Sociological Methods &Research, Ahead of Print.
A linear model is often used to find the effect of a binary treatment [math] on a noncontinuous outcome [math] with covariates [math]. Particularly, a binary [math] gives the popular “linear probability model (LPM),” but the linear model is untenable if [math] 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 [math]-conditional heterogeneous effect plus a bias. Under the condition that [math] is equal to the linear projection of [math] on [math], the bias becomes zero, and the OLS estimates the “overlap-weighted average” of the [math]-conditional effect. Although the condition does not hold in general, specifying the [math]-part of the LPM such that the [math]-part predicts [math] well, not [math], 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” [math]. An empirical analysis demonstrates our points.