# Tobit model

The Tobit model is a statistical model proposed by James Tobin (1958)[1] to describe the relationship between a non-negative dependent variable $y_i$ and an independent variable (or vector) $x_i$. The term Tobit was derived from Tobin's name by truncating and adding -it by analogy with the probit model.[2]

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:

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

## Consistency

If the relationship parameter $\beta$ is estimated by regressing the observed $y_i$ on $x_i$, the resulting ordinary least squares regression estimator is inconsistent. It will yield a downwards-biased estimate of the slope coefficient and an upward-biased estimate of the intercept. Takeshi Amemiya (1973) has proven that the maximum likelihood estimator suggested by Tobin for this model is consistent.

## Interpretation

The $\beta$ coefficient should not be interpreted as the effect of $x_i$ on $y_i$, as one would with a linear regression model; this is a common error. Instead, it should be interpreted as the combination of (1) the change in $y_i$ of those above the limit, weighted by the probability of being above the limit; and (2) the change in the probability of being above the limit, weighted by the expected value of $y_i$ if above.[3]

## Variations of the Tobit model

Variations of the Tobit model can be produced by changing where and when censoring occurs. Amemiya (1985, p. 384) classifies these variations into five categories (Tobit type I - Tobit type V), where Tobit type I stands for the first model described above. Schnedler (2005) provides a general formula to obtain consistent likelihood estimators for these and other variations of the Tobit model.

### Type I

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:

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

Another example is censoring of values above $y_U$.

$y_i = \begin{cases} y_i^* & \textrm{if} \; y_i^*

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

The rest of the models will be presented as being bounded from below at 0, though this can be generalized as we have done for Type I.

### Type II

Type II Tobit models introduce a second latent variable.

$y_{2i} = \begin{cases} y_{2i}^* & \textrm{if} \; y_{1i}^* >0 \\ 0 & \textrm{if} \; y_{1i}^* \leq 0. \end{cases}$

Heckman (1987) falls into the Type II Tobit. In Type I Tobit, the latent variable absorb both the process of participation and 'outcome' of interest. Type II Tobit allows the process of participation/selection and the process of 'outcome' to be independent, conditional on x.

### Type III

Type III introduces a second observed dependent variable.

$y_{1i} = \begin{cases} y_{1i}^* & \textrm{if} \; y_{1i}^* >0 \\ 0 & \textrm{if} \; y_{1i}^* \leq 0. \end{cases}$
$y_{2i} = \begin{cases} y_{2i}^* & \textrm{if} \; y_{1i}^* >0 \\ 0 & \textrm{if} \; y_{1i}^* \leq 0. \end{cases}$

The Heckman model falls into this type.

### Type IV

Type IV introduces a third observed dependent variable and a third latent variable.

$y_{1i} = \begin{cases} y_{1i}^* & \textrm{if} \; y_{1i}^* >0 \\ 0 & \textrm{if} \; y_{1i}^* \leq 0. \end{cases}$
$y_{2i} = \begin{cases} y_{2i}^* & \textrm{if} \; y_{1i}^* >0 \\ 0 & \textrm{if} \; y_{1i}^* \leq 0. \end{cases}$
$y_{3i} = \begin{cases} y_{3i}^* & \textrm{if} \; y_{1i}^* >0 \\ 0 & \textrm{if} \; y_{1i}^* \leq 0. \end{cases}$

### Type V

Similar to Type II, in Type V we only observe the sign of $y_{1i}^*$.

$y_{2i} = \begin{cases} y_{2i}^* & \textrm{if} \; y_{1i}^* >0 \\ 0 & \textrm{if} \; y_{1i}^* \leq 0. \end{cases}$
$y_{3i} = \begin{cases} y_{3i}^* & \textrm{if} \; y_{1i}^* >0 \\ 0 & \textrm{if} \; y_{1i}^* \leq 0. \end{cases}$

## The likelihood function

Below are the likelihood and log likelihood functions for a type I Tobit. This is a Tobit that is censored from below at $y_L$ when the latent variable $y_j^* \leq y_L$. In writing out the likelihood function, we first define an indicator function $I(y_j)$ where:

$I(y_j) = \begin{cases} 0 & \textrm{if} \; y_j = y_L \\ 1 & \textrm{if} \; y_j \neq y_L. \end{cases}$

Next, we mean $\Phi$ to be the standard normal cumulative distribution function and $\phi$ to be the standard normal probability density function. For a data set with N observations the likelihood function for a type I Tobit is

$\mathcal{L}(\beta, \sigma) = \prod _{j=1}^N \left(\frac{1}{\sigma}\phi \left(\frac{y_j-X_j\beta }{\sigma }\right)\right)^{I\left(y_j\right)} \left(1-\Phi \left(\frac{X_j\beta-y_L}{\sigma}\right)\right)^{1-I\left(y_j\right)}$

and the log likelihood is given by

$\log \mathcal{L}(\beta, \sigma) = \sum^n_{j = 1} I(y_j) \log \left( \frac{1}{\sigma} \phi\left( \frac{y_j - X_j\beta}{\sigma} \right) \right) + (1 - I(y_j)) \log\left( 1- \Phi\left( \frac{X_j \beta - y_L}{\sigma} \right) \right)$