Instrumental variables for binary treatments with heterogeneous treatment effects: a simple exposition

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