  In Regression Discontinuity (RD) design, self-selection leads to different
distributions of covariates on two sides of the policy intervention, which
essentially violates the continuity of potential outcome assumption. The
standard RD estimand becomes difficult to interpret due to the existence of
some indirect effect, i.e. the effect due to self selection. We show that the
direct causal effect of interest can still be recovered under a class of
estimands. Specifically, we consider a class of weighted average treatment
effects tailored for potentially different target populations. We show that a
special case of our estimands can recover the average treatment effect under
the conditional independence assumption per Angrist and Rokkanen (2015), and
another example is the estimand recently proposed in Fr\"olich and Huber
(2018). We propose a set of estimators through a weighted local linear
regression framework and prove the consistency and asymptotic normality of the
estimators. Our approach can be further extended to the fuzzy RD case. In
simulation exercises, we compare the performance of our estimator with the
standard RD estimator. Finally, we apply our method to two empirical data sets:
the U.S. House elections data in Lee (2008) and a novel data set from Microsoft
Bing on Generalized Second Price (GSP) auction.
