Correcting for truncation bias caused by a latent truncation variable

We discuss estimation of the model Y_i=X_ib_Y+e_Yi, T_i=X_ib_T+ e_Ti, when data on the continuous dependent variable Y and on the independent variables X are observed iff the ‘truncation variable’ T>0 and when T is latent. This case is distinct from both (i) the‘censored sample’ case, in which Y data are available iff T>0, T is latent and X data are available for all observations, and (ii) the ‘observed truncation variable’ case, in which both Y and X are observed iff T>0 and in which the actual value of T is observed whenever T>0. We derive a maximum-likelihood procedure for estimating this model and discuss identification and estimation.