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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.

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

### Keywords:

survey; censored regression; truncated regression; nonlinear estimation; Tobit models; probit models of regression; maximum likelihood; least squares; two-step least squares### References:

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