Generalized Tobit Models
As stated in Section 10.1, we can classify Tobit models into five common types according to similarities in the likelihood function. Type 1 is the standard Tobit model, which we have discussed in the preceding sections. In the following sections we shall define and discuss the remaining four types of Tobit models.
It is useful to characterize the likelihood function of each type of model schematically as in Table 10.2, where each yjf j = 1, 2, and 3, is assumed to be distributed as N(Xjfij, aj), and P denotes a probability or a density or a combination thereof. We are to take the product of each P over the observations that belong to a particular category determined by the sign of yx. Thus, in Type 1 (standard Tobit model), P(yx < 0) • P(yx) is an abbreviated notation for П0Р(у* < 0) • nju(yu), where fu is the density of N(x’u0x, <r?). This
Table 10.2 Likelihood functions of the five types of Tobit models
expression can be rewritten as (10.2.5) after dropping the unnecessary subscript 1.
Another way to characterize the five types is by the classification of the three dependent variables that appear in Table 10.3. In each type of model, the sign of yx determines one of the two possible categories for the observations, and a censored variable is observed in one category and unobserved in the other. Note that when yx is labeled C, it plays two roles: the role of the variable the sign of which determines categories and the role of a censored variable.
We allow for the possibility that there are constraints among the parameters of the model (ftj, aj),j= 1, 2, or 3. For example, constraints will occur if the original model is specified as a simultaneous equations model in terms of yx, y2, and y3. Then the fi’s denote the reduced-form parameters.
We shall not discuss models in which there is more than one binary variable and, hence, models the likelihood function of which consists of more than two components. Such models are computationally more burdensome because they involve double or higher-order integration of joint normal densities. The only exception occurs in Section 10.10.6, which includes models that are
Table 10.3 Characterization of the five types of Tobit models Dependent variables
Type yx y2 y3
1 C — —
2 В C —
3 C C —
4 С С C
5 В С C
Note: C = censored; В = binary.
obvious generalizations of the Type 5 Tobit model. Neither shall we discuss a simultaneous equations Tobit model of Amemiya (1974b). The simplest two – equation case of this model is defined by yu = тах(у, у2, + xufii + Mu,0)and y2l = max (у2Уи + x2iP2 + w2,, 0)> where (uu, u2i) are bivariate normal and у, y2 < 1 must be assumed for the model to be logically consistent.10 A schematic representation of the likelihood function of this two-equation model is
Р(Уі, Уі)‘ Р(Уі < 0> Уз) • Р(Уі < 0, у4) • Р(Уз < 0. Уа < 0) with y’s appropriately defined.