Testing copulas to model financial dependence

Copulas offer economic agents facing uncertainty a powerful and flexible tool to model dependence between random variables and are preferable to the traditional, correlation-based approach. In this paper we show how standard tests for the fit of a distribution can be extended to copulas. Because they can be applied to any copula and because they are based on a direct comparison of a given copula with observed data, these tests are preferable to existing, indirect tests. We illustrate the tests by selecting a copula to manage the risk of a well diversified portfolio consisting of stocks, bonds and real estate. They provide clear evidence in favor of the Student’s t copula, and reject both the correlation-based Gaussian and the extreme value-based Gumbel copula. A detailed inspection of the tails reveals that the Student’s t copula accurately captures the risk of joint downside movements, while it is underestimated by the Gaussian and overestimated by the Gumbel copula. Because existing tests that focus on bivariate tail dependence fail to unambiguously select from these three alternatives, the results indicate the superiority of our approach to test and select copulas for modelling dependence.