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.