key: cord-0747881-mbdnmeke authors: Castillo, Brenda; León, Ángel; Ñíguez, Trino-Manuel title: Backtesting VaR under the COVID-19 sudden changes in volatility date: 2021-03-18 journal: Financ Res Lett DOI: 10.1016/j.frl.2021.102024 sha: edeae04d341ff506e1a70adffcaf9920bb31c11f doc_id: 747881 cord_uid: mbdnmeke We analyze the impact of the COVID-19 pandemic on the conditional variance of stock returns. We look at this effect from a global perspective, so we employ series of major stock market and sector indices. We use the Hansen’s Skewed-t distribution with EGARCH extended to control for sudden changes in volatility. We oversee the COVID-19 effect on measures of downside risk such as the Value-at-Risk. Our results show that there is a significant sudden shift up in the return distribution variance post the announcement of the pandemic, which must be explained properly to obtain reliable measures for financial risk management. The COVID-19 outbreak has caused the most severe economic crisis since The Great Depression (Caggiano et al., 2020) . In particular, global financial markets have experienced extensive and massive uncertainty with volatility at record levels in recent history (Baker et al., 2020; Shehzad et al., 2020; Goodell, 2020; Zhang et al., 2020, and Baig et al., 2021) . New research has shown that connectedness in financial markets has increased during this period of due-to-COVID-19 uncertainty; see, among others, (Bouri et al., 2020a; 2020b) ; (Izzeldin et al., 2021; Bouri et al., 2021; Gupta et al., 2021) ; and (Shahzad et al., 2021) . In this paper we contribute to this literature with an investigation on the impact of the COVID-19 crisis on the time-varying variances of major stock market and sector index returns. We follow the methodology based on GARCH models with shift dummy variables of Lamoureux and Lastrapes (1990) ; Aggarwal et al. (1999) and (Mikosch and Stȃricȃ, 2004) ; see also Malik et al. (2005) ; Kang et al. (2009) ; Ewing and Malik (2017) and Anjum and Malik (2020) for recent empirical studies. For this purpose we use the exponential GARCH (EGARCH) model of Nelson (1991) augmented with a sudden shift dummy variable to incorporate the COVID-19 effect on volatility. For the skewed and heavy-tailed distribution of the standardized returns, we employ the popular Skewed-t (ST) of Hansen (1994) . Hereafter, this model is referred to as EGARCH-D-ST. In our empirical exercise, we show evidence that incorporating the COVID-19 abrupt shift has an important impact on the accuracy of estimating volatility dynamics and forecasting future Value-at-Risk (VaR). In line with previous results, we also find clear evidence on that accounting for the sudden change reduces the persistence in the EGARCH model. The performance of the previous model is compared with that of the model without the dummy variable through the unconditional backtesting procedure of Kupiec (1995) for the pandemic period. Since the asymptotic distribution of the Kupiec's backtesting test is not adequate for our small sample size, we have obtained Monte Carlo p-values according to Christoffersen (2011) . The remainder of the paper is organized as follows. In Section 2 we present the EGARCH-D-ST model for asset returns. Section 3 provides an empirical application to forecast the VaR of major stock and sector returns with a backtesting analysis. Section 4 gathers the conclusions. Let the asset return r t be a process characterized by the sequence of conditional densities f(r t |I t− 1 ; ψ), where I t− 1 denotes the information set available prior to the realization of r t , ψ = (θ, ξ) is the vector of unknown parameters such that θ is the subset characterizing both the conditional mean and variance of r t , i.e. μ t (θ) = μ(I t− 1 ; θ) and σ 2 t (θ) = σ 2 (I t− 1 ; θ), and finally, ξ is the subset characterizing the shape of the distribution of the innovations, z t . Thus, we assume that So, Eq. (1) decomposes the return at time t into a conditional mean which is assumed to be constant, μ t = μ, and the term ε t defined as the product between the conditional standard deviation, σ t , and the innovation (or standardized return), z t , with zero mean and unit variance. It is assumed that {z t } is a sequence of independent identically distributed (iid) random variables driven by the ST distribution with parameter set ξ = (λ, υ) where λ ∈ (− 1, 1) and υ > 2 control, respectively, for skewness and kurtosis, and denoted as where D t = 1 if the return observation belongs to after the 31th of December 2019 as the starting date of the COVID-19 period when the first case was reported to the World Health Organization (2020) by China. We analyze the time-series behavior of 17 major stock market and 27 world sector indices. The data employed were daily percentage log returns, which were computed as r t = 100log(P t /P t− 1 ) from daily closing prices (in $). The time period for {r t } T t=1 series used comprises from January 2, 2017 to May 25, 2020, for a total number of T = 886 observations. Table 1 provides the list of the series. All data series were downloaded from Datastream. The world sector indices data are supplied by Morgan Stanley Capital International (MSCI) Barra. The MSCI world sector indices capture the large and mid-cap companies across 23 developed markets countries around the world. All securities in each index are classified in the corresponding sector as per the Global Industry Classification Standard. The stock market indices analyzed are selected to represent major stock markets across the world. Table 1 also reports the standard deviations of daily returns before and after December 31, 2019. These statistics confirm that the pandemic has had a great influence on the stock markets and as a result, an increase in the volatility in all cases. This evidence suggests a possible This table presents the names and sample standard deviations of the stock market and sector indices used in the empirical analysis of this article. Both s b and s a denote the sample standard deviations of the series before and after 31/12/2019, respectively. structural change in the unconditional volatility that should be considered in modeling the conditional variance in the spirit of Lamoureux and Lastrapes (1990) . The parameters of our EGARCH-D-ST model were estimated using maximum likelihood (ML). Table 2 presents the estimation results. The unconditional mean parameter, μ, is not significant for many return series. The parameter estimates of the conditional variance Eq. (2) show that, for all series, the model correctly captures the asset returns stylized features of (i) clustering and high persistence in volatility, and (ii) asymmetric response of volatility to positive and negative shocks. Indeed, the parameter β, which is related to the persistence for the EGARCH, is rather high for all series with mean estimates of 0.944 and 0.958 for stock market and world sector indices, respectively. Also, asymmetric response, γ, is significant for all series. The ST asymmetry parameter, λ, is significant for 13 out the 17 stock market indices, and 19 out of the 27 world sector indices. So, there is evidence for asymmetry for most standardized returns series. Note also that the ST degrees of freedom parameter, υ, estimates indicate that the cross-sectional means for the stock market and sector indices exhibit kurtosis levels of 6.2 and 7.1, which are far away from the Normal distribution (i.e., large value of υ). In short, the previous results suggest that the standardized returns are not normally distributed. The dummy parameter, denoted as δ, is significant for 15 and 25 stock market and world sector indices, respectively, indicating an important due-to-COVID sudden change in volatility across these indices. In order to analyse more precisely when the shift in volatility starts to become relevant, we estimate our model for four different subsamples across the whole sample period. The results, presented in Table 2 (Panel 2), indicate that for most of the series the sudden-change dummy variable effect kicks in March 2020 as δ becomes statistically significant. The sector index volatilities that reacted faster to the COVID shock seem to be those related to online activity. Namely, IT technology, IT services and Software series showed the dummy parameter is significant already in January 2020. As regards the stock market indices, predominantly those from less developed markets picked up the shock faster. The dummy parameter is significant in February 2020 for the Chinese, Mexican and Russian series, although it is also for NASDAQ. From our results we can infer high connectedness among the majority of sector and stock index volatilities in the response to the COVID crisis, as most series showed a shift in volatility in March 2020. We find that the magnitudes of the dummy coefficients become larger as well as significant at lower levels as we move through the out-of-sample (OOS) period from February 3, 2020 to May 25, 2020. Fig. 1 shows the plots of coefficient estimates and t-statistics over the OOS period for NASDAQ and Banks return series as representative examples. 1 For the OOS analysis, we are interested in the VaR-backtesting performance comparison between the EGARCH-D-ST model, which does consider the sudden change in volatility due to the COVID-19 effect, and the EGARCH-ST model which is nested in the former when δ = 0 in (2) and does not account for the previous effect. The backtest implementation involves the first T -N observations for the first in-sample window and the OOS period of length N = 81 from February 3, 2020 to May 25, 2020, using a constant-sized rolling window. For every window we estimate the model parameters by ML and obtain a one-day-ahead forecast of the conditional variance, σ 2 t+1 . We have done this for all return series presented above under several coverage levels (denoted as α): 1%, 2.5%, 5%. The one-day-ahead VaR for the α-quantile is given by VaR t+1 (α) = μ +σ t+1 F − 1 z (α; ξ) where F − 1 z (α; ξ) represents the α-quantile of the ST(ξ) distribution for the random variable z t obtained through the inverse of its cumulative distribution function (cdf), and denoted as F z (⋅; ξ). Let denote the violation or hit variable. We obtain the quadratic loss function, which incorporates the exception magnitude and provides useful information to discriminate among similar models in terms of the unconditional coverage criterion. Thus, We estimate the sample averages for the daily estimations of (3) and (4) corresponding to the daily violations in (3) and the daily quadratic losses in (4) for the OOS period of N = 81 days. The probability P(r t+1 < VaR t+1 (α)|I t ) = α suggests that violations are Bernoulli variables with mean α. The null hypothesis for the unconditional backtest, H 0 : E[h t+1 (α)] = α, corresponds to the following likelihood ratio (LR) test statistic initially proposed by Kupiec (1995) : where L(α) is the likelihood of an i.i.d. Bernoulli (α) hit sequence, i.e. L(α) = (1 − α) N0 α N1 such that N 0 and N 1 are the number of zeroes and ones (or hits) in the sample, and π = N 1 /N is the sample average of the hit sequence in (3) for the whole OOS period such that ĥ t+1 (α) = 1(r t+1