Selecting criterion of multivariate Copula based on the conditional probability integral transformation

In the field of financial risk management, there is still no effective solution for the selection and goodness-of-fit test of multivariate Copula functions. This paper proposes a selection criterion for Copula function based on the method of conditional probability integral transformation (CPIT). By comparing the statistics under the CPIT, we discuss the goodness-of-fit of Gaussian Copula, T-Copula and Clayton Copula in different sample sizes and dimensions. By using daily data of 3 stock indices in North America stock market, we compare the test statistics based on CPIT method with those based on maximum likelihood and kernel density estimates. The results show that the CPIT test can help effectively select the appropriate multivariate Copula function, and results in more accurate and stable statistics in the goodness-of-fit test. Test based on maximum likelihood estimate is unstable while test based on kernel density estimate is less stable for small samples.