Are Archimedean copulas useless for representing multivariate data?

Question

Following Hofert et al.'s paper "Likelihood inference for Archimedean copulas in high dimensions under known margins," (http://dl.acm.org/citation.cfm?id=2263953) I wrote a script in Matlab to produce estimates of Archimidean copulas in high dimensions. The procedure in their paper allows computing the derivatives of the generator functions without resorting to iterative methods; just closed forms. So I thought it a would idea to employ it as an alternative to the use of elliptical copulas for some multivariate modeling.

I used it to represent data with extreme dependence using Gumbel copulas. However, when I employed it to represent more than say four variates, the estimates weren't as good as with elliptical copulas! In a paper that I subsequently wrote a reviewer made me notice that this was because Gumbel copula has only one parameter, which turns it out to be useless to represent high dimensional data. I should employ, he/she said a vine copula in those cases (with added complexity); not a one-parmeter copula.

Thereafter my question is, Are Archimidean copulas really useful to represent high dimensional data? Are they useless except for maybe homogeneous dependence structures, where the dependence might be equal between pairs of variates? Any example of a good representation of multivariate data?

At the request of @kiran-k, here you can find the code to compute the copula PDF: http://goo.gl/q1b2ny

Answer 1

among the two most intensively discussed issue of Archimedean copulas are:

• exchangeability, i.e. in a two dimensional case $C(u_1,u_2)=C(u_2,u_1)$,
• not enough degrees of freedom.

The issue you seem to mention is the second one. The most direct solutions to this are:

1. use vine copula - you seem to have done research on the subject already -

2. put a hierarchical structure on your data, i.e. :

   • make small groups of data that make sense to link with Archimedean copula,
   • model dependency between groups with other dependence functions, for example with Archimedean copulas again in which case you would be in the company of nested Archimedean copula (see the R-package docu for an easy-to-read first intro)

Two are viable alternatives you should look at but be aware that the first one is rarely used in practise (if at all) since the calibration and the simulation parts of vines copulas are not pieces of cake. Practitioners tend to favor the second option, in particular for its interpretability.