On convergence and singularity of conditional copulas of multivariate Archimedean copulas, and estimating conditional dependence

TL;DR

This paper derives explicit formulas for conditional distributions of Archimedean copulas, explores their convergence and singularity properties, and shows how to estimate their generators and dependence measures conditionally.

Contribution

It generalizes convergence results for multivariate Archimedean copulas and links generator estimation to conditional copula estimation, providing new tools for dependence analysis.

Findings

Convergence of Archimedean copulas implies convergence of their conditional kernels.

An Archimedean copula is singular iff almost all its conditional kernels are singular.

Estimating the generator of an Archimedean copula also estimates its conditional copula generator.

Abstract

The present contribution derives an explicit expression for (a version of) every uni- and multi-variate conditional distribution (i.e., Markov kernel) of Archimedean copulas and uses this representation to generalize a recently established result, saying that in the class of multivariate Archimedean copulas standard uniform convergence implies weak convergence of almost all univariate Markov kernels, to arbitrary multivariate Markov kernels. Moreover, we prove that an Archimedean copula is singular if, and only if, almost all uni- and multivariate Markov kernels are singular. These results are then applied to conditional Archimedean copulas which are reintroduced largely from a Markov kernel perspective and it is shown that convergence, singularity and conditional increasingness carry over from Archimedean copulas to their conditional copulas. As consequence the surprising fact is established that estimating (the generator of) an Archimedean copula directly yields an estimator of (the generator of) its conditional copula. Building upon that, we sketch the use and estimation of a conditional version of a recently introduced dependence measure as alternative to well-known conditional versions of association measures in order to study the dependence behaviour of Archimedean models when fixing covariate values.