Intraday conditional value at risk: A periodic mixed‐frequency generalized autoregressive score approach

Gribisch, Bastian; Eckernkemper, Tobias Article — Published Version Intraday conditional value at risk: A periodic mixed‐ frequency generalized autoregressive score approach Journal of Forecasting Provided in Cooperation with: John Wiley & Sons Suggested Citation: Gribisch, Bastian; Eckernkemper, Tobias (2021) : Intraday conditional value at risk: A periodic mixed‐frequency generalized autoregressive score approach, Journal of Forecasting, ISSN 1099-131X, Wiley, Hoboken, NJ, Vol. 40, Iss. 5, pp. 883-910, https://doi.org/10.1002/for.2744 This Version is available at: https://hdl.handle.net/10419/233643 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc-nd/4.0/ Received: 19 June 2019 Revised: 12 October 2020 Accepted: 20 November 2020 DOI: 10.1002/for.2744 RESEARCH ARTICLE Intraday conditional value at risk: A periodic mixed-frequency generalized autoregressive score approach Tobias Eckernkemper Bastian Gribisch Institute of Econometrics and Statistics, University of Cologne, Cologne, Germany Correspondence Bastian Gribisch, Institute of Econometrics and Statistics, University of Cologne, Universitätsstr. 22a, D-50937 Cologne, Germany. Email: [email protected]oeln.de [Correction added on 3 March 2021, after first online publication: The order of authors in the author byline has been corrected.] Abstract We propose a copula-based periodic mixed frequency generalized autoregressive (GAS) framework in order to model and forecast the intraday exposure conditional value at risk (ECoVaR) for an intraday asset return and the corresponding market return. In particular, we analyze GAS models that account for long-memory-type of dependencies, periodicities, asymmetric nonlinear dependence structures, fat-tailed conditional return distributions, and intraday jump processes for asset returns. We apply our framework in order to analyze the ECoVaR forecasting performance for a large data set of intraday asset returns of the S&P500 index. KEYWORDS CoVaR, dynamic copulas, intraday, systemic risk JEL CLASSIFICATION C32; C51; C58G17 1INTRODUCTION Intraday trading on financial markets has become increasingly important over the last decades and increased the need for traders and heads of trading desks to have within-day access to market information in order to make well-informed decisions. This finding spurred research on modeling the time-series characteristics of intraday returns, where recent contributions typically focus on univariate volatility modeling; see, for example, Engle and Sokalska (2012), Stroud and Johannes (2014), Rossi and Fantazzini (2015), and Bekierman and Gribisch (2017). Precise intraday volatility estimates are of considerable importance for intraday risk management, that is, analyzing and predicting intraday risk measures like the value at risk (VaR) as an ingredient for portfolio selection, hedging, and placing limit orders. As, for example, noted by Gourieroux and Jasiak (1997), intraday risk management is constantly used by commercial banks in order to monitor their internal trading desks. For example, traders have to be able to constantly provide estimates of their risks during the trading day. Although daily risk measures may be adequate from a reporting perspective, traders typically need up-to-date information on their risks in order to have competitive advantages by reacting on risks in real time (see also Liu & Tse, 2015). In addition to the pure volatility aspect, financial risk management increasingly focusses on the assessment of dependencies in the tails of the bivariate return distribution of the asset and the market: while the popular VaR measure focusses on the single institution, more up-to-date approaches incorporate systemic tail risk effects via the connection of the institution with the whole financial system. Prominent examples are the exposure conditional VaR (ECoVaR) and the ΔECoVaR of Adrian and Brunnermeier (2016). These measures are important ingredients for the assessment of spillover and contagion effects and the connectedness in network analysis and This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. © 2020 The Authors. Journal of Forecasting published by John Wiley & Sons Ltd. wileyonlinelibrary.com/journal/for Journal of Forecasting. 2021;40:883–910. 883 ECKERNKEMPER AND GRIBISCH allow to reveal those institutions that are most at risk if a financial crisis occurs (see, e.g., Bernardi & Catania, 2016; Chen et al., 2019; Girardi & Ergün, 2013; Reboredo & Ugolini, 2015). According risk estimates are clearly of interest from an intraday monitoring perspective, where the single trader is, for example, asked to provide an estimate of the within-day vulnerability of his portfolio to systemic effects. While aspects of intraday VaR modeling have already been discussed in the literature, we are not aware of contributions on the intraday modeling of systemic risk measures. In fact, approaches based on daily data may easily oversee important risk spillovers within the trading day, as, for example, a zero open to close return on a given trading day might still be the result of heavy return variation within the day with eventually hidden systemic effects for the market as a whole. Derived intraday asset linkages within the tails of the distribution might also be of interest for high-frequency contagion and network analysis, which become more and more important since crisis often reveal patterns of high-speed information transmission through markets. In this paper, we propose a mixed frequency generalized autoregressive score (MF-GAS) framework (see Creal et al., 2013) in order to model and forecast the intraday (Δ)ECoVaR measure. The (Δ)ECoVaR captures tail-specific risk spillovers for the bivariate relationship of an intraday asset return and the market by providing information on the VaR of the individual asset conditional on the market being in distress. The computation of this systemic risk measure requires a bivariate modeling of the intraday asset and market return while accounting for long-memory type of dependence patterns in the volatility and dependence processes, seasonalities, and potential asymmetries in tail dependence. In particular, we propose to model the bivariate asset–market relationship in a dynamic seasonal copula framework with time-varying volatilities and (a)symmetric tail dependencies. Based on the copula specification, we obtain forecasts of the (Δ)ECoVaR in a straightforward way by using the approach of Mainik and Schaanning (2014) and Reboredo and Ugolini (2015). The copula approach allows us to separate the joint distribution into the dependence process and the margins. The modeling of the marginal distribution of intraday asset returns is challenging for several reasons: the returns feature heavy tails, discrete jump components, and long-memory type of persistencies and periodicity in the volatility process. Standard generalized autoregressive conditional heteroskedasticity (GARCH) models (Bollerslev, 1986; Engle, 1982) have therefore proven to be unsatisfactory for intraday return dynamics (see, e.g., Andersen & Bollerslev, 1997). A popular approach to intraday volatility modeling is to factorize the conditional variance into a product of daily and intraday components (see, e.g., Andersen & Bollerslev, 1997, 1998), where the daily component captures the long-range dependence in the volatility series. Engle and Sokalska (2012) use commercially available volatility forecasts in order to approximate the daily volatility part. The remaining intraday fluctuations are then modeled by short-memory GARCH processes, where periodic intraday volatility patterns can be estimated individually or approximated by flexible Fourier forms (see, e.g., Payne, 1996).1An alternative approach, which has, for example, been analyzed by Rossi and Fantazzini (2015), builds on periodic fractional integrated exponential GARCH (FI-PEGARCH) models for the intraday return series. A similar (though nonperiodic) approach has been used by Janus et al. (2014) in order to model long-memory dynamics in the copula dependence parameters for daily asset returns. A fractionally integrated GAS (FIGAS) approach for daily returns is analyzed by Opschoor and Lucas (2019). ARFIMA-type processes model ‘true’ long memory but tend to induce complications and instability in model estimation and suffer from problems related to the initialization of the long-memory process. Our approach in contrast builds on mixed frequency GAS dynamics for the individual asset log-volatilities and the dynamic copula parameters. The mixed frequency structure offers a flexible and robust approximation to long-memory while preserving stationarity of the volatility and dependence components and computational ease of estimation. The model builds on latent short-term and long-term components that move at different frequencies. Here, we assume that the long-term component moves at the daily and the short-term component at the intraday frequency, which results in a framework that is similar in spirit to the Engle and Sokalska (2012) approach but explicitly models the long-term component instead of using exogenously determined commercial forecasts. The model also shows similarities to the popular mixed data sampling (MIDAS) schemes for volatility modeling (see, e.g., Colacito et al., 2011; Engle et al., 2013; Ghysels et al., 2006) but is more close to the stochastic volatility (SV) literature due do the flexible GAS assumption (see, e.g., Koopman et al., 2016). Although not formally belonging to the class of long-memory models, multiple component models are well known to reproduce highly persistent long-memory type of dependence structures by aggregating independent autoregressive dynamics at different frequencies (see, 1Related studies on the component-modeling of intraday volatilities are found in Andersen and Bollerslev (1997, 1998), Bekierman and Gribisch (2017), and Beltratti and Morana (1999). 884 ECKERNKEMPER AND GRIBISCH e.g., Corsi, 2009; Granger, 1980; LeBaron, 2001). A further advantage of using latent volatility components is that the dynamic mixed frequency GAS structure can easily be applied in order to model the dynamics of the copula dependence parameter, where, for example, MIDAS schemes or the Engle and Sokalska model are not directly applicable. The GAS approach optimally approximates the innovation terms of the latent state processes by scaled likelihood scores, which makes the model very flexible but still simple to implement.2The framework has also the advantage that the optimal frequency for the long-term component can be selected fully data-driven by comparing the log-likelihood values obtained for different frequencies and choosing the frequency associated with the highest log-likelihood value. We finally extend our basic model specification by considering conditional fat-tailed distributions and GAS-driven autoregressive Poisson jump processes for the marginal return specifications. For the copula models, we consider the Gaussian copula, the Student tcopula and a mixture of the Clayton and the rotated Clayton copula in order to allow for (a)symmetric and possibly periodic tail dependence. In fact, the application of GAS models with mixed frequencies is not new to the literature. Gorgi et al. (2019) propose a MIDAS-GAS model for economic time series like inflation or GDP growth, where a low-frequency economic variable is to be forecasted based on high-frequency financial information using a weighted sum of the high-frequency GAS innovations. This setting differs from our MF-GAS approach and the high-frequency financial data perspective: the MIDAS-GAS of Gorgi et al. (2019) models mixed frequency data using latent GAS-driven components at the lower frequency, while the MF-GAS considers lowand high-frequency GAS components in order to model data, which is observed at a single frequency. Our intraday copula approach also shows some similarities to recent work of Koopman et al. (2018). The authors propose a GAS driven copula framework for U.S. financial stocks at the tick-by-tick frequency and are particularly interested in analyzing the intraday dependence structure. In contrast to our approach, the model of Koopman et al. (2018) does not allow for long-memory type of dependence patterns and does neither consider periodicity in the tail-dependence nor model the intraday periodicity jointly with the GAS dynamics (which turns 2GAS dynamics (Creal et al., 2013) have been intensively used in recent years in order to approximate the dynamics of unobserved component models. Theoretical justifications for the use of the conditional score as a flexible updating mechanism are, e.g., provided by Harvey (2013), Creal et al. (2013), and Blasques et al. (2015). Applications of score driven dynamics are, e.g., found in Creal et al. (2011), Janus et al. (2014), Harvey and Sucarrat (2014), and Eckernkemper (2018). out to be crucial for intraday (Δ)ECoVaR forecasting; see below). The model is furthermore fitted separately for each trading day, which stands in contrast to our mixedfrequency approach for intraday volatility/dependence across trading days. We provide an empirical application to a data set of the 378 most liquid stocks from the S&P500, which have been jointly traded between 2004 and 2012. We focus on analyzing the mixed frequency structure, long memory-type of dependencies and seasonalities in intraday variances, correlations, tail-dependencies, and jumps as well as their effects on the resulting (Δ)ECoVaR estimates. We find strong evidence of long memory and seasonalities in the intraday variance and dependence processes. Residual diagnostics show that the proposed mixed frequency GAS model successfully captures these effects. Furthermore, we find that those model specifications that allow for fat-tailed conditional return distributions and symmetric intraday tail dependencies are preferred. We also perform an in-sample and out-of-sample analysis in order to analyze (i) the time series behavior of the intraday (Δ)ECoVaR and (ii) the effect of seasonalities in the volatility and dependence processes on the (Δ)ECoVaR forecasts. Our results show significant intraday variation and periodicity of the (Δ)ECoVaR and that the modeling of seasonalities in the volatility and dependence process is essential in order to obtain reliable (Δ)ECoVaR forecasts. The remainder of the paper is organized as follows. Section 2 gives an overview on the ECoVaR measure. Section 3 presents the mixed-frequency GAS model for intraday volatility and dependence including the model extension to price jumps. The application of the mixedfrequency GAS model to a large dataset of intraday asset returns of the S&P500 index is provided in Section 4. Finally, Section 5 summarizes the results and concludes. 2 (Δ)ECoVaR The CoVaR of the market (or the financial system) conditional on institution iis defined as the VaR of the market return conditional on the event that institution iis in distress (see Adrian & Brunnermeier, 2016; Girardi & Ergün, 2013).3 Let ri,𝜏denote the return of institution iin period 𝜏and rm,𝜏the corresponding return of the market. The CoVaR is then the 𝛽-quantile of the conditional distribution of rm,𝜏: Pr (rm,𝜏 ≤CoVaRm|i 𝛼,𝛽,𝜏 |ri,𝜏 ≤VaRi 𝛼,𝜏,𝜏−1)=𝛽. (1) 3Here and in the following sections, we will for simplicity refer to a single institution i. We however note that for risk management applications, the single institution is typically replaced by a portfolio of institutions. 885 ECKERNKEMPER AND GRIBISCH The VaR for institution i,VaRi 𝛼,𝜏, is defined as the 𝛼-quantile of the conditional return distribution of asset i in period 𝜏,Pr(ri,𝜏 ≤VaRi 𝛼,𝜏)|𝜏−1)=𝛼,where𝜏denotes a filtration of the returns of the asset and the market up to period 𝜏. Inordertomeasurethepartofthesystemicrisk,whichis caused by the distress of institution i,wemaycomputethe ΔCoVaR as defined by Adrian and Brunnermeier (2016) and Girardi and Ergün (2013): ΔCoVaRm|i 𝛼,𝛽,𝜏 = CoVaRm|i 𝛼,𝛽,𝜏 −CoVaRm|benchi 𝛽,𝜏 CoVaRm|benchi 𝛽,𝜏 ,(2) that is, the additional effect on the VaR of the market if institution iis in distress, relative to the situation, where the return of institution iis at its benchmark level. Here, the benchmark level is defined as one standard deviation about the mean event, conditional on 𝜏−1: Pr (rm,𝜏 ≤CoVaRm|benchi 𝛽,𝜏 | 𝜇i,t−𝜎i,t≤ri,𝜏 ≤𝜇i,t+𝜎i,t,𝜏−1)=𝛽, (3) where 𝜇i,tand 𝜎i,tdenote the mean and standard deviation of the conditional distribution of ri,𝜏given 𝜏−1.The ΔCoVaR measures the systemic risk part which comoves with the distress of institution iand can therefore be interpreted as a statistical tail dependence measure (see Adrian & Brunnermeier, 2016). While the CoVaR itself is highly dependent on the conditional variance of the market return, the ΔCoVaR is mainly driven by the covariance between the market and the institution. The (Δ)CoVaR as defined above is typically used in order to measure systemic risk contributions. In this paper, however, we are particularly interested in the risk management perspective. Here, it appears useful to reverse the conditioning of the CoVaR in order to obtain the exposure CoVaR (labeled ECoVaR) as proposed by Adrian and Brunnermeier (2016): Pr (ri,𝜏 ≤ECoVaRi|m 𝛼,𝛽,𝜏 |rm,𝜏 ≤VaRm 𝛼,𝜏,𝜏−1)=𝛽, (4) where VaRm 𝛼,𝜏 denotes the VaR of the market. According to Equation (2), the ΔECoVaR is then defined as ΔECoVaRi|m 𝛼,𝛽,𝜏 = ECoVaRi|m 𝛼,𝛽,𝜏 −ECoVaRi|benchm 𝛽,𝜏 ECoVaRi|benchm 𝛽,𝜏 (5) and measures the institution's (or the portfolio's) exposure to system-wide distress relative to a normal market situation. In the remainder of the paper, we will focus on the (Δ)ECoVaR. 3MF-GAS MODELING OF INTRADAY (Δ)ECOVAR The computation and forecasting of the ECoVaR requires a joint conditional distribution for the assetand market return. A flexible way of modeling this distribution is provided by the copula approach, where the joint conditional distribution is decomposed into the marginal distributions and the dependence structure, which is captured by a copula function (see, e.g., Nelsen, 2006, for an introduction to copulas). An attractive feature of the copula approach is that it allows to compute the ECoVaR in a straight-forward way (see Mainik & Schaanning, 2014; Reboredo & Ugolini, 2015). In particular, we can rewrite Equation (4) as Pr (ri,𝜏 ≤ECoVaRi|m 𝛼,𝛽,𝜏,rm,𝜏 ≤VaRm 𝛼,𝜏 |𝜏−1) Pr (rm,𝜏 ≤VaRm 𝛼,𝜏 |𝜏−1)=𝛽, (6) resulting in Pr (ri,𝜏 ≤ECoVaRi|m 𝛼,𝛽,𝜏,rm,𝜏 ≤VaRm 𝛼,𝜏 | 𝜏−1)=Fri,𝜏 ,rm,𝜏 (ECoVaRi|m 𝛼,𝛽,𝜏,VaRm 𝛼,𝜏|𝜏−1)=𝛼𝛽, (7) where Frm,𝜏 ,ri,𝜏 denotes the joint cumulative distribution function of ri,𝜏and rm,𝜏given all lagged return information. Following the Sklar (1959) theorem, we now express the joint cumulative distribution function by a conditional copula function, C(ui,𝜏,um,𝜏|𝜏−1)=𝛼𝛽, (8) with ui,𝜏 =Fri,𝜏 (ECoVaRi|m 𝛼,𝛽,𝜏|𝜏−1)and um,𝜏 = Frm,𝜏 (VaRm 𝛼,𝜏|𝜏−1)=𝛼. Given known (or estimated) marginal distribution functions and copula the ECoVaR can be computed via a two-step procedure: 1. Solve C(ui,𝜏,𝛼|𝜏−1)=𝛼𝛽 for ui,𝜏; 2. Compute ECoVaRi|m 𝛼,𝛽,𝜏 via ECoVaRi|m 𝛼,𝛽,𝜏 =F−1 ri,𝜏 (ui,𝜏|𝜏−1). For the computation and forecasting of the ECoVaR, we employ parametric specifications for the conditional return distributions and the bivariate copula function. In the upcoming two sections, we therefore develop flexible dynamic volatility models for intraday asset returns and dynamic copula specifications, which account for the time-varying and potentially nonlinear dependence between the intraday asset and market return. 3.1 Volatility 3.1.1 The V-MF-GAS Model Let Pt,𝓁denote the intraday price of a particular asset at day t,t=1,…,T, and intraday period 𝓁,𝓁=1,…,S, where Sis the total number of intraday periods per trading day (e.g., S=390 for minute-returns). For the ease of notation, we define the overall period index 𝜏=(t−1)S+𝓁 886 ECKERNKEMPER AND GRIBISCH FIGURE 1 Total volatility and its components. Total log-volatility (solid black), intraday (dashed dark blue), and diurnal (dotted light blue) volatility components fluctuating around the daily (dashed red) volatility level for S=26 intraday periods (corresponding to 15-min returns) and four trading days [Colour figure can be viewed at wileyonlinelibrary.com] with 𝜏=1,…,S·T. The demeaned continuously compounded intraday return is then obtained as r𝜏=100 × [ r𝜏−(1∕(S·T)) ∑S·T 𝜏=1 r𝜏],where r𝜏=[log P𝜏−log P𝜏−1].4 We then model the stochastic evolution of the intraday asset return r𝜏via r𝜏=√h𝜏𝜂𝜏,where h𝜏=Var(r𝜏|𝜏−1),(9) with 𝜏={r𝜏,rt−𝜏,···}and 𝜂𝜏 iid ∼F(0,1),whereF(0, 1) refers to a known parametric distribution with zero mean and unit variance. Popular candidates are the normal and the standardized Student's tdistribution. We follow Engle and Sokalska (2012) in assuming a mixed frequency three-component structure for the log-volatility process in order to capture the strong persistence and periodicity of the intraday volatility dynamics: h𝜏=exp{𝜔𝓁+z𝜏+lt}.(10) The period-specific intercepts 𝜔𝓁capture the well-known u-shaped periodicity in intraday volatilities (the “volatility smile”; see, e.g., Andersen & Bollerslev, 1997) and z𝜏and ltare both short-memory latent volatility components, which are realized at frequencies 𝜏 and t. More precisely, z𝜏takes new values every period 𝜏, while ltchanges its value every Speriods and stays constant for the next S−1 periods, until it changes again. Note that, although being driven by short-memory stochastic processes, the mixed-frequency GAS approach is able to generate long-memory type of dependence patterns at the intraday frequency by aggregating independent dynamic stochastic volatility components at different frequencies (see, e.g., Corsi, 2009; Granger, 1980). The dynamics of the three volatility components are schematically illustrated in Figure 1. 4Note that we skip the index ifor the institution for notational convenience. Also note that we do not model the mean dynamics of the intraday returns, since the data set illustrated in Section 4 does not show significant serial correlation in the return levels. We assume a GAS-driven autoregressive framework for the stochastic evolution of z𝜏and ltover time, that is, z𝜏=𝛼(z) 1z𝜏−1+𝛼(z) 2𝜉(z) 𝜏−1,(11) lt=𝛼(l) 1lt−1+𝛼(l) 2𝜉(l) t−1,(12) where the innovations 𝜉(z) 𝜏and 𝜉(l) tare defined as the scaled scores of log-likelihood contributions at the respective frequencies: 𝜉(z) 𝜏=s(z) 𝜏∇(z) 𝜏,∇(z) 𝜏=𝜕log 𝑓(r𝜏|𝜏−1) 𝜕z𝜏 ,(13) 𝜉(l) t=s(l) t∇(l) t, ∇(l) t=𝜕log 𝑓(r(t−1)·S+1,r(t−1)·S+2,…,rt·S|(t−1)·S) 𝜕lt .(14) While the definition of the innovation 𝜉(z) 𝜏in Equation (13) is standard in the context of GAS models, the definition of 𝜉(l) tin Equation (14) reflects the mixed frequency structure of the model by taking into account the informational content of all return information at frequency tfor the evolution of the process for lt.We denote the framework as volatility mixed frequency GAS (V-MF-GAS) model. A typical choice for the scaling coefficients s(z) 𝜏and s(l) t is the square-root of the inverse Fisher information, s(z) 𝜏= (E[(∇(z) 𝜏)2|𝜏−1])−0.5and s(l) t=(E[(∇(l) t)2|t−1])−0.5,which allows for a straight forward analysis of weak stationarity of the stochastic processes for z𝜏and lt. The analytical derivation of the Fisher information is however often problematic for complex stochastic models, such that a unit scaling appears to be an obvious choice in such situations. For the GAS specifications proposed in the following sections, we will employ analytical Fisher scaling whenever possible and unit scaling in all cases where analytical expressions for the Fisher information are not available. Our basic specification of the V-MF-GAS model (labeled V-MF-GAS-N) assumes a normal distribution for the 887 ECKERNKEMPER AND GRIBISCH conditional return innovation 𝜂𝜏in Equation (9). The Gaussian log-likelihood contribution is given by log 𝑓(r𝜏|𝜏−1)=−0.5log(2𝜋)−0.5log(h𝜏)−0.5r2 𝜏h−1 𝜏.(15) For the scores and Fisher information, we then obtain ∇(z) 𝜏=0.5(r2 𝜏h−1 𝜏−1),E[(∇(z) 𝜏)2|𝜏−1]=0.5,(16) ∇(l) t=0.5 S ∑ 𝑗=1(r2 (t−1)·S+𝑗h−1 (t−1)·S+𝑗−1), E[(∇(l) t)2|t−1]=0.5S. (17) Since intraday asset return data are typically characterized by heavy tails, it may be reasonable to replace the normal by a standardized Student's tdistribution. The Student tlog-likelihood contribution is given by log 𝑓(r𝜏|𝜏−1)=log Γ(𝜈+1 2)−log Γ(𝜈 2) −1 2log(𝜈−2)−1 2log(h𝜏) −(𝜈+1 2)log (1+r2 𝜏 (𝜈−2)h𝜏), (18) where 𝜈denotes the d.o.f. parameter. We then obtain ∇(z) 𝜏=(𝜈+1) 2(1+r2 𝜏 (𝜈−2)h𝜏)−1r2 𝜏 (𝜈−2)h𝜏 −1 2,(19) ∇(l) t= S ∑ 𝑗=1⎡⎢⎢⎣ (𝜈+1) 2(1+ r2 (t−1)·S+𝑗 (𝜈−2)h(t−1)·S+𝑗)−1 × r2 (t−1)·S+𝑗 (𝜈−2)h(t−1)·S+𝑗 −1 2⎤⎥⎥⎦ , (20) and E[(∇(z) 𝜏)2|𝜏−1]= 𝜈 2(𝜈+3),E[(∇(l) t)2|t−1]= S𝜈 2(𝜈+3). (21) The Student's tspecification of the V-MF-GAS model is labeled V-MF-GAS-t. Given the GAS innovations defined above, the stochastic processes for z𝜏and ltin Equations (11) and (12) are weakly stationary if |𝛼(z) 1|<1 and |𝛼(l) 1|<1, respectively. The V-MF-GAS models assume a mixed frequency volatility process where the “long-term” volatility component ltchanges at the daily frequency. Although the daily frequency is in line with the original approach of Engle and Sokalska (2012), the V-MF-GAS framework trivially allows to change the frequency to any desired change point. For example, we could allow for changing volatility levels every hour or every half-day and so on. In fact, we can also estimate the optimal change point by likelihood comparison over several hypothetical values for S (see Section 3.3 for details on the ML parameter estimation of the V-MF-GAS models). Initial estimations, however, showed that a typical ML estimate for the change point is the daily frequency, which led us to fix the frequency of the long-term component ltto the daily one. This choice is consistent with the work of Engle and Sokalska (2012), offers a good fit to the data (see Section 4), and simplifies model estimation without loss of flexibility. The literature offers several approaches for modeling the intraday periodicity {𝜔𝓁}S 𝓁=1: One could estimate all S intercepts individually or apply splines or Fourier transforms in order to save parameters (compare, e.g., Deo et al., 2006; Koopman et al., 2017; Payne, 1996). In our case, the data set comprises Sintraday return observations for a total of Ttrading days. Hence, we can easily estimate all Sintercepts individually in order to preserve a maximum of flexibility without unduly increasing overall estimation error.5 We finally note that it is also possible to model the autoregressive parameters of the z𝜏process periodically as in Rossi and Fantazzini (2015). However, we refrain to follow this approach since it dramatically increases the number of model parameters and an initial investigation did not indicate significant gains in model fit. 3.1.2 The V-MF-GAS-J Model It is widely documented that intraday return data may be affected by discrete price jumps induced by unusual news events like earnings surprises. These events cause infrequent large moves in the returns, which tend to cluster together. For example, market crashes can be realized as a series of jumps over a short period of time (see, e.g., Maheu & McCurdy, 2004). In order to account for intraday price jumps, we extend the V-MF-GAS approach to a mixed GAS-Jump model (labeled as V-MF-GAS-J) based on a compound Poisson jump process in the spirit of Maheu and McCurdy (2004): We decompose the return process into two components, 𝜖1𝜏and 𝜖2𝜏,where𝜖1𝜏represents “normal” news events inducing smooth price changes, and 𝜖2𝜏denotes news “surprises,” which cause relatively infrequent large price changes. Under the basic assumption of Gaussian distributed return innovations, we obtain r𝜏=𝜖1𝜏+𝜖2𝜏,(22) where 𝜖1𝜏=√h𝜏e𝜏,e𝜏 iid ∼N(0,1),(23) 5Note that each 𝜔𝓁is estimated using roughly Tobservations (e.g., T= 2265 in the empirical application; see Section 4). 888 ECKERNKEMPER AND GRIBISCH 𝜖2𝜏= n𝜏 ∑ 𝑗=1 c𝑗𝜏 −𝜃𝜆𝜏,c𝑗𝜏 iid ∼N(𝜃,𝛿2),(24) and n𝜏|𝜏−1∼Poi(𝜆𝜏)with time-varying news arrival intensity 𝜆𝜏. Here, we assume that the conditional volatility h𝜏is driven by the V-MF-GAS process of the previous section. Under Gaussian innovations e𝜏and cj𝜏,weobtain 𝑓(r𝜏|n𝜏=𝑗,𝜏−1)∼N(𝜃(𝑗−𝜆𝜏),h𝜏+𝑗𝛿2).(25) The corresponding volatility forecast is given by Var(r𝜏|𝜏−1)=h𝜏+𝜆𝜏(𝛿2+𝜃2). We model the jump intensity by an autoregressive GAS process, where 𝜆𝜏=exp{ 𝑓𝜏}  𝑓𝜏=𝜙0+𝜙1 𝑓𝜏−1+𝜙2𝜉(𝑗) 𝜏−1, with 𝜉(𝑗) 𝜏=s(𝑗) 𝜏∇(𝑗) 𝜏,∇(𝑗) 𝜏=𝜕log 𝑓(r𝜏|𝜏−1) 𝜕 𝑓𝜏 . We then obtain the likelihood contribution 𝑓(r𝜏|𝜏−1)= ∞ ∑ 𝑗=0 𝑓(r𝜏,n𝜏=𝑗|𝜏−1) = ∞ ∑ 𝑗=0 exp{−𝜆𝜏}𝜆𝑗 𝜏 𝑗!√2𝜋(h𝜏+𝑗𝛿2) exp {−(r𝜏+𝜃𝜆𝜏−𝜃𝑗)2 2(h𝜏+𝑗𝛿2)}. (26) The jump-adjusted scores are given by ∇(z) 𝜏=h𝜏 𝑓(r𝜏|𝜏−1) ∞ ∑ 𝑗=0 𝑓(r𝜏,n𝜏=𝑗|𝜏−1) ×[(r𝜏+𝜃𝜆𝜏−𝜃𝑗)2 2(h𝜏+𝑗𝛿2)2−1 2(h𝜏+𝑗𝛿2)], ∇(l) t= S ∑ 𝑗=1 ∇(z) (t−1)·S+𝑗, ∇(𝑗) 𝜏=𝜆𝜏 𝑓(r𝜏|𝜏−1) ∞ ∑ 𝑗=0 𝑓(r𝜏,n𝜏=𝑗|𝜏−1) ×[[𝑗 𝜆𝜏 −1]−(r𝜏+𝜃𝜆𝜏−𝜃𝑗)𝜃 h𝜏+𝑗𝛿2]. The infinite sums are truncated at 𝑗=20 for likelihood evaluation.6Since it appears difficult to derive a closed-form expression for the Fisher information under the V-MF-GAS-Jump framework, we follow the standard approach in the literature and set all scaling coefficients to one (s(z) 𝜏=s(l) t=s(𝑗) 𝜏=1). The resulting jump model is labeled V-MF-GAS-N-J. 6In our empirical application, we seldom encountered nonzero probability mass for more the 10 jumps (see also Maheu & McCurdy, 2004, for a similar truncation). Our results also appear to be robust to increasing truncation limits. While the V-MF-GAS-N-J model builds on Gaussian return innovations in the spirit of Maheu and McCurdy (2004), it appears natural to replace the normal in Equation (25) by its Student's tanalog. While the resulting V-MF-GAS-t-J specification is straightforward to implement, the practical estimation turned out to be unstable for the majority of the 378 time series in our empirical application (see Section 4) with frequent convergence problems during the numerical optimization of the log-likelihood. We attribute this instability to problems with model identification: both the jump process and the Student t innovations account for the heavy tails of the intraday return data, and it appears to be hard to discriminate between the contribution of the jump-part and the return innovation part. We therefore focus on Gaussian innovation processes. Details on the Student tjump specification are available upon request. 3.2 Dependence 3.2.1 The C-MF-GAS Model We employ a dynamic copula approach in order to model the time-varying dependence between the market and the individual asset return. In particular, we follow the Sklar (1959) theorem, which states that the joint conditional distribution can be decomposed into the marginal conditional distribution functions and the conditional copula. Consider the bivariate time series process {ri,𝜏,rm,𝜏}T·S 𝜏=1, where r·,𝜏follows one of the V-MF-GAS models of Section 3.1. The standardized return residuals obtain as 𝜂i,𝜏 =ri,𝜏 √Var(ri,𝜏|𝜏−1) and 𝜂m,𝜏 =rm,𝜏 √Var(rm,𝜏|𝜏−1) , (27) with the conditional distribution of 𝜂𝜏=(𝜂i,𝜏 ,𝜂 m,𝜏)′given 𝜏−1denoted by 𝜂𝜏|𝜏−1∼F𝜂i,𝜏 𝜂m,𝜏 (𝜂i,𝜏,𝜂 m,𝜏|𝜏−1).(28) According to Sklar's theorem, we can now write the joint distribution as F𝜂i,𝜏 𝜂m,𝜏 (𝜂i,𝜏,𝜂 m,𝜏|𝜏−1)=C(ui,𝜏,um,𝜏;𝜃𝜏|𝜏−1),(29) where C(·) denotes a conditional copula function with n-dimensional dynamic dependence parameter 𝜃𝜏= (𝜃1,𝜏,…,𝜃 n,𝜏)′and corresponding copula density c(·). The copula is defined as a distribution function on the two-dimensional hypercube with uniform marginals. The arguments ui,𝜏and um,𝜏of the copula function are obtained by the probability integral transform ui,𝜏 =F𝜂i,𝜏 (𝜂i,𝜏|𝜏−1)and um,𝜏 =F𝜂m,𝜏 (𝜂m,𝜏|𝜏−1),(30) with F𝜂·,𝜏 (𝜂·,𝜏|𝜏−1)being conditional distribution functions given 𝜏−1, as implied by the volatility models for ri,𝜏 and rm,𝜏.FortheV-MF-GAS-NandV-MF-GAS-tmodels, 889 ECKERNKEMPER AND GRIBISCH the marginal distributions F𝜂·,𝜏 are standard Gaussian and Student's t, respectively (compare the model specifications in Section 3.1.1). For the jump specifications, the distributions obtain as mixtures over the stochastic numbers of jumps (see Equation 26) and can be approximated by the empirical cdf of the estimated residual series {𝜂i,𝜏} and {𝜂m,𝜏}obtained after ML estimation of the model parameters. Empirical applications of conditional copula models for daily asset returns typically report evidence of long-memory type of dependence structures for the dependence parameters (see, e.g., Janus et al., 2014; Grossmass & Poon, 2015). We therefore employ the MF-GAS process of Section 3.1 in order to model the copula dynamics. Let 𝜃𝜏=g(𝜓𝜏),with (31) 𝜓𝜏=𝜔𝓁+ z𝜏+ lt,(32) where 𝜔𝓁=(𝜔1,𝓁,…,𝜔n,𝓁)′, z𝜏=(  z1,𝜏,…, zn,𝜏)′and  lt=(  l1,𝜏,…, ln,𝜏)′are n-dimensional dynamic intraday and daily dependence components and gis an n-dimensional monotonously increasing function, which maps the dependence components into the domain of the copula parameters. Note that we include periodic constants 𝜔𝓁in order to account for potential seasonality in the copula parameters. Let  z𝑗,𝜏 =𝛼(z) 𝑗,1 z𝑗,𝜏−1+𝛼(z) 𝑗,2 𝜉(z) 𝑗,𝜏−1(33)  l𝑗,t=𝛼(l) 𝑗,1l𝑗,t−1+𝛼(l) 𝑗,2 𝜉(l) 𝑗,t−1,(34) for 𝑗=1,…,n. The GAS innovations  𝜉(z) 𝑗,𝜏 and  𝜉(l) 𝑗,tare defined analogously to Equations (13) and (14): 𝜉( z) 𝑗,𝜏 =s( z) 𝑗,𝜏∇( z) 𝑗,𝜏,∇( z) 𝑗,𝜏 =𝜕log c(u𝜏|𝜏−1) 𝜕 z𝑗,𝜏 ,(35) 𝜉( l) 𝑗,t=s( l) 𝑗,t∇( l) 𝑗,t,∇( l) 𝑗,t =𝜕log c(u(t−1)·S+1,u(t−1)·S+2,…,ut·S|(t−1)·S) 𝜕 l𝑗,t ,(36) with u𝜏=(ui,𝜏 ,um,𝜏 )′. According to the previous section, we denote the framework as Copula(C)-MF-GAS. We investigate three popular copula functions, which account for different kinds of dependencies: 1. The bivariate Gaussian copula with time-varying correlation parameter 𝜃𝜏=𝜌𝜏∈(−1,1). The copula density is given by c(ui,𝜏,um,𝜏|𝜏−1)= 1 √1−𝜌2 𝜏 exp {− 𝜌2 𝜏(x2 i,𝜏 +x2 m,𝜏)−2𝜌𝜏xi,𝜏xm,𝜏 2(1−𝜌2 𝜏)}, (37) where xi,𝜏 =Φ −1(ui,𝜏)and xm,𝜏 =Φ −1(um,𝜏)with Φ−1(·) denoting the inverse of the Gaussian cdf. We define the link function gas 𝜌𝜏=g(𝜓𝜏)=exp(𝜓𝜏)−1 exp(𝜓𝜏)+1(38) and obtain ∇( z) 𝜏= 𝜌𝜏(1−x2 i,𝜏 −x2 m,𝜏)+(1+𝜌2 𝜏)xi,𝜏xm,𝜏 −𝜌3 𝜏 (𝜌2 𝜏−1)2 . 𝜌𝜏, (39) ∇( l) t= S ∑ 𝑗=1 ∇( z) (t−1)·q+𝑗,(40) where . 𝜌𝜏=𝜕𝜌𝜏∕𝜕 z𝜏. We use the square-root of the inverse Fisher information for the scaling of the GAS innovations with the according formulas given in Appendix A. 2. The bivariate Student's tcopula with time-varying correlation parameter 𝜃𝜏=𝜌𝜏∈(−1,1)and time-constant degrees of freedom parameter 𝜅>4. The copula density is given by c(ui,𝜏,um,𝜏|𝜏−1)= Γ(𝜅+2 2)Γ(𝜅 2) Γ(𝜅+1 2)√1−𝜌2 𝜏 ·[1+𝜅−1(1−𝜌2 𝜏)−1(x2 i,𝜏 +x2 m,𝜏 −2𝜌𝜏xi,𝜏xm,𝜏)]−(𝜅+2)∕2 [(1+x2 i,𝜏∕𝜅)(1+x2 m,𝜏∕𝜅)]−(𝜅+1)∕2, (41) where xi,𝜏=t−1(ui,𝜏;𝜅)andxm,𝜏=t−1(um,𝜏;𝜅)witht−1(·) denoting the inverse of the Student tcdf. The link function gis defined as in Equation (38), and we obtain ∇( z) 𝜏=(1−𝜌2 𝜏)−2 ×[(1+𝜌2 𝜏)(𝜋𝜏xi,𝜏xm,𝜏 −𝜌𝜏)−𝜌𝜏(𝜋𝜏x2 i,𝜏 +𝜋𝜏x2 m,𝜏 −2)]. 𝜌𝜏, (42) ∇( l) t= S ∑ 𝑗=1 ∇( z) (t−1)·q+𝑗,(43) where 𝜋𝜏=(𝜅+2)(𝜅+m𝜏)−1, m𝜏=(1−𝜌2 𝜏)−2(x2 i,𝜏 +x2 m,𝜏 −2𝜌𝜏xi,𝜏xm,𝜏), . 𝜌𝜏=𝜕𝜌𝜏∕𝜕 z𝜏. (44) We use the square-root of the inverse Fisher information for the scaling of the GAS innovations with the according formulas given in Appendix A. 3. A mixture of Clayton and rotated Clayton copula (labeled CrC Copula) with time-varying copula parameters 𝜃1𝜏>0, 𝜃2𝜏>0 and time-constant mixture weight w 890 ECKERNKEMPER AND GRIBISCH FIGURE 4 Mean correlation estimates 𝜌𝓁, averaged over time (together with the according pattern estimates 𝜔𝓁), the estimated periodic d.o.f. 𝜅𝓁, and the mean estimate  𝜆𝓁obtained by averaging over the time-varying correlations for each of the 𝓁=1,…,26 trading periods. All estimates are obtained under the C-MF-GAS-t𝜅𝓁model and the V-MF-GAS-tmodel for the margins. 𝜅estimates >40 are truncated to 40 for better visibility. The Student tcopula for more than 40 d.o.f. is virtually undistinguishable from the Gaussian case d.o.f.: Figure 4 depicts the mean correlation estimates 𝜌𝓁, averaged over trading days (together with the according pattern estimates 𝜔𝓁), the estimated periodic d.o.f. 𝜅𝓁, and the mean estimate  𝜆𝓁obtained by averaging over the time-varying correlations for each of the 𝓁=1,…,26 trading periods. We observe a correlation pattern similar to an inverted Ushape with less correlation at the beginning than at the end of the active trading hours and an overall increasing correlation trend. We also find an increasing level of tail dependence in the second half of the trading day which is caused by the high correlation level at the end of trading together with an associated decrease of the d.o.f. estimate. The correlation level and the tail dependence measure vary within (0.4, 0.66) and (0.02, 0.22) ((0.37, 0.64) and (0.00, 0.14)) for AXP (MSFT), respectively. We analyze the significance of the patterns for AXP and MSFT by Wald tests for the d.o.f. 𝜅𝓁and periodicity Ftests for the time series of correlations and tail dependence measures.10 All test results are significant at the 1% level with the exception of the d.o.f. estimates for MSFT. The variation of tail dependence over the trading day is particularly pronounced for 10The Wald test considers the null of equal d.o.f. 𝜅1=𝜅2=…=𝜅𝓁.The seasonality test is computed as a standard Ftest for a regression of the correlations and tail dependence measures on a set of 26 dummies, one dummy for each intraday trading period. AXP—a finding that is also reflected by the model fit: The AIC selects the C-MF-GAS-t𝜅model for MSFT and the C-MF-GAS-t𝜅𝓁model for AXP as the best fitting specifications. The Gaussian and the CrC copula are clearly rejected by the data. Figure 5 depicts estimates of the time-varying copula GAS processes for the two stocks and the four model specifications. The time series show a high degree of persistence and distinct patterns for AXP and MSFT. The daily copula component evolves smoothly over time and captures a major part of the overall variation in the dependence parameters. We observe frequent negative correlation peaks under the Gaussian copula, which are somewhat dampened by the Student tspecifications. Under the CrC copula, the two dependence processes for upperand lower tail dependence (CRC1and CRC2, respectively) evolve overall similar to the correlations of the Student t specifications. The aggregated estimation results for the complete set of 378 asset-market combinations are provided in the right panel of Table 2. The C-MF-GAS-t𝜅model is AIC preferred for 76% of the time series and the Gaussian and the CrC copulas are clearly rejected by the data. This result is in contrast to the findings of Koopman et al. (2018), who select the Gaussian copula as the best fitting specification within their copula approach on the tick-by-tick frequency. 897 ECKERNKEMPER AND GRIBISCH FIGURE 5 Estimates of the time-varying copula GAS parameters for the AXP and MSFT stocks and the C-MF-GAS-G, C-MF-GAS-t𝜅, C-MF-GAS-t𝜅𝓁, and C-MF-GAS-CrC models. All estimates are obtained under the V-MF-GAS-tmodel for the margins. Dotted grey line:   z𝜏+  lt; black line:   lt.CrCi:i-th time-varying parameter of the CrC copula The C-MF-GAS-t𝜅𝓁model is AIC preferred for a subset of 91 assets (about 24%), which are dominated by the Industrials, IT, and Materials sectors. We analyze the in-sample fit of the four copula models via the Anderson–Darling (AD) approach, which tests the null of independence of the Rosenblatt transformed 𝜂i,𝜏and 𝜂m,𝜏residuals, that is, the null of correct specification of the dependence structure (see Manner & Reznikova, 2012). The results are provided in the last line of Table 2. The in-sample residual analyses for the Student tcopulas with constant and periodic d.o.f show remarkably good results with 2.9% and 2.4% rejections while the Gaussian and CrC copulas are rejected in 14.6% and even 53.2% of the cases. Figure S2 reports sample averages of the estimated intraday correlation periodicities 𝜔𝓁,theStudenttd.o.f. 𝜅𝓁,and the average tail-dependence measures 𝜆𝓁,computedover the assets in each of the 11 industry sectors. All estimates are obtained under the C-MF-GAS-t𝜅𝓁model. The correlation pattern show the inverted Ushape and increasing correlation trend, which have already been found for the AXP and MSFT stocks. This structure appears consistent over all industry sectors. The correlation “break-downs” at the beginning and the end of the active trading hours are accompanied by comparatively low 𝜅estimates at 4:00 p.m. The high correlation at the afternoon together with a slight tendency of decreasing 𝜅𝓁results in a common pattern of increasing tail dependence over the trading day. For all industry sectors, the tail-dependence coefficient reaches its maximum at 4 p.m. with a maximum value of 0.16 for the industrial stocks. The positive correlation trend along with the relatively low correlation levels at the beginning and the end of the active trading hours can be explained by the information flow over the 24-h cycle: the rather low correlation level at the start of trading is explained by a relatively large idiosyncratic information component, which is generated by the incorporation of overnight information. During the day however, available pricing-relevant information can directly be processed into stock prices. At the end of active trading, it can be expected that many traders unwind their positions in order to limit the overnight risk. This induces again a rise in the idiosyncratic information component and generates a decrease in the correlations, which induces the Ushape (see also the discussion in Koopman et al., 2018). To summarize our results, we find that the C-MF-GAS-t𝜅 and C-MF-GAS-t𝜅𝓁models provide a good fit to the time-varying and highly persistent intraday dependence processes. We also find an inverted Ushaped intraday correlation pattern with a positive common trend over the trading day and a positive trend for the tail-dependence 898 ECKERNKEMPER AND GRIBISCH coefficient. The correlation pattern is consistent with the findings reported by Bibinger et al. (2019) and Koopman et al. (2018). 4.4 (Δ)ECoVaR We now select the flexible and in-sample preferred V-MF-GAS-t-andC-MF-GAS-t𝜅𝓁copula models for the margins and the dependence structure in order to compute model-based forecasts for the ECoVaR measure. These forecasts are obtained in a straight-forward fashion by applying the approach of Mainik and Schaanning (2014) and Reboredo and Ugolini (2015) as detailed in Section 3 and plugging in the parameter estimates and dynamic volatility and dependence forecasts from the GAS-recursions. In our application, we consider both the ECoVaR (denoted ECoVaRi|m 𝛼,𝛽,𝜏)andtheΔECoVaR (denoted ΔECoVaRi|m 𝛼,𝛽,𝜏) as defined in Equation (5) of Section 2. All forecasts are computed for the VaR levels 𝛼=𝛽=0.05. The ECoVaR reflects the absolute risk level of the asset in extreme market situations and is therefore closely related to the asset's VaR and the individual volatility dynamics. The ΔECoVaR in contrast measures the exposure of asset ito system-wide distress relative to normal market conditions. It can therefore be interpreted as a measure of the robustness of the asset's VaR to turbulent market conditions. For example, an individual asset (or a portfolio) could feature a high absolute ECoVaR, but alowΔECoVaR which means that the asset has a high risk level but is rather insensitive to market conditions turning from normal to distress. 4.4.1 In-sample Analysis Figure 6 depicts the time-series of in-sample ECoVaRand ΔECoVaR forecasts for the bivariate asset-market relationships of AXP and MSFT. We observe strong serial dependence for both measures and the ΔECoVaR appears more noisy and less affected by the financial crisis episode of 2008 and 2009 relative to the ECoVaR. Comparing the time-series plots in Figure 6 to the dynamic volatility and dependence estimates in Figures 2 and 5 reveals a strong correspondence of the ECoVaRand ΔECoVaR dynamics to the volatility and dependence pattern respectively. As expected, we find that episodes of extreme risk like the financial crisis are typically accompanied by high values of the ΔECoVaR (e.g., up to 160% additional ECoVaR risk for AXP relative to a normal market situation). But we also find situations where the ΔECoVaR suddenly drops down while the ECoVaR itself persists on a rather high level (see, e.g., AXP in 2009). Such situations are generated by a sudden decrease in the asset-market dependence FIGURE 6 In-sample ECoVaR (lower panel) and ΔECoVaR (upper panel) forecasts for the bivariate asset-market relationships of AXP and MSFT. The forecasts are generated under the V-MF-GAS-tand C-MF-GAS-t𝜅𝓁models for the margins and the dependence structure. The computation of the forecasts is based on the copula approach of Mainik and Schaanning (2014) and Reboredo and Ugolini (2015) (see Section 3.1) conditional on the full sample parameter estimates and the copula parameter forecasts from the GAS recursions 899 ECKERNKEMPER AND GRIBISCH while the individual volatility level remains persistently high (compare the dependence pattern in Figure 5). The dependence of the ECoVaR and ΔECoVaR measures on the intraday volatility and dependence process suggests the existence of intraday seasonalities in the ECoVaR forecasts. These seasonalities are indirectly modeled by the periodicities 𝜔𝓁and 𝜔𝓁in the MF-GAS recursions for the volatilities and dynamic copula parameters. The knowledge of such regularities in the ECoVaR measures is important for the active portfolio manager since regular risk-minimizing shiftings of portfolio components due to neglected seasonal effects might induce unnecessary managing costs. Figure 7 depicts mean values of the in-sample ECoVaR and ΔECoVaR estimates for AXP and MSFT averaged over time for each of the 26 intraday periods of the trading day. We find a significant inverse Ushape in the ECoVaR estimates and a positive trend for the ΔECoVaR. These patterns are generated by the volatility and correlation periodicities analyzed in Sections 4.2 and 4.3 and confirm the volatility and correlation sensitivity of the ECoVaR and ΔECoVaR measures, respectively. Figure S3 depicts heat-plots in order to illustrate the variation of the intraday periodicity in the ECoVaR and ΔECoVaR measures over the 378 assets of the complete data set. We observe an overall increasing trend of the ΔECoVaR, which reaches its peak between 2:15 and 3:45 PM and finally fades out at a reduced level at 4:00 PM. The highest ΔECoVaR levels are obtained for the Financial, Industrial , IT , and Material sectors, which show a particularly high risk level at the afternoon hours. Almost all ECoVaR estimates exhibit the inverted Ushape generated by the volatility process. The lowest level (highest risk) of the ECoVaR is always obtained at the starting of active trading at 9:45 a.m. The ECoVaR then increases and finally slightly decreases again after 3:45 p.m. The lowest average ECoVaRs are found for the Consumer Staples and Utilities Sectors. 4.4.2 Out-of-sample Analysis We now turn to an analysis of the out-of-sample (Δ)ECoVaR forecasting performance. Here, we focus on the importance of modeling the intraday volatility, correlation, and nonlinear dependence patterns. In particular, we investigate the performance of four different models (copula and margins), which are all based on the V-MF-GAS-tand the C-MF-GAS-t𝜅𝓁specifications but account for different aspects of intraday seasonality, that is, seasonality in the volatilities, seasonality in the FIGURE 7 Estimated intraday periodicities of the ECoVaRand ΔECoVaR forecasts for the bivariate asset-market relationships of AXP (left panel) and MSFT (right panel). The estimated periodicities are obtained as sample averages over the T=2,265 observations for each intraday period 𝓁=1,…,S=26. The forecasts are generated under the V-MF-GAS-tand C-MF-GAS-t𝜅𝓁models for the margins and the dependence structure. The computation of the forecasts is based on the copula approach of Mainik and Schaanning (2014) and Reboredo and Ugolini (2015) (see Section 3.1) conditional on the full sample parameter estimates and the copula parameter forecasts from the GAS recursions 900 ECKERNKEMPER AND GRIBISCH Margins Copula Periodicity in Periodicity in Periodicity in Volatility Correlation Degrees of Freedom Model 1 √√√ Model 2 √ √ × Model 3 √×× Model 4 × × × TABLE 4 Overview on the competing model specifications for the out-of-sample forecasting evaluation correlations, and seasonality in the d.o.f. Table 4 gives an overview on the models. Because the ECoVaR is a special case of a standard VaR measure, we can readily apply the classical backtesting approaches of the VaR literature in order to compare the out-of-sample performance of various model specifications and periodic structures. Here, we condition on those data points, where the market exceeds its VaR level (see Girardi & Ergün, 2013). In particular, we rely on the standard Kupiec (1995) and Christoffersen (1998) VaR hit-rate tests for those periods, where rm,𝜏 ≤VaRm 𝛼,𝜏.Wedefinethe “hit sequence” of ECoVaR violations as Ii|m 𝜏∗={1ifri,𝜏∗≤ECoVaRi|m 𝛼,𝛽,𝜏∗ 0 else, where the index 𝜏∗refers to the subsample of observations of size T∗,whererm,𝜏 ≤VaRm 𝛼,𝜏.Adequateforecasts of the ECoVaR should satisfy unconditional coverage, that is, P(Ii|m 𝜏∗=1)=𝛼, which can be tested by the likelihood-ratio (LR) test of Kupiec (1995). A sensible ECoVaR forecasting approach should however also account for the temporal dependence in the ECoVaR estimates. The null of independence in the hit-rate sequence can be tested by the LR independence test proposed by Christoffersen (1998) against the alternative of first order Markov dependence. The conditional coverage test of Christoffersen (1998) then jointly tests the null of unconditional coverage and independence by combining the two individual LR testing procedures. Unfortunately, it is not possible to design direct backtesting devices in order to investigate the effect of neglected periodicities on the ΔECoVaR. The key problem is that we are not able to observe the “true” ΔECoVaR or any series of hits that can be used for backtesting. We therefore follow an alternative approach and test independence, unconditional coverage and conditional coverage jointly for the two constituents of the ΔECoVaR: the ECoVaR, ECoVaRi|m 𝛼,𝛽,𝜏, and the benchmark ECoVaR, ECoVaRi|benchm 𝛽,𝜏 (see Equation 5). Our joint level-𝛼test is conducted via separate Kupiec (1995) or Christoffersen (1998) hit-rate test for the ECoVaR and the benchmark ECoVaR with Bonferroni correction. The resulting test tends to be conservative with significance level 𝛼∗≤𝛼, which accommodates the data-rich environment and the related overfitting issue since the huge intraday sample sizes easily drive the individual test statistics into significance. We obtain the (Δ)ECoVaR forecasts for all 378 asset-market combinations by splitting the data in the middle and reserving the second half of the time series as the forecasting window. We consider two separate forecasting periods: the period from July 2, 2008, to December 31, 2009, with comparatively high market volatility triggered by the financial crisis, and the relatively calm post-crisis period from January 2, 2010, to December 31, 2012. The two periods together cover a total of 1133 ×26 =29458 periods. Forecasts are generated iteratively with a rolling window scheme11 and the models are re-estimated at the end of each month. Hence, we obtain overall 1133 forecasts for each of the S=26 intraday periods. As a natural competitor for our MF-GAS approach, we consider the fractionally integrated GAS (FIGAS) model of Opschoor and Lucas (2019). The model was originally proposed for the joint modeling of daily asset return vectors and realized kernels but is easily adjusted to intraday return series. Let r∗ 𝜏=(ri,𝜏,rm,𝜏)′denote the bivariate return vector comprising the individual asset return and the market return. Under the FIGAS structure, we obtain r∗ 𝜏|𝜏−1∼t2(V𝜏,𝛾 𝓁),V𝜏=Ω𝓁 1−𝛽+(1−(1−L)d(1−𝜙L) 1−𝛽L)S∗ 𝜏, (53) where t2(V𝜏,𝛾𝓁) denotes a bivariate Student's tdistribution with zero mean, covariance matrix V𝜏=(Vi𝑗,𝜏),and(possibly) periodic scalar-valued d.o.f. parameter 𝛾𝓁.Ldenotes the lag-operator, Ω𝓁is a p.d. parameter matrix representing the unconditional covariance matrix of r∗ 𝜏,0<𝛽<1, d≥𝛽, and S∗ 𝜏is a p.d. GAS innovation matrix with S∗ 𝜏=S𝜏+V𝜏,S𝜏=S𝜏 𝜕log 𝑓(r∗ 𝜏|V𝜏) 𝜕V𝜏 S′ 𝜏, where S𝜏=√2V𝜏.S∗ 𝜏then obtains as S∗ 𝜏=𝜔𝜏r∗ 𝜏r∗ 𝜏′,with w𝜏=(𝛾𝓁+2)(𝛾𝓁−2+r∗ 𝜏′V−1 𝜏r∗ 𝜏)−1. See Opschoor and Lucas (2019) for details on the model, its derivation, and the ML estimation of the model parameters. The FIGAS accounts for “true” long-memory, 11The estimation window size is 1132 ×26 =29432 periods. 901 ECKERNKEMPER AND GRIBISCH TABLE 5 Out-of-sample (Δ)ECoVaR backtesting results: July 2, 2008, to December 31, 2012 ECoVaR ΔECoVaR Specification (i) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 22 22 17 323 369 372 367 378 (5.8%) (5.8%) (4.5%) (85.4%) (97.6%) (98.4%) (97.1%) (100.0%) Ind-test 25 22 21 5 57 53 50 66 (6.6%) (5.8%) (5.6%) (1.3%) (15.1%) (14.0%) (13.2%) (17.5%) CC-test 28 30 26 276 295 298 285 378 (7.4%) (7.9%) (6.9%) (73.0%) (78.0%) (78.8%) (75.4%) (100.0%) 𝛼=0.05 UC-test 49 48 46 353 377 377 376 378 (13.0%) (12.7%) (12.2%) (93.4%) (99.7%) (99.7%) (99.5%) (100.0%) Ind-test5047552181828397 (13.2%) (12.4%) (14.6%) (5.6%) (21.4%) (21.7%) (22.0%) (25.7%) CC-test 59 60 62 334 317 320 313 378 (15.6%) (15.9%) (16.4%) (88.4%) (83.9%) (84.7%) (82.8%) (100.0%) Specification (ii) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 76 69 50 72 364 370 367 375 (20.1%) (18.3%) (13.2%) (19.0%) (96.3%) (97.9%) (97.1%) (99.2%) Ind-test 38 42 45 96 114 109 114 309 (10.1%) (11.1%) (11.9%) (25.4%) (30.2%) (28.8%) (30.2%) (81.7%) CC-test 80 81 76 136 337 340 334 378 (21.2%) (21.4%) (20.1%) (36.0%) (89.2%) (89.9%) (88.4%) (100.0%) 𝛼=0.05 UC-test 134 119 98 111 375 375 372 377 (35.4%) (31.5%) (25.9%) (29.4%) (99.2%) (99.2%) (98.4%) (99.7%) Ind-test 86 90 100 156 150 148 152 331 (22.8%) (23.8%) (26.5%) (41.3%) (39.7%) (39.2%) (40.2%) (87.6%) CC-test 141 128 124 202 351 351 349 378 (37.3%) (33.9%) (32.8%) (53.4%) (92.9%) (92.9%) (92.3%) (100.0%) Specification (iii) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 308 313 319 378 345 351 357 378 (81.5%) (82.8%) (84.4%) (100.0%) (91.3%) (92.9%) (94.4%) (100.0%) Ind-test 9 9 13 2 20 21 21 213 (2.4%) (2.4%) (3.4%) (0.5%) (5.3%) (5.6%) (5.6%) (56.3%) CC-test 269 296 317 378 287 310 320 378 (71.2%) (78.3%) (83.9%) (100.0%) (75.9%) (82.0%) (84.7%) (100.0%) 𝛼=0.05 UC-test 330 335 349 378 358 367 368 378 (87.3%) (88.6%) (92.3%) (100.0%) (94.7%) (97.1%) (97.4%) (100.0%) Ind-test28334316413846245 (7.4%) (8.7%) (11.4%) (4.2%) (10.8%) (10.1%) (12.2%) (64.8%) CC-test 328 331 339 378 330 342 353 378 (86.8%) (87.6%) (89.7%) (100.0%) (87.3%) (90.5%) (93.4%) (100.0%) Specification (iv) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 18 17 22 169 378 376 376 378 (4.8%) (4.5%) (5.8%) (44.7%) (100.0%) (99.5%) (99.5%) (100.0%) Ind-test 15 21 27 8 44 48 53 213 (4.0%) (5.6%) (7.1%) (2.1%) (11.6%) (12.7%) (14.0%) (56.3%) CC-test 29 27 38 156 362 351 355 378 (7.7%) (7.1%) (10.1%) (41.3%) (95.8%) (92.9%) (93.9%) (100.0%) 𝛼=0.05 UC-test 52 50 53 246 378 378 378 378 (13.8%) (13.2%) (14.0%) (65.1%) (100.0%) (100.0%) (100.0%) (100.0%) Ind-test49526539727782242 (13.0%) (13.8%) (17.2%) (10.3%) (19.0%) (20.4%) (21.7%) (64.0%) CC-test 65 67 78 225 370 363 365 378 (17.2%) (17.7%) (20.6%) (59.5%) (97.9%) (96.0%) (96.6%) (100.0%) Note: The table shows the number and percentage of rejections of the unconditional coverage (UC), independence (Ind) and conditional coverage (CC) test for all 378 institutions at the 1% and the 5% significance level. Model 1 to Model 4 refer to the periodic model structures giveninTable4. 902 ECKERNKEMPER AND GRIBISCH TABLE 6 Out-of-sample (Δ)ECoVaR backtesting results: July 2, 2008, to December 31, 2009 ECoVaR ΔECoVaR Specification (i) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 23 19 18 26 357 354 346 363 (6.1%) (5.0%) (4.8%) (6.9%) (94.4%) (93.7%) (91.5%) (96.0%) Ind-test 12 8 9 2 36 35 29 42 (3.2%) (2.1%) (2.4%) (0.5%) (9.5%) (9.3%) (7.7%) (11.1%) CC-test 20 19 18 16 297 295 278 336 (5.3%) (5.0%) (4.8%) (4.2%) (78.6%) (78.0%) (73.5%) (88.9%) 𝛼=0.05 UC-test 58 52 48 69 366 367 363 374 (15.3%) (13.8%) (12.7%) (18.3%) (96.8%) (97.1%) (96.0%) (98.9%) Ind-test 33 34 27 4 63 62 57 72 (8.7%) (9.0%) (7.1%) (1.1%) (16.7%) (16.4%) (15.1%) (19.0%) CC-test 48 39 40 52 337 341 330 363 (12.7%) (10.3%) (10.6%) (13.8%) (89.2%) (90.2%) (87.3%) (96.0%) Specification (ii) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 216 208 204 263 374 375 374 377 (57.1%) (55.0%) (54.0%) (69.6%) (98.9%) (99.2%) (98.9%) (99.7%) Ind-test20222322 61 61 59 88 (5.3%) (5.8%) (6.1%) (5.8%) (16.1%) (16.1%) (15.6%) (23.3%) CC-test 189 184 179 224 374 374 374 378 (50.0%) (48.7%) (47.4%) (59.3%) (98.9%) (98.9%) (98.9%) (100.0%) 𝛼=0.05 UC-test 273 270 260 293 375 375 375 377 (72.2%) (71.4%) (68.8%) (77.5%) (99.2%) (99.2%) (99.2%) (99.7%) Ind-test 54 51 65 57 97 93 104 127 (14.3%) (13.5%) (17.2%) (15.1%) (25.7%) (24.6%) (27.5%) (33.6%) CC-test 253 245 239 285 376 376 376 378 (66.9%) (64.8%) (63.2%) (75.4%) (99.5%) (99.5%) (99.5%) (100.0%) Specification (iii) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 107 110 128 216 313 321 315 370 (28.3%) (29.1%) (33.9%) (57.1%) (82.8%) (84.9%) (83.3%) (97.9%) Ind-test4371 68757 (1.1%) (0.8%) (1.9%) (0.3%) (1.6%) (2.1%) (1.9%) (15.1%) CC-test 98 104 123 191 221 226 227 358 (25.9%) (27.5%) (32.5%) (50.5%) (58.5%) (59.8%) (60.1%) (94.7%) 𝛼=0.05 UC-test 180 183 201 289 355 358 355 377 (47.6%) (48.4%) (53.2%) (76.5%) (93.9%) (94.7%) (93.9%) (99.7%) Ind-test22272410 24 23 26 97 (5.8%) (7.1%) (6.3%) (2.6%) (6.3%) (6.1%) (6.9%) (25.7%) CC-test 161 166 184 269 278 289 289 372 (42.6%) (43.9%) (48.7%) (71.2%) (73.5%) (76.5%) (76.5%) (98.4%) Specification (iv) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 7 16 5 4 378 378 378 378 (1.9%) (4.2%) (1.3%) (1.1%) (100.0%) (100.0%) (100.0%) (100.0%) Ind-test11108 2 20191456 (2.9%) (2.6%) (2.1%) (0.5%) (5.3%) (5.0%) (3.7%) (14.8%) CC-test 10 14 8 3 378 378 378 378 (2.6%) (3.7%) (2.1%) (0.8%) (100.0%) (100.0%) (100.0%) (100.0%) 𝛼=0.05 UC-test 31 40 29 23 378 378 378 378 (8.2%) (10.6%) (7.7%) (6.1%) (100.0%) (100.0%) (100.0%) (100.0%) Ind-test34353710 41 42 42 75 (9.0%) (9.3%) (9.8%) (2.6%) (10.8%) (11.1%) (11.1%) (19.8%) CC-test 35 42 28 16 378 378 378 378 (9.3%) (11.1%) (7.4%) (4.2%) (100.0%) (100.0%) (100.0%) (100.0%) Note: The table shows the number and percentage of rejections of the unconditional coverage (UC), independence (Ind), and conditional coverage (CC) test for all 378 institutions at the 1% and the 5% significance level. Model 1 to Model 4 refer to the periodic model structures given in Table 4. 903 ECKERNKEMPER AND GRIBISCH TABLE 7 Out-of-sample (Δ)ECoVaR backtesting results: January 2, 2010, to December 31, 2012 ECoVaR ΔECoVaR Specification (i) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 5 13 9 327 251 258 244 375 (1.3%) (3.4%) (2.4%) (86.5%) (66.4%) (68.3%) (64.6%) (99.2%) Ind-test 11 14 15 5 43 47 44 58 (2.9%) (3.7%) (4.0%) (1.3%) (11.4%) (12.4%) (11.6%) (15.3%) CC-test 12 16 15 310 137 134 128 375 (3.2%) (4.2%) (4.0%) (82.0%) (36.2%) (35.4%) (33.9%) (99.2%) 𝛼=0.05 UC-test 30 30 36 365 303 310 304 378 (7.9%) (7.9%) (9.5%) (96.6%) (80.2%) (82.0%) (80.4%) (100.0%) Ind-test36363217 66727195 (9.5%) (9.5%) (8.5%) (4.5%) (17.5%) (19.0%) (18.8%) (25.1%) CC-test 38 39 47 349 183 201 189 376 (10.1%) (10.3%) (12.4%) (92.3%) (48.4%) (53.2%) (50.0%) (99.5%) Specification (ii) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 12 17 28 361 134 139 126 372 (3.2%) (4.5%) (7.4%) (95.5%) (35.4%) (36.8%) (33.3%) (98.4%) Ind-test 7 12 10 5 70 70 73 272 (1.9%) (3.2%) (2.6%) (1.3%) (18.5%) (18.5%) (19.3%) (72.0%) CC-test 14 17 24 352 118 123 129 377 (3.7%) (4.5%) (6.3%) (93.1%) (31.2%) (32.5%) (34.1%) (99.7%) 𝛼=0.05 UC-test 42 54 74 374 207 216 232 378 (11.1%) (14.3%) (19.6%) (98.9%) (54.8%) (57.1%) (61.4%) (100.0%) Ind-test 32 31 32 29 101 106 100 306 (8.5%) (8.2%) (8.5%) (7.7%) (26.7%) (28.0%) (26.5%) (81.0%) CC-test 44 63 75 368 165 173 180 377 (11.6%) (16.7%) (19.8%) (97.4%) (43.7%) (45.8%) (47.6%) (99.7%) Specification (iii) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 293 311 324 378 304 317 328 378 (77.5%) (82.3%) (85.7%) (100.0%) (80.4%) (83.9%) (86.8%) (100.0%) Ind-test 3 5 4 3 16 15 17 234 (0.8%) (1.3%) (1.1%) (0.8%) (4.2%) (4.0%) (4.5%) (61.9%) CC-test 278 296 309 378 264 279 301 378 (73.5%) (78.3%) (81.7%) (100.0%) (69.8%) (73.8%) (79.6%) (100.0%) 𝛼=0.05 UC-test 345 350 361 378 341 347 355 378 (91.3%) (92.6%) (95.5%) (100.0%) (90.2%) (91.8%) (93.9%) (100.0%) Ind-test15192113 292641264 (4.0%) (5.0%) (5.6%) (3.4%) (7.7%) (6.9%) (10.8%) (69.8%) CC-test 325 338 350 378 312 327 338 378 (86.0%) (89.4%) (92.6%) (100.0%) (82.5%) (86.5%) (89.4%) (100.0%) Specification (iv) Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 𝛼=0.01 UC-test 18 10 27 253 253 232 231 374 (4.8%) (2.6%) (7.1%) (66.9%) (66.9%) (61.4%) (61.1%) (98.9%) Ind-test 12 10 14 8 39 45 49 244 (3.2%) (2.6%) (3.7%) (2.1%) (10.3%) (11.9%) (13.0%) (64.6%) CC-test 19 11 29 239 184 159 173 378 (5.0%) (2.9%) (7.7%) (63.2%) (48.7%) (42.1%) (45.8%) (100.0%) 𝛼=0.05 UC-test 48 44 64 324 295 273 284 378 (12.7%) (11.6%) (16.9%) (85.7%) (78.0%) (72.2%) (75.1%) (100.0%) Ind-test43535224 757585279 (11.4%) (14.0%) (13.8%) (6.3%) (19.8%) (19.8%) (22.5%) (73.8%) CC-test 71 71 85 315 226 208 224 378 (18.8%) (18.8%) (22.5%) (83.3%) (59.8%) (55.0%) (59.3%) (100.0%) Note: The table shows the number and percentage of rejections of the unconditional coverage (UC), independence (Ind) and conditional coverage (CC) test for all 378 institutions at the 1% and the 5% significance level. Model 1 to Model 4 refer to the periodic model structures given in Table 4. 904 ECKERNKEMPER AND GRIBISCH while the MF-GAS approximates long-memory via a mixed-frequency component framework, which combines short-memory GAS processes at different frequencies. Following the Sklar (1959) theorem, the conditional bivariate Student's tdistribution in (53) can be decomposed into univariate Student's tmarginals coupled with the Student tcopula with conditional correlation parameter 𝜌𝜏(see, e.g., Nelsen, 2006), similar to the MF-GAS setting. According to the FIGAS model outlined above, we obtain ri,𝜏|𝜏−1∼t1(V11,𝜏,𝛾 𝓁),rm,𝜏|𝜏−1∼t1(V22,𝜏,𝛾 𝓁) and the Student tcopula C(ui,𝜏,um,𝜏 ;𝜌𝜏,𝛾 𝓁|𝜏−1)with 𝜌𝜏=V12,𝜏∕√V11,𝜏V22,𝜏 and (ui,𝜏,um,𝜏) being the probability integral transforms of (ri,𝜏,rm,𝜏) based on the marginal Student tcdf (compare Equation 30 and Equation 41 for details on the Student tcopula). Note that the parameter matrix Ω𝓁drives the unconditional covariance process of r∗ 𝜏. Hence, we can directly apply models 1–4 of Table 4 for modeling the periodic structures of the FIGAS model. For this purpose, we separate intraday periodicity in volatilities (marginals) and correlations (copula) via a variance/correlation decomposition of Ω𝓁, whose elements are then estimated individually for each intraday trading period 𝓁, similar to the MF-GAS setting. We consider four different specifications for out-of-sample forecasting of the (Δ)ECoVaR: (i) The in-sample preferred V-MF-GAS-t-and C-MF-GAS-t𝜅𝓁copula models for the margins and the dependence structure. (ii) Similar to Specification (i) but without the intraday volatility/dependence components z𝜏and  z𝜏(z𝜏=  z𝜏=0∀𝜏). The model structure then implies a GAS setting, where intraday variation is restricted to the periodicities. (iii) Similar to Specification (i) but without the daily volatility/dependence components l𝜏and  l𝜏(lt=  lt=0∀t). The model structure then implies a standard short-memory GAS setting without mixed frequency component. (iv) The FIGAS model of Opschoor and Lucas (2019) detailed above. For each of the specifications (i)-(iv) we consider the four seasonal structures of models 1–4 in Table 4. Tables 5–7 provide the results on the out-of-sample (Δ)ECoVaR backtesting application. The tables report the absolute and relative number of rejections of the null of unconditional coverage (UC), independence of the hit-rate sequence (Ind), and conditional coverage (CC) at the 1% and the 5% significance level for all 378 asset-market combinations. For the discussion of the test results, we focus on the 5% level and the conditional coverage test, which aggregates both unconditional coverage and independence. Table 5 shows results for the complete out-of-sample period from 2008 to 2012. The overall best ECoVaR performance is achieved for the MF-GAS setting of Specification (i) and Model 1. Hence, a flexible dynamic modeling with periodicity in both, the volatilities and the dependence structure, appears important for ECoVaR forecasting. The FIGAS approach comes relatively close to the MF-GAS setting but is overall outperformed. The worst results are obtained for Specification (iii). We therefore conclude that long-term persistence as generated by the daily volatiltiy/dependence component appears to be crucial for ECoVaR forecasting. Turning to the ΔECoVaR, we observe a sharp increase in the rejection rates, which is explained by the joint testing for the ECoVaR and the benchmark ECoVaR, where the hit-rate tests for the latter involve thousands of data points, which easily drive the test statistics into significance. The best results are again obtained for the MF-GAS setting under Specification (i). From the discussion of Section 4.4.1, we would expect that the dependence pattern matters most for the ΔECoVaR. However, the overall best performance is achieved for Model 3, which neither contains periodicity in the correlation, nor the d.o.f. parameter. Nevertheless, Models 1 and 2 perform overall similar to Model 3 and only Model 4, which contains neither volatility nor dependence patterns, is clearly rejected by the data. The results for the crisis period from 2008 to 2009 in Table 6 show a really good ECoVaR performance for the FIGAS model of Specification (iv) (Model 4) while the MF-GAS of Specification (i) is second best under Model 2. For the ΔECoVaR however the FIGAS is clearly outperformed with 100% rejection rates for all periodic model structures. The best ΔECoVaR performance is achieved for Specification (iii) under Model 1, that is, a standard GAS model without daily component and with periodicity in volatility and dependence—a result that may be explained by short-lived volatility and dependence shocks generated by single crisis events. The best results for the calm period from 2010 to 2012 are obtained under the full periodic structure of Model 1, that is, periodicity for volatility and dependence (see Table 7). While the MF-GAS of Specification (i) performs best for the ECoVaR, the best ΔECoVaR forecasts are obtained under Specification (ii), which only contains a daily volatility component and no intraday dynamics. Overall, we can conclude that the MF-GAS approach provides a solid out-of-sample (Δ)ECoVaR forecasting performance, where Models 1 and 2 with periodicity in both, the correlation and the volatility process, are typically the best performing models for the ECoVaR and the (Δ)ECoVaR measure. Except for the crisis period, the worst performance is typically obtained for Model 4, 905 ECKERNKEMPER AND GRIBISCH which contains no periodicity at all. Hence, accounting for seasonal patterns in intraday volatilities appears to matter most for a solid (Δ)ECoVaR forecasting performance. Interestingly, while we would expect that the volatility (correlation) pattern is most important for the ECoVaR (ΔECoVaR), we cannot clearly identify separate effects of the respective periodicities in the hit-rate based backtesting application. In fact, accounting for periodicities in volatilities and correlations appears to be important for both measures. Our hit-rate based backtesting experiment gives insights on the forecasting performance for a huge empirical data set. However, the analysis appears restrained due to the unavailability of the “true” (Δ)ECoVaR as a solid benchmark for assessing the forecasting performance. The results further represent an aggregation over various asset/market relations with different degrees of periodicity in the volatility and dependence structure. In order to obtain a deeper insight into the effects of the periodic structures in volatilities and dependencies on the (Δ)ECoVaR forecasts, we conduct an additional simulation-based forecasting experiment based on an artificial data set, which shows significant periodic effects in both, volatilities and dependence, and for which we observe the true simulated (Δ)ECoVaR measures. In particular, we use our most flexible model specification with V-MF-GAS-t-and C-MF-GAS-t𝜅𝓁processes for the margins and the copula (Specification (i)) in order to simulate an intraday return series of length equal to our empirical data (T=2265, S= 26). For the parametrization, we choose our estimates for the PFG stock, which resembles data with significant intraday periodicity in all three cases: volatilities, correlations, and tail-dependence measures. The periodic patterns used for the simulation are depicted in Figure S4. We then conduct an out-of-sample forecasting experiment identical to the one outlined above but restricted to the V-MF-GAS-tand C-MF-GAS-t𝜅𝓁processes of Specification (i). Because we observe the true ECoVaR and ΔECoVaR measures from thesimulation,weareabletoanalyzetheresultingforecasting errors directly. Figure 8 uses boxplots in order to depict the distribution of the ECoVaR forecasting errors for the 26 intraday periods and the four periodic model structures of Table 4. The ECoVaR forecasts stay relatively unaffected by neglected intraday patterns in (non-)linear dependencies (Model 1 to Model 3) but are clearly influenced by the volatility pattern. The resulting biases for Model 4 reflect the Ushaped seasonality in the volatilities: positive biases in the morning turn to negative biases after 10:15 a.m. until the bias finally vanishes in the afternoon. The forecasting results FIGURE 8 Distribution of the forecast errors of the out-of-sample ECoVaR forecasts for the simulated data set as detailed in Section 4.4.2 for each of the 26 intraday periods. The out-of-sample results are obtained by splitting the data set in the middle and reserving the second half of the time series as the forecasting window. The forecasting period then starts at July 2, 2008, and ends on December 31, 2012, covering a total of 1133×26 =29,458 periods. Forecasts are generated iteratively with a rolling window scheme and the models are re-estimated at the end of each trading day. Hence, we obtain overall 1133 forecasts for each intraday period. M1 to M4 refer to the periodic model structures given in Table 4 906