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Bayesian Endogenous Tobit Quantile Regression

This study proposes $p$-th Tobit quantile regression models with endogenous variables. In the first stage regression of the endogenous variable on the exogenous variables, the assumption that the $α$-th quantile of the error term is zero is introduced. Then, the residual of this regression model is included in the $p$-th quantile regression model in such a way that the $p$-th conditional quantile of the new error term is zero. The error distribution of the first stage regression is modelled around the zero $α$-th quantile assumption by using parametric and semiparametric approaches. Since the value of $α$ is a priori unknown, it is treated as an additional parameter and is estimated from the data. The proposed models are then demonstrated by using simulated data and real data on the labour supply of married women.