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Say that I have a bivariate random variable $X=(X_1,X_2)$ with known marginal distributions $F_1$ and $F_2$ and a known covariance matrix $S$. However, I do not know the joint distribution of $(X_1,X_2)$ and therefore not its copula.

  1. If I decide to use a Gaussian copula to simulate from the joint distribution, how do I choose which Gaussian copula to use and what exactly is the simulation algorithm?

  2. Is it correctly understood that because I do not know the joint distribution and therefore not the "correct" copula, that this simulation algorithm will only result in random variables that have the correct marginal distributions and covariance matrix, but not necessarily the correct joint distribution?

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    $\begingroup$ Your question is confusing because a Gaussian copula together with the marginal distributions determines the joint distribution, yet you state you don't know the joint distribution. Do you mean that you don't know the parameters of the Gaussian copula? If so, doesn't that make your question pointless, given that different Gaussian copulas can produce such a huge range of joint distributions? If not, then how are we to resolve the apparent contradictions in your description? $\endgroup$ – whuber Apr 14 '18 at 18:48
  • $\begingroup$ I don't know the joint distribution and therefore I don't know the "correct" copula (however, I have historical data to estimate the covariance matrix S). If I now decide to use a Gaussian copula, how do I choose which one to use and what is the simulation algorithm? $\endgroup$ – arni Apr 15 '18 at 2:51
  • $\begingroup$ Since that is quite a different question than you have posted, please edit your post so that it articulates what you want to know. $\endgroup$ – whuber Apr 15 '18 at 14:59
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    $\begingroup$ +1 Thank you. It might be worth pointing out that usually in statistics "known" means specified with perfect mathematical accuracy whereas you describe a situation in which the $F_i$ and $S$ are only estimated: they are not "known" at all. If you indeed have historical data sufficient to estimate these quantities with excellent accuracy, then why not use the same data to estimate the entire joint distribution? $\endgroup$ – whuber Apr 15 '18 at 19:37
  • $\begingroup$ Yes, that is a very good point. I modified the question again. $\endgroup$ – arni Apr 15 '18 at 22:00
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Since you have empirical data to estimate the copula, you have really two questions: 1) How to estimate a copula from empirical data? and 2) How to simulate from some estimated (gaussian) copula. Both question have been asked here before, separately.

Question 2 is here: How to simulate from a Gaussian copula? with a good answer. You can find other relevant posts by searching this site for "copula simulation". For question 1 I cannot find a post with some really good answers, this is the best one: Estimating joint distributions using copula package in R. But there are some good information on the internet, have a look at https://firsttimeprogrammer.blogspot.no/2016/03/how-to-fit-copula-model-in-r-heavily.html and https://www.r-bloggers.com/how-to-fit-a-copula-model-in-r-heavily-revised-part-2-fitting-the-copula/

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