Tobit model

From Wikipedia, the free encyclopedia

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

The model supposes that there is a latent (i.e. unobservable) variable y_i^*. This variable linearly depends on xi via a parameter (vector) β which determines the relationship between the independent variable (or vector) xi and the latent variable y_i^* (just as in a linear model). In addition, there is a normally distributed error term ui to capture random influences on this relationship. The observable variable yi 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 β is estimated by regressing the observed yi on xi, 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 xi is observable. A common variation of the Tobit model is censoring at a value yL 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 yU.

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 yi 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.

[edit] 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.
Languages