This note provides a simple exposition of what IV can and cannot estimate in a model with binary treatment variable and heterogenous treatment effects. It shows how linear IV is essentially a misspecification of functional form and the reason why linear IV estimates will generally depend on the instrument used is because of this misspecification. It shows that if one can estimate the correct functional form then the treatment effects are independent of the instruments used. However, the data may not be rich enough in practice to be able to identify these treatments effects without strong distributional assumptions. In this case, one will have to settle for estimations of treatment effects that are instrument-dependent.
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