Asymptotic Bias and Optimal Convergence Rates for Semiparametric Kernel Estimators in the Regression Discontinuity Model

The regression discontinuity model has recently become a commonly applied framework for empirical work in economics. Hahn, Todd, and Van der Klaauw (2001) provide a formal development of the identification of a treatment effect in this framework and also note the potential bias problems in its estimation. This bias difficulty is the result of a particular feature of the regression discontinuity treatment effect estimation problem that distinguishes it from typical semiparametric estimation problems where smoothness is lacking. Here, the discontinuity is not simply an obstacle to overcome in estimation; instead, the size of discontinuity is itself the object of estimation interest. In this paper, I derive the optimal rate of convergence for estimation of the regression discontinuity treatment effect. The optimal rate suggests that the appropriate choice of estimator the bias difficulties are no worse than would be found in the usual nonparametric conditional mean estimation problem (at an interior point of the covariate support). Two estimators are proposed that attain the optimal rate under varying conditions. One estimator is based on Robinson's (1988) partially linear estimator. The other estimator uses local polynomial estimation and is optimal under a broader set of conditions.