Simple and Honest Confidence Intervals in Nonparametric Regression

TL;DR

This paper develops honest confidence intervals for nonparametric regression parameters that account for bias, leading to more efficient inference than traditional methods, with practical guidelines for bandwidth and critical value selection.

Contribution

It introduces a bias-aware approach for constructing honest CIs in nonparametric regression, deriving optimal bandwidths and critical values for improved efficiency.

Findings

Honest CIs outperform undersmoothing-based CIs in coverage and efficiency.

Optimal bandwidth minimizes maximum MSE and yields nearly efficient CIs.

Critical value for 95% CIs with $n^{-2/5}$ rate is 2.18, not 1.96.

Abstract

We consider the problem of constructing honest confidence intervals (CIs) for a scalar parameter of interest, such as the regression discontinuity parameter, in nonparametric regression based on kernel or local polynomial estimators. To ensure that our CIs are honest, we use critical values that take into account the possible bias of the estimator upon which the CIs are based. We show that this approach leads to CIs that are more efficient than conventional CIs that achieve coverage by undersmoothing or subtracting an estimate of the bias. We give sharp efficiency bounds of using different kernels, and derive the optimal bandwidth for constructing honest CIs. We show that using the bandwidth that minimizes the maximum mean-squared error results in CIs that are nearly efficient and that in this case, the critical value depends only on the rate of convergence. For the common case in which the rate of convergence is $n^{-2/5}$, the appropriate critical value for 95% CIs is 2.18, rather than the usual 1.96 critical value. We illustrate our results in a Monte Carlo analysis and an empirical application.