A New Graphical Tool for Copula Selection

Abstract

The selection of copulas is an important aspect of dependence modeling. In many practical applications, only a limited number of copulas is tested, and the modeling applications usually are restricted to the bivariate case. One explanation is the fact that no graphical copula tool exists that allows us to assess the goodness-of-fit of a large set of (possible higher-dimensional) copula functions at once. This article seeks to overcome this problem by developing a new graphical tool for the copula selection, based on a statistical analysis technique called “principal coordinate analysis.” The advantage is three-fold. First, when projecting the empirical copula of a modeling application on a two-dimensional (2D) copula space, it allows us to visualize the fit of a whole collection of multivariate copulas at once. Second, the visual tool allows us to identify “search” directions for potential fit improvements (e.g., through the use of copula transforms). Finally, the tool makes it also possible to give a 2D visual overview of a large number of known copula families, leading to a better understanding and a more efficient use of the different copula families. The robustness of the new graphical tool is investigated by means of a small simulation study, and the practical use of the tool is demonstrated for two 2D and two 3D (three-dimensional) fitting examples. MATLAB code through the examples is available online in the supplementary materials.

Key Words:

• Copulas
• Dependence models
• Kendall's tau
• Multidimensional scaling

ACKNOWLEDGMENTS

Part of this work was financially supported by the Research Foundation Flanders (FWO) grant G.0125.08 and by the Special Research Fund (BOF – University of Antwerp) grant NOI.20916. The authors thank the referees for their appropriate and relevant comments. This article gained a lot because of their helpful and constructive advice.

Notes

¹Note that copula family C ₁₉ is only appropriate in the case of weak dependence.

The complete set of pictures corresponding to τ = −0.9 up to τ = +0.9, with increments of 0.1, is available online in the supplementary material, see Appendix B, Section B.1.

Figure 2 Two-dimensional mapping of bivariate comparable test spaces.

This fact can also be assured by making λ-plots.

For the other combinations, the 2D representations are available online in the supplementary material, see Appendix B, Section B.2.

Figure 3 Simulated goodness-of-fit procedure for τ = +0.5.

For a definition of the empirical copula, see Appendix A, Section A.3.

Note that copula family C ₁₉ is only appropriate in the case of weak dependence.

Families C ₈ and C ₁₀ are left out because of their very poor performance.

Figure 6 Two-dimensional representation of trivariate financial data fitting example.