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 $D$ 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.