Identification of Nonseparable Models Using Instruments With Small Support

I consider nonparametric identification of nonseparable instrumental variables models with continuous endogenous variables. If both the outcome and first stage equations are strictly increasing in a scalar unobservable, then many kinds of continuous, discrete, and even binary instruments can be used to point‐identify the levels of the outcome equation. This contrasts sharply with related work by Imbens and Newey, 2009 that requires continuous instruments with large support. One implication is that assumptions about the dimension of heterogeneity can provide nonparametric point‐identification of the distribution of treatment response for a continuous treatment in a randomized controlled experiment with partial compliance.