Instrumental Variables with Unrestricted Heterogeneity and Continuous Treatment

This article discusses identification in continuous triangular systems without restrictions on heterogeneity or functional form. 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. Under these conditions alone, and given rich enough support of the data, potential outcome distributions, the average structural function, and quantile structural functions are point identified. 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 multi-dimensional, additional exclusion restrictions yield identification. The set-up discussed in this article 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 systems of structural equations.