Two-Way Exclusion Restrictions in Models with Heterogeneous Treatment Effects

In this paper, we propose a novel method to identify the conditional average treatment effect partial derivative (CATE-PD) in an environment in which the treatment is endogenous, the treatment effect is heterogeneous, the candidate 'instrumental variables' can be correlated with latent errors, and the treatment selection does not need to be (weakly) monotone. We show that CATE-PD is point-identified under mild conditions if two-way exclusion restrictions exist: (a) an outcome-exclusive variable, which affects the treatment but is excluded from the potential outcome equation, and (b) a treatment-exclusive variable, which affects the potential outcome but is excluded from the selection equation. We also propose an asymptotically normal two-step estimator and illustrate our method by investigating how the return to education varies across regions at different levels of development in China.