Model-Free Local Partial Correlation

Abstract

In the simple linear regression context, partial correlation measures the linear association between two variables, with the linear effects of a third control variable removed. In this paper, we investigate the local partial correlation using a kernel smoothing approach. We show that for trivariate data with kernel weights assigned by the third control variable, the kernel-weighted product–moment correlation measures the strength of local linearity with the nonparametric effects of the third variable removed, is a nonparametric analogue of the local partial correlation and has connections to varying coefficient models. In addition, in a model-free setting, the local partial correlation is asymptotically the conditional correlation, conditioned on the third variable. The asymptotic properties are derived under three scenarios, namely the model-free setting, nonparametric regression models, and varying coefficient models, and the optimal orders of the bandwidth are provided. Simulated examples confirm the asymptotic theory. An application to mortality data demonstrates the advantages of using the local partial correlation as a unitless measure of local association.