key: cord-0874274-ce7bkbex
authors: Prakash, Navendu; Srivastava, Bhavya; Singh, Shveta; Sharma, Seema; Jain, Sonali
title: Effectiveness of social distancing interventions in containing COVID-19 incidence: International evidence using Kalman filter
date: 2021-12-02
journal: Econ Hum Biol
DOI: 10.1016/j.ehb.2021.101091
sha: feaf6b27ae2d03af5a01b6513de376accbc21576
doc_id: 874274
cord_uid: ce7bkbex

The epidemiological literature has widely documented the importance of social distancing interventions in containing the spread of the COVID-19 pandemic. However, the epidemiological measure of virus reproduction, R(0), provides a myopic view of containment, especially when the absolute number of cases is still high. The paper investigates cross-country variations concerning the impact of social distancing interventions on COVID-19 incidence by employing a statistical measure of containment, which models the daily number of cases as a structural time-series, state-space vector. Countries that adopt strict lockdown policies and provide economic support in the form of income augmentations and debt relief improve the response towards the pandemic. Countries like China and South Korea have been most influential in containing the spread of infections. European nations of France, Italy, Spain and the UK are witnessing a second wave of the virus, indicating that re-opening the European economy perhaps has instigated an exponential spread.

Sweden, however, followed voluntary social distancing with bars, restaurants and elementary schools kept open.

In this paper, we put forward an argument that the epidemiological measure of containment, based on the reproduction rate of the virus, may provide a myopic view of the situation as what matters is not only that the rate of reproduction of the strain is less than one but also whether the incremental increase in the number of reported cases is under control.

Focusing on this limitation, we propose a new, statistical measure of containment by modelling the daily number of reported cases as a structural time-series state-space model with two latent vectors. A Kalman filter methodology is employed to recursively obtain the conditional mean and variance of the time-series for 20 select countries across the European Union, North America, South America, Asia-pacific, and Africa. Based on the work of Moosa (2020) , we establish seven categories of containment, ranging from a situation beyond control to a situation where the magnitude and the spread of infection are under control.

Containment is defined as a situation wherein the level and the slope of the state-space model are kept insignificant for a sufficient period of time (Moosa, 2020) .

Further, the spread of the virus depends on a host of factors, of which the government's policy response is of principal importance. To this end, we examine the impact of pandemicinduced policies, such as the degree of stringency in mandatory lockdowns, containment policies, health policies, and economic policies on the spread and mortality rate of COVID-19 infections after controlling for the influence of several country-specific factors. To date, there is not much evidence on the quantitative impact of these policies. While most countries have adopted stringent policy measures in response to the pandemic, it remains unclear why countries differ in containing the spread of the virus. In the present study, we thus, attempt to address this knowledge gap in the literature and explain observed divergence across countries concerning the evolution of the pandemic post the implementation of pandemic-induced measures. To analyze these differences, we determine how the influence of various policy measures on COVID-19 incidence varies across a matrix of country-specific characteristics such as demographic, environmental and health dimensions. These characteristics are related to divergence in behavioural response and divergence in resource availability to governments that likely forms a requisite for policy enforcement. Therefore, we examine the moderating effect of population density, the proportion of the elderly population, and countries' overall health scenario in influencing the associations between government policies and COVID-19 J o u r n a l P r e -p r o o f incidence. Pandemic-induced policies may be less favourable in highly populated countries in congruence with the view that higher population density augments the recurrence of human interactions and places more demand on resources for compliance with these policies.

Further, while voluntary social distancing before the enactment of interventions is likely to be less evident in countries with lower proportions of elderly and access to better public health systems on account of lower perceived risk, pandemic-induced policies may be more advantageous in countries with significant proportions of youngsters and availability of better health systems.

The rest of the paper is organized as follows. Section 2 critically evaluates the available literature on the influence of non-pharmaceutical interventions in containing the pandemic.

Section 3 illustrates the modelling strategy and data sources. Section 4 provides a detailed analysis of the results. Section 5 contains an array of checks to lend robustness to the results.

Section 6 concludes the study.

The importance of social distancing interventions can be traced back to previous pandemics. Glass et al. (2006) (Milne et al., 2008; Lee et al., 2010; Mao, 2011; Zhang et al., 2012; Milne et al., 2013) .

The literature on the influence of social distancing measures in controlling the pandemic is also rapidly expanding. Employing the daily data on confirmed COVID-19 cases for a set of 10 countries, Moosa (2020) advocates that social distancing measures, in general, are beneficial in controlling the spread of the virus. The author reports an insignificant level and slope of the stochastic trend for countries such as Australia and China, highlighting that they have successfully contained the infection's spread over the sampled timeframe by imposing social distancing measures when the reported cases were lower. South Korea, J o u r n a l P r e -p r o o f however, is cited as an anomaly. Even in the absence of government-imposed restrictions, the country is observed to perform reasonably well, perhaps attributable to the intrusive surveillance by the Korean government. Furthermore, the author observes a significant level of the stochastic trend for countries such as the UK and the USA, highlighting no containment of the virus over the period analyzed given the non-timely and less stringent imposition of lockdown. On similar lines, Wong et al. (2020) highlight the importance of implementing stringent social distancing measures immediately upon confirmation of the first case in controlling the spread of the virus in Hong Kong. The authors further cite several social distancing measures, including the closure of primary and secondary schools, work from home arrangements, closure of leisure facilities, banning non-HongKong residents from overseas countries, the prohibition of public gatherings, mandatory 14-day quarantine for people entering from the mainland, mandatory self-quarantine for those who have come in close contact with infected persons together with close surveillance using electronic wristband among others, which have played an essential role in the controlling the infection.

Using daily data on reported cases and real-time data on containment measures, Deb et al. (2020) attempt to assess the relationship between containment policies and the spread of the virus for a sample of 129 countries. The authors demonstrate that containment measures are successful in flattening the pandemic curve. Furthermore, the authors explore the crosscountry heterogeneity in baseline associations by employing several country-specific factors and document that the influence of containment measures in minimizing the spread is more pronounced in countries with faster implementation of containment measures, lower population density and a larger proportion of elderly in the population. Cowling et al. (2020) examine the influence of non-pharmaceutical interventions on COVID-19 transmission in Hong Kong SAR and report that social distancing measures lead to a substantial drop in the virus transmission rate. Employing a standard SIR epidemiological model, Brotherhood et al. (2020) analyze the influence of quarantine measures. Their findings suggest that confining mobility for the young can extend the pandemic as herd immunity is delayed while exposing the elderly to risk. In a quasiexperimental framework, Alimohamadi et al. (2020b) empirically analyze the implications of social distancing measures on COVID-19 incidence in Iran. The authors observe that while new cases and mortalities exhibit an increasing trend before the intervention period, a declining trend is observed post-intervention. Overall, the findings highlight that irrespective of the economic and psychological influence of NPIs, their importance in containing the J o u r n a l P r e -p r o o f COVID-19 incidence is undeniable. Employing a network-based SEIR model, Lai et al. (2020) undertake a quantitative assessment of the impact of NPIs in managing incidence in China. The authors predict that in the absence of NPIs, by February 29, 2020, COVID-19 cases would have been higher by 67-fold. The authors further identify three NPIs implemented in China: intercity travel restrictions, early detection and isolation of cases, and contact reduction. The findings suggest that while NPIs successfully contain the spread, a quantitative comparison of the three highlights that early detection and isolation of cases are far more effective in infection prevention than travel restrictions and contact reduction.

Further, commenting upon the timing of interventions, the authors report that early implementation of NPIs could have reduced the cases significantly. The authors cite that three weeks earlier implementation of NPIs vis-à-vis the actual implementation date could have reduced COVID-19 cases by 95 percent in China. Thus, the authors advocate the importance of proactively planning NPIs and earlier implementation to maximize the benefits and minimize the social and economic costs of NPIs. Koh et al. (2020) empirically assess the impact of social distancing measures in containing the virus transmission for a sample of 142 countries. While the study employs several social distancing measures broadly divided into three categories (international travel restrictions, lockdown-type measures, and cancellation of mass gatherings), an average of the basic reproduction number for two weeks after the 100 th reported case is used to capture the viral transmission. The findings suggest that stringent social distancing measures are beneficial in reducing the reproduction number. Further, the authors demonstrate that for countries with a stringency score of less than 50, the imposition of social distancing interventions cannot bring the reproduction number below one in two weeks after the 100 th reported case. Further, commenting upon the three different categories of social distancing measures that vary in terms of intensity and implementation time, the authors note that lockdown measures and complete travel bans effectively contain the COVID-19 crisis. The authors further highlight the importance of early implementation of social distancing measures (computed using the observed global median timing of implementation of intervention across countries) in confining the spread of the virus. Kucharski et al. (2020) identify that stringent travel restrictions are instrumental in containing COVID-19 transmission in China. Duc Huynh (2020) highlights that strong public health messaging, combined with lockdown measures, has effectively slowed down the J o u r n a l P r e -p r o o f spread of the COVID-19 virus in Vietnam. Sposato (2020) reports that voluntary physical distancing measures, including wearing masks, maintaining silence while travelling in public transport, and avoiding handshakes, are critical in containing the infection. Using data from Italy, France, China, South Korea, Iran and the USA, Hsiang et al. (2020) analyze the impact of anti-contagion policies in containing the growth rate of infections. Applying reduced-form econometric methodology, the authors observe that policy interventions significantly reduce the infection growth rate. For a panel of 69 countries, Ullah and Ajala (2020) empirically investigate the association between lockdown measures and COVID-19 transmission.

Employing the two-step system Generalized Method of Moments (GMM) estimator, the authors report that policy interventions effectively contain viral transmission. Further, the findings suggest that while it takes seven days for lockdown measures to reduce cases significantly, it takes around 21 days for testing to impact confirmed cases after implementation. Fang et al. (2020) analyze the quantitative impact of lockdown in Wuhan, China, on human mobility and containment of the virus. Employing the difference-indifferences approach, the authors observe that the lockdown significantly reduced human mobility inside, outside and within Wuhan. Findings suggest that the lockdown substantially reduced virus transmission outside of Wuhan, even though other cities delayed imposing social distancing policies. The authors further document that had the lockdown not been imposed in Wuhan, cases outside Hubei province would have been higher by 64.81 percent.

While several studies have highlighted the significance of NPIs in managing the spread of the virus, Barro (2020) , in contrast, demonstrates the failure of NPIs in reducing overall deaths during the Spanish Influenza pandemic in 1918. The author attributes the findings to the view that the interventions were not maintained for a long time. Barro et al. (2020) state that with everything else held constant, the mortality rate of 2.1 percent during the 1918 Spanish Influenza pandemic is estimated to result in 150 million deaths worldwide during the recent COVID-19 pandemic, which in turn, eventually corresponds to a decline of 8 percent in private consumption and 6 percent in the gross domestic product (GDP). Carlsson-Szlezak et al. (2020) expect the economic recovery post the COVID-19 pandemic not to be straightforward, unlike the 'V-shaped' recoveries during the past pandemics, as the employment effects owing to the social distancing interventions are expected to be much larger. Gourinchas (2020) argues that half of the working population may suffer from transitory unemployment. Baldwin (2020) describes the role of COVID-19 in disrupting the income flows in the economy. Lu et al. (2020) report the negative influence of social J o u r n a l P r e -p r o o f distancing interventions on the psychological well-being of individuals through financial loss, exclusion by neighbours, boredom and lack of availability of essential supplies. Tubadji et al. (2020) confirm that the diffusion of toll statistics on public death negatively impacts public mental health. Given the profound social and economic impact of NPIs, the present study attempts to empirically test the impact of NPIs on COVID-19 spread and associated mortalities and explain the reasons for the observed divergence across countries.

We collect data on the daily number of infections for 20 select countries spanning the European Union, North America, South America, Asia-pacific, and Africa. The countries, in alphabetical order, are Argentina, Australia, Brazil, China, Colombia, France, Germany, India, Italy, Japan, Mexico, New Zealand, Russia, South Africa, South Korea, Spain, Sweden, Turkey, United Kingdom, and The United States of America. Some of these countries have been the epicenter of the COVID-19 pandemic. The data is collected from the European Centre for Disease Prevention and Control, which reports data for daily cases and deaths of countries all over the world. For empirical analysis, data for each country is collected from the date when the first case was reported to November 30, 2020 (cut-off date).

The epidemiological measure of containment does not provide an exact representation of containment, especially when the number of cases is high (Moosa, 2020) . The measure also distorts across demographics, as highly populated countries fulfilling the condition (R<1) may continue to depict a swift growth in the magnitude of cases. Further, the measure does not accurately enumerate the prevalence of the infection in a country, thereby underestimating the severity of the pandemic. To circumvent this problem, we propose a statistical measure of containment based on the number of daily reported cases. Following Harvey (1989) , we propose a structural time-series (STS) model represented as: 

where, follows a first-order autoregressive process, represented as:

The above model is based upon an assumption that 1 and 2 are normally distributed 2 , and mutually and serially uncorrelated with one another. 

The idea behind state-space representation is to capture the dynamics of estimates of an observable time-series through an unobservable, latent state vector of equations (Anderson and Moore, 1979; Brockwell and Davis, 1991 Jong, 1988; De Jong and Chu-Chun-Lin, 1994) . The estimates of and , representing the level and the slope of the state-space model, are analogous to the intercept and the slope term in a conventional regression model, respectively. For this analysis, COVID-19 is considered to be under control if the level and the slope of the STS model are statistically insignificant and continue to remain so for a sufficient period of time (Moosa, 2020) .

Modelling variables in the form of a stochastic STS model has some attractive advantages.

Traditionally, the unobserved components (trend, cycle, seasonality, or irregular) of any time series were modelled in a deterministic manner. However, the presence of high-frequency time series makes it challenging to assume that the series depicts a fixed pattern across time (Brockwell and Davis, 1991) . Deterministic models also fixate assumptions on the time series, contaminating the underlying data generating process (Hamilton, 1994) . In this context, STS models allow for introducing the unobservable components in a stochastic manner. Specifically, these models are built upon an assumption that the components of a time series are stochastic processes having random disturbances (Harvey, 1989) . Further, modelling time series components as stochastic, iterative processes that update as more information becomes available is a superior method vis-à-vis a deterministic trend. This is achieved through the prediction and correction mechanism of the Kalman filter. Under a state-space representation, the filter operates a recursive least squares estimation of the state's parameters and identifies a new state by adding to the previous estimation a proportional correction term to the prediction error, such that the latter is minimized (Jalles, 2009 ). The recursive estimation of the parameters using the Kalman filter provides a distinct advantage as the estimates are affected by the distant history of the series and are continuously updated with the incorporation of new observations. This is effective in modelling highly dynamic and unobservable phenomena, such as the spread of COVID-19 5 .

The Kalman filter encompasses a set of mathematical equations that aims at the recursive prediction of the state vector at time , based on the values of time − 1, and updates these estimates with the additional information available at time . A notable advantage of applying the Kalman filter is its ability to estimate the past, present, and future state of the state vector, even when the precise nature of the modelled system is unknown, a feature not possible in conventional forecasting applications such as ARIMA modelling. In the absence of a priori knowledge about the nature of the time series, ARIMA models are susceptible in the sense that it is easy to select a wrong model. This is often true of elaborate, overparameterized models that usually pass diagnostic tests but do not provide reliable forecasts. Further, ARIMA assumes that a series can achieve stationarity through differencing. In contrast, STS models estimated by the Kalman filter lay less emphasis on differencing transformations for achieving stationarity. In the presence of high-frequency time series, the estimates of the Kalman filter are continuously updated with new information, and hence are more reliable than those provided by ARIMA modelling.

The spread and subsequent containment of the COVID-19 pandemic is a continuously evolving process. The response towards the pandemic is often dependent upon the prevalence and severity of infections in a country. Considering the possible economic, social, and psychological costs of mandatory lockdowns, the need for such strict measures is continuously evaluated with respect to the magnitude of incremental cases. It is observed that Modelling the daily number of cases in the form of a state-space vector of equations provides valuable information for explaining cross-country differences in COVID-19

incidence. However, the measure in itself is limited as it does not identify the impact of specific policy measures in response to the pandemic. The success (or lack of success) of a country in controlling COVID-19 incidence depends on a host of factors. Some of these are related to the country's demographics. For instance, the mortality rate resulting from COVID-19 is higher in countries with a greater proportion of the aging population (Deb et al., 2020) . 

where, is the response variable representing the natural logarithm of daily cases and daily deaths in separate regression models. represents the index of the ℎ type, based on the OxCGRT dataset, is a vector of control variables, is the number of lags, and is the stochastic disturbance term.

A classic econometric issue in estimating the above regression model is reverse causality, as many countries strengthen (or weaken) their policy measures in response to the degree of spread of the virus. For instance, India imposed a nationwide lockdown from March 23, 2020, to contain the spread of the virus. As the spread appeared under control after two months, it adopted a policy of staggered re-opening of public places and transport systems. In the meantime, the country responded to the pandemic by concentrating on policy measures that circumvent additional stress on the healthcare system. The presence of an endogenous response may create an upward bias in estimates (Deb et al., 2020) . To circumvent this problem, we introduce lags in the estimation process. Specifically, we hypothesize that there is a sufficient lag as countries attempt to modify their policy measures to respond to the current spread of the virus. To this end, we provide for a seven-day and a fourteen-day lag response 7 . Providing for lags reduces endogeneity in government interventions in response to the spread of the virus. We also include country dummies to address endogeneity concerns.

Nevertheless, some endogeneity can also be traced to a misspecification bias. To this end, we control for population density, population age structure, economic growth, temperature, and the overall effectiveness of public health care systems of countries. However, some endogeneity may still be present in the regression model. In the absence of suitable instrumental variables, it is challenging to take care of endogeneity. Further, generalized moment estimators, such as those provided by Arellano and Bond (1991) and Blundell and Bond (1998) , are only suitable for micro panel (large N, small T) models. In addition, serial correlation and cross-sectional dependence create significant estimation issues for long panels, such as in this analysis. To this end, we employ a feasible generalized least squares (FGLS) estimator with an AR (1) autoregressive process. Further, we provide for a heteroscedastic error structure with a cross-sectional correlation to resolve cross-sectional dependence. The estimator is suitable for small N, large T models and provides efficient estimates corrected for heteroscedasticity and autocorrelation.

As a preliminary analysis, Figure 1 and Figure 2 illustrate seven-day moving averages of daily cases and daily deaths reported in 20 countries when the first case was reported until

November 30, 2020. Figure for examining whether the series has been generated from a stationary process. Consistent with the specified state-space model, the statistic has been calculated assuming a null hypothesis that the series follows a random walk with a nonzero drift.

As per the definition of containment given by Moosa (2020) have also been witnessed in the USA, which has been the most severely affected country. The position is also alarming for highly populated countries like the USA, Russia, and Turkey, as the trend continues to increase without any evidence of stabilization. India, however, presents a unique case. The country is reporting a high level of infections, chiefly because of a vast population and high population density. However, the country presently lies in the 4 th category, implying that the infection spread is stabilizing at a quicker pace. The result may be due to the policy actions undertaken by the Indian government in response to the pandemic, specifically increasing the testing capacity and rapid contact tracing and testing measures. J o u r n a l P r e -p r o o f China J o u r n a l P r e -p r o o f Table 4 reports the FGLS results with the natural logarithm of daily cases as the dependent variable, while Table 5 In line with the above discussion, we study cross-country variations by focusing on the moderating effect of demographic, economic, and health dimensions on the association between pandemic-induced measures and the spread of COVID-19 infections. The following discussion highlights the effect of country-specific differences, which, to a greater extent, dictate the selection and the influence of specific policies adopted by J o u r n a l P r e -p r o o f nations. Panels (B), (C), and (D) of Table 4 and Table 5 report such results with a seven-day and fourteenday lag period. Concerning demographic dimensions, we hypothesize that countries with high population density result in individuals increasing the frequency of social interactions, and hence, the effect of government-imposed lockdowns might be less effective in containing the spread of the virus. Higher population density also affects compliance to government-imposed lockdowns, making NPIs ineffective. An opposite sign on the interaction term indicates that the benefits accruing to a country by imposing stringent lockdown measures get weakened for densely populated nations. In other words, social distancing measures are more beneficial in reducing the spread of the virus for countries with lower population density. While social distancing is a more behavior-oriented process than a policy-oriented one, ipso facto, countries with higher population density such as India, Brazil, and Mexico must devote substantial resources to ensure 

The statistical measure of containment, as proposed in this paper, attempts to model the pandemic through the daily number of reported cases. However, case data are a function of testing and reporting quality, both of which were very poor early on and continued to be deficient in many countries throughout the study period. For instance, the UK conducted as many as 6, Likewise, in addition to daily cases and daily deaths, we also employ case positivity rate and moving averages of cases and deaths as dependent variables to gauge the association between the government's policy response and the spread of COVID-19 infections and mortalities. Table 7 reports FGLS regressions with seven-day and fourteen-day lags in policy response. A negative sign on the containment indices indicates that, on average, NPIs have resulted in reductions in case positivity rate. The result also holds for a 14-day lagged response. Interestingly, the policy estimates of columns (3)-(6) of Table 7 report lower standard errors vis-à-vis those provided in Table 4 and In line with the direction of the paper in identifying the impact of social distancing measures, we also examine the impact of specific containment policies on COVID-19 incidence, deaths, and case positivity rate. Specifically, we analyze the influence of four 10 containment measures: (1) Closure of schools and universities, (2) Closure of workplaces, (3) restrictions on internal movement and (4) Restrictions on public transport. Table 8 reports these results. We observe that a few estimates are significant, albeit small in terms of magnitude. Hence, the results must be interpreted with caution. It is quite challenging to draw any substantial conclusion concerning any specific containment policy. Nonetheless, a general interpretation from the findings reported in Table 8 is that containment policies are advantageous in reducing COVID-19

incidence, mortalities and case positivity rate. To address reverse causality and resulting endogenous response, we provide for seven-day and fourteen-day lags in policy measures. ***, **, and * indicate significance at 1, 5, and 10 per cent, respectively. Heteroscedasticity corrected standard errors in parentheses.

The COVID-19 pandemic has raised serious epidemiological, health, economic, and social consequences for nations all over the world. The government's response towards the pandemic has been variable across countries, which, to a large extent, explains observed divergence across countries in terms of containing the spread of infections and associated mortalities. However, most countries have adopted some form of social The study has some limitations. First, the statistical measure of containment is based upon the reported case data. Many countries had inadequate testing procedures at the onset of the pandemic, resulting in the early under-reporting of cases. Further, discrepancies in testing policies (such as lower testing on weekends and holidays) and reporting policies (such as whether tests are assigned to the date when they are conducted or when the report is available) across countries tend to contaminate the statistical significance of coefficients. In addition, there have been suspicions of massive under-reporting of COVID-19 cases and J o u r n a l P r e -p r o o f deaths in some countries. Another econometric issue may also occur as countries with more noise in the time-series will be less likely to be classified in categories where parameters are statistically significant. The study attempts to circumvent these issues by employing an array of observable variables in the form of daily deaths, moving average of cases and moving average of deaths under different specifications. Nevertheless, all estimates are contingent upon the accuracy of the reported data. The study also employs the case positivity rate as another outcome measure that better reflects the prevalence of infection relative to the reported case data by addressing divergence in testing figures across nations. However, the case positivity rate may not provide a true incidence of infection unless there is random sample population testing. This was rare across countries. Further, it is recognized that as testing strategies not only vary across countries but also over the course of the pandemic, the case positivity rate does not reflect the actual prevalence of infection. Second, as with any epidemiological study on COVID-19, the present study attempts to model the pandemic, a highly dynamic phenomenon. For instance, at the onset of the outbreak, many policymakers did not have an accurate understanding of the speed and severity of the virus. In the absence of a vaccine, social distancing measures were considered a preliminary action by many countries to counter the spread of infection. As the virus propagated to other countries with a lag, advances in epidemiological research may have allowed those countries to behave more rationally in implementing social distancing measures (for example, it has been widely acknowledged that the creation of localized containment zones where infection have spiraled are more optimal from an economic perspective rather than imposing nationwide lockdowns).

Further, the mere presence of a pharmacological intervention may lessen the need for a country to rely on NPIs, which are only optimal in a finite game framework. In an ideal scenario, the results would have been more robust when the pandemic had ended, as we would have had more information on its antecedents.

Although the findings suggest that social distancing is a beneficial mitigation strategy to control the propagation of the COVID-19 virus and the related mortalities, it bears economic, social and psychological consequencesan appalling situation that can be dealt with the recent pharmacological intervention.

Therefore, it is also essential to discuss the findings in a post-vaccine setting. With the recent development of vaccines, it is acknowledged that while the spread of cases continues, severe illness and death are heavily hand, we provide evidence that the impact of social distancing measures is dependent upon a host of demographic, environmental, and economic dimensions. Hence, the matrix of government policies adopted as a pandemic-induced response must be correlated with these dimensions to determine an effective strategy that not only controls the magnitude and speed of infections but also minimizes COVID-19 induced deaths.

The authors have no competing interests to declare.

Here, and are ( × 1) and ( × 1) vectors of observed variables and unobserved state variables, respectively. and are ( × 1) and ( × 1) vectors of exogenous variables, respectively. is a ( × 1) vector of state-error terms ( ≤ ); and is an ( × 1) vector of observation-error terms ( ≤ ).

Further, at its basic form, the error terms are assumed to be normally distributed and mutually and serially uncorrelated with one another:

Based on these assumptions, the Kalman filter produces a conditional state vector, | and a conditional covariance matrix Ω | , which includes information up to and including time . For each :

The residuals and the mean square error (MSE) matrix of the prediction error can be specified as:

Once the filter is initialized, the estimates of the conditional state vector and conditional covariance matrix are updated with information from time : The concept of non-linearities in state-space models is different from that of standard regression models.

The matrices A, B, C, and Q in the state equations and D, F, and R in observable equations are known as system matrices. These matrices are usually assumed to be non-stochastic; hence even though the parameters may change with time, they do so in a manner that is predetermined (Harvey, 1989) . As a result, the state-space representation becomes linear, and for all , can be expressed as linear combinations of the initial state vector, 0 and present and past values of and . This, however, is a restrictive assumption.

One way to give flexibility to the model is to assume that the system matrices are stochastic so that they are influenced by the information available at time − 1, albeit having Gaussian disturbances. These are known as conditionally Gaussian models and can be expressed in the form of:

The paper employs the abovementioned model for estimating the parameters. A notable advantage follows in the sense that, assuming Gaussian disturbances, these models are still solvable by the Kalman filter as the likelihood function can be obtained via prediction error decomposition (Jalles, 2009 (Harrison and Stevens, 1976) and multiplicative models (Harvey, 1989) .

Further, the state-space model is specified, assuming that the disturbances are normally distributed. Such an assumption allows the Kalman filter to provide the conditional mean of the state vector, which is also the MMSE of the state vector. However, if the disturbances are assumed to be non-Gaussian, then out of the estimators which reflect linear combinations of observations, the Kalman filter estimator is still the one that minimizes the MSE matrix. Hence, the Kalman filter is the minimum mean square linear estimator (MMSLE) of the state vector. However, in the non-Gaussian case, assumptions need to be made about the distributions of the error terms. Harvey (1989) discusses the approximations of the Kalman filter with

Poisson and binomial distributions. However, these apply only to local-level STS models and are often difficult to compute with a reasonable degree of precision. Hence, the assumption of Gaussian disturbances is more of a necessity than a sufficiency while estimating local linear trend models through the Kalman J o u r n a l P r e -p r o o f filter. Nevertheless, the assumption of normality is somewhat relaxed in this paper through the use of the Quasi-Maximum likelihood (QML) estimator provided by Hamilton (1994 

In our study, we employ the mean daily temperature as a control variable in the FGLS regressions.

Likewise, recent studies have examined the effect of seasonality on the spread of COVID-19 (see, for example, Kassem, 2020; Wang et al., 2021; Abouk and Heydari, 2021) . We account for seasonality by introducing a stochastic-seasonal component in the STS model:

J o u r n a l P r e -p r o o f

where is the seasonal component with zero mean and finite variance, 3 , such that + −1 + ⋯ + −( −1) = 3 , and and follow distributions mentioned in Equation (4). In other words, the seasonal effects sum to a random variable defined as: (Jalles, 2009 ). An alternative is to employ spline functions, which the authors feel is beyond the scope of this paper.

J o u r n a l P r e -p r o o f Appendix E: Impact of policy measures on reproduction rate (R 0 )

As an additional robustness check, we also examine the impact of NPIs on the reproduction rate of the virus. deaths. 2 Note: The above table describes the level and slope estimates of the state-space model across 90-day rolling windows (R 1 to R 9 ) to track country-specific movements in rankings over the course of the pandemic. Equally spaced windows (except the last one for each country) are constructed using 30-day increments. For non-converging models, the estimation was limited to 1000 iterations. ***, **, and * indicate significance at 1, 5, and 10 per cent, respectively. Robust standard errors in parentheses.

European Centre for Disease Prevention and Control: An agency of the European Union

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Forecasting

The Effect of Large-Scale Anti-Contagion Policies on the COVID-19 Pandemic

Rolling Window Selection for Out-of-sample Forecasting with Time-varying Parameters

Structural Time Series Models and the Kalman Filter: A Concise Review. FEUNL Working Paper No. 541, Faculty of Economics and Politics

Does Temperature Affect COVID-19 Transmission? Front. Public Health

Projecting the Transmission Dynamics of SARS-Cov-2 through the Postpandemic Period

Estimating the Impact of Physical Distancing Measures in Containing COVID-19: An Empirical Analysis

Early Dynamics of Transmission and Control of Covid-19: A Mathematical Modelling Study

Effect of Non-pharmaceutical Interventions to Contain Covid-19 in China

The Psychological Burden of the COVID-19 Pandemic Severity

Effectiveness of Public Health Measures in Mitigating Pandemic Influenza Spread: A Prospective Sero-Epidemiological Cohort Study

On a Measure of Lack of Fit in Time Series Models

2020) Do Quarantine Experiences and Attitudes Towards COVID-19 Affect the Distribution of Psychological Outcomes in China? A Quantile Regression Analysis (WP512)

Agent-Based Simulation for Weekend-Extension Strategies to Mitigate Influenza Outbreaks

Pandemic Influenza in Papua New Guinea: A Modelling Study Comparison with Pandemic Spread in a Developed Country

A Small Community Model for the Transmission of Infectious Diseases: Comparison of School Closure as an Intervention in Individual-Based Models of an Influenza Pandemic

The Effectiveness of Social Distancing in Containing Covid-19

The Effect of Control Strategies to Reduce Social Mixing on Outcomes of the Covid-19 Epidemic in Wuhan, China: A Modelling Study

Japan's Halfhearted Coronavirus Measures are Working Anyway. Foreign Policy

Narrative Economics, Public Policy and Mental Health

Efficacy of Control Measures in the Control of Ebola

Do Lockdown and Testing Help in Curbing Covid-19 Transmission? Covid Economics

Impact of Temperature and Relative Humidity on the Transmission of COVID-19: A Modelling Study in China and the United States

A Heteroscedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroscedasticity

What can Countries Learn from Hong Kong's Response to the COVID-19 pandemic?

Evaluating Temporal Factors in Combined Interventions of Workforce Shift and School Closure for Mitigating the Spread of Influenza. PLoS One. 7, e32203. CRediT Author statement Navendu Prakash: Conceptualization, Methodology, Formal analysis, Investigation, Writing -Original draft, Visualization. Bhavya Srivastava: Conceptualization, Methodology, Formal analysis, Investigation, Writing -Original draft, Data Curation. Shveta Singh: Writing -Review and Editing, Supervision, Project administration. Seema Sharma: Writing -Review and Editing, Supervision, Project administration

The authors would like to thank the editor and two anonymous reviewers for their time and valuable suggestions. Navendu Prakash acknowledges the support of the Junior Research Fellowship from the University Grants Commission (UGC), India (926/NET-NOV2017) as a part of his Ph.D. program. Bhavya Srivastava acknowledges the support of the Junior Research Fellowship from the University Grants Commission (UGC), India (988/NET-JULY2018) as a part of her Ph.D. program. However, UGC was not involved in study design; in the collection, analysis and interpretation of data; in the writing of the report; and in the decision to submit the article for publication, for either of the authors. 

The below discussion provides a technical account of the recursive derivation of the Kalman filter and the impact of non-linearity and non-normality on estimation. Setting a state-space model with exogenous variables:

Intuitively, a trend that models COVID-19 cases as a stochastic phenomenon using an iterative process of the Kalman filter is known to be more efficient than a deterministic non-linear trend (Jalles, 2009 namely mean absolute deviations (MAD) and root mean square deviations (RMSD) from the actual trend in order to compare the fit of stochastic trend (employed in this study) with two deterministic non-linear polynomial trends of order 4 and 5. For all the countries, the deviations of the stochastic model are much lower than the two deterministic trends, implying a better fit. It is unlikely that any other deterministic model will provide a better fit than a stochastic model, primarily due to the time-varying nature of the latter. 

While this paper primarily focuses on the impact of non-pharmaceutical interventions, we also analyze the partial impact of a pharmaceutical intervention (vaccination) in containing the spread of infection. The introduction of pharmaceutical interventions can significantly revolutionize the fight against the pandemic. This is because social distancing measures are sub-optimal from a long-term policy perspective, involving significant social, economic, and psychological costs, and hence, cannot be implemented indefinitely.Further, NPIs also create the problem of leaving a reservoir of persons prone to infection. Whilst the quest for creating an effective vaccine is in progress, preliminary pharmacological data indicate that vaccines are indeed effective in reducing the possibility of COVID-19 deaths. To this end, we extend our dataset tillMarch 31, 2021, to determine the impact of vaccinations and vaccination policy on COVID-19 cases and