Optimal Bandwidth Choice for the Regression Discontinuity Estimator

Abstract

We investigate the choice of the bandwidth for the regression discontinuity estimator. We focus on estimation by local linear regression, which was shown to have attractive properties (<cross-ref type="bib" refid="bib23">Porter, J. 2003</cross-ref>, “Estimation in the Regression Discontinuity Model” (unpublished, Department of Economics, University of Wisconsin, Madison)). We derive the asymptotically optimal bandwidth under squared error loss. This optimal bandwidth depends on unknown functionals of the distribution of the data and we propose simple and consistent estimators for these functionals to obtain a fully data-driven bandwidth algorithm. We show that this bandwidth estimator is optimal according to the criterion of Li (1987, “Asymptotic Optimality for <it>C</it><inf><it>p</it></inf>, <it>C</it><inf><it>L</it></inf>, Cross-validation and Generalized Cross-validation: Discrete Index Set”, <it>Annals of Statistics</it>, <b>15</b>, 958–975), although it is not unique in the sense that alternative consistent estimators for the unknown functionals would lead to bandwidth estimators with the same optimality properties. We illustrate the proposed bandwidth, and the sensitivity to the choices made in our algorithm, by applying the methods to a data set previously analysed by Lee (2008, “Randomized Experiments from Non-random Selection in U.S. House Elections”, <it>Journal of Econometrics</it>, <b>142</b>, 675–697) as well as by conducting a small simulation study.