A copula goodness-of-fit test based on the probability integral transform

Copulae is one of the main ways of modelling dependence. However, to check whether the dependency structure of a data set is appropriately modelled by a chosen family of copulae, there is no recommended method agreed upon. We introduce a new goodnessof-fit test, based on the probability integral transform. The test is consistent, meaning that no deviations from the null hypothesis are neglected. The test also incorporates a weighting functionality, allowing us to increase the influence of some specific areas of the copulae, for example the tails. Results show that our test has good power at distinguishing tail heavinessand skewness properties. When we add tail weight, the power at distinguishing tail heaviness increases dramatically. The results also show that a different proposed test, that is not consistent, performs well for tail heaviness but very poor for skewness. Simulations are used to obtain the empirical behaviour of the tests. Applied to a collection of stock portfolios our test strongly rejects the Gaussian and the Clayton copulae, while the Student’s t copula provides a good fit.