Tobit model

The Tobit Model is an econometric, biometric model proposed by James Tobin (1958) to describethe 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 dependson $x\_i$ via a parameter (vector) which determines therelationship 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 distributederror term $u\_i$ to capture random influences on thisrelationship.The observable variable $y\_i$ is defined to be equal to the latent variable wheneverthe latent variable is above zero and zero otherwise.

where $y\_i^*$ is a latent variable:

If the relationship parameter 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:

Another example is censoring of values above $y\_U$.

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

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.

Bibliography

* Amemiya, Takeshi (1973). "Regression analysis when the dependent variable is truncated normal". "Econometrica" 41 (6), 997–1016.

* Amemiya, Takeshi (1984). "Tobit models: A survey". "Journal of Econometrics" 24 (1-2), 3-61.

* Amemiya, Takeshi (1985). "Advanced Econometrics". Basil Blackwell. Oxford.

* Schnedler, Wendelin (2005). "Likelihood estimation for censored random vectors". "Econometric Reviews" 24 (2),195–217.

* Tobin, James (1958). "Estimation for relationships with limited dependent variables". "Econometrica" 26 (1), 24–36.