Maximum likelihood estimation of skew-t copulas with its applications to stock returns

The multivariate Student-t copula family is used in statistical finance and other areas when there is tail dependence in the data. It often is a good-fitting copula but can be improved on when there is tail asymmetry. Multivariate skew-t copula families can be considered when there is tail dependence and tail asymmetry, and we show how a fast numerical implementation for maximum likelihood estimation is possible. For the copula implicit in the multivariate skew-t distribution of Azzalini and Capitanio (2003), the fast implementation makes use of (i) monotone interpolation of the univariate marginal quantile function and (ii) a reparametrization of the correlation matrix. The same techniques apply to the generalized hyperbolic skew-t copula. Our numerical approach is tested with simulated data with realistic parameters. A real data example involves the daily returns of three stock indices: the Nikkei225, S&P500, and DAX. We investigate both unfiltered returns and GARCH/EGARCH filtered returns comparing with the Azzalini–Capitanio skew-t, generalized hyperbolic skew-t, non-skewed Student-t, skew-Normal, and Normal copulas.