How simplifying and flexible is the simplifying assumption in pair-copula constructions -- analytic answers in dimension three and a glimpse beyond

TL;DR

This paper investigates the flexibility and limitations of the simplifying assumption in pair-copula constructions, revealing that simplified copulas are dense under uniform metrics but not under stronger notions, and highlighting the discontinuity and approximation issues of partial vine copulas.

Contribution

It provides a detailed analytic study of the simplifying assumption in pair-copula models, addressing open questions and extending results beyond three dimensions.

Findings

Simplified copulas are dense in three dimensions under the uniform metric.

Partial vine copulas are not optimal approximations and can have large errors.

The mapping to partial vine copulas is discontinuous under the uniform metric.

Abstract

Motivated by the increasing popularity and the seemingly broad applicability of pair-copula constructions underlined by numerous publications in the last decade, in this contribution we tackle the unavoidable question on how flexible and simplifying the commonly used `simplifying assumption' is from an analytic perspective and provide answers to two related open questions posed by Nagler and Czado in 2016. Aiming at a simplest possible setup for deriving the main results we first focus on the three-dimensional setting. We prove that the family of simplified copulas is flexible in the sense that it is dense in the set of all three-dimensional co\-pulas with respect to the uniform metric $d_\infty$ - considering stronger notions of convergence like the one induced by the metric $D_1$, by weak conditional convergence, by total variation, or by Kullback-Leibler divergence, however, the family even turn out to be nowhere dense and hence insufficient for any kind of flexible approximation. Furthermore, returning to $d_\infty$ we show that the partial vine copula is never the optimal simplified copula approximation of a given, non-simplified copula $C$, and derive examples illustrating that the corresponding approximation error can be strikingly large and extend to more than 28\% of the diameter of the metric space. Moreover, the mapping $\psi$ assigning each three-dimensional copula its unique partial vine copula turns out to be discontinuous with respect to $d_\infty$ (but continuous with respect to $D_1$ and to weak conditional convergence), implying a surprising sensitivity of partial vine copula approximations. The afore-mentioned main results concerning $d_\infty$ are then extended to the general multivariate setting.