Research 

This paper develops a control function estimator for a binary response model that includes a continuous endogenous regressor in the absence of exclusion restrictions. The proposed parametric and semiparametric approaches utilize heteroscedasticity to construct a control variable to address the endogeneity issue. In an application using the Health and Retirement Study data, the semiparametric estimation method is employed to estimate the effect of education on health and health behaviors without introducing an instrumental variable for the endogenous variable for education. This paper establishes the large sample properties of the proposed estimator. In Monte-Carlo simulations, it performs quite well in finite samples.