The partial vine copula: A dependence measure and approximation based on the simplifying assumption

TL;DR

This paper introduces the partial vine copula (PVC), a new dependence measure for high-dimensional copula models, and analyzes its properties and approximation capabilities under the simplifying assumption.

Contribution

The paper proposes the PVC as a novel multivariate dependence measure and explores its role in approximating multivariate distributions with simplified vine copulas.

Findings

PVC does not minimize Kullback-Leibler divergence from the true copula.

Step-wise estimators of pair-copula constructions converge to the PVC.

PVC is the best feasible SVC approximation in practice.

Abstract

Simplified vine copulas (SVCs), or pair-copula constructions, have become an important tool in high-dimensional dependence modeling. So far, specification and estimation of SVCs has been conducted under the simplifying assumption, i.e., all bivariate conditional copulas of the vine are assumed to be bivariate unconditional copulas. We introduce the partial vine copula (PVC) which provides a new multivariate dependence measure and which plays a major role in the approximation of multivariate distributions by SVCs. The PVC is a particular SVC where to any edge a j-th order partial copula is assigned and constitutes a multivariate analogue of the bivariate partial copula. We investigate to what extent the PVC describes the dependence structure of the underlying copula. We show that the PVC does not minimize the Kullback-Leibler divergence from the true copula and that the best approximation satisfying the simplifying assumption is given by a vine pseudo-copula. However, under regularity conditions, step-wise estimators of pair-copula constructions converge to the PVC irrespective of whether the simplifying assumption holds or not. Moreover, we elucidate why the PVC is the best feasible SVC approximation in practice.