Models for construction of multivariate dependence

In this article we review models for construction of higher-dimensional dependence that have arisen recent years. A multivariate data set, which exhibit complex patterns of dependence, particularly in the tails, can be modelled using a cascade of lower-dimensional copulae. We examine two such models that differ in their construction of the dependency structure, namely the nested Archimedean constructions and the pair-copula constructions (also referred to as vines). The constructions are compared, and estimation- and simulation techniques are examined. The fit of the two constructions is tested on two different four-dimensional data sets; precipitation values and equity returns, using a state of the art copula goodness-of-fit procedure. The nested Archimedean construction is strongly rejected for both our data sets, while the pair-copula construction provides an appropriate fit. Through VaR calculations, we show that the latter does not overfit data, but works very well even out-of-sample.