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

Myoung-jae Lee, Goeun Lee and Jin-young Choi

Sociological Methods & Research, 2025, vol. 54, issue 1, 173-186

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.

Keywords: linear probability model; propensity-score residual; overlap weight (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:sae:somere:v:54:y:2025:i:1:p:173-186

DOI: 10.1177/00491241231176850

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