  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.
