Journal
SOCIOLOGICAL METHODS & RESEARCH
Volume -, Issue -, Pages -Publisher
SAGE PUBLICATIONS INC
DOI: 10.1177/00491241231176850
Keywords
linear probability model; propensity-score residual; overlap weight
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This article investigates the use of linear models to analyze noncontinuous outcomes resulting from binary treatments. The results show that the OLS estimator provides a weighted average of the X-conditional effects, but with some bias.
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.
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