Simplified Pair Copula Constructions --- Limits and Extensions

Abstract: So called pair copula constructions (PCCs), specifying multivariate distributions only in terms of bivariate building blocks (pair copulas), constitute a flexible class of dependence models. To keep them tractable for inference and model selection, the simplifying assumption that copulas of conditional distributions do not depend on the values of the variables which they are conditioned on is popular. In this paper, we show for which classes of distributions such a simplification is applicable, significantly extending the discussion of Hob{\ae}k Haff et al. (2010). In particular, we show that the only Archimedean copula in dimension d \geq 4 which is of the simplified type is that based on the gamma Laplace transform or its extension, while the Student-t copula is the only one arising from a scale mixture of Normals. Further, we illustrate how PCCs can be adapted for situations where conditional copulas depend on values which are conditioned on.