Advanced Econometrics Takeshi Amemiya
Gronau’s Model
Gronau (1973) assumed that the offered wage W° is given to each housewife independently of hours worked (H), rather than as a schedule W°(H). Given W°, a housewife maximizes her utility function U(C, X) subject to X = W°H + V and C + H = T, where C is time spent at home for childcare, X represents all other goods, T is total available time, and V is other income. Thus a housewife does not work if
(10.7.10)
 |
and works if the inequality in (10.7.10) is reversed. If she works, the hours of work H and the actual wage rate W must be such that
Gronau called the left-hand side of (10.7.10) the housewife’s value of time or, more commonly, the reservation wage, denoted W1.12
Assuming that both W° and Wx can be written as linear combinations of independent variables plus error terms, his model may be statistically described as follows:
Щ = ХгА + «2/
W] — z|a + Vj
W, = Wf if Wf > W]
= 0 if WJ, /- 1,2,... , л,
where (u2i, vt) are i. i.d. drawings from a bivariate normal distribution with zero mean, variances and a, and covariance am. Thus the model can be written in the form of( 10.7.1) by putting WJ — W] = yftand FFj^yJ-.Note that H (hours worked) is not explained by this statistical model although it is
Tobit Models 3,89*
•t' »
determined by Gronau’s theoretical model. A statistical model explaining as well as W was developed by Heckman (1974) and will be discussed,'^ Section 10.8.2.
Because the model (10.7.11) can be transformed into the form (10.7.1) ш» such a way that the parameters of (10.7.11) can be determined from the parameters of (10.7.1), all the parameters of the model are identifiable except V( Wf — Wf), which can be set equal to 1 without loss of generality. If, however, at least one element of x2l is not included in zt, all the parameters are identifiable.13 They can be estimated by the MLE or Heckman’s two-step estimator by procedures described in Section 10.7.1. We can also use the probit MLE (the first step of Heckman’s two-step) to estimate a certain subset of the parameters.14
