Model of Lee
In the model of Lee (1978), y2i represents the logarithm of the wage rate of the rth worker in case he or she joins the union and y3i represents the same in case he or she does not join the union. Whether or not the worker joins the union is determined by the sign of the variable
Уи = УІіУІ + *іа + і>і
Because we observe only y2i if the worker joins the union and y3i if the worker does not, the logarithm of the observed wage, denoted yt, is defined by
Уі = Уи if Тн>0
= y’ti if j/J) s 0.
Lee assumed that x2 and x3 (the independent variables in the y* and y3 equations) include the individual characteristics of firms and workers such as regional location, city size, education, experience, race, sex, and health, whereas z includes certain other individual characteristics and variables that represent the monetary and nonmonetary costs of becoming a union member. Because yf is unobserved except for the sign, the variance ofyfcan be assumed to be unity without loss of generality.
Lee estimated his model by Heckman’s twostep method applied separately to the у * and y3 equations. In Lee’s model simultaneity exists only in the y* equation and hence is ignored in the application of Heckman’s twostep method. Amemiya’s LS or GLS, which accounts for the simultaneity, will, of course, work for this model as well, and the latter will yield more efficient estimates—although, of course, not as fully efficient as the MLE.
10.10.2 Type 5 Model of Heckman
The Type 5 model of Heckman (1978) is a simultaneous equations model consisting of two equations
У и = УіУіі + x’uPi + <*1 wi + «II 
(10.10.5) 
and 

Ун = УгУи + *2 іРг + S2wt + u2h 
(10.10.6) 
where we observe y2i, X!,, x2i, and wt defined by 

о A * II 
(10.10.7) 
= 0 if yl S 0.
There are no empirical results in the 1978 article, but the same model was estimated by Heckman (1976b); in this application у & represents the average income of black people in the ith state, y* the unobservable sentiment toward blacks in the ith state, and wt = 1 if an antidiscrimination law is instituted in the ith state.
When we solve (10.10.5) and (10.10.6) for yft, the solution should not depend upon wh for that would clearly lead to logical inconsistencies. Therefore we must assume
У& + 6 , = 0 (10.10.8)
for Heckman’s model to be logically consistent. Using the constraint (10.10.8), we can write the reducedform equations (although strictly speaking not reducedform because of the presence of w, ) of the model as
yt> = X/X + vu (10.10.9)
and
y2i = S2Wi – І – х’іЩ + v2i, (10.10.10)
where we can assume Vvu = 1 without loss of generality. Thus Heckman’s model is a special case of Type 5 with just a constant shift between у J and y* (that is, y* = X(7T2 + v2i and y* = S2 + x’tn2 + v2i). Moreover, if S2 = 0, it is a special case of Type 5 where y*= y*.
Let us compare Heckman’s reducedform model defined by (10.10.9) and
(10.10.10) with Lee’s model. Equation (10.10.9) is essentially the same as (10.10.3) of Lee’s model. Equation (10.10.4) of Lee’s model can be rewritten as
Уі = Щ(*гА + M2i) + (1 – щ№зіР3 + u3l) (10.10.11)
= x’3(03 + u3i + Wiix’vfa + u2i – x’3i03 – u3i).
By comparing (10.10.10) and (10.10.11), we readily see that Heckman’s reducedform model is a special case of Lee’s model in which the coefficient multiplied by w( is a constant.
Heckman proposed a sequential method of estimation for the structural parameters, which can be regarded as an extension of Heckman’s simultaneous equations twostep estimation discussed in Section 10.8.3. His method consists of the following steps:
Step 1. Estimate by applying the probit MLE to (10.10.9). Denote the estimator щ and define Pt = F{x’i‘kl).
Step 2. Insert (10.10.9) into (10.10.6), replace Яі with nv and wt with Ft, and then estimate y2, 03, and S2 by least squares applied to (10.10.6).
Step 3. Solve (10.10.5) for y2i, eliminate y* by (10.10.9), and then apply least squares to the resulting equation after replacing Jt, by я, and wt by Ft to estimate УГ1, 77lPi, and y7l<5t.
Amemiya (1978c) derived the asymptotic variancecovariance matrix of Heckman’s estimator defined in the preceding paragraph and showed that Amemiya’s GLS (defined in Section 10.8.4) applied to the model yields an asymptotically more efficient estimator in the special case of S1 = S2 — 0. As pointed out by Lee (1981), however, Amemiya’s GLS can also be applied to the model with nonzero <J’s as follows:
Step 1. Estimate л, by the probit MLE я, applied to (10.10.9).
Step 2. Estimate S2 and n2 by applying the instrumental variables method to (10.10.10), using Ft as the instrument for wt. Denote these estimators as S2 and n2.
Step 3. Derive the estimates of the structural parameters y,, 0и*і, Уг,02, and S2 from щ, п2, and S2, using the relationship between the reducedform parameters and the structural parameters as well as the constraint (10.10.8) in the manner described in Section 10.8.4.
The resulting estimator can be shown to be asymptotically more efficient than Heckman’s estimator.
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