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

ABSTRACT

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 a multivariate skew-t distribution, the fast implementation makes use of (i) monotone interpolation of the univariate marginal quantile function and (ii) a re-parametrization of the correlation matrix. Our numerical approach is tested with simulated data with data-driven parameters. A real data example involves the daily returns of three stock indices: the Nikkei225, S&P500 and DAX. With both unfiltered returns and GARCH/EGARCH filtered returns, we compare the fits of the Azzalini–Capitanio skew-t, generalized hyperbolic skew-t, Student-t, skew-Normal and Normal copulas.

KEYWORDS:

• Skew-t distribution
• copula
• maximum likelihood estimation
• tail asymmetry
• tail dependence
• generalized hyperbolic distribution

AMS SUBJECT CLASSIFICATIONS:

• 62E17
• 62H10
• 62H20
• 65C60

Acknowledgements

The author deeply appreciates Harry Joe who gave a lot of substantial suggestions including the idea of using a monotone interpolator to calculate quantiles quickly. The author would like to thank the three anonymous reviewers for helpful constructive comments and suggestions on the earlier version of the manuscript. The author is also grateful to Adelchi Azzalini, Hironori Fujisawa, Tsunehiro Ishihara, Shogo Kato, Satoshi Kuriki, Alexander J. McNeil, Gareth Peters, Pavel V. Shevchenko, Hideatsu Tsukahara, Toshiaki Watanabe and Satoshi Yamashita for their helpful comments. The views expressed here are those of the author and do not necessarily reflect the official views of the Bank of Japan.

Disclosure statement

No potential conflict of interest was reported by the author.