I am fitting copula to log returns data for my undergraduate thesis, and comparing the quality of the fit with AIC. One interesting thing that I found is that the Clayton copula, which has negative tail dependence, provides a worse fit then the Normal and Gumbel copulas, which have no and positive tail dependence, respectively. It seems that a copula family which has negative tail dependence should have a better fit because markets are more correlated on crashes.
I looked at the support set of the Clayton copula, and it has a large degree of "spread" on the positive tail. I believe that this is the reason why the Clayton copula fails to fit the returns data well. However, I can't find a copula-based metric that explains this. Tail dependence would be the most obvious candidate, but it doesn't really explain this behavior (compare the negative tails of a Gumbel copula with the positive tail of a Clayton copula, both have 0 dependence at those tails but the spread is vastly different). I was wondering if there was any alternative metric that could be used to explain this. Thanks!