[gov3009-l] Applied Statistics Workshop: Maximilian Kasy on Wed., Oct. 3

without restrictions on heterogeneity or functional form. In particular, we do not assume separability of structural functions, restrictions on the dimensionality of unobservables, or monotonicity in unobservables. We do maintain monotonicity of the first stage relationship in the instrument. We show that under this condition alone, and given rich enough support of the data, we can achieve point identification of potential outcome distributions, and in particular of the average structural function. If the support of the continuous instrument is not large enough potential outcome distributions are partially identified. If the instrument is discrete identification fails completely. The setup discussed in this paper covers important cases not covered by existing approaches such as conditional moment restrictions (c.f. Newey and Powell, 2003) and control variables (c.f. Imbens and Newey, 2009). It covers, in particular, random coe fficient models, as well as models arising as the reduced form of a system of structural equations.