# Metric for measuring the “spread” of a copula

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!

• What is the frequency of your data? if you fit a copula then to how many stocks do you fit it? I assume two stocks - which 2? Could just be that those two do not really have negative tail dependence. Wich period do you work on? – Richard Apr 5 at 6:32
• I am fitting the daily log returns data of the SP and RUT indices from 1995-2015. Good point that I should check tail dependence, when I fit a rotated Gumbel (negative tail dependence), I get a better AIC then any of the other copulas, so maybe that is evidence that the two indices have negative tail dependence. – Jason Apr 5 at 7:01

I assume with "spread of a copula" you mean the spread/dispersion of points in a bivariate scatter plot of the copula. This spread is related to the density of the copula. In an area where the copula is near uniform, scatter will be somewhat uniform. Obviously in areas where the density is concentrated, scatter will be concentrated.

You can verify this by comparing the density of the copula conditional on being in the upper tail with the uniform density, i.e. a constant.