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Title: How simplifying and flexible is the simplifying assumption in pair-copula constructions Authors:  Thomas Mroz - University of Salzburg (Austria)
Sebastian Fuchs - University of Salzburg (Austria) [presenting]
Wolfgang Trutschnig - University of Salzburg (Austria)
Abstract: Motivated by the popularity and the seemingly broad applicability of pair-copula constructions underlined by numerous publications in the last decade, we tackle the unavoidable question of how flexible and simplifying the commonly used `simplifying assumption' is from an analytic perspective and provide answers to two open questions. Aiming at the simplest possible setup for deriving the main results we first focus on the three-dimensional setting. We prove that the family of simplified copulas is flexible in the sense that it is dense in the set of all copulas with respect to the uniform metric. Considering stronger notions of convergence like the one induced by the metric $D_1$, by weak conditional convergence, by total variation, or by Kullback-Leibler divergence, however, the family even turns out to be nowhere dense and hence insufficient for any kind of flexible approximation. Furthermore, returning to the uniform metric we show that the partial vine copula is never the optimal simplified copula approximation of a given, non-simplified copula, and derive examples illustrating that the corresponding approximation error can be strikingly large. Moreover, the mapping assigning each copula its unique partial vine copula turns out to be discontinuous with respect to the uniform metric, implying a surprising sensitivity of partial vine copula approximations. The aforementioned main results are then extended to the general multivariate setting.