Jumpy or Kinky? Regression Discontinuity without the Discontinuity

Regression Discontinuity (RD) models identify local treatment effects by associating a discrete change in an outcome with a corresponding discrete change in the probability of treatment at a known threshold of a running variable. This paper shows that it is possible to identify RD model treatment effects without a discontinuity. The intuition is that identification can come from a slope change (a kink) instead of a discrete level change (a jump) in the treatment probability. Formally this can be shown using L'hopital's rule. I also interpret the identification results intuitively using instrumental variable models. Estimators are proposed that can be applied in the presence or absence of a discontinuity, by exploiting either a jump or a kink.