Linear Probability Model Revisited: Why It Works and How It Should Be Specified

Abstract

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.