  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 $\alpha$-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 $\alpha$-th quantile
assumption by using parametric and semiparametric approaches. Since the value
of $\alpha$ 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.
