AbstractThe empirical joint distribution of return pairs on stock indices displays high tail‐dependence in the lower tail and low tail‐dependence in the upper tail. The presence of tail‐dependence is not compatible with the assumption of (conditional) joint normality. The presence of asymmetric tail‐dependence is not compatible with the assumption of a joint student‐t distribution. A general test for one dependence structure versus another via the profile likelihood is described and employed in a bivariate GARCH model, where the joint distribution of the disturbances is split into its marginals and its copula. The copula used in the paper is such that it allows for the existence of lower tail‐dependence and for asymmetric tail‐dependence, and is such that it encompasses the normal or t‐copula, depending on the benchmark tested. The model is estimated using bivariate data on a set of European stock indices. We find that the assumption of normal or student‐t dependence is easily rejected in favour of an asymmetrically tail‐dependent distribution. Copyright © 2002 John Wiley & Sons, Ltd.