Dependence patterns across financial markets: a mixed copula approach

Abstract

This paper studies the modelling and estimation of dependence across international financial markets, with a focus on the structure of dependence. A new approach is proposed based on a mixed copula model and the model is constructed so that it can capture various patterns of dependence structures. The marginal distribution of asset returns in each market is estimated non-parametrically and a quasi-ML method is used to estimate the mixed copula. The methodology is applied to estimate the dependence across several international stock markets. The empirical findings are shown to have some implications that are important for a wide range of multivariate studies in Economics and Finance.

Acknowledgements

An early version of the paper appeared under the title ‘Dependence Patterns across Financial Markets: Methods and Evidence’.

Notes

¹ For more properties of a copula and detailed explanations of the concepts introduced in this section, please refer to Joe (1997), Nelson (1999) or Bouye et al. (2000).

² The survival function needs to be distinguished from the survival copula. For a copula C(u, v), the relation between its survival copula C _S(u, v) and its survival function is

which can be seen from the numerical example in the next section.

³ We set Kendall's τ to be 0.3 in all the plots in Figs 1 and 2.

⁴ Engle (2002) proposes a multivariate GARCH model with dynamic conditional correlations. A two-stage procedure is proposed in the ML estimation. First, a GARCH filter is applied to each individual sequence; second, the parameters involved in the dynamics of the correlation are estimated using the residuals from the first step. This is similar to the pre-filtering we adopted in this paper.

⁵ In this experiment, the mean of the three cases are 0.5000 (i.i.d.), 0.5022 (filtered), 0.5070 (GARCH but unfiltered).

⁶ There is also a large literature documenting and discussing negative skewness in financial return series; see Harvey and Siddique (2000), Chen et al. (2001).

⁷ In the table, if there is no weight estimate ‘W’, then either Gaussian or Gumbel Survival takes unit weight. The number in parenthesis are standard errors of the association parameters.

⁸ The World Bank has a website for financial contagion which collects many interesting works on theoretic models and empirical evidences of contagion. The website address is http://www1.worldbank.org/economicpolicy/managing%20volatility/contagion/index.html

⁹ Embrechts et al. (2003) is a general reference on using copulas in risk management.

¹⁰The variables thatare controlled in the study include market beta, the size effect, and the book-to-market effect.