Multivariate Generalizations
By a multivariate generalization of Type 5, we mean a model in which y* and у fj in (10.10.1) are vectors, whereas y* is a scalar variable the sign of which is observed as before. Therefore the FairJafFee model with likelihood function characterized by (10.10.14) is an example of this type of model.
In Lee’s model (1977) the y*( equation is split into two equations and
T% = zb<*2 + v2, (10.10.16)
where CJ, and T*t denote the cost and the time incurred by the rth person traveling by a private mode of transportation, and, similarly, the cost and the time of traveling by a public mode are specified as
Lee assumed that C* and Т% are observed if the ith person uses a private mode and C% and T% are observed if he or she uses a public mode. A private mode is used if y*i > 0, where y* is given by
y*u = s’A + S2n + S3n + S4(Ct – C£) + є,. (10.10.19)
Lee estimated his model by the following sequential procedure:
Step 1. Apply the probit MLE to (10.10.19) after replacing the starred variables with their respective righthand sides.
Step 2. Apply LS to each of the four equations (10.10.15) through (10.10.18) after adding to the righthand side of each the estimated hazard from step 1.
Step 3. Predict the dependent variables of the four equations (10.10.15) through (10.10.18), using the estimates obtained in step 2; insert the predictors into (10.10.19) and apply the probit MLE again.
Step 4. Calculate the MLE by iteration, starting from the estimates obtained at the end of the step 3.
Willis and Rosen (1979) studied earnings differentials between those who went to college and those who did not, using a more elaborate model than that of Kenny et al. (1979), which was discussed in Section 10.9.2. In the model of Kenny et al., y* (the desired years of college education, the sign of which determines whether an individual attends college) is specified not to depend directly on yl and y*t (the earnings of the collegegoer and the noncollege – goer, respectively). The first inclination of a researcher might be to hypothesize у * = уі — y3i. However, this would be an oversimplification because the decision to go to college should depend on the difference in expected lifetime earnings rather than in current earnings.
Willis and Rosen solved this problem by developing a theory of the maximization of discounted, expected lifetime earnings, which led to the following model:




Ri = s’iy + e„ і =1,2,…, и
where і*і and G2i denote the initial earnings (in logarithm) and the growth rate of earnings for the collegegoer, /* and Gl denote the same for the noncollegegoer, and Rt denotes the discount rate. It is assumed that the ith person goes to college if yl > 0 where
У и ~ 12i И + <5o "I" ^iG*i + S2G + S3Ri
and that the variables with subscript 2 are observed if y* > 0, those with subscript 3 are observed if у ft ^ 0, and Rt is never observed. Thus the model is formally identical to Lee’s model (1977). Willis and Rosen used an estimation method identical to that of Lee, given earlier in this subsection.
Boijas and Rosen (1980) used the same model as Willis and Rosen to study the earnings differential between those who changed jobs and those who did not within a certain period of observation.
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