We define a hierarchy of parametric and semiparametric specifications for censored regression models that is ordered according to the strength of the assumptions that are made. We review the estimation techniques and specification tests that are available at each level of the hierarchy. The estimation and testing procedures are illustrated by applying them to a model of strike duration. Several tests (some graphical) are able to detect errors in standard parametric specifications even in the presence of fairly heavy censoring, but two distinct semiparametric specifications leading to different substantive inferences cannot be rejected. In addition, the conditional mean as a prediction of the dependent variable is highly sensitive to specification errors, whereas the conditional median is not.