Tobit regression is used to estimate a linear regression model when the dependent variable is censored, i.e. when it is only observed over an interval of its support.
Tobit regression deals with censored dependent variables for which standard regression models would yield inconsistent estimates. Examples for such cases are expenditure on automobiles as many individuals will have 0 expenditure for a given year (left censoring) or confidentiality issues as in the Current Population Survey where high incomes are top-coded (right censoring). Left and right censoring may occur at the same time.
For $d=1$ non-censored and $d=0$ censored observations with the censoring point $\gamma$, Tobit regressions estimate $$\begin{equation} f(y_i) = \left[ \frac{1}{\sqrt{2\pi \sigma^2}} exp \left(-\frac{1}{2\sigma^2}(y_i-x'_i\beta)^2\right) \right]^{d_i} \left[\Phi \left(\frac{\gamma - x'_i\beta}{\sigma}\right) \right]^{1-d_i} \end{equation}$$ via maximum likelihood (which requires that the errors are $\epsilon_i \sim N(0,\sigma^2)$). Under less strict assumptions it is also possible to use a two-step estimator in which the first step uses a Probit model to predict the censored outcomes (see Cameron and Trivedi (2009) for details).
Tobit regression can deal with unknown constant censoring points (see Carson and Sun (2007)) but becomes problematic with unknown changing censoring points.
