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

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.

Automated summary

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.