When we use gaussian copula, are we implying that the underlying marginals are gaussian? Since we can plug anything consistent with a cdf, I would like to clarify whether we are implying that each of the underlying marginals are gaussian when we use gaussian copula?
thanks
 A: Definitely not! The name (I assume) comes from the fact that it is the copula of the multivariate Gaussian distribution.
Feel free to use other marginals. I’d encourage you to try some simulations where you have the same Gaussian copula with high correlation while you vary the marginal distributions.
A: Marginal distributions are dependent only on that subset of data without reference to any other. Conditional distributions are influenced by other variables.
Copulas create a conditional distribution between variables (e.g. through rank correlation) but do not determine the local marginal distribution of individual variables. The marginal distribution of each variable can be set independently for each variable in the copula by applying a transformation to the result of the copula.
In single variable random number generation you typically start by generating a uniformly distributed set of random numbers. This is then transformed to gaussian by applying a transform. If we do this for two variables independently we get a diamond shaped joint distribution. This is not desirable - the numbers are no longer uniformly distributed and will not transform the same as the single variable case. The marginal distribution will be influenced by the conditional one in this case.
The gaussian copula creates two sets of locally (within variable) uniformly distributed numbers that are conditionally distributed. This means then you can apply the same transforms as for single uniformly distributed numbers, but have a built in rank correlation. This will be preserved when you apply different marginal transformations (e.g. one variable can become gaussian and the other binomial)
