The Tobit model is a statistical model proposed by James Tobin to describe the relationship
between a non-negative dependent variable and an independent variable . The term Tobit
was derived from Tobin's name by truncating and adding -it by analogy with the probit
model. The model supposes that there is a latent
variable . This variable linearly depends on via a parameter which determines the relationship
between the independent variable and the latent variable . In addition, there is a normally
distributed error term to capture random influences on this relationship. The observable variable
is defined to be equal to the latent variable whenever the latent variable is above zero
and zero otherwise.
where is a latent variable:
Consistency If the relationship parameter is estimated
by regressing the observed on , 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 has proven that the maximum likelihood estimator
suggested by Tobin for this model is consistent. Interpretation
The coefficient should not be interpreted as the effect of on , as one would with a
linear regression model; this is a common error. Instead, it should be interpreted as
the combination of the change in of those above the limit, weighted by the probability
of being above the limit; and the change in the probability of being above the limit,
weighted by the expected value of if above. Variations of the Tobit model
Variations of the Tobit model can be produced by changing where and when censoring occurs.
Amemiya classifies these variations into five categories, where Tobit type I stands for
the first model described above. Schnedler 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 cannot always be observed while the independent
variable is observable. A common variation of the Tobit model is censoring at a value
different from zero:
Another example is censoring of values above .
Yet another model results when is censored from above and below at the same time.
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.
Heckman 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.
The Heckman model falls into this type. Type IV
Type IV introduces a third observed dependent variable and a third latent variable.
Type V Similar to Type II, in Type V we only observe
the sign of .
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 when the latent
variable . In writing out the likelihood function, we first define an indicator function where:
Next, we mean to be the standard normal cumulative distribution function and to be the standard
normal probability density function. For a data set with N observations the likelihood
function for a type I Tobit is
and the log likelihood is given by
See also Generalized Tobit
Limited dependent variable Truncated regression model
References
Further reading Amemiya, Takeshi. "Regression analysis when
the dependent variable is truncated normal". Econometrica 41: 997–1016. doi:10.2307/1914031.
JSTOR 1914031. Amemiya, Takeshi. "Tobit models: A survey".
Journal of Econometrics 24: 3–61. doi:10.1016/0304-4076(84)90074-5. Amemiya, Takeshi. Advanced Econometrics. Oxford:
Basil Blackwell. ISBN 0-631-13345-3. Schnedler, Wendelin. "Likelihood estimation
for censored random vectors". Econometric Reviews 24: 195–217. doi:10.1081/ETC-200067925.
Tobin, James. "Estimation of relationships for limited dependent variables". Econometrica
26: 24–36. doi:10.2307/1907382. JSTOR 1907382.