Classification of Copulas

[coffee_achiever]

I'm having a hard time trying to understand the classification of copulas. Here is my attempt; any corrections / guidance would be appreciated.

At the top level we have empirical vs non-empirical copulas, which is kind of like non-parametric vs parametric VaR.

A) Empirical copulas are completely data driven and you *construct* the copula based on empirical joint data you may have. I don't know much else about empirical copulas other than the obvious (you're going to need a lot of data to get the dependencies done right, and if you've got sparse joint data, you're in trouble. I'm guessing you probably use an MLE with your data to get an empirical copula.

B) Non-Empirical copulas assume a functional form for the copula, and your job here is not to construct the copula, but rather, to fit the data to the copula using parameters. The point is you don't construct, but rather, fit.

The next level are sub-levels of class B (non-empirical copulas)

B1) Elliptical Copulas: These are obtained using "elliptical distributions", for example, the Gaussian and Student-T distributions have corresponding copulas. I don't know what an elliptical distribution is (can someone shed light on that?) Elliptical copulas always have symmetric tails (in fact, they're always radially symmetric), which is often unsuited for finance because most peoples' marginal utilities are not symmetric (most people don't like losing money more than they like making lots of money). Since the dependence structure of the copula should match the dependence structure of whatever it is you're modeling, and very few things show symmetry, elliptical copulas are often a poor choice. You obtain the functional form of elliptical copulas using Sklar's Theorem (or you look it up on the Internet...) Also, elliptical copulas don't have closed-form representations, which is always a hassle. For example, every joint event in a bivariate distribution requires a double integral of something that looks like the error function. I'm guessing that distributions that are "Gaussian-like" have their own elliptical copulas as well, like the Cauchy distribution.

B2) Archimedean Copulas: These are "fiat copulas" in that if it looks like a copula, smells like a copula, and sounds like a copula, it must be a copula. They are popular because unlike the elliptical copulas, they do often have closed form representations and they are not necessarily radially symmetric. Some example of Archimedean copulas are the Frank copula, Gumbel copula, and the Clayton copula. As with the elliptical copula, you fit Archimedean copulas using parameters (which you probably get using some goodness of fit estimation). Rather than appealing to Sklar's Theorem, you obtain the functional form of an Archimedean copula using something called a generating function, which I assume is something like how you can get Legendre polynomials from some generating function by taking multiple derivatives or some such thing.

OK, how close to reality did I come? This was pretty much a brain dump of what I know.

What exactly is an elliptical distribution? And if there are elliptical distributions with elliptical copulas, are there hyperbolic and parabolic distributions with hyperbolic and parabolic copulas?

Thanks!
Pete

 

[Aleksander Hansen]

I'm having a hard time trying to understand the classification of copulas. Here is my attempt; any corrections / guidance would be appreciated.

B) Non-Empirical copulas assume a functional form for the copula, and your job here is not to construct the copula, but rather, to fit the data to the copula using parameters. The point is you don't construct, but rather, fit.

You still need to construct the copula

1)I don't know what an elliptical distribution is (can someone shed light on that?)
2) Elliptical copulas always have symmetric tails (in fact, they're always radially symmetric), which is often unsuited for finance 3)Also, elliptical copulas don't have closed-form representations, which is always a hassle. For example, every joint event in a bivariate distribution requires a double integral[...] I'm guessing that distributions that are "Gaussian-like" have their own elliptical copulas as well, like the Cauchy distribution.

1) Easiest just to Google: several equivalent characterizations. A useful thing to keep in mind though, is that among the elliptical distributions you have the normal and the Cauchy distribution which are stable, additive and infinitely divisible. Then, you have the uniform and the student distributions, which are 'just' infinitely divisible (and elliptical of course).
2) Not necessarily
3) some have closed form solutions. A common 'mistake'. In general, you need to integrate for each marginal so you can easily get say an equation with 5000 integrals when trying to simulate something. That sounds inefficient, and that's because it is. It's the common approach though. But we can do better:

We can take two steps: separation, to decompose arbitrary joint distributions into their copula and marginals; and combination, to glue arbitrary copulas and marginals into new joint distributions.
Unlike traditional copula techniques this is a) not restricted to few parametric copulas such as elliptical or Archimedean; b) never requires the explicit computation of marginal cdf’s or quantile functions; c) does not assume equal probabilities for all the scenarios, and thus allows for advanced techniques such as importance sampling or entropy pooling; d) allows for arbitrary transformations of copulas. Furthermore, the implementation is also computationally very efficient in arbitrary large dimensions. (don't need those integrals floating around)