Regression Discontinuity Design under Self-selection

TL;DR

This paper addresses the challenge of self-selection in Regression Discontinuity designs by proposing weighted estimands and estimators that recover the direct causal effect, with theoretical validation and empirical applications.

Contribution

It introduces a class of weighted average treatment effect estimands and estimators that account for self-selection, extending RD analysis to more realistic scenarios.

Findings

Estimators are consistent and asymptotically normal.

Performance comparison shows improvements over standard RD estimators.

Applications to real datasets demonstrate practical utility.

Abstract

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.