Simplified vine copula models: Approximations based on the simplifying assumption

In the last decade, simplified vine copula models (SVCMs), or pair-copula constructions, have become an important tool in high-dimensional dependence modeling. So far, specification and estimation of SVCMs has been conducted under the simplifying assumption, i.e., all bivariate conditional copulas of the data generating vine are assumed to be bivariate unconditional copulas. For the first time, we consider the case when the simplifying assumption does not hold and an SVCM acts as an approximation of a multivariate copula. Several results concerning optimal simplified vine copula approximations and their properties are established. Under regularity conditions, step-by-step estimators of pair-copula constructions converge to the partial vine copula approximation (PVCA) if the simplifying assumption does not hold. The PVCA can be regarded as a generalization of the partial correlation matrix where partial correlations are replaced by j-th order partial copulas. We show that the PVCA does not minimize the KL divergence from the true copula and that the best approximation satisfying the simplifying assumption is given by a vine pseudo-copula. Moreover, we demonstrate how spurious conditional (in)dependencies may arise in SVCMs.