Dynamic Tail Dependence in Copula-GARCH Models: an Application to Stock-Index Returns

1. Methods and application Several studies in empirical finance literature have highlighted the importance of allowing for skewness, tail-fatness, non normality of returns for asset allocation and pricing models. Moreover, the dependence between returns, that can impact portfolio decisions, often exhibits nonlinear structures and asymmetric extremal behavior that the usual correlation coefficient is not sufficient to describe. In particular, tail dependence plays an increasing relevant role in optimal assets allocation. Intuitively, tail dependence exists when there is a positive probability that some extreme events can jointly occur. Roughly speaking, there is lower (upper) tail dependence between two assets when one asset presents extremely negative (positive) returns given that the other one has yielded extremely negative (positive) returns. Two popular measures of tail dependence are the well-known lower and upper tail coefficients. They have an easy expression via copula function which is a flexible tool for modeling the dependence structure in multivariate distribution (Nelsen, 1999). We investigate the joint behavior of stock returns of S&P500 component companies which belong to 10 sectors according to the Global Industry Classification Standard (i.e. Energy, Health care etc). The data are daily 10-sector stock index return series for 9 years up to 2005. In describing the bivariate distribution of all pairs of stock return indices we must specify three models, two for margins and one for the copula. We use models of the stock returns that can capture the empirically observed time-varying means and variances. Further we employ models of the dependence structure that allow also for change in this dependence structure through time following Patton (2006). All the considered series exhibit skewness and excess kurtosis. Jarque-Bera test confirms departure from normality. The Liung-Box statistic for up to tenth order serial correlation of squared returns is highly significant at any level suggesting the presence of strong nonlinear dependence in the data. Nonlinear dependence and heavy tails may be due to autoregressive heteroskedasticity. Such behavior can be captured by incorporating GARCH structures in the model. Over the last two decades in financial studies, different GARCH models have been introduced to explain and predict patterns in volatility. We use the GARCH models to describe the conditional distributions of the stock index returns. In particular, we adapt to marginal series a class of GARCH models, namely Asymmetric Power GARCH, Ding et al. (1993), and some variants that are nested within this wide class, as TGARCH (threshold GARCH) including …