Abstract:

This paper discusses identification in continuous triangular systems 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 and consider the case of real-valued treatment. We show that under these conditions 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 and quantile structural functions. 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. If treatment is multidimensional, additional exclusion restrictions allow to achieve identification. The setup discussed in this paper covers important cases not covered by existing approaches such as conditional moment restrictions (cf. Newey and Powell 2003) and control variables (cf. Imbens and Newey 2009). It covers, in particular, random coefficient models, as well as models arising as the reduced form of a system of structural equations.

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Original version: January 2012