Efficient Bayesian inference for stochastic time-varying copula models

There is strong empirical evidence that dependence in multivariate financial time series varies over time. To incorporate this effect we suggest a time varying copula class, which allows for stochastic autoregressive (SCAR) copula time dependence. For this we introduce latent variables which are analytically related to Kendall’s τ , specifically we introduce latent variables that are the Fisher transformation of Kendall’s τ allowing for easy comparison of different copula families such as the Gaussian, Clayton and Gumbel copula. The inclusion of latent variables renders maximum likelihood estimation computationally infeasible, therefore a Bayesian approach is followed. Such an approach also enables credibility intervals to be easily computed in addition to point estimates. We design two sampling approaches in a Markov Chain Monte Carlo (MCMC) framework. The first is a näıve approach based on Metropolis-Hastings in Gibbs while the second is a more efficient coarse grid sampler using ideas of Liu and Sabatti (2000). The performance of these samplers are investigated in a large simulation study and are applied to two data sets involving financial stock indices. It is shown that time varying dependence is present for these data sets and can be quantified by estimating time varying Kendall’s τ with point-wise credible intervals over the series.