Bayesian inference for high dimensional factor copula models

Abstract: Copulas have been applied to many research areas as multivariate probability distributions for non-linear dependence structures. However, extending copula functions in high dimensions is challenging due to the increase of model parameters and computational intensity. Fortunately, in many circumstances, high dimensional dependence can be explained by a few common factors. This dissertation focuses on using factor copula models to analyze the high dimensional dependence structure of random variables. Different factor copula models are proposed as a solution for the curse of dimensionality. Then, a parallel Bayesian inference or a Variational Inference (VI) is employed to speed up the computation time. Chapter 2 concentrates on a dynamic one factor model for group generalized hyperbolic skew Student-t copulas. Chapter 3 and 4 extend the multi-factor copula models to suit with different high dimensional data sets. These models have applications in a wide variety of disciplines, such as financial stock returns, spatial time series, and economic time series, among others. Chapter 2 develops a class of dynamic one factor copula models for tackling the curse of dimensionality. The asymmetric dependence is taken into account by group generalized hyperbolic skew Student-t copulas. The study is influenced by Creal and Tsay [2015], Oh and Patton [2017b], but instead, the dynamic factor loading equation follows a generalized autoregresive score process which depends on the copula density conditional on the factor rather than the unconditional copula density, as proposed in Oh and Patton [2017b]. As the conditional posterior distributions of parameters in groups can be inferred independently due to model specifications, a parallel Bayesian inference is employed. This reduces the time of computation for a sizable problem from several days to one hour using a personal computer. The model is illustrated for 140 firms listed in v the S&P500 index and the optimal portfolio allocation is obtanied based on minimum Conditional Value at Risk (CVaR). The major content of this chapter resulted into a paper by Nguyen et al. [2019] which had been accepted for publication in Journal of Financial Econometrics. Chapter 3 takes advantage of the static structured factor copula models proposed by Krupskii and Joe [2015a] for the dependence of homogenerous variables in different groups. To extend one factor copula models, Krupskii and Joe [2015a] assume a hierarchical structure for the latent factors and model the dependence of the observables through a serial of bivariate copula links between the observables and the latent variables. This topology stems from vine copulas and becomes very flexible to capture both asymmetric tail dependence as well as correlation among variables. The VI is used to estimate the different specifications of structured factor copula models. VI aims to approximate the joint posterior distribution of model parameters by a simpler distribution which resuls in a very fast inference algorithm in comparison to the MCMC approach. Secondly, an automated procedure is proposed to recover the dependence linkages. By taking advantage of the posterior modes of the latent variables, the initial assumptions of bivariate copula functions are inspected and replaced for better copula functions based on the Bayesian information criterion (BIC). Chapter 3 shows an empirical example where the structured factor copula models help to predict the missing temperatures of 24 locations among 479 stations in Germany. The major content of this chapter resulted into a working paper by Nguyen et al. [2018]. Chapter 4 supplements the factor copula model with a combination of a factor copula model at the first tree level and a truncated vine copula structure at a higher tree level. The model is not only suitable to capture different behaviors at the tail of the distribution but also remains parsimonious with interpretable economic meanings. The truncated factor vine copula models can outperform the multi-factor copula model in cases that there is weak dependence among variables in higher tree levels and the inference of group latent factors becomes inaccurate. The VI strategy is used and the dependence structure can be recovered with a similar copula selection procedure. Chapter 4 compares the statistical criteria of different factor models for the dependence structure of stock returns from 218 companies listed in 10 different European countries.