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Are copulas good tool to model the dependence between the two uncorrelated variables. I have X and Y datasets with 260 data points each with Pearson's correlation=-0.06 and Kendall rank correlation=0.1093. Will copulas be able to capture the dependence between the two variables? I tried fitting Arhimedian copulas and found that the Gumbel copula is the best fit. When I am finding conditional copula distribution to obtain Function C(P<=y|X=x), the results are not good. I could not figure out where the problem lies? Is it because I chose copula for uncorrelated variables or I am missing something in copula analysis.

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  • $\begingroup$ what is $P$ in the conditional copula? Can you show the Gumbel, and the one that was not good $\endgroup$ – develarist Sep 9 at 19:18
  • $\begingroup$ "Copula" is just another word for "dependence between variables". So they can model any type of dependence by definition. But this is theory. Whether you are able to find the "right" dependence from data depends on many things. For example the dependence in your case simply might not be Archimedian. I think it is a good start to ask yourself what kind of dependence you would expect from your knowledge of the underlying data and then try to find a copula(family) which reflects this. $\endgroup$ – g g Sep 10 at 8:31
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Uncorrelated does not imply independent, hence a copula could capture the dependence for very small correlations if there is any. As an example, the student t copula with a degree of freedom of 0.4 and a correlation of 0 even has a tail-dependence of 0.4 and you can see the structure in the scatter plot of a sample. However, if there is not any structure in your data, the conditional copula C(v|u=u_0) will result in a uniform distribution on [0,1] (= no matter which value u takes, v can still be anything). You can use the copulatheque.org to play around with several copula families to better assess and understand their properties.

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