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Tobit models: A survey. (English) Zbl 0539.62121
Tobit models are probit models of regression used by J. Tobin in econometrics, where the range of the dependent variable is constrained in some way. A classification is made into five basic types of these models from a wide variety of applications. These five types depend on the form of the likelihood function as the criterion of classification.
For the basic Tobit model, the regression model is $$y^*_ i=x^ t_ i\beta +u_ i$$, $$i=1,2,...,n$$ and $$y_ i=y^*_ i$$ for $$y^*_ i>0$$ and $$y_ i=0$$ for $$y^*_ i\leq 0$$. Here $$\{u_ i\}$$ are i.i.d. drawings from a normal $$N(0,\sigma^ 2)$$, $$\{y_ i\}$$, $$\{x_ i\}$$ are observed for $$i=1,2,...,n$$ but $$\{y^*_ i\}$$ are unobserved if $$y^*_ i\leq 0.$$
The likelihood function L can be written as $$L=\prod_{0}F_ i(y_{0i})\prod_{1}f_ i(y_ i)$$, where $$F_ i$$ and $$f_ i$$ are distribution and density function, respectively, of $$y^*_ i$$, $$\prod_{0}$$ means the product over those i for which $$y^*_ i\leq y_{0i}$$ and $$\prod_{1}$$ means the product over those i for which $$y^*_ i>y_{0i}.$$
By using the notation $$P(y_ 1<0).P(y_ 1)$$ for $$\prod_{0}F_ i(y_{0i})\prod_{1}f_ i(y_ i)$$, the other four types of Tobit models may be denoted by $$P(y_ 1<0).P(y_ 1>0,y_ 2)$$; $$P(y_ 1<0).P(y_ 1,y_ 2)$$ and $$P(y_ 1<0,y_ 3).P(y_ 1,y_ 2)$$; $$P(y_ 1<0,y_ 3).P(y_ 1>0,y_ 2)$$. Five types of estimators: maximum likelihood, least squares and two-step least squares are surveyed.
Reviewer: J.K.Sengupta

##### MSC:
 62P20 Applications of statistics to economics 62J05 Linear regression; mixed models
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