  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.
