A control function approach to estimating switching regression models with endogenous explanatory variables and endogenous switching

We derive simple, multi-step estimation methods for a linear model with heterogeneous coefficients when there are both continuous and discrete endogenous explanatory variables. We consider both cross-sectional and panel data settings. When we extend our model to panel data, we use the Chamberlain-Mundlak device to allow heterogeneity to be correlated with time-varying explanatory variables. We apply the panel data methods we propose to estimation of a housing budget share equation where a homeownership dummy variable plays the role of the endogenous regime, and total expenditure plays the role of a continuous endogenous explanatory variable. We find that the constant coefficient model seems sufficient, and that the estimation methods we propose produce rather plausible estimates of the model parameters.