Pair-copula constructions of multivariate copulas

The famous Sklar’s theorem (see [54]) allows to build multivariate distributions using a copula and marginal distributions. For the basic theory on copulas see the first chapter ([14]) or the books on copulas by Joe ([32]) and Nelson ([51]). Much emphasis has been put on the bivariate case and in [32] and [51] many examples of bivariate copula families are given. However the class of multivariate copulas utilized so far has been limited. Especially financial applications need flexible multivariate dependence structures in the center of the distribution as well as in tails. For value at risk (for a definition see for example [44]) calculations we need flexibility in the tails. One such measure are the upper and lower tail dependence parameter (for a definition see [15]), which coincide for (reflection) symmetric distributions. For example the Gaussian copula allows for an arbitrary correlation matrix with zero tail dependence, while the the multivariate t-copula has only a single degree of freedom parameter which drives the tail dependence parameter. Both the Gaussian and the t-copula are examples of an elliptical copula (see for example [18] and [20]). In addition to elliptical copulas attention has focused on multivariate extensions of the Archimedian copulas. In this class we have fully and partially nested Archimedian copulas as discussed in [32], [56] and [52]. Hierarchical Archimedian copulas are considered in [52], while multiplicative Archimedian copulas are proposed in [50] and [43]. However these extensions require additional parameter restrictions and thus result in reduced flexibility for modeling dependence structures. The first topic of this chapter is to present a general construction method for multivariate copulas using only bivariate copulas, which is called a pair-copula construction (PCC). This includes a simple derivation of PCC models such as D-vines and canonical vines. More general PCC’s such as regular vines (see [4], [5] and