in 2020. Accordingly, coefficient \(\lambda _{1,2020,2}\) indicates the effect of a state of emergency in 2020 on the online purchases two weeks after the state of emergency. \(Q_{p}\) is the set of prefectures neighboring *p*. Therefore, \(\sum_{q \in Q_{p}} \mathrm{CASE}_{qtw}\) represents the sum of the COVID-19 cases in neighboring prefectures; thus, its coefficient, \(\mu _{w}\), indicates the spillover effect from neighboring prefectures in wave *w*. Semistates of emergency, which are less restrictive than are states of emergency (refer to the previous section), are also included in the analysis but are not distinguished by declaration month for simplicity.

### 3.3 Econometric issues

Several econometric issues remain in estimating Equation (2). First, from April 16 to May 14, 2020, all 47 prefectures in Japan were in a state of emergency. Because we cannot compare prefectures with and without a state of emergency during this period, we drop this time period from the sample [39].

Second, when the timing of the treatments varies, the estimate of the treatment variables without any consideration of the time variation may be biased. We alleviate this problem by incorporating the time-variant effects of the COVID-19 shocks, as explained earlier. Moreover, bias may be minimal in this study because the prefectures that declared a state of emergency earlier than others lifted it later than did the others in all three stages(Fig. 4). Therefore, our estimation avoids the biases arising from a comparison among the prefectures that were in a state of emergency but are currently not and the prefectures that are currently in a state of emergency, assuming that the former would not have been treated [42].

Finally, an important identifying assumption of our estimation is a parallel trend prior to the treatment. If any difference arises in the prepandemic trend between the prefectures that were severely hit by COVID-19 and thus declared a state of emergency for a long time and the other prefectures, *β*s in Equation (2) may not simply be the effect of the pandemic shocks but may include the systematic difference between the two prefecture types [41]. We test this possibility using the data prior to the pandemic, i.e., from January to December 2019. Specifically, we regress the purchase amount, the number of buyers, and the purchase amount per buyer on the dummy variable for the last half of 2019; the dummy variable for the prefectures that declared the state of emergency for more days than the median prefecture (47 days); and the interaction term between the two dummies. SI Table 5 shows that the interaction term is not significantly different from zero in any of the regressions. Therefore, the pretreatment parallel-trend assumption is satisfied.

### 3.4 Heterogeneity across products

The analysis at the prefecture–product level estimates an equation similar to Equation (2) but assumes the heterogeneous effects of the COVID-19 cases and states of emergency on the different products, using the four product categories broadly defined in the previous section: health-related products, daily essentials, nonessential products used mostly at home, and nonessential products used mostly outside the home (SI Table 4). Conceptually, however, because of increasing needs for essential products, particularly health products and health foods, the effect on these products is predicted to be greater than that on the nonessential products. To test this hypothesis, all the independent variables except for the fixed effects are interacted with the dummy variables for the product categories. Product fixed effects are also included, and city fixed effects are replaced with prefecture fixed effects. Standard errors are clustered at the prefecture and date levels, controlling for a correlation between the error terms across products. Specifically, the estimation equation for the prefecture–product level estimates is given by the following:

$$\begin{aligned} \ln Y_{pjt} =&\sum _{j} \sum_{w} \beta _{1jw} \ln \mathrm{CASE}_{ptw} \times D_{j} + \sum_{j} \sum_{m} \beta _{2m} \mathrm{SoE}_{ptm} \times D_{j} \\ &{}+ \sum_{j} \sum _{y=2020}^{2021} \sum_{k=1}^{4} \lambda _{1jyk} \mathrm{PreSoE}_{ptyk} \times D_{j} + \sum_{j} \sum _{y=2020}^{2021} \sum_{k=1}^{4} \lambda _{2jyk} \mathrm{PostSoE}_{ptyk} \times D_{j} \\ &{}+ \sum_{j} \sum _{w} \mu _{j}w \ln \biggl( \sum _{q \in Q_{p}} \mathrm{CASE}_{qtw} \biggr) \times D_{j} + \sum_{j} \sum _{m} \mu _{j}m \sum _{q \in Q_{p}} \mathrm{SoE}_{qtm} \times D_{j} \\ &{}+ \delta _{p} + \delta _{j} + \delta _{t} + \delta _{d} + \epsilon _{pjt}, \end{aligned}$$

(3)

where *j* represents product categories and \(D_{j}\) is the dummy variable for product *j*.