Bayesian inference for latent factor copulas and application to financial risk forecasting

Schamberger, Benedikt; Gruber, Lutz F.; Czado, Claudia Article Bayesian inference for latent factor copulas and application to financial risk forecasting Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Schamberger, Benedikt; Gruber, Lutz F.; Czado, Claudia (2017) : Bayesian inference for latent factor copulas and application to financial risk forecasting, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 5, Iss. 2, pp. 1-23, https://doi.org/10.3390/econometrics5020021 This Version is available at: https://hdl.handle.net/10419/171917 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ econometrics Article Bayesian Inference for Latent Factor Copulas and Application to Financial Risk Forecasting Benedikt Schamberger †, Lutz F. Gruber †and Claudia Czado * Department of Mathematics, Technical University of Munich, 85748 Garching, Germany; schamberger[email protected] (B.S.); [email protected] (L.F.G.) *Correspondence: [email protected]; Tel.: +49-89-289-17428 † These authors contributed equally to this work. Academic Editor: Jean-David Fermanian Received: 25 November 2016; Accepted: 8 May 2017; Published: 23 May 2017 Abstract: Factor modeling is a popular strategy to induce sparsity in multivariate models as they scale to higher dimensions. We develop Bayesian inference for a recently proposed latent factor copula model, which utilizes a pair copula construction to couple the variables with the latent factor. We use adaptive rejection Metropolis sampling (ARMS) within Gibbs sampling for posterior simulation: Gibbs sampling enables application to Bayesian problems, while ARMS is an adaptive strategy that replaces traditional Metropolis-Hastings updates, which typically require careful tuning. Our simulation study shows favorable performance of our proposed approach both in terms of sampling efficiency and accuracy. We provide an extensive application example using historical data on European financial stocks that forecasts portfolio Value at Risk (VaR) and Expected Shortfall (ES). Keywords: Bayesian inference; dependence modeling; factor copulas; factor models; factor analysis; latent variables; MCMC; portfolio risk; value at risk; expected shortfall JEL Classification: C11; C31; C38; C51; C58; C63 1. Introduction Copulas (Sklar 1959) are powerful models of multivariate dependence. Copula modeling has had profound impact on the field of financial econometrics by substantially improving the quality of key metrics of financial risk of portfolios such as Value at Risk (VaR) and Expected Shortfall (ES) (see Embrechts et al. (1999)). Even in the simplest multivariate Gaussian model the number of correlation parameters increases quadratically in the number of variables d , and the quadratic growth in dependence parameters also extends to more general copula models. This illustrates that sparse modeling of multivariate dependence becomes critical for robust estimation and forecasting in increasingly high-dimensional problems. One approach to achieve parsimony is to use dynamic latent factors to capture dependence induced by common underlying economic activity components. Dynamic factor models, also known as index models, were first suggested by Geweke (1978) and Sargent and Sims (1977) based on ideas of Burns and Mitchell (1946), and later generalized by Forni et al. (2000,2012), among others. Factor modeling has become popular in credit risk modeling (see for example Gordy (2000); Crouhy et al. (2000) ), with the multivariate Gaussian and t copulas serving as the backbone of many credit risk models (Frey and McNeil 2003). We present a Bayesian latent factor copula model that reduces the quadratic growth rate of the number of parameters to a linear one. Factor copulas have previously been successfully employed to scale the dimension of multivariate dependence models (Murray et al. (2013); Krupskii and Joe (2013, 2015); Oh and Patton (2017)). The pair copula construction-based factor model includes the Gaussian Econometrics 2017,5, 21; doi:10.3390/econometrics5020021 www.mdpi.com/journal/econometrics Econometrics 2017,5, 21 2 of 23 factor model as a special case, and allows much more flexible fine-tuning to capture elaborate dependence characteristics than Murray et al. (2013) and Oh and Patton (2017)’s factor copula models. Our proposed model builds on Krupskii and Joe (2013)’s, and shares their strategy to use bivariate pair copulas to couple the marginal variables with a latent factor. In contrast to their work, we present a Bayesian learning strategy that replaces frequentist uniform integration over the latent factor with Bayesian posterior simulation. A key advantage of Bayesian modeling over frequentist modeling is that credible intervals of the predictive distributions of derived quantities such as value at risk or expected shortfall can be easily obtained. Copula models are typically estimated in two stages (see, for example, (Joe and Xu 1996) and (Joe 2005)): first, univariate marginal models are estimated to make the observed data independent and identically distributed on each of the margins by applying the cumulative distribution function of the marginal models to the observed data; second, a (multivariate) copula is estimated based on the transformed, marginally i.i.d. input data. The generally unfavorable view of two stage estimation procedures must be suspended in the context of copula estimation: the definition a copula requires that the marginal distributions of a copula be uniform on the unit interval; hence marginal models must be estimated to obtain said marginally i.i.d. uniform data before any copula modeling can be performed. Indeed, Czado et al. (2011)’s study confirms that joint Bayesian estimation of the marginal models and copula does not lead to improved estimates; the Markov chain Monte Carlo (MCMC) iterations until convergence of the marginal models produce non-uniform marginal input data for the copula, which leads to substantially increased MCMC runtime. An added benefit of estimating the marginal and dependence models separately is that inference methods for the copula can be applied universally as they do not depend on characteristics of the marginal data (which is marginally i.i.d. uniform, by definition). We use adaptive rejection Metropolis sampling (ARMS) within Gibbs sampling (Gilks et al. 1995) for posterior simulation to eliminate the need for lengthy pilot runs to fine-tune proposal parameters: this strategy retains the Gibbs sampler’s main conceptual idea to recursively simulate from the full conditional distributions, but uses ARMS to generate samples from full conditionals that cannot be drawn from directly. This way, there is no need for Metropolis-Hastings updates ( Metropolis et al. (1953) ; Hastings (1970)), which typically require substantial effort from the user to carefully tune the proposals. The proposed approach is then illustrated to forecast the VaR and ES of a portfolio of European bank stocks. The Bayesian factor copula model combines univariate daily one-day ahead from the marginal return time series, which are modeled by dynamic linear models (DLMs) with ARMA(1,1) structure (West and Harrison 1997). The paper is organized as follows: we define the latent factor copula in Section 2, and present Bayesian theory as well as several MCMC-based strategies for posterior simulation in Section 3. Section 4presents a simulation study that investigates the sampling performance of our simulation strategies, and Section 5discusses an application to financial time series data. The paper concludes with summary comments in Section 6. 2. Latent Factor Copulas A latent factor copula models the dependence among d variables uj∼Unif( 0, 1 ) , j= 1 :d , by a linking copula cj(uj , v ; θj) with copula parameter θj for each variable uj and one common latent variable v . Each of the linking copulas cj can be any bivariate copula; a detailed discussion of many bivariate copulas can be found in Joe (1997). Conditionally on the latent variable v , the density function of the latent factor copula is c(u1, . . . , ud|v) = ∏ j=1:d cj(uj,v;θj). (1) Econometrics 2017,5, 21 3 of 23 Linking Copulas We will use the bivariate Gaussian and Gumbel copulas (Joe 1997) as candidate linking copulas; these are the same linking copulas used by Krupskii and Joe (2013). The Gaussian copula is tail-independent and symmetric, while the Gumbel copula is asymmetric and lower tail-dependent. Density functions and properties about these copulas are summarized in Table 1, while Table 2shows pairs plots for several different parameter values. To further reduce model complexity, we assume a common linking copula family for all margins, but allow for a margin-specific copula parameter, that is, cj(uj,ν,θj) = c(uj,ν,θj). Table 1. Density functions and Kendall’s τ as a function of the parameter of the Gaussian and Gumbel pair copulas. Copula Density Function τ=h(θ) Gaussian cN(u1 , u2 ; θ) = 1 √1−δ22πφ(Φ−1(u1))φ(Φ−1(u2)) × exp −Φ−1(u1)+Φ−1(u2)−2δΦ−1(u1)Φ−1(u2) 2(1−δ2)hN(θ) = 2 πarcsin(θ) Gumbel cG(u1 , u2 ; θ) = 1 u1u2(x1x2)θ−1×exp −x1θ−x2θ θ× (1−θ)−x1θ−x2θ1 θ−2+−x1θ−xθ 22 θ−2 , where x1=−ln u1and x2=−ln u2 hG(θ) = 1−1 θ Survival Gumbel cSG(u1,u2;θ) = cG(1−u1, 1 −u2;θ)hSG(θ) = 1−1 θ Table 2. Pairs plots for the Gaussian, Gumbel and survival Gumbel copulas for different parameter values. As the Gumbel and survival Gumbel copulas only exhibit positive dependence, the 90 ◦ and 270◦rotations of the Gumbel copulas are shown for negative Fisher zparameters. Fisher’s z z =−1.2 z=−0.5 z=0.7 z=1 Gaussian ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 Gumbel ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 Survival Gumbel ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 u1 u2 We parameterize the linking copulas by the Fisher z-transformation of Kendall’s τ, given by Econometrics 2017,5, 21 4 of 23 z=z(τ) = 1 2ln 1+τ 1−τ,τ∈(−1, 1), (2) τ=z−1(z) = 1−2 e2z+1,z∈R. (3) This is a more convenient parameterization for analysis across different copula families than the copulas’ natural parameters, because it makes the parameters comparable across different copula families. The domains of the Fisher z -transformation depend on the copula family; for example, the Fisher z parameter of a Gumbel copula takes values in ( 0, ∞) , while that of the Gaussian copulas takes values in (−∞,∞). Likelihood Let u= (utj)t=1:T,j=1:d be T i.i.d. d -dimensional observations ut= (ut1 , . . . , utd) , t= 1 :T , in the copula domain [ 0, 1 ]d ; and denote the states of the latent factor by v= (vt)t=1:T . The joint likelihood function `(θ,v;u)of the latent factor vand copula parameters θ= (θj)j=1:dfollows from (1) as `(θ,v;u) = ∏ t=1:T ∏ j=1:d cj(utj,vt;θj). (4) Substituting the Fisher z parameters z= (z1 , . . . , zd) , where zi:=z(hj(θj) (see (2) for z(·) and Table 1for h(·) ), for the natural copula parameters θ , the joint likelihood of the latent factor v and Fisher zparameters z= (zj)j=1:dfollows as `(z,v;u) = ∏ t=1:T ∏ j=1:d cjutj,vt;h−1 jz−1(zj). (5) Frequentist Inference Frequentist inference (Krupskii and Joe 2013) integrates over the latent variable v to obtain the unconditional density function of the latent factor copulas as c(u1, . . . , ud) = Z1 0∏ j=1:d cj(uj,v;θj)dv, (6) and then maximizes the unconditional likelihood function with respect to the copula parameters. 3. Bayesian Posterior Analysis In Bayesian analyses, prior information about the unknown parameters is quantified in a “prior distribution.” Once data is observed, the prior distribution and likelihood are combined to derive the posterior distribution—the distribution of the unknown parameters given the observed data. As the posterior distribution typically can not be derived in a closed form, Markov chain Monte Carlo (MCMC) methods are used to simulate from the posterior distribution. 3.1. Prior Distributions Prior for Copula Parameters z Each linking copula’s Fisher z parameter can take values in DSG =DG= ( 0, ∞) or DN= (−∞ , ∞) , depending on whether it is a Gaussian or Gumbel copula. We propose normally-distributed priors on the Fisher z parameter, truncated to the allowable parameter range DSG =DG and DN , respectively. Furthermore, we assume that the priors are independent over all linking copulas j=1:d: Econometrics 2017,5, 21 5 of 23 zj∼NDj(0, σ2 j), π(z) = ∏ j=1:d nDj(zj; 0, σ2 j), (7) where ND(µ , σ2) denotes a normal distribution with mean µ and variance σ2 truncated to D with corresponding probability density function nD(·;µ,σ2). Prior for Latent Factor v In line with the frequentist assumption that integrates out the latent factor v using a uniform density, we choose a uniform prior for v, independent across observations t=1:T, vt∼Unif(0, 1), π(v) = ∏ t=1:T π(vt)≡1. (8) Given that the latent factor vt feeds directly into a copula distribution, the uniform distribution on the unit interval is the only natural choice that does not violate the requirement of uniform copula margins. Joint Prior for zand v We assume no dependence between the latent factor v and copula parameters z , and choose the joint prior density π(z,v)of zand vas the product of the individual prior densities, π(z,v) = π(z)π(v). (9) Independence priors are a common choice in Bayesian modeling, and they do not prevent the posterior from exhibiting data-induced dependence; more comments below after the posterior density (10). 3.2. Posterior Distribution and Full Conditionals Posterior Density An analytical expression of the joint posterior density p(z , v|u) of the copula parameters z and latent factor v, given observations ucan only be obtained up to a normalizing constant, p(z,v|u)∝`(z,v;u)π(z,v). (10) Indeed, this posterior density cannot be factorized into independent marginal distributions for z and v , given that the likelihood function does not exhibit such a product form and thus induces multivariate dependencies in the posterior distribution. Full Conditionals The full conditionals of the latent factor v= (vt)t=1:T and Fisher z parameters z= (zj)j=1:d can be derived by straightforward calculation. The joint posterior density p(z,v|u)of zand v, given observations ucan be written as p(z,v|u)∝`(z,v;u)π(z,v) = `(z,v;u)π(z)π(v) = `(z,v;u)∏ j=1:d NDj(zj) = ∏ t=1:T ∏ j=1:d cjutj,vt;h−1 jz−1(zj)!∏ j=1:d nDj(zj; 0, σ2 j); (11) Econometrics 2017,5, 21 6 of 23 here we used the likelihood from (5) and priors from (7) to (9). Then, for all t= 1 :T , the full conditional of vt , given all other latent factor states v−t:= (v1 , . . . , vt−1 , vt+1 , . . . , vT) , observations u and Fisher z copula parameters zis p(vt|v−t,u,z) = p(z,v|u) p(z,v−t|u)∝p(z,v|u)∝∏ j=1:d cj(utj,vt;zj), (12) where the prior densities of vtand zare constant factors independent of vt. Similarly to (12) , the full conditional of zl , given all other Fisher z copula parameters z−j , observations uand latent factor v, can be derived as p(zj|z−j,u,v)∝ ∏ t=1:T cj(utj,vt;zj)!nDj(zj; 0, σ2 j). (13) 3.3. MCMC Methods for Posterior Simulation The complex forms of the posterior and full conditional densities render direct sampling infeasible. We investigate the performance characteristics of several different sampling strategies: • Metropolis-Hastings within Gibbs sampling (Metropolis et al. (1953); Hastings (1970); Gelfand and Smith (1990)): – MCM: Truncated normal proposals for z and Beta proposals for v ; proposal mode and curvature match those of the full conditionals; – EVM: Gamma proposals for z and Beta proposals for v ; proposal expectation and variance match those of the full conditionals; – IRW: Truncated normal random walk proposals for z and uniform independence proposals v; and • ARMGS: Adaptive rejection Metropolis sampling within Gibbs sampling ( Gelfand and Smith (1990) ; Gilks et al. (1995)). Details of the sampling schemes are given in Appendix A.1. We will investigate these strategies in a simulation study using Gumbel linking copulas (Section 4), and then adapt the best-performing method to also work for Gaussian linking copulas. Here the main difference will be the extension from only non-negative Fisher zvalues to positive and negative values. 4. Simulation Study We test our four different MCMC sampling strategies in strong, weak, and mixed dependence scenarios. The goal is to verify that these samplers provide good estimates, and to determine if some perform better than others. 4.1. Simulation Setup Scenarios To explore the behavior of the Gibbs sampler under different dependence characteristics between the marginal data uand the latent factor v, we consider three different scenarios: in the first scenario, all linking copulas show weak dependence, with Kendall’s τ values of 0.1 ≤τ≤ 0.2; in the second scenario, we use high Kendall’s τ values of 0.5 ≤τ≤ 0.8; and in the third scenario, we use a combination of strong and weak dependence between the marginal data and latent factor with Kendall’s τ ’s of the linking copulas varying between 0.1 ≤τ≤ 0.8. Table 3summarizes the Kendall’s τ’s, copula parameters θ, and Fisher zparameters zof all scenarios. Econometrics 2017,5, 21 7 of 23 Table 3. Copula parameters used in the simulation to model the dependence between each marginal series u:,j:= (u1j, . . . , uTj)and the latent factor v. c1c2c3c4c5 Low τ τ0.10 0.12 0.15 0.18 0.20 θ1.11 1.14 1.18 1.21 1.25 z0.10 0.13 0.15 0.18 0.20 High τ τ0.50 0.57 0.65 0.73 0.80 θ2.00 2.35 2.86 3.64 5.00 z0.55 0.65 0.78 0.92 1.10 Mixed τ τ0.10 0.28 0.45 0.62 0.80 θ1.11 1.38 1.82 2.67 5.00 z0.10 0.28 0.48 0.73 1.10 Simulation Data We investigate N= 100 data sets from the latent factor copula model in three different scenarios. Each simulation data set contains T= 200 observations of dimension d= 5. We use Gumbel linking copulas for each pair c1 , . . . , c5 . As an asymmetric extreme value copula with upper tail dependence and lower tail independence, the bivariate Gumbel copula can be a good fit to model financial data such as large negative stock returns: the probability of many stocks selling off simultaneously tends to be much higher than the probability of many stocks all making outsize gains on the same day. The density of the bivariate Gumbel copula quickly tends to infinity for (u , u)↓( 0, 0 ) or (u , u)↑( 1, 1 ) , allowing to test the robustness of our proposed sampling routines for linking copulas that show difficult numerical behavior. Posterior Simulation For each of the N= 100 simulation data sets from the three scenarios described in Section 4.1, we generated 10, 000 MCMC samples with each sampling method from Section 3.3, using the starting values from our heuristic of Appendix A.2 and prior variances σ2 j= 100 2 for the Fisher zcopula parameters. The proposal variance for our IRW method were tuned in pilot runs with 1, 000 iterations each to achieve acceptance rates between 20% and 30% for each Fisher z parameter z1 , . . . , z5 ; we specified this range of desirous acceptance rates in line with Roberts et al. (1997)’s suggestion for tuning to 23%. Uniform Unif( 0, 1 ) proposals for the latent factor v= (v1 , . . . , vT) lead to acceptance rates around 20%, which we accepted as good enough. These settings performed well across all three scenarios. 4.2. Results Kendall’s τPair Copula Parameters For ease of interpretation, we analyze the results posterior samples in terms of Kendall’s τ , which is a simple transformation from Fisher’s z . Posterior trace plots show immediate convergence to an equilibrium state as well as good mixing behavior (see Figure 1); the latter is confirmed by posterior density plots. The runtimes of our different sampling strategies varied substantially: the slowest method (EVM) took 14 times as long as the fastest (IRW) to generate 10, 000 posterior samples (Table 4); the faster methods, IRW and ARMGS, also out-performed the slower ones in terms of effective sample size (ESS; Thiébaux and Zwiers (1984)) per minute. Table 4shows that Metropolis-Hastings within Gibbs sampling with IRW updates was marginally faster than adaptive rejection Metropolis Econometrics 2017,5, 21 8 of 23 sampling within Gibbs sampling (ARMGS), but the latter’s coverage of credible intervals was better. Metropolis-Hastings within Gibbs Sampling with MCM updates provided the smallest point estimation errors (both mean absolute deviation and mean squared errors) of all analyzed Bayesian procedures, but it still trails frequentist maximum likelihood estimation (MLE). Overall, our ARMGS and IRW performed substantially better than MCM and EVM. As an adaptive procedure, ARMGS requires less effort of the user than IRW, making it the grand winner. 0 2000 4000 6000 8000 10000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Iteration τ1 ARMGS True value Posterior mode 0.00 0.05 0.10 0.15 0.20 0.25 0.30 02468 τ1 Density ARMGS True value Posterior mode 0 2000 4000 6000 8000 10000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Iteration τ1 MCM True value Posterior mode 0.00 0.05 0.10 0.15 0.20 0.25 0.30 02468 τ1 Density MCM True value Posterior mode 0 2000 4000 6000 8000 10000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Iteration τ1 EVM True value Posterior mode 0.00 0.05 0.10 0.15 0.20 0.25 0.30 02468 τ1 Density EVM True value Posterior mode 0 2000 4000 6000 8000 10000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Iteration τ1 IRW True value Posterior mode 0.00 0.05 0.10 0.15 0.20 0.25 0.30 02468 τ1 Density IRW True value Posterior mode Figure 1. Trace plots and density plots of a single run from each of our four sampling methods on Kendall’s τscale. The output is based on 10, 000 posterior samples of z1in the mixed τscenario. Econometrics 2017,5, 21 15 of 23 2006 2008 2010 2012 2014 0.4 0.6 0.8 1.0 1.2 1.4 Relative portfolio value Year Relative portfolio value 2006 2008 2010 2012 2014 −0.15 −0.05 0.05 0.15 Portfolio daily log returns Year Daily log return Empirical 90% VaR Empirical 90% ES Figure 5. Historical relative portfolio value of a constant mix strategy with equal weights in the 8 bank stocks ACA, BBVA, BNP, CBK, DBK, GLE, ISP and SAN for years 2006 to 2013. Portfolio weights are readjusted daily. The 90% empirical VaR is shown in blue and the 90% empirical ES is in green. Value at Risk Forecasts Figure 6shows the dynamic VaR forecasts provided by our Gaussian latent factor copula (in conjunction with marginal time series DLMs); our VaR forecasts are the 10% quantiles of each day’s 1, 000 simulated portfolio returns. This figure shows that our dynamic combined multivariate time series model adapts quickly to changing market volatilities and eliminates the clustered occurrence of VaR violations found in an empirical constant volatility model (see right sub-figure of Figure 5). The realized frequency of VaR violations at the predicted levels was closest to the theoretical value of 10% for forecasts from the Bayesian Gumbel latent factor copula (10.17%), followed by the frequentist Gumbel latent factor copula (10.60%), and the regular vine copula (9.64%); furthermore, the Bayesian version of the latent factor copula always out-performed the frequentist version with the same choice of linking copula (Table 8). 2006 2008 2010 2012 2014 −0.14 −0.10 −0.06 −0.02 Latent−factor copula model with Gaussian copulas Year Daily log return 90 % VaR forecast 90 % ES forecast Figure 6. Daily log-returns of the equally weighted constant mix portfolio with (negative) 90% VaR (blue line) and ES (green line) forecasts. In further analyses, we utilized the conditional coverage test suggested by (Christoffersen 2011, Chapter 13) to evaluate the performance of our models’ VaR forecasts. This test jointly analyzes the rate of VaR violations as well as whether they occur independently. Specifically, the null hypothesis of this test is that the hit sequence of ( 1 −α % ) VaR violations is a first-order Markov chain, where violations happen with probability α and are independent of the current state of the chain. Again, the Econometrics 2017,5, 21 16 of 23 Bayesian latent factor copula with Gumbel linking copulas fares best, showing the highest p -value of 0.44 (Table 8). Most noteworthy is that all frequentist latent factor copula models can be rejected at the 10% level, while all Bayesian latent factor copulas and the Bayesian regular vine copula pass. Table 8. Frequency of 90% VaR violations and 90% ES of an equals weights portfolio of all eight selected financial stocks, and p -values of conditional coverage test of VaR violations. RVR denotes the multivariate regular vine copula model; the other columns are for the dynamic factor copula model with the respective linking copula families. The best values are emphasized in bold. Gumbel Gaussian Survival Gumbel RVRARMGS MLE ARMGS MLE ARMGS MLE 90% VaR viol. 9.64% 10.17% 10.60% 9.59% 9.36% 9.35% 9.17% 90% ES 4.07% 4.00% 3.88% 4.10% 4.08% 4.13% 4.08% p-value, cond. coverage test 0.24 0.44 0.07 0.30 0.01 0.10 0.03 Although regulators often require higher confidence levels such as 99%, 99.5% or even 99.9%, producing robust estimates at these levels is not statistically feasible unless unrealistically big training data is available (see, e.g., McNeil et al. 2005, Example 7.15). In our study, no reliable results could be obtained about such extreme quantiles. Expected Shortfall Forecasts The empirical 90% ES of our equal weights portfolio from 2006 through 2014 was 4.58%, and the forecasts range between 4.00% and 4.13% for Bayesian latent factor copulas, 3.88% and 3.08% for frequentist latent factor copulas, and 4.07% for the regular vine copula (Table 8). There are no universally accepted backtests for ES, as academic and regulatory discussions are ongoing (Embrechts et al. 2014). Part of the problem is that ES is not elicitable, which means that there is no obvious way to compare different ES forecasts. In this paper, we compare ES forecasts with the observed ES at the 90% level instead: using this decision criterion, our Bayesian latent factor model with survival Gumbel linking copulas yields performed best. Again, all frequentist latent factor copulas are out-performed by their Bayesian peers (Table 8). 6. Summary Comments Latent factor copulas offer a simple and elegant way to parsimoniously model multivariate dependence and achieve scalability to high dimensions. Allowing for different choices of linking copulas, they provide sufficient flexibility to be used in a wide range of multivariate applications. Our proposed Bayesian inference strategy utilizes adaptive rejection Metropolis sampling within Gibbs sampling for posterior simulation (Section 3). A main benefit to the end-user is that no tuning of MCMC proposals is required, given that ARMS is an adaptive method. At the same time, our analyses showed that adaptive rejection Metropolis sampling within Gibbs sampling (“ARMGS” in earlier sections) is a potent strategy that out-performed regular Metropolis-Hastings samplers with ease (see Section 4). Our case study on forecasting the portfolio Value at Risk (VaR) and Expected Shortfall (ES) demonstrated that Bayesian latent factor copulas generated substantially improved risk forecasts than frequentist ones, and also out-performed the benchmark regular vine copula in terms of forecast accuracy (Section 5). We expect that these findings will impact industry best practices as well as regulatory requirements to potentially increase resilience of world financial markets. Methods for model selection of linking copulas as well as time-varying extensions of the latent factor copula model are the subject of ongoing research. For example, we could potentially expect that the dependence parameters of the linking copulas change over time. Furthermore, Bayesian extensions allowing for several latent factors and/or structured factor copulas as proposed in Krupskii and Joe (2015) are future research directions. Econometrics 2017,5, 21 17 of 23 Acknowledgments: The third author is supported by the German Research Foundation (DFG grant CZ 86/4-1). The authors are grateful to the Academic Editor and two anonymous referees for their detailed comments on the original version of the paper. Their suggestions were most relevant in revision and in defining the final version. Author Contributions: All authors contributed equally. Conflicts of Interest: The authors declare no conflict of interest. Appendix A. MCMC Sampling Methods Appendix A.1. Details on MCMC Sampling Schemes Mode and Curvature Matching (MCM) This strategy uses Beta(αt , βt) -distributed ( αt , βt> 0) proposal distributions for each latent factor vt∈( 0, 1 ) , and a truncated normal distribution N[a,b](µj , σ2 j) for the Fisher z parameters zj of each linking copula. The mode and curvature of the Beta(αt,βt)distribution with density f(vt;αt,βt)are modevt(αt,βt) = αt−1 αt+βt−2for αt,βt>1; and curvvt(vt,αt,βt):=∂2f(vt;αt,βt) ∂v2 t =1 B(αt,βt)vαt−3 t(1−vt)βt−3α2 t+v2 t(αt+βt−3)(αt+βt−2) −2(αt−1)vt(αt+βt−3)−3αt+2)) . The mode and curvature of the truncated normal N[a,b](µj,σ2 j)proposals for zjare modezj(µj) = µjfor a≤µj≤b; and curvzj(zj,µj,σj) = e−(zj−µj)2 2σ2 j(µ2 j−σ2 j+z2 j−2µjzj) √2π σ5 jΦb−zj σj−Φa−zj σjfor zj∈[a,b]. Numerically unstable behavior of the copula density functions at perfect dependence requires truncation of the allowable parameter range of the linking copulas. We limit the range of allowable Kendall’s τto [0, 0.99], which translates to allowable Fisher zparameters zj∈[z(0),z(0.99)] =:[a,b]. The proposal parameters αt , βt , µj and σ2 j are chosen such that the mode and curvature of the proposal distributions agree with the ones of the respective full conditional distributions. The calculations are performed using numerical methods. Expectation and Variance Matching (EVM) This strategy is basically the same as the previous one, but it matches the proposals’ and full conditionals’ expectation and variances instead of their modes and curvatures. In another deviation from the previous approach, we use Gamma -distributed proposals for the Fisher z parameters here instead of truncated normal proposals used above. The expectation and variance of the Beta(αt,βt)proposals for the latent factor vtare Evt(αt,βt) = αt αt+βt; and Vvt(αt,βt) = αtβt (αt+βt)2(αt+βt+1). Econometrics 2017,5, 21 18 of 23 Given expectation 0 <Evt< 1 and variance 0 <Vvt<∞ of the full conditional distribution of vt , this simple functional form enables analytic solution for the parameters αt and βt of the proposal distribution: αt=−Evt(E2 vt−Evt+Vvt) Vvt ; and βt=αt1 Evt−1. For non-negative Fisher z parameters zj , j= 1 :d , of the linking copulas, we utilize Gamma(sj,rj)-distributed proposals, sj,rj>0, with expectation and variance Ezj(sj,rj) = sj rj; and Vzj(sj,rj) = sj r2 j . Given expectation 0 <Ezj<∞ and variance 0 <Vzj<∞ of the full conditional distribution of zj, the proposal parameters sjand rjfollow as sj= E2 zj Vzj ; and rj=sj Ezj . Again, the expectation and variance of the full conditional distributions are evaluated using a numerical scheme. Independence and Random Walk Samplers (IRW) A numerically less complex alternative to the previous methods, this strategy employs an independence sampler for the latent factors vt , t= 1 :T , and a random walk sampler for the Fisher zparameters zj,j=1 : d. Proposals for each latent factor vt are drawn from a Unif( 0, 1 ) distribution, and are independent of the current state of the sampling chain. The proposal for the (r+ 1 ) -st iteration zr+1 j of parameter zj is drawn from a truncated normal distribution N[a,b](µr+1 j , σ2 j) with random walk means µr+1 j=zr j . The proposal variances σ2 j are chosen such that the acceptance rate is roughly 23%, which is the optimal acceptance rate for multivariate normal random walk proposals (Roberts et al. 1997). Adaptive Rejection Metropolis within Gibbs Sampling (ARMGS) The last sampling method is adaptive rejection Metropolis sampling (ARMS) from the full conditionals of each variable vt , t= 1 :T , and zj , j= 1 :d ; this strategy is called adaptive rejection Metropolis sampling within Gibbs sampling (ARMGS; Gilks et al. (1995)). ARMS extends acceptance rejection sampling by a adaptive exponential envelope and a Metropolis-Hastings acceptance/rejection step. The acceptance/rejection step ensures proper sampling in case of a density that is not fully covered by the exponential envelope. A major advantage of ARMS is that it adapts automatically to characteristics of the target distribution. For univariate log-concave target densities, ARMS creates an envelope for rejection sampling, while for other univariate target densities, ARMS uses acceptance/rejection sampling. In spite of being applicable to sample from general univariate target distributions, ARMS demonstrates Econometrics 2017,5, 21 19 of 23 good performance in many situations and is particularly appealing in the context of Gibbs sampling with complex full conditional distributions. Block ARMS is a multivariate extension to ARMS. Instead of sampling each vt , t= 1 :T , and zj , j= 1 :d , individually, some or all of them can be sampled jointly from their multivariate block full conditional distribution. As this strategy did not perform well in our initial analyses (Schamberger 2015, Chapter 8) , we will not discuss it here. Appendix A.2. Strategy to Find Starting Values for MCMC All methods discussed above require starting values for all parameters v and z . Good starting values can improve convergence speed of these MCMC methods and can also be used as an initial values for maximum likelihood estimation. We propose a two-step heuristic that first chooses starting values for the Fisher z parameters z of the linking copulas, and then chooses starting values for the latent factor v. Starting Values for z: In order to estimate z , one series j∈ { 1, . . . , d} is chosen as a proxy for latent factor v . We do this by selecting the 1-truncated c-vine copula that maximizes the sum of absolute Kendall’s τ ’s of all linking copulas and choosing the root node variable j∗ of this c-vine as the proxy for v . The starting values of the Fisher z parameters z−j∗= (zj)j∈{1:d}\{j∗} are set to the Fisher z parameter of the pair copula linking the corresponding series u:,j , j∈ { 1 :d}\{j∗} , with the root node variable u:,j∗ . For the Fisher z parameter zj∗ of the linking copula that connects the root node variable u:,j∗ to the latent factor v , we use the highest absolute value of Fisher z parameters appearing in the c-vine as the starting value. Starting Values for v: Given starting values for the Fisher z parameters z , we set the latent factors v= (v1 , . . . , vT) to the modes of their full conditional densities. Formal Procedure: We generate starting values for posterior sampling of z and v , given T observations of d-dimensional data u= (ut,:)t=1:Twith uniform marginals as follows. 1. Calculate the empirical d×d Kendall’s τ matrix T= (τij)i=1:d,j=1:d , where τij denotes the empirical Kendall’s τbetween series u:,iand u:,j. 2. Find the column j∗of Tthat maximizes the sum of absolute Kendall’s τ’s, j∗=arg max j=1:d(∑ i=1:d|τij|) 3. Set the starting values z0= (z0 1, . . . , z0 d)for the Fisher zparameters zof the linking copulas to z0 i=zτij∗for i∈ {1 : d}\{j∗} z0 j∗=z max i={1:d}\{j∗}τij∗! 4. Set the starting values v0= (v0 t)t=1:Tof the latent factor vto v0 t=arg max v∈(0,1)np(v|ut,z0)o, where pis the full conditional density of vtgiven by (12). Econometrics 2017,5, 21 20 of 23 Appendix B. Detailed Simulation Results Table A1. Mean absolute deviation (MAD), mean squared errors (MSE), effective sample size (ESS) per minute, and coverage of 90% and 95% credible intervals for Kendall’s τposterior sample. Results are averages over 100 independent replications of the analysis. Low τHigh τMixed τ ARMGS MCM EVM IRW MLE ARMGS MCM EVM IRW MLE ARMGS MCM EVM IRW MLE τ1 MAD 0.0829 0.0762 0.0898 0.0829 0.0656 0.0359 0.0352 0.0359 0.0357 0.0260 0.0431 0.0427 0.0432 0.0431 0.0321 MSE 0.0203 0.0104 0.0269 0.0203 0.0099 0.0020 0.0020 0.0020 0.0020 0.0011 0.0029 0.0028 0.0029 0.0029 0.0016 ESS/min 9.5 0.5 0.3 13.7 n/a 37.2 2.5 2.1 35.5 n/a 46.8 3.1 1.6 60.2 n/a 90% C.I. 0.92 0.65 0.93 0.92 n/a 0.89 0.88 0.89 0.88 n/a 0.97 0.96 0.97 0.97 n/a 95% C.I. 0.97 0.76 0.96 0.95 n/a 0.94 0.94 0.94 0.94 n/a 0.98 0.98 0.97 0.98 n/a τ2 MAD 0.0936 0.0832 0.0831 0.0916 0.0652 0.0334 0.0338 0.0335 0.0336 0.0259 0.0487 0.0486 0.0483 0.0492 0.0388 MSE 0.0246 0.0116 0.0156 0.0229 0.0076 0.0017 0.0018 0.0017 0.0018 0.0010 0.0037 0.0037 0.0036 0.0038 0.0022 ESS/min 7.9 0.6 0.2 11.6 n/a 33.2 2.2 1.8 31.5 n/a 37.6 2.9 1.8 46.7 n/a 90% C.I. 0.91 0.58 0.92 0.89 n/a 0.83 0.81 0.84 0.81 n/a 0.85 0.85 0.87 0.82 n/a 95% C.I. 0.96 0.67 0.96 0.93 n/a 0.92 0.88 0.92 0.92 n/a 0.92 0.91 0.92 0.92 n/a τ3 MAD 0.1070 0.0913 0.1144 0.0930 0.0811 0.0276 0.0279 0.0276 0.0277 0.0198 0.0406 0.0399 0.0407 0.0406 0.0290 MSE 0.0298 0.0131 0.0354 0.0163 0.0137 0.0012 0.0012 0.0012 0.0012 0.0006 0.0026 0.0025 0.0026 0.0026 0.0013 ESS/min 5.1 0.5 0.2 8.1 n/a 25.9 1.8 1.5 23.9 n/a 19.1 2.2 1.5 31.1 n/a 90% C.I. 0.91 0.50 0.86 0.88 n/a 0.90 0.87 0.90 0.90 n/a 0.87 0.87 0.86 0.84 n/a 95% C.I. 0.96 0.61 0.95 0.93 n/a 0.95 0.93 0.96 0.95 n/a 0.93 0.93 0.93 0.92 n/a τ4 MAD 0.1142 0.0983 0.1144 0.1231 0.0787 0.0252 0.0239 0.0254 0.0253 0.0172 0.0404 0.0401 0.0407 0.0401 0.0288 MSE 0.0302 0.0145 0.0301 0.0369 0.0120 0.0010 0.0009 0.0010 0.0010 0.0005 0.0026 0.0025 0.0026 0.0026 0.0013 ESS/min 4.7 0.6 0.2 6.8 n/a 15.7 1.2 1.0 14.4 n/a 2.8 0.8 0.4 6.8 n/a 90% C.I. 0.88 0.42 0.89 0.83 n/a 0.86 0.86 0.86 0.87 n/a 0.84 0.74 0.86 0.82 n/a 95% C.I. 0.94 0.49 0.91 0.89 n/a 0.93 0.90 0.92 0.93 n/a 0.89 0.79 0.89 0.86 n/a τ5 MAD 0.1463 0.1008 0.1507 0.1251 0.0787 0.0241 0.0249 0.0244 0.0245 0.0170 0.0817 0.0459 0.0855 0.0801 0.0406 MSE 0.0524 0.0150 0.0557 0.0346 0.0094 0.0009 0.0010 0.0010 0.0010 0.0004 0.0099 0.0034 0.0106 0.0094 0.0023 ESS/min 3.3 0.5 0.1 5.6 n/a 6.7 0.4 0.4 5.8 n/a 0.8 0.4 0.1 1.0 n/a 90% C.I. 0.94 0.49 0.91 0.92 n/a 0.97 0.80 0.98 0.97 n/a 0.81 0.48 0.82 0.71 n/a 95% C.I. 0.98 0.51 0.93 0.94 n/a 1.00 0.92 1.00 1.00 n/a 0.93 0.57 0.92 0.81 n/a Econometrics 2017,5, 21 21 of 23 Appendix C. Sequential Learning of DLMs Priors at time t: The prior for the states (θjt,λjt)given time t−1 information set Dt−1is normal-gamma (θjt,λjt)|Dt−1∼NG(ajt,Rjt,rjt,cjt)(A1) with parameters ajt ∈R3,Rjt ∈R3×3,rjt,cjt >0. The normal-gamma distribution is θjt|λjt,Dt−1∼N(ajt,Rjt/(cjtλjt)) (A2) λjt|Dt−1∼G(rjt/2, rjtcjt/2). (A3) Forecasts at time t: The unconditional forecast distribution of yjt given time t− 1 information set Dt−1 is obtained by integrating observation Equation (14) over the prior distribution of θjt and λjt: yjt|Dt−1∼Trjt (fjt,qjt), (A4) with degrees of freedom rjt , mean fjt =F0 jtajt and variance factor qjt =F0 jtRjtFjt +cjt . Here Trjt (fjt , qjt) denotes a non-standard t distribution, which is a location-scale transformation of a t distribution with rjt degrees of freedom, Trjt : Trjt (fjt,qjt) = fjt +pqjtTrjt . (A5) Posteriors at time t: Upon observation of yjt , the posterior distribution of (θjt , λjt) given information set Dt:= {Dt−1,yjt}is normal-gamma (θjt,λjt)|Dt∼NG(mjt,Cjt,njt,sjt)(A6) with parameters mjt =ajt +Ajtejt , Cjt = (Rjt −Ajt A0jtqjt)zjt , njt =rjt + 1, sjt =zjtcjt ; these can be calculated using the adaptive coefficient vector Ajt =RjtFjt/qjt , forecast error ejt =yjt −fjt , and volatility update factor zjt = (rjt +e2 jt/qjt)/njt. Evolution to time t+1: The step-ahead prior for (θj,t+1,λj,t+1)given time tinformation set Dtis normal-gamma (θj,t+1,λj,t+1)|Dt∼NG(aj,t+1,Rj,t+1,rj,t+1,cj,t+1), (A7) where aj,t+1=mjt , Rj,t+1=Cjt/δ , rj,t+1=βnjt , cj,t+1=sjt , and β , δ∈( 0, 1 ) are discount factors. This normal-gamma distribution is obtained via normal evolution of θjt in system Equation (A8) and beta discount evolution of λjt in system Equation (A10). 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