Minimally capturing heterogeneous complier effect of endogenous treatment for any outcome variable

When a binary treatment D is possibly endogenous, a binary instrument δ is often used to identify the "effect on compliers."If covariates X affect both D and an outcome Y, X should be controlled to identify the "X-conditional complier effect."However, its nonparametric estimation leads to the well-known dimension problem. To avoid this problem while capturing the effect heterogeneity, we identify the complier effect heterogeneous with respect to only the one-dimensional "instrument score"E (δ|X) for non-randomized δ. This effect heterogeneity is minimal, in the sense that any other "balancing score"is finer than the instrument score. We establish two critical "reduced-form models"that are linear in D D or δ, even though no parametric assumption is imposed. The models hold for any form of Y (continuous, binary, count, ...). The desired effect is then estimated using either single index model estimators or an instrumental variable estimator after applying a power approximation to the effect. Simulation and empirical studies are performed to illustrate the proposed approaches.