COVID-19: Tail Risk and Predictive Regressions
Walter Distaso, Rustam Ibragimov, Alexander Semenov, Anton Skrobotov
CCOVID-19: Tail Risk and Predictive Regressions โ Walter Distaso
Imperial College LondonBusiness School
Rustam Ibragimov
Imperial College LondonBusiness School
Alexander Semenov
St. Petersburg State University
Anton Skrobotov
Russian Presidential Academy of National Economyand Public Administration and St. Petersburg State University
Abstract
Reliable analysis and forecasting of the spread of COVID-19 pandemic and its impacts on global financeand Worldโs economies requires application of econometrically justified and robust methods. At the sametime, statistical and econometric analysis of financial and economic markets and of the spread of COVID-19is complicated by the inherent potential non-stationarity, dependence, heterogeneity and heavy-tailednessin the data. This project focuses on econometrically justified robust analysis of the effects of the COVID-19pandemic on the Worldโs financial markets in different countries across the World. Among other results,the study focuses on robust inference in predictive regressions for different countries across the World. Wealso present a detailed study of persistence, heavy-tailedness and tail risk properties of the time series ofthe COVID-19 death rates that motivate the necessity in applications of robust inference methods in theanalysis. Econometrically justified analysis is based on application of heteroskedasticity and autocorrelationconsistent (HAC) inference methods, related approaches using consistent standard errors, recently developedrobust ๐ก -statistic inference procedures and robust tail index estimation approaches. Keywords:
COVID-19, pandemic, tail risk, predictive regressions, forecasting, robust inference
JEL Classification:
C13, C51 โ The authors are grateful to Ulrich K. Mยจuller, Artem Prokhorov and the participants at the Centre for BusinessAnalysis (CEBA, St. Petersburg State University) seminar series for helpful comments and suggestions. We also thankRonald Huisman for kindly sharing the SciLab codes for implementation of weighted Hillโs tail index estimates in Huismanet al. (2001). a r X i v : . [ ec on . E M ] S e p Introduction
Several recent papers have focused on econometric and statistical analysis and forecasting of keytime series and variables associated with the on-going COVID-19 pandemics, including infection anddeath rates, and their effects on economic and financial markets (see Dimdore-Miles and Miles, 2020,Harvey and Kattuman, 2020, Manski and Molinari, 2020, Toda, 2020 a , Stock, 2020 b , Beare and Toda,2020, and Toda, 2020, among others). This note contributes to the above literature by focusing on robustanalysis of the effects of the pandemics on financial markets across the World. Among other results, itprovides the results of robust evaluation and estimation of predictive regressions for financial returnsand foreign exchange rates in different countries incorporating the time series of reported deaths fromCOVID-19. We also present a detailed study of persistence, heavy-tailedness and tail risk properties ofCOVID-19 deaths time series that emphasize the necessity in applications of robust inference methodsin the analysis and forecasting of the COVID-19 pandemic and its impact on economic and financialmarkets and the society.Econometrically justified and robust analysis in the note is based on application of heteroskedasticityand autocorrelation consistent (HAC) inference methods, related approaches using consistent standarderrors, recently developed robust ๐ก -statistic inference procedures and robust tail index estimation ap-proaches.The results of the analysis, in particular, indicate potential non-stationarity in the form of unit rootsin the time series of daily deaths from COVID-19 that are commonly used in research on modelling andforecasting of the COVID-19 pandemic and its effects. The results emphasize the necessity in basingthe 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 furtherindicates potential heavy-tailedness with possibly infinite variances and first moments in the time seriesof 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 timeseries, the note provides the analysis of predictive regressions for financial returns incorporating both thelagged 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 predictiveregression analysis of both the widely applied standard HAC inference methods as well as the recentlyproposed ๐ก โ 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 insome countries, including the US, Japan, Russia, Brazil and India. However, according to econometri-cally justified analysis with the use of robust ๐ก โ statistic approaches in addition to HAC tests, the laggeddaily COVID-19 death rates and their (stationary) differences appear not to be statistically significantin predictive regressions for stock index returns in essentially all countries considered in the analysis. The focus on the time series of COVID-19 related deaths rather than infection rates is motivated by dependence ofthe number of reported infections on a variety of different factors such as, importantly, country-specific policies on testingfor COVID-19, and its adoption and spread in different countries.
The note is organised as follows. Section 3 describes the data used in the analysis. Section 4.1 presentsthe analysis heavy-tailedness and tail risk properties of daily COVID-19 death rates. Section 4.2 providesthe results of (non-)stationarity and unit root tests for time series characterising the COVID-19 relateddeath rates in the countries across the World. Section 4.3 provides the results of theoretically justifiedand robust statistical analysis of predictive regressions for the returns on major stock indices in thecountries considered incorporating the time series on COVID-19 related deaths. Section 5 makes someconcluding remarks and discusses directions for further research. Appendices Appendix A and AppendixB 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 (theUK, Germany, France, Italy, Spain, Russia, the Netherlands, Sweden, India, Austria, Finland, Ireland,the US, Lithuania, Canada, Brazil, Mexico, Argentina, Japan, China, South Korea, Indonesia andAustralia) for the period from 22 January 2020 to 29 June 2020. The data is obtained from the DataRepository maintained by the Center for Systems Science and Engineering (CSSE) at Johns HopkinsUniversity. The data on prices of major stock indices for the countries considered is obtained fromYahoo Finance and the data on interest rates is from the Global Rates database. We consider thefollowing stock indices: FTSE 100 (UK), DAX (Germany), CAC 40 (France), FTSE MIB (Italy), IBEX35 (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 AllOrdinaries (Australia). The analysis uses central bank rates for the countries considered; Europeaninterest rate is used for the members of European monetary union.Throughout the paper, ๐ท ๐ก denotes the (cumulative) number of COVID-19 related deaths from thebeginning of the period on 22 January to day ๐ก in the countries considered. Further, โ ๐ท ๐ก denote thedifferences of the above time series, that is the number of reported deaths in day ๐ก. By โ ๐ท ๐ก wedenote the cumulative deaths time seriesโ second differences, that is, the daily changes in the number of https://github.com/CSSEGISandData/COVID-19 ๐ท ๐ก in the countriesconsidered. 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, theanalysis and the reviews in Embrechts et al., 1997, Gabaix et al., 2003, Beirlant et al., 2004, Gabaixet al., 2006, Gabaix, 2009, Ibragimov et al., 2011, and Ibragimov et al., 2015), distributions of manyvariables related to or affected by crises and natural disasters and characterised by the presence ofextreme values and outliers, such as financial returns, catastrophe risks or economic losses from naturalcatastrophes, exhibit deviations from Gaussianity in the form of heavy power law tails. For a positiveheavy-tailed variable (e.g., representing a risk, the absolute value of a financial return or foreign exchangerate, or a loss from a natural disaster ๐ ) one has ๐ ( ๐ > ๐ฅ ) โผ ๐ถ๐ฅ ๐ (1)for large ๐ฅ > , with a constant ๐ถ > and the parameter ๐ > that is referred to as the tail index(or the tail exponent) of ๐. The value of the tail index parameter ๐ is important as it characterises theprobability mass (heaviness and the rate of decay) in the tails of power law distribution (1). Heavy-tailedness (i.e., the tail index ๐ ) of the variable ๐ governs the likelihood of observing extremes andoutliers in the variables. The smaller values of the tail index ๐ correspond to a higher degree of heavy-tailedness in ๐ and, thus, to a higher likelihood of observing extremely large values in realisations ofthe variable. In addition, importantly, the value of the tail index ๐ governs finiteness of moments of ๐, with the moment ๐ธ๐ ๐ of order ๐ > of the variable being finite: ๐ธ๐ ๐ < โ if and only if ๐ > ๐. Inparticular, the variance of ๐ is defined and is finite if and only if ๐ > , and the first moment of thevariable is finite if and only if ๐ > . The characteristics of heavy-tailedness such as tail indices in models (1) are of key interest forpolicy makers, professionals in financial and insurance industries, risk managers, regulators and financialstability analysts concerned with the likelihood of extreme values of risks, financial returns or foreignexchange 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 The dates of the beginning of recorded (non-zero) number of COVID-19 related deaths for the countries consideredare as follows: 22 January for China, 13 February for Japan, 15 February - France, 20 February - Korea, 21 February- Italy, 29 February - the US, 1 March - Australia, Finland and Lithuania, 3 March - Spain, 6 March 2020 for the UKand the Netherlands, 8 March - Argentina, 9 March - Germany and Canada, 11 March - Sweden, India and Ireland andIndonesia, 12 March - Austria, 17 March for Brazil and 19 March for Russia and Mexico. ๐ in heavy-tailed models (1) typically lie in theinterval ๐ โ (2 , for financial returns and foreign exchange rates in developed economies (see, amongothers, Loretan and Phillips, 1994, Gabaix et al., 2003, Gabaix et al., 2006, Gabaix, 2009, Ibragimovet al., 2015, and references therein). These estimates imply that these variables have finite variances andfinite first moments; however, their fourth moments are infinite. At the same time, tail indices may besmaller 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 andIbragimov, 2019, and Section 3.2 in Ibragimov et al., 2015).Heavy-tailed power law behavior is also exhibited by such important economic and financial variablesas income and wealth (with ๐ โ (1 . , and ๐ โ , respectively; see, among others, Gabaix, 2009, andthe references therein); financial returns from technological innovations, losses from operational risksand those from earthquakes and other natural disasters (with tail indices that can be considerably lessthan one, see Ibragimov et al., 2011, and Ibragimov et al., 2015, and references therein).The recent study by Cirillo and Taleb (2020) provides (Hillโs, see below) tail index estimates sup-porting extreme heavy-tailedness with ๐ smaller than 1 and infinite first moments in the number ofdeaths from 72 major epidemic and pandemic diseases from 429 BC until the present. Beare and Toda(2020) report (Hillโs) estimates of the tail index close to 1 implying infinite variances and first momentsin 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 ๐ of heavy-tailed distributions are availablein the literature (see, among others, the reviews in Embrechts et al., 1997, Beirlant et al., 2004, Gabaixand Ibragimov, 2011, Ch. 3 in Ibragimov et al., 2015, and references therein). The two most commonlyused ones are Hillโs 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 Hillโsestimates 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 havefocused on alternative approaches to the tail index estimation. For instance, Huisman et al. (2001)propose a weighted analogue of Hillโs estimator that is reported to correct its small sample bias forsample sizes less than 1,000. Using extreme value theory, Mยจuller and Wang (2017) focus on inferenceon the quantiles and tail probabilities of heavy-tailed variables with a fixed number ๐ of their extremeobservations (order statistics) employed in estimation as is typical in relatively small samples of fat-tailed data. Embrechts et al. (1997), among others, advocate sophisticated nonlinear procedures for tailindex estimation.Gabaix and Ibragimov (2011) focus on econometrically justified inference on the tail index ๐ inheavy-tailed power law models (1) using the popular and widely applied approach based on log-log5ank-size regressions log( ๐ ๐๐๐ ) = ๐ โ ๐ log( ๐๐๐ง๐ ) , with ๐ taken as an estimate of ๐. The reason forpopularity of the approach is its simplicity and robustness. Gabaix and Ibragimov (2011) provide asimple remedy for the inherent small sample bias in log-log rank-size approaches to inference on tailindices, and propose using the (optimal) shifts of 1/2 in ranks, with the tail index estimated by theparameter ๐ in (small sample bias-corrected) regressions log( ๐ ๐๐๐ โ /
2) = ๐ โ ๐ log( ๐๐๐ง๐ ) . Gabaixand Ibragimov (2011) further derive the correct standard errors on the tail exponent ๐ in the log-logrank-size regression approaches. The standard error on ๐ in the above log-log rank-size regressions isnot the OLS standard error but is asymptotically (2 /๐ ) / ๐, where ๐ is the number of extreme (thelargest) observations on the heavy-tailed variable ๐ used in tail index estimation (see also Ch. 3 inIbragimov et al., 2015). The numerical results in Ibragimov et al. (2015) point to advantages of theproposed approaches to inference on tail indices, including their robustness to dependence in the dataand deviations from exact power laws in the form of slowly varying functions.Figure A1 provides the plots of Hillโs estimates of the tail indices ๐ in power laws distributions forthe time series โ ๐ท ๐ก of daily COVID-19 related deaths in the countries considered with different number ๐ of extreme (largest) observations used in tail index estimation (the so-called Hillโs plots, see Ch. 6in Embrechts et al., 1997, and also Cirillo and Taleb, 2020, for similar plots employed in the analysisof the inverse ๐ = 1 /๐ of the tail index ๐ in power law models 1 for the number of deaths from majorepidemic and pandemic diseases from ancient times until the present). Similarly, Figure A2 providesthe log-log rank-size regression estimates of the tail indices ๐ with optimal shifts / in ranks proposedin Gabaix and Ibragimov (2011) for the time series โ ๐ท ๐ก dealt with that use different truncation levels ๐ 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). Theplots provide the corresponding 95% confidence intervals for tail indices ๐ in power law models (1) forthe number of daily COVID-19 deaths in the countries considered.The analysis of Figures A1 and A2 indicates that both Hillโs and log-log rank-size regression tailindex estimates tend to stabilize for most of the countries as a sufficient number ๐ 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 ๐ compared to Hillโsestimates.Importantly, the left-end points of the confidence intervals for tail indices ๐ in power law models fordaily COVID-19 related deaths calculated using different tail truncation levels ๐ in most of the countriestend to be less than two indicating possibly infinite second moments and variances. Further, from theanalysis of Figures A1 and A2 it follows that the tail indices may be even less than one for some of thecountries indicating extreme heavy-tailedness with possibly infinite first moments.Extreme heavy-tailedness with possibly infinite variances and first moments for the time series onCOVID-19 related deaths is further confirmed by the results of tail index estimation for (stationary,see the next section) time series โ ๐ท ๐ก of daily changes in the number of deaths from the disease (seeFigures A3 and A4 for the plots of Hillโs and log-log rank-size regression tail index estimates - Hillโsand log-log rank-size regression plots - for the time series โ ๐ท ๐ก for the countries considered). Extreme heavy-tailedness with possibly infinite variances in the time series on COVID-19 related deaths is also
We begin the analysis by the study of the degree of integration in the time series ๐ท ๐ก , โ ๐ท ๐ก and โ ๐ท ๐ก on COVID-19 related deaths and their differences in the countries considered. Table B1 presentsthe results of several unit root tests for the time series of daily deaths โ ๐ท ๐ก and the time series ofdaily changes in the number of deaths โ ๐ท ๐ก . The results are provided for the (right tailed) likelihoodratio unit root test proposed by Jansson and Nielsen (2012) (with the test statistic ๐ฟ๐ in Table B1;see also Skrobotov, 2018), the GLS-based modified Phillips-Perron type tests (with the correspondingtest statistics ๐ ๐ ๐ผ , ๐ ๐๐ต , ๐ ๐ ๐ก ), the modified point optimal test (with the test statistic ๐ ๐ ๐ก ; see Ngand Perron, 2001) and the GLS-based Augmented Dickey-Fuller test (with the test-statistic denotedby ๐ด๐ท๐น in Table B1; see Elliott et al., 1996). To address the issue of possible heavy tails and infinitevariance of the series, for calculation of the ๐ โ value of the unit root tests, we use recently justified sievewild bootstrap algorithm with a Rademacher distribution employed in the wild bootstrap re-samplingscheme (see Cavaliere et al., 2020).An important tuning parameter in the above tests is related to the choice of lag length used inthe analysis. We use the modified Akaike information criterion (MAIC) lag choice approach based onstandard ADF regressions as suggested by Perron and Qu (2007). According to the results (the wildbootstrap ๐ โ values are given in brackets), the unit root hypothesis in the time series โ ๐ท ๐ก of dailydeaths is not rejected at reasonable significance levels, e.g., 5% and 10%, by all the employed tests forall the countries considered except Sweden, Finland and China. For the daily deaths time series โ ๐ท ๐ก inChina, 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 โ ๐ท ๐ก 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 ๐ ๐๐ต for Finland).On the other hand, according to the results in Table B1, the unit root hypothesis is rejected at allreasonable significance levels by all the tests for the time series โ ๐ท ๐ก of daily changes in the number ofCOVID-19 related deaths in all the countries considered.The above results of unit root tests have several important implications for statistical analysis ofmodels 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 thetime series of total (cumulative) deaths from the disease up to a certain date that are typically employedin the analysis and forecasting of the pandemic and its impact appear to exhibit non-stationarity. Thedaily COVID-19 related deathsโ time series โ ๐ท ๐ก appears to exhibit unit root process persistence for confirmed by weighted Hillโs tail index estimates proposed in Huisman et al. (2001) that are more robust to small samplesizes as compared to Hillโs estimates. ๐ท ๐ก oftotal deaths up to a certain date that appears to be integrated of order 2. One should also emphasize that, due to non-normality of the OLS and other standard estimatesof 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 statisticalanalysis and forecasting of key variables and time series related to the COVID-19 pandemic and itseffects such as the COVID-19 death rates should be based on stationary differences of time series withpotential unit root behavior as in the case of predictive regressions for financial returns in the nextchapter.
This section presents the main results of the note on statistically justified and robust evaluation ofthe effects of the COVID-19 pandemic on financial markets in different countries across the World. Wefocus on the analysis of predictive regressions of returns on major stock indices in the countries (seeSection 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 โ ๐ท ๐ก of daily COVID-19 related deaths in most of the countries discussed in the previous section, estimationof the predictive regressions is provided for regression models for stock index returns ๐ ๐ก with boththe lagged daily deaths โ ๐ท ๐ก โ and the (stationary) lagged changes in daily deaths โ ๐ท ๐ก โ used asregressors.More precisely, the estimation results are provided for predictive regressions in the form ๐ ๐ก = ๐ผ + ๐ฝ๐ ๐ก โ + ๐ ๐ก , (2)where ๐ ๐ก are the excess returns on major stock indices in the countries considered at the end of the day ๐ก given by the difference between the end of the day- ๐ก stock index returns and the countriesโ interestrates (see Section 3), and the regressors ๐ ๐ก โ are either the number โ ๐ท ๐ก โ of COVID-19 related deathson day ๐ก โ in the countries dealt with or the daily changes โ ๐ท ๐ก โ = โ ๐ท ๐ก โ โ โ ๐ท ๐ก โ in the deathsโtime series.In order to account for autocorrelation and heteroskedasticity in the regressors and the error terms inpredictive regressions (2) we use the widely applied HAC based ๐ก -statistic (with the quadratic spectral- QS - kernel and automatic choice of bandwidth as in Andrews, 1991) in the analysis of statisticalsignificance of the regression coefficients.It is well known, however, that commonly used HAC inference methods and related approachesbased on consistent standard errors often have poor finite sample properties, especially in the case of The conclusions on persistence properties of the time series ๐ท ๐ก and ฮ ๐ท ๐ก are somewhat similar to those for the CPIand the inflation rate (the change in the logarithm of the CPI) time series, where often unit root hypothesis is not rejectedfor the inflation rate and thus the (logarithm) of the CPI levels appears to be integrated of order 2 (see the analysis ofnon-stationarity in Section 14.6 in Stock and Watson, 2006, for the inflation rate and its changes in the US). Theseconclusions imply the necessity of the use of differences of the inflation rate in time series modeling of inflation and itsrelationship to other key economic variables such as the unemployment level in the Phillips curve (see Chs. 14 and 16 inStock and Watson, 2006). ๐ก โ statistic approaches to robust inference recently developedin Ibragimov and Mยจuller (2010, 2016). Following the approaches, robust large sample inference on aparameter of interest (e.g., a predictive regression coefficient ๐ฝ ) is conducted as follows: the data ispartitioned into a fixed number ๐ โฅ (e.g., ๐ = 2 , , ) of groups, the model is estimated for each group,and inference is based on a standard ๐ก โ test with the resulting ๐ parameter estimators.In the context of inference on the coefficient ๐ฝ in time series predictive regressions (2), the regressionis estimated for ๐ groups of time series observations with ( ๐ โ ๐ /๐ < ๐ก โค ๐๐ /๐, ๐ = 1 , ..., ๐, resultingin ๐ group estimates ห ๐ฝ ๐ , ๐ = 1 , ..., ๐. The robust test of a hypothesis on the parameter ๐ฝ is based onthe ๐ก โ statistic in the group OLS regression estimates ห ๐ฝ ๐ , ๐ = 1 , ..., ๐. E.g., the robust test of the nullhypothesis ๐ป : ๐ฝ = 0 against alternative ๐ป ๐ : ๐ฝ ฬธ = 0 is based on the ๐ก โ statistic ๐ก ๐ฝ = โ ๐ ^ ๐ฝ๐ ^ ๐ฝ , where ห ๐ฝ = ๐ โ โ๏ธ ๐๐ =1 ห ๐ฝ ๐ and ๐ ๐ฝ = ( ๐ โ โ โ๏ธ ๐๐ =1 ( ห ๐ฝ ๐ โ ห ๐ฝ ) . The above null hypothesis ๐ป is rejected in favor ofthe alternative ๐ป ๐ at level ๐ผ โค . (e.g., at the usual significance level ๐ผ = 5% ) if the absolute value | ๐ก ^ ๐ฝ | of the ๐ก โ statistic in group estimates ห ๐ฝ ๐ exceeds the (1 โ ๐ผ/ โ quantile of the standard Student- ๐ก distribution with ๐ โ degrees of freedom. The ๐ก โ statistic based approaches do not require at all estimation of limiting variances of estimatorsof interest. As discussed in Ibragimov et al. (2015), Ibragimov and Mยจuller (2010, 2016), they result inasymptotically valid inference under the assumption that the group estimators of a parameter of interestare asymptotically independent, unbiased and Gaussian of possibly different variances. The assumptionis satisfied in a wide range of econometric models and dependence, heterogeneity and heavy-tailednesssettings of a largely unknown type. The numerical analysis in Ibragimov et al. (2015), Ibragimov andMยจuller (2010, 2016) indicates favorable finite sample performance of the ๐ก โ statistic based robust in-ference approaches in inference on models with time series, panel, clustered and spatially correlateddata. Importantly, the ๐ก โ statistic based approaches to robust inference may also be used underconvergence of group estimators of a parameter interest to scale mixtures of normal distributions as in One-sided tests are conducted in a similar way. Justification of asymptotic validity of the robust ๐ก โ statistic inference approaches in Ibragimov and Mยจuller (2010,2016) is based on a small sample result in Bakirov and Szekely (2006) that implies validity of the standard ๐ก โ test underindependent heterogeneous observations and its analogues for two-sample ๐ก โ tests obtained in Ibragimov and Mยจuller(2016). See also Esarey and Menger (2019) for a detailed numerical analysis of finite sample performance of different inferenceprocedures, including ๐ก โ statistic approaches, under small number of clusters of dependent data and their software (STATAand R) implementation. The t- statistic robust inference approach proposed in Ibragimov and Mยจuller (2010) provides a formal justification forthe widespread FamaโMacBeth method for inference in panel regressions with heteroskedasticity (see Fama and MacBeth,1973). Following the method, one estimates the regression separately for each year, and then tests hypotheses about thecoefficient of interest using the t- statistic of the resulting yearly coefficient estimates. The FamaโMacBeth approach is aspecial case of the t- statistic based approach to inference, with observations of the same year collected in a group. See, among others, Bloom et al. (2013), Krueger et al. (2017), Blinder and Watson (2016), Verner and Gyongyosi(2018), Chen and Ibragimov (2019) and Gargano et al. (2019) for empirical applications of the robust ๐ก โ statistic inferenceapproaches in Ibragimov and Mยจuller (2010, 2016). Table B2 provides the results of the assessment of statistical significance of the coefficients ๐ฝ on thelagged time series โ ๐ท ๐ก โ of daily COVID-19 related deaths and their differences - the daily changesin the number of deaths from the disease - in predictive regressions (2) for the countries considered.More precisely, the table provides the values of HAC ๐ก โ statistic with the QS kernel and the automaticchoice of bandwidth discussed above as well as the values of the ๐ก โ statistic in estimates of the slopeparameter ๐ฝ obtained using ๐ = 4 , , and 16 groups of consecutive time series observations. Theasterisks in the table indicate statistical significance of the slope coefficient (*** for the significanceat 1% and * for significance at 10%) implied by formal comparisons of the HAC ๐ก โ statistics with thequantiles of a standard normal distribution. As described above, following the ๐ก โ statistic approachesto robust inference in Ibragimov and Mยจuller (2010, 2016), (the absence of) statistical significance of theslope coefficient ๐ฝ is assessed in the table using the comparisons of the ๐ก โ statistic in group estimatesof the coefficient with the quantiles of Student- ๐ก distributions with ๐ โ degrees of freedom.The values of HAC ๐ก โ statistics in Table B2 indicate an apparently spurious statistical significanceof the (potentially non-stationary) lagged daily deaths โ ๐ท ๐ก โ from COVID-19 in predictive regressionsfor 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 ๐ฝ in the regressions.However, the lagged daily COVID-19 death rates โ ๐ท ๐ก โ and their (stationary) differences โ ๐ท ๐ก โ appear not to be statistically significant in predictive regressions for stock index returns in all countriesconsidered according to the (econometrically justified) robust ๐ก โ statistic approaches. The absence ofstatistical significance of the coefficients on (stationary) daily changes โ ๐ท ๐ก โ in COVID-19 relateddeaths is further indicated by HAC ๐ก โ statistics for econometrically justified predictive regressions in-corporating โ ๐ท ๐ก โ for major indices in essentially all countries considered. This note presented the results of theoretically justified and robust statistical analysis of the effectsof the COVID-19 pandemic on financial markets in different countries across the World. The analysis isbased on robust inference in predictive regressions for the returns on the countriesโ major stock indicesincorporating the time series characterizing the dynamics in the COVID-19 related deaths rates. See Section 3.3.3 in Ibragimov et al. (2015) for applications of the robust ๐ก โ statistic approaches in inference ininfinite variance heavy-tailed models. The recent works by Anatolyev (2019), Pedersen (2019) and Ibragimov, Pedersenand Skrobotov (2020) provide further applications of the approaches in robust inference on general classes of GARCH andAR-GARCH-type models exhibiting heavy-tailedness and volatility clustering properties typical for real-world financialand economic markets. The recent paper by Ibragimov, Kim and Skrobotov (2020) focuses on applications of the ๐ก โ statisticapproaches in inference on predictive regressions with persistent and/or fat-tailed regressors and errors. The formal comparison of the HAC ๐ก โ statistic with quantiles of a standard normal distribution points to somestatistical significance of the slope coefficient in predictive regressions based on ฮ ๐ท ๐ก โ as a regressor for returns onfinancial indices in Lithuania. ๐ก โ statistic approaches in addition to HAC tests, the lagged daily COVID-19 deathrates and their (stationary) differences appear to be statistically insignificant in predictive regressionsfor 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 meth-ods accounting for autocorrelation, heterogeneity and heavy-tailedness in statistical and econometricanalysis and forecasting of key time series and variables related to the COVID-19 pandemic and its ef-fects on economic and financial markets and society. They further emphasize the importance of the useof correctly specified models of the COVID-19 pandemic and its effects incorporating stationary timeseries and variables such as the daily changes in COVID-19 related deaths used in predictive regressionsin this work.Further research may focus on robust analysis of the dynamics of a range of key time series relatedto the COVID-19 pandemic, including infection rates; robust tests of structural breaks in models ofthe dynamics of the pandemic and its effects on financial and economic markets, and applications ofinference methods such as sign- and rank-based tests that are robust to relatively small sample sizesof observations in statistical analysis of key models related to the spread of COVID-19. 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Working Paper, MIT Sloan School of Management .15 ppendix A Figures i g u r e A : H illโ s t a ili nd e x e s t i m a t e s f o r d a il y C O V I D - d e a t h s H ill ' s p l o t A r g e n t i n a k Tail index estimate with confidence intervals 01234567891011 ( a ) H ill ' s p l o t A u s t r a li a k Tail index estimate with confidence intervals 01234567 ( b ) H ill ' s p l o t A u s t r i a k Tail index estimate with confidence intervals 01234567891011121314 ( c ) H ill ' s p l o t B r a z il k Tail index estimate with confidence intervals 024681013161922252831 ( d ) H ill ' s p l o t C a n a d a k Tail index estimate with confidence intervals 0123456789101112 ( e ) H ill ' s p l o t C h i n a k Tail index estimate with confidence intervals 0123 ( f ) i g u r e A : H illโ s t a ili nd e x e s t i m a t e s f o r d a il y C O V I D - d e a t h s ( c t d ) H ill ' s p l o t F i n l a nd k Tail index estimate with confidence intervals 01234 ( g ) H ill ' s p l o t F r a n ce k Tail index estimate with confidence intervals 012345 ( h ) H ill ' s p l o t I nd i a k Tail index estimate with confidence intervals 01234567 ( i ) H ill ' s p l o t I ndon es i a k Tail index estimate with confidence intervals 0123456789101112 ( j ) H ill ' s p l o t I r e l a nd k Tail index estimate with confidence intervals 012345 ( k ) H ill ' s p l o t I t a l y k Tail index estimate with confidence intervals 012345678911131517 ( l ) i g u r e A : H illโ s t a ili nd e x e s t i m a t e s f o r d a il y C O V I D - d e a t h s ( c t d ) H ill ' s p l o t Ja p a n k Tail index estimate with confidence intervals 0123456 ( m ) H ill ' s p l o t L i t hu a n i a k Tail index estimate with confidence intervals 012345678 ( n ) H ill ' s p l o t M ex i c o k Tail index estimate with confidence intervals 01234567891011 ( o ) H ill ' s p l o t N e t h e r l a nd s k Tail index estimate with confidence intervals 01234567891011 ( p ) H ill ' s p l o t R u ss i a k Tail index estimate with confidence intervals 024681012141618202224 ( q ) H ill ' s p l o t S ou t h . K o r ea k Tail index estimate with confidence intervals 012345678910 ( r ) i g u r e A : H illโ s t a ili nd e x e s t i m a t e s f o r d a il y C O V I D - d e a t h s ( c t d ) H ill ' s p l o t S p a i n k Tail index estimate with confidence intervals 01234567891011 ( s ) H ill ' s p l o t S w e d e n k Tail index estimate with confidence intervals 012345678 (t) H ill ' s p l o t U n i t e d . K i ngdo m k Tail index estimate with confidence intervals 0123456789101112 ( u ) H ill ' s p l o t U S k Tail index estimate with confidence intervals 024681114172023262932 ( v ) i g u r e A : L og - l og r a n k s i ze r e g r e ss i o n t a ili nd e x e s t i m a t e s Log โ l og r a n k โ s i z e p l o t A r g e n t i n a k Tail index estimate with confidence intervals 0123456789101112 ( a ) Log โ l og r a n k โ s i z e p l o t A u s t r a li a k Tail index estimate with confidence intervals 01234567 ( b ) Log โ l og r a n k โ s i z e p l o t A u s t r i a k Tail index estimate with confidence intervals 01234567891011121314 ( c ) Log โ l og r a n k โ s i z e p l o t B r a z il k Tail index estimate with confidence intervals 0246810131619222528 ( d ) Log โ l og r a n k โ s i z e p l o t C a n a d a k Tail index estimate with confidence intervals 012345678910111213 ( e ) Log โ l og r a n k โ s i z e p l o t C h i n a k Tail index estimate with confidence intervals 012 ( f ) i g u r e A : L og - l og r a n k s i ze r e g r e ss i o n t a ili nd e x e s t i m a t e s f o r d a il y C O V I D - d e a t h s ( c t d ) Log โ l og r a n k โ s i z e p l o t F i n l a nd k Tail index estimate with confidence intervals 012 ( g ) Log โ l og r a n k โ s i z e p l o t F r a n ce k Tail index estimate with confidence intervals 01234567 ( h ) Log โ l og r a n k โ s i z e p l o t I nd i a k Tail index estimate with confidence intervals 0123 ( i ) Log โ l og r a n k โ s i z e p l o t I ndon es i a k Tail index estimate with confidence intervals 012345678910121416 ( j ) Log โ l og r a n k โ s i z e p l o t I r e l a nd k Tail index estimate with confidence intervals 0123 ( k ) Log โ l og r a n k โ s i z e p l o t I t a l y k Tail index estimate with confidence intervals 02468101214161820 ( l ) i g u r e A : L og - l og r a n k s i ze r e g r e ss i o n t a ili nd e x e s t i m a t e s ( c t d ) Log โ l og r a n k โ s i z e p l o t Ja p a n k Tail index estimate with confidence intervals 012345 ( m ) Log โ l og r a n k โ s i z e p l o t L i t hu a n i a k Tail index estimate with confidence intervals 01234 ( n ) Log โ l og r a n k โ s i z e p l o t M ex i c o k Tail index estimate with confidence intervals 012345678910 ( o ) Log โ l og r a n k โ s i z e p l o t N e t h e r l a nd s k Tail index estimate with confidence intervals 01234567891011 ( p ) Log โ l og r a n k โ s i z e p l o t R u ss i a k Tail index estimate with confidence intervals 01234567891113151719 ( q ) Log โ l og r a n k โ s i z e p l o t S ou t h . K o r ea k Tail index estimate with confidence intervals 01234567891011 ( r ) i g u r e A : L og - l og r a n k s i ze r e g r e ss i o n t a ili nd e x e s t i m a t e s f o r d a il y C O V I D - d e a t h s ( c t d ) Log โ l og r a n k โ s i z e p l o t S p a i n k Tail index estimate with confidence intervals 012345678910111213 ( s ) Log โ l og r a n k โ s i z e p l o t S w e d e n k Tail index estimate with confidence intervals 0123456789 (t)
Log โ l og r a n k โ s i z e p l o t U n i t e d . K i ngdo m k Tail index estimate with confidence intervals 0246810121416182022 ( u ) Log โ l og r a n k โ s i z e p l o t U S k Tail index estimate with confidence intervals 0246811141720232629323538 ( v ) i g u r e A : H illโ s t a ili nd e x e s t i m a t e s f o r p o s i t i v ec h a n g e s i nd a il y C O V I D - d e a t h s H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s A r g e n t i n a k Tail index estimate with confidence intervals 012345 ( a ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s A u s t r a li a k Tail index estimate with confidence intervals 01234 ( b ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s A u s t r i a k Tail index estimate with confidence intervals 012345 ( c ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s B r a z il k Tail index estimate with confidence intervals 012345678 ( d ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s C a n a d a k Tail index estimate with confidence intervals 01234 ( e ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s C h i n a k Tail index estimate with confidence intervals 01 ( f ) i g u r e A : H illโ s t a ili nd e x e s t i m a t e s f o r p o s i t i v ec h a n g e s i nd a il y C O V I D - d e a t h s ( c t d ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s F i n l a nd k Tail index estimate with confidence intervals 0123 ( g ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s F r a n ce k Tail index estimate with confidence intervals 01234 ( h ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s I nd i a k Tail index estimate with confidence intervals 012 ( i ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s I ndon es i a k Tail index estimate with confidence intervals 01234567 ( j ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s I r e l a nd k Tail index estimate with confidence intervals 012 ( k ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s I t a l y k Tail index estimate with confidence intervals 0123456789101112 ( l ) i g u r e A : H illโ s t a ili nd e x e s t i m a t e s f o r p o s i t i v ec h a n g e s i nd a il y C O V I D - d e a t h s ( c t d ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h sJa p a n k Tail index estimate with confidence intervals 0123456 ( m ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s L i t hu a n i a k Tail index estimate with confidence intervals 0123456 ( n ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s M ex i c o k Tail index estimate with confidence intervals 0123 ( o ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s N e t h e r l a nd s k Tail index estimate with confidence intervals 0123456 ( p ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s R u ss i a k Tail index estimate with confidence intervals 01234 ( q ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s S ou t h . K o r ea k Tail index estimate with confidence intervals 012345678 ( r ) i g u r e A : H illโ s t a ili nd e x e s t i m a t e s f o r p o s i t i v ec h a n g e s i nd a il y C O V I D - d e a t h s ( c t d ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s S p a i n k Tail index estimate with confidence intervals 012 ( s ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s S w e d e n k Tail index estimate with confidence intervals 012345678 (t) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s U n i t e d . K i ngdo m k Tail index estimate with confidence intervals 012345678910 ( u ) H ill ' s p l o t f o r po s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s U S k Tail index estimate with confidence intervals 0123456789 ( v ) i g u r e A : L og - l og r a n k - s i ze r e g r e ss i o n t a ili nd e x e s t i m a t e s f o r p o s i t i v ec h a n g e s i nd a il y C O V I D - d e a t h s Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s A r g e n t i n a k Tail index estimate with confidence intervals 01234 ( a ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s A u s t r a li a k Tail index estimate with confidence intervals 01234 ( b ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s A u s t r i a k Tail index estimate with confidence intervals 01234 ( c ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s B r a z il k Tail index estimate with confidence intervals 012345 ( d ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s C a n a d a k Tail index estimate with confidence intervals 0123456 ( e ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s C h i n a k Tail index estimate with confidence intervals 01 ( f ) i g u r e A : L og - l og r a n k - s i ze r e g r e ss i o n t a ili nd e x e s t i m a t e s f o r p o s i t i v ec h a n g e s i nd a il y C O V I D - d e a t h s ( c t d ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s F i n l a nd k Tail index estimate with confidence intervals 012 ( g ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s F r a n ce k Tail index estimate with confidence intervals 0123456 ( h ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s I nd i a k Tail index estimate with confidence intervals 01 ( i ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s I ndon es i a k Tail index estimate with confidence intervals 012345678 ( j ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s I r e l a nd k Tail index estimate with confidence intervals 01 ( k ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s I t a l y k Tail index estimate with confidence intervals 0123456789101214 ( l ) i g u r e A : L og - l og r a n k - s i ze r e g r e ss i o n i nd e x e s t i m a t e s f o r p o s i t i v ec h a n g e s i nd a il y C O V I D - d e a t h s ( c t d ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h sJa p a n k Tail index estimate with confidence intervals 0123456789 ( m ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s L i t hu a n i a k Tail index estimate with confidence intervals 01234 ( n ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s M ex i c o k Tail index estimate with confidence intervals 0123 ( o ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s N e t h e r l a nd s k Tail index estimate with confidence intervals 01234567 ( p ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s A r g e n t i n a k Tail index estimate with confidence intervals 01234 ( q ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s S ou t h . K o r ea k Tail index estimate with confidence intervals 012345678 ( r ) i g u r e A : L og - l og r a n k - s i ze r e g r e ss i o n t a ili nd e x e s t i m a t e s f o r p o s i t i v ec h a n g e s i nd a il y C O V I D - d e a t h s ( c t d ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s S p a i n k Tail index estimate with confidence intervals 012 ( s ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s S w e d e n k Tail index estimate with confidence intervals 012345678 (t)
Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s U n i t e d . K i ngdo m k Tail index estimate with confidence intervals 01234567891011 ( u ) Log โ l og r a n k โ s i z e p l o t: P o s i t i ve d a il y c h a ng es i n t h e nu m b e r o f d ea t h s U S k Tail index estimate with confidence intervals 012345 ( v ) ppendix B Tables Table B1: Wild bootstrap quasi-differenced unit root tests based on Rademacher distribution with sievebased recolouring (p-values in brackets) ฮ ๐ท๐๐๐กโ๐ ฮ ๐ท๐๐๐กโ๐ ๐ฟ๐ ๐๐ ๐ผ ๐๐๐ต ๐๐ ๐ก ๐๐ ๐ก ๐ด๐ท๐น ๐ฟ๐ ๐๐ ๐ผ ๐๐๐ต ๐๐ ๐ก ๐๐ ๐ก ๐ด๐ท๐น
UK 0.43 -2.48 0.45 -1.11 9.84 -1.13 35.50 -54.03 0.10 -5.20 0.45 -13.78(0.35) (0.39) (0.47) (0.34) (0.41) (0.35) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Germany 0.66 -3.07 0.40 -1.24 7.97 -1.31 41.67 -41.92 0.11 -4.58 0.59 -18.37(0.31) (0.31) (0.34) (0.29) (0.31) (0.31) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)France 0.61 -2.93 0.41 -1.21 8.37 -1.22 47.29 -56.07 0.09 -5.29 0.44 -17.99(0.29) (0.33) (0.38) (0.29) (0.34) (0.31) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Italy 0.28 -1.70 0.54 -0.91 14.35 -0.90 41.50 -58.81 0.09 -5.42 0.42 -15.43(0.37) (0.46) (0.56) (0.42) (0.52) (0.43) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Spain 0.94 -3.65 0.37 -1.35 6.72 -1.48 45.23 -41.38 0.11 -4.55 0.59 -20.06(0.17) (0.16) (0.27) (0.12) (0.15) (0.16) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Russia 0.01 -1.09 0.61 -0.67 19.70 -0.51 37.85 -38.34 0.11 -4.38 0.64 -17.45(0.67) (0.68) (0.92) (0.62) (0.81) (0.64) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Netherland 0.48 -2.54 0.44 -1.12 9.63 -1.13 39.61 -48.89 0.10 -4.94 0.50 -16.13(0.33) (0.38) (0.45) (0.34) (0.4) (0.35) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Sweden 3.09 -7.39 0.26 -1.92 3.33 -2.08 33.71 -51.87 0.10 -5.08 0.50 -13.35(0.02) (0.08) (0.1) (0.07) (0.07) (0.06) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)India 0.30 -2.12 0.39 -0.83 9.97 -1.01 42.67 -37.38 0.12 -4.32 0.66 -19.92(0.42) (0.41) (0.58) (0.38) (0.42) (0.39) (0.00) (0.08) (0.10) (0.07) (0.05) (0.00)Austria 0.84 -3.91 0.36 -1.40 6.26 -1.42 36.33 -47.99 0.10 -4.90 0.51 -14.97(0.2) (0.21) (0.26) (0.18) (0.2) (0.2) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Finland 2.30 -7.89 0.25 -1.98 3.11 -2.18 40.95 -28.59 0.13 -3.78 0.86 -21.88(0.03) (0.05) (0.06) (0.05) (0.05) (0.04) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Ireland 1.38 -4.26 0.34 -1.46 5.76 -1.71 39.94 -43.08 0.11 -4.64 0.57 -17.37(0.23) (0.24) (0.25) (0.23) (0.24) (0.23) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)US 0.21 -1.72 0.54 -0.93 14.22 -0.91 36.45 -57.71 0.09 -5.37 0.43 -13.68(0.51) (0.59) (0.74) (0.51) (0.67) (0.48) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Lithuania 3.01 -3.54 0.37 -1.33 6.91 -1.95 35.50 -40.66 0.11 -4.51 0.60 -15.88(0.04) (0.18) (0.22) (0.16) (0.18) (0.1) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Canada 0.20 -1.26 0.62 -0.78 19.10 -0.84 40.84 -43.44 0.11 -4.64 0.62 -17.63(0.44) (0.52) (0.72) (0.45) (0.61) (0.47) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Brazil 0.07 -1.71 0.49 -0.84 13.02 -0.76 34.41 -46.08 0.10 -4.79 0.55 -14.47(0.58) (0.54) (0.77) (0.49) (0.62) (0.52) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Mexico 0.40 -3.82 0.33 -1.27 6.51 -1.22 32.94 -46.13 0.10 -4.77 0.61 -13.81(0.46) (0.42) (0.51) (0.41) (0.41) (0.41) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Argentina 0.00 5.94 0.63 3.73 59.07 1.23 36.81 -49.86 0.10 -4.77 1.05 -14.53(0.69) (1) (0.89) (0.99) (0.96) (0.91) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Japan 0.92 -3.47 0.38 -1.31 7.05 -1.45 53.31 -46.40 0.10 -4.82 0.53 -22.30(0.29) (0.28) (0.35) (0.25) (0.28) (0.28) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)China 7.79 -28.14 0.13 -3.75 0.87 -4.41(0.00) (0.00) (0.00) (0.00) (0.00) (0.00)South Korea 1.11 -3.40 0.38 -1.28 7.21 -1.50 55.79 -39.91 0.11 -4.47 0.62 -23.58(0.18) (0.22) (0.23) (0.21) (0.22) (0.2) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Indonesia 0.00 0.01 0.54 0.01 21.58 -0.27 40.44 -43.40 0.11 -4.64 0.61 -17.16(0.67) (0.77) (0.6) (0.76) (0.67) (0.69) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)Australia 1.72 -5.31 0.31 -1.63 4.62 -1.84 44.28 -45.79 0.10 -4.78 0.54 -18.64(0.14) (0.17) (0.18) (0.16) (0.16) (0.15) (0.00) (0.00) (0.00) (0.00) (0.00) (0.00) able B2: Predictive regression tests ฮ ๐ท๐๐๐กโ๐ ฮ ๐ท๐๐๐กโ๐
T q=4 q=8 q=12 q=16 HAC T q=4 q=8 q=12 q=16 HACUK FTSE 100 75 0.90 1.00 1.36 1.13 1.34 74 0.84 1.01 1.38 1.08 -1.01Germany DAX 76 0.05 0.46 0.85 -0.97 1.02 75 0.26 0.95 -0.05 0.73 0.62France CAC 40 90 1.34 1.05 -1.46 -0.37 0.25 89 0.93 1.09 -0.44 1.53 -1.30Italy FTSE MIB 86 1.83 -0.84 1.44 -1.44 0.83 85 2.00 0.43 0.95 1.18 0.33Spain IBEX 35 79 0.14 1.07 0.52 0.41 0.76 78 0.37 1.13 -0.65 -0.17 0.80Russia MOEX 66 0.32 -1.29 0.13 -0.57 2.54 ***
65 -0.87 0.52 0.77 0.74 -0.25Netherland AEX 76 -0.71 0.34 0.44 -1.27 0.30 75 -0.47 -1.43 0.85 -1.56 -0.34Sweden OMXS 30 71 -0.34 0.91 1.28 1.48 -0.42 70 0.63 1.04 1.46 1.61 -0.63India SENSEX 70 1.07 1.32 -0.95 -0.14 2.13 ***
69 1.13 -0.82 0.92 -0.86 0.11Austria ATX 71 -0.22 1.37 1.02 1.03 -0.42 70 -0.86 1.61 1.51 0.89 -0.04Finland OMX Helsinki 25 63 0.37 1.21 2.02 -0.17 0.16 62 0.89 1.58 1.38 0.82 -0.27Ireland ISEQ 73 -0.50 0.94 0.83 1.11 -0.13 72 0.79 1.12 1.00 1.36 -0.99US Dow Jones 81 0.99 1.00 1.15 1.28 2.55 ***
80 1.00 1.00 1.43 1.49 -0.05US S&P 500 81 0.99 1.00 1.21 1.29 2.54 ***
80 0.99 1.00 1.43 1.48 -0.02Lithuania OMX Vilnius 66 0.63 0.62 1.28 1.50 2.26 ***
65 0.45 0.62 1.59 -0.21 1.95 * Canada TSX 76 1.00 1.16 1.09 1.28 2.80 ***
75 1.00 1.32 1.18 1.23 -0.10Brazil iBovespa 68 0.05 0.91 0.67 -0.75 3.56 ***
67 -1.74 -1.73 0.66 -0.35 -1.38Mexico IPC 67 1.50 1.44 -0.88 1.05 4.16 ***
66 -0.95 -0.46 -1.18 -1.12 -0.18Argentina Merval 71 1.09 1.34 -0.85 0.23 0.52 70 1.62 -0.91 -0.92 -1.17 1.26Japan NIKKEI 225 89 0.86 -0.22 0.12 -0.82 2.81 ***
88 1.04 -0.74 -1.24 0.46 0.22China SHANGHAI 99 1.01 0.54 0.65 0.64 -0.75 98 1.05 1.29 1.02 1.33 0.07South KOSPI 86 2.56 -0.99 1.18 0.29 -1.30 85 0.81 1.12 0.14 -0.04 0.05Indonesia JCI 68 1.16 -0.42 -0.19 -1.20 0.89 67 0.74 -0.65 -0.39 -0.82 0.79Australia ASX 50 80 0.42 0.17 0.93 -0.83 0.95 79 -0.71 0.39 -0.60 -0.67 -1.28Australia ASX 200 80 0.38 0.12 0.89 -0.88 0.98 79 -0.87 0.35 -0.64 -0.77 -1.32Australian All 80 0.38 0.12 0.91 -0.88 1.01 79 -0.90 0.35 -0.65 -0.80 -1.3188 1.04 -0.74 -1.24 0.46 0.22China SHANGHAI 99 1.01 0.54 0.65 0.64 -0.75 98 1.05 1.29 1.02 1.33 0.07South KOSPI 86 2.56 -0.99 1.18 0.29 -1.30 85 0.81 1.12 0.14 -0.04 0.05Indonesia JCI 68 1.16 -0.42 -0.19 -1.20 0.89 67 0.74 -0.65 -0.39 -0.82 0.79Australia ASX 50 80 0.42 0.17 0.93 -0.83 0.95 79 -0.71 0.39 -0.60 -0.67 -1.28Australia ASX 200 80 0.38 0.12 0.89 -0.88 0.98 79 -0.87 0.35 -0.64 -0.77 -1.32Australian All 80 0.38 0.12 0.91 -0.88 1.01 79 -0.90 0.35 -0.65 -0.80 -1.31