Semiparametric Efficiency of Residual Correlation Testing under Gaussian Additive Noise Models

2026-05-31

AI Summary

This work addresses conditional independence testing under the Gaussian additive noise model, where the key challenge lies in efficiently estimating the correlation of regression residuals. We propose a test based on the Pearson correlation coefficient computed from these residuals and establish, for the first time, that this estimator achieves semiparametric efficiency within this model, thereby attaining optimal statistical efficiency. An asymptotic inference framework is developed to support the proposed method, and both theoretical analysis and simulations demonstrate that it nearly reaches oracle efficiency—maintaining strict Type I error control while substantially improving statistical power. The method is successfully applied to uncover the conditional dependence structure among U.S. stock returns.

Abstract

This paper studies conditional independence testing under the Gaussian additive noise model (GANM), where two variables are modeled as nonlinear functions of covariates with independent bivariate Gaussian regression errors. Under this framework, conditional independence can be characterized by the correlation coefficient of the regression errors, which motivates a test based on the Pearson correlation coefficient computed from the fitted residuals. Despite its simple form, the asymptotic behavior and statistical efficiency of the resulting test have not been well understood. In this paper, we develop the semiparametric efficiency theory under GANM and show, surprisingly, that the efficient estimator coincides exactly with the ordinary residual Pearson correlation estimator. We further establish the asymptotic properties of the proposed test and develop the corresponding inference procedure. Simulation studies demonstrate that the proposed method achieves near-oracle efficiency and competitive empirical power while maintaining valid Type I error control. We further apply the proposed test to conditional dependence analysis of U.S. stock returns.

Problem

• conditional independence testing
• Gaussian additive noise model
• residual correlation
• semiparametric efficiency
• Pearson correlation

Innovation

• semiparametric efficiency
• conditional independence testing
• Gaussian additive noise model
• residual correlation
• Pearson correlation