Testing identifying assumptions in fuzzy regression discontinuity designs

We propose a new specification test for assessing the validity of fuzzy regression discontinuity designs (FRD-validity). We derive a new set of testable implications, characterized by a set of inequality restrictions on the joint distribution of observed outcomes and treatment status at the cut-off. We show that this new characterization exploits all of the information in the data that is useful for detecting violations of FRD-validity. Our approach differs from and complements existing approaches that test continuity of the distributions of running variables and baseline covariates at the cut-off in that we focus on the distribution of the observed outcome and treatment status. We show that the proposed test has appealing statistical properties. It controls size in a large sample setting uniformly over a large class of data generating processes, is consistent against all fixed alternatives, and has non-trivial power against some local alternatives. We apply our test to evaluate the validity of two FRD designs. The test does not reject FRD-validity in the class size design studied by Angrist and Lavy (1999) but rejects it in the insurance subsidy design for poor households in Colombia studied by Miller, Pinto, and Vera-Hernández (2013) for some outcome variables. Existing density continuity tests suggest the opposite in each of the two cases.