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.

翻译：本文研究高斯加性噪声模型（GANM）下的条件独立性检验问题，其中两个变量被建模为协变量的非线性函数，且回归误差服从独立二元高斯分布。在该框架下，条件独立性可通过回归误差的相关系数刻画，这启发了一种基于拟合残差计算Pearson相关系数的检验方法。尽管形式简洁，但该检验的渐近行为与统计效率尚未得到充分研究。本文建立了GANM下的半参数效率理论，并令人惊讶地发现：有效估计量恰好与普通残差Pearson相关估计量一致。我们进一步推导了所提检验的渐近性质，并开发了相应的推断流程。仿真研究表明，该方法在有效控制第一类错误的同时，实现了近乎最优的效率与有竞争力的经验功效。最后，我们将所提检验应用于美国股票收益的条件依赖性分析。