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Suppose we have a bunch of marginal distributions

$$F_1,...,F_n$$

which have some unknown joint distribution.

My understanding of the importance of copulas is twofold:

  1. By always using the Probability Integral Transform to map our marginals to a multivariate distribution on $[0,1]^n$ with uniform marginals. It means that we need only spend our time compiling a list of joint distributions with uniform marginals, rather than the seemingly insurmountable task of cataloging many different lists of joint distributions, one list for each possible combination of marginals.

  2. Copulas allow us to decouple the computation of the marginals from the computation of their dependence structure. This is important in higher dimensions when there is enough data to approximate the marginals non-parametrically, but insufficient data to approximate the joint distribution non-parametrically. Thus we instead choose an appropriate parametric copula to go with our non-parametrically estimated marginals.

We then simply take the appropriate copula and form

$$C(F_1(x),...,F_n(x))$$

in order to recover a joint distribution with the appropriate marginals.

Now my question is, what exactly makes the Gaussian copula so important among all the possible choices of copulas?

Suppose our marginals are normal, and suppose that their joint distribution is not a multivariate normal. Then my understanding is that since marginals are decoupled from the copula, that their joint distribution (being non-normal) cannot have a Gaussian copula. But will they still be in some sense 'well approximated' by a Gaussian copula?

However I'm not entirely sure of the previous paragraph, as there is a family of distributions which are referred to as meta-gaussian, which I don't fully understand.

Furthermore, what if our marginals are not normal, now it seems even less justifiable to use a Gaussian copula, rather than just choosing the copula that fits the data the best in $[0,1]^n$ space.

Thus I'm left to conclude that the preeminence of the Gaussian copula is really about tractability, rather than stemming from any theoretical justification.

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Now my question is, what exactly makes the Gaussian copula so important among all the possible choices of copulas?

Is it? What makes you say it's especially important?

suppose that their joint distribution is not a multivariate normal. Then my understanding is that since marginals are decoupled from the copula, that their joint distribution (being non-normal) cannot have a Gaussian copula.

Correct; if it did have a Gaussian copula it would be multivariate normal.

But will they still be in some sense 'well approximated' by a Gaussian copula?

It depends on the copula they do have, but in general, no.

Furthermore, what if our marginals are not normal, now it seems even less justifiable to use a Gaussian copula,

It depends on why it's being used. It may be a reasonable approximation or it may not.

rather than just choosing the copula that fits the data the best in [0,1]n space.

If you have no particular reason to choose a Gaussian copula it may be a convenient - but often not - ideal choice.

Could you also speak to the family of meta-Gaussian distributions?

There's no distinction between "distributions with a Gaussian copula" and "meta Gaussian distributions". So we've been discussing the dependence structure of the family of meta-Gaussian distributions all along.

In some areas people are tempted to use the Gaussian copula in multivariate situations because more generally copulas are more work once you move beyond the bivariate case and in some ways the Gaussian case is easy to work with (if you transform the margins to normal you can just fit a multivariate Gaussian). However, there are vine copulas, for example.

The Gaussian copula is frequently inadequate -- it can't model tail dependence, for example, making it unsuitable for the many situations where tail dependence exists. This stuff is pretty well documented in basic books and papers on copulas though. Indeed, misuse of the Gaussian copula to model dependence among debt defaults was credited with making the global financial crisis worse (precisely because as you condition on being in the upper tail, the Gaussian copula does essentially the exact opposite of what's needed for describing the dependence).

The Gaussian copula is most popular when dealing with elliptical distributions, for which there's at least some argument for considering it, since the correlation coefficients still have a relatively direct interpretation. Otherwise, they're just parameters of the dependence structure, and it would usually be better to consider the actual characteristics of the dependence you have (or at least the most essential characteristics of it).

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  • $\begingroup$ Thanks Gleb_b for your detailed answer. Could you also speak to the family of meta-Gaussian distributions? My understanding is that these are a class of multivariate distributions which exhibit identical dependency structure to a multivariate Gaussian, but do not have normal marginals. And they therefore map to Gaussian copula (under the probability integral transform), since copula extract the dependency structure while discarding the marginal information. $\endgroup$ – Thoth Oct 3 '16 at 13:07
  • $\begingroup$ I added some comments $\endgroup$ – Glen_b Oct 3 '16 at 16:02
  • $\begingroup$ Actually, I now see that much of what I was trying to say there is already at the wikipedia page on copulas anyway... en.wikipedia.org/wiki/… $\endgroup$ – Glen_b Oct 3 '16 at 16:06
  • $\begingroup$ So basically you're saying meta-Gaussian distributions are a superset of the family of multivariate normal distributions which also contain any other multivariate distributions with a Gaussian dependence structure but with non-normal marginals? $\endgroup$ – Thoth Oct 3 '16 at 21:47
  • $\begingroup$ That seems to be an unnecessarily complicated way of restating it, but it is the case -- obviously the multivariate Gaussian is included in the class of distributions with Gaussian copula. $\endgroup$ – Glen_b Oct 3 '16 at 21:55

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