  Regression discontinuity (RD) designs are often interpreted as local
randomized experiments: a RD design can be considered as a randomized
experiment for units with a realized value of a so-called forcing variable
falling around a pre-fixed threshold. Motivated by the evaluation of Italian
university grants, we consider a fuzzy RD design where the receipt of the
treatment is based on both eligibility criteria and a voluntary application
status. Resting on the fact that grant application and grant receipt statuses
are post-assignment (post-eligibility) intermediate variables, we use the
principal stratification framework to define causal estimands within the Rubin
Causal Model. We propose a probabilistic formulation of the assignment
mechanism underlying RD designs, by re-formulating the Stable Unit Treatment
Value Assumption (SUTVA) and making an explicit local overlap assumption for a
subpopulation around the threshold. A local randomization assumption is invoked
instead of more standard continuity assumptions. We also develop a model-based
Bayesian approach to select the target subpopulation(s) with adjustment for
multiple comparisons, and to draw inference for the target causal estimands in
this framework. Applying the method to the data from two Italian universities,
we find evidence that university grants are effective in preventing students
from low-income families from dropping out of higher education.
