The analysis of multivariate returns via asymmetric archimedean copulae

Financial asset returns are characterized by a complex set of relationships, so that a multivariate approach is generally able to significantly improve the analysis. Nevertheless, a growing number of jointly modelled variables produces an increasing formal and computational complexity, which rapidly becomes hard to manage. For this reason a preliminary variable selection is crucial in this context. Given P assets, we could desire to choose the p assets having the lowest negative tail dependences with a reference asset. The choice can be made by means of a data mining technique, able to select the most important predictors in a classification problem, using tree-based learning ensembles. In a statistical perspective, the multivariate analysis of financial asset returns have suffered from the (ab)use of the hypothesis of Gaussianity. Copula functions have become a significant quantitative tool because of their aptitude to analyse, separately, the behaviour of the marginal distribution and the specification of the whole dependence structure. The most popular copula families are the Elliptical and the Archimedean. However, in the most prominent Elliptical copulae, the Gaussian and the t-copula, the dependence in the lower and upper tail of the joint distribution is, respectively, absent or symmetric. Archimedean copulae do have the advantage of allowing possibly different lower and upper tail dependence. However, in the multivariate case, they have the undesirable exchangeability property, implying the same tail dependence between any pair of the analysed variables. An asymmetric multivariate Archimedean copula function, constructed by a hierarchical structure, allows to overcome this drawback. After coupling the variables with the strongest degree of dependence through a copula function, this is joined with another variable and so on, in a continuous process of assembling. A case study is presented with geographical MSCI equity indices.