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I'm looking into using a Gaussian copula to model correlation between two r.v. in a Monte Carlo simulation. I opt for the Gaussian copula as 1) I understand it and 2) it's the most easy one to explain to senior management.

However when reading literature on the subject I got confused. When I read [1, slide 9] it states that "When the underlying cdf is multivariate normal, then the copula is Gaussian", do they refer to the new joint distribution? (that would make sense). In this document [2, page 24 below the Key concept on Sklar’s theorem ] the authors states that you need to estimate the copula's cdf via maximum likelihood (thus making me think I can also make assumptions and just choose one?). Finally when looking at this implementation [3] it looks like there either are no implicit assumptions or they disregard them.

So my question to you, are there any assumptions that come with choosing e.g. a gaussian or t-copula when simulating correlation of two distributions in monte carlo simulation?

[1]: http://www.stat.ncsu.edu/people/bloomfield/courses/st810j/slides/copula.pdf '' NC state university lecture on copulas"

[2]: http:// www.garp.org/media/691726/quant_classroom_oct2011.pdf "A Short, Comprehensive, Practical Guide to Copulas"

[3]: https://dahtah.wordpress.com/2011/10/28/hello-world/ "Copula's made easy''

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The assumptions that you pose on your model can best be seen from the different copula densities, see the copulatheque for an interactive visualization. Gaussian and t-Copulas are for instance both elliptically symmetric (the typical (squeezed) bell shape if you add Gaussian margins in the Gaussian case). Many other families (i.e. Archimedean copulas) are symmetric about the first main diagonal (assuming positive dependence). However, there are also asymmetric families such as the Tawn family. A further property/implicit assumption of the Gaussian copula is the lack of upper or lower tail dependence (the occurrence of an extreme on one margin is independent of the other margin).

To your statements:

1) Any multivariate Gaussian CDF has a Gaussian copula, but not any multivariate CDF with Gaussian margins must have a Gaussian copula as you can put Gaussian margins into any other copula as well. Likewise, you can use a Gaussian copula with any other set of different marginal CDFs. Hence, The multivariate Gaussian CDF is the special case of a Gaussian copula with Gaussian margins.

2) As in the univariate case, one can choose between a set of distributions (i.e. copulas) and tools like AIC, BIC or GOF-test need to be applied to select the right family for the data set at hand.

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  • $\begingroup$ Hi Ben, Thanks for the answer, just to check to see if I'm correct: It's best to fit a copula using e.g. AIC, and if you choose to assume a copula, you basically make assumptions about how the dependence between 2 variables is modeled. Thus if I choose a gaussian copula, I assume the dependence between two variables follows a gaussian distribution eventhough the r.v. themselves might follow some other distribution. Lastly, for t- and Archimedean copulas I can extract negative loglikelihood / AIC from matlab, any tips on a code to fit gaussian copula and find neg loglikelihood / AIC score? $\endgroup$ – Mr. N Sep 20 '17 at 12:36
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    $\begingroup$ Yes: the choice of copulas determines how two RV depend on each other; adding margins will complete the entire multivariate distribution. I am a R user and have no experience with copulas in MatLab. $\endgroup$ – Ben Sep 26 '17 at 8:41

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