A Goodness-of-fit Test for Copulas

Abstract

We propose a new rank-based goodness-of-fit test for copulas. It uses the information matrix equality and so relates to the White (1982) specification test. The test avoids parametric specification of marginal distributions, it does not involve kernel weighting, bandwidth selection, or any other strategic choices, it is asymptotically pivotal with a standard distribution, and it is simple to compute compared to available alternatives. The finite-sample size of this type of tests is known to deviate from their nominal size based on asymptotic critical values, and bootstrapping critical values could be a preferred alternative. A power study shows that, in a bivariate setting, the test has reasonable properties compared to its competitors. We conclude with an application in which we apply the test to two stock indices.

Keywords:

• Copula
• Goodness-of-fit
• Information matrix
• MLE

JEL Classification:

• C13

ACKNOWLEDGMENT

Helpful comments from Prosper Dovonon, Gordon Fisher, Christian Genest, Robert Kohn, Adrian Pagan, Valentyn Panchenko, Peter Schmidt, seminar participants at University of Sherbrooke (Department of Mathematics), and participants of the 2009 International Conference on Econometrics and World Economy in Fukuoka, the 2009 Quantitative Methods in Finance Conference in Sydney, the 2009 Conference on Modelling and Analysis of Safety and Risk in Complex Systems in St. Petersburg, and the 2010 International Symposium on Econometrics of Specification Tests in Xiamen are gratefully acknowledged.

Notes

The regularity conditions are listed in many papers on semiparametric copula estimation (see, e.g., Chen and Fan, 2006a,b; Genest et al., 1995; Hu, 1998; Shih and Louis, 1995; Tsukahara, 2005). They include compactness of the parameter set, smoothness of the marginals, existence and continuity of the relevant log-density derivatives and some other conditions that guarantee consistency of CMLE. Verification of these conditions for commonly used copula families is beyond the scope of this paper. For many copulas, including those we use, this has already been done elsewhere (Hu, 1998, see, e.g.,][Chapter 5).

For test consistency, it is important to differentiate between the H _o as stated in (1) and the null of a specific copula family. The test may not be consistent against false copula densities such that ℍ(θ _o ) + ℂ(θ _o ) = 0. This seems to be a feature of all information matrix based tests. We thank a referee for pointing this out to us.

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lecr