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

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.