Advanced Econometrics Takeshi Amemiya
Type 4 Tobit Model: P(y, < 0, y3) • P(y,, y2)
10.9.1 Definition and Estimation
The Type 4 Tobit model is defined as follows:
У и = x'uA + Uu
У*і = *2.7*2 + “2,
Уіі — хЗіРз + U3i Уи ~ У*і if Уи>0
= 0 if yf, ^ 0
Уи = Уіі if У и > 0
= 0 if yf, S0
Узі = У*і if Уи = 0
= 0 if yf,>0, /= 1,2,. . . ,/1,
where {uu, и2/. «зЛ are i. i.d. drawings from a trivariate normal distribution.
This model differs from Type 3 defined by (10.8.1) only by the addition of yf,, which is observed only if yf, £ 0. The estimation of this model is not significantly different from that of Type 3. The likelihood function can be written as
(10.9.2)
where f3 ( • , •) is the joint density of yf, and Тз< and^( •, •) is the joint density of у * and y*. Heckman’s two-step method for this model is similar to the method for the preceding model. However, we must deal with three conditional expectation equations in the present model. The equation for y3l - will be slightly different from the other two because the variable is nonzero when yf, is nonpositive. We obtain
Е(УзіУи = 0) = ХзіРз ~
We shall discuss three examples of the Type 4 Tobit model in the following subsections: the model of Kenny et al. (1979); the model of Nelson and Olson (1978); and the model of Tomes (1981). In the first two models the y* equations are written as simultaneous equations, like Heckman’s model (1974), for which the reduced-form equations take the form of (10.9.1). Tomes’ model has a slight twist. The estimation of the structural parameters of such models can be handled in much the same way as the estimation of Heckman’s model (1974), that is, by either Heckman’s simultaneous equations two-step method (and its Lee-Maddala-Trost extension) or by Amemiya’s LS and GLS, both of which were discussed in Section 10.8.
In fact, these two estimation methods can easily accommodate the following very general simultaneous equations Tobit model:
Г'у? = B'x, + u„ / = 1, 2,. . . , n, (10.9.4)
where the elements of the vector yf contain the following three classes of variables: (1) always completely observable, (2) sometimes completely observable and sometimes observed to lie in intervals, and (3) always observed to lie in intervals. Note that the variable classified as Cin Table 10.3 belongs to class 2 and the variable classified as В belongs to class 3. The models of Heckman (1974), Kenny et al. (1979), and Nelson and Olson (1978), as well as a few more models discussed under Type 5, such as that of Heckman (1978), are all special cases of the model (10.9.4).
