Tobit model

The Tobit Model is an econometric, biometric model proposed by James Tobin (1958) to describe the relationship between a non-negative dependent variable $$y_i$$ and an independent variable (or vector) $$x_i$$.

The model supposes that there is a latent (i.e. unobservable) variable $$y_i^*$$. This variable linearly depends on $$x_i$$ via a parameter (vector) $$\beta$$ which determines the relationship between the independent variable (or vector) $$x_i$$ and the latent variable $$y_i^*$$ (just as in a linear model). In addition, there is a normally distributed error term $$u_i$$ to capture random influences on this relationship. The observable variable $$y_i$$ is defined to be equal to the latent variable whenever the latent variable is above zero and zero otherwise.

$$ y_i = \begin{cases} y_i^* & \textrm{if} \; y_i^* >0 \\ 0    & \textrm{if} \; y_i^* \leq 0 \end{cases}$$

where $$y_i^*$$ is a latent variable:

$$ y_i^* = \beta x_i + u_i, u_i \sim N(0,\sigma^2) $$

If the relationship parameter $$\beta$$ is estimated by regressing the observed $$ y_i $$ on $$ x_i $$, the resulting ordinary least squares estimator is inconsistent. Takeshi Amemiya (1973) has proven that the likelihood estimator suggested by Tobin for this model is consistent.

The Tobit model is a special case of a censored regression model, because the latent variable $$y_i^*$$ cannot always be observed while the independent variable $$ x_i $$ is observable. A common variation of the Tobit model is censoring at a value $$ y_L$$ different from zero:

$$ y_i = \begin{cases} y_i^* & \textrm{if} \; y_i^* >y_L \\ 0    & \textrm{if} \; y_i^* \leq y_L. \end{cases}$$

Another example is censoring of values above $$ y_U$$.

$$ y_i = \begin{cases} y_i^* & \textrm{if} \; y_i^* <y_U \\ 0    & \textrm{if} \; y_i^* \geq y_U. \end{cases}$$

Yet another model results when $$ y_i $$ is censored from above and below at the same time.

$$ y_i = \begin{cases} y_i^* & \textrm{if} \; y_L<y_i^* <y_U \\ 0    & \textrm{if} \; y_i^* \leq y_L \text{ or } y_i^* \geq y_U. \end{cases}$$

Such generalizations are typically also called Tobit model. Depending on where and when censoring occurs, other variations of the Tobit model can be obtained. Amemiya (1985) classifies these variations into five categories (Tobit type I - Tobit type V), where Tobit type I stands for the model described above. Schnedler (2005) provides a general formula to obtain consistent likelihood estimators for these and other variations of the Tobit model.