The attractiveness of the Regression Discontinuity Design (RDD) in its sharp formulation rests on close similarities with a formal experimental design. On the other hand, it is of limited applicability since rarely individuals are assigned to the treatment group on the basis of a pre-program measure observable to the analyst. Besides, it only allows to identify the mean impact of the program for a very specific sub-population of individuals. In this paper we show that the sharp RDD straightforwardly generalizes to the instances in which the eligibility for the program is established with respect to an observable pre-program measure with eligible individuals self-selecting into the treatment group according to an unknown process. This set-up also turns out very convenient to define a specification test on conventional non-experimental estimators of the program effect needed to identify the mean impact away from the threshold for eligibility. Data requirements are made explicit.