Models for construction of multivariate dependence – a comparison study

Abstract

A multivariate data set, which exhibit complex patterns of dependence, particularly in the tails, can be modelled using a cascade of lower-dimensional copulae. In this paper, we compare two such models that differ in their representation of the dependency structure, namely the nested Archimedean construction (NAC) and the pair-copula construction (PCC). The NAC is much more restrictive than the PCC in two respects. There are strong limitations on the degree of dependence in each level of the NAC, and all the bivariate copulas in this construction has to be Archimedean. Based on an empirical study with two different four-dimensional data sets; precipitation values and equity returns, we show that the PCC provides a better fit than the NAC and that it is computationally more efficient. Hence, we claim that the PCC is more suitable than the NAC for hich-dimensional modelling.

Keywords:

• nested Archimedean copulas
• pair-copula constructions
• equity returns
• precipitation values
• goodness-of-fit
• out-of-sample validation

Acknowledgements

Daniel Berg's work is supported by the Norwegian Research Council, grant number 154079/420 and Kjersti Aas’ part is sponsored by the Norwegian fund Finansmarkedsfondet. We are very grateful to Cornelia Savu, Institute for Econometrics, University of Münster, Germany, for providing us with her code for the NAC's along with helpful comments. In addition we would like to express our deep gratitude for assistance on the GARCH-NIG filtration to Professor J.H. Venter, Centre of Business Mathematics and Informatics, North–West University, Potchefstroom, South Africa. We also thank two anonymous referees and Professor C.J. Adcock (the editor) for comments and suggestions that helped to improve the article. Finally, we thank participants at the conference on copulae and multivariate probability distributions at Warwick Business School in September 2007 for their valuable comments, in particular Professor Alexander McNeil.

Notes

The Laplace transform of a distribution function G on ℛ⁺ satisfying G(0)=0 is .

Personal communication with Harry Joe.

The experiments were run on an Intel(R) Pentium(R) 4 CPU 2.80 GHz PC.