Direction Estimation in a General Regression Model with Discrete Predictors

Consider a general regression model, where the response Y depends on discrete predictors X only through the index βX. It is well-known that the ordinary least squares (OLS) estimator can recover the underlying direction β exactly if the link function between Y and X is linear. Li and Duan (1989) showed that the OLS estimator can recover β proportionally if the predictors satisfy the linear conditional mean (LCM) condition. For discrete predictors, we demonstrate that the LCM condition generally does not hold. To improve the OLS estimator in the presence of discrete predictors, we model the conditional mean E(X | βX) as a polynomial function of βX and use the central solution space (CSS) estimator. The superior performances of the CSS estimators are confirmed through numerical studies.