The Locally Gaussian Partial Correlation

TL;DR

The paper introduces the local Gaussian partial correlation (LGPC), a new measure for conditional dependence that extends partial correlation to non-Gaussian populations and can detect local departures from independence.

Contribution

It proposes the LGPC, a novel local measure of conditional dependence that generalizes partial correlation and applies to a broad class of distributions.

Findings

LGPC reduces to partial correlation in Gaussian cases

LGPC distinguishes positive and negative dependence

It enables local tests for conditional independence and causality

Abstract

It is well known that the dependence structure for jointly Gaussian variables can be fully captured using correlations, and that the conditional dependence structure in the same way can be described using partial correlations. The partial orrelation does not, however, characterize conditional dependence in many non-Gaussian populations. This paper introduces the local Gaussian partial correlation (LGPC), a new measure of conditional dependence. It is a local version of the partial correlation coefficient that characterizes conditional dependence in a large class of populations. It has some useful and novel properties besides: The LGPC reduces to the ordinary partial correlation for jointly normal variables, and it distinguishes between positive and negative conditional dependence. Furthermore, the LGPC can be used to study departures from conditional independence in specific parts of the distribution. We provide several examples of this, both simulated and real, and derive estimation theory under a local likelihood framework. Finally, we indicate how the LGPC can be used to construct a powerful test for conditional independence, which, again, can be used to detect Granger causality in time series.