Dependence in elliptical partial correlation graphs

Abstract

The Gaussian model equips strong properties that facilitate studying and interpreting graphical models. Specifically it reduces conditional independence and the study of positive association to determining partial correlations and their signs. When Gaussianity does not hold partial correlation graphs are a useful relaxation of graphical models, but it is not clear what information they contain (besides the obvious lack of linear association). We study elliptical and transelliptical distributions as middle-ground between the Gaussian and other families that are more flexible but either do not embed strong properties or do not lead to simple interpretation. We characterize the meaning of zero partial correlations in the elliptical family and transelliptical copula models and show that it retains much of the dependence structure from the Gaussian case. Regarding positive dependence, we prove impossibility results to learn (trans)elliptical graphical models, including that an elliptical distribution that is multivariate totally positive of order two for all dimensions must be essentially Gaussian. We then show how to interpret positive partial correlations as a relaxation, and obtain important properties related to faithfulness and Simpson's paradox. We illustrate the transelliptical model potential to study tail dependence in S&P500 data, and of positivity to improve regularized inference.