Modeling coskewness with zero correlation and correlation with zero coskewness

TL;DR

This paper demonstrates that correlation and coskewness can vary independently, showing no general link between them even with symmetric variables or arbitrary distributions.

Contribution

It establishes that correlation and coskewness are not necessarily linked, providing theoretical results and generalizing to arbitrary distributions.

Findings

Zero correlation can coexist with any coskewness value.

Zero coskewness can occur at any correlation level.

No universal relationship exists between correlation and coskewness.

Abstract

This paper shows that one needs to be careful when making statements on potential links between correlation and coskewness. Specifically, we first show that, on the one hand, it is possible to observe any possible values of coskewness among symmetric random variables but zero pairwise correlations of these variables. On the other hand, it is also possible to have zero coskewness and any level of correlation. Second, we generalize this result to the case of arbitrary marginal distributions showing the absence of a general link between rank correlation and standardized rank coskewness.