Advanced Econometrics Takeshi Amemiya
Other Applications of the Type 2 Tobit Model
Nelson (1977) noted that a Type 2 Tobit model arises if y0 in (10.2.1) is assumed to be a random variable with its mean equal to a linear combination of independent variables. He reestimated Gronau’s model by the MLE.
In the study of Westin and Gillen (1978), yrepresents the parking cost with x2 including zonal dummies, wage rate (as a proxy for value of walking time), and the square of wage rate. A researcher observes y* = y2 if y2 < C where C represents transit cost, which itself is a function of independent variables plus an error term.
10.8 Type 3 Tobit Model: P(y, < 0) • Р(ул, y2)
10.8.1 Definition and Estimation
The Type 3 Tobit model is defined as follows:
ytt-*ufii + uu 00.8.1)
Уіі = + M2<
Уи = У и if У* > 0 = 0 if у S 0
Уи = Узі if У*>0
= 0 if yf, S0, і" — 1,2,. . . , и, where {м1(, m2i) are i. i.d. drawings from a bivariate normal distribution with
zero mean, variances a and a, and covariance <t12. Note that this model differs from Type 2 only in that in this model y* is also observed when it is positive.
Because the estimation of this model can be handled in a manner similar to the handling of Type 2, we shall discuss it only briefly. Instead, in the following we shall give a detailed discussion of the estimation of Heckman’s model (1974), which constitutes the structural equations version of the model (10.8.1).
The likelihood function of the model (10.8.1) can be written as
£ = П Wf = 0) П Ллі, Уи). (10.8.2)
о і
where/( •, •) is the joint density of у * and у J-. Because у * is observed when it is positive, all the parameters of the model are identifiable, including a.
Heckman’s two-step estimator was originally proposed by Heckman (1976a) for this model. Here we shall obtain two conditional expectation equations,(10.4.11)and(10.7.8),foryi and.^«respectively. [Add subscript 1 to all the variables and the parameters in (10.4.11) to conform to the notation of this section.] In the first step of the method, a, is estimated by the
probit MLE a,. In the second step, least squares is applied separately to
(10.4.11) and (10.7.8) after replacingat by a,. The asymptotic variance-covariance matrix of the resulting estimates of (/?,, cr,) is given in (10.4.22) and that for (/?2, <7l2oTl) can be similarly obtained. The latter is given by Heckman (1979). A consistent estimate of a2 can be obtained using the residuals of Eq. (10.7.8). As Heckman (1976a) suggested and as was noted in Section
10.4.3, a more efficient WLS can be used for each equation in the second step of the method. An even more efficient GLS can be applied simultaneously to the two equations. However, even GLS is not fully efficient compared to MLE, and the added computational burden of MLE may be sufficiently compensated for by the gain in efficiency. A two-step method based on unconditional means of yx and y2, which is a generalization of the method discussed in Section 10.4.3, can also be used for this model.
Wales and Woodland (1980) compared the LS estimator, Heckman’s two - step estimator, probit MLE, conditional MLE (using only those who worked), MLE, and another inconsistent estimator in a Type 3 Tobit model in a simulation study with one replication (sample size 1000 and 5000). The particular model they used is the labor supply model of Heckman (1974), which will be discussed in the next subsection.15 The LS estimator was found to be poor, and all three ML estimators were found to perform well. Heckman’s two-step estimator was ranked somewhere between LS and MLE.
