This paper analyzes the linear regression model with a conditional median assumption med () = 0, where is a vector of exogenous instrument random variables. We study inference on the parameter when is censored and is endogenous. We treat the censored model as a model with interval observation on an outcome, thus obtaining an model with inequality restrictions on conditional median regressions. We analyze the identified features of the model and provide conditions for point identification of the parameter . We use a minimum distance estimator to consistently estimate the identified features of the model. We show that under point identification conditions and additional regularity conditions, the estimator based on inequality restrictions is normal and we derive its asymptotic variance. One can use our setup to treat the identification and estimation of endogenous linear median regression models with no censoring. A Monte Carlo analysis illustrates our estimator in the censored and the uncensored case.