Unsupervised Learning Applied to the Grouped t-Copula or the Modeling of Real-Life Dependence

Abstract

Grouped t-copulas were introduced by Embrechts et al. (1999) and Fang et al. (2002) to address the inability of Gaussian copulas to model non-linear dependencies and of t-copulas to model heterogeneous tail-dependencies. These heterogeneous tail-dependencies can be observed in many fields (finance, hydrology, meteorology). Nonetheless, the use of grouped t-copulas comes at the price of a higher number of parameters to fit, and the necessity to form a priori unknown groups which variables' tail-dependencies are the same. This paper takes up these two challenges by providing an unsupervised method based on the bootstrapped estimates of individual t-copulas to form the groups, and a procedure to fit the grouped t-copula once the groups are known by combining the four-step procedures introduced in Brin et Xu (2016) with a bootstrap on the MLE of the grouped t-copula. This methodology gives good results on simulated data sets as soon as the number of observations is large enough (above 1000).