Copulas Can Capture Dependence Beyond Correlation

Summary

This exchange considers whether copulas can model dependence between variables with little or no linear correlation, and why a fitted conditional distribution may appear uninformative. The answer emphasizes that zero or near-zero correlation does not establish independence. A copula can represent nonlinear or tail dependence that Pearson correlation misses; a Student t copula with zero correlation is offered as an illustration of possible tail association.

The response also explains the alternative: when the data contain no dependence structure, conditioning on one variable leaves the other uniformly distributed on the copula scale. In that case, an apparently flat conditional result may reflect independence rather than an error in the calculation. The question mentions a modest sample, rank correlation, and a fitted Archimedean family, but the answer does not validate that fit or diagnose the specific conditional calculation. Model choice and evidence of dependence therefore remain open, and the example illustrates a possibility rather than a conclusion about the supplied dataset.

Key ideas

• Uncorrelated variables can still be dependent, so correlation alone cannot determine whether a copula is useful.
• Copulas can represent nonlinear and tail dependence that linear correlation misses.
• If the variables are independent, conditioning on one leaves the other uniform on the copula scale.
• The response does not assess the specific copula fit or conditional calculation in the question.

Tags

# Modelling dependance between the two uncorrelated variables using copulas


# Modelling dependance between the two uncorrelated variables using copulas












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.

## Answer by Ben (score 1)

https://quant.stackexchange.com/a/57914

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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