Tobit model

Tobit model

The Tobit Model is an econometric, biometric model proposed by James Tobin (1958) to describethe relationship between a non-negative dependent variabley_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) eta 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.

y_i = egin{cases} y_i^* & extrm{if} ; y_i^* >0 \ 0 & extrm{if} ; y_i^* leq 0end{cases}

where y_i^* is a latent variable:

y_i^* = eta x_i + u_i, u_i sim N(0,sigma^2)

If the relationship parameter eta 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 = egin{cases} y_i^* & extrm{if} ; y_i^* >y_L \ 0 & extrm{if} ; y_i^* leq y_L.end{cases}

Another example is censoring of values above y_U.

y_i = egin{cases} y_i^* & extrm{if} ; y_i^*

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

y_i = egin{cases} y_i^* & extrm{if} ; y_L

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.


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