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 D on a noncontinuous outcome Y with covariates X . Particularly, a binary Y gives the popular “linear probability model (LPM),†but the linear model is untenable if X 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 X -conditional heterogeneous effect plus a bias. Under the condition that E ( D | X ) is equal to the linear projection of D on X , the bias becomes zero, and the OLS estimates the “overlap-weighted average†of the X -conditional effect. Although the condition does not hold in general, specifying the X -part of the LPM such that the X -part predicts D well, not Y , 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†D − E ( D | X ) . An empirical analysis demonstrates our points.