Does two variables have same correlation before and after applying copula?

I was reading a blog about copula. The post starts with two variables XX, YY sampling from a multivariate Gaussian with some kind of positive correlation, let’s say ρ\rho. Then applying copula (basically doing two times inverse cumulative function trick), we have new variables X′=F−1(Φ(X))X’=F^{-1}(\Phi(X)) and Y′=G−1(Φ(X))Y’=G^{-1}(\Phi(X)), where Φ(x)\Phi(x) is normal cumulative distribution function. While I understand X′X’ and Y′Y’ then have marginal distribution F(x)F(x) and G(x)G(x), my question is does corr(X′,Y′)corr(X’, Y’) equal ρ\rho?

My guess is not necessarily, then does any method help correct variant correlation in case we want to maintain same correlation after applying copula?

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1 Answer
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Well it depends what you define as correlation. Linear correlation is not invariant under monotone reparametrisation of marginals, so it is not a property of the copula. Kendalls tau, for example, is. The blog you cited seems to be pretty uninformed about these issues. In general linear correlation is relevant only if you are in the Gaussian (or elliptical) setting. If you are not, linear correlation is pretty useless, maybe not even defined (e.g. for random variables without second moments). So if you want to look into copulas, you are not in the Gaussian setting, almost by definition, hence you should not be concerned about linear correlation. For a good intro to dependence modelling using copulas see here (as a shocker have a look at Example 3.2) For interesting (and shocking as well) reading about the dangers of linear correlation see this