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

Myoung Jae Lee, Goeun Lee, Jin Young Choi

Research output: Contribution to journalArticlepeer-review


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

Original languageEnglish
JournalSociological Methods and Research
Publication statusAccepted/In press - 2023

Bibliographical note

Funding Information:
Jin-young Choi's research was supported by the Hankuk University of Foreign Studies Research Fund of 2023. Myoung-jae Lee’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2020R1A2C1A01007786).

Publisher Copyright:
© The Author(s) 2023.


  • linear probability model
  • overlap weight
  • propensity-score residual

ASJC Scopus subject areas

  • Social Sciences (miscellaneous)
  • Sociology and Political Science


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