# Heckman’s Model

Heckman’s model (Heckman, 1974) differs from Gronau’s model (10.7.11) in that Heckman included the determination of hours worked (H) in his model.16 Like Gronau, Heckman assumes that the offered wage W° is given independently of Я; therefore Heckman’s W° equation is the same as Gronau’s:

W° = Х2І02 "I" u2i – (10.8.3)

Heckman defined WT = (dU/dC)/(dU/dX) and specified17

Wti = yHt + z’ta + vi. (10.8.4)

It is assumed that the fth individual works if

W]{H, = 0) = z’a + v,< W°t (10.8.5)

and then the wage Щ and hours worked H, are determined by solving (10.8.3) and (10.8.4) simultaneously after putting Wf= W = W,. Thus we can de­fine Heckman’s model as

Щ = хЬ02 + и» (10.8.6)

and

Wt = yH( + z-a + v( (10.8.7)

for those і for which desired hours of work

Hf = x’ufit + uu > 0, (10.8.8)

where x’ufii = f1 — 2,-a) and uu = y~l (u2i — vt). Note that (10.8.5) and

(10.8.8) are equivalent because у > 0.

Call (10.8.6) and (10.8.7) the structural equations; then (10.8.6) and the identity part of (10.8.8) constitute the reduced-form equations. The reduced – form equations of Heckman’s model can be shown to correspond to the Type 3 Tobit model (10.8.1) if we put H* = yf, H = yx, W° = yf, and W= y2.

We have already discussed the estimation of the reduced-form parameters in the context of the model (10.8.1), but we have not discussed the estimation of the structural parameters. Heckman (1974) estimated the structural param­eters by MLE. In the next two subsections we shall discuss three alternative methods of estimating the structural parameters.