Simplified vine copula models: Approximations based on the simplifying assumption

Vine copulas, or pair-copula constructions, have become an important tool in high-dimensional dependence modeling. Commonly, it is assumed that the data generating copula can be represented by a simplified vine copula (SVC). In this paper, we study the simplifying assumption and investigate the approximation of multivariate copulas by SVCs. We introduce the partial vine copula (PVC) which is a particular SVC where to any edge a $j$-th order partial copula is assigned. The PVC generalizes the partial correlation matrix and plays a major role in the approximation of copulas by SVCs. We investigate to what extent the PVC describes the dependence structure of the underlying copula. We show that, in general, the PVC does not minimize the Kullback-Leibler divergence from the true copula if the simplifying assumption does not hold. However, under regularity conditions, stepwise 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 often the best feasible SVC approximation in practice.