Dynamic dependencies between the Tunisian stock market and other international stock markets: GARCH-EVT-Copula approach

Abstract

We propose a time-varying copula model to analyse the comovement between the Tunisian stock market and three stock markets: American, French and Moroccan. The model is implemented with a GJR- GARCH-EVT-Copula, which allows capturing nonlinear dependency, tails behaviour and offers significant advantages over econometric techniques in analysing the comovement of financial time series. To capture this dependency structure, we use two time-varying copulas: symmetrized Joe Clayton and Clayton. The time dynamics of the dependency parameter follow those proposed by Patton (2006). We first extract the filtered residuals from each return series with an asymmetric GARCH model, and then we construct the sample marginal cumulative distribution function of each index return using a Gaussian kernel estimate for the interior and a generalized Pareto distribution estimate for the upper and lower tails. A time-varying copula is then fit to the data and used to induce correlation between the simulated residuals of each asset. Empirical results show that the Tunisian stock exchange and the American markets have the greatest dependencies with the French market. Therefore, the managers of portfolios that include assets from these pairs of countries should be particularly concerned about downside risk exposure.

Keywords:

• extreme value theory
• copulas theory
• dependency structure
• dynamic copulas
• generalized pareto distribution

JEL Classification:

• G15
• C22
• C10
• C50
• C59

Notes

¹ Note that shocks from the US financial market are often considered as a common factor in the treatment of volatility across countries.

² Illustrations of this statistic are provided in Appendix 1.

³ The copula theory has been extended to the conditional case: the notion of the conditional copula, introduced by Patton (2006) who defines the conditional copula as a multivariate distribution of variables that are each distributed as U(0,1) conditional on F_t – 1.

⁴ The Glosten, Jegannathan and Runkle generalized autoregressive conditional heteroscedasticity (GJR-GARCH) model takes account of the effects of asymmetric information.

⁵ The definition of the unconditional and conditional copulas is given in Appendix 2.

⁶ The Clayton copulas are used to capture left tail dependencies.

⁷ The symmetrized Joe Clayton (SJC) copula is more general, because it allows the tail dependencies to be either symmetric or asymmetric.

⁸ Kendall’s tau for the Clayton copula, upper and lower tail dependency coefficients for the SJC copula.

⁹ The data source is from http://www.msci.com

¹⁰ The skewed-t density is given in Appendix 3.

¹¹  The log-likelihood functions of the copulas models are provided in Appendix 4.