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Suppose that I have bivariate data, and I need to model the bivariate dependence structure using a copula. Suppose further, I do not know what the best-fit copula family to my data is. Hence, I can use any selection method such as AIC or MLE and so on. To use these methods, I need the estimated value of each copula family (the families to choose from). I could estimate a copula's parameter(s) using Kendall's tau or any other estimation method. In all copula articles which I read, the authors said that they selected the best family and then estimated its parameter(s)!! My question is, how did they select the family and then estimate its parameter(s)? That is, did they already have the parameter(s) estimated? Where do they plug in the selection method!! Any help, please?

Please see this the sequential estimate and this second paper

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  • $\begingroup$ Two ways to go about this (there are probably more): 1.) There are only a few parametric copula families, so assume one, find the copula parameter, and evaluate the fit. Do this for all the families and find the one that fits best. If you don't have a ton of data, this may be feasible. $\endgroup$ – Kiran K. Oct 4 '16 at 12:33
  • $\begingroup$ 2.) Since you have bivariate data, you can simply look at a scatter plot of the data. Between the follow copula families (Gaussian, Frank, Gumbel, Clayton), only the Gaussian allows for negative dependence, so if you have negative dependence, your choice is almost made for you. If you have positive dependence, then look at the tail dependence and see which copula would make the most sense. Gaussian copulas do not model tail dependence, a Clayton copula has lower tail dependence, etc ..., so that might be something to look for before deciding a copula family to use. $\endgroup$ – Kiran K. Oct 4 '16 at 12:34
  • $\begingroup$ They are many selection methods. However, they select copula before estimating its parameter!!! If they use AIC, then they must plug the estimation of the parameter. The articles that I read said select via AIC then estimate. If we suppose they select based on the shape of data, it is make sense but how?!! $\endgroup$ – user130885 Oct 4 '16 at 12:37
  • $\begingroup$ What you said is correct if that is so clear. However, if we use AIC or MLE, then the models will based on the parameters to select the best fit copula. $\endgroup$ – user130885 Oct 4 '16 at 12:40
  • $\begingroup$ If you have a link to the article it would be helpful. $\endgroup$ – Kiran K. Oct 5 '16 at 11:56
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Use empirical Copula to see the governing distribtution of 2 sets of variables and compare the shapes with official copula paterns. Of course, if you are not sure afterwards you can go for Goodness of fit values as well.

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    $\begingroup$ please provide full reference for your link in case it dies in the future $\endgroup$ – Antoine Oct 1 '19 at 9:42
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We can select the best fit copula families before estimating their parameters using scatter plots as each copula has its own shape. Not only Gaussian copula can model negative dependency but also Frank copula which also becomes very close to Gaussian if the degree of the dependency structures is small.

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