A Simple Adjustment for Bandwidth Snooping

TL;DR

This paper introduces a simple adjustment method for confidence intervals in kernel-based estimators used in regression discontinuity designs, correcting for bandwidth snooping bias and ensuring proper coverage.

Contribution

It proposes a critical value adjustment based on kernel and bandwidth ratio, providing a practical solution for valid inference after bandwidth selection.

Findings

Conventional 95% CIs often under-cover, with actual coverage between 70-90%.

The adjustment increases the critical value from 1.96 to between 2.2 and 2.8.

The method applies to various settings involving trimmed data and overlap.

Abstract

Kernel-based estimators such as local polynomial estimators in regression discontinuity designs are often evaluated at multiple bandwidths as a form of sensitivity analysis. However, if in the reported results, a researcher selects the bandwidth based on this analysis, the associated confidence intervals may not have correct coverage, even if the estimator is unbiased. This paper proposes a simple adjustment that gives correct coverage in such situations: replace the normal quantile with a critical value that depends only on the kernel and ratio of the maximum and minimum bandwidths the researcher has entertained. We tabulate these critical values and quantify the loss in coverage for conventional confidence intervals. For a range of relevant cases, a conventional 95% confidence interval has coverage between 70% and 90%, and our adjustment amounts to replacing the conventional critical value 1.96 with a number between 2.2 and 2.8. Our results also apply to other settings involving trimmed data, such as trimming to ensure overlap in treatment effect estimation. We illustrate our approach with three empirical applications.