Censored and Truncated Regression Models
A censored regression model is appropriate when the dependent variable is unavailable is above or below some threshold.
A truncated regression model is appropriate when cases are systemically not collected/unreported when the dependent variable is above or below some threshold.
The Tobit model, named for Tobin (1958), is a special case of a censored regression model.
Description
This is a modification of the OLS model, where the dependent variable Y is related to the independent variable(s) X as Yi = bXi + Ui.
Univariate
Suppose that the variable of interest is unobserved if it is less than zero. The expected value is then expressed as E[Yi|Xi,Yi≥0]. Substituting Yi with the model equation yields E[bXi + Ui|Xi,bXi + Ui≥0], and because the expectation is conditioned on a given Xi this simplifies to bXi + E[Ui|Xi,bXi + Ui≥0]. Algebraically this is rewritten as:
where σ is the standard deviation of the error term Ui. The insertion of that standard deviation term transforms this into a formula that is easily decomposed into terms of the p.d.f. and c.d.f. of the standard normal distribution. Altogether, the expected value is:
The hazard ratio or inverse Mills' ratio (IMR) is notated as λ here. Sometimes λ evaluated for a given bXi/σ is notated as λi.
Provided that the sample is censored (i.e., not truncated), it should be possible to estimate λi using a probit model. This reveals that selection bias seen in the initial model can be treated as omitted variable bias, and can be corrected by using the model Yi = bXi + σλi + Vi.
Bivariate
Suppose the variable of interest is unobserved if a second variable is less than zero, and suppose that these are drawn from a joint normal distribution. In other words, the model is specified as:
Y1i = bXi + U1i
Y2i = γZi + U2i
Xi and Zi can be the same, but often the system is only solvable when Zi has more predictors.
Following the same procedures above, it can be demonstrated that:
