On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Inference

Nonparametric methods play a central role in empirical work. While they provide inference procedures that are robust to parametric misspecification bias, they may be sensitive to tuning parameter choices. We study the effects of bias correction on confidence interval coverage in the context of kernel density and local polynomial regression estimation, and prove that bias correction can be preferred to undersmoothing for minimizing coverage error. This result is achieved using a novel, yet simple, Studentization, which leads to a new way of constructing kernel-based bias-corrected confidence intervals. For local polynomial regression, we show that, as with point estimation, coverage error adapts to boundary points automatically. We derive coverage error optimal bandwidths and discuss easy-to-implement bandwidth selectors. In particular, we show that the MSE-optimal bandwidth delivers the fastest coverage error decay rate only at interior points when second-order kernels are employed, but is otherwise suboptimal at interior and boundary points. All results are established using valid Edgeworth expansions and illustrated with simulated data. Our findings have important consequences for empirical work as they indicate that bias-corrected confidence intervals, coupled with appropriate standard errors, have smaller coverage errors and are less sensitive to tuning parameter choices. To illustrate the applicability of our results, we study inference in regression discontinuity designs, where we establish the same coverage error and robustness improvements for bias-corrected confidence intervals and give a rule-of-thumb bandwidth choice for implementation via correcting the MSE-optimal choice. For example, for the local-linear estimator and a sample size of $n=500$, shrinking the MSE-optimal bandwidth by $27\%$ yields bias-corrected confidence intervals with the best coverage error decay rate.