endogenous treatment, complier effect, instrument score, propensity score, single index model, instrumental variable estimator, 62g05, 62p10, 62p25, Mathematics, QA1-939, Probabilities. Mathematical statistics, QA273-280

When a binary treatment DD is possibly endogenous, a binary instrument δ\delta is often used to identify the “effect on compliers.” If covariates XX affect both DD and an outcome YY, XX should be controlled to identify the “XX-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)E\left(\delta | X) for non-randomized δ\delta . 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 DD or δ\delta , even though no parametric assumption is imposed. The models hold for any form of YY (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.