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

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.

Monthly Streamflow Simulation for the Headwater Catchment of the Yellow River Basin With a Hybrid Statistical‐Dynamical Model

“…Copulas functions are able to capture the dependence structure separately from their marginal distributions (Yin et al, ). Regular vine copulas further extend the dependence model to the multivariate framework by building a hierarchical set of bivariate and conditional bivariate copulas (Killiches et al, ). Z Liu et al () developed a streamflow prediction model using the conditional functions of canonical vine copulas to generate streamflow of 1 month ahead.…”

Pair-Copula Constructions for Financial Applications: A Review

“…The work in [13] shows that the approximation in fact may be a good one, even when the simplifying assumption is far from being fulfilled. This subject has also been investigated by Stöber et al [14], Killiches et al [15] and Spanhel and Kurz [16].…”

Estimating non-simplified vine copulas using penalized splines

“…The data set contains 655 observations which measure the log-concentration of 7 chemicals in water samples from the Montrose quadrangle of western Colorado. This particular data set is often discussed in articles related to copulas, see for instance Gijbels et al (2012) or Killiches et al (2016) who both investigate the simplifying assumption for three-dimensional subsets. Both studies detect that a non-simplified vine copula improves the fit of the data subset.…”