The General Tail Dependence Function in the Marshall-Olkin and Other Parametric Copula Models with an Application to Financial Time Series

Abstract
The accurate understanding of the dependence structure implied by the parametric models studied in statistical and financial literature has drawn growing attention in recent times. In particular, tail dependence is crucial in this analysis. We study the general tail dependence function in some of the most common copula models found in literature, in which this function has not been obtained. These models are used by statisticians and practitioners alike. With the general tail dependence function, we cover positive and non-positive tail dependence often overlooked. The use of the general tail dependence function generalises the well known approach of using the survival copula to tackle upper tail dependence. We present relevant results regarding tail dependence related functions. We include a broad guide for bivariate families and Hierarchical Archimedean copulas in dimensions 3 and 4. In the multivariate case we study the Marshall-Olkin copula, examples of models based on Max-Id distributions and Extreme Value copulas among others. In an empirical section we exemplify with real data the usefulness of our results by modelling arbitrary types of tail dependence. Furthermore we show how our approach can yield better estimates of Value at Risk than other standard approaches.