A discontinuity test for identification in triangular nonseparable models

This paper presents a test for the validity of control variable approaches to identification in triangular nonseparable models. Assumptions commonly imposed to justify such methods include full independence of instruments and disturbances and existence of a reduced form that is strictly monotonic in a scalar disturbance. We show that if the data has a particular structure, namely that the distribution of the endogenous variable has a mass point at the lower (or upper) boundary of its support, validity of the control variable approach implies a continuity condition on an identified function, which can be tested empirically.