Exploiting nonlinear difference-in-difference assumptions in a regression discontinuity design

The paper tackles the problem of identification of treatment effects in a regression-discontinuity-design (RDD) in the presence of heterogeneous effects. A RDD allows identification of average treatment effects only for a subset of individuals around the threshold for the participation status. The paper shows how a sharp RDD may provide: a) additional ways to define specification tests for the continuity assumptions at the discontinuity point on which identification usually rests; b) additional ways to test the performance of alternative non-experimental estimators of programme effects away from the threshold; c) alternative identification assumptions, similar to those on which nonlinear difference-in-difference estimators rest, which can be partially tested, and allow to extend estimation results away from the threshold. The considered set-up is one where a budget-constraint induced threshold splits the relevant population into two groups, the ex-post eligible and ineligible individuals, and application in both groups is determined according to rules potentially unknown to the researcher, so application (or participation) is not mandatory but voluntary. The proposed tools are applied to the evaluation of Italian university grants. Applicants meeting some ex-ante eligibility criteria receive a grant if their family economic indicator S is below a threshold ˆs. Results show that, at the threshold, the grant is an effective tool to prevent students from low income families from dropping out of higher education. However, under some relatively weak nonlinear difference-in-difference type of assumptions, results show that moving below the threshold, thus for less well-off (poorer) students, the impact of the grant becomes smaller and not significant.