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?
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.
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.