Testing the simplifying assumption in high-dimensional vine copulas

Abstract

Testing the simplifying assumption in high-dimensional vine copulas is a difficult task because tests must be based on estimated observations and amount to checking constraints on high-dimensional distributions.So far, corresponding tests have been limited to single conditional copulas with a low-dimensional set of conditioning variables. We propose a novel testing procedure that is computationally feasible for highdimensional data sets and that exhibits a power that decreases only slightly with the dimension. By discretizing the support of the conditioning variables and incorporating a penalty in the test statistic, we mitigate the curse of dimensions by looking for the possibly strongest deviation from the simplifying assumption. The use of a decision tree renders the test computationally feasible for large dimensions. We derive the asymptotic distribution of the test and analyze its finite sample performance in an extensive simulation study. The utility of the test is demonstrated by its application to 10 data sets with up to 49 dimensions.

Simplified vine copula models: Approximations based on the simplifying assumption

“…But such examples do not provide any information about the appropriateness of the simplifying assumption in practice. To investigate whether the simplifying assumption is true and the PVC is a good approximation in applications, one can use Lemma 3.1 to develop tests for the simplifying assumption, see Kurz and Spanhel [21]. Moreover, even in cases where the simplifying assumption is strongly violated, an estimator of the PVC can yield an approximation that is superior to competing approaches.…”

“…Thus, if the simplifying assumption holds, then C 1:d = C PVC 1:d , i.e., higher-order partial copulas, partial copulas and conditional copulas coincide. This insight is used by [21] to develop tests for the simplifying assumption in high-dimensional vine copulas.…”

“…Nevertheless, due to the prohibitive computational burden or simply because only a stepwise model selection and estimation is possible, the PVC is the best feasible SVC approximation in practice. Moreover, the PVC is used by [20] to construct a new non-parametric estimator of a multivariate distribution that can outperform classical non-parametric approaches and by [21] to test the simplifying assumption in high-dimensional vine copulas. All in all, these facts highlight the great practical importance of the PVC for multivariate dependence modeling.…”

Model selection in sparse high-dimensional vine copula models with an application to portfolio risk

“…This simplifying assumption is often valid for financial data and even if it is violated, it may serve as a useful approximation to the truth. For further discussions, see [3,23,25,27,34,36,37], and references therein. Since tractability is vital in very high dimensions, we shall assume that the simplifying assumption holds for the remainder of this paper.…”

Testing for equality between conditional copulas given discretized conditioning events

“…the choice of the pointwise conditioning event, in particular for inference purpose ( [12]). This is the famous so-called "simplifying assumption", that is standard for vine models in particular ( [3,15]). It can be written as follow:…”