  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.
