A trivariate non-Gaussian copula having 2-dimensional Gaussian copulas as margins

Arthur Charpentier (see Arthur's blog) was recently contacted by some researchers willing to test if a multivariate copula is - or not - Gaussian. They use a test proposed in Malevergne and Sornette (2003) stating that one should simply test for pairwise normality. This test may be of importance in finance, in actuarial science, and in risk management in general: for example, given 120 financial assets, in order to test whether or not some 120-dimensional random vector of interest in finance admits a Gaussian copula, can one restrict the Gaussian copula hypothesis test to pairs of assets? This short note proves that it is not the case, and provides a simple counter-example based on some multivariate EFGM copula. This confirms the intuition that one cannot only consider all pairs of the studied random variables and that one cannot avoid to study the full vector to test whether a random vector admits a Gaussian copula. An earlier counter-example, discovered after writing this note, is also mentioned.