## Abstract

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 language | English |
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Journal | Sociological Methods and Research |

DOIs | |

Publication status | Accepted/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.

## Keywords

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

## ASJC Scopus subject areas

- Social Sciences (miscellaneous)
- Sociology and Political Science