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

When a binary treatment D is possibly endogenous, a binary instrument d 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(d|X) for non-randomized d. 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 or d, 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.