Modeling Systemic Risk: Time-Varying Tail Dependence When Forecasting Marginal Expected Shortfall
Eckernkemper,, Tobias
2018-01-01 00:00:00
Abstract In this article, a copula-based model is proposed to estimate the marginal expected shortfall. The model is based on a dynamic mixture copula. The proposed model captures time-varying nonlinear dependence, which is assumed to be constant in alternative approaches. The time-varying copula parameters are endowed with generalized autoregressive score dynamics. For the institutions of the Dow Jones Industrial Average, several variations of the proposed model are considered and compared with alternative, competing models. It is shown that the proposed model outperforms standard benchmarks and produces reasonable findings regarding the risk contributions of the sectors of the Dow Jones Industrial Average. Introduction The 2008–9 financial crisis painfully emphasized the importance of managing systemic risk. The International Monetary Fund, Bank for International Settlements and Financial Stability Board (2009) define systemic risk as “… a risk of disruption to financial services that is (i) caused by an impairment of all or parts of the financial system and (ii) has the potential to have serious negative consequences of the real economy.” As a consequence of the financial crisis, the Basel Committee on Banking Supervision (2010) passed the objective to improve the stability of the banking sector by monitoring and regulating systemic risk which is usually referred to as Basel III. For econometricians, the main questions are how to quantify systemic risk and how to produce reliable forecasts. Accordingly, the literature on systemic risk has rapidly increased during recent years and several systemic risk measures have been proposed.1 A popular systemic risk measure is SRISK (see Brownlees and Engle 2016) which is used to provide rankings for systemic institutions. SRISK is a function of the institutions size, leverage, and risk. The risk of the institution is a key component of SRISK and it is measured among others by the marginal expected shortfall (MES). The MES is defined as the expected return of an institution under the condition that the entire market return is less or equal to a given threshold. In this way, it allows the researcher to quantify the sensitivity of the institution with respect to the market. In this article, a copula-based model is proposed to estimate the MES. It is denoted as the dynamic-mixture-copula MES (DMC-MES) and presents an alternative framework to the one-period ahead MES estimation developed in the working paper version of Brownlees and Engle (2016). The DMC-MES embeds the main elements of the Brownlees-Engle MES (BE-MES) in terms of volatility, linear, and nonlinear dependence but the model explicitly allows for time-varying nonlinear dependence which is assumed to be time-invariant in the BE-MES approach (see Brownlees and Engle 2012). That it is too restrictive to impose such a time invariance is illustrated in Figure 1 for the Dow Jones Industrial Average and four institutions. The figure is based on rolling window estimates for the tail dependence parameter of a t copula.2 It reveals that the nonlinear dependence in terms of tail dependence changes over time and provides similar yet distinct dynamics. In comparison to the BE-MES, which essentially captures time-varying correlation and then maps it to its different parameters, the DMC-MES is based on a dynamic two component mixture copula for the market and each institution. The dynamic copula parameters are modeled by the generalized autoregressive score (GAS) approach of Creal et al. (2013) which endows each parameter with its own dynamics. The proposed model is free in the choice of the parametric class of copulas. To provide maximal flexibility in approximating the dependence structure, I use an asymmetric copula and its rotated version as the two components in the mixture copula. In particular, I consider a mixture of a rotated Clayton and a Clayton and, alternatively, a mixture of a rotated Gumbel and a Gumbel. This mixture approach has two key advantages. First, it is possible to model asymmetric dependencies with a clear separation of nonlinear dependence structures into lower and upper tail dependence. Second, due to mixing these asymmetric copulas and its rotated versions, symmetric dependence structures are also nested which provides high flexibility to model asymmetric and symmetric dependencies together in one framework. Figure 1. Open in new tabDownload slide Tail dependence. Note: The presented lines show the tail dependence parameter λ (solid lines) between the Dow Jones Industrial Average and four selected institutions together with the corresponding 95% confidence intervals (dotted lines). The results are generated by maximum likelihood estimation for a static t copula and a rolling window of 3 years. Figure 1. Open in new tabDownload slide Tail dependence. Note: The presented lines show the tail dependence parameter λ (solid lines) between the Dow Jones Industrial Average and four selected institutions together with the corresponding 95% confidence intervals (dotted lines). The results are generated by maximum likelihood estimation for a static t copula and a rolling window of 3 years. The benefits of the proposed model are shown in an application to the institutions of the Dow Jones Industrial Average. The application is based on daily observations for the period January 1, 2000 to December 31, 2014. There the MES obtained under the dynamic mixture copulas is compared to its counterparts resulting from alternative copula models as well as alternative parametric and nonparametric MES estimators which are not based on copulas. The alternative copula models include static versions of the mixture copulas as well as a skew t copula (Demarta and McNeil 2005) with static and dynamic parameters. In-sample, the parameter estimates of the mixture copulas reveal strong evidence for an asymmetric dependence structure between the market and its institutions. Furthermore, it can be seen that the amount of upper and lower tail dependence is quite heterogenous across the institutions and that the mixture copula reveals a higher weighting for lower than for upper tail dependence. The parameter estimates of the skew t copula indicate fat-tailed joint distributions which imply a high degree of tail dependence. The adequacy of the copula models is evaluated by different goodness-of-fit tests based on the Rosenblatt transformations (Rosenblatt 1952). It turns out that the static and dynamic copulas typically provide a good approximation to the observed dependence structure. It also reveals that there is no essential difference between the in-sample performance of the proposed rotated Clayton and Clayton (RCaC) copula as well as the rotated Gumbel and Gumbel (RGaG) copula. An extensive out-of-sample forecast exercise compares the predictive performance of the dynamic mixture copula with those of their competitors. The out-of-sample analysis is based on one-period ahead forecasts for the evaluation period January 1, 2007 to December 12, 2014. Furthermore, two different scenarios are considered which belong both to a relatively high market downturn. From the results the following conclusions can be drawn. First, time-varying nonlinear dependence should be considered while modeling the MES because the DMC-MES based on the RCaC copula performs better than the BE-MES without time-varying tail dependence for both considered scenarios. Second, for both scenarios the dynamic skew t copula is a serious competitor of the dynamic mixture copula due to its high degree of flexibility. However, the dynamic mixture copula has benefits in practice compared to the dynamic skew t copula because the dynamic skew t copula suffers from relatively complex construction regarding its cdf and gradient which are needed, for example, for the MES and GAS framework. Third, the static nonparametric exponentially-weighted MES (EW-MES) based on the exponentially weighted moving average (EWMA) model of Risk Metrics (see Longerstaey and Spencer 1996) is a reliable and simple alternative to all dynamic MES frameworks especially in the scenario associated with a more extreme market downturn. Finally, I use the proposed DMC-MES to analyze the risk contributions of the individual Dow Jones institutions as a function of their respective sectoral affiliations. It can be seen that the risk contributions of the individual institutions are highly dependent on the institutions’ area of activity, so that certain sectors are less influenced by extreme market drops, whereas other sectors are highly affected by extreme market drops. In more detail, it turns out that the sector Financials followed by Industrials & Materials provides the largest risk contribution and the sector Consumer Staples followed by Health Care the smallest one, whereas the sectors IT & Telecommunication Services, Consumer Discretionary and Energy are in between. The remainder of the article is organized as follows. The concept of the MES as well as the BE-MES are introduced in Section 1. The proposed DMC-MES is considered in Section 2. Section 3 presents the empirical application based on the institutions of the Dow Jones Industrial Average. Section 4 concludes. 1 MES Let {ri,t,rm,t}t=1T be a bivariate time series process where rm,t denotes the return of the entire market m and ri,t the return of an institution i, both at time t. The market return rm,t is given by a linear combination of institution weights wi,t and institution returns ri,t, that is, rm,t=∑i=1Nwi,tri,t . The expected shortfall (ES) of the market is used to measure the risk of the market. It is defined as ESm,t=Et−1(rm,t | rm,t≤ct) with threshold ct. Plugging rm,t=∑i=1Nwi,tri,t into ESm,t yields ESm,t=Et−1(∑i=1Nwi,tri,t | rm,t≤ct)=∑i=1Nwi,t Et−1(ri,t | rm,t≤ct) , (1.1) and partial differentiation with respect to weight wi,t gives the formal definition of the MES for institution i at time t: MESi,t=∂ESm,t∂wi,t=Et−1(ri,t | rm,t≤ct) . (1.2) A typical choice for ct is the value at risk (VaR) of the market return. The MES allows for two interpretations. First, the MES is equal to a conditional expectation, which measures the expected return of institution i under the condition that the return of market m is less or equal to the threshold ct. This allows a researcher to predict the behavior of an institution if the market is under stress. Second, as the partial derivative of ESm,t with respect to wi,t, the MES measures the changes of the market’s ES if the weight of institution i within the market portfolio will marginally increase. Thus, the MES measures the change of the market risk if institution i would have a higher weight on the market. Both interpretations provide insights into the risk contribution of an institution and its importance for the considered market. Brownlees and Engle (2012) embed the MES in a bivariate time series framework which allows to model volatility, correlation, and nonlinear dependence over time. The model for the market and institution return processes with zero means is defined as ri,t=σi,t (ρt εm,t+1−ρt2 ξi,t) , (1.3) rm,t=σm,t εm,t , (1.4) with conditional institution and market standard deviation σi,t and σm,t, conditional market and institution correlation ρt as well as institution and market innovation ξi,t and εm,t . The conditional standard deviations are modeled by Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroscedasticity (GJR-GARCH) models (Glosten et al. 1993) and the conditional correlation as in a dynamic conditional correlation (DCC) model (Engle 2002,2009). Under the return Equations (1.3) and (1.4), the MES in Equation (1.2) is given by BE-MESi,t=Et−1(σi,t ρt εm,t+σi,t1−ρt2 ξi,t | σm,tεm,t≤ct)=σi,t ρt Et−1(εm,t | εm,t≤κ)+σi,t 1−ρt2 Et−1(ξi,t | εm,t≤κ) , (1.5) with κ=ctσm,t .3 To estimate the one-period ahead MES in Equation (1.5), Brownlees and Engle (2012) propose a two-step procedure. First, the fitted values σ^i,t and σ^m,t of GJR-GARCH(1,1,1) specifications and the fitted value ρ^t of a DCC(1,1) specification are used. Second, the conditional tail expectations are approximated by the following nonparametric kernel estimation (Scaillet 2005) based on the residuals {ε^m,τ,ξ^i,τ}τ=1t−1 : E^t−1(zt | εm,t≤κ)=∑τ=1t−1z^τ K(κ^−ε^m,τh)∑τ=1t−1K(κ^−ε^m,τh) , (1.6) where z^t stands for ε^m,t or ξ^i,t . The function K(·) is a Gaussian kernel with positive bandwidth h and κ^=ctσ^m,t . Both models, the GJR-GARCH and DCC, are dynamic models, whereas the nonparametric kernel estimation in Equation (1.6) implies a static model. In this way, it is assumed that the nonlinear dependence between the institution and market is constant over time. 2 DMC-MES This section presents the concept of the DMC-MES. First, a brief review on copulas is given.4 Second, the framework and model setup of the DMC-MES are presented. Third, the model components and an estimation procedure of the DMC-MES are provided. 2.1 Preliminaries Let Ft denote the joint distribution of the institution and market innovation, εi,t and εm,t . The institution and market innovation are assumed to be individually iid. The resulting static marginal distributions are denoted by Fi and Fm. Sklar (1959) points out that a joint distribution function can be represented by its marginal distributions, Fi and Fm, combined via a copula function Ct, that is, P(εi,t≤εi,εm,t≤εm)=Ft (εi,εm)=Ct (Fi(εi)︸=ui, Fm(εm)︸=um ; θt) , (2.1) which can be solved for Ct (ui,um ; θt)=Ft(Fi−1(εi), Fm−1(εm)) , (2.2) with ui,um∈[0,1], εm,εi∈ℝ , dynamic copula parameter θt and quantile functions Fi−1 and Fm−1 . The main advantage of the copula approach is seen from Equation (2.1). The marginal distributions Fi and Fm contain individual information of εi,t and εm,t without interconnectedness of εi,t and εm,t , whereas the copula function Ct contains the whole dependence structure between εi and εm . In this way, it is possible to model a wide range of joint distribution functions with individual behaviors of the marginals and a certain dependence structure. As proposed in different studies (see, e.g., Chollete et al. 2009; Okimoto 2008; Rodriguez 2007; and Hu 2006), it can be sensible to use a mixture copula to model asymmetric dependencies. A mixture copula can combine different dependence patterns which induces a higher flexibility relative to a copula presented by Equation (2.2). A mixture of K copula functions is defined as Ct (ui,um ; θt)=∑k=1Kwk Ck,t(ui,um ; θk,t) , ∑k=1Kwk=1 , (2.3) where the kth copula function is denoted by Ck,t with weight wk∈[0,1] and dynamic parameter θk,t. By choosing different copula families with different properties, it is possible to combine their individual properties within the copula function Ct. Finally, a conditional expectation of the form in Equation (1.2) as a function of the copula Ct obtains as Et−1(εi,t | εm,t≤κ)=1α∫01Fi−1(ui)·∂Ct(ui,α ; θt)∂ui dui , (2.4) and for the mixture copula as given by Equation (2.3) as Et−1(εi,t | εm,t≤κ)=∑k=1Kwk 1α∫01Fi−1(ui)·∂Ck,t(ui,α ; θk,t)∂ui dui , (2.5) where α = Fm(κ).5 The conditional expectation in Equation (2.5) represents the basis of the DMC-MES in the following section and will be discussed there in more detail. 2.2 Framework The framework of the DMC-MES embeds the main elements of the BE-MES in Equation (1.5) but allows for dynamic effects in the nonlinear dependence structure. The conditional marginal distributions of the institution and market return processes are modeled as in Equation (1.4) and the conditional standard deviations are assumed to follow univariate GARCH models (Engle 1982; Bollerslev 1986). For the mutual dependence of the institution and market innovations, I assume a dynamic copula function. While choosing a copula function with a high degree of flexibility for modeling dependencies, there is no need to consider the decomposition of the MES which Brownlees and Engle (2012) use in their framework. Overall, the assumptions are as follows: εm,t and εi,t are each iid with unspecified, static distributions Fm and Fi, E(εm,t)=E(εi,t)=0, Var(εm,t)=Var(εi,t)=1 and P (εi,t≤εi, εm,t≤εm)=Ct (Fi(εi)︸=ui, Fm(εm)︸=um ; θt) with dynamic copula parameter θt. Plugging rm,t=σm,t εm,t and ri,t=σi,t εi,t in Equation (1.2) yields MESi,t=σi,t Et−1(εi,t | εm,t≤κ) , (2.6) with κ=ctσm,t . Using Equation (2.4), the MES in Equation (2.6) obtains as DC‐MESi,t=σi,tα∫01Fi−1(ui) ∂Ct(ui,α ; θt)∂ui dui , (2.7) with quantile function Fi−1 , copula of institution and market innovation Ct and dynamic copula parameter θt. The DC-MES depends on (i) individual institution effects through σi,t and Fi−1 as well as (ii) interdependencies between the market and institution through the copula function Ct. As mentioned in Section 2.1, recent studies show that a copula function, that only depends on a single parameter is often unable to fully capture the observed dependence structure. Therefore, I propose to mix different copula functions by choosing a mixture of two dynamic copulas with static weights. Using Equation (2.5) for K = 2, the MES according to Equation (2.6) can be written as DMC‐MESi,t=w˜ σi,tα∫01Fi−1(ui) ∂C1,t(ui,α ; θ1,t)∂ui dui +(1−w˜) σi,tα∫01Fi−1(ui) ∂C2,t(ui,α ; θ2,t)∂ui dui , (2.8) where w˜ denotes the copula weight. The DMC-MES is free in the choice of copula parameterization and it divides the dependence structure of the MES into the contributions of C1,t and C2,t . To create a flexible dependence structure, two asymmetric copulas with different properties are chosen for C1,t and C2,t . In particular, I propose to use (i) a mixture of a RCaC copula and (ii) a mixture of a RGaG copula. This has two key advantages. First, it is possible to account for asymmetric dependencies with a clear separation of nonlinear dependence structures into lower and upper tail dependence because the Clayton and rotated Gumbel copula allow for lower tail dependence, whereas the Gumbel and rotated Clayton copula allow for upper tail dependence. The Clayton and Gumbel copula differ (among others) by their respective degree of asymmetry. The Clayton copula is more asymmetric than the Gumbel copula for comparable parameter settings, which allows it to approximate potentially complex dependence structures. Second, it is possible to allow for symmetric dependencies for the parameter settings w˜=0.5 and θ1,t=θ2,t . In this way, symmetric dependence structures are nested within the proposed framework and it is possible to model asymmetric and symmetric dependencies together in one framework.6 As a single copula benchmark for the proposed dynamic mixture copula approach, I consider the skew t copula of Demarta and McNeil (2005) which represents a flexible asymmetric copula indexed by multiple parameters. This specification and alternative parameterizations of the DC-MES and DMC-MES are considered in Section 3. Figure 2 provides contour plots for the mixture copula pdf of the RCaC and RGaG. The contour plots on the left illustrate symmetric dependence structures, whereas the contour plots on the right depict asymmetric dependence structures with different amount of tail dependence. Furthermore, it can be seen that the asymmetry of the Clayton copula is more distinct than that of the Gumbel copula. For an empirical application, the DMC-MES approach provides high flexibility because it is able to capture asymmetric and symmetric dependencies with an overall consideration of time-varying aspects. Figure 2. Open in new tabDownload slide Contour plots. Note: The presented contour plots are based on the mixture copula pdf of a Clayton and rotated Clayton (upper) as well as a Gumbel and rotated Gumbel (lower) with different lower and upper tail dependence parameters λL and λU. The copula weight is set to w˜=0.5 . Figure 2. Open in new tabDownload slide Contour plots. Note: The presented contour plots are based on the mixture copula pdf of a Clayton and rotated Clayton (upper) as well as a Gumbel and rotated Gumbel (lower) with different lower and upper tail dependence parameters λL and λU. The copula weight is set to w˜=0.5 . 2.3 Model Components and Estimation In the following, the model components of the DMC-MES and its parameterization are discussed as well as an estimation procedure for the DMC-MES is proposed. The main focus will be the parameterization of the two-component mixture copula in Equation (2.8). First, it is assumed that the market and institution return processes follow independent AR(1)-GARCH(1,1) specifications (Engle 1982; Bollerslev 1986) presented by Equations (2.12) and (2.13). In this way, the presence of autocorrelation and conditional heteroscedasticity are captured by the model. Second, for the dynamic copula parameters θ1,t and θ2,t , I use the GAS recursion of Creal et al. (2013) given in Equations (2.15)–(2.17). The GAS specification is an observation-driven model where the score function acts as an observation driven part. The GAS framework allows to model more than one dynamic parameter simultaneously and independently of each other because the score function provides information for each time-varying parameter. Classical frameworks for modeling dynamic parameters, like Patton (2006), turned out to be less suitable since they do not address the specific properties of each copula. A general overview of further concepts to model dynamic copula parameters is given, for example, in Manner and Reznikova (2012). The model for the market and institution return rt=(rm,t,ri,t)′ is specified as rt=μt+σt⊙εt , (2.9) μt=(μm,t,μi,t)′ , σt=(σm,t,σi,t)′ , εt=(εm,t,εi,t)′ , (2.10) εm,t∼iid Fm(εm), εi,t∼iid Fi(εi) , (2.11) where ⊙ denotes the Hadamard product and the jth elements of μt and σt are given by μj,t=αj,0+αj,1rj,t−1 , (2.12) σj,t2=ωj+αjrj,t−12+βjσj,t−12 . (2.13) The joint cdf for εm,t and εi,t is specified as Ft(εm,εi)=w˜ C1,t(um,ui ; θ1,t(ψ1,t))+(1−w˜) C2,t(um,ui ; θ2,t(ψ2,t)) , (2.14) ψt+1=ω+A st+B ψt , ψt=(ψ1,t,ψ2,t)′ , (2.15) st=St·∇t , (2.16) ∇t=∂ log ct (ui,t,um,t;θt(ψt),w˜ | Ft−1)∂ψt , (2.17) log ct (ui,t,um,t;θt(ψt),w˜ | Ft−1)=log [w˜·c1,t(ui,t,um,t;θ1,t(ψ1,t))+(1−w˜)·c2,t(ui,t,um,t;θ2,t(ψ2,t))] . (2.18) The information set at period t – 1 is Ft−1={umt−1,uit−1,Θt−1} with uit={ui,1,...,ui,t},umt={um,1,...,um,t} and Θt={θ0,θ1,...,θt} . The specifications of the GAS(1,1) model follow Creal et al. (2013) where st is the score function, St a scaling matrix, ∇t the gradient, ω a vector of constants and A, B parameter matrices. The matrices A and B are assumed to be diagonal. It implies the reasonable assumption that the parameter ψ1,t ( ψ2,t ) does not depend on the lagged parameter ψ2,t−1 ( ψ1,t−1 ) and lagged score s2,t−1 (s1,t−1) . This guarantees a clear separation of modeling lower and upper tail dependence. Further details on the GAS framework, including admissible parameter regions and asymptotic properties, are presented in Blasques et al. (2014b) and Blasques et al. (2014a). The estimation of the presented framework is based on the two-stage estimation method proposed by Joe (2005) and Joe (1997). At the first stage, the marginals are estimated and the fitted values are used to estimate the innovations according to ε^i,t=ri,t−μ^i,tσ^i,t and ε^m,t=rm,t−μ^m,tσ^m,t . At the second stage, the copula parameters are estimated. The gradient ∇t=(∇1,t,∇2,t)′ as defined in Equation (2.17) is given by ∇t=(w˜·∂c1,t(ui,t,um,t;θ1,t(ψ1,t))∂ψ1,t·1(w˜·c1,t(ui,t,um,t;θ1,t(ψ1,t))+(1−w˜)·c2,t(ui,t,um,t;θ2,t(ψ2,t)))(1−w˜)·∂c2,t(ui,t,um,t;θ2,t(ψ2,t))∂ψ2,t·1(w˜·c1,t(ui,t,um,t;θ1,t(ψ1,t))+(1−w˜)·c2,t(ui,t,um,t;θ2,t(ψ2,t)))) . (2.19) The first derivatives of the copula densities, needed in Equation (2.19), can be derived analytically with MATLAB. The scaling matrix St can be approximated by different methods. An overview is listed in Creal et al. (2013). One possibility is to use the inverse of the Fisher information matrix. However, this requires an appropriate estimate of the Fisher information which is not convenient for the mixture copula. Therefore, to keep the framework as simple as possible, the scaling matrix is set to St = I. To link the copula parameters to the framework, the transformations θ1,t=1+exp(ψ1,t) and θ2,t=1+exp(ψ2,t) are used to restrict the range of the Gumbel and rotated Gumbel parameters θ1,t,θ2,t∈[1,∞) and the transformations θ1,t=exp(ψ1,t) and θ2,t=exp(ψ2,t) are used to restrict the range of the Clayton and rotated Clayton parameters θ1,t,θ2,t∈(0,∞) . Finally, the unknown parameters in ω, A, B and the copula weight w˜ are estimated by maximum likelihood. The maximization problem is {ω^,A^,B^,w˜^}=arg maxω,A,B,w˜∑t=1Tlog ct (ui,t,um,t;θt(ψt),w˜ | Ft−1) , (2.20) with implementation of the parameter updating in Equation (2.15). 3 Empirical Application This section presents an empirical application to the constituent firms of the Dow Jones Industrial Average. First, an in-sample analysis is performed where the marginal distributions and different copula parameterization for the DMC-MES are analyzed. Second, nonparametric and parametric MES benchmark estimators are presented. Third, an out-of-sample analysis is performed where the DMC-MES is compared with the alternative, competing models. Finally, a brief risk analysis is presented which can be used to quantify the risk contributions of the individual institutions and the corresponding sectors of the Dow Jones Industrial Average. 3.1 In-Sample Analysis The analysis is based on the components of the Dow Jones Industrial Average. The dataset contains daily returns with 3772 observations from January 1, 2000 to December 31, 2014. The stock prices are taken from Yahoo Finance (adjusted closing prices) and converted to log returns multiplied by 100. Table 1 provides a list of the stocks.7 Table 1. Institutions of the Dow Jones Industrial Average Abbreviations Institutions GICS sector AAPL Apple Information technology AXP American Express Financials BA Boeing Industrials CAT Caterpillar Industrials CSCO Cisco Information technology CVX Chevron Energy DD DuPont Materials DIS Disney Consumer discretionary GE General Electric Industrials GS Goldman Sachs Financials HD Home Depot Consumer discretionary IBM IBM Information technology INTC Intel Information technology JNJ Johnson & Johnson Health care JPM JPMorgan Chase Financials KO Coca-Cola Consumer staples MCD McDonald’s Consumer discretionary MMM 3M Industrials MRK Merck & Co. Health care MSFT Microsoft Information technology NKE Nike Consumer discretionary PFE Pfizer Health care PG Procter & Gamble Consumer staples TRV Travelers Financials UNH United Health Health care UTX United Technologies Industrials VZ Verizon Communications Telecommunications services WMT Wal-Mart Consumer staples XOM ExxonMobil Energy Abbreviations Institutions GICS sector AAPL Apple Information technology AXP American Express Financials BA Boeing Industrials CAT Caterpillar Industrials CSCO Cisco Information technology CVX Chevron Energy DD DuPont Materials DIS Disney Consumer discretionary GE General Electric Industrials GS Goldman Sachs Financials HD Home Depot Consumer discretionary IBM IBM Information technology INTC Intel Information technology JNJ Johnson & Johnson Health care JPM JPMorgan Chase Financials KO Coca-Cola Consumer staples MCD McDonald’s Consumer discretionary MMM 3M Industrials MRK Merck & Co. Health care MSFT Microsoft Information technology NKE Nike Consumer discretionary PFE Pfizer Health care PG Procter & Gamble Consumer staples TRV Travelers Financials UNH United Health Health care UTX United Technologies Industrials VZ Verizon Communications Telecommunications services WMT Wal-Mart Consumer staples XOM ExxonMobil Energy Note: The sectors are based on the Global Industry Classfication Standard (GICS). Table 1. Institutions of the Dow Jones Industrial Average Abbreviations Institutions GICS sector AAPL Apple Information technology AXP American Express Financials BA Boeing Industrials CAT Caterpillar Industrials CSCO Cisco Information technology CVX Chevron Energy DD DuPont Materials DIS Disney Consumer discretionary GE General Electric Industrials GS Goldman Sachs Financials HD Home Depot Consumer discretionary IBM IBM Information technology INTC Intel Information technology JNJ Johnson & Johnson Health care JPM JPMorgan Chase Financials KO Coca-Cola Consumer staples MCD McDonald’s Consumer discretionary MMM 3M Industrials MRK Merck & Co. Health care MSFT Microsoft Information technology NKE Nike Consumer discretionary PFE Pfizer Health care PG Procter & Gamble Consumer staples TRV Travelers Financials UNH United Health Health care UTX United Technologies Industrials VZ Verizon Communications Telecommunications services WMT Wal-Mart Consumer staples XOM ExxonMobil Energy Abbreviations Institutions GICS sector AAPL Apple Information technology AXP American Express Financials BA Boeing Industrials CAT Caterpillar Industrials CSCO Cisco Information technology CVX Chevron Energy DD DuPont Materials DIS Disney Consumer discretionary GE General Electric Industrials GS Goldman Sachs Financials HD Home Depot Consumer discretionary IBM IBM Information technology INTC Intel Information technology JNJ Johnson & Johnson Health care JPM JPMorgan Chase Financials KO Coca-Cola Consumer staples MCD McDonald’s Consumer discretionary MMM 3M Industrials MRK Merck & Co. Health care MSFT Microsoft Information technology NKE Nike Consumer discretionary PFE Pfizer Health care PG Procter & Gamble Consumer staples TRV Travelers Financials UNH United Health Health care UTX United Technologies Industrials VZ Verizon Communications Telecommunications services WMT Wal-Mart Consumer staples XOM ExxonMobil Energy Note: The sectors are based on the Global Industry Classfication Standard (GICS). At the first stage, the marginal distributions of the innovations in Equation (2.11) are modeled. By applying the Jarque–Bera test, the null hypothesis of normally distributed innovations is clearly rejected for the market and each institution. As proposed by different studies (see, e.g., Patton 2004 or Jondeau and Rockinger 2003), I use a skew t distribution for the marginals with degrees of freedom ν and skewness parameter λ (see Hansen 1994). The maximum likelihood parameter estimates for the AR-GARCH models are reported in Table 2 (see also Appendix A.4, Table A1). As expected from the initial Jarque–Bera test, the results show evidence of fat-tailed distributions with 6.0414 as the median of the estimates of degrees of freedom. This is a well-known result for financial stock returns. Most skewness parameters are negative indicating left-skewed distributions. The sum of the GARCH parameters α and β is close to unity, which indicates highly persistent volatilities. All together, the results are in line with other copula studies (see, e.g., Manner and Reznikova 2012 or Avdulaj and Barunik 2015). The adequacy of the fitted AR-GARCH models is evaluated by the Ljung–Box test. The test is applied to the innovations as well as to the squared innovations. Different lag lengths provide similar results. The p-values for a lag length of 50 (not reported here) show that the null hypothesis of a serially uncorrelated process in the first- and second-order moments cannot be rejected on a 1% significance level. Table 2. Estimation results of marginal distributions AR(1) GARCH(1,1) dof Skewness αj,0 αj,1 ωj αj βj νj λj 1st quartile 0.0139 −0.0468 0.0141 0.0452 0.9206 5.2023 −0.0343 Median 0.0245 −0.0305 0.0190 0.0573 0.9341 6.0414 −0.0202 3rd quartile 0.0422 −0.0039 0.0223 0.0729 0.9492 7.0975 −0.0089 AR(1) GARCH(1,1) dof Skewness αj,0 αj,1 ωj αj βj νj λj 1st quartile 0.0139 −0.0468 0.0141 0.0452 0.9206 5.2023 −0.0343 Median 0.0245 −0.0305 0.0190 0.0573 0.9341 6.0414 −0.0202 3rd quartile 0.0422 −0.0039 0.0223 0.0729 0.9492 7.0975 −0.0089 Note: The reported numbers are the 1st quartile, median, and 3rd quartile of the empirical distribution of the parameter estimates of the univariate AR(1) and GARCH(1,1) models across the 29 institutions and the Dow Jones (DJ). Time period 1/2000–12/2014. Table 2. Estimation results of marginal distributions AR(1) GARCH(1,1) dof Skewness αj,0 αj,1 ωj αj βj νj λj 1st quartile 0.0139 −0.0468 0.0141 0.0452 0.9206 5.2023 −0.0343 Median 0.0245 −0.0305 0.0190 0.0573 0.9341 6.0414 −0.0202 3rd quartile 0.0422 −0.0039 0.0223 0.0729 0.9492 7.0975 −0.0089 AR(1) GARCH(1,1) dof Skewness αj,0 αj,1 ωj αj βj νj λj 1st quartile 0.0139 −0.0468 0.0141 0.0452 0.9206 5.2023 −0.0343 Median 0.0245 −0.0305 0.0190 0.0573 0.9341 6.0414 −0.0202 3rd quartile 0.0422 −0.0039 0.0223 0.0729 0.9492 7.0975 −0.0089 Note: The reported numbers are the 1st quartile, median, and 3rd quartile of the empirical distribution of the parameter estimates of the univariate AR(1) and GARCH(1,1) models across the 29 institutions and the Dow Jones (DJ). Time period 1/2000–12/2014. At the second stage, the parameters of the copulas between the Dow Jones Industrial Average and its components are estimated. The estimation results for the dynamic mixture copula are reported in the upper panel of Table 3 (see also Appendix A.4, Tables A2-A3). It can be seen that the GAS parameter estimates are similar to those obtained by other applications in the GAS literature (see, e.g., Oh and Patton 2016, Avdulaj and Barunik 2015 or De Lira Salvatierra and Patton 2015). The first and third quartile of the copula weights show that at least 50% of the estimates are between 0.3926 and 0.4539 for the dynamic mixture of RCaC as well as between 0.5692 and 0.6269 for the dynamic mixture of RGaG. This provides strong evidence for an asymmetric dependence structure. Furthermore, it shows that the estimated weights for the Clayton component (in the mixture of the RCaC) as well as for the rotated Gumbel component (in the mixture of the RGaG) are typically larger than those for the rotated Clayton and Gumbel component. Hence, the importance of lower tail dependence seems to be larger than that of upper tail dependence. Table 3. Estimation results of dynamic and static mixture copula Panel A: dynamic mixture copula (static weights) Weight GAS(1,1) w˜ ω1 ω2 A11 A22 B11 B22 RCaC 1st quartile 0.3926 0.0015 0.0010 0.0967 0.0914 0.9867 0.9604 Median 0.4293 0.0033 0.0021 0.1380 0.1410 0.9899 0.9894 3rd quartile 0.4539 0.0057 0.0065 0.2618 0.2530 0.9955 0.9950 RGaG 1st quartile 0.5692 −0.0116 −0.0084 0.1076 0.1664 0.9211 0.9792 Median 0.5978 −0.0022 −0.0011 0.1761 0.2465 0.9918 0.9890 3rd quartile 0.6269 −0.0002 0.0005 0.2985 0.4582 0.9951 0.9949 Panel B: static mixture copula (static weights) Weight Copula w˜ θ1 θ2 RCaC 1st quartile 0.4051 1.1623 1.1271 – – – – Median 0.4383 1.5043 1.4978 – – – – 3rd quartile 0.4605 1.9405 1.7288 – – – – RGaG 1st quartile 0.5578 1.6354 1.5564 – – – – Median 0.5890 1.8566 1.8126 – – – – 3rd quartile 0.6341 2.0074 2.0674 – – – – Panel C: static mixture copula (dynamic weights) Copula GAS(1,1) θ1 θ2 ω A B RCaC 1st quartile 1.1499 1.1338 −0.0445 −0.1137 0.6513 – – Median 1.5254 1.4634 −0.0099 0.0582 0.8902 – – 3rd quartile 2.0093 1.8329 −0.0019 0.2206 0.9901 – – RGaG 1st quartile 1.5435 1.3315 0.0010 0.2594 0.9842 – – Median 1.9233 1.7536 0.0019 0.3153 0.9902 – – 3rd quartile 2.2546 2.2035 0.0033 0.4411 0.9956 &nda