Tobit model
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The Tobit Model is an econometric model proposed by James Tobin (1958) to describe the relationship between a dependent variable [y_i] which cannot take on values smaller than zero and an independent variable (or vector) [x_i].
The model supposes that there is a latent 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 else.
[ y_i = \begin y_i^* & \textrm \; y_i^* >0 \\ 0 & \textrm \; y_i^* \leq 0\end]
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 [\beta] is estimated by regressing the observed [ y_i ] on [ x_i ], the resulting ordinary least squares estimator is inconsistent. 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 truncated regression model, because the latent variable [y_i^*] cannot always be observed. Other examples are the Tobit type II-IV model.
Bibliography
- Amemiya, Takeshi (1973). "Regression analysis when the dependent variable is truncated normal". Econometrica 41 (6), 997–1016.
- Tobin, James (1958). "Estimation for relationships with limited dependent variables". Econometrica 26 (1), 24–36.
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