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Journal of Financial Econometrics
, Volume Advance Article – Apr 16, 2018

32 pages

/lp/ou_press/realized-wishart-garch-a-score-driven-multi-asset-volatility-model-uUcqCSV90Z

- Publisher
- Oxford University Press
- Copyright
- © The Author(s), 2018. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com
- ISSN
- 1479-8409
- eISSN
- 1479-8417
- D.O.I.
- 10.1093/jjfinec/nby007
- Publisher site
- See Article on Publisher Site

Abstract We propose a novel multivariate GARCH model that incorporates realized measures for the covariance matrix of returns. The joint formulation of a multivariate dynamic model for outer-products of returns, realized variances, and realized covariances leads to a feasible approach for analysis and forecasting. The updating of the covariance matrix relies on the score function of the joint likelihood function based on Gaussian and Wishart densities. The dynamic model is parsimonious while the analysis relies on straightforward computations. In a Monte Carlo study, we show that parameters are estimated accurately for different small sample sizes. We illustrate the model with an empirical in-sample and out-of-sample analysis for a portfolio of 15 U.S. financial assets. Modeling conditional dependency structure of financial assets through time-varying covariance matrices is typically based on multivariate extensions of generalized autoregressive conditional heteroskedasticity (GARCH) models and stochastic volatility (SV) models for daily returns. These classes of models aim to extract time-varying covariance matrices from vector time series of financial returns. The dynamic process for multivariate volatility (variances, covariances, and correlations) is typically specified as a vector autoregressive moving average (VARMA) process. Various multivariate GARCH and SV models have been developed and applied in recent years. For a comprehensive overview of multivariate GARCH models, we refer to Bauwens, Laurent, and Rombouts (2006), Silvennoinen and Teräsvirta (2009), and Audrino and Trojani (2011). Reviews of multivariate SV models are provided by Asai, McAleer, and Yu (2006) and Jungbacker and Koopman (2006). These developments in financial econometrics are also related with the theoretical developments in finance and in particular with the literature on option pricing, optimal portfolio modeling, and term structure modeling. For example, Driessen, Maenhout, and Vilkov (2009) investigate individual volatility risk premia differences (typically in relation to a portfolio or index) and they explain them by a high correlation risk premium. Buraschi, Porchia, and Trojani (2010) adopt a Wishart specification for modeling optimal portfolio choice with correlation risk. More recently, the study of Buraschi, Trojani, and Vedolin (2018) focuses on a priced disagreement risk that explains returns of option volatility and correlation in trading strategies. In all such studies, the multivariate GARCH and SV models for volatilities and correlations in multiple asset returns are of key importance. The main shortcoming of traditional multivariate GARCH and SV models is that they solely rely on daily returns to infer the current level of multivariate volatility. Given the increasing availability of high-frequency intraday data for a vast range of financial assets, the use of only low-frequency daily data appears inefficient for making statistical inference on time-varying multivariate volatility. One important consequence is that models based on daily data do not adapt quickly enough to changes in volatilities which is key to track the financial risk in a timely manner; see Andersen et al. (2003) for a more detailed discussion. The relevance of these issues in the context of discrete volatility models, possibly with leverage effects, and their relations to option pricing models have been discussed and reviewed recently in Khrapov and Renault (2016). Various attempts have been made to use high-frequency intraday data into the modeling and analysis of volatility. For instance, information from high-frequency data can be incorporated by adding it in the form of an explanatory variable to the GARCH or SV volatility dynamics; see Engle (2002b) and Koopman, Jungbacker, and Hol (2005). With the advent of high-frequency data, one can estimate ex-post daily return variation with the so-called realized variance (or realized volatility) measures; see Andersen and Bollerslev (1998), Andersen et al. (2001), and Barndorff-Nielsen and Shephard (2002). Inherent to high-frequency data is the microstructure noise (bid-ask bounce, decimal misplacement, etc.), which leads to bias and inconsistency of standard measures. A number of related measures have been developed to restore the consistency; see Aït-Sahalia, Mykland, and Zhang (2005), Barndorff-Nielsen et al. (2008), Jacod et al. (2009), Hansen and Horel (2009), and references therein. In the case of multiple assets, realized measures of asset covariance have also been proposed and considered; see Christensen, Kinnebrock, and Podolskij (2010), Barndorff-Nielsen et al. (2011a), Griffin and Oomen (2011), and references therein. Andersen et al. (2001) have explored the use of autoregressive models to analyze time series of realized volatilities. They have found considerable improvements in volatility forecasts over standard GARCH models. More recently, some new promising models have been proposed that rely on time series of realized measures. Gourieroux, Jasiak, and Sufana (2009) have proposed (noncentral) Wishart autoregressive model for realized covariance matrix. Asai and So (2013) and Golosnoy, Gribisch, and Liesenfeld (2012) have proposed alternative dynamic formulation for covariance parameters with the underlying Wishart distribution. Chiriac and Voev (2011) and Bauer and Vorkink (2011) have proposed models for realized covariances using appropriate transformations to ensure the positive definiteness of the covariance matrix. In our study, we also rely on the Wishart distribution but we propose a novel conditional model formulation for the covariance matrix. For the updating of the conditional covariance matrix, we use daily and intra-daily financial returns. An approach that combines possibly several measures of volatility based on low- and high-frequency data is recently proposed by Engle and Gallo (2006). They model jointly close-to-close returns, range and realized variance with the multiplicative error model (MEM) where each measure has its own dynamics for the update of latent volatility augmented with lagged values of other two measures. Engle and Gallo (2006) find that combination of these three noisy measures of volatility brings gains when making medium-run volatility forecasts. Shephard and Sheppard (2010) explore a similar model structure and refer to it as the HEAVY model, which was extended to the multivariate setting in Noureldin, Shephard, and Sheppard (2012). Then a further extension based on the use of more heavy-tailed distributions is proposed by Opschoor et al. (2017). In the aforementioned models, a time-varying parameter is introduced for every realized measure that is included in the model. An alternative approach is the Realized GARCH framework by Hansen, Huang, and Shek (2012) where daily returns and realized measures of volatility are both associated with the same latent volatility, which circumvents the need for additional latent variables. The Realized GARCH framework has been developed further in Hansen, Lunde, and Voev (2014). A Realized SV model is proposed by Koopman and Scharth (2013). Our present work introduces an extension of the Realized GARCH model to the multivariate case and the use of a score-driven framework for the time-varying conditional covariance matrix. Our primary aim is to specify a model for the daily time-varying covariance matrix and to extract it by using both low- and high-frequency data. For this purpose, we propose a specification for the unobserved daily covariance matrix as a function of realized measures of daily covariance matrices and past outer-products of daily return vectors. The challenge is to suitably weight these different variance and covariance signals. For this purpose, we adopt the score-driven framework of Creal, Koopman, and Lucas (2013). Our joint modeling framework relies on a Wishart distribution for realized covariance matrices and on a Gaussian distribution for vectors of daily returns. The updating of the time-varying covariance matrix is driven by the scaled score of the predictive joint likelihood function; Blasques, Koopman, and Lucas (2015) have argued that such updating is locally optimal in a Kullback–Leibler sense. The score function turns out to be a weighted combination of the outer-product of daily returns and the actual realized measures; the weighting relies on the number of degrees of freedom in the Wishart distribution. We refer to our resulting model as the Realized Wishart-GARCH (RWG) model. In our empirical illustration for a portfolio of 15 U.S. financial assets, the parameter estimates imply that the realized measures receive more weight than the outer-product of the vector of daily returns. We confirm that the realized measure is a more accurate measure of the covariance matrix as it exploits intraday high-frequency data. In an out-of-sample study, we show that our modeling framework can lead to accuracy improvements in forecasting, especially those for the density in daily returns. The remainder of the article is organized as follows. In Section 1, we introduce the RWG model for multivariate conditional volatility. In Section 2, we conduct a Monte Carlo study to verify the performance of likelihood-based estimation. Section 3 presents the results of our in-sample and out-of-sample empirical study for a portfolio of fifteen equities that are listed at the New York Stock Exchange (NYSE). It includes a thorough forecasting comparison of our model against several other competitive models and methods. Section 4 concludes. The Appendices provide some matrix algebra results, proofs of the main results, and additional estimation results. 1 The RWG Model The development of our model for the time-varying conditional covariance matrix starts with the assumption that for each trading day and for a selection of assets, we have a data vector of daily returns and a measure (or possibly several measures) of the daily realized covariance matrix. We build a model for these data sources and implicitly use both low- and high-frequency data. The proposed structure of the model permits the use of several realized measures that are based on different sampling frequencies. In this section we discuss our modeling assumptions. We then describe the modeling strategy and we provide technical details of our new model for multivariate conditional volatility. Some matrix notation and preliminary results are presented in Appendix A and proofs are collected in Appendix B. 1.1 Modeling assumptions Let rt∈Rk denote a k×1 vector of daily (demeaned) log returns for k assets and let the Xt∈Rk×k denote a k × k realized covariance matrix of k assets on day t, with t=1,…,T. Let Ft−1 be the sigma field generated by the past values of rt and Xt, that is Ft−1=σ(rs,Xs;s=1,…,t−1). We assume the following conditional densities rt|Ft−1∼Nk(0,Ht), (1) Xt|Ft−1∼Wk(Vt/ν,ν), (2) with nonsingular k × k covariance matrix Ht of the zero-mean multivariate normal distribution Nk(0,Ht) and nonsingular k × k covariance matrix Vt as the mean of the k-th dimensional Wishart distribution Wk(Vt/ν,ν) with degrees of freedom ν≥k. The covariance matrices Ht and Vt are both measurable with respect to Ft−1. The variables rt and Xt in (1) and (2) are conditionally independent of each other. The (unconditional) dependence between rt and Xt is assumed to rely only on the dependence between Ht and Vt. The coefficient ν encapsulates the precision by which Xt measures Vt. A larger value of ν implies a more accurate measurement Xt for Vt. The normal density function for rt|Ft−1 is given by 1(2π)k2|Ht|12exp{−12tr(Ht−1rtr′t)}, (3) and the density function of the k-variate standard Wishart distribution for Xt|Ft−1 is given by |Xt|(ν−k−1)/22(νk)/2ν−(νk)/2|Vt|ν/2Γk(ν2)exp{−ν2tr(Vt−1Xt)}, (4) with multivariate Gamma function Γk(a)=πk(k−1)4∏i=1kΓ(a+(1−i)/2) for any a > 0. The measurement equations can be formally given by rt|Ft−1=Ht1/2ɛt, Xt|Ft−1=Vt1/2ηtVt1/2, (5) where A1/2 denotes the square root matrix of A and where the measurement innovations are assumed to be, mutually and serially, identically and independently distributed (iid) random variables, that is ɛt∼Nk(0,Ik), ηt∼Wk(Ik / ν,ν), with k×1 random vector ɛt and k × k random matrix ηt with property E(ɛt ηs′)=0, for t,s=1,…,T. We assume that realized covariance Xt is available on each day t as it can be measured consistently by the multivariate realized kernel of Barndorff-Nielsen et al. (2011a) or related measures described by Griffin and Oomen (2011). The distributional Assumption (1) implies that the outer product of the daily returns vector is distributed as rtrt′|Ft−1∼Wks(Ht,1), (6) where Wks(Ht,1) is the singular Wishart distribution with mean Ht and one degree of freedom, see Uhlig (1994) and Srivastava (2003). We notice that the covariance matrix Ht is nonsingular; the distinctive feature of the singular Wishart is that ν<k and in (6) we have ν = 1 while for the Wishart we have ν>k. Given the specification in (6), we can formulate the measurement equations alternatively as rtrt′|Ft−1=Ht1/2ζtHt1/2, Xt|Ft−1 = Vt1/2ηtVt1/2, with ζt∼Wks(Ik,1), and where ζt and ηt are, serially and mutually, iid processes of k × k stochastic matrices. In this representation, the measurement equations are expressed in terms of variances and covariances. The developments in our study are based on the assumption that the conditional covariance matrix of (daily) returns and the conditional mean of the realized covariance matrix share the same dynamic processes. Specifically, we let the covariance matrix Ht to be fully dependent on Vt, and vice-versa, that is Ht=ΛVtΛ′, (7) where Λ=(λij) is a k × k nonsingular matrix. Due to the quadratic form in (7), a sign restriction on Λ needs to be imposed to ensure identifiability. For this purpose, we impose the sign restriction λ11>0. The specific role and economic interpretation of Λ depends on whether daily returns are computed as close-to-close or open-to-close; we refer to the empirical study for a discussion. Our model specification implies that the conditional statistical properties of the measurements can be expressed in terms of Vt and Λ, that is E[rtr′t|Ft−1] = ΛVtΛ′, E[Xt|Ft−1] = Vt, (8) Var[vec(rtr′t)|Ft−1]=(Ik2+Kk)(Λ⊗Λ)(Vt⊗Vt)(Λ′⊗Λ′), (9) Var[vec(Λ−1rtr′t(Λ′)−1)|Ft−1]=(Ik2+Kk)(Vt⊗Vt), (10) Var[vec(Xt)|Ft−1]=ν−1(Ik2+Kk)(Vt⊗Vt), (11) where Kk is the k2×k2 commutation matrix as discussed in detail in Magnus and Neudecker (1979) from which also the results of (9) and (11) follow directly. The result in (8) corresponds to the conditional second moment, while the results in (9) and (10) correspond to the conditional fourth moment (kurtosis) of returns. It is a convenient feature of our modeling framework that conditional second moments of realized covariance (11) provides model-implied volatilities-of-volatilities and volatility cross-asset effects (also known as spillover effects). We introduce the time-varying vector process ft for which the details of its dynamic model specification are given below. We assume that Vt is a function of ft, that is Vt=Vt(ft) for t=1,…,T. This flexible specification can accommodate a covariance matrix Vt that is only partly time-varying. But it can also allow for specifications that lead to a fully time-varying matrix Vt. In our study we consider the specification ft=vech(Vt) where the operator vech(Vt) stacks the diagonal and lower-triangular elements of the covariance matrix Vt into a vector. 1.2 Score-driven dynamics In this section, we discuss how the dynamic properties of the time-varying parameter ft can be specified. We provide details of how the model formulation is derived taking into account the measurement densities that are introduced in the previous section. We adopt the score-driven approach to time-varying parameters as developed by Creal, Koopman, and Lucas (2013). They construct a general dynamic modeling framework in which the local score function (at time t) of the conditional or predictive likelihood function is used for updating time-varying parameters. Given that the conditional score function is a function of past observations, the model belongs to the class of observation-driven models; see Cox (1981). Consider the set Zt consisting of m vector or matrix variables, we have Zt={Zt1,…,Ztm}, for which observations or measurements are available for t=1,…,T. For our RWG model, we have m = 2, Zt1=rtrt′ and Zt2=Xt. It is a straightforward extension to include more variables into Zt, such as other realized measures that can possibly provide more information on Vt=Vt(ft). The measurement distribution for the i-th variable in Zt is given by Zti∼ϕi(Zti|ft,Ft−1;ψ), i=1,…,m, t=1,…,T, (12) where ft is the d×1 vector of time-varying parameters, Ft−1=σ(Zs;s=1,…,t−1) is the sigma field generated by all observations up to time t – 1, and ψ is a vector of (unknown) static model parameters. In this framework, the individual distribution ϕi may correspond to different families of distributions. All distributions, however, depend partially on the same time-varying parameter vector ft. For our RWG model with conditional distributions (1) and (2), and with specification (7), return vector rt and realized covariance matrix Xt have different distributions but are assumed to be propelled by the common covariance matrix Vt=Vt(ft). Finally, the distribution ϕi in (12) may depend on exogenous variables; we omit this extension for simplicity in notation. We assume that the m variables in Zt are conditionally independent, conditional on both ft and the information set Ft−1. We further assume that the distributional functions ϕi are at least differentiable up to the first order with respect to ft. The log-likelihood function is then given by L(ψ)=∑t=1T∑i=1mlog ϕi(Zti|ft,Ft−1;ψ). (13) The time-varying parameter ft is updated via the recursive equation ft+1=ω+∑i=1pBift−i+1+∑j=1qAjst−j+1, (14) where ω is an d×1 vector of constants, st is a mean-zero and finite variance martingale difference sequence, Bi and Aj are d × d matrices of coefficients. The unknown parameters in ω, B1,…,Bp, A1,…,Aq and those associated with the measurement equations, such as the number of degrees of freedom in the Wishart distribution, are collected in the static parameter vector ψ. The VARMA representation (14) proves convenient for understanding the statistical dynamic properties of the ft process but also for parameter estimation. The specification (14) can be extended to incorporate some exogenous variables or other functions of lagged endogenous variables, or one could also consider long-memory specification of (14). Given the linear updating in (14), the main challenge is to formulate the martingale innovation st. Here we adopt an observation-driven approach in which we formulate the innovation term st as a function of directly observable variables. Our modeling approach follows Creal, Koopman, and Lucas (2013) by setting the innovation st equal to the scaled score of the predictive likelihood function. Under the assumption of correct model specification, the score has the convenient property that it forms a martingale difference sequence. In particular, the score vector takes an additive form given by ∇t=∑i=1m∇i,t=∑i=1m∂log ϕi(Zti|ft,Ft−1;ψ)∂ft, (15) which corresponds to the sum of individual scores. The existence of ∇t relies on the assumption of differentiability of ϕi with respect to ft up to the first order. The scaling term is based on the Fisher information matrix and can also be expressed in additive form, It=∑i=1mIi,t=∑i=1mE[∇i,t∇i,t′|Ft−1]. (16) The existence of It relies on the assumption of differentiability of ϕi with respect to ft up to the second order. The innovation term is now defined as st=It−1∇t, (17) where the invertibility of It is assumed but is often simply implied by the choices of distribution ϕi, for i=1,…,m. Further, the martingale property of ∇t implies that E[st|Ft−1]=0. In this approach, the one-step ahead prediction of the time-varying parameter vector, ft+1, is primarily based on the scaled score that exploits the full likelihood contribution at time t. The score-driven time-varying parameter Equations (14) and (17) are formulated as in Creal, Koopman, and Lucas (2013), for the case of the measurement distributions in (12). The details for the RWG model are given next. In the remainder of this treatment, we consider the updating Equation (14) with p=q=1 to obtain ft+1=ω+Bft+Ast, (18) with A = A1 and B = B1. 1.3 The details for the RWG model We provide the details of the score-driven model as introduced above for the RWG model with the time-varying covariance matrix Vt=Vt(ft) for the specification that ft simply represents all elements of Vt. In particular, we require expressions for the score function and the Fisher information matrix. Given the conditional independence assumption for the variables in Zt, in our case Zt1=rtrt′ and Zt2=Xt, we can decompose the contribution of the log-likelihood function (13) at time t in two parts, that is L(ψ)=∑t=1TLt(ψ), Lt(ψ)=Lr,t+LX,t, with the log-likelihood parts given by Lr,t=12dr(k)−12log |ΛVtΛ′|−12tr((ΛVtΛ′)−1rtr′t), (19) LX,t=12dX(k,ν)+ν−k−12log |Xt|−ν2log |Vt|−ν2tr(Vt−1Xt), (20) where dr(k)=−klog (2π), dX(k,ν)=νklog (ν/2)−2log Γk(ν/2) and Γk() is the multivariate Gamma function for dimension k. In case of the RWG model, the two log-likelihood expressions follow immediately since the distribution ϕ1=Wks(Ht,1) is the singular Wishart distribution and ϕ2=Wk(Vt/ν,ν) is the k-th dimensional Wishart distribution. Our aim is to specify a dynamic model for the matrix Vt and the time-varying parameter vector ft is therefore simply defined as ft=vech(Vt), (21) such that ft is a k*×1 vector with k*=k(k+1)/2. For the updating Equation (14), we require the score vector and Fisher information matrix that we obtain as described in Section 1.2. Theorem 1 For the measurements densities (1) and (2), the score vector of dimension k*×1 is given by ∇t=12Dk′(Vt−1⊗Vt−1)(ν·[vec(Xt)−vec(Vt)]+[vec(Λ−1rtr′t(Λ′)−1)−vec(Vt)]),where Dk is the duplication matrix as discussed in detail by Magnus and Neudecker (1979). □ Given the statistical properties in (8), it follows that E[∇t|Ft−1]=0 under correct model specification; it implies that ∇t forms a martingale difference sequence. The expression for the score shows that for the updating of ft, and hence Vt, information from the deviations of realized covariance Xt from its mean Vt receives a weight ν, whereas information from deviations of rtrt′ from Vt receives a weight of one. This model feature is pertinent as the outer-product of daily returns typically contains a weak signal about the current covariance of assets as it does not exploit intraday information. Theorem 2 For the measurements densities (1) and (2), the conditional Fisher information matrix of dimension k*×k* is given by It=E[∇t∇t′|Ft−1]=1+ν2D′k(Vt−1⊗Vt−1)Dk. □ The inverse of the conditional information matrix exists since we have assumed that Vt is nonsingular. This inverse matrix will be used to scale the score vector. Theorem 3 For the measurements densities (1) and (2), the scaled score vector st=It−1∇t is given by st=1ν+1(νvech(Xt)+vech(Λ−1rtr′t(Λ′)−1))−vech(Vt). □ The proofs of Theorems 1, 2, and 3 are given in Appendix A. For the updating of the time-varying parameter vector ft in (18), and to avoid the curse of dimensionality, we can consider specifications with diagonal matrices for A=diag(α1,…,αk*) and B=diag(β1,…,βk*), or with even more simpler scalar versions that have A=αIk* and B=βIk*. We need to impose some constraints on the parameters to guarantee that the covariance matrix Vt is positive definite with probability 1. For the scalar specification, the conditions α≥0 and β−α≥0 are sufficient to ensure that Vt is positive definite. Other constraints are needed for the diagonal specification that are discussed in more detail in Appendix C. 1.4 The RWG model with multiple measures The results in Theorems 1–3 hold for our model with the two measurement Equations (1) and (2). However, it is straightforward to extend our RWG modeling framework to incorporate several noisy measures of the daily equity covariance matrix Vt. For example, let Xti=Vt1/2ηtiVt1/2, ηti∼Wk(Ik,νi), i=1,…,G, where Xti is a noisy measure of the daily realized covariance matrix, for i=1,…,G, with G∈N. We define ν*=∑i=1Gνi and we have ∇t=12D′k(Vt−1⊗Vt−1)∑i=1Gνi[vec(Xti)−vec(Vt)], It=E[∇t∇t′|Ft−1]=D′k(Vt−1⊗Vt−1)Dkν*2, and st=(∑i=1Gνiν*vech(Xti))−vech(Vt), where the numbers of degrees of freedom ν1,ν2,…,νG are estimated along with other model static parameters. We notice that νi≡1 if Xti=rtrt′ or for any matrix Xti that has rank one. 2 Estimation procedure and Monte Carlo study We discuss the maximum likelihood estimation procedure and present simulation evidence for the statistical small-sample properties of the maximum likelihood estimation method for our model. We study estimation performance for varying sample size T and number of assets k. 2.1 Estimation procedure The log-likelihood function is given by L(ψ)=∑t=1T(Lr,t+LX,t), (22) where Lr,t and LX,t are given in (19) and (20), respectively. The time-variation of Vt is determined by the score recursion (14) and parameterization (21). The static parameter vector is given by ψ=(vec(Λ)′,ω′,vec(A)′,vec(B)′)′, and contains at least k2+k(k+1)/2 elements for ω and Λ and more elements depending on the specification of A and B; the number of parameters is therefore of order O(k2). The computation of the log-likelihood function (13) requires the updating Equation (18) that needs to be initialized. It is natural to set s0=0 and f0 either to the unconditional first moment estimated from the data or it can be added to the vector of parameters ψ. In our empirical analysis, we set f0 to be (the vec of) the sample average of the realized covariance matrices X1,…,XT. For a given parameter vector ψ, the log-likelihood function can be evaluated in a straightforward manner. In practice, ψ is unknown and estimation of all parameters is carried out via the numerical maximization of (13) with respect to ψ. The maximization relies typically on a standard quasi-Newton numerical optimization procedure; the initial values for ψ can be determined through a grid search method. For both the simulation study and the empirical application, the model parameters are estimated using numerical derivatives. As the dimension k increases, parameter estimation can become computationally demanding. A possible approach to reduce the number of parameters can be based on covariance targeting as proposed by Engle and Mezrich (1996) for GARCH models. Since the updating Equation (18) admits a VARMA representation, an analytical expression for the intercept can be provided, if stationarity conditions are satisfied. When we replace ω in (18) by its unconditional mean, we obtain ft+1=(Ik*−B)E[ft]+Bft+Ast, where E[ft] is replaced by vech(T−1∑t=1TXt). The introduction of targeting leads to a two-step approach in estimation. We first remove the vector of constants by replacing it through some consistent estimator of the unconditional mean. Then maximize the log-likelihood function with respect to the remaining parameters. To avoid the curse of dimensionality further, parameter reductions can be achieved by setting A and B as the diagonal matrices or to scalars. 2.2 Monte Carlo study We study properties of the likelihood-based estimation method by means of simulation exercises. We consider a dimension of k∈{2,5,10} and we simulate a series of T∈{250,500,1000} daily returns and daily covariance matrices. For simplicity, we study the scalar specification for the time-varying parameter (18) with A=αIk* and B=βIk*. We further consider that all elements of Λ are the same, that is λi,j=λ, for i,j=1,…,k. The Monte Carlo data generation process has adopted the following parameter values ν=k+10, ω=0.10 vech(Ik), β=0.97, α=0.30, λ=1. (23) These parameter values are roughly in line with the empirical estimates that we present in Section 3. A close-to-unity value for the autoregressive coefficient β=0.97 is typically found in many volatility studies. We simulate 5000 datasets in our Monte Carlo study. For each generated dataset, we maximize the likelihood and we collect the estimates of parameters (23). We estimate the parameters without constraints but with covariance targeting. We emphasize that we do not simulate intraday prices as we do not analyze the properties of high-frequency realized measures but we only aim to validate the estimation procedures for our model. In Figure 1, we present the density kernel estimates of the histograms of the 5000 estimates for each parameter in ψ. Each graph contains three densities which are associated with the three time series dimensions 250, 500, and 1000. For an increasing sample size T, the estimates concentrate more at their true values while the densities become more symmetric. We find some more skewness and heavy tails in the densities of the estimates obtained from the smaller sample size T = 250. In particular, the density for the memory parameter β is skewed to the left and the mode is shifted to the left near β=0.97. This bias for β in small samples is somewhat expected since autoregressive coefficients require generally a relatively longtime series for its estimation. Moreover, it is likely that the adhoc treatment of the initial value f0 will require some strong adjustments for ft in the first part of the sample. This will cause a (negative) bias in the estimation of β for relatively small samples. For an increasing sample size, this initial estimation bias will vanish. The number of degrees of freedom of the Wishart distribution ν is estimated rather accurately, even for moderate sample sizes. This finding is promising but somewhat surprising given that ν is a highly nonlinear parameter. Figure 1. View largeDownload slide Parameter estimate densities from the Monte Carlo study. Figure 1. View largeDownload slide Parameter estimate densities from the Monte Carlo study. By increasing k, this is the number of assets in our simulation study, the shapes of the densities become considerably more symmetric and more peaked around their true values; in particular, compare the panels for k = 2 and k = 10. We notice that in the Monte Carlo study our parameterization is parsimonious and therefore increasing k will lead to more pooling for the estimation of the parameters. Also, the data size increases with k2 while the number of parameters increases with k. The improvement is however remarkable for parameters α and β. We may conclude overall that the maximum likelihood method is successful in the accurate estimation of model parameters. 3 Empirical illustration 3.1 Dataset: open-to-close daily returns and realized covariance matrices In our empirical study for a portfolio of equities, we aim to measure the variation across firms and across market conditions. The equities consist of fifteen Dow Jones Industrial Average components with ticker symbols AA, AXP, BA, CAT, GE, HD, HON, IBM, JPM, KO, MCD, PFE, PG, WMT, and XOM. The empirical study is based on consolidated trades (transaction prices) extracted from the Trade and Quote (TAQ) database through the Wharton Research Data Services (WRDS) system. The time stamp precision is one second. The sample period spans ten years, from January 2, 2001 to December 31, 2010, with a total of T = 2515 trading days for all equities. We analyze these fifteen equities using the RWG model for different dimensions of k∈{2,5,15}. To conserve space, we will present results for a randomly selected set of ten bivariate models and ten 5-variate models among the fifteen equities; the random selection is justified as our primary aim is to verify estimation results, to understand their implications and to detect similarities. We also present results for our model with all fifteen equities included, which requires the modeling of a 15 × 15 conditional covariance matrix. The sample period 2001–2010 represents two characteristic periods: first a period of low volatility and then a period of high or even extreme volatility due to the “financial crises.” The length of a ten-year period is rather standard in the GARCH literature. In our study, we have followed the standard practice of excluding the overnight return for the computation of realized measures while daily asset returns can be based on both open-to-close and close-to-close returns. The vector of daily asset returns rt is taken as open-to-close returns in our study. The conditional covariance matrix Ht therefore measures the intraday variations and covariations. Hence the covariance matrices Ht and Vt contain similar information. Given the specification Ht=ΛVtΛ′ in (7), we may expect matrix Λ to be close to an identity matrix. However, the diagonal elements may be close to unity, the off-diagonal elements may reveal some interesting information on cross-asset or spillover effects. When we would have considered close-to-close returns, the overnight market risk, specific for each individual stock, would have been accounted for by the parameter matrix Λ; this overnight effect is of key interest to many market players such as liquidity providers or market makers who generally want to minimize this risk and hedge it effectively. Before we compute the realized measures, we carry out cleaning procedures to the raw transaction data. The importance of tick-by-tick data cleaning is highlighted by Hansen and Lunde (2006) and Barndorff-Nielsen et al. (2009) who provide a guideline on cleaning procedures based on the TAQ qualifiers that are included in the files (see TAQ User’s Guide from WRDS). In particular, we carry out the following steps: (i) we delete entries with a time stamp outside the 9:30 a.m.–4:00 p.m. window; (ii) we delete entries with transaction price equal to zero; (iii) we retain entries originating from a single exchange (NYSE in our application); (iv) we delete entries with corrected trades (trades with a correction indicator, “CORR” ≠0); (v) we delete entries with abnormal sale condition (trades with “COND” has a letter code, except for “E” and “F”); (vi) we use the median price for multiple transactions with the same time stamp; (vii) we delete entries with prices that are above the ask plus the bid-ask spread. For the computation of the realized covariance matrices, we adopt a kernel that is based on a subsampling scheme. We use an overall sample frequency of five minutes and adopt the refresh sampling scheme of Barndorff-Nielsen et al. (2011b). The refresh sampling scheme refers to the irregular sampling over time: a time interval ends when at least one realization is recorded for all considered k stocks. By shifting the starting time by one-second increments, we obtain 300 different estimates in five-minutes interval; the average is our subsampled realized covariance measure. Table 1 provides the number of observations and Table 2 provides the data fractions that we have retained in constructing the refresh sampling scheme. Given the dimension k, we record the resulting daily number of price observations. These statistics are averaged for each year in our sample. We observe that for the 2 × 2 datasets we retain on average of around 60−65% observations; this fraction is somewhat robust over time and across equities. The average number of refresh time observations is around 2800 and it moderately varies in time with higher volatility during the financial crisis period of 2007–2009. For the 5 × 5 case the data loss is more pronounced. We retain around 35−40% and we have 1800 refresh observations on average. For the 15 × 15 case, the overall average of fraction of retained observations equals around 22% while the average number of observations is around 950. Table 1. Average daily number of high-frequency observations maintained by the refresh sampling scheme of Barndorff-Nielsen et al. (2011b) Equities 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2x2 AA/CAT 803 1043 1340 1899 1919 2458 3538 3730 2810 2006 AXP/PFE 1805 2081 2486 2198 2372 2413 4007 4355 3527 2108 AXP/WMT 1508 1760 1865 2062 2323 2449 3994 4816 3900 2827 BA/HON 959 1248 1665 1719 2036 2171 3111 3069 2407 2154 CAT/KO 831 1144 1516 1934 2059 2382 3469 3809 3049 2585 GE/PFE 2064 2753 3061 3135 3156 3201 5105 5374 3514 1935 HD/JPM 1657 2022 2421 2329 2523 2817 4706 5454 3693 2906 IBM/PG 1566 1971 2390 2618 2659 3017 4252 4549 3493 2895 JPM/XOM 1476 1980 2516 2607 3044 3531 6187 7799 5747 4169 MCD/PG 1147 1516 1847 1969 2397 2517 3531 4330 3315 2442 5x5 AA/AXP/IBM/JPM/WMT 827 940 1048 1304 1405 1553 2632 3074 2210 1526 AA/BA/CAT/GE/KO 570 736 933 1172 1247 1466 2340 2584 1790 1266 AXP/CAT/IBM/KO/XOM 671 885 1141 1272 1352 1520 2521 2787 2239 1924 BA/HD/JPM/PFE/PG 847 1060 1336 1332 1472 1639 2665 2920 2039 1395 BA/HD/MCD/PG/XOM 748 990 1232 1238 1462 1596 2483 2834 2009 1620 CAT/GE/KO/PFE/WMT 680 887 1055 1367 1481 1646 2625 2912 2070 1333 CAT/HON/IBM/MCD/WMT 626 783 951 1172 1332 1440 2186 2342 1857 1614 GE/IBM/JPM/PG/XOM 947 1256 1548 1586 1709 1915 3283 3773 2616 1863 HD/HON/KO/MCD/PG 662 868 1066 1136 1371 1414 2196 2443 1768 1432 HON/IBM/MCD/WMT/XOM 745 940 1079 1266 1537 1585 2408 2602 1994 1669 15x15 AA/… /XOM 430 530 649 759 856 951 1613 1779 1267 894 Equities 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2x2 AA/CAT 803 1043 1340 1899 1919 2458 3538 3730 2810 2006 AXP/PFE 1805 2081 2486 2198 2372 2413 4007 4355 3527 2108 AXP/WMT 1508 1760 1865 2062 2323 2449 3994 4816 3900 2827 BA/HON 959 1248 1665 1719 2036 2171 3111 3069 2407 2154 CAT/KO 831 1144 1516 1934 2059 2382 3469 3809 3049 2585 GE/PFE 2064 2753 3061 3135 3156 3201 5105 5374 3514 1935 HD/JPM 1657 2022 2421 2329 2523 2817 4706 5454 3693 2906 IBM/PG 1566 1971 2390 2618 2659 3017 4252 4549 3493 2895 JPM/XOM 1476 1980 2516 2607 3044 3531 6187 7799 5747 4169 MCD/PG 1147 1516 1847 1969 2397 2517 3531 4330 3315 2442 5x5 AA/AXP/IBM/JPM/WMT 827 940 1048 1304 1405 1553 2632 3074 2210 1526 AA/BA/CAT/GE/KO 570 736 933 1172 1247 1466 2340 2584 1790 1266 AXP/CAT/IBM/KO/XOM 671 885 1141 1272 1352 1520 2521 2787 2239 1924 BA/HD/JPM/PFE/PG 847 1060 1336 1332 1472 1639 2665 2920 2039 1395 BA/HD/MCD/PG/XOM 748 990 1232 1238 1462 1596 2483 2834 2009 1620 CAT/GE/KO/PFE/WMT 680 887 1055 1367 1481 1646 2625 2912 2070 1333 CAT/HON/IBM/MCD/WMT 626 783 951 1172 1332 1440 2186 2342 1857 1614 GE/IBM/JPM/PG/XOM 947 1256 1548 1586 1709 1915 3283 3773 2616 1863 HD/HON/KO/MCD/PG 662 868 1066 1136 1371 1414 2196 2443 1768 1432 HON/IBM/MCD/WMT/XOM 745 940 1079 1266 1537 1585 2408 2602 1994 1669 15x15 AA/… /XOM 430 530 649 759 856 951 1613 1779 1267 894 Note: The averages are over the days in each year of our sample. Table 1. Average daily number of high-frequency observations maintained by the refresh sampling scheme of Barndorff-Nielsen et al. (2011b) Equities 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2x2 AA/CAT 803 1043 1340 1899 1919 2458 3538 3730 2810 2006 AXP/PFE 1805 2081 2486 2198 2372 2413 4007 4355 3527 2108 AXP/WMT 1508 1760 1865 2062 2323 2449 3994 4816 3900 2827 BA/HON 959 1248 1665 1719 2036 2171 3111 3069 2407 2154 CAT/KO 831 1144 1516 1934 2059 2382 3469 3809 3049 2585 GE/PFE 2064 2753 3061 3135 3156 3201 5105 5374 3514 1935 HD/JPM 1657 2022 2421 2329 2523 2817 4706 5454 3693 2906 IBM/PG 1566 1971 2390 2618 2659 3017 4252 4549 3493 2895 JPM/XOM 1476 1980 2516 2607 3044 3531 6187 7799 5747 4169 MCD/PG 1147 1516 1847 1969 2397 2517 3531 4330 3315 2442 5x5 AA/AXP/IBM/JPM/WMT 827 940 1048 1304 1405 1553 2632 3074 2210 1526 AA/BA/CAT/GE/KO 570 736 933 1172 1247 1466 2340 2584 1790 1266 AXP/CAT/IBM/KO/XOM 671 885 1141 1272 1352 1520 2521 2787 2239 1924 BA/HD/JPM/PFE/PG 847 1060 1336 1332 1472 1639 2665 2920 2039 1395 BA/HD/MCD/PG/XOM 748 990 1232 1238 1462 1596 2483 2834 2009 1620 CAT/GE/KO/PFE/WMT 680 887 1055 1367 1481 1646 2625 2912 2070 1333 CAT/HON/IBM/MCD/WMT 626 783 951 1172 1332 1440 2186 2342 1857 1614 GE/IBM/JPM/PG/XOM 947 1256 1548 1586 1709 1915 3283 3773 2616 1863 HD/HON/KO/MCD/PG 662 868 1066 1136 1371 1414 2196 2443 1768 1432 HON/IBM/MCD/WMT/XOM 745 940 1079 1266 1537 1585 2408 2602 1994 1669 15x15 AA/… /XOM 430 530 649 759 856 951 1613 1779 1267 894 Equities 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2x2 AA/CAT 803 1043 1340 1899 1919 2458 3538 3730 2810 2006 AXP/PFE 1805 2081 2486 2198 2372 2413 4007 4355 3527 2108 AXP/WMT 1508 1760 1865 2062 2323 2449 3994 4816 3900 2827 BA/HON 959 1248 1665 1719 2036 2171 3111 3069 2407 2154 CAT/KO 831 1144 1516 1934 2059 2382 3469 3809 3049 2585 GE/PFE 2064 2753 3061 3135 3156 3201 5105 5374 3514 1935 HD/JPM 1657 2022 2421 2329 2523 2817 4706 5454 3693 2906 IBM/PG 1566 1971 2390 2618 2659 3017 4252 4549 3493 2895 JPM/XOM 1476 1980 2516 2607 3044 3531 6187 7799 5747 4169 MCD/PG 1147 1516 1847 1969 2397 2517 3531 4330 3315 2442 5x5 AA/AXP/IBM/JPM/WMT 827 940 1048 1304 1405 1553 2632 3074 2210 1526 AA/BA/CAT/GE/KO 570 736 933 1172 1247 1466 2340 2584 1790 1266 AXP/CAT/IBM/KO/XOM 671 885 1141 1272 1352 1520 2521 2787 2239 1924 BA/HD/JPM/PFE/PG 847 1060 1336 1332 1472 1639 2665 2920 2039 1395 BA/HD/MCD/PG/XOM 748 990 1232 1238 1462 1596 2483 2834 2009 1620 CAT/GE/KO/PFE/WMT 680 887 1055 1367 1481 1646 2625 2912 2070 1333 CAT/HON/IBM/MCD/WMT 626 783 951 1172 1332 1440 2186 2342 1857 1614 GE/IBM/JPM/PG/XOM 947 1256 1548 1586 1709 1915 3283 3773 2616 1863 HD/HON/KO/MCD/PG 662 868 1066 1136 1371 1414 2196 2443 1768 1432 HON/IBM/MCD/WMT/XOM 745 940 1079 1266 1537 1585 2408 2602 1994 1669 15x15 AA/… /XOM 430 530 649 759 856 951 1613 1779 1267 894 Note: The averages are over the days in each year of our sample. Table 2. Average ratio of the data maintained by the refresh sampling scheme of Barndorff-Nielsen et al. (2011b) Equities 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2x2 AA/CAT 0.599 0.588 0.587 0.601 0.584 0.579 0.625 0.640 0.612 0.543 AXP/PFE 0.646 0.625 0.627 0.565 0.579 0.572 0.625 0.653 0.620 0.548 AXP/WMT 0.637 0.629 0.600 0.576 0.584 0.570 0.631 0.666 0.652 0.625 BA/HON 0.616 0.601 0.615 0.603 0.598 0.586 0.627 0.636 0.629 0.632 CAT/KO 0.583 0.573 0.577 0.595 0.584 0.585 0.636 0.651 0.643 0.626 GE/PFE 0.655 0.642 0.655 0.640 0.640 0.624 0.668 0.663 0.617 0.548 HD/JPM 0.644 0.625 0.635 0.615 0.621 0.607 0.652 0.636 0.575 0.582 IBM/PG 0.579 0.646 0.648 0.636 0.626 0.628 0.662 0.672 0.654 0.642 JPM/XOM 0.626 0.618 0.629 0.620 0.584 0.566 0.672 0.732 0.699 0.668 MCD/PG 0.643 0.628 0.624 0.610 0.621 0.597 0.634 0.662 0.643 0.637 5x5 AA/AXP/IBM/JPM/WMT 0.338 0.338 0.322 0.347 0.354 0.348 0.396 0.407 0.371 0.329 AA/BA/CAT/GE/KO 0.314 0.288 0.308 0.336 0.334 0.339 0.385 0.394 0.375 0.334 AXP/CAT/IBM/KO/XOM 0.305 0.324 0.338 0.348 0.322 0.313 0.374 0.398 0.394 0.400 BA/HD/JPM/PFE/PG 0.357 0.348 0.363 0.345 0.354 0.354 0.400 0.395 0.360 0.328 BA/HD/MCD/PG/XOM 0.373 0.353 0.365 0.352 0.337 0.319 0.369 0.399 0.373 0.373 CAT/GE/KO/PFE/WMT 0.296 0.290 0.303 0.330 0.340 0.348 0.393 0.404 0.384 0.331 CAT/HON/IBM/ MCD/WMT 0.305 0.326 0.330 0.333 0.339 0.336 0.382 0.396 0.385 0.388 GE/IBM/JPM/PG/XOM 0.358 0.366 0.384 0.371 0.361 0.352 0.416 0.426 0.392 0.362 HD/HON/KO/MCD/PG 0.359 0.347 0.354 0.353 0.362 0.348 0.393 0.405 0.385 0.389 HON/IBM/MCD/ WMT/XOM 0.333 0.340 0.333 0.335 0.337 0.316 0.357 0.374 0.366 0.370 15x15 AA/… /XOM 0.197 0.189 0.195 0.206 0.208 0.207 0.247 0.253 0.234 0.210 Equities 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2x2 AA/CAT 0.599 0.588 0.587 0.601 0.584 0.579 0.625 0.640 0.612 0.543 AXP/PFE 0.646 0.625 0.627 0.565 0.579 0.572 0.625 0.653 0.620 0.548 AXP/WMT 0.637 0.629 0.600 0.576 0.584 0.570 0.631 0.666 0.652 0.625 BA/HON 0.616 0.601 0.615 0.603 0.598 0.586 0.627 0.636 0.629 0.632 CAT/KO 0.583 0.573 0.577 0.595 0.584 0.585 0.636 0.651 0.643 0.626 GE/PFE 0.655 0.642 0.655 0.640 0.640 0.624 0.668 0.663 0.617 0.548 HD/JPM 0.644 0.625 0.635 0.615 0.621 0.607 0.652 0.636 0.575 0.582 IBM/PG 0.579 0.646 0.648 0.636 0.626 0.628 0.662 0.672 0.654 0.642 JPM/XOM 0.626 0.618 0.629 0.620 0.584 0.566 0.672 0.732 0.699 0.668 MCD/PG 0.643 0.628 0.624 0.610 0.621 0.597 0.634 0.662 0.643 0.637 5x5 AA/AXP/IBM/JPM/WMT 0.338 0.338 0.322 0.347 0.354 0.348 0.396 0.407 0.371 0.329 AA/BA/CAT/GE/KO 0.314 0.288 0.308 0.336 0.334 0.339 0.385 0.394 0.375 0.334 AXP/CAT/IBM/KO/XOM 0.305 0.324 0.338 0.348 0.322 0.313 0.374 0.398 0.394 0.400 BA/HD/JPM/PFE/PG 0.357 0.348 0.363 0.345 0.354 0.354 0.400 0.395 0.360 0.328 BA/HD/MCD/PG/XOM 0.373 0.353 0.365 0.352 0.337 0.319 0.369 0.399 0.373 0.373 CAT/GE/KO/PFE/WMT 0.296 0.290 0.303 0.330 0.340 0.348 0.393 0.404 0.384 0.331 CAT/HON/IBM/ MCD/WMT 0.305 0.326 0.330 0.333 0.339 0.336 0.382 0.396 0.385 0.388 GE/IBM/JPM/PG/XOM 0.358 0.366 0.384 0.371 0.361 0.352 0.416 0.426 0.392 0.362 HD/HON/KO/MCD/PG 0.359 0.347 0.354 0.353 0.362 0.348 0.393 0.405 0.385 0.389 HON/IBM/MCD/ WMT/XOM 0.333 0.340 0.333 0.335 0.337 0.316 0.357 0.374 0.366 0.370 15x15 AA/… /XOM 0.197 0.189 0.195 0.206 0.208 0.207 0.247 0.253 0.234 0.210 Note: The averages are over the days in each year of our sample. Table 2. Average ratio of the data maintained by the refresh sampling scheme of Barndorff-Nielsen et al. (2011b) Equities 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2x2 AA/CAT 0.599 0.588 0.587 0.601 0.584 0.579 0.625 0.640 0.612 0.543 AXP/PFE 0.646 0.625 0.627 0.565 0.579 0.572 0.625 0.653 0.620 0.548 AXP/WMT 0.637 0.629 0.600 0.576 0.584 0.570 0.631 0.666 0.652 0.625 BA/HON 0.616 0.601 0.615 0.603 0.598 0.586 0.627 0.636 0.629 0.632 CAT/KO 0.583 0.573 0.577 0.595 0.584 0.585 0.636 0.651 0.643 0.626 GE/PFE 0.655 0.642 0.655 0.640 0.640 0.624 0.668 0.663 0.617 0.548 HD/JPM 0.644 0.625 0.635 0.615 0.621 0.607 0.652 0.636 0.575 0.582 IBM/PG 0.579 0.646 0.648 0.636 0.626 0.628 0.662 0.672 0.654 0.642 JPM/XOM 0.626 0.618 0.629 0.620 0.584 0.566 0.672 0.732 0.699 0.668 MCD/PG 0.643 0.628 0.624 0.610 0.621 0.597 0.634 0.662 0.643 0.637 5x5 AA/AXP/IBM/JPM/WMT 0.338 0.338 0.322 0.347 0.354 0.348 0.396 0.407 0.371 0.329 AA/BA/CAT/GE/KO 0.314 0.288 0.308 0.336 0.334 0.339 0.385 0.394 0.375 0.334 AXP/CAT/IBM/KO/XOM 0.305 0.324 0.338 0.348 0.322 0.313 0.374 0.398 0.394 0.400 BA/HD/JPM/PFE/PG 0.357 0.348 0.363 0.345 0.354 0.354 0.400 0.395 0.360 0.328 BA/HD/MCD/PG/XOM 0.373 0.353 0.365 0.352 0.337 0.319 0.369 0.399 0.373 0.373 CAT/GE/KO/PFE/WMT 0.296 0.290 0.303 0.330 0.340 0.348 0.393 0.404 0.384 0.331 CAT/HON/IBM/ MCD/WMT 0.305 0.326 0.330 0.333 0.339 0.336 0.382 0.396 0.385 0.388 GE/IBM/JPM/PG/XOM 0.358 0.366 0.384 0.371 0.361 0.352 0.416 0.426 0.392 0.362 HD/HON/KO/MCD/PG 0.359 0.347 0.354 0.353 0.362 0.348 0.393 0.405 0.385 0.389 HON/IBM/MCD/ WMT/XOM 0.333 0.340 0.333 0.335 0.337 0.316 0.357 0.374 0.366 0.370 15x15 AA/… /XOM 0.197 0.189 0.195 0.206 0.208 0.207 0.247 0.253 0.234 0.210 Equities 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2x2 AA/CAT 0.599 0.588 0.587 0.601 0.584 0.579 0.625 0.640 0.612 0.543 AXP/PFE 0.646 0.625 0.627 0.565 0.579 0.572 0.625 0.653 0.620 0.548 AXP/WMT 0.637 0.629 0.600 0.576 0.584 0.570 0.631 0.666 0.652 0.625 BA/HON 0.616 0.601 0.615 0.603 0.598 0.586 0.627 0.636 0.629 0.632 CAT/KO 0.583 0.573 0.577 0.595 0.584 0.585 0.636 0.651 0.643 0.626 GE/PFE 0.655 0.642 0.655 0.640 0.640 0.624 0.668 0.663 0.617 0.548 HD/JPM 0.644 0.625 0.635 0.615 0.621 0.607 0.652 0.636 0.575 0.582 IBM/PG 0.579 0.646 0.648 0.636 0.626 0.628 0.662 0.672 0.654 0.642 JPM/XOM 0.626 0.618 0.629 0.620 0.584 0.566 0.672 0.732 0.699 0.668 MCD/PG 0.643 0.628 0.624 0.610 0.621 0.597 0.634 0.662 0.643 0.637 5x5 AA/AXP/IBM/JPM/WMT 0.338 0.338 0.322 0.347 0.354 0.348 0.396 0.407 0.371 0.329 AA/BA/CAT/GE/KO 0.314 0.288 0.308 0.336 0.334 0.339 0.385 0.394 0.375 0.334 AXP/CAT/IBM/KO/XOM 0.305 0.324 0.338 0.348 0.322 0.313 0.374 0.398 0.394 0.400 BA/HD/JPM/PFE/PG 0.357 0.348 0.363 0.345 0.354 0.354 0.400 0.395 0.360 0.328 BA/HD/MCD/PG/XOM 0.373 0.353 0.365 0.352 0.337 0.319 0.369 0.399 0.373 0.373 CAT/GE/KO/PFE/WMT 0.296 0.290 0.303 0.330 0.340 0.348 0.393 0.404 0.384 0.331 CAT/HON/IBM/ MCD/WMT 0.305 0.326 0.330 0.333 0.339 0.336 0.382 0.396 0.385 0.388 GE/IBM/JPM/PG/XOM 0.358 0.366 0.384 0.371 0.361 0.352 0.416 0.426 0.392 0.362 HD/HON/KO/MCD/PG 0.359 0.347 0.354 0.353 0.362 0.348 0.393 0.405 0.385 0.389 HON/IBM/MCD/ WMT/XOM 0.333 0.340 0.333 0.335 0.337 0.316 0.357 0.374 0.366 0.370 15x15 AA/… /XOM 0.197 0.189 0.195 0.206 0.208 0.207 0.247 0.253 0.234 0.210 Note: The averages are over the days in each year of our sample. 3.2 Estimation results We present the parameter estimation results from the RWG model when applied to the datasets as described. The dynamic specification for the covariance matrix Vt is based on the updating Equation (18) for ft=vech(Vt) with A=αIk* and B=βIk*. In Appendix C, we consider the estimation results for a less parsimonious specification that allows for different dynamics for the variances (αv and βv) and covariances (αc and βc). The additional results do not suggest that a more flexible specification provides better results compared with those for the basic specification. We also investigate the presence of cross-effects by having Λ as a diagonal matrix and as a full matrix. When off-diagonal elements of Λ are estimated to be significantly different from zero, it implies that cross-effects are present. Table 3 presents the maximum likelihood estimation results for the parameters in the RWG model for k = 2. We report the estimates of the models for a full matrix Λ (first panel) and for a diagonal matrix Λ (second panel). The estimates of the diagonal elements of Λ tend to be close-to-unity but most have estimated values just below unity, the smallest estimate is 0.88 and the largest is 1.03. Many off-diagonal elements are estimated as not being significantly different from zero, only 5 out of 20 appear to have some statistical impact. The significantly estimated off-diagonal elements of Λ are all positive and range from 0.03 to 0.23. Although in most cases, the Akaike information criterion (AIC) points weakly toward a model specification with a full Λ matrix, other aspects of our analyses, including the estimates of ν, β and α, are not affected when we restrict Λ to be diagonal. Table 4 presents the results for the model with k = 5 and Table 5 presents those for the model with k = 15, both with a diagonal matrix Λ. Table 3. Maximum likelihood estimates for the 2 × 2 models Equities ν β α λ11 λ22 λ12 λ21 logL AIC 2×2 AA/CAT 12.428 0.977 0.331 0.893 1.022 0.226 −0.032 −20,171.5 40,356.9 (0.189) (0.002) (0.011) (0.034) (0.026) (0.073) (0.050) AXP/PFE 10.876 0.991 0.378 1.032 0.918 −0.018 0.078 −17,107.3 34,228.5 (0.164) (0.001) (0.012) (0.016) (0.016) (0.030) (0.021) AXP/WMT 11.907 0.993 0.360 1.018 0.887 0.033 0.025 −15,347.7 30,709.4 (0.180) (0.001) (0.012) (0.017) (0.015) (0.033) (0.017) BA/HON 10.681 0.975 0.354 0.986 0.894 0.026 0.104 −17,860.8 35,735.5 (0.161) (0.002) (0.011) (0.028) (0.029) (0.053) (0.055) CAT/KO 12.829 0.977 0.354 0.986 0.928 0.095 −0.037 −14,227.6 28,469.1 (0.195) (0.002) (0.011) (0.022) (0.017) (0.074) (0.030) GE/PFE 11.015 0.984 0.405 0.943 0.911 0.016 0.072 −15,622.3 31,258.7 (0.166) (0.001) (0.013) (0.017) (0.018) (0.030) (0.029) HD/JPM 12.458 0.988 0.447 0.953 0.944 0.020 0.125 −18,481.1 36,976.2 (0.189) (0.001) (0.013) (0.018) (0.020) (0.031) (0.036) IBM/PG 12.409 0.977 0.383 0.984 0.866 −0.025 0.030 −10,961.1 21,936.2 (0.189) (0.002) (0.012) (0.020) (0.020) (0.052) (0.035) JPM/XOM 13.086 0.989 0.441 0.987 0.930 0.012 0.034 −16,082.0 32,178.0 (0.199) (0.001) (0.012) (0.016) (0.016) (0.032) (0.018) MCD/PG 10.427 0.979 0.310 0.919 0.880 0.039 0.015 −12,645.9 25,305.8 (0.157) (0.002) (0.011) (0.018) (0.017) (0.049) (0.027) AA/CAT 12.424 0.977 0.333 0.952 0.978 – – −20,201.6 40,413.3 (0.189) (0.002) (0.011) (0.013) (0.013) AXP/PFE 10.876 0.991 0.377 1.013 0.940 – – −17,118.6 34,247.2 (0.164) (0.001) (0.012) (0.014) (0.013) AXP/WMT 11.908 0.993 0.360 1.016 0.890 – – −15,353.2 30,716.3 (0.180) (0.001) (0.012) (0.014) (0.012) BA/HON 10.680 0.975 0.354 0.969 0.915 – – −17,883.1 35,776.3 (0.161) (0.002) (0.011) (0.013) (0.012) CAT/KO 12.829 0.977 0.354 1.007 0.913 – – −14,228.4 28,466.8 (0.195) (0.002) (0.011) (0.014) (0.013) GE/PFE 11.013 0.984 0.405 0.931 0.926 – – −15,634.8 31,279.6 (0.166) (0.001) (0.013) (0.013) (0.013) HD/JPM 12.455 0.988 0.445 0.937 0.968 – – −18,509.2 37,028.4 (0.189) (0.001) (0.013) (0.013) (0.013) IBM/PG 12.409 0.977 0.383 0.974 0.877 – – 10,961.7 21,933.3 (0.189) (0.002) (0.012) (0.014) (0.012) JPM/XOM 13.088 0.989 0.441 0.979 0.939 – – −16,087.1 32,184.1 (0.199) (0.001) (0.012) (0.014) (0.013) MCD/PG 10.426 0.979 0.310 0.921 0.879 – – −12,649.5 25,309.0 (0.157) (0.002) (0.011) (0.013) (0.012) Equities ν β α λ11 λ22 λ12 λ21 logL AIC 2×2 AA/CAT 12.428 0.977 0.331 0.893 1.022 0.226 −0.032 −20,171.5 40,356.9 (0.189) (0.002) (0.011) (0.034) (0.026) (0.073) (0.050) AXP/PFE 10.876 0.991 0.378 1.032 0.918 −0.018 0.078 −17,107.3 34,228.5 (0.164) (0.001) (0.012) (0.016) (0.016) (0.030) (0.021) AXP/WMT 11.907 0.993 0.360 1.018 0.887 0.033 0.025 −15,347.7 30,709.4 (0.180) (0.001) (0.012) (0.017) (0.015) (0.033) (0.017) BA/HON 10.681 0.975 0.354 0.986 0.894 0.026 0.104 −17,860.8 35,735.5 (0.161) (0.002) (0.011) (0.028) (0.029) (0.053) (0.055) CAT/KO 12.829 0.977 0.354 0.986 0.928 0.095 −0.037 −14,227.6 28,469.1 (0.195) (0.002) (0.011) (0.022) (0.017) (0.074) (0.030) GE/PFE 11.015 0.984 0.405 0.943 0.911 0.016 0.072 −15,622.3 31,258.7 (0.166) (0.001) (0.013) (0.017) (0.018) (0.030) (0.029) HD/JPM 12.458 0.988 0.447 0.953 0.944 0.020 0.125 −18,481.1 36,976.2 (0.189) (0.001) (0.013) (0.018) (0.020) (0.031) (0.036) IBM/PG 12.409 0.977 0.383 0.984 0.866 −0.025 0.030 −10,961.1 21,936.2 (0.189) (0.002) (0.012) (0.020) (0.020) (0.052) (0.035) JPM/XOM 13.086 0.989 0.441 0.987 0.930 0.012 0.034 −16,082.0 32,178.0 (0.199) (0.001) (0.012) (0.016) (0.016) (0.032) (0.018) MCD/PG 10.427 0.979 0.310 0.919 0.880 0.039 0.015 −12,645.9 25,305.8 (0.157) (0.002) (0.011) (0.018) (0.017) (0.049) (0.027) AA/CAT 12.424 0.977 0.333 0.952 0.978 – – −20,201.6 40,413.3 (0.189) (0.002) (0.011) (0.013) (0.013) AXP/PFE 10.876 0.991 0.377 1.013 0.940 – – −17,118.6 34,247.2 (0.164) (0.001) (0.012) (0.014) (0.013) AXP/WMT 11.908 0.993 0.360 1.016 0.890 – – −15,353.2 30,716.3 (0.180) (0.001) (0.012) (0.014) (0.012) BA/HON 10.680 0.975 0.354 0.969 0.915 – – −17,883.1 35,776.3 (0.161) (0.002) (0.011) (0.013) (0.012) CAT/KO 12.829 0.977 0.354 1.007 0.913 – – −14,228.4 28,466.8 (0.195) (0.002) (0.011) (0.014) (0.013) GE/PFE 11.013 0.984 0.405 0.931 0.926 – – −15,634.8 31,279.6 (0.166) (0.001) (0.013) (0.013) (0.013) HD/JPM 12.455 0.988 0.445 0.937 0.968 – – −18,509.2 37,028.4 (0.189) (0.001) (0.013) (0.013) (0.013) IBM/PG 12.409 0.977 0.383 0.974 0.877 – – 10,961.7 21,933.3 (0.189) (0.002) (0.012) (0.014) (0.012) JPM/XOM 13.088 0.989 0.441 0.979 0.939 – – −16,087.1 32,184.1 (0.199) (0.001) (0.012) (0.014) (0.013) MCD/PG 10.426 0.979 0.310 0.921 0.879 – – −12,649.5 25,309.0 (0.157) (0.002) (0.011) (0.013) (0.012) Note: Standard errors are shown in parentheses. Table 3. Maximum likelihood estimates for the 2 × 2 models Equities ν β α λ11 λ22 λ12 λ21 logL AIC 2×2 AA/CAT 12.428 0.977 0.331 0.893 1.022 0.226 −0.032 −20,171.5 40,356.9 (0.189) (0.002) (0.011) (0.034) (0.026) (0.073) (0.050) AXP/PFE 10.876 0.991 0.378 1.032 0.918 −0.018 0.078 −17,107.3 34,228.5 (0.164) (0.001) (0.012) (0.016) (0.016) (0.030) (0.021) AXP/WMT 11.907 0.993 0.360 1.018 0.887 0.033 0.025 −15,347.7 30,709.4 (0.180) (0.001) (0.012) (0.017) (0.015) (0.033) (0.017) BA/HON 10.681 0.975 0.354 0.986 0.894 0.026 0.104 −17,860.8 35,735.5 (0.161) (0.002) (0.011) (0.028) (0.029) (0.053) (0.055) CAT/KO 12.829 0.977 0.354 0.986 0.928 0.095 −0.037 −14,227.6 28,469.1 (0.195) (0.002) (0.011) (0.022) (0.017) (0.074) (0.030) GE/PFE 11.015 0.984 0.405 0.943 0.911 0.016 0.072 −15,622.3 31,258.7 (0.166) (0.001) (0.013) (0.017) (0.018) (0.030) (0.029) HD/JPM 12.458 0.988 0.447 0.953 0.944 0.020 0.125 −18,481.1 36,976.2 (0.189) (0.001) (0.013) (0.018) (0.020) (0.031) (0.036) IBM/PG 12.409 0.977 0.383 0.984 0.866 −0.025 0.030 −10,961.1 21,936.2 (0.189) (0.002) (0.012) (0.020) (0.020) (0.052) (0.035) JPM/XOM 13.086 0.989 0.441 0.987 0.930 0.012 0.034 −16,082.0 32,178.0 (0.199) (0.001) (0.012) (0.016) (0.016) (0.032) (0.018) MCD/PG 10.427 0.979 0.310 0.919 0.880 0.039 0.015 −12,645.9 25,305.8 (0.157) (0.002) (0.011) (0.018) (0.017) (0.049) (0.027) AA/CAT 12.424 0.977 0.333 0.952 0.978 – – −20,201.6 40,413.3 (0.189) (0.002) (0.011) (0.013) (0.013) AXP/PFE 10.876 0.991 0.377 1.013 0.940 – – −17,118.6 34,247.2 (0.164) (0.001) (0.012) (0.014) (0.013) AXP/WMT 11.908 0.993 0.360 1.016 0.890 – – −15,353.2 30,716.3 (0.180) (0.001) (0.012) (0.014) (0.012) BA/HON 10.680 0.975 0.354 0.969 0.915 – – −17,883.1 35,776.3 (0.161) (0.002) (0.011) (0.013) (0.012) CAT/KO 12.829 0.977 0.354 1.007 0.913 – – −14,228.4 28,466.8 (0.195) (0.002) (0.011) (0.014) (0.013) GE/PFE 11.013 0.984 0.405 0.931 0.926 – – −15,634.8 31,279.6 (0.166) (0.001) (0.013) (0.013) (0.013) HD/JPM 12.455 0.988 0.445 0.937 0.968 – – −18,509.2 37,028.4 (0.189) (0.001) (0.013) (0.013) (0.013) IBM/PG 12.409 0.977 0.383 0.974 0.877 – – 10,961.7 21,933.3 (0.189) (0.002) (0.012) (0.014) (0.012) JPM/XOM 13.088 0.989 0.441 0.979 0.939 – – −16,087.1 32,184.1 (0.199) (0.001) (0.012) (0.014) (0.013) MCD/PG 10.426 0.979 0.310 0.921 0.879 – – −12,649.5 25,309.0 (0.157) (0.002) (0.011) (0.013) (0.012) Equities ν β α λ11 λ22 λ12 λ21 logL AIC 2×2 AA/CAT 12.428 0.977 0.331 0.893 1.022 0.226 −0.032 −20,171.5 40,356.9 (0.189) (0.002) (0.011) (0.034) (0.026) (0.073) (0.050) AXP/PFE 10.876 0.991 0.378 1.032 0.918 −0.018 0.078 −17,107.3 34,228.5 (0.164) (0.001) (0.012) (0.016) (0.016) (0.030) (0.021) AXP/WMT 11.907 0.993 0.360 1.018 0.887 0.033 0.025 −15,347.7 30,709.4 (0.180) (0.001) (0.012) (0.017) (0.015) (0.033) (0.017) BA/HON 10.681 0.975 0.354 0.986 0.894 0.026 0.104 −17,860.8 35,735.5 (0.161) (0.002) (0.011) (0.028) (0.029) (0.053) (0.055) CAT/KO 12.829 0.977 0.354 0.986 0.928 0.095 −0.037 −14,227.6 28,469.1 (0.195) (0.002) (0.011) (0.022) (0.017) (0.074) (0.030) GE/PFE 11.015 0.984 0.405 0.943 0.911 0.016 0.072 −15,622.3 31,258.7 (0.166) (0.001) (0.013) (0.017) (0.018) (0.030) (0.029) HD/JPM 12.458 0.988 0.447 0.953 0.944 0.020 0.125 −18,481.1 36,976.2 (0.189) (0.001) (0.013) (0.018) (0.020) (0.031) (0.036) IBM/PG 12.409 0.977 0.383 0.984 0.866 −0.025 0.030 −10,961.1 21,936.2 (0.189) (0.002) (0.012) (0.020) (0.020) (0.052) (0.035) JPM/XOM 13.086 0.989 0.441 0.987 0.930 0.012 0.034 −16,082.0 32,178.0 (0.199) (0.001) (0.012) (0.016) (0.016) (0.032) (0.018) MCD/PG 10.427 0.979 0.310 0.919 0.880 0.039 0.015 −12,645.9 25,305.8 (0.157) (0.002) (0.011) (0.018) (0.017) (0.049) (0.027) AA/CAT 12.424 0.977 0.333 0.952 0.978 – – −20,201.6 40,413.3 (0.189) (0.002) (0.011) (0.013) (0.013) AXP/PFE 10.876 0.991 0.377 1.013 0.940 – – −17,118.6 34,247.2 (0.164) (0.001) (0.012) (0.014) (0.013) AXP/WMT 11.908 0.993 0.360 1.016 0.890 – – −15,353.2 30,716.3 (0.180) (0.001) (0.012) (0.014) (0.012) BA/HON 10.680 0.975 0.354 0.969 0.915 – – −17,883.1 35,776.3 (0.161) (0.002) (0.011) (0.013) (0.012) CAT/KO 12.829 0.977 0.354 1.007 0.913 – – −14,228.4 28,466.8 (0.195) (0.002) (0.011) (0.014) (0.013) GE/PFE 11.013 0.984 0.405 0.931 0.926 – – −15,634.8 31,279.6 (0.166) (0.001) (0.013) (0.013) (0.013) HD/JPM 12.455 0.988 0.445 0.937 0.968 – – −18,509.2 37,028.4 (0.189) (0.001) (0.013) (0.013) (0.013) IBM/PG 12.409 0.977 0.383 0.974 0.877 – – 10,961.7 21,933.3 (0.189) (0.002) (0.012) (0.014) (0.012) JPM/XOM 13.088 0.989 0.441 0.979 0.939 – – −16,087.1 32,184.1 (0.199) (0.001) (0.012) (0.014) (0.013) MCD/PG 10.426 0.979 0.310 0.921 0.879 – – −12,649.5 25,309.0 (0.157) (0.002) (0.011) (0.013) (0.012) Note: Standard errors are shown in parentheses. Table 4. Maximum likelihood estimates for the 5 × 5 models Equities ν β α λ11 λ22 λ33 λ44 λ55 logL 5×5 AA/AXP/IBM/JPM/WMT 18.515 0.991 0.302 0.968 0.979 0.970 0.955 0.891 −43,769.0 (0.120) (0.000) (0.005) (0.013) (0.012) (0.013) (0.012) (0.012) AA/BA/CAT/GE/KO 17.727 0.986 0.266 0.958 0.984 0.985 0.925 0.923 −44,903.2 (0.115) (0.001) (0.005) (0.013) (0.013) (0.013) (0.012) (0.013) AXP/CAT/IBM/KO/XOM 19.185 0.990 0.296 0.998 0.991 0.977 0.919 0.946 −32,878.6 (0.125) (0.001) (0.004) (0.013) (0.013) (0.013) (0.012) (0.012) BA/HD/JPM/PFE/PG 17.856 0.987 0.300 0.986 0.949 0.968 0.941 0.889 −42,213.9 (0.116) (0.001) (0.005) (0.013) (0.013) (0.013) (0.013) (0.012) BA/HD/MCD/PG/XOM 18.178 0.981 0.273 0.987 0.962 0.932 0.888 0.954 −36,404.4 (0.118) (0.001) (0.005) (0.013) (0.013) (0.013) (0.012) (0.013) CAT/GE/KO/PFE/WMT 18.084 0.985 0.283 1.004 0.933 0.921 0.944 0.902 −33,962.4 (0.117) (0.001) (0.005) (0.013) (0.012) (0.012) (0.013) (0.012) CAT/HON/IBM/MCD/WMT 17.252 0.981 0.268 0.997 0.928 0.982 0.948 0.901 −38,326.4 (0.111) (0.001) (0.004) (0.013) (0.012) (0.013) (0.013) (0.012) GE/IBM/JPM/PG/XOM 19.650 0.989 0.342 0.912 0.969 0.958 0.879 0.940 −30,238.6 (0.129) (0.001) (0.005) (0.011) (0.013) (0.012) (0.012) (0.012) HD/HON/KO/MCD/PG 17.018 0.980 0.276 0.956 0.937 0.915 0.939 0.890 −34,545.2 (0.109) (0.001) (0.005) (0.013) (0.012) (0.012) (0.013) (0.012) HON/IBM/MCD/WMT/XOM 18.139 0.982 0.279 0.933 0.984 0.946 0.903 0.956 −33,552.8 (0.118) (0.001) (0.004) (0.012) (0.013) (0.013) (0.012) (0.013) Equities ν β α λ11 λ22 λ33 λ44 λ55 logL 5×5 AA/AXP/IBM/JPM/WMT 18.515 0.991 0.302 0.968 0.979 0.970 0.955 0.891 −43,769.0 (0.120) (0.000) (0.005) (0.013) (0.012) (0.013) (0.012) (0.012) AA/BA/CAT/GE/KO 17.727 0.986 0.266 0.958 0.984 0.985 0.925 0.923 −44,903.2 (0.115) (0.001) (0.005) (0.013) (0.013) (0.013) (0.012) (0.013) AXP/CAT/IBM/KO/XOM 19.185 0.990 0.296 0.998 0.991 0.977 0.919 0.946 −32,878.6 (0.125) (0.001) (0.004) (0.013) (0.013) (0.013) (0.012) (0.012) BA/HD/JPM/PFE/PG 17.856 0.987 0.300 0.986 0.949 0.968 0.941 0.889 −42,213.9 (0.116) (0.001) (0.005) (0.013) (0.013) (0.013) (0.013) (0.012) BA/HD/MCD/PG/XOM 18.178 0.981 0.273 0.987 0.962 0.932 0.888 0.954 −36,404.4 (0.118) (0.001) (0.005) (0.013) (0.013) (0.013) (0.012) (0.013) CAT/GE/KO/PFE/WMT 18.084 0.985 0.283 1.004 0.933 0.921 0.944 0.902 −33,962.4 (0.117) (0.001) (0.005) (0.013) (0.012) (0.012) (0.013) (0.012) CAT/HON/IBM/MCD/WMT 17.252 0.981 0.268 0.997 0.928 0.982 0.948 0.901 −38,326.4 (0.111) (0.001) (0.004) (0.013) (0.012) (0.013) (0.013) (0.012) GE/IBM/JPM/PG/XOM 19.650 0.989 0.342 0.912 0.969 0.958 0.879 0.940 −30,238.6 (0.129) (0.001) (0.005) (0.011) (0.013) (0.012) (0.012) (0.012) HD/HON/KO/MCD/PG 17.018 0.980 0.276 0.956 0.937 0.915 0.939 0.890 −34,545.2 (0.109) (0.001) (0.005) (0.013) (0.012) (0.012) (0.013) (0.012) HON/IBM/MCD/WMT/XOM 18.139 0.982 0.279 0.933 0.984 0.946 0.903 0.956 −33,552.8 (0.118) (0.001) (0.004) (0.012) (0.013) (0.013) (0.012) (0.013) Note: Standard errors are shown in parentheses. Table 4. Maximum likelihood estimates for the 5 × 5 models Equities ν β α λ11 λ22 λ33 λ44 λ55 logL 5×5 AA/AXP/IBM/JPM/WMT 18.515 0.991 0.302 0.968 0.979 0.970 0.955 0.891 −43,769.0 (0.120) (0.000) (0.005) (0.013) (0.012) (0.013) (0.012) (0.012) AA/BA/CAT/GE/KO 17.727 0.986 0.266 0.958 0.984 0.985 0.925 0.923 −44,903.2 (0.115) (0.001) (0.005) (0.013) (0.013) (0.013) (0.012) (0.013) AXP/CAT/IBM/KO/XOM 19.185 0.990 0.296 0.998 0.991 0.977 0.919 0.946 −32,878.6 (0.125) (0.001) (0.004) (0.013) (0.013) (0.013) (0.012) (0.012) BA/HD/JPM/PFE/PG 17.856 0.987 0.300 0.986 0.949 0.968 0.941 0.889 −42,213.9 (0.116) (0.001) (0.005) (0.013) (0.013) (0.013) (0.013) (0.012) BA/HD/MCD/PG/XOM 18.178 0.981 0.273 0.987 0.962 0.932 0.888 0.954 −36,404.4 (0.118) (0.001) (0.005) (0.013) (0.013) (0.013) (0.012) (0.013) CAT/GE/KO/PFE/WMT 18.084 0.985 0.283 1.004 0.933 0.921 0.944 0.902 −33,962.4 (0.117) (0.001) (0.005) (0.013) (0.012) (0.012) (0.013) (0.012) CAT/HON/IBM/MCD/WMT 17.252 0.981 0.268 0.997 0.928 0.982 0.948 0.901 −38,326.4 (0.111) (0.001) (0.004) (0.013) (0.012) (0.013) (0.013) (0.012) GE/IBM/JPM/PG/XOM 19.650 0.989 0.342 0.912 0.969 0.958 0.879 0.940 −30,238.6 (0.129) (0.001) (0.005) (0.011) (0.013) (0.012) (0.012) (0.012) HD/HON/KO/MCD/PG 17.018 0.980 0.276 0.956 0.937 0.915 0.939 0.890 −34,545.2 (0.109) (0.001) (0.005) (0.013) (0.012) (0.012) (0.013) (0.012) HON/IBM/MCD/WMT/XOM 18.139 0.982 0.279 0.933 0.984 0.946 0.903 0.956 −33,552.8 (0.118) (0.001) (0.004) (0.012) (0.013) (0.013) (0.012) (0.013) Equities ν β α λ11 λ22 λ33 λ44 λ55 logL 5×5 AA/AXP/IBM/JPM/WMT 18.515 0.991 0.302 0.968 0.979 0.970 0.955 0.891 −43,769.0 (0.120) (0.000) (0.005) (0.013) (0.012) (0.013) (0.012) (0.012) AA/BA/CAT/GE/KO 17.727 0.986 0.266 0.958 0.984 0.985 0.925 0.923 −44,903.2 (0.115) (0.001) (0.005) (0.013) (0.013) (0.013) (0.012) (0.013) AXP/CAT/IBM/KO/XOM 19.185 0.990 0.296 0.998 0.991 0.977 0.919 0.946 −32,878.6 (0.125) (0.001) (0.004) (0.013) (0.013) (0.013) (0.012) (0.012) BA/HD/JPM/PFE/PG 17.856 0.987 0.300 0.986 0.949 0.968 0.941 0.889 −42,213.9 (0.116) (0.001) (0.005) (0.013) (0.013) (0.013) (0.013) (0.012) BA/HD/MCD/PG/XOM 18.178 0.981 0.273 0.987 0.962 0.932 0.888 0.954 −36,404.4 (0.118) (0.001) (0.005) (0.013) (0.013) (0.013) (0.012) (0.013) CAT/GE/KO/PFE/WMT 18.084 0.985 0.283 1.004 0.933 0.921 0.944 0.902 −33,962.4 (0.117) (0.001) (0.005) (0.013) (0.012) (0.012) (0.013) (0.012) CAT/HON/IBM/MCD/WMT 17.252 0.981 0.268 0.997 0.928 0.982 0.948 0.901 −38,326.4 (0.111) (0.001) (0.004) (0.013) (0.012) (0.013) (0.013) (0.012) GE/IBM/JPM/PG/XOM 19.650 0.989 0.342 0.912 0.969 0.958 0.879 0.940 −30,238.6 (0.129) (0.001) (0.005) (0.011) (0.013) (0.012) (0.012) (0.012) HD/HON/KO/MCD/PG 17.018 0.980 0.276 0.956 0.937 0.915 0.939 0.890 −34,545.2 (0.109) (0.001) (0.005) (0.013) (0.012) (0.012) (0.013) (0.012) HON/IBM/MCD/WMT/XOM 18.139 0.982 0.279 0.933 0.984 0.946 0.903 0.956 −33,552.8 (0.118) (0.001) (0.004) (0.012) (0.013) (0.013) (0.012) (0.013) Note: Standard errors are shown in parentheses. Table 5. Maximum likelihood estimates for the 15 × 15 model 15×15 AA/…/XOM ν 29.260 (0.060) β 0.990 (0.000) α 0.187 (0.001) λ11 0.966 (0.012) λ22 0.978 (0.012) λ33 0.992 (0.013) λ44 0.990 (0.012) λ55 0.908 (0.010) λ66 0.943 (0.012) λ77 0.922 (0.011) λ88 0.985 (0.012) λ99 0.951 (0.012) λ1010 0.928 (0.012) λ1111 0.954 (0.013) λ1212 0.954 (0.013) λ1313 0.900 (0.012) λ1414 0.903 (0.011) λ1515 0.952 (0.012) logL −61,418.1 15×15 AA/…/XOM ν 29.260 (0.060) β 0.990 (0.000) α 0.187 (0.001) λ11 0.966 (0.012) λ22 0.978 (0.012) λ33 0.992 (0.013) λ44 0.990 (0.012) λ55 0.908 (0.010) λ66 0.943 (0.012) λ77 0.922 (0.011) λ88 0.985 (0.012) λ99 0.951 (0.012) λ1010 0.928 (0.012) λ1111 0.954 (0.013) λ1212 0.954 (0.013) λ1313 0.900 (0.012) λ1414 0.903 (0.011) λ1515 0.952 (0.012) logL −61,418.1 Note: Standard errors are shown in parentheses. Table 5. Maximum likelihood estimates for the 15 × 15 model 15×15 AA/…/XOM ν 29.260 (0.060) β 0.990 (0.000) α 0.187 (0.001) λ11 0.966 (0.012) λ22 0.978 (0.012) λ33 0.992 (0.013) λ44 0.990 (0.012) λ55 0.908 (0.010) λ66 0.943 (0.012) λ77 0.922 (0.011) λ88 0.985 (0.012) λ99 0.951 (0.012) λ1010 0.928 (0.012) λ1111 0.954 (0.013) λ1212 0.954 (0.013) λ1313 0.900 (0.012) λ1414 0.903 (0.011) λ1515 0.952 (0.012) logL −61,418.1 15×15 AA/…/XOM ν 29.260 (0.060) β 0.990 (0.000) α 0.187 (0.001) λ11 0.966 (0.012) λ22 0.978 (0.012) λ33 0.992 (0.013) λ44 0.990 (0.012) λ55 0.908 (0.010) λ66 0.943 (0.012) λ77 0.922 (0.011) λ88 0.985 (0.012) λ99 0.951 (0.012) λ1010 0.928 (0.012) λ1111 0.954 (0.013) λ1212 0.954 (0.013) λ1313 0.900 (0.012) λ1414 0.903 (0.011) λ1515 0.952 (0.012) logL −61,418.1 Note: Standard errors are shown in parentheses. Taking all results together, the estimates of the parameters among the different stock combinations are very similar. In general, we find that the estimates of β are close-to-unity from which we can infer that the time-varying process of the covariance matrix is highly persistent. We also observe that the dynamics of Vt rely more on the realized kernel measures given the highly significant estimates of ν. Furthermore, we find that for a higher dimension k, the estimates of ν become higher and more significant. It implies that for models with more stocks, more reliance is given to the realized measures. We emphasize that the degrees of freedom ν needs to grow with the dimension k to ensure that the Wishart covariance matrix does not become nonsingular; see Seber (1998, Section 2.3). However, when the dimension of k is fixed, a larger value for ν implies that the information coming from the realized measure is given more prominence in our RWG model. The estimates of ν appear to be higher in relation to the dimension k and we therefore conclude that the realized measures play a considerable role in our analysis. 3.3 Forecasting study: other forecasting models and methods In our forecasting study, we compare the out-of-sample performance of the RWG model against four alternative forecasting models and methods. Our model allows for a joint analysis of daily returns and realized variance variables. In our comparisons, we consider two forecasting approaches for daily returns and two for realized measures. The two models for the vector of daily returns are the dynamic conditional correlation (DCC) model of Engle (2002a) and the so-called BEKK model of Engle and Kroner (1995). The model-based forecasting framework for the realized covariance matrix is the conditional autoregressive Wishart (CAW) model of Golosnoy, Gribisch, and Liesenfeld (2012) while the nonparametric forecasting method is based on the exponentially weighted moving average (EWMA) scheme. In the forecasting study, we consider the scalar specifications for the updating of the conditional covariance matrix in the RWG model but also, where applicable, for the DCC, BEKK, and CAW models. Finally, we assume matrix Λ to be diagonal in the RWG model. A short practical introduction to each model is provided next. The CAW model assumes that the conditional distribution of the realized variance is Wishart with scale matrix Vtc and degrees of freedom νc, we simply have Xt|Ft−1∼Wk(Vtc/νc,νc). The updating of the conditional covariance matrix is also subject to covariance targeting and to the scalar specification, that is Vt+1c=(1−βc−αc)X¯+βc Vtc+αc Xt, βc≥0, αc>0, αc+βc<1, for t=1,…,T and with X¯=(1/T)∑t=1TXt. The EWMA method is the one-step ahead forecasting scheme applied to the realized variance series; it is the default method used by practitioners and regulators; see, for example, RiskMetrics as described by Morgan (1996). The updating equation also has a scalar specification and is given by Vt+1e=βe·Vte+(1−βe)·Xt, 0<βe<1, where we treat βe as a fixed smoothing constant that we set equal to βe=0.96. In our implementation, we can regard EWMA as a special or limiting case of CAW with αc=0.04 and βc=βe=0.96. The DCC model assumes that the daily returns vector is conditionally normally distributed as rt|Ft−1∼N(0,Vtd) with its covariance matrix given by Vt+1d=DtRtDt where Dt is a diagonal matrix with its i-th diagonal element given by hi,t and where Rt is the conditional correlation matrix with Rt=diag[Qt]−1/2Qtdiag[Qt]−1/2, for t=1,…,T. The updating of hi,t and Qt takes place in two different steps. It is assumed that hi,t follows the GARCH(1,1) process as given by hi,t+1=ωid+βid hi,t+αid ri,t2, ωid>0, βid≥0, αid>0, αid+βid<1, for i=1,…,k and where ri,t is the i-th element of daily return vector rt. The scalar updating equation with covariance targeting for Qt is given by Qt=(1−β+−α+)Q¯+β+ Qt+α+εtεt′, β+≥0, α+>0, α++β+<1, where εt is the GARCH residual vector with its i-th element given by εi,t=ri,t / hi,t, for i=1,…,k, and Q¯=T−1∑t=1Tεtεt′. The BEKK model assumes that rt|Ft−1∼N(0,Vtb) and the covariance matrix of the vector of asset returns is driven by the outer-products of daily returns. The scalar updating equation with covariance targeting is given by Vt+1b=(1−βb−αb)V¯+βbVtb+αbrtrt′, βb≥0, αb>0, αb+βb<1, where V¯=T−1∑t=1Trtrt′ is the sample covariance matrix of daily returns, and ab and bb are unknown coefficients. 3.4 Forecasting study: design and forecast loss functions We split our original dataset in two subsamples: the in-sample data consists of the years 2001–2008 and the out-of-sample consists of the years 2009–2010. We consider these last two years as our forecasting evaluation period. The years 2009–2010 are somewhat representative of financial markets. In 2009 many large equity recovery operations have taken place in the United States while 2010 has shown a return to a modest market risk. The estimation of the static parameter vector, for all model specifications, is done only once for the in-sample data. The one-step ahead forecasts are generated for the out-of-sample data (without the re-estimation of static parameters), for all model specifications. The evaluation of the out-of-sample forecasts is based on the Diebold–Mariano (DM) test to assess the statistical significance of the superiority of the forecasting performance of a specific model; see Diebold and Mariano (1995). In our study, we test whether our RWG model has a significantly smaller out-of-sample loss compared with the loss of the other considered models in our forecasting study. For this purpose, we measure the performance of the models by means of two loss functions: the root mean squared error (RMSE) based on the matrix norm given by RMSE(Vt,St)=||St−Vt||1/2=[∑i,j(Sij,t−Vij,t)2]1/2, and the quasi-likelihood (QL) loss function as given by QL(Vt,St)=log |Vt|+tr(Vt−1St), where St is an observed measure of the covariance matrix and Vt is the covariance matrix as predicted by the model or method. Given that we jointly analyze rt and Xt with our RWG model, we evaluate the performances of all models in forecasting the daily returns density and the realized variances and covariances. Therefore St=Xt for the forecasting of the realized covariance matrix and St=rtrt′ for the forecasting of the density in daily returns. We notice that in case of daily returns with St=rtrt′, the QL loss is equivalent to the log-score criterion for a Gaussian distribution. The log-score criterion is widely used in density forecast comparisons between different models; see Geweke and Amisano (2011). 3.5 Forecasting study: empirical results The results of our forecasting study are summarized in Tables 6 and 7: in Table 6 we report the forecasting results for the realized covariance matrix and in Table 7 for the density in daily returns. Both tables display the relative value of the loss function for our RWG model against the other models. We measure the relative performance by the ratio between the loss for a given model and the loss for the RWG model. When a model has a relative performance larger than unity, the implication is that it underperforms the RWG model. The opposite is also true. When the relative performance is smaller than unity, the model outperforms the RWG model. Table 6. Out-of-sample RMSE loss and QL loss for the realized covariance matrix RMSE loss QL loss RWG CAW EWMA BEKK DCC RWG CAW EWMA BEKK DCC 2x2 AA/CAT 6.08 1.01 1.38 *** 1.32 *** 1.19 *** 4.74 1.00 1.02 *** 1.03 *** 1.02 *** AXP/PFE 3.86 0.99 1.55 *** 1.84 *** 1.76 *** 4.14 1.00 * 1.03 *** 1.06 *** 1.04 *** AXP/WMT 3.41 1.00 1.56 *** 1.86 *** 1.71 *** 3.40 1.00 1.03 *** 1.05 *** 1.04 *** BA/HON 3.20 1.00 1.40 *** 1.23 *** 1.17 ** 3.41 1.00 1.03 *** 1.03 *** 1.03 *** CAT/KO 2.51 1.01 1.48 *** 1.15 *** 1.10 * 3.18 1.00 1.03 *** 1.03 *** 1.03 *** GE/PFE 4.11 1.02 1.57 *** 1.25 *** 1.23 *** 3.98 1.00 1.04 *** 1.05 *** 1.04 *** HD/JPM 4.75 1.00 1.60 *** 1.78 *** 1.55 *** 4.12 1.00 1.03 *** 1.06 *** 1.04 *** IBM/PG 1.62 1.01 1.54 *** 1.13 1.06 2.09 1.00 1.09 ** 1.24 1.21 JPM/XOM 4.19 1.00 1.63 *** 1.79 *** 1.59 *** 3.53 1.00 1.05 *** 1.09 ** 1.05 ** MCD/PG 1.33 1.00 1.49 *** 1.14 ** 1.18 *** 2.03 1.00 1.07 * 1.17 1.24 5x5 AA/AXP/IBM/JPM/WMT 20.77 1.00 1.54 *** 2.16 *** 1.57 *** 8.59 1.00 1.03 *** 1.07 *** 1.05 *** AA/BA/CAT/GE/KO 20.04 1.01 * 1.46 *** 2.31 *** 1.16 *** 9.05 1.00 1.02 *** 1.07 *** 1.02 *** AXP/CAT/IBM/KO/XOM 13.29 1.01 1.59 *** 1.90 *** 1.43 *** 6.91 1.00 1.04 *** 1.09 *** 1.05 ** BA/HD/JPM/PFE/PG 13.34 1.00 1.53 *** 1.77 *** 1.34 *** 8.26 1.00 1.03 *** 1.11 * 1.08 * BA/HD/MCD/PG/XOM 9.73 1.01 1.54 *** 1.33 *** 1.09 6.16 1.00 1.04 * 1.10 * 1.09 CAT/GE/KO/PFE/WMT 12.40 1.01 1.54 *** 1.42 *** 1.18 *** 7.38 1.00 1.03 *** 1.06 *** 1.03 *** CAT/HON/IBM/MCD/WMT 10.61 1.01 * 1.53 *** 1.38 *** 1.12 * 5.97 1.00 1.03 *** 1.06 *** 1.05 *** GE/IBM/JPM/PG/XOM 15.79 1.00 1.65 *** 1.98 *** 1.45 *** 7.10 1.00 1.06 *** 1.16 * 1.11 HD/HON/KO/MCD/PG 9.03 1.01 1.57 *** 1.42 *** 1.12 * 5.63 1.00 1.04 ** 1.11 * 1.11 HON/IBM/MCD/WMT/XOM 8.68 1.01 1.59 *** 1.41 *** 1.10 5.00 1.00 1.04 ** 1.07 *** 1.06 *** 15x15 AA/…/XOM 119.91 1.00 1.47 *** 2.06 *** 1.21 *** 20.09 1.00 1.03 *** 1.10 *** 1.06 ** RMSE loss QL loss RWG CAW EWMA BEKK DCC RWG CAW EWMA BEKK DCC 2x2 AA/CAT 6.08 1.01 1.38 *** 1.32 *** 1.19 *** 4.74 1.00 1.02 *** 1.03 *** 1.02 *** AXP/PFE 3.86 0.99 1.55 *** 1.84 *** 1.76 *** 4.14 1.00 * 1.03 *** 1.06 *** 1.04 *** AXP/WMT 3.41 1.00 1.56 *** 1.86 *** 1.71 *** 3.40 1.00 1.03 *** 1.05 *** 1.04 *** BA/HON 3.20 1.00 1.40 *** 1.23 *** 1.17 ** 3.41 1.00 1.03 *** 1.03 *** 1.03 *** CAT/KO 2.51 1.01 1.48 *** 1.15 *** 1.10 * 3.18 1.00 1.03 *** 1.03 *** 1.03 *** GE/PFE 4.11 1.02 1.57 *** 1.25 *** 1.23 *** 3.98 1.00 1.04 *** 1.05 *** 1.04 *** HD/JPM 4.75 1.00 1.60 *** 1.78 *** 1.55 *** 4.12 1.00 1.03 *** 1.06 *** 1.04 *** IBM/PG 1.62 1.01 1.54 *** 1.13 1.06 2.09 1.00 1.09 ** 1.24 1.21 JPM/XOM 4.19 1.00 1.63 *** 1.79 *** 1.59 *** 3.53 1.00 1.05 *** 1.09 ** 1.05 ** MCD/PG 1.33 1.00 1.49 *** 1.14 ** 1.18 *** 2.03 1.00 1.07 * 1.17 1.24 5x5 AA/AXP/IBM/JPM/WMT 20.77 1.00 1.54 *** 2.16 *** 1.57 *** 8.59 1.00 1.03 *** 1.07 *** 1.05 *** AA/BA/CAT/GE/KO 20.04 1.01 * 1.46 *** 2.31 *** 1.16 *** 9.05 1.00 1.02 *** 1.07 *** 1.02 *** AXP/CAT/IBM/KO/XOM 13.29 1.01 1.59 *** 1.90 *** 1.43 *** 6.91 1.00 1.04 *** 1.09 *** 1.05 ** BA/HD/JPM/PFE/PG 13.34 1.00 1.53 *** 1.77 *** 1.34 *** 8.26 1.00 1.03 *** 1.11 * 1.08 * BA/HD/MCD/PG/XOM 9.73 1.01 1.54 *** 1.33 *** 1.09 6.16 1.00 1.04 * 1.10 * 1.09 CAT/GE/KO/PFE/WMT 12.40 1.01 1.54 *** 1.42 *** 1.18 *** 7.38 1.00 1.03 *** 1.06 *** 1.03 *** CAT/HON/IBM/MCD/WMT 10.61 1.01 * 1.53 *** 1.38 *** 1.12 * 5.97 1.00 1.03 *** 1.06 *** 1.05 *** GE/IBM/JPM/PG/XOM 15.79 1.00 1.65 *** 1.98 *** 1.45 *** 7.10 1.00 1.06 *** 1.16 * 1.11 HD/HON/KO/MCD/PG 9.03 1.01 1.57 *** 1.42 *** 1.12 * 5.63 1.00 1.04 ** 1.11 * 1.11 HON/IBM/MCD/WMT/XOM 8.68 1.01 1.59 *** 1.41 *** 1.10 5.00 1.00 1.04 ** 1.07 *** 1.06 *** 15x15 AA/…/XOM 119.91 1.00 1.47 *** 2.06 *** 1.21 *** 20.09 1.00 1.03 *** 1.10 *** 1.06 ** Notes: The out-of-sample window is two years. The best configurations are identified in bold. The RWG is the benchmark model. The average loss is reported for the benchmark model while the relative loss is reported for the other models. The relative loss is the ratio between the loss of a model and the loss of the benchmark. The asterisks *, ** and *** indicate a significance level of 95, 99 and 99.9%, respectively, for the Diebold-Mariano test with the alternative hypothesis that a model has a different average loss than the benchmark. Table 6. Out-of-sample RMSE loss and QL loss for the realized covariance matrix RMSE loss QL loss RWG CAW EWMA BEKK DCC RWG CAW EWMA BEKK DCC 2x2 AA/CAT 6.08 1.01 1.38 *** 1.32 *** 1.19 *** 4.74 1.00 1.02 *** 1.03 *** 1.02 *** AXP/PFE 3.86 0.99 1.55 *** 1.84 *** 1.76 *** 4.14 1.00 * 1.03 *** 1.06 *** 1.04 *** AXP/WMT 3.41 1.00 1.56 *** 1.86 *** 1.71 *** 3.40 1.00 1.03 *** 1.05 *** 1.04 *** BA/HON 3.20 1.00 1.40 *** 1.23 *** 1.17 ** 3.41 1.00 1.03 *** 1.03 *** 1.03 *** CAT/KO 2.51 1.01 1.48 *** 1.15 *** 1.10 * 3.18 1.00 1.03 *** 1.03 *** 1.03 *** GE/PFE 4.11 1.02 1.57 *** 1.25 *** 1.23 *** 3.98 1.00 1.04 *** 1.05 *** 1.04 *** HD/JPM 4.75 1.00 1.60 *** 1.78 *** 1.55 *** 4.12 1.00 1.03 *** 1.06 *** 1.04 *** IBM/PG 1.62 1.01 1.54 *** 1.13 1.06 2.09 1.00 1.09 ** 1.24 1.21 JPM/XOM 4.19 1.00 1.63 *** 1.79 *** 1.59 *** 3.53 1.00 1.05 *** 1.09 ** 1.05 ** MCD/PG 1.33 1.00 1.49 *** 1.14 ** 1.18 *** 2.03 1.00 1.07 * 1.17 1.24 5x5 AA/AXP/IBM/JPM/WMT 20.77 1.00 1.54 *** 2.16 *** 1.57 *** 8.59 1.00 1.03 *** 1.07 *** 1.05 *** AA/BA/CAT/GE/KO 20.04 1.01 * 1.46 *** 2.31 *** 1.16 *** 9.05 1.00 1.02 *** 1.07 *** 1.02 *** AXP/CAT/IBM/KO/XOM 13.29 1.01 1.59 *** 1.90 *** 1.43 *** 6.91 1.00 1.04 *** 1.09 *** 1.05 ** BA/HD/JPM/PFE/PG 13.34 1.00 1.53 *** 1.77 *** 1.34 *** 8.26 1.00 1.03 *** 1.11 * 1.08 * BA/HD/MCD/PG/XOM 9.73 1.01 1.54 *** 1.33 *** 1.09 6.16 1.00 1.04 * 1.10 * 1.09 CAT/GE/KO/PFE/WMT 12.40 1.01 1.54 *** 1.42 *** 1.18 *** 7.38 1.00 1.03 *** 1.06 *** 1.03 *** CAT/HON/IBM/MCD/WMT 10.61 1.01 * 1.53 *** 1.38 *** 1.12 * 5.97 1.00 1.03 *** 1.06 *** 1.05 *** GE/IBM/JPM/PG/XOM 15.79 1.00 1.65 *** 1.98 *** 1.45 *** 7.10 1.00 1.06 *** 1.16 * 1.11 HD/HON/KO/MCD/PG 9.03 1.01 1.57 *** 1.42 *** 1.12 * 5.63 1.00 1.04 ** 1.11 * 1.11 HON/IBM/MCD/WMT/XOM 8.68 1.01 1.59 *** 1.41 *** 1.10 5.00 1.00 1.04 ** 1.07 *** 1.06 *** 15x15 AA/…/XOM 119.91 1.00 1.47 *** 2.06 *** 1.21 *** 20.09 1.00 1.03 *** 1.10 *** 1.06 ** RMSE loss QL loss RWG CAW EWMA BEKK DCC RWG CAW EWMA BEKK DCC 2x2 AA/CAT 6.08 1.01 1.38 *** 1.32 *** 1.19 *** 4.74 1.00 1.02 *** 1.03 *** 1.02 *** AXP/PFE 3.86 0.99 1.55 *** 1.84 *** 1.76 *** 4.14 1.00 * 1.03 *** 1.06 *** 1.04 *** AXP/WMT 3.41 1.00 1.56 *** 1.86 *** 1.71 *** 3.40 1.00 1.03 *** 1.05 *** 1.04 *** BA/HON 3.20 1.00 1.40 *** 1.23 *** 1.17 ** 3.41 1.00 1.03 *** 1.03 *** 1.03 *** CAT/KO 2.51 1.01 1.48 *** 1.15 *** 1.10 * 3.18 1.00 1.03 *** 1.03 *** 1.03 *** GE/PFE 4.11 1.02 1.57 *** 1.25 *** 1.23 *** 3.98 1.00 1.04 *** 1.05 *** 1.04 *** HD/JPM 4.75 1.00 1.60 *** 1.78 *** 1.55 *** 4.12 1.00 1.03 *** 1.06 *** 1.04 *** IBM/PG 1.62 1.01 1.54 *** 1.13 1.06 2.09 1.00 1.09 ** 1.24 1.21 JPM/XOM 4.19 1.00 1.63 *** 1.79 *** 1.59 *** 3.53 1.00 1.05 *** 1.09 ** 1.05 ** MCD/PG 1.33 1.00 1.49 *** 1.14 ** 1.18 *** 2.03 1.00 1.07 * 1.17 1.24 5x5 AA/AXP/IBM/JPM/WMT 20.77 1.00 1.54 *** 2.16 *** 1.57 *** 8.59 1.00 1.03 *** 1.07 *** 1.05 *** AA/BA/CAT/GE/KO 20.04 1.01 * 1.46 *** 2.31 *** 1.16 *** 9.05 1.00 1.02 *** 1.07 *** 1.02 *** AXP/CAT/IBM/KO/XOM 13.29 1.01 1.59 *** 1.90 *** 1.43 *** 6.91 1.00 1.04 *** 1.09 *** 1.05 ** BA/HD/JPM/PFE/PG 13.34 1.00 1.53 *** 1.77 *** 1.34 *** 8.26 1.00 1.03 *** 1.11 * 1.08 * BA/HD/MCD/PG/XOM 9.73 1.01 1.54 *** 1.33 *** 1.09 6.16 1.00 1.04 * 1.10 * 1.09 CAT/GE/KO/PFE/WMT 12.40 1.01 1.54 *** 1.42 *** 1.18 *** 7.38 1.00 1.03 *** 1.06 *** 1.03 *** CAT/HON/IBM/MCD/WMT 10.61 1.01 * 1.53 *** 1.38 *** 1.12 * 5.97 1.00 1.03 *** 1.06 *** 1.05 *** GE/IBM/JPM/PG/XOM 15.79 1.00 1.65 *** 1.98 *** 1.45 *** 7.10 1.00 1.06 *** 1.16 * 1.11 HD/HON/KO/MCD/PG 9.03 1.01 1.57 *** 1.42 *** 1.12 * 5.63 1.00 1.04 ** 1.11 * 1.11 HON/IBM/MCD/WMT/XOM 8.68 1.01 1.59 *** 1.41 *** 1.10 5.00 1.00 1.04 ** 1.07 *** 1.06 *** 15x15 AA/…/XOM 119.91 1.00 1.47 *** 2.06 *** 1.21 *** 20.09 1.00 1.03 *** 1.10 *** 1.06 ** Notes: The out-of-sample window is two years. The best configurations are identified in bold. The RWG is the benchmark model. The average loss is reported for the benchmark model while the relative loss is reported for the other models. The relative loss is the ratio between the loss of a model and the loss of the benchmark. The asterisks *, ** and *** indicate a significance level of 95, 99 and 99.9%, respectively, for the Diebold-Mariano test with the alternative hypothesis that a model has a different average loss than the benchmark. Table 7. Out-of-sample RMSE loss and QL loss for the density in daily returns RMSE loss QL loss RWG CAW EWMA BEKK DCC RWG CAW EWMA BEKK DCC 2x2 AA/CAT 19.31 1.03 *** 1.11 *** 1.12 *** 1.08 *** 4.67 1.00 1.01 1.00 1.01 AXP/PFE 13.76 1.01 1.12 *** 1.15 *** 1.14 *** 4.10 1.00 1.02 * 1.02 1.03 * AXP/WMT 11.73 1.01 ** 1.13 *** 1.13 *** 1.10 *** 3.35 1.01 1.03 * 1.03 1.03 BA/HON 8.91 1.05 *** 1.15 *** 1.06 ** 1.03 3.17 1.01 1.03 ** 1.02 1.03 * CAT/KO 7.20 1.01 *** 1.13 *** 1.03 1.01 3.02 1.00 1.03 1.02 1.02 GE/PFE 9.68 1.07 *** 1.23 *** 1.08 * 1.08 * 3.54 1.01 1.05 *** 1.02 * 1.02 * HD/JPM 14.92 1.02 ** 1.15 *** 1.19 *** 1.16 *** 4.01 1.00 1.03 ** 1.02 * 1.03 ** IBM/PG 3.48 1.08 *** 1.26 *** 1.06 1.03 1.52 1.04 *** 1.15 *** 1.09 ** 1.08 ** JPM/XOM 13.34 1.01 * 1.14 *** 1.18 *** 1.14 * 3.32 1.00 1.05 *** 1.02 1.03 ** MCD/PG 3.11 1.10 *** 1.25 *** 1.06 * 1.07 *** 1.27 1.09 *** 1.17 *** 1.06 * 1.05 5x5 AA/AXP/IBM/JPM/WMT 73.86 1.02 *** 1.13 *** 1.26 *** 1.15 *** 8.48 1.00 1.02 * 1.03 ** 1.03 ** AA/BA/CAT/GE/KO 57.63 1.03 *** 1.16 *** 1.38 *** 1.06 *** 8.62 1.00 1.02 ** 1.05 *** 1.02 * AXP/CAT/IBM/KO/XOM 44.77 1.01 *** 1.12 *** 1.19 *** 1.08 *** 6.44 1.00 1.04 *** 1.05 *** 1.04 ** BA/HD/JPM/PFE/PG 39.74 1.03 *** 1.15 *** 1.20 *** 1.09 *** 7.45 1.01 *** 1.04 *** 1.03 ** 1.02 ** BA/HD/MCD/PG/XOM 23.44 1.04 *** 1.24 *** 1.15 *** 1.00 4.95 1.03 *** 1.07 *** 1.05 *** 1.04 *** CAT/GE/KO/PFE/WMT 32.58 1.04 *** 1.18 *** 1.14 *** 1.04 * 6.55 1.01 * 1.05 *** 1.03 ** 1.03 ** CAT/HON/IBM/MCD/WMT 28.56 1.04 *** 1.19 *** 1.14 *** 1.01 5.20 1.02 *** 1.05 *** 1.06 *** 1.05 *** GE/IBM/JPM/PG/XOM 44.24 1.03 *** 1.18 *** 1.27 *** 1.13 *** 5.93 1.02 *** 1.08 *** 1.06 *** 1.04 *** HD/HON/KO/MCD/PG 21.33 1.07 *** 1.27 *** 1.20 *** 1.02 4.61 1.02 *** 1.06 *** 1.04 * 1.04 ** HON/IBM/MCD/WMT/XOM 21.11 1.05 *** 1.24 *** 1.17 *** 1.02 4.12 1.02 *** 1.05 *** 1.07 *** 1.06 *** 15x15 AA/…/XOM 333.91 1.03 *** 1.16 *** 1.31 *** 1.08 *** 18.18 1.01 ** 1.03 *** 1.06 *** 1.03 *** RMSE loss QL loss RWG CAW EWMA BEKK DCC RWG CAW EWMA BEKK DCC 2x2 AA/CAT 19.31 1.03 *** 1.11 *** 1.12 *** 1.08 *** 4.67 1.00 1.01 1.00 1.01 AXP/PFE 13.76 1.01 1.12 *** 1.15 *** 1.14 *** 4.10 1.00 1.02 * 1.02 1.03 * AXP/WMT 11.73 1.01 ** 1.13 *** 1.13 *** 1.10 *** 3.35 1.01 1.03 * 1.03 1.03 BA/HON 8.91 1.05 *** 1.15 *** 1.06 ** 1.03 3.17 1.01 1.03 ** 1.02 1.03 * CAT/KO 7.20 1.01 *** 1.13 *** 1.03 1.01 3.02 1.00 1.03 1.02 1.02 GE/PFE 9.68 1.07 *** 1.23 *** 1.08 * 1.08 * 3.54 1.01 1.05 *** 1.02 * 1.02 * HD/JPM 14.92 1.02 ** 1.15 *** 1.19 *** 1.16 *** 4.01 1.00 1.03 ** 1.02 * 1.03 ** IBM/PG 3.48 1.08 *** 1.26 *** 1.06 1.03 1.52 1.04 *** 1.15 *** 1.09 ** 1.08 ** JPM/XOM 13.34 1.01 * 1.14 *** 1.18 *** 1.14 * 3.32 1.00 1.05 *** 1.02 1.03 ** MCD/PG 3.11 1.10 *** 1.25 *** 1.06 * 1.07 *** 1.27 1.09 *** 1.17 *** 1.06 * 1.05 5x5 AA/AXP/IBM/JPM/WMT 73.86 1.02 *** 1.13 *** 1.26 *** 1.15 *** 8.48 1.00 1.02 * 1.03 ** 1.03 ** AA/BA/CAT/GE/KO 57.63 1.03 *** 1.16 *** 1.38 *** 1.06 *** 8.62 1.00 1.02 ** 1.05 *** 1.02 * AXP/CAT/IBM/KO/XOM 44.77 1.01 *** 1.12 *** 1.19 *** 1.08 *** 6.44 1.00 1.04 *** 1.05 *** 1.04 ** BA/HD/JPM/PFE/PG 39.74 1.03 *** 1.15 *** 1.20 *** 1.09 *** 7.45 1.01 *** 1.04 *** 1.03 ** 1.02 ** BA/HD/MCD/PG/XOM 23.44 1.04 *** 1.24 *** 1.15 *** 1.00 4.95 1.03 *** 1.07 *** 1.05 *** 1.04 *** CAT/GE/KO/PFE/WMT 32.58 1.04 *** 1.18 *** 1.14 *** 1.04 * 6.55 1.01 * 1.05 *** 1.03 ** 1.03 ** CAT/HON/IBM/MCD/WMT 28.56 1.04 *** 1.19 *** 1.14 *** 1.01 5.20 1.02 *** 1.05 *** 1.06 *** 1.05 *** GE/IBM/JPM/PG/XOM 44.24 1.03 *** 1.18 *** 1.27 *** 1.13 *** 5.93 1.02 *** 1.08 *** 1.06 *** 1.04 *** HD/HON/KO/MCD/PG 21.33 1.07 *** 1.27 *** 1.20 *** 1.02 4.61 1.02 *** 1.06 *** 1.04 * 1.04 ** HON/IBM/MCD/WMT/XOM 21.11 1.05 *** 1.24 *** 1.17 *** 1.02 4.12 1.02 *** 1.05 *** 1.07 *** 1.06 *** 15x15 AA/…/XOM 333.91 1.03 *** 1.16 *** 1.31 *** 1.08 *** 18.18 1.01 ** 1.03 *** 1.06 *** 1.03 *** Notes: The out-of-sample window is two years. The best configurations are identified by bold font. The RWG is the benchmark model. The average loss is reported for the benchmark model while the relative loss is reported for the other models. The relative loss is the ratio between the loss of a model and the loss of the benchmark. The asterisks *, ** and *** indicate a significance level of 95, 99 and 99.9%, respectively, for the Diebold-Mariano test with the alternative hypothesis that a model has a different average loss than the benchmark. Table 7. Out-of-sample RMSE loss and QL loss for the density in daily returns RMSE loss QL loss RWG CAW EWMA BEKK DCC RWG CAW EWMA BEKK DCC 2x2 AA/CAT 19.31 1.03 *** 1.11 *** 1.12 *** 1.08 *** 4.67 1.00 1.01 1.00 1.01 AXP/PFE 13.76 1.01 1.12 *** 1.15 *** 1.14 *** 4.10 1.00 1.02 * 1.02 1.03 * AXP/WMT 11.73 1.01 ** 1.13 *** 1.13 *** 1.10 *** 3.35 1.01 1.03 * 1.03 1.03 BA/HON 8.91 1.05 *** 1.15 *** 1.06 ** 1.03 3.17 1.01 1.03 ** 1.02 1.03 * CAT/KO 7.20 1.01 *** 1.13 *** 1.03 1.01 3.02 1.00 1.03 1.02 1.02 GE/PFE 9.68 1.07 *** 1.23 *** 1.08 * 1.08 * 3.54 1.01 1.05 *** 1.02 * 1.02 * HD/JPM 14.92 1.02 ** 1.15 *** 1.19 *** 1.16 *** 4.01 1.00 1.03 ** 1.02 * 1.03 ** IBM/PG 3.48 1.08 *** 1.26 *** 1.06 1.03 1.52 1.04 *** 1.15 *** 1.09 ** 1.08 ** JPM/XOM 13.34 1.01 * 1.14 *** 1.18 *** 1.14 * 3.32 1.00 1.05 *** 1.02 1.03 ** MCD/PG 3.11 1.10 *** 1.25 *** 1.06 * 1.07 *** 1.27 1.09 *** 1.17 *** 1.06 * 1.05 5x5 AA/AXP/IBM/JPM/WMT 73.86 1.02 *** 1.13 *** 1.26 *** 1.15 *** 8.48 1.00 1.02 * 1.03 ** 1.03 ** AA/BA/CAT/GE/KO 57.63 1.03 *** 1.16 *** 1.38 *** 1.06 *** 8.62 1.00 1.02 ** 1.05 *** 1.02 * AXP/CAT/IBM/KO/XOM 44.77 1.01 *** 1.12 *** 1.19 *** 1.08 *** 6.44 1.00 1.04 *** 1.05 *** 1.04 ** BA/HD/JPM/PFE/PG 39.74 1.03 *** 1.15 *** 1.20 *** 1.09 *** 7.45 1.01 *** 1.04 *** 1.03 ** 1.02 ** BA/HD/MCD/PG/XOM 23.44 1.04 *** 1.24 *** 1.15 *** 1.00 4.95 1.03 *** 1.07 *** 1.05 *** 1.04 *** CAT/GE/KO/PFE/WMT 32.58 1.04 *** 1.18 *** 1.14 *** 1.04 * 6.55 1.01 * 1.05 *** 1.03 ** 1.03 ** CAT/HON/IBM/MCD/WMT 28.56 1.04 *** 1.19 *** 1.14 *** 1.01 5.20 1.02 *** 1.05 *** 1.06 *** 1.05 *** GE/IBM/JPM/PG/XOM 44.24 1.03 *** 1.18 *** 1.27 *** 1.13 *** 5.93 1.02 *** 1.08 *** 1.06 *** 1.04 *** HD/HON/KO/MCD/PG 21.33 1.07 *** 1.27 *** 1.20 *** 1.02 4.61 1.02 *** 1.06 *** 1.04 * 1.04 ** HON/IBM/MCD/WMT/XOM 21.11 1.05 *** 1.24 *** 1.17 *** 1.02 4.12 1.02 *** 1.05 *** 1.07 *** 1.06 *** 15x15 AA/…/XOM 333.91 1.03 *** 1.16 *** 1.31 *** 1.08 *** 18.18 1.01 ** 1.03 *** 1.06 *** 1.03 *** RMSE loss QL loss RWG CAW EWMA BEKK DCC RWG CAW EWMA BEKK DCC 2x2 AA/CAT 19.31 1.03 *** 1.11 *** 1.12 *** 1.08 *** 4.67 1.00 1.01 1.00 1.01 AXP/PFE 13.76 1.01 1.12 *** 1.15 *** 1.14 *** 4.10 1.00 1.02 * 1.02 1.03 * AXP/WMT 11.73 1.01 ** 1.13 *** 1.13 *** 1.10 *** 3.35 1.01 1.03 * 1.03 1.03 BA/HON 8.91 1.05 *** 1.15 *** 1.06 ** 1.03 3.17 1.01 1.03 ** 1.02 1.03 * CAT/KO 7.20 1.01 *** 1.13 *** 1.03 1.01 3.02 1.00 1.03 1.02 1.02 GE/PFE 9.68 1.07 *** 1.23 *** 1.08 * 1.08 * 3.54 1.01 1.05 *** 1.02 * 1.02 * HD/JPM 14.92 1.02 ** 1.15 *** 1.19 *** 1.16 *** 4.01 1.00 1.03 ** 1.02 * 1.03 ** IBM/PG 3.48 1.08 *** 1.26 *** 1.06 1.03 1.52 1.04 *** 1.15 *** 1.09 ** 1.08 ** JPM/XOM 13.34 1.01 * 1.14 *** 1.18 *** 1.14 * 3.32 1.00 1.05 *** 1.02 1.03 ** MCD/PG 3.11 1.10 *** 1.25 *** 1.06 * 1.07 *** 1.27 1.09 *** 1.17 *** 1.06 * 1.05 5x5 AA/AXP/IBM/JPM/WMT 73.86 1.02 *** 1.13 *** 1.26 *** 1.15 *** 8.48 1.00 1.02 * 1.03 ** 1.03 ** AA/BA/CAT/GE/KO 57.63 1.03 *** 1.16 *** 1.38 *** 1.06 *** 8.62 1.00 1.02 ** 1.05 *** 1.02 * AXP/CAT/IBM/KO/XOM 44.77 1.01 *** 1.12 *** 1.19 *** 1.08 *** 6.44 1.00 1.04 *** 1.05 *** 1.04 ** BA/HD/JPM/PFE/PG 39.74 1.03 *** 1.15 *** 1.20 *** 1.09 *** 7.45 1.01 *** 1.04 *** 1.03 ** 1.02 ** BA/HD/MCD/PG/XOM 23.44 1.04 *** 1.24 *** 1.15 *** 1.00 4.95 1.03 *** 1.07 *** 1.05 *** 1.04 *** CAT/GE/KO/PFE/WMT 32.58 1.04 *** 1.18 *** 1.14 *** 1.04 * 6.55 1.01 * 1.05 *** 1.03 ** 1.03 ** CAT/HON/IBM/MCD/WMT 28.56 1.04 *** 1.19 *** 1.14 *** 1.01 5.20 1.02 *** 1.05 *** 1.06 *** 1.05 *** GE/IBM/JPM/PG/XOM 44.24 1.03 *** 1.18 *** 1.27 *** 1.13 *** 5.93 1.02 *** 1.08 *** 1.06 *** 1.04 *** HD/HON/KO/MCD/PG 21.33 1.07 *** 1.27 *** 1.20 *** 1.02 4.61 1.02 *** 1.06 *** 1.04 * 1.04 ** HON/IBM/MCD/WMT/XOM 21.11 1.05 *** 1.24 *** 1.17 *** 1.02 4.12 1.02 *** 1.05 *** 1.07 *** 1.06 *** 15x15 AA/…/XOM 333.91 1.03 *** 1.16 *** 1.31 *** 1.08 *** 18.18 1.01 ** 1.03 *** 1.06 *** 1.03 *** Notes: The out-of-sample window is two years. The best configurations are identified by bold font. The RWG is the benchmark model. The average loss is reported for the benchmark model while the relative loss is reported for the other models. The relative loss is the ratio between the loss of a model and the loss of the benchmark. The asterisks *, ** and *** indicate a significance level of 95, 99 and 99.9%, respectively, for the Diebold-Mariano test with the alternative hypothesis that a model has a different average loss than the benchmark. We learn from Table 6 that the RWG and CAW models are the best performing models in forecasting the realized measures. Their performances are very similar in relative terms and, except for a few cases, there is not a statistically significant difference. This finding is to be expected given that the daily returns are not very informative to forecast the realized measures. Therefore, the RWG model is not expected to outperform the CAW model by a large amount. However, from Table 7 we can conclude that the RWG model is by far the best performing model in forecasting the density in daily returns. The outperformance is in relative terms as well as in statistical terms because the reported DM tests are clearly significant in most cases. Here the RWG is able to outperform the DCC and BEKK convincingly. The reason is obvious since it exploits additional information as provided by the realized measures. In a similar fashion, the RWG model outperforms the CAW model and the EWMA method since our preferred model analyzes the daily returns jointly with the realized measures. On the other hand, the CAW model and the EWMA method only consider the realized measures. We can therefore conclude that the factor structure of the RWG model is particularly useful in exploiting the realized measures for the forecasting of the density in daily returns. 4 Conclusions We have proposed a new model for the joint modeling and forecasting of daily time series of returns and realized covariance matrices of financial assets: the RWG model. There are many distinguishing features of our model when compared with alternative frameworks. First, the model relies both on low- (daily) and on high-frequency (intraday) information. It turns out that the high-frequency measures are given most weight since they exploit intraday data of financial assets to infer about the underlying covariance structures. Several noisy measures that are based on different sampling frequencies can be considered in the analysis. Second, the time-varying features of the RWG model are driven by updates of the covariance matrix that exploit full-likelihood information. The model relies on standard parsimonious formulations, which is a convenient property for multivariate conditional volatility models. In particular, the model is closely connected with the multivariate GARCH literature and the dynamics are related with VARMA models. Third, the model parameters can be interpreted straightforwardly. An example is that overnight market risk can be measured directly via the parameter matrix Λ when daily close-to-close returns are considered in the analysis. Fourth, the modeling framework is flexible: it can be extended easily when more realized measures are considered. The multivariate model can also be used to simulate realistic dynamic paths for portfolios to facilitate the validation of investment strategies. Fifth, the likelihood function is available analytically and hence estimation is easy; nonetheless computer code is made available for its use. Finally, in an empirical study for a portfolio of fifteen NYSE equities, we have studied the RWG model and its different specifications. We have provided in-sample evidence that our basic specification can be effective in extracting the salient features in the data. In an out-of-sample forecasting study, we compare our model performance against four competitive models and methods. The ability of our model to jointly capture the daily returns vector and the realized covariance matrix appears in particular to benefit the accuracy in forecasting the density of daily returns. Appendices A: Matrix notation and preliminary results The results in this article make use of the following matrix notation and definitions. Let A and B be k × k matrices, then A⊗B denotes the Kronecker product, which is a k2×k2 block matrix {aijB} where aij is the (i, j) element of matrix A. The vec(A) operator stacks the columns of matrix A consecutively into the k2×1 column vector, while vech(A) stacks the lower triangular part including diagonal into k*×1 column vector, with k*=k(k+1)/2. The k × k identity matrix is denoted by Ik. We define the k2×k2 commutation matrix Kk, the k2×k* duplication matrix Dk, and the k*×k2 elimination matrix Lk, by the identities Kkvec(B)=vec(B′), Dkvech(A)=vec(A), and Lkvec(A)=vech(A), where B is an arbitrary k × k matrix and A is an arbitrary symmetric k × k matrix. Here Lk=(Dk′Dk)−1D′k is the Moore–Penrose inverse of the duplication matrix Dk. Additional properties and results related to these matrices can be found in Magnus and Neudecker (2007) and Seber (2007). The proofs in the next appendix make use of the following results in matrix calculus. For a k × k symmetric matrix X, the derivative of vec(X) with respect to vech(X) is given by ∂vec(X)∂vech(X)′=Dk, where the duplication matrix Dk is defined above. For all k × k nonsingular matrices A, X and B, we have ∂log|AXB|∂vec(X)′=vec[(X−1)′]′,∂vec(X−1)∂vec(X)=−(X−1)′⊗X−1,∂tr(AXB)∂vec(X)=vec(A′B′)′. (24) Finally, for all k × k matrices A, B and C, we have vec(ABC)=(C′⊗A)vec(B). (25) B: Proofs Proof of Theorem 1 We derive the score vector of which the general form is given by (15). From the Equations (19) and (20), the relevant parts of log-likelihoods for the score vector derivation can be explicitly given as in Lr,t=cr−12(log |ΛVtΛ′|+tr((ΛVtΛ′)−1rtr′t)), (26) LX,t=cX−ν2(log |Vt|+tr(Vt−1Xt)), (27) where cr and cX are nonrelevant constants. We consider the covariance matrix Vt and parameter vector ft, given by (21), as two unknown, nonrandom variables. Using the chain rule for vector differentiation, the score functions for the individual measurements associated with (1) and (2) can be expressed by ∂log ϕi(Zti|ft,Ft−1;ψ)∂ft′=∂log ϕi(Zti|ft,Ft−1;ψ)∂vec(Vt)′∂vec(Vt)∂ft′. We first differentiate the measurement density for returns (26). Using (24) and (25), together with noting that Vt is symmetric and Vt−1=Vt−1VtVt−1, we obtain ∂Lr,t∂vec(Vt)′=−12[vec(Vt−1)′−vec(Λ−1rtrt′(Λ′)−1)′(Vt−1⊗Vt−1)]=−12[vec(Vt)′(Vt−1⊗Vt−1)−vec(Λ−1rtrt′(Λ′)−1)′(Vt−1⊗Vt−1)]=12[vec(Λ−1rtrt′(Λ′)−1)′−vec(Vt)′](Vt−1⊗Vt−1), (28) and similarly we differentiate the measurement density for the realized covariance (27), we have ∂LX,t∂vec(Vt)′=−ν2[vec(Vt−1)′−vec(Xt)′(Vt−1⊗Vt−1)]=−ν2[vec(Vt)′(Vt−1⊗Vt−1)−vec(Xt)′(Vt−1⊗Vt−1)]=ν2[vec(Xt)−vec(Vt)]′(Vt−1⊗Vt−1). (29) Therefore, given the results (28) and (29), combined with the fact that ∂vec(Vt)/∂ft′=Dk and with the score defined in (15), we conclude that the proof of Theorem 1 is completed. □ Proof of Theorem 2 We derive the Fisher information matrix whose general form is given by (16). Using the results from the proof of Theorem 1, the individual score functions are given by ∇r,t=12D′k(Vt−1⊗Vt−1)[vec(Λ−1rtr′t(Λ′)−1)−vec(Vt)],∇X,t=ν2D′k(Vt−1⊗Vt−1)[vec(Xt)−vec(Vt)], for the measurement densities of the vector of returns and of the covariance matrix, respectively. By taking E[∇i,t∇i,t′|Ft−1], we obtain Ir,t=14D′k(Vt−1⊗Vt−1)var[vec(Λ−1rtr′t(Λ′)−1)−vec(Vt)|Ft−1](Vt−1⊗Vt−1)Dk,IX,t=ν24D′k(Vt−1⊗Vt−1)var[vec(Xt)−vec(Vt)|Ft−1](Vt−1⊗Vt−1)Dk. Using the results (10) and (11), and given that (Vt−1⊗Vt−1)(Vt⊗Vt)=Ik2, we have Ir,t=14D′k(Vt−1⊗Vt−1)(Ik2+Kk)Dk,IX,t=ν4D′k(Vt−1⊗Vt−1)(Ik2+Kk)Dk. Finally, considering that Ik2+Kk=2DkLk (see Theorem 12 in Chapter 3 of Magnus and Neudecker (2007)) and that LkDk=Ik*, we obtain Ir,t=12D′k(Vt−1⊗Vt−1)Dk,IX,t=ν2D′k(Vt−1⊗Vt−1)Dk, which combined with (16) completes the proof. □ Proof of Theorem 3 The score ∇t can be written as ∇t=12Dk′(Vt−1⊗Vt−1)DkLk(ν[vec(Xt)−vec(Vt)]+[vec(Λ−1rtr′t(Λ′)−1)−vec(Vt)]), since Dk′(Vt−1⊗Vt−1)DkLk=Dk′(Vt−1⊗Vt−1); see Theorem 13 in Chapter 3 of Magnus and Neudecker (2007). Together with the expression of the conditional Fisher information It=ν+12D′k(Vt−1⊗Vt−1)Dk and the equality (Dk′(Vt−1⊗Vt−1)Dk)−1Dk′(Vt−1⊗Vt−1)Dk=Ik*, we have completed the proof for Theorem 3. □ C: Additional estimation results In this Appendix, we consider a less parsimonious dynamic specification for the covariance matrix Vt: we allow variances and covariances to have different persistency levels. The empirical results do not suggest that the more general specification leads to improvements in terms of in-sample goodness-of-fit. We consider matrices A and B in (18) to be diagonal matrices where the coefficients αi and βi corresponding to a conditional variance are set equal to αv and βv, respectively. The coefficients αi and βi corresponding to a conditional covariance are set equal to αc and βc, respectively. The matrices A and B can also be defined as A=diag(vech(A˜)) and B=diag(vech(B˜)). The matrix A˜ is a k × k matrix with diagonal elements equal to αv and outer diagonal elements equal to αc. Similarly, the matrix B˜ is a k × k matrix with diagonal elements equal to βv and outer diagonal elements equal to βc. This specification allows us to explore whether the variances and covariances have different dynamic properties. We impose the additional parameter constraints αv≥αc≥0 and βv−αv≥βc−αc≥0 to ensure that Vt is positive definite with probability 1. These constraints can be easily obtained when we notice that the covariance matrix Vt can be expressed as Vt+1=E[Vt](Ik−B˜)+(B˜−A˜)⊙Vt+A˜⊙(1v+1(vXt+Λ−1rtr′t(Λ′)−1)), where ⊙ denotes the Hadamard product. Therefore we impose that B˜−A˜ and A˜ are positive definite, which leads to the parameter constraints as stated above. Imposing B˜−A˜ and A˜ to be positive definite also guarantees that Vt is positive definite by an application of the Schur product theorem. We estimate the parameters for the 2 × 2 models of Table 3 and consider both the case where Λ is a full matrix and the case where Λ is a diagonal matrix. The results are reported in Table 8. The results suggest that the variances and covariances have the same dynamics, that is, αv=αc and βv=βc. This can be concluded since the estimates of αv and αc, as well as βv and βc, are not significantly different from each other. Finally, we notice that imposing αv=αc and βv=βc leads to the scalar models that are estimated in Table 3. Table 8. Maximum likelihood estimates for the 2 × 2 models Equities ν βv βc αv αc λ11 λ22 λ12 λ21 logL AIC 2×2 AA/CAT 12.428 0.977 0.977 0.331 0.331 0.893 1.022 0.226 −0.032 −20,171.5 40,360.9 (0.190) (0.002) (0.004) (0.011) (0.017) (0.035) (0.026) (0.073) (0.051) AXP/PFE 10.876 0.991 0.991 0.378 0.378 1.032 0.918 −0.018 0.078 −17,107.3 34,232.5 (0.164) (0.001) (0.003) (0.012) (0.023) (0.016) (0.016) (0.030) (0.021) AXP/WMT 11.909 0.993 0.993 0.360 0.360 1.018 0.887 0.032 0.025 −15,347.7 30,713.4 (0.181) (0.001) (0.002) (0.012) (0.020) (0.017) (0.015) (0.033) (0.017) BA/HON 10.684 0.975 0.970 0.355 0.351 0.985 0.894 0.026 0.102 −17,859.4 35,736.9 (0.161) (0.002) (0.005) (0.011) (0.019) (0.028) (0.029) (0.053) (0.055) CAT/KO 12.826 0.977 0.973 0.355 0.351 0.986 0.928 0.093 −0.037 −14,226.8 28,471.6 (0.196) (0.002) (0.006) (0.011) (0.022) (0.022) (0.017) (0.074) (0.030) GE/PFE 11.016 0.984 0.984 0.405 0.405 0.943 0.911 0.016 0.072 −15,622.7 31,263.5 (0.166) (0.001) (0.004) (0.013) (0.021) (0.017) (0.018) (0.030) (0.029) HD/JPM 12.458 0.988 0.988 0.447 0.447 0.953 0.944 0.020 0.125 −18,481.1 36,980.2 (0.190) (0.001) (0.003) (0.013) (0.021) (0.018) (0.020) (0.031) (0.036) IBM/PG 12.407 0.977 0.974 0.383 0.381 0.985 0.866 −0.026 0.030 −10,960.8 21,939.7 (0.189) (0.002) (0.005) (0.012) (0.020) (0.020) (0.020) (0.052) (0.036) JPM/XOM 13.067 0.989 0.984 0.444 0.440 0.988 0.928 0.006 0.036 −16,081.5 32,181.0 (0.199) (0.001) (0.003) (0.012) (0.020) (0.016) (0.016) (0.032) (0.018) MCD/PG 10.432 0.978 0.972 0.311 0.305 0.919 0.880 0.037 0.015 −12,645.5 25,308.9 (0.157) (0.002) (0.006) (0.011) (0.021) (0.018) (0.017) (0.049) (0.027) AA/CAT 12.424 0.977 0.977 0.333 0.333 0.952 0.978 – – −20,201.6 40,417.3 (0.190) (0.002) (0.004) (0.011) (0.017) (0.013) (0.013) AXP/PFE 10.876 0.991 0.991 0.377 0.377 1.014 0.940 – – −17,118.6 34,251.2 (0.164) (0.001) (0.003) (0.012) (0.022) (0.014) (0.013) AXP/WMT 11.908 0.993 0.992 0.360 0.360 1.016 0.890 – – −15,353.2 30,720.3 (0.181) (0.001) (0.002) (0.012) (0.020) (0.014) (0.012) BA/HON 10.681 0.974 0.969 0.355 0.351 0.968 0.914 – – −17,881.1 35,776.3 (0.161) (0.002) (0.005) (0.011) (0.019) (0.013) (0.012) CAT/KO 12.838 0.977 0.972 0.355 0.350 1.007 0.913 – – −14,227.6 28,469.3 (0.196) (0.002) (0.006) (0.011) (0.022) (0.014) (0.013) GE/PFE 11.013 0.984 0.982 0.406 0.404 0.931 0.926 – – −15,635.9 31,285.9 (0.166) (0.002) (0.004) (0.013) (0.020) (0.013) (0.013) HD/JPM 12.454 0.988 0.987 0.446 0.445 0.937 0.968 – – −18,509.2 37,032.3 (0.190) (0.001) (0.003) (0.013) (0.021) (0.013) (0.013) IBM/PG 12.412 0.977 0.973 0.383 0.380 0.974 0.877 – – −10,961.6 21,937.0 (0.189) (0.002) (0.005) (0.012) (0.020) (0.014) (0.012) JPM/XOM 13.085 0.989 0.985 0.442 0.439 0.979 0.939 – – 16,085.9 32,185.8 (0.200) (0.001) (0.003) (0.012) (0.020) (0.014) (0.013) MCD/PG 10.426 0.979 0.974 0.311 0.307 0.921 0.879 – – −12,648.8 25,311.5 (0.157) (0.002) (0.006) (0.011) (0.021) (0.013) (0.012) Equities ν βv βc αv αc λ11 λ22 λ12 λ21 logL AIC 2×2 AA/CAT 12.428 0.977 0.977 0.331 0.331 0.893 1.022 0.226 −0.032 −20,171.5 40,360.9 (0.190) (0.002) (0.004) (0.011) (0.017) (0.035) (0.026) (0.073) (0.051) AXP/PFE 10.876 0.991 0.991 0.378 0.378 1.032 0.918 −0.018 0.078 −17,107.3 34,232.5 (0.164) (0.001) (0.003) (0.012) (0.023) (0.016) (0.016) (0.030) (0.021) AXP/WMT 11.909 0.993 0.993 0.360 0.360 1.018 0.887 0.032 0.025 −15,347.7 30,713.4 (0.181) (0.001) (0.002) (0.012) (0.020) (0.017) (0.015) (0.033) (0.017) BA/HON 10.684 0.975 0.970 0.355 0.351 0.985 0.894 0.026 0.102 −17,859.4 35,736.9 (0.161) (0.002) (0.005) (0.011) (0.019) (0.028) (0.029) (0.053) (0.055) CAT/KO 12.826 0.977 0.973 0.355 0.351 0.986 0.928 0.093 −0.037 −14,226.8 28,471.6 (0.196) (0.002) (0.006) (0.011) (0.022) (0.022) (0.017) (0.074) (0.030) GE/PFE 11.016 0.984 0.984 0.405 0.405 0.943 0.911 0.016 0.072 −15,622.7 31,263.5 (0.166) (0.001) (0.004) (0.013) (0.021) (0.017) (0.018) (0.030) (0.029) HD/JPM 12.458 0.988 0.988 0.447 0.447 0.953 0.944 0.020 0.125 −18,481.1 36,980.2 (0.190) (0.001) (0.003) (0.013) (0.021) (0.018) (0.020) (0.031) (0.036) IBM/PG 12.407 0.977 0.974 0.383 0.381 0.985 0.866 −0.026 0.030 −10,960.8 21,939.7 (0.189) (0.002) (0.005) (0.012) (0.020) (0.020) (0.020) (0.052) (0.036) JPM/XOM 13.067 0.989 0.984 0.444 0.440 0.988 0.928 0.006 0.036 −16,081.5 32,181.0 (0.199) (0.001) (0.003) (0.012) (0.020) (0.016) (0.016) (0.032) (0.018) MCD/PG 10.432 0.978 0.972 0.311 0.305 0.919 0.880 0.037 0.015 −12,645.5 25,308.9 (0.157) (0.002) (0.006) (0.011) (0.021) (0.018) (0.017) (0.049) (0.027) AA/CAT 12.424 0.977 0.977 0.333 0.333 0.952 0.978 – – −20,201.6 40,417.3 (0.190) (0.002) (0.004) (0.011) (0.017) (0.013) (0.013) AXP/PFE 10.876 0.991 0.991 0.377 0.377 1.014 0.940 – – −17,118.6 34,251.2 (0.164) (0.001) (0.003) (0.012) (0.022) (0.014) (0.013) AXP/WMT 11.908 0.993 0.992 0.360 0.360 1.016 0.890 – – −15,353.2 30,720.3 (0.181) (0.001) (0.002) (0.012) (0.020) (0.014) (0.012) BA/HON 10.681 0.974 0.969 0.355 0.351 0.968 0.914 – – −17,881.1 35,776.3 (0.161) (0.002) (0.005) (0.011) (0.019) (0.013) (0.012) CAT/KO 12.838 0.977 0.972 0.355 0.350 1.007 0.913 – – −14,227.6 28,469.3 (0.196) (0.002) (0.006) (0.011) (0.022) (0.014) (0.013) GE/PFE 11.013 0.984 0.982 0.406 0.404 0.931 0.926 – – −15,635.9 31,285.9 (0.166) (0.002) (0.004) (0.013) (0.020) (0.013) (0.013) HD/JPM 12.454 0.988 0.987 0.446 0.445 0.937 0.968 – – −18,509.2 37,032.3 (0.190) (0.001) (0.003) (0.013) (0.021) (0.013) (0.013) IBM/PG 12.412 0.977 0.973 0.383 0.380 0.974 0.877 – – −10,961.6 21,937.0 (0.189) (0.002) (0.005) (0.012) (0.020) (0.014) (0.012) JPM/XOM 13.085 0.989 0.985 0.442 0.439 0.979 0.939 – – 16,085.9 32,185.8 (0.200) (0.001) (0.003) (0.012) (0.020) (0.014) (0.013) MCD/PG 10.426 0.979 0.974 0.311 0.307 0.921 0.879 – – −12,648.8 25,311.5 (0.157) (0.002) (0.006) (0.011) (0.021) (0.013) (0.012) Note: Standard errors are shown in parentheses. Table 8. Maximum likelihood estimates for the 2 × 2 models Equities ν βv βc αv αc λ11 λ22 λ12 λ21 logL AIC 2×2 AA/CAT 12.428 0.977 0.977 0.331 0.331 0.893 1.022 0.226 −0.032 −20,171.5 40,360.9 (0.190) (0.002) (0.004) (0.011) (0.017) (0.035) (0.026) (0.073) (0.051) AXP/PFE 10.876 0.991 0.991 0.378 0.378 1.032 0.918 −0.018 0.078 −17,107.3 34,232.5 (0.164) (0.001) (0.003) (0.012) (0.023) (0.016) (0.016) (0.030) (0.021) AXP/WMT 11.909 0.993 0.993 0.360 0.360 1.018 0.887 0.032 0.025 −15,347.7 30,713.4 (0.181) (0.001) (0.002) (0.012) (0.020) (0.017) (0.015) (0.033) (0.017) BA/HON 10.684 0.975 0.970 0.355 0.351 0.985 0.894 0.026 0.102 −17,859.4 35,736.9 (0.161) (0.002) (0.005) (0.011) (0.019) (0.028) (0.029) (0.053) (0.055) CAT/KO 12.826 0.977 0.973 0.355 0.351 0.986 0.928 0.093 −0.037 −14,226.8 28,471.6 (0.196) (0.002) (0.006) (0.011) (0.022) (0.022) (0.017) (0.074) (0.030) GE/PFE 11.016 0.984 0.984 0.405 0.405 0.943 0.911 0.016 0.072 −15,622.7 31,263.5 (0.166) (0.001) (0.004) (0.013) (0.021) (0.017) (0.018) (0.030) (0.029) HD/JPM 12.458 0.988 0.988 0.447 0.447 0.953 0.944 0.020 0.125 −18,481.1 36,980.2 (0.190) (0.001) (0.003) (0.013) (0.021) (0.018) (0.020) (0.031) (0.036) IBM/PG 12.407 0.977 0.974 0.383 0.381 0.985 0.866 −0.026 0.030 −10,960.8 21,939.7 (0.189) (0.002) (0.005) (0.012) (0.020) (0.020) (0.020) (0.052) (0.036) JPM/XOM 13.067 0.989 0.984 0.444 0.440 0.988 0.928 0.006 0.036 −16,081.5 32,181.0 (0.199) (0.001) (0.003) (0.012) (0.020) (0.016) (0.016) (0.032) (0.018) MCD/PG 10.432 0.978 0.972 0.311 0.305 0.919 0.880 0.037 0.015 −12,645.5 25,308.9 (0.157) (0.002) (0.006) (0.011) (0.021) (0.018) (0.017) (0.049) (0.027) AA/CAT 12.424 0.977 0.977 0.333 0.333 0.952 0.978 – – −20,201.6 40,417.3 (0.190) (0.002) (0.004) (0.011) (0.017) (0.013) (0.013) AXP/PFE 10.876 0.991 0.991 0.377 0.377 1.014 0.940 – – −17,118.6 34,251.2 (0.164) (0.001) (0.003) (0.012) (0.022) (0.014) (0.013) AXP/WMT 11.908 0.993 0.992 0.360 0.360 1.016 0.890 – – −15,353.2 30,720.3 (0.181) (0.001) (0.002) (0.012) (0.020) (0.014) (0.012) BA/HON 10.681 0.974 0.969 0.355 0.351 0.968 0.914 – – −17,881.1 35,776.3 (0.161) (0.002) (0.005) (0.011) (0.019) (0.013) (0.012) CAT/KO 12.838 0.977 0.972 0.355 0.350 1.007 0.913 – – −14,227.6 28,469.3 (0.196) (0.002) (0.006) (0.011) (0.022) (0.014) (0.013) GE/PFE 11.013 0.984 0.982 0.406 0.404 0.931 0.926 – – −15,635.9 31,285.9 (0.166) (0.002) (0.004) (0.013) (0.020) (0.013) (0.013) HD/JPM 12.454 0.988 0.987 0.446 0.445 0.937 0.968 – – −18,509.2 37,032.3 (0.190) (0.001) (0.003) (0.013) (0.021) (0.013) (0.013) IBM/PG 12.412 0.977 0.973 0.383 0.380 0.974 0.877 – – −10,961.6 21,937.0 (0.189) (0.002) (0.005) (0.012) (0.020) (0.014) (0.012) JPM/XOM 13.085 0.989 0.985 0.442 0.439 0.979 0.939 – – 16,085.9 32,185.8 (0.200) (0.001) (0.003) (0.012) (0.020) (0.014) (0.013) MCD/PG 10.426 0.979 0.974 0.311 0.307 0.921 0.879 – – −12,648.8 25,311.5 (0.157) (0.002) (0.006) (0.011) (0.021) (0.013) (0.012) Equities ν βv βc αv αc λ11 λ22 λ12 λ21 logL AIC 2×2 AA/CAT 12.428 0.977 0.977 0.331 0.331 0.893 1.022 0.226 −0.032 −20,171.5 40,360.9 (0.190) (0.002) (0.004) (0.011) (0.017) (0.035) (0.026) (0.073) (0.051) AXP/PFE 10.876 0.991 0.991 0.378 0.378 1.032 0.918 −0.018 0.078 −17,107.3 34,232.5 (0.164) (0.001) (0.003) (0.012) (0.023) (0.016) (0.016) (0.030) (0.021) AXP/WMT 11.909 0.993 0.993 0.360 0.360 1.018 0.887 0.032 0.025 −15,347.7 30,713.4 (0.181) (0.001) (0.002) (0.012) (0.020) (0.017) (0.015) (0.033) (0.017) BA/HON 10.684 0.975 0.970 0.355 0.351 0.985 0.894 0.026 0.102 −17,859.4 35,736.9 (0.161) (0.002) (0.005) (0.011) (0.019) (0.028) (0.029) (0.053) (0.055) CAT/KO 12.826 0.977 0.973 0.355 0.351 0.986 0.928 0.093 −0.037 −14,226.8 28,471.6 (0.196) (0.002) (0.006) (0.011) (0.022) (0.022) (0.017) (0.074) (0.030) GE/PFE 11.016 0.984 0.984 0.405 0.405 0.943 0.911 0.016 0.072 −15,622.7 31,263.5 (0.166) (0.001) (0.004) (0.013) (0.021) (0.017) (0.018) (0.030) (0.029) HD/JPM 12.458 0.988 0.988 0.447 0.447 0.953 0.944 0.020 0.125 −18,481.1 36,980.2 (0.190) (0.001) (0.003) (0.013) (0.021) (0.018) (0.020) (0.031) (0.036) IBM/PG 12.407 0.977 0.974 0.383 0.381 0.985 0.866 −0.026 0.030 −10,960.8 21,939.7 (0.189) (0.002) (0.005) (0.012) (0.020) (0.020) (0.020) (0.052) (0.036) JPM/XOM 13.067 0.989 0.984 0.444 0.440 0.988 0.928 0.006 0.036 −16,081.5 32,181.0 (0.199) (0.001) (0.003) (0.012) (0.020) (0.016) (0.016) (0.032) (0.018) MCD/PG 10.432 0.978 0.972 0.311 0.305 0.919 0.880 0.037 0.015 −12,645.5 25,308.9 (0.157) (0.002) (0.006) (0.011) (0.021) (0.018) (0.017) (0.049) (0.027) AA/CAT 12.424 0.977 0.977 0.333 0.333 0.952 0.978 – – −20,201.6 40,417.3 (0.190) (0.002) (0.004) (0.011) (0.017) (0.013) (0.013) AXP/PFE 10.876 0.991 0.991 0.377 0.377 1.014 0.940 – – −17,118.6 34,251.2 (0.164) (0.001) (0.003) (0.012) (0.022) (0.014) (0.013) AXP/WMT 11.908 0.993 0.992 0.360 0.360 1.016 0.890 – – −15,353.2 30,720.3 (0.181) (0.001) (0.002) (0.012) (0.020) (0.014) (0.012) BA/HON 10.681 0.974 0.969 0.355 0.351 0.968 0.914 – – −17,881.1 35,776.3 (0.161) (0.002) (0.005) (0.011) (0.019) (0.013) (0.012) CAT/KO 12.838 0.977 0.972 0.355 0.350 1.007 0.913 – – −14,227.6 28,469.3 (0.196) (0.002) (0.006) (0.011) (0.022) (0.014) (0.013) GE/PFE 11.013 0.984 0.982 0.406 0.404 0.931 0.926 – – −15,635.9 31,285.9 (0.166) (0.002) (0.004) (0.013) (0.020) (0.013) (0.013) HD/JPM 12.454 0.988 0.987 0.446 0.445 0.937 0.968 – – −18,509.2 37,032.3 (0.190) (0.001) (0.003) (0.013) (0.021) (0.013) (0.013) IBM/PG 12.412 0.977 0.973 0.383 0.380 0.974 0.877 – – −10,961.6 21,937.0 (0.189) (0.002) (0.005) (0.012) (0.020) (0.014) (0.012) JPM/XOM 13.085 0.989 0.985 0.442 0.439 0.979 0.939 – – 16,085.9 32,185.8 (0.200) (0.001) (0.003) (0.012) (0.020) (0.014) (0.013) MCD/PG 10.426 0.979 0.974 0.311 0.307 0.921 0.879 – – −12,648.8 25,311.5 (0.157) (0.002) (0.006) (0.011) (0.021) (0.013) (0.012) Note: Standard errors are shown in parentheses. 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Journal of Financial Econometrics – Oxford University Press

**Published: ** Apr 16, 2018

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