Copula approach to fitting bivariate time series

Abstract: We apply the GARCH-copula method to estimate Value at Risk (VaR) for European and Stockholm stock indices. First, marginal distributions are estimated by the ARMA-GARCH model with normal, Student-t, and skewed t distributions. Then we investigate the tails of innovations of ARMA-GARCH models using the Peaks over thresholds method and find that the distributions of stock returns are asymmetric with heavier left tails than right. In order to analyze the dependence between the time series, we try elliptical copulas (Gaussian, t) and Archimedean copulas (Gumbel, Frank, and Clayton) to model the dependence structure between two time series' returns. Parameter estimation is based on the so-called inference for margins, which is a two-step method. Moreover, we adopt backtesting to test the goodness-of-fit of different copulas using Monte-Carlo simulations. Our empirical results show that ARMA(1,1)-GARCH(1,1)-t distribution is proven to be the best fits for margins and Student's t copula gives the highest log-likelihood of the model and best VaR estimation.