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A linear model is often used to find the effect of a binary treatment [Formula: see text] on a noncontinuous outcome [Formula: see text] with covariates [Formula: see text]. Particularly, a binary [Formula: see text] gives the popular “linear probability model (LPM),” but the linear model is untenable if [Formula: see text] contains a continuous regressor. This raises the question: what kind of treatment effect does the ordinary least squares estimator (OLS) to LPM estimate? This article shows that the OLS estimates a weighted average of the [Formula: see text]-conditional heterogeneous effect plus a bias. Under the condition that [Formula: see text] is equal to the linear projection of [Formula: see text] on [Formula: see text], the bias becomes zero, and the OLS estimates the “overlap-weighted average” of the [Formula: see text]-conditional effect. Although the condition does not hold in general, specifying the [Formula: see text]-part of the LPM such that the [Formula: see text]-part predicts [Formula: see text] well, not [Formula: see text], minimizes the bias counter-intuitively. This article also shows how to estimate the overlap-weighted average without the condition by using the “propensity-score residual” [Formula: see text]. An empirical analysis demonstrates our points.

Regression discontinuity is popular in finding treatment/policy effects when the treatment is determined by a continuous variable crossing a cutoff. Typically, a local linear regression (LLR) estimator is used to find the effects. For binary response, however, LLR is not suitable in extrapolating the treatment, as in doubling/tripling the treatment dose/intensity. The reason is that doubling/tripling the LLR estimate can give a number out of the bound [Formula: see text], despite that the effect should be a change in probability. We propose local maximum likelihood estimators which overcome these shortcomings, while giving almost the same estimates as the LLR estimator does for the original treatment. A simulation study and an empirical analysis for effects of an income subsidy program on religion demonstrate these points.

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