Several recent papers have focused on econometric and statistical analysis and forecasting of key time series and variables associated with the on-going COVID-19 pandemics, including infection and death rates, and their effects on economic and financial markets (see Dimdore-Miles and Miles, 2020 , Harvey and Kattuman, 2020 , Manski and Molinari, 2020 , Toda, 2020a , Stock, 2020b , Beare and Toda, 2020 , and Toda, 2020 . This note contributes to the above literature by focusing on robust analysis of the effects of the pandemics on financial markets across the World. Among other results, it provides the results of robust evaluation and estimation of predictive regressions for financial returns and foreign exchange rates in different countries incorporating the time series of reported deaths from COVID-19. 1 We also present a detailed study of persistence, heavy-tailedness and tail risk properties of COVID-19 deaths time series that emphasize the necessity in applications of robust inference methods in the analysis and forecasting of the COVID-19 pandemic and its impact on economic and financial markets and the society.
Econometrically justified and robust analysis in the note is based on application of heteroskedasticity and autocorrelation consistent (HAC) inference methods, related approaches using consistent standard errors, recently developed robust -statistic inference procedures and robust tail index estimation approaches.
The results of the analysis, in particular, indicate potential non-stationarity in the form of unit roots in the time series of daily deaths from COVID-19 that are commonly used in research on modelling and forecasting of the COVID-19 pandemic and its effects. The results emphasize the necessity in basing the analysis of models incorporating the COVID-19-related time series such as daily death rates on (stationary) differences of the latter. The analysis using a range of tail index inference methods further indicates potential heavy-tailedness with possibly infinite variances and first moments in the time series of daily deaths from COVID-19 and their differences in countries across the World.
In order to account for the problems of potential non-stationarity in the daily COVID-19 deaths time series, the note provides the analysis of predictive regressions for financial returns incorporating both the lagged daily deaths from COVID-19 and their differences. Further, the properties of autocorrelation, heavy-tailedness and heterogeneity in the time series are accounted for by the use in the predictive regression analysis of both the widely applied standard HAC inference methods as well as the recently proposed −statistic approaches to robust inference under the above problems in the data.
The standard HAC inference methods indicate statistical significance of the (potentially non-stationary) lagged daily deaths from COVID-19 in predictive regressions for returns on the main stock indices in some countries, including the US, Japan, Russia, Brazil and India. However, according to econometrically justified analysis with the use of robust −statistic approaches in addition to HAC tests, the lagged daily COVID-19 death rates and their (stationary) differences appear not to be statistically significant in predictive regressions for stock index returns in essentially all countries considered in the analysis. 1 The focus on the time series of COVID-19 related deaths rather than infection rates is motivated by dependence of the number of reported infections on a variety of different factors such as, importantly, country-specific policies on testing for COVID-19, and its adoption and spread in different countries.
Overall, the main message of the results in the note is that statistical and econometric analyses and forecasts of the on-going COVID-19 pandemic and its impacts on economic and financial markets and the society should be based on theoretically justified robust inference methods. The methods used in the analysis and the forecasting of the pandemic and its effects should account, in particular, for the problems of potential non-stationarity, autocorrelation, heavy-tailedness and heterogeneity in the key time series and variables related to COVID-19, including the deaths and infection time series.
The note is organised as follows. Section 3 describes the data used in the analysis. Section 4.1 presents the analysis heavy-tailedness and tail risk properties of daily COVID-19 death rates. Section 4.2 provides the results of (non-)stationarity and unit root tests for time series characterising the COVID-19 related death rates in the countries across the World. Section 4.3 provides the results of theoretically justified and robust statistical analysis of predictive regressions for the returns on major stock indices in the countries considered incorporating the time series on COVID-19 related deaths. Section 5 makes some concluding remarks and discusses directions for further research. Appendices Appendix A and Appendix B provide the diagrams and tables on the results of statistical analysis in the note.
The analysis in the note uses the data on COVID-19 in different countries across the World (the UK, Germany, France, Italy, Spain, Russia, the Netherlands, Sweden, India, Austria, Finland, Ireland, the US, Lithuania, Canada, Brazil, Mexico, Argentina, Japan, China, South Korea, Indonesia and Australia) for the period from 22 January 2020 to 29 June 2020. The data is obtained from the Data Repository maintained by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University. 2 The data on prices of major stock indices for the countries considered is obtained from Yahoo Finance and the data on interest rates is from the Global Rates database. 3 We consider the following stock indices: FTSE 100 (UK), DAX (Germany), CAC 40 (France), FTSE MIB (Italy), IBEX 35 (Spain), MOEX (Russia), AEX (Netherlands), OMXS 30 (Sweden), SENSEX (India), ATX (Austria), OMX Helsinki 25 (Finland), ISEQ (Ireland), Dow Jones, S&P 500 (USA), OMX Vilnius (Lithuania), TSX (Canada), iBovespa (Brazil), IPC Mexico (Mexico), Merval (Argentina), NIKKEI 225 (Japan), SHANGHAI (China), KOSPI (South Korea), JCI (Indonesia), ASX 50, ASX 200 and Australian All Ordinaries (Australia). The analysis uses central bank rates for the countries considered; European interest rate is used for the members of European monetary union.
Throughout the paper, denotes the (cumulative) number of COVID-19 related deaths from the beginning of the period on 22 January to day in the countries considered. Further, ∆ denote the differences of the above time series, that is the number of reported deaths in day . By ∆ 2 we denote the cumulative deaths time series' second differences, that is, the daily changes in the number of COVID-19 related deaths in the countries dealt with. The estimation and testing in the note is based on the periods with positive values of the number of COVID-19 related deaths in the countries considered. 4 The sample sizes of daily time series used in the analysis range from 63 (Finland) to 99 (China) observations (see Table B2 ).
As indicated in many empirical and theoretical works in the literature (see, among others, the analysis and the reviews in Embrechts et al., 1997 , Gabaix et al., 2003 , Beirlant et al., 2004 , Gabaix et al., 2006 , Gabaix, 2009 , Ibragimov et al., 2011 , and Ibragimov et al., 2015 , distributions of many variables related to or affected by crises and natural disasters and characterised by the presence of extreme values and outliers, such as financial returns, catastrophe risks or economic losses from natural catastrophes, exhibit deviations from Gaussianity in the form of heavy power law tails. For a positive heavy-tailed variable (e.g., representing a risk, the absolute value of a financial return or foreign exchange rate, or a loss from a natural disaster ) one has
for large > 0, with a constant > 0 and the parameter > 0 that is referred to as the tail index (or the tail exponent) of . The value of the tail index parameter is important as it characterises the probability mass (heaviness and the rate of decay) in the tails of power law distribution (1). Heavytailedness (i.e., the tail index ) of the variable governs the likelihood of observing extremes and outliers in the variables. The smaller values of the tail index correspond to a higher degree of heavytailedness in and, thus, to a higher likelihood of observing extremely large values in realisations of the variable. In addition, importantly, the value of the tail index governs finiteness of moments of , with the moment of order > 0 of the variable being finite: < ∞ if and only if > . In particular, the variance of is defined and is finite if and only if > 2, and the first moment of the variable is finite if and only if > 1.
The characteristics of heavy-tailedness such as tail indices in models (1) are of key interest for policy makers, professionals in financial and insurance industries, risk managers, regulators and financial stability analysts concerned with the likelihood of extreme values of risks, financial returns or foreign exchange rates in consideration, their tail risk and the related risk measures.
Further, naturally, the degree of heavy-tailedness and finiteness of variances for variables dealt with, such as economic and financial indicators like financial returns and exchange rates or risks and losses 4 from natural disasters, is crucial for applicability of standard statistical and econometric approaches, including regression and least squares methods. Similarly, the problem of potentially infinite fourth moments of (economic and financial) time series dealt with needs to be taken into account in applications of autocorrelation-based methods and related inference procedures in their analysis (see the discussion in Cont, 2001 , Ch. 1 in Ibragimov et al., 2015 , Ibragimov, Pedersen and Skrobotov, 2020 .
Many recent studies argue that the tail indices in heavy-tailed models (1) typically lie in the interval ∈ (2, 4) for financial returns and foreign exchange rates in developed economies (see, among others, Loretan and Phillips, 1994 , Gabaix et al., 2003 , Gabaix et al., 2006 , Gabaix, 2009 , Ibragimov et al., 2015 . These estimates imply that these variables have finite variances and finite first moments; however, their fourth moments are infinite. At the same time, tail indices may be smaller than two for financial returns and foreign exchange rates in emerging and developed economies, thus implying possibly infinite variances (see Ibragimov et al., 2013 , Gu and Ibragimov, 2018 , Chen and Ibragimov, 2019 , and Section 3.2 in Ibragimov et al., 2015 .
Heavy-tailed power law behavior is also exhibited by such important economic and financial variables as income and wealth (with ∈ (1.5, 3) and ≈ 3, respectively; see, among others, Gabaix, 2009, and the references therein); financial returns from technological innovations, losses from operational risks and those from earthquakes and other natural disasters (with tail indices that can be considerably less than one, see Ibragimov et al., 2011, and Ibragimov et al., 2015, and references therein) .
The recent study by Cirillo and Taleb (2020) provides (Hill's, see below) tail index estimates supporting extreme heavy-tailedness with smaller than 1 and infinite first moments in the number of deaths from 72 major epidemic and pandemic diseases from 429 BC until the present. Beare and Toda (2020) report (Hill's) estimates of the tail index close to 1 implying infinite variances and first moments in the distribution of COVID-19 infections across the US counties at the beginning of the pandemic.
Several approaches to the inference about the tail index of heavy-tailed distributions are available in the literature (see, among others, the reviews in Embrechts et al., 1997 , Beirlant et al., 2004 , Gabaix and Ibragimov, 2011 , Ch. 3 in Ibragimov et al., 2015 . The two most commonly used ones are Hill's estimates and the OLS approach using the log-log rank-size regression.
It was reported in a number of studies that inference on the tail index using widely applied Hill's estimates suffers from several problems, including sensitivity to dependence and small sample sizes (see, among others, Ch. 6 in Embrechts et al., 1997) . Motivated by these problems, several studies have focused on alternative approaches to the tail index estimation. For instance, Huisman et al. (2001) propose a weighted analogue of Hill's estimator that is reported to correct its small sample bias for sample sizes less than 1,000. Using extreme value theory, Müller and Wang (2017) focus on inference on the quantiles and tail probabilities of heavy-tailed variables with a fixed number of their extreme observations (order statistics) employed in estimation as is typical in relatively small samples of fattailed data. Embrechts et al. (1997) , among others, advocate sophisticated nonlinear procedures for tail index estimation. Gabaix and Ibragimov (2011) focus on econometrically justified inference on the tail index in heavy-tailed power law models (1) using the popular and widely applied approach based on log-log rank-size regressions log( ) = − log( ), with taken as an estimate of . The reason for popularity of the approach is its simplicity and robustness. Gabaix and Ibragimov (2011) provide a simple remedy for the inherent small sample bias in log-log rank-size approaches to inference on tail indices, and propose using the (optimal) shifts of 1/2 in ranks, with the tail index estimated by the parameter in (small sample bias-corrected) regressions log( − 1/2) = − log( ). Gabaix and Ibragimov (2011) further derive the correct standard errors on the tail exponent in the log-log rank-size regression approaches. The standard error on in the above log-log rank-size regressions is not the OLS standard error but is asymptotically (2/ ) 1/2 , where is the number of extreme (the largest) observations on the heavy-tailed variable used in tail index estimation (see also Ch. 3 in Ibragimov et al., 2015) . The numerical results in Ibragimov et al. (2015) point to advantages of the proposed approaches to inference on tail indices, including their robustness to dependence in the data and deviations from exact power laws in the form of slowly varying functions. Figure A1 provides the plots of Hill's estimates of the tail indices in power laws distributions for the time series ∆ of daily COVID-19 related deaths in the countries considered with different number of extreme (largest) observations used in tail index estimation (the so-called Hill's plots, see Ch. 6 in Embrechts et al., 1997, and also Cirillo and Taleb, 2020 , for similar plots employed in the analysis of the inverse = 1/ of the tail index in power law models 1 for the number of deaths from major epidemic and pandemic diseases from ancient times until the present). Similarly, Figure A2 provides the log-log rank-size regression estimates of the tail indices with optimal shifts 1/2 in ranks proposed in Gabaix and Ibragimov (2011) for the time series ∆ dealt with that use different truncation levels for the largest values of daily deaths from COVID-19 used in inference (see Ibragimov et al., 2013 , for the analysis of such log-log rank-size plots for foreign exchange rates in emerging economies). The plots provide the corresponding 95% confidence intervals for tail indices in power law models (1) for the number of daily COVID-19 deaths in the countries considered.
The analysis of Figures A1 and A2 indicates that both Hill's and log-log rank-size regression tail index estimates tend to stabilize for most of the countries as a sufficient number of extreme (largest) observations (order statistics) on the daily COVID-19 related deaths is used in inference. As expected, log-log rank-size regression estimates tend to be less sensitive to the choice of compared to Hill's estimates.
Importantly, the left-end points of the confidence intervals for tail indices in power law models for daily COVID-19 related deaths calculated using different tail truncation levels in most of the countries tend to be less than two indicating possibly infinite second moments and variances. Further, from the analysis of Figures A1 and A2 it follows that the tail indices may be even less than one for some of the countries indicating extreme heavy-tailedness with possibly infinite first moments.
Extreme heavy-tailedness with possibly infinite variances and first moments for the time series on COVID-19 related deaths is further confirmed by the results of tail index estimation for (stationary, see the next section) time series ∆ 2 of daily changes in the number of deaths from the disease (see Figures A3 and A4 for the plots of Hill's and log-log rank-size regression tail index estimates -Hill's and log-log rank-size regression plots -for the time series ∆ 2 for the countries considered). 5
The conclusions on heavy-tailedness in the COVID-19 related deaths time series are important as, according to the above discussion, they point to high likelihood of observing extremely large values of daily deaths from COVID-19. They further emphasise the necessity in the use of robust methods in statistical analysis and forecasting of the dynamics of the COVID-19 pandemic and its impacts, including the approaches robust to the problems of heavy-tailedness and heterogeneity in the data.
We begin the analysis by the study of the degree of integration in the time series , ∆ and ∆ 2 on COVID-19 related deaths and their differences in the countries considered. Table B1 presents the results of several unit root tests for the time series of daily deaths ∆ and the time series of daily changes in the number of deaths ∆ 2 . The results are provided for the (right tailed) likelihood ratio unit root test proposed by Jansson and Nielsen (2012) (with the test statistic in Table B1 ; see also Skrobotov, 2018) , the GLS-based modified Phillips-Perron type tests (with the corresponding test statistics , , ), the modified point optimal test (with the test statistic ; see Ng and Perron, 2001) and the GLS-based Augmented Dickey-Fuller test (with the test-statistic denoted by in Table B1 ; see Elliott et al., 1996) . To address the issue of possible heavy tails and infinite variance of the series, for calculation of the −value of the unit root tests, we use recently justified sieve wild bootstrap algorithm with a Rademacher distribution employed in the wild bootstrap re-sampling scheme (see Cavaliere et al., 2020) .
An important tuning parameter in the above tests is related to the choice of lag length used in the analysis. We use the modified Akaike information criterion (MAIC) lag choice approach based on standard ADF regressions as suggested by Perron and Qu (2007) . According to the results (the wild bootstrap −values are given in brackets), the unit root hypothesis in the time series ∆ of daily deaths is not rejected at reasonable significance levels, e.g., 5% and 10%, by all the employed tests for all the countries considered except Sweden, Finland and China. For the daily deaths time series ∆ in China, the rejection of the unit root hypothesis is on every reasonable significance level (even at 1%). The unit root hypothesis is rejected for the time series ∆ in Finland and Sweden at 10% by all tests (the hypothesis is also rejected at 5% by the likelihood ratio test for Sweden, and by all the tests except for Finland). On the other hand, according to the results in Table B1 , the unit root hypothesis is rejected at all reasonable significance levels by all the tests for the time series ∆ 2 of daily changes in the number of COVID-19 related deaths in all the countries considered.
The above results of unit root tests have several important implications for statistical analysis of models and key time series related to the COVID-19 pandemic and its effects. According to the results, in most of the countries across the World, the time series of daily COVID-19 related deaths and thus the time series of total (cumulative) deaths from the disease up to a certain date that are typically employed in the analysis and forecasting of the pandemic and its impact appear to exhibit non-stationarity. The daily COVID-19 related deaths' time series ∆ appears to exhibit unit root process persistence for confirmed by weighted Hill's tail index estimates proposed in Huisman et al. (2001) that are more robust to small sample sizes as compared to Hill's estimates.
most of the countries considered. This, in turn, implies very high persistence in the time series of total deaths up to a certain date that appears to be integrated of order 2. 6 One should also emphasize that, due to non-normality of the OLS and other standard estimates of parameters in (e.g., regression) models incorporating nonstationary variables (e.g., regressors; see, among others, the discussion in Sections 14.6 and 16.4 in Stock and Watson, 2006) , the statistical analysis and forecasting of key variables and time series related to the COVID-19 pandemic and its effects such as the COVID-19 death rates should be based on stationary differences of time series with potential unit root behavior as in the case of predictive regressions for financial returns in the next chapter.
This section presents the main results of the note on statistically justified and robust evaluation of the effects of the COVID-19 pandemic on financial markets in different countries across the World. We focus on the analysis of predictive regressions of returns on major stock indices in the countries (see Section 3) on the time series characterizing the death rates from COVID-19 in the countries considered. Importantly, due to the problems of nonstationarity and the unit root dynamics in the time series ∆ of daily COVID-19 related deaths in most of the countries discussed in the previous section, estimation of the predictive regressions is provided for regression models for stock index returns with both the lagged daily deaths ∆ −1 and the (stationary) lagged changes in daily deaths ∆ 2 −1 used as regressors.
More precisely, the estimation results are provided for predictive regressions in the form
where are the excess returns on major stock indices in the countries considered at the end of the day given by the difference between the end of the day-stock index returns and the countries' interest rates (see Section 3), and the regressors −1 are either the number ∆ −1 of COVID-19 related deaths on day − 1 in the countries dealt with or the daily changes ∆ 2 −1 = ∆ −1 − ∆ −2 in the deaths' time series.
In order to account for autocorrelation and heteroskedasticity in the regressors and the error terms in predictive regressions (2) we use the widely applied HAC based -statistic (with the quadratic spectral -QS -kernel and automatic choice of bandwidth as in Andrews, 1991) in the analysis of statistical significance of the regression coefficients.
It is well known, however, that commonly used HAC inference methods and related approaches based on consistent standard errors often have poor finite sample properties, especially in the case of 6 The conclusions on persistence properties of the time series and Δ are somewhat similar to those for the CPI and the inflation rate (the change in the logarithm of the CPI) time series, where often unit root hypothesis is not rejected for the inflation rate and thus the (logarithm) of the CPI levels appears to be integrated of order 2 (see the analysis of non-stationarity in Section 14.6 in Stock and Watson, 2006 , for the inflation rate and its changes in the US). These conclusions imply the necessity of the use of differences of the inflation rate in time series modeling of inflation and its relationship to other key economic variables such as the unemployment level in the Phillips curve (see Chs. 14 and 16 in Stock and Watson, 2006) . pronounced dependence, heterogeneity and heavy-tailedness in the data (see the discussion and the analysis in Ibragimov and Müller, 2010 , Section 3.3 in Ibragimov et al., 2015 . To account for these problems, we also provide the analysis of statistical significance of predictive regression coefficients using the −statistic approaches to robust inference recently developed in Müller (2010, 2016) . Following the approaches, robust large sample inference on a parameter of interest (e.g., a predictive regression coefficient ) is conducted as follows: the data is partitioned into a fixed number ≥ 2 (e.g., = 2, 4, 8) of groups, the model is estimated for each group, and inference is based on a standard −test with the resulting parameter estimators.
In the context of inference on the coefficient in time series predictive regressions (2), the regression is estimated for groups of time series observations with ( − 1) / < ≤ / , = 1, ..., , resulting in group estimatesˆ, = 1, ..., . The robust test of a hypothesis on the parameter is based on the −statistic in the group OLS regression estimatesˆ, = 1, ..., . E.g., the robust test of the null hypothesis 0 : = 0 against alternative : ̸ = 0 is based on the −statistic = √^^, wherê
The above null hypothesis 0 is rejected in favor of the alternative at level ≤ 8.3% (e.g., at the usual significance level = 5%) if the absolute value |^| of the −statistic in group estimatesˆexceeds the (1 − /2)−quantile of the standard Studentdistribution with − 1 degrees of freedom. 7
The −statistic based approaches do not require at all estimation of limiting variances of estimators of interest. As discussed in Ibragimov et al. (2015) , Müller (2010, 2016) , they result in asymptotically valid inference under the assumption that the group estimators of a parameter of interest are asymptotically independent, unbiased and Gaussian of possibly different variances. 8 The assumption is satisfied in a wide range of econometric models and dependence, heterogeneity and heavy-tailedness settings of a largely unknown type. The numerical analysis in Ibragimov et al. (2015) , Müller (2010, 2016) indicates favorable finite sample performance of the −statistic based robust inference approaches in inference on models with time series, panel, clustered and spatially correlated data. 91011 Importantly, the −statistic based approaches to robust inference may also be used under convergence of group estimators of a parameter interest to scale mixtures of normal distributions as in 7 One-sided tests are conducted in a similar way. 8 Justification of asymptotic validity of the robust −statistic inference approaches in Müller (2010, 2016 ) is based on a small sample result in Bakirov and Szekely (2006) that implies validity of the standard −test under independent heterogeneous observations and its analogues for two-sample −tests obtained in Ibragimov and Müller (2016) . the case of models under heavy-tailedness with infinite variances and in regressions with non-stationary exogenous regressors. 12 Table B2 provides the results of the assessment of statistical significance of the coefficients on the lagged time series ∆ −1 of daily COVID-19 related deaths and their differences -the daily changes in the number of deaths from the disease -in predictive regressions (2) for the countries considered. More precisely, the table provides the values of HAC −statistic with the QS kernel and the automatic choice of bandwidth discussed above as well as the values of the −statistic in estimates of the slope parameter obtained using = 4, 8, 12 and 16 groups of consecutive time series observations. The asterisks in the table indicate statistical significance of the slope coefficient (*** for the significance at 1% and * for significance at 10%) implied by formal comparisons of the HAC −statistics with the quantiles of a standard normal distribution. As described above, following the −statistic approaches to robust inference in Müller (2010, 2016) , (the absence of) statistical significance of the slope coefficient is assessed in the table using the comparisons of the −statistic in group estimates of the coefficient with the quantiles of Student-distributions with − 1 degrees of freedom.
The values of HAC −statistics in Table B2 indicate an apparently spurious statistical significance of the (potentially non-stationary) lagged daily deaths ∆ −1 from COVID-19 in predictive regressions for returns on the main stock indices in some countries, namely, for the US, Japan, Russia, Brazil, India, Mexico, Canada and Lithuania, with unexpected positive signs of the estimates of the slope coefficient in the regressions. However, the lagged daily COVID-19 death rates ∆ −1 and their (stationary) differences ∆ 2 −1 appear not to be statistically significant in predictive regressions for stock index returns in all countries considered according to the (econometrically justified) robust −statistic approaches. The absence of statistical significance of the coefficients on (stationary) daily changes ∆ 2 −1 in COVID-19 related deaths is further indicated by HAC −statistics for econometrically justified predictive regressions incorporating ∆ 2 −1 for major indices in essentially all countries considered. 13
This note presented the results of theoretically justified and robust statistical analysis of the effects of the COVID-19 pandemic on financial markets in different countries across the World. The analysis is based on robust inference in predictive regressions for the returns on the countries' major stock indices incorporating the time series characterizing the dynamics in the COVID-19 related deaths rates. 12 See Section 3.3.3 in Ibragimov et al. (2015) for applications of the robust −statistic approaches in inference in infinite variance heavy-tailed models. The recent works by Anatolyev (2019), Pedersen (2019) and Ibragimov, Pedersen and Skrobotov (2020) provide further applications of the approaches in robust inference on general classes of GARCH and AR-GARCH-type models exhibiting heavy-tailedness and volatility clustering properties typical for real-world financial and economic markets. The recent paper by Ibragimov, Kim and Skrobotov (2020) focuses on applications of the −statistic approaches in inference on predictive regressions with persistent and/or fat-tailed regressors and errors. 13 The formal comparison of the HAC −statistic with quantiles of a standard normal distribution points to some statistical significance of the slope coefficient in predictive regressions based on Δ 2 −1 as a regressor for returns on financial indices in Lithuania.
The note further presented the results of the statistical analysis of (non-)stationarity, heavy-tailedness and tail risk in the time series on death rates from COVID-19 in the countries considered. The obtained results point to non-stationary unit root dynamics and pronounced heavy-tailedness with possibly infinite variances and fist moments in the time series of daily COVID-19 related deaths in most of the countries dealt with.
According to the results in the note, the standard HAC inference methods indicate apparently spurious statistical significance of the (potentially non-stationary) lagged daily deaths from COVID-19 in predictive regressions for returns on the major stock indices in some countries, including the US, Japan, Russia, Brazil and India. On the other hand, according to statistically justified analysis with the use of robust −statistic approaches in addition to HAC tests, the lagged daily COVID-19 death rates and their (stationary) differences appear to be statistically insignificant in predictive regressions for stock index returns in essentially all countries considered in the analysis.
The analysis and conclusions in the note emphasize the necessity in the use of robust inference methods accounting for autocorrelation, heterogeneity and heavy-tailedness in statistical and econometric analysis and forecasting of key time series and variables related to the COVID-19 pandemic and its effects on economic and financial markets and society. They further emphasize the importance of the use of correctly specified models of the COVID-19 pandemic and its effects incorporating stationary time series and variables such as the daily changes in COVID-19 related deaths used in predictive regressions in this work.
Further research may focus on robust analysis of the dynamics of a range of key time series related to the COVID-19 pandemic, including infection rates; robust tests of structural breaks in models of the dynamics of the pandemic and its effects on financial and economic markets, and applications of inference methods such as sign-and rank-based tests that are robust to relatively small sample sizes of observations in statistical analysis of key models related to the spread of COVID-19. It would also be of interest to apply further estimation approaches for heavy-tailed models for time series associated with the pandemic that are robust to small samples, including the recently developed fixed-inference approaches for power-law models (1) in Müller and Wang (2017) . The analysis in these directions is currently under way by the authors and co-authors. (v) 