Standard Errors for Nonparametric Regression

This paper proposes five pointwise consistent and asymptotic normal estimators of the asymptotic variance function of the Nadaraya-Watson kernel estimator for nonparametric regression. The proposed estimators are constructed based on the first-stage nonparametric residuals, and their asymptotic properties are established under the assumption that the same bandwidth sequences are used throughout, which mimics what researchers do in practice while making derivations more complicated instead. A limited Monte Carlo experiment demonstrates that the proposed estimators possess smaller pointwise variability in small samples than the pair and wild bootstrap estimators which are commonly used in practice.

本文针对非参数回归（nonparametric regression）中的Nadaraya-Watson核估计器（Nadaraya-Watson kernel estimator）的渐近方差函数，提出了五种逐点相合且渐近正态的估计量。所提估计量基于第一阶段非参数残差（first-stage nonparametric residuals）构建，其渐近性质是在全程使用相同带宽序列（bandwidth sequences）的假设下推导得到的——该设置贴合研究者的实际操作习惯，但也使得推导过程更为复杂。一项有限的蒙特卡洛（Monte Carlo）试验结果表明，相较于实际中常用的配对自助法与野自助法估计量，本文所提估计量在小样本场景下具备更小的逐点变异性。