Examination and visualisation of the simplifying assumption for vine copulas in three dimensions

TL;DR

This paper compares simplified and non-simplified vine copulas in three dimensions, highlighting their differences in shape complexity and demonstrating the necessity of non-simplified models for capturing complex dependencies in real data.

Contribution

It provides a detailed visual and conceptual comparison of simplified and non-simplified vine copulas, emphasizing the importance of non-simplified models for complex dependence structures.

Findings

Non-simplified vine copulas can have arbitrarily irregular shapes.

Simplified vine copulas are smooth and extend bivariate margins.

Real data analysis shows non-simplified copulas better capture complex dependencies.

Abstract

Vine copulas are a highly flexible class of dependence models, which are based on the decomposition of the density into bivariate building blocks. For applications one usually makes the simplifying assumption that copulas of conditional distributions are independent of the variables on which they are conditioned. However this assumption has been criticised for being too restrictive. We examine both simplified and non-simplified vine copulas in three dimensions and investigate conceptual differences. We show and compare contour surfaces of three-dimensional vine copula models, which prove to be much more informative than the contour lines of the bivariate marginals. Our investigation shows that non-simplified vine copulas can exhibit arbitrarily irregular shapes, whereas simplified vine copulas appear to be smooth extrapolations of their bivariate margins to three dimensions. In addition to a variety of constructed examples, we also investigate a three-dimensional subset of the well-known uranium data set and visually detect that a non-simplified vine copula is necessary to capture its complex dependence structure.