Time-varying joint distribution through copulas

This paper deals with the analysis of temporal dependence in multivariate time series. The dependence structure between the marginal series is modelled through the use of copulas which, unlike the correlation matrix, give a complete description of the joint distribution. The parameters of the copula function vary through time following certain evolution equations depending on their previous values and the historical data. The marginal time series follow standard univariate GARCH models. We develop full Bayesian inference where the whole set of model parameters is estimated simultaneously. This represents an essential difference with previous approaches in the literature where the marginal and the copula parameters are estimated separately in two consecutive steps. Moreover, we propose a Bayesian procedure for the estimation of the Value-at-Risk (VaR) of a portfolio of assets, providing point estimates and predictive intervals. The proposed copula model allows us to capture the dependence structure between the individual assets which strongly influences the portfolio VaR. Finally, we also address the problem of optimal portfolio selection based on the estimation of mean-VaR efficient frontiers. The proposed approach is illustrated with simulated and real financial time series.