Estimating Non-Simplified Vine Copulas Using Penalized Splines

TL;DR

This paper introduces a novel non-parametric estimator for non-simplified vine copulas using penalized hierarchical B-splines, enabling flexible dependence modeling in high dimensions and testing the simplifying assumption.

Contribution

It presents the first data-driven non-simplified vine copula estimator that allows for varying conditional copulas and addresses the curse of dimensionality with principal component approximation.

Findings

Significant improvement in out-of-sample Kullback-Leibler divergence when rejecting the simplified model.

Effective application to uranium data and eye state classification datasets.

Demonstrates the benefit of modeling conditional copulas explicitly.

Abstract

Vine copulas (or pair-copula constructions) have become an important tool for high-dimensional dependence modeling. Typically, so called simplified vine copula models are estimated where bivariate conditional copulas are approximated by bivariate unconditional copulas. We present the first non-parametric estimator of a non-simplified vine copula that allows for varying conditional copulas using penalized hierarchical B-splines. Throughout the vine copula, we test for the simplifying assumption in each edge, establishing a data-driven non-simplified vine copula estimator. To overcome the curse of dimensionality, we approximate conditional copulas with more than one conditioning argument by a conditional copula with the first principal component as conditioning argument. An extensive simulation study is conducted, showing a substantial improvement in the out-of-sample Kullback-Leibler divergence if the null hypothesis of a simplified vine copula can be rejected. We apply our method to the famous uranium data and present a classification of an eye state data set, demonstrating the potential benefit that can be achieved when conditional copulas are modeled.