Goodness-of-fit tests for copulas: A review and a power study

Abstract

Many proposals have been made recently for goodness-of-fit testing of copula models. After reviewing them briefly, the authors concentrate on “blanket tests”, i.e., those whose implementation requires neither an arbitrary categorization of the data nor any strategic choice of smoothing parameter, weight function, kernel, window, etc. The authors present a critical review of these procedures and suggest new ones. They describe and interpret the results of a large Monte Carlo experiment designed to assess the effect of the sample size and the strength of dependence on the level and power of the blanket tests for various combinations of copula models under the null hypothesis and the alternative. To circumvent problems in the determination of the limiting distribution of the test statistics under composite null hypotheses, they recommend the use of a double parametric bootstrap procedure, whose implementation is detailed. They conclude with a number of practical recommendations.

Introduction

Consider a continuous random vector $X=(X_1,\dots ,X_d)$ with joint cumulative distribution function $H$ and margins $F_1,\dots ,F_d$. The copula representation of $H$ is given by $H(x_1,\dots ,x_d)=C\{F_1\left(x_1\right),\dots ,F_d\left(x_d\right)\}$, where $C$ is a unique cumulative distribution function having uniform margins on $(0,1)$. A copula model for $X$ arises when $C$ is unknown but assumed to belong to a class

$${\mathcal{C}}_0=\{C_{\theta }:\theta \in \mathcal{O}\}\text{,}$$

where $\mathcal{O}$ is an open subset of ${\mathbb{R}}^p$ for some integer $p\ge 1$. The books of Joe (1997) and Nelsen (2006) provide handy compendiums of the most common parametric families of copulas.

Copula modeling has found many successful applications of late, notably in actuarial science, survival analysis and hydrology; see, e.g., Frees and Valdez (1998), Cui and Sun (2004) and Genest and Favre (2007) and references therein. However, nowhere has the methodology been adopted and used with greater intensity than in finance. Ample illustrations are provided in the books of Cherubini et al. (2004) and McNeil et al. (2005), notably in the context of asset pricing and credit risk management.

Given independent copies $X_1=(X_{11},\dots ,X_{1d}),\dots ,X_n=(X_{n1},\dots ,X_{nd})$ of $X$, the problem of estimating $\theta$ under the assumption

$$H_0:C\in {\mathcal{C}}_0$$

has already been the object of much work; see, e.g., Genest et al. (1995), Shih and Louis (1995), Joe, 1997, Joe, 2005, Tsukahara (2005) or Chen et al. (2006). However, the complementary issue of testing $H_0$ is only beginning to draw attention.

The situation is evolving rapidly but at this point in time, the literature on the subject can be divided broadly into three groups:

• Procedures developed for testing specific dependence structures such as the Normal copula (Malevergne and Sornette, 2003) or the equally popular Clayton family, also referred to as the gamma frailty model in survival analysis (Shih, 1998, Glidden, 1999, Cui and Sun, 2004).

• Statistics that can be used to test the goodness-of-fit of any class of copulas but whose implementation involves:

  • an arbitrary parameter, as in the rank-based statistic due to Wang and Wells (2000);

  • kernels, weight functions and associated smoothing parameters, as in Berg and Bakken (2005), Fermanian (2005), Panchenko (2005) and Scaillet (2007);

  • ad hoc categorization of the data into a multiway contingency table in order to apply an analogue of the standard chi-squared test, along the lines of Genest and Rivest (1993), Klugman and Parsa (1999), Andersen et al. (2005), Dobrić and Schmid (2005) or Junker and May (2005).

• “Blanket tests”, i.e., those applicable to all copula structures and requiring no strategic choice for their use. Included in this category are variants of the Wang–Wells approach due to Genest et al. (2006), but also the procedures investigated or used by Breymann et al. (2003), Genest and Rémillard (in press) and Dobrić and Schmid (2007).

And then there are authors who, in applied work, use standard goodness-of-fit statistics as a tool for choosing between several copulas, but without attempting to formally test whether the selected model is appropriate, in the light of a $P$-value. See, e.g., the analysis of stock index returns by Ané and Kharoubi (2003).

The purpose of this paper is to present a critical review of the blanket goodness-of-fit tests proposed to date, to suggest variants or improvements, and to compare the relative power of these procedures through a Monte Carlo study involving a large number of copula alternatives and dependence conditions. After some general considerations given in Section 2, existing tests are described in Section 3 and new statistics are proposed in Section 4. Listed in Section 5 are the factors considered in the study designed to assess the level and compare the power of the selected tests. Results are reported and discussed in Section 6. Finally, various observations and methodological recommendations are made in the Conclusion.