Score manipulation, density continuity and intent-to-treat effect for regression discontinuity

Regression discontinuity (RD) is widely used in many disciplines of science to find treatment effect when the treatment is determined by an underlying running variable (‘score') (Formula presented.) crossing a cutoff (Formula presented.) or not. The main attraction of RD is local randomization around (Formula presented.), which is, however, often ruined by manipulation on (Formula presented.). To detect manipulation, the continuity of score density function (Formula presented.) at (Formula presented.) is routinely tested in practice. In this paper, we examine how informative (Formula presented.) is for RD, and show the following. First, for incumbency effect in election to which RD has been heavily applied, (Formula presented.) may have no information content. Second, for RD in general, the (Formula presented.) continuity is neither necessary nor sufficient for RD validity. Third, if the treatment cannot be implemented without manipulation of (Formula presented.), then the manipulation had better be considered as part of the treatment effect, much as in ‘intent-to-treat effect’ for clinical trials. These findings call for relying less on (Formula presented.) continuity tests and, instead, thinking more about how subjects react to the treatment to modify their (Formula presented.), how to design the treatment to lessen manipulation, and what to take as the desired treatment effect.