# Appropriate number of degrees of freedom in t-Copula

In a consultation paper (EBA/CP/2014/08) the European Banking Authority (EBA) wrote: “it is proposed […] that Gaussian or Normal like Copulas are not to be used for operational risk modelling. For instance a T-Student copula with few degrees of freedom (eg. 3 or 4) in most cases appears more appropriate to capture the dependencies between the operational risk events.”

So far I always thought that degrees of freedom (dof) equal the number of observations minus the parameters to be estimated and are thus well defined. The EBA paper, however, gives the impression that the dof—at least in the case of copulas—are the choice of the modeller. If so, what would be a suitable measure/argument to choose the degrees of freedom of a t-Copula?

So far I always thought that degrees of freedom (dof) equal the number of observations minus the parameters to be estimated and are thus well defined.

You're confusing two very different things.

a) Let's start with the family of (univariate) t-distributions. There's a degrees of freedom parameter, $\nu$, that is just that - a parameter - freely definable if we want to use a t-model for something. We don't have to have any data, we're not necessarily doing any sampling, but the $t_3$, the $t_6$ and the $t_{14.38}$ distributions (each with their own densities) are all perfectly sensible objects.

$\,$ More generally, the bivariate-$t$ (and beyond that, the multivariate $t$) are also families of distributions, which has a particular degrees of freedom parameter - and also a correlation parameter (parameters in the multivariate case) - that between them describe the distribution (and hence, the dependence structure among the variates in the corresponding copula).

b) Now, under the case where we're sampling from a normal distribution with mean $\mu$ such that the observations are independent and identically distributed, if we calculate $T=\frac{\bar{x}-\mu}{s/\sqrt{n}}$ then it turns out that $T$ has a $t$ distribution where $\nu=n-1$. Similarly, in a regression situation, $\frac{\hat{\beta}-\beta}{\text{s.e.}(\hat{\beta})}$ has a $t$ distribution where $\nu=n-p$ where $p$ is the total number of mean-parameters estimated. The $t$ also turns up in a number of other similar sampling situations.

$\,$ At no point in this copula problem are we dealing with creating a sampling statistic like those.

Instead we're looking at bivariate (or more generally multivariate) distributions of variables (which we're typically attempting to estimate from data), which underlying variables have some copula. We're then using as a model for that copula the one based on the copula for the bivariate-t (/ multivariate-t). That is, we're describing the dependence structure in terms of some (more-or-less) simple mathematical construction that is sometimes useful at describing the tail-dependence among variables. So our model is chosen so that $\nu$ (and $\rho$) give it the properties we want to see; they might be estimated (e.g. via ML), or they might be selected, or one might be estimated (e.g. by matching a nonparametric correlation in in the sample) and the other selected to yield a certain level of tail dependence.

In some cases (and operational risk is sometimes one of those) you may not have much ability to use data to assess tail dependence; then it's quite common to simply choose some suitable copula. The Gaussian copula has a correlation matrix but doesn't have tail dependence. The t-copula has in addition a single parameter (the d.f. parameter) that allows you to increase or decrease the amount of tail dependence (in effect, the joint-riskiness) of your variables in your model.

In that case, you might - just as they suggest - simply choose some $\nu$ to express some desired amount of tendency to "move together" in that high upper tail. Suggested values are right there in the text you quote.