Relation between tail-dependence and correlation in t-copula I have data for two variables, say X and Y and I fit a bivariate t-copula to this data. The fitting of a t-copula gave me two values: a degree-of-freedom (nu = 4.5) and a correlation matrix. From this correlation matrix, I got the (estimated) correlation between X and Y is 0.70. From this information, can I say that the tail-dependence between X and Y is also 0.70? In other words, how does the correlation, in this case, relate to their tail-dependence? Is there any way to calculate tail-dependence from the correlation coefficient?
 A: Given that you refer to the copula parameter $\theta$ of the t-copula (which is in general not the correlation of the two random variables $X$ and $Y$ - if that is not clear, check the copula literature for explanations), there is a relationship between the tail-dependence and the combination of $\theta$ and the  degrees of freedom $\nu$ (Demarta and McNeil, 2004):
$$
\lambda_{\nu,\theta} = 2 t_{\nu+1} \left( -{\sqrt{(1+\nu)(1-\theta)}\over\sqrt{1+\theta}}\right) 
$$
where $t_{\nu+1}$ is the univariate central student t-distribution with $\nu+1$-degrees of freedom. The plot for a few degrees of freedom $\nu$ can be seen below:

Interestingly, a t-copula can have positive tail-dependence although the "overall" association is negative $(\theta < 0)$. And only a t-Copula with larger $\nu$ will tend to $0$ tail-dependence even though the correlation is $0$ (another example for uncorrelated does not imply independent). The figure below (generated through the copulatheque) shows a t-copula with $\theta = -0.3$ and $\nu = 1$ yielding a Kendall's tau of $-0.19$ and upper and lower tail-dependence of $\lambda = 0.19$.

