Regression as Best Linear Prediction: The Case of Discrete Regressors

This paper examines the properties of the ordinary least squares (OLS) estimator when applied to a model with a non-linear relationship between outcome and a discrete regressor. I investigate what parameters OLS estimates in such a case, focusing on both level and incremental effects. The analysis reveals that the OLS estimand is a convex average of incremental effects, but weights can be negative for level effects and in the presence of neglected heterogeneity. An empirical application to a wage equation demonstrates these issues, highlighting the importance of using unrestricted models or carefully considering the limitations of OLS estimates in similar situations.