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Abstract This paper revisits several existing volatility models by the light of extremal dependence, that is, serial dependence in extreme returns. First, we investigate the extremal properties of different high-frequency-based volatility processes and show that only a subset of them can generate dependence in the extremes. Second, we corroborate the empirical evidence on extremal dependence in financial returns, showing that extreme returns present strong and persistent correlation and that extreme negative returns are much more correlated than positive ones. Finally, a large empirical analysis suggests that only models exhibiting extremal dependence and endowed with a leverage component can appropriately explain extreme events. A large body of the financial economics literature has been devoted to the study of the autocorrelation of asset returns, particularly for its relevance in connection with the Efficient Market Hypothesis (Conrad and Kaul, 1988; Lo and MacKinlay, 1988). The empirical evidence suggests that daily returns present a mild and often not significant positive autocorrelation that varies according to other variables, such as the volume of trades (Campbell, Grossman, and Wang, 1993) and the volatility of asset returns (LeBaron, 1992). In contrast, the volatility of asset returns is a highly predictable quantity (Engle, 1982) which exhibits a strong degree of persistence. While the serial correlation of stock returns and volatility has attracted a lot of attention, serial dependence in extreme returns has been unexpectedly overlooked. Understanding the behavior of extreme observations is crucial as they are informative of tail risk. Are daily extreme observations in financial returns isolated events or do they tend to cluster? Does the occurrence of an extreme event increase the probability of observing a subsequent event of similar magnitude? Are financial econometrics models capable of accurately capturing the observed dependence in the extremes of asset returns? These questions are of interest for both regulatory purposes and financial risk management, and this paper attempts to provide some answers. The left panel of Figure 1 reports the price path recorded by the S&P500 index in the last part of 2008, in the aftermath of the Lehman Brothers collapse. Noted in red are the days from September 26 to October 10 on which the index plummeted from 1200 points to 900, losing almost one fourth of its capitalization. The figure emphasizes that this crash was not due to one single observation but to a sequence of extreme negative observations. In particular, 5 out of 10 days experienced a daily return that was three standard deviations below the mean. The right panel of Figure 1 reports a similar scenario for the Euro Stoxx 50, an index of Eurozone blue-chips stocks, during the European sovereign debt crisis in the fall of 2011. In this case, the index lost 20% of its value in 2 weeks, and 5 out of 12 days registered negative values that were two standard deviations below the mean. These two examples warn that extreme returns may be dependent and it is thus important to investigate this behavior and understand which models are eventually able to capture it appropriately. Figure 1 View largeDownload slide Stock market crashes. The left panel shows the S&P500 (SP500), while the right panel shows the EuroStoxx50 (ESX). Figure 1 View largeDownload slide Stock market crashes. The left panel shows the S&P500 (SP500), while the right panel shows the EuroStoxx50 (ESX). A definition of extremal dependence can be obtained from multivariate extreme value theory. The latter examines the joint limiting behavior of the largest (or smallest) observations in a sample (Sibuya, 1959). Let {Xt} be a stationary sequence and denote the conditional tail probability of the random vector (Xt,Xt+h) at level x as c(x,h)=Pr(Xt+h>x|Xt>x), h∈N, (1) then (Xt,Xt+h) are asymptotically independent if c(x,h)→0 as x→∞, while they are asymptotically dependent, and thus present extremal dependence, if c(x,h)→C>0 as x→∞. The extremal properties of traditional classes of models for the daily returns, such as the GARCH of Bollerslev (1986) and the stochastic volatility (SV) pioneered by Clark (1973), have been widely studied in Mikosch and Starica (2000), Davis and Mikosch (2009a), and Mikosch and Rezapur (2013). These authors show that GARCH processes exhibit extremal dependence while SV processes present extremal dependence only if the volatility component satisfies certain conditions (Mikosch and Rezapur, 2013). With the availability of HF data, new classes of volatility models are quickly becoming the new standard. These models improve the accuracy of volatility forecasts exploiting the so-called realized measures, non-parametric estimates of the daily asset price variation obtained from intra-daily returns (Barndorff-Nielsen and Shephard, 2002). Hansen and Lunde (2011) classify the existing approaches in two classes. The reduced-form class provides time-series models for the realized measures. Within this class, the heterogeneous autoregressive (HAR) model of Corsi (2009) has probably been the most successful. The model-based class jointly models the returns and the realized measures of volatility. Notable examples within this class are the multiplicative error model (MEM) introduced by Engle and Gallo (2006) and the HEAVY model of Shephard and Sheppard (2010). In this paper, we consider models in this model-based class and refer to them as HF-based volatility processes. Our first contribution is to provide results for the existence of HF-based volatility processes that are asymptotically independent and processes that exhibit extremal dependence. Empirical assessment of the extremal dependence in financial returns is limited to a few small examples (Davis and Mikosch, 2009b; Liu and Tawn, 2013). Our second contribution is to provide a comprehensive empirical investigation, based on 17 international equity indices, of the extremal dependence in daily returns. Relying on two different test statistics, we document strong and persistent dependence in daily extreme returns, with negative extremes being more dependent than positive ones, and in the extremes of the return variance. The compelling evidence of extremal dependence in real data brings us to question whether volatility models can account for it. Our third contribution is to assess which HF-based volatility model better explains the dependence in extreme returns. We use a test of goodness of fit on the spectral density of the extremes provided by Mikosch and Zhao (2015) to compare six different models on 17 international equity indices. For comparison purposes with more standard models, we also include a GJR-GARCH model (Glosten, Jagannathan, and Runkle, 1993) in the comparison. We find that volatility models presenting extremal dependence and leverage are essential to capture the behavior of extreme events. In particular, a HEAVY model with leverage and the GJR-GARCH provide the best performance. To appreciate the implication of this result from a risk management perspective, we corroborate the results of the statistical test with a scenario analysis on the S&P500 during the period of turmoil depicted in Figure 1. The analysis confirms that leveraged models implying extremal dependence provide more conservative estimates of the probability of tail risk. The remainder of the paper is organized as follows: Section 1 describes two different estimators of the extremal dependence; Section 2 presents results concerning the extremal properties of different HF-based volatility processes; Section 3 provides evidence of dependence in the extremes of several stock markets and test which HF-based volatility model can explain it better; Section 4 concludes. An (Online) Supplementary Appendix contains additional details and results. 1 Measuring Extremal Dependence Measuring and estimating the extremal dependence in a time series is a rather challenging problem. Since financial time series are not Gaussian processes, the autocorrelation function is not well-suited for describing the dependence structure in the extremes. In what follows, we present two different but related measures of extremal dependence and their estimators: the extremal index, verifying whether a time series presents extremal dependence, and the extremogram, measuring the strength of such dependence and its persistence. 1.1 Extremal index The extremal index is a measure of the degree of local dependence in the extremes of a stationary process, and characterizes the clustering of extreme events (Leadbetter, Lindgren, and Rootzén, 1983). Let {Xt} be a strictly stationary sequence with marginal distribution function F, finite or infinite right endpoint ω=sup{x:F(x)<1} and survival function F¯=1−F. Let X1,…,Xn be a random sample from F and denote the sample maxima as Mn=max{X1,…,Xn}. The process {Xt} is said to have extremal index θ∈[0,1] if for integer n≥1 and for each τ>0 there exists a sequence of real numbers {un} such that as n→∞ the following are equivalent, nF¯(un)→τ, (2) Pr(Mn≤un)→e−θτ. (3) This equivalence establishes a close relationship between the tail behavior of F(x) and the limiting distribution of the sample maxima (Embrechts, Klüppelberg, and Mikosch, 1997). When θ = 1, extremes are independent and the standard convergence result for maxima of independent sequences holds, that is, Pr(Mn≤un)→e−τ. When θ<1, the limiting distribution of M(x) is equivalent to that of the maxima of an independent sequence with marginal distribution F, but location and scaling are affected by θ. In particular, consider an independent sequence X˜1,…,X˜n coming from F with corresponding sample maxima M˜n=max{X˜1,…,X˜n}, for u large we have that Pr(Mn≤u)≈Prθ(M˜n≤u)=Fnθ(u). The distribution of the maximum of n observations from a time series with extremal index θ can thus be approximated by the distribution of the maximum of nθ<n observations from the associated independent sequence. This entails that for θ<1, extreme observations tend to cluster and nθ can be interpreted as the number of independent clusters in n observations. The role of θ in characterizing the dependence in the extremes is further exemplified in Ledford and Tawn (2003) who established a bound relationship between the conditional probability in Equation (1) and the extremal index. In particular, they show that when {Xt} is asymptotically dependent for at least one lag, then θ<1. Conversely, if the process is asymptotically independent at all lags, theoretical results suggest that θ = 1. To estimate the extremal index, we use the approach of Ferro and Segers (2003). They show that properly normalized times between consecutive exceedances converge to a random variable which is zero with probability 1−θ and exponential with mean θ−1 with probability θ. They use this characterization to propose a moment estimator of θ that they call intervals estimator. Let X1,…,Xn be a random sample from F and u a high threshold. Let N(u)=∑t=1nI(Xt>u), with I(·) the indicator function, be the number of observations exceeding u, and let 1≤S1<⋯<Sn≤n be the exceedance times. The observed interexceedance times are Ei=Si+1−Si for i=1,…,n−1. The intervals estimator is a consistent estimator of the extremal index and is defined as θ˜(u)={1∧θ^(u)ifmax{Ei:1≤i≤n−1}≤21∧θ^*(u)ifmax{Ei:1≤i≤n−1}>2, (4) where θ^(u)=2(∑i=1n−1Ei)2(n−1)∑i−1n−1Ei2 and θ^*(u)=2{∑i−1n−1(Ei−1)}2(n−1)∑i−1n−1(Ei−1)(Ei−2) are obtained equating the first two theoretical moments of the interexceedance times limiting distribution to their empirical counterparts. In sum, Ferro and Segers (2003) note that the asymptotic distribution of times between threshold exceedances is uniquely characterized by the extremal index θ. In particular, with probability θ an arbitrary exceedance is the last of a cluster, and the time to the next exceedance has an exponential distribution with mean 1/θ, otherwise the next exceedance belongs to the same cluster with probability 1−θ. For a random sample {Xt}t=1T, one can define the threshold u as a high quantile of the sample.1 Assuming that the asymptotic characterization holds for the exceedances of u, I(Xt>u), an estimate of the extremal index can be obtained with the efficiently combined estimator θ˜(u). 1.2 Extremogram Davis and Mikosch (2009b) propose another tool for measuring the extremal dependence in a stationary series: the extremogram. It provides information on the degree of extremal dependence at several lags and can be viewed as the extreme-value analog of the autocorrelation function of a stationary process. For a strictly stationary Rd-valued time series Xt, the extremogram is defined as a limiting sequence given by ρAB(h)=limn→∞corr(I[an−1Xt∈A],I[an−1Xt+h∈B]), (5) where I[·] is the indicator function, an is a suitably chosen normalization sequence, and A, B are two fixed sets bounded away from zero. For example, setting d = 1 and A=B=[1,+∞), one obtains the extremogram for a univariate time series, limn→∞corr(I[Xt>an],I[Xt+h>an]), that will be used throughout in the paper. A natural estimator for the extremogram can be obtained by replacing the limiting sequence an in Equation (5) with a high quantile of the process. Defining am as the mth upper order statistics of the sample X1,…,Xn, the sample extremogram is given by ρ^AB(h)=∑t=1n−hI[am−1Xt∈A]I[am−1Xt+h∈B]∑t=1nI[am−1Xt∈A]. In order to have a consistent result, m=mn→∞ with m/n→0 as n→∞. Under suitable mixing conditions and other distributional assumptions that ensure the existence of the limit in Equation (5), Davis and Mikosch (2009b) show that the following central limit theorem holds, n/m(ρ^AB−ρAB:m(h))→dN(0,σAB2(h)), (6) where ρAB:m(h)=Pr(am−1Xt+h∈B|am−1Xt∈A) (7) is called pre-asymptotic extremogram and can be considered a finite approximation of its asymptotic limit ρAB(h). Davis and Mikosch (2009b) show that ρAB(h) can be substituted to ρAB:m(h) when nm|ρAB:m(h)−ρAB(h)|→0, otherwise one faces a bias issue. Note however that, as claimed by Davis, Mikosch, and Cribben (2012), the pre-asymptotic extremogram is still a conditional probability of extreme events and it is a quantity of interest per se. Apart from studying the extremal dependence, the extremogram can also be used to perform model selection based on the ability of explaining the dependence in the extremes of a real time series. In particular, Mikosch and Zhao (2015) propose a test of goodness of fit in frequency domain to do so, and we use this test in Section 3.2 to compare the ability of different volatility models to explain the dependence in the extremes. In what follows, we outline the asymptotic results of the test. The extremogram is the autocorrelation function of some stationary process, therefore, in agreement with classical time series analysis, it is possible to study its spectral properties. Let define the extremal spectral density as hA(λ)=∑h∈ZγA(h)exp(−ihλ) λ∈[0,π]=Π, where γA(h)=γAA(h) is the extremogram expressed in terms of autocovariance instead of autocorrelation. Then, a natural estimator is the extremal periodogram, In,A(λ)=mn|∑t=1nI{am−1Xt∈A}exp(−itλ)|2=γ^A(0)+2∑h=1n−1γ^A(h)cos(hλ), λ∈Π, with γ^A(h) the sample extremogram expressed in autocovariance terms. Mikosch and Zhao (2015) use the integrated version of these two quantities to construct goodness of fit test on the null hypothesis that the extremes of a sample comes from a time series model M0. Define the integrated extremal spectral density JA(x) and the integrated extremal periodogram Jn,A(x), respectively, as JA(x)=∫0xhA(λ)g(λ)dλ=c0(x)γA(0)+2∑h=1∞ch(x)γA(h), and Jn,A(x)=∫0xIn,A(λ)g(λ)dλ=c0(x)γ^A(0)+2∑h=1n−1ch(x)γ^A(h), where ch=∫0xcos(hλ)g(λ)dλ, with g(λ) a weight function and x∈(0,π). Mikosch and Zhao (2015) show that (nm)(Jn,A−EJn,A)→dG n→∞, (8) where G=c0Z0+2∑h=1∞chZh, with (Zh) a mean zero Gaussian sequence, and EJn,A(x) is the pre-asymptotic centering for JA(x). As for the extremogram, EJn,A(x) can be considered a finite sample approximation of JA(x) which cannot always be substituted by the latter in the asymptotic result because of an asymptotic bias problem. Using Equation (8), Mikosch and Zhao (2015) establish the limiting results of the following Grenander–Rosenblatt statistic (GRS), (nm)supx∈(0,π)|Jn,A(x)−EJn,A(x)|→dsupx∈(0,π)|G(x)|. (9) This can be used for testing the goodness of fit of the extremal spectral density of model M0. Under the null hypothesis that the model is correct, one can simulate the quantiles of the GRS from the theoretical model. Then, compute the difference between the integrated extremal periogram from the data and the pre-asymptotic integrated extremal spectral density of M0. If the supremum of this difference exceeds the 1−α quantile of the GRS, the null hypothesis that the extremes come from the model M0 is rejected at the α level. 2 Models and Theory Standard classes of time-varying volatility models, such as GARCH and SV, can nicely reproduce several aspects of asset returns, but they can differ substantially in their extremal behavior. Let rt be the logarithmic return of an asset at time t, the GARCH(p,q) model is defined through the following equations, rt=σtZt, σt2=α0+∑i=1pαirt−i2+∑j=1qβjσt−j2, (10) where Zt is an iid random variable with mean zero and unit variance. Basrak, Davis, and Mikosch (2002) use the theory of stochastic recurrence equations to show that, under suitable assumptions, GARCH(p,q) processes present extremal dependence, and the extent of such dependence increases in α and β. To grasp the intuition behind this result, take for instance the GARCH(1,1) model. One can note that the variance process σt2=ω+(αZt−12+β)σt−12 can become locally non-stationary ( αZt−12+β>1) when Zt−12 is occasionally large. Thus, a large shock at time t produces a burst of volatility at time t + 1, which in turn generates a large return. This asymptotic result holds true regardless the assumption on the innovation term Zt. Although it is very common to assume that Zt is Gaussian, empirical findings suggest the use of innovations with heavier tails (Bollerslev, 1987). Assuming that Zt follows a Student’s t distribution, Laurini and Tawn (2012) show that, for fixed parameters of the GARCH model, the extremal dependence weakens the heavier the tails of the Student’s t. In contrast to GARCH models, the extremal behavior of SV processes depends on the assumptions characterizing both returns and volatility innovations. Let consider the SV model introduced by Taylor (1982), rt=σtZt, ln(σt2)=ω+∑i=1pψiln(σt−i2)+νt (11) where Zt and νt are mean zero iid random variables with variance one and σν2, respectively. Under the assumption of normality of νt, Davis and Mikosch (2009a) show that the extremes of rt are independent when Zt is both light-tailed (Gaussian) and heavy-tailed (regularly varying2), meaning that SV processes have extremal index θ = 1. However, Mikosch and Rezapur (2013) show that SV processes may exhibit extremal clustering provided that σt is regularly varying with index α and the iid noise Zt has α+cth moment for some c > 0. Essentially, the only way to have a SV model with extremal dependence is to induce extremal clustering in the volatility process. Gaussian stationary sequences do no allow to do that, but under certain conditions, a stationary regular varying sequence can generate extremal dependence. Models based on HF data outperform the standard GARCH and SV models in terms of forecasting ability (Engle and Gallo, 2006; Brownlees and Gallo, 2010; Shephard and Sheppard, 2010; Hansen, Huang, and Shek, 2012), yet nothing is known about their extremal behavior. To study the extremal properties of a class of HF-based volatility processes, we set up an environment where a general realized measure characterizes the conditional returns distribution. Letting rt and xt be the return and the realized measure at time t, respectively, we consider the following general framework, rt=σtZt, (12) σt2=ω+κxt−1, ω,κ≥0, (13) xt=m(xt−1,…,xt−p,rt−1,…,rt−q;β,ηt), (14) where Zt is a random variable with mean zero and unit variance, m(·) is a Ft−1-measurable function depending on the vector of parameters β and the ηt∼N(0,ση2), with Ft−1=σ(xt−1,rt−1,xt−2,rt−2,…). We refer throughout to these three equations as the return equation, the variance equation, and the realized equation. Equation (13) nests inside the HEAVY structure of Shephard and Sheppard (2010), as they define σt2=ω+κRVt−1+δσt−12. The processes defined by Equations (12)–(14) can be considered latent volatility processes, but the fact that we rely on the observable xt represents a crucial difference with the standard class of SV models. The extremal properties of a process defined by Equations (12)–(14) depend on the specification of the function m(·) and of the error term ηt. In Section 2.1, we define a class of asymptotically independent HF-based volatility processes, while in Section 2.2 we use the results of Mikosch and Rezapur (2013) to establish a class of HF-based volatility processes that presents extremal dependence. 2.1 Asymptotically independent HF-based volatility processes The first specification we consider for the realized equation is a logarithmic HAR model (Andersen, Bollerslev, and Diebold, 2007; Corsi, 2009) with Gaussian innovations ηt∼N(0,ση2). This simple three-factor model has attracted a lot of attention in the financial econometrics literature because of its ability to capture the long-range pattern in the volatility decay. Casting this model in the environment defined by Equations (12)–(14), we obtain the HAR process, rt=σtZt,σt2=ω+κRVt−1,lnRVt=β0+βdlnRVt−1+βwlnRVt−1(5)+βmlnRVt−1(22)+ηt, (15) where lnRVt−1(h)=(1/h)∑j=1hlnRVt−j with lnRVt−1(1)≡lnRVt−1, and (βd+βw+βm)<1 to guarantee stationarity. A logarithmic specification for RVt is used because it does not require positivity constraints on the parameters, and it is preferable from an econometric perspective as lnRVt exhibits a bell-shaped distribution. A leveraged version of this model due to Maheu and McCurdy (2011) defines the levHAR process, rt=σtZt,σt2=ω+κRVt−1,lnRVt=β0+βdlnRVt−1+βwlnRVt−1(5)+βmlnRVt−1(22)+γZt−1+ηt, (16) with (βd+βw+βm)<1 to guarantee stationarity. The parameter γ accounts for the asymmetric feedback between the innovations and the volatility. Proposition 1 Consider the stationary process in Equation (15), then the sequences {RVt}, {σt}, and {|rt|} are all asymptotically independent according to Equation (1). Proof The realized equation of the process in Equation (15) can be written as an autoregressive process of order 22, lnRVt=β0+βdlnRVt−1+βwlnRVt−1(5)︸15lnRVt−1+⋯+15lnRVt−5+βmlnRVt−1(22)︸122lnRVt−1+⋯+122lnRVt−22+ηt =β0︸β˜0+(βd+15βw+122βm)︸β˜1lnRVt−1+(15βw+122βm)︸β˜2lnRVt−2+⋯+122βm︸β˜22lnRVt−22+ηt=β˜0+β˜1lnRVt−1+⋯+β˜22lnRVt−22+ηt. Thus, lnRVt is a Gaussian linear process. Note that lnσt2 is equal to lnRVt−1 up to a shift in scale and location; therefore, the asymptotic behavior of lnσt2 is not affected by (ω,κ). We can set the constants (ω,κ)=(0,1) so that lnσt2=lnRVt−1 and we can write lnRVt−1 as an infinite sum of Gaussian shocks. Then, the asymptotic independence of RVt, σt, and |rt| follows from Breidt and Davis (1998). Arguments similar to those used in the proof of Proposition 1 can be used to prove that the levHAR process in Equation (16) is asymptotically independent. The problem for these HAR-type processes is that the volatility is driven by a finite-order Markov process with homogeneous Gaussian errors. Gaussian sequences do not present clustering of the extremes and thus cannot generate extremal dependence in the returns regardless of the assumption on the innovation Zt. 2.2 Asymptotically dependent HF-based volatility models We have seen that the HAR and levHAR processes are asymptotically independent. We propose two different approaches to obtain extremal dependence in a HF-based volatility process. First, we consider a recurrence structure for the realized equation and use the results of Mikosch and Rezapur (2013) to show that this approach generates extremal dependence. Second, we consider a recurrence equation to model the variance of the variance, ση2. In this case, we are not able to formally establish the asymptotic dependence, but use the results of Borkovec (2000) to conjecture that this approach leads to extremal dependence. We consider the stationary HEAVY process which takes the form, rt=σtZt,σt2=ω+κRVt−1+δσt−12,RVt=μtηt2,μt=β0+β1RVt−1+β2μt−1, (17) where ηt∼N(0,1), ω,κ,δ≥0, δ∈(0,1], β0,β1,β2≥0, and β1+β2∈[0,1). The realized equation is defined as a MEM (Engle and Gallo, 2006) with the mean following a recurrence equation. This structure was proposed by Shephard and Sheppard (2010), who also add the autoregressive term σt−12 in the variance equation. We include it to be consistent with their framework, but it is not relevant to prove the extremal dependence property in Proposition 2. Shephard and Sheppard (2010) also suggest to add a second realized equation modeling the realized semi-variance to obtain a leveraged version of the HEAVY model. We consider also this extended model in Section 3 and referred to it as the levHEAVY model. Proposition 2 Consider the stationary process in Equation (17), then the sequences {RVt}, {σt}, and {|rt|} present extremal dependence according to Equation (1). Proof μt can be represented as a stochastic recurrence equation, μt=β0+μt−1(β1ηt−12+β2). Assume, without loss of generality, that (ω,κ,δ)=(0,1,0) so that σt2=RVt−1. Extremal dependence of RVt, σt, and |rt| directly follows from Theorem 4.4 of Mikosch and Rezapur (2013). The intuition behind this result is close to that of GARCH processes. The conditional variance σt2 is driven by RVt which in turn inherits its dynamics from μt. An occasional large realization of ηt−1 makes μt locally unstable, that is, (β1ηt−12+β2)>1. These occasional episodes of locally explosive dynamics cause clustering in the extremes of RVt, which translates in extremal dependence of σt2 and thus rt. The validity of this result is independent of the distributional assumption on Zt, though this may affect the degree of dependence in the extremes. In a simulation experiment (available in the Online Supplementary Appendix), we investigate the extremal behavior of the HEAVY model with both Gaussian and Student’s t innovations. Analogously to the GARCH process, we find that the heavy-tailed distribution induces less dependence in the extremes. An alternative strategy that we conjecture to lead to extremal-dependent processes consists in modeling the variance of the variance with a stochastic recurrence equation. Corsi et al. (2008) find strong evidence of time-variation and dependence in the variance of the variance and propose to model this behavior. Following their intuition, we relax the assumption that ηt∼iid(0,ση2) and propose the HAR-G process, rt=σtZt,σt2=ω+κRVt−1,lnRVt=β0+βdlnRVt−1+βwlnRVt−1(5)+βmlnRVt−1(22)+ηt,ση,t2=ψ0+ψ1ηt−12+ψ2ση,t−12, (18) with ηt∼N(0,ση,t2). To guarantee stationarity of the process we require βd+βw+βm<1 and ψ1+ψ2<1. Analogous extension of the levHAR process leads to the levHAR-G process. We do not present a formal proof for the extremal behavior of this process, but, based on the result of Borkovec (2000) showing that an AR(1)–ARCH(1) process exhibits extremal dependence, we conjecture that the extremes of lnRVt are asymptotically dependent and that this property is inherited by σt2 and |rt| as a consequence of Theorem 4.4 of Mikosch and Rezapur (2013). Again, the intuition is akin to the one of the HEAVY process. Episodes of locally explosive dynamics in ση,t2 generate extremal clustering in ηt that in turn affects the extremal behavior of RVt, σt2, and rt. To support our claim, we investigate the extremal behavior of the HAR and HAR-G processes by simulations. Results reported in the Online Supplementary Appendix confirm that the HAR-G process produces stronger dependence in the extremes than the HAR. 3 Empirical Analysis This section answers two questions regarding extremal dependence: do financial returns exhibit extremal dependence? What model is actually able to account for the extremal behavior observed in financial returns? To address the first question, we investigate whether the upper and lower tails of the daily returns exhibit extremal dependence. Furthermore, given the relevance of volatility in the financial econometrics literature, we think that insights on its extremal dependence are important from a modeling perspective. To this end, we inspect the serial dependence in the extreme observations of the realized variance (RV). Let pt denote the log-price of an asset at time t and rt,Δ=pt−pt−Δ the discretely sampled Δ-period return, the RV on day t is then defined as RVt=∑j=11/Δrt−1+(j·Δ)2. To answer the second question, we assess which HF-based volatility process better reproduces the strength of dependence observed in the extremes using the test of goodness of fit outline in Section 1.2. Moreover, we run a scenario analysis to exemplify the risk management implications of the different models when a large shock hits the market. The empirical analysis is based on the Oxford-Man Institute “Realized Library” version 0.2 (Heber et al., 2009). We consider 17 different stock indices from the beginning of 2000 to the end of 2014. To ease the exposition, we report the details only for a sample of four indices: Eurostoxx50 (ESX), FTSE100 (FT), NASDAQ (NSQ), and S&P500 (SPX). These results are representative of the whole study, and the outcomes for the other indices (available in the Online Supplementary Appendix) lead to equivalent conclusions. 3.1 Do financial returns exhibit extremal dependence? To ascertain the presence of extremal dependence, we estimate the extremal index on the upper tail of the following time series: the daily log-returns rt, the negated returns lt=−rt, and RVt. We use the intervals estimator described in Section 1.1, setting the threshold u at the 97th empirical quantile of each series, and compute the 95%-confidence bounds with the bootstrap procedure described in Ferro and Segers (2003). Recall that an extremal index θ<1 implies extremal dependence. The estimates θ^ and the corresponding bootstrap upper confidence bound reported in Table 1 are well below one for each series of extremes across the different assets. In sum, evidence of extremal dependence is overwhelming in both tails and in the variance process. Table 1 Extremal index estimates Upper tail Lower tail RV est. q0.975 est. q0.975 est. q0.975 ESX 0.18 0.64 0.29 0.51 0.10 0.28 FT 0.24 0.46 0.19 0.45 0.16 0.48 NSQ 0.09 0.49 0.14 0.49 0.07 0.24 SPX 0.13 0.59 0.15 0.62 0.08 0.50 Upper tail Lower tail RV est. q0.975 est. q0.975 est. q0.975 ESX 0.18 0.64 0.29 0.51 0.10 0.28 FT 0.24 0.46 0.19 0.45 0.16 0.48 NSQ 0.09 0.49 0.14 0.49 0.07 0.24 SPX 0.13 0.59 0.15 0.62 0.08 0.50 Notes: Estimated values (est.) of θ at the threshold level corresponding to the 97th quantile and upper bound of the two-sided bootstrap confidence interval computed at the 5% level ( q0.975). Table 1 Extremal index estimates Upper tail Lower tail RV est. q0.975 est. q0.975 est. q0.975 ESX 0.18 0.64 0.29 0.51 0.10 0.28 FT 0.24 0.46 0.19 0.45 0.16 0.48 NSQ 0.09 0.49 0.14 0.49 0.07 0.24 SPX 0.13 0.59 0.15 0.62 0.08 0.50 Upper tail Lower tail RV est. q0.975 est. q0.975 est. q0.975 ESX 0.18 0.64 0.29 0.51 0.10 0.28 FT 0.24 0.46 0.19 0.45 0.16 0.48 NSQ 0.09 0.49 0.14 0.49 0.07 0.24 SPX 0.13 0.59 0.15 0.62 0.08 0.50 Notes: Estimated values (est.) of θ at the threshold level corresponding to the 97th quantile and upper bound of the two-sided bootstrap confidence interval computed at the 5% level ( q0.975). We now turn our attention to the magnitude and persistence of such dependence. By means of the sample extremogram introduced in Section 1.2, we check the degree of dependence in the extremes up to 100 lags. The first two columns of Figure 2 report the sample extremograms for the upper and lower tails, obtained at the threshold level u corresponding to the 97th quantile of rt and lt, respectively. We also report the upper limit of the 99%-confidence interval obtained under the assumption of independence with the permutation procedure described in Davis, Mikosch, and Cribben (2012). Values of the sample extremogram extending beyond this bound support evidence of extremal dependence at the corresponding lag. The series present a degree of extremal dependence that persists for several lags. Moreover, the extremal dependence in the two tails is strongly different, with the lower tail exhibiting stronger extremal dependence. The third column of Figure 2 shows the sample extremogram at the 97th quantile for RV. Dependence in the extremes of the second moment is even stronger than that observed at the return level. All the series exhibit a high degree of dependence that lasts for more than 50 lags. This evidence on the return variance confirms the argument of Mikosch and Rezapur (2013) that extremal dependence in the volatility process is crucial to generate extremal dependence in the return process. Figure 2 View largeDownload slide Sample Extremograms. Colored caption, the points show the values of the sample extremogram at the 97th quantile up to 100 lags. The 99%-confidence bound in red is equal to 0.07 in each panel. Blue points are those exceeding the confidence bound and highlight the significance of the dependence at the corresponding lag. Gray points correspond to non significant lags. Gray-scale caption, the points show the values of the sample extremogram at the 97th quantile up to 100 lags. The gray line represents the 99%-confidence bound and it is equal to 0.07 in each panel. Light-gray points are those exceeding the confidence bound and highlight the significance of the dependence at the corresponding lag. Dark-gray points correspond to non-significant lags. Figure 2 View largeDownload slide Sample Extremograms. Colored caption, the points show the values of the sample extremogram at the 97th quantile up to 100 lags. The 99%-confidence bound in red is equal to 0.07 in each panel. Blue points are those exceeding the confidence bound and highlight the significance of the dependence at the corresponding lag. Gray points correspond to non significant lags. Gray-scale caption, the points show the values of the sample extremogram at the 97th quantile up to 100 lags. The gray line represents the 99%-confidence bound and it is equal to 0.07 in each panel. Light-gray points are those exceeding the confidence bound and highlight the significance of the dependence at the corresponding lag. Dark-gray points correspond to non-significant lags. To check that our findings are robust to the choice of the threshold level, we repeat the study at the 98th quantile. The results, available in the Online Supplementary Appendix, lead to equivalent conclusions. One potential explanation for the observed serial dependence in the extremes is that it spuriously arises as a consequence of stale prices. Dating back to the work of Lo and MacKinlay (1988), it is well known that infrequent or nonsynchronous trading accounts for a large portion of the serial correlation found in the returns of a stock index. The common intuition is that small firms are traded less frequently than large firms. Information updates at the close of the trading day are quickly absorbed by heavily traded (large) stocks, pushing them to move in the same direction that afternoon. In contrast, thinly traded (small) stocks only adjust the following morning. This lag generates spurious positive autocorrelation as a byproduct of the index construction. Our analysis is based on stock market indices; therefore, we need to consider stale prices as an explanatory factor. However, the degree of extremal dependence uncovered in the analysis is very similar across the stock indices. Yet, these indices vary greatly in their composition, and one would expect much greater differences in the extent of such dependence. Evidence that positive and negative extreme returns present positive and statistically significant autocorrelation clashes with the efficient market hypothesis, for which returns are completely unpredictable given the available information. A possible mechanism explaining this effect follows Bollerslev, Tauchen, and Zhou (2009). They develop a general equilibrium model where large price moves can generate a small amount of predictability in accordance with the volatility risk premium. Within this framework, a key role is recognized to the time-varying volatility of volatility of consumption growth as the driving factor of the equity premium and the determinant of return predictability. They show theoretically that increasing the persistence of this process increases both the extent and the horizon of return predictability. In our view, a persistent volatility of volatility process generates clusters of extreme volatility, which can explain the persistent extremal dependence in the returns. An alternative explanation may be a fractionally integrated volatility process (Bollerslev and Mikkelsen, 1996). Under this circumstance, periods of extreme volatility would be very persistent inducing persistent dependence in the extremes. Bollerslev, Sizova, and Tauchen (2012) develop a general equilibrium model with time-varying volatility of volatility of consumption growth accounting for long-memory in return volatility, leverage, and volatility feedback effects, and this mechanism could explain the emergence of extremal dependence. However, long-range dependence in volatility seems to be a less relevant factor. Unlike GARCH, it is well known that HAR models, though not fractionally integrated, accurately capture the seemingly long-memory pattern of volatility. Nonetheless, they fail to explain extremal dependence in the returns, while GARCH do. Our findings contribute to a consistent body of literature investigating the short- and long-term reaction of the stock market to large price movements (Brown, Harlow, and Tinic, 1988; Atkins and Dyl, 1990; Cox and Peterson, 1994; Lasfer, Melnik, and Thomas, 2003). While we focus on the relationship between current and future large returns, these works study the predictability of expected returns following large price changes. Analyzing stocks listed in the New York Stock Exchange, Brown, Harlow, and Tinic (1988) find that both positive and negative large shocks generate positive expected returns, while Atkins and Dyl (1990) support evidence of price reversals in both directions. Lasfer, Melnik, and Thomas (2003) find short-term positive (negative) abnormal returns following positive (negative) shocks in several stock market indices. Return predictability suggests profitable trading strategies, but this literature finds highly unlikely for arbitrageurs to earn excess profits exploiting the predictable pattern after a large shock (Atkins and Dyl, 1990; Cox and Peterson, 1994). Whether or not trading strategies exploiting serial correlation in the extremes can yield profits is an interesting question for future research. 3.2 What model explains extreme events? Section 2 establishes the existence of both asymptotically dependent and asymptotically independent HF-based volatility processes. Given the results of Section 3.1, we should favor models in the first class, but a deeper investigation is needed. Having a large set of models at hand, it is worth performing an empirical analysis to assess which HF-based volatility process better reproduces the dependence in the extremes. We consider the following models: HAR, levHAR, HEAVY, levHEAVY, HAR-G, and levHAR-G. For comparison purposes, we also consider a GJR-GARCH(1,1) process (Glosten, Jagannathan, and Runkle, 1993) that falls in the category of the asymptotically dependent models. We use the GRS test of goodness of fit on the integrated extremal spectral density in Equation (9). The test is performed on the exceedances beyond the 97th quantile of each time series considered, that is, rt, lt, and RVt. Table 2 reports the p-values of the GRS obtained from the different volatility models. The results show that the GJR-GARCH(1,1) model performs extremely well on both the upper and lower tails. Among the HF-based volatility models, only the levHEAVY model presents performance closed to that of the GARCH. Following, the levHAR-G and the HEAVY present similar rejection rates. Although these are asymptotically dependent models, they cannot fully account for the observed dependence. The HAR, levHAR, and HAR-G models all perform poorly in the lower tail, confirming that the extremal dependence property and the leverage effect are both necessary to account for the dependence in the negative extremes. Table 2 Goodness of fit Upper Lower RV Upper Lower RV ESX (i) 0.426 0.384 NSQ (i) 0.358 0.250 (ii) 0.348 0.029 0.406 (ii) 0.205 0.006 0.514 (iii) 0.021 0.011 0.021 (iii) 0.234 0.021 0.494 (iv) 0.060 0.102 0.584 (iv) 0.060 0.023 0.259 (v) 0.056 0.204 0.145 (v) 0.227 0.162 0.231 (vi) 0.496 0.096 0.367 (vi) 0.276 0.011 0.687 (vii) 0.329 0.263 0.448 (vii) 0.416 0.615 0.761 FT (i) 0.072 0.819 SPX (i) 0.404 0.646 (ii) 0.033 0.024 0.499 (ii) 0.349 0.020 0.200 (iii) 0.027 0.137 0.824 (iii) 0.219 0.043 0.055 (iv) 0.050 0.040 0.877 (iv) 0.349 0.235 0.170 (v) 0.041 0.394 0.386 (v) 0.141 0.420 0.050 (vi) 0.036 0.014 0.792 (vi) 0.355 0.027 0.142 (vii) 0.023 0.034 0.587 (vii) 0.340 0.599 0.052 Upper Lower RV Upper Lower RV ESX (i) 0.426 0.384 NSQ (i) 0.358 0.250 (ii) 0.348 0.029 0.406 (ii) 0.205 0.006 0.514 (iii) 0.021 0.011 0.021 (iii) 0.234 0.021 0.494 (iv) 0.060 0.102 0.584 (iv) 0.060 0.023 0.259 (v) 0.056 0.204 0.145 (v) 0.227 0.162 0.231 (vi) 0.496 0.096 0.367 (vi) 0.276 0.011 0.687 (vii) 0.329 0.263 0.448 (vii) 0.416 0.615 0.761 FT (i) 0.072 0.819 SPX (i) 0.404 0.646 (ii) 0.033 0.024 0.499 (ii) 0.349 0.020 0.200 (iii) 0.027 0.137 0.824 (iii) 0.219 0.043 0.055 (iv) 0.050 0.040 0.877 (iv) 0.349 0.235 0.170 (v) 0.041 0.394 0.386 (v) 0.141 0.420 0.050 (vi) 0.036 0.014 0.792 (vi) 0.355 0.027 0.142 (vii) 0.023 0.034 0.587 (vii) 0.340 0.599 0.052 Notes: p-Values of the GRS in Equation (9) performed on the extremes at the 97th quantile. The models are: (i) GJR-GARCH(1,1); (ii) HAR; (iii) levHAR; (iv) HEAVY; (v) levHEAVY; (vi) HAR-G; and (vii) levHAR-G. Rejections at the 5% level are in bold. Table 2 Goodness of fit Upper Lower RV Upper Lower RV ESX (i) 0.426 0.384 NSQ (i) 0.358 0.250 (ii) 0.348 0.029 0.406 (ii) 0.205 0.006 0.514 (iii) 0.021 0.011 0.021 (iii) 0.234 0.021 0.494 (iv) 0.060 0.102 0.584 (iv) 0.060 0.023 0.259 (v) 0.056 0.204 0.145 (v) 0.227 0.162 0.231 (vi) 0.496 0.096 0.367 (vi) 0.276 0.011 0.687 (vii) 0.329 0.263 0.448 (vii) 0.416 0.615 0.761 FT (i) 0.072 0.819 SPX (i) 0.404 0.646 (ii) 0.033 0.024 0.499 (ii) 0.349 0.020 0.200 (iii) 0.027 0.137 0.824 (iii) 0.219 0.043 0.055 (iv) 0.050 0.040 0.877 (iv) 0.349 0.235 0.170 (v) 0.041 0.394 0.386 (v) 0.141 0.420 0.050 (vi) 0.036 0.014 0.792 (vi) 0.355 0.027 0.142 (vii) 0.023 0.034 0.587 (vii) 0.340 0.599 0.052 Upper Lower RV Upper Lower RV ESX (i) 0.426 0.384 NSQ (i) 0.358 0.250 (ii) 0.348 0.029 0.406 (ii) 0.205 0.006 0.514 (iii) 0.021 0.011 0.021 (iii) 0.234 0.021 0.494 (iv) 0.060 0.102 0.584 (iv) 0.060 0.023 0.259 (v) 0.056 0.204 0.145 (v) 0.227 0.162 0.231 (vi) 0.496 0.096 0.367 (vi) 0.276 0.011 0.687 (vii) 0.329 0.263 0.448 (vii) 0.416 0.615 0.761 FT (i) 0.072 0.819 SPX (i) 0.404 0.646 (ii) 0.033 0.024 0.499 (ii) 0.349 0.020 0.200 (iii) 0.027 0.137 0.824 (iii) 0.219 0.043 0.055 (iv) 0.050 0.040 0.877 (iv) 0.349 0.235 0.170 (v) 0.041 0.394 0.386 (v) 0.141 0.420 0.050 (vi) 0.036 0.014 0.792 (vi) 0.355 0.027 0.142 (vii) 0.023 0.034 0.587 (vii) 0.340 0.599 0.052 Notes: p-Values of the GRS in Equation (9) performed on the extremes at the 97th quantile. The models are: (i) GJR-GARCH(1,1); (ii) HAR; (iii) levHAR; (iv) HEAVY; (v) levHEAVY; (vi) HAR-G; and (vii) levHAR-G. Rejections at the 5% level are in bold. To appreciate the implications of the different models from a risk management perspective, we propose a scenario analysis during the last financial crisis. Consider a financial risk manager at the close of the trading day on Monday September 29, 2008. The S&P500 just lost 7% of its capitalization and he needs to provide risk assessment over the next 2 weeks. To have a perception of the models behavior, we simulate 10,000 paths of 9 days for each model and, on each day, we compute the probability that the random observation falls below the S&P500 return. We use the 1000 days prior September 29 as the training sample, and obtain estimates of the model parameters by Maximum Likelihood. To enable a better fit, we endow each model with a conditional Student’s t distribution, warranting once again that this does not affect the asymptotic result on the extremes of the models. According to Section 2 and the results from the GRS test, models implying extremal dependence and endowed with leverage should generate more extreme observations after a large negative shock, and thus be able to provide more conservative estimates of tail risk. Table 3 reports the results. Excluding h = 1, where the index bounced back after the Monday shock, the S&P500 records a loss each of the following 8 days. One can see that for these days, the GJR-GARCH(1,1) and the levHEAVY models present higher probabilities of exceeding the realized loss, leading to more prudent estimates of risk. A curious aspect is that the levHAR is more conservative than the HAR-G, though the latter is asymptotically dependent. We explore this issue in a simulation experiment (available in the Online Supplementary Appendix), and find that at finite quantiles the levHAR can generate more dependence in the extremes than the HAR-G model thanks to the leverage parameter. However, things change increasing the quantiles at very extreme levels, with the extremal dependence implied by the levHAR decreasing toward zero while it stays positive for the HAR-G. Table 3 Risk management example h 1 2 3 4 5 6 7 8 9 Return 0.0466 −0.0018 −0.0406 −0.0141 −0.0325 −0.0581 −0.0047 −0.0745 0.0024 GARCH 0.92 0.46 0.10 0.31 0.15 0.04 0.43 0.02 0.54 HAR 0.87 0.46 0.04 0.23 0.08 0.02 0.40 0.01 0.56 levHAR 0.82 0.47 0.06 0.25 0.09 0.03 0.41 0.01 0.54 HEAVY 0.91 0.46 0.08 0.28 0.11 0.03 0.41 0.02 0.53 levHEAVY 0.92 0.47 0.12 0.33 0.16 0.05 0.44 0.03 0.54 HAR-G 0.86 0.46 0.04 0.23 0.07 0.01 0.40 0.00 0.55 levHAR-G 0.84 0.45 0.07 0.24 0.10 0.04 0.40 0.02 0.60 h 1 2 3 4 5 6 7 8 9 Return 0.0466 −0.0018 −0.0406 −0.0141 −0.0325 −0.0581 −0.0047 −0.0745 0.0024 GARCH 0.92 0.46 0.10 0.31 0.15 0.04 0.43 0.02 0.54 HAR 0.87 0.46 0.04 0.23 0.08 0.02 0.40 0.01 0.56 levHAR 0.82 0.47 0.06 0.25 0.09 0.03 0.41 0.01 0.54 HEAVY 0.91 0.46 0.08 0.28 0.11 0.03 0.41 0.02 0.53 levHEAVY 0.92 0.47 0.12 0.33 0.16 0.05 0.44 0.03 0.54 HAR-G 0.86 0.46 0.04 0.23 0.07 0.01 0.40 0.00 0.55 levHAR-G 0.84 0.45 0.07 0.24 0.10 0.04 0.40 0.02 0.60 Notes: For day t + h: the S&P500 returns and, for each model, the fraction of times that the generated observation are below the S&P return. Table 3 Risk management example h 1 2 3 4 5 6 7 8 9 Return 0.0466 −0.0018 −0.0406 −0.0141 −0.0325 −0.0581 −0.0047 −0.0745 0.0024 GARCH 0.92 0.46 0.10 0.31 0.15 0.04 0.43 0.02 0.54 HAR 0.87 0.46 0.04 0.23 0.08 0.02 0.40 0.01 0.56 levHAR 0.82 0.47 0.06 0.25 0.09 0.03 0.41 0.01 0.54 HEAVY 0.91 0.46 0.08 0.28 0.11 0.03 0.41 0.02 0.53 levHEAVY 0.92 0.47 0.12 0.33 0.16 0.05 0.44 0.03 0.54 HAR-G 0.86 0.46 0.04 0.23 0.07 0.01 0.40 0.00 0.55 levHAR-G 0.84 0.45 0.07 0.24 0.10 0.04 0.40 0.02 0.60 h 1 2 3 4 5 6 7 8 9 Return 0.0466 −0.0018 −0.0406 −0.0141 −0.0325 −0.0581 −0.0047 −0.0745 0.0024 GARCH 0.92 0.46 0.10 0.31 0.15 0.04 0.43 0.02 0.54 HAR 0.87 0.46 0.04 0.23 0.08 0.02 0.40 0.01 0.56 levHAR 0.82 0.47 0.06 0.25 0.09 0.03 0.41 0.01 0.54 HEAVY 0.91 0.46 0.08 0.28 0.11 0.03 0.41 0.02 0.53 levHEAVY 0.92 0.47 0.12 0.33 0.16 0.05 0.44 0.03 0.54 HAR-G 0.86 0.46 0.04 0.23 0.07 0.01 0.40 0.00 0.55 levHAR-G 0.84 0.45 0.07 0.24 0.10 0.04 0.40 0.02 0.60 Notes: For day t + h: the S&P500 returns and, for each model, the fraction of times that the generated observation are below the S&P return. Overall, these results are coherent with the findings in Section 3.1 and the results of Section 2. Financial returns present strong and persistence extremal dependence of different magnitude in the two tails. This requires volatility models that generate asymptotically dependent observations and at the same time account for the asymmetric dependence in the extremes. Given the superior performance of HF-based volatility models in forecasting volatility and risk measures, the levHEAVY model should be recommended as a reference model in the future. 4 Conclusions Although empirical investigations of autocorrelations in returns and volatility have a long history in financial economics and econometrics, dependence in the extremes has been neglected. Understanding the extremal properties of financial time series and related models is of central importance to gauge financial risk, and this paper makes a step forward in this direction. First, we study the extremal behavior of several HF-based volatility processes. We establish the existence of asymptotically independent models, such as the widely used HAR of Corsi (2009), and models exhibiting extremal dependence, such as the HEAVY of Shephard and Sheppard (2010). Second, we add new evidence on the extremal behavior of daily stock returns, finding that dependence in the extremes is strong and persistent in both tails of the return distribution, and also in the volatility process. In particular, the strength of the dependence in the lower tail is higher than in the upper tail. Finally, we perform a large empirical study to asses the ability of both HF-based volatility models and traditional GARCH models of explaining the dependence in the extremes of 17 international indices. The results reveal that only models exhibiting extremal dependence and endowed with a leverage component can appropriately explain extreme events. Future research on this topic should address questions regarding the multivariate extremal dependence. Although tail dependence among assets is widely documented (Poon, Rockinger, and Tawn, 2003), less is known about the clustering in time of co-extreme observations between two or more assets. Supplementary Data Supplementary data are available at Journal of Financial Econometrics online. Footnotes * Part of this work was conducted while the author was a post-doctoral fellow at HEC Montréal and financial support is gratefully acknowledged. I am indebted to the editor George Tauchen and two anonymous referees that helped improve an earlier version of the manuscript. I am grateful to Marco Bee, Christian Brownlees, Debbie J. Dupuis, Fabrizio Lillo, and Roberto Renò for their helpful comments. 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Journal of Financial Econometrics – Oxford University Press
Published: Apr 1, 2018
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