Advanced Econometrics Takeshi Amemiya

Models with Heterogeneity

We shall study the problem of specifying a panel data QR model by consider­ing a concrete example of sequential labor force participation by married women, following the analysis of Heckman and Willis (1977). As is customary with a study of labor force participation, it is postulated that a person works if the offered wage exceeds the reservation wage (shadow price of time). The offered wage depends on the person’s characteristics such as her education, training, intelligence, and motivation, as well as on local labor market condi­tions. The reservation wage depends on the person’s assets, her husband’s wage rate, prices of goods, interest rates, expected value of future wages and prices, and the number and age of her children. Let y„ = 1 if the ith person works at time t and =0 otherwise and let xu be a vector of the observed independent variables (a subset of the variables listed above). Then for fixed t it seems reasonable to specify a binary QR model

Р(.Уи = 1) = FWffiU і = 1, 2,. . . , n, (9.7.1)

where F is an appropriate distribution function.

However, with panel data, (9.7.1) is not enough because it specifies only a marginal probability for fixed t and leaves the joint probability P(yn, уa,. . . , yiT) unspecified. (We assume (y/t) are independent over /.) The simplest way to specify a joint probablity is to assume independence and

specify Р(Уц, Уа, ■ ■ , У гг) = П£.і-Р(уі(). Then we obtain the binary QR

model studied in Section 9.2, the only difference being that we have nT observations here.

The independence assumption would imply P(yu= 1|У/,»-і = f)= P(yit =1). In other words, once we observe xft, whether or not a woman worked in the previous year would offer no new information concerning her status this year. Surely, such a hypothesis could not be supported empirically.

There are two major reasons why we would expect P(yu = 1 yt,-i — 1) Ф Р(Уи= 1):

1. Heterogeneity. There are unobservable variables that affect people dif­ferently in regard to their tendency to work.23

2. True State Dependence. For each person, present status influences future status. For example, once you start doing something, it becomes habitual.

Heckman and Willis (1977) attempted to take account of only heterogeneity in their study. We shall consider how to model true state dependence in Section 9.7.2.

A straightforward approach to take account of heterogeneity is to specify Р(Уи = 1|M() = P«P + Щ), (9.7.2)

/=1,2,. . . ,n, /=1, 2,. . . , Г,

and assume {y(,} are serially independent (that is, over t) conditional on ut. Therefore we have (suppressing subscript i)

P(yt = 1 Уі— = 1) - P(y, = 1) (9.7.3)

E[F(x't/J + u)F(x't-J + и)]

--------- EF(x'.J+u)--------------- EF(X'0+u)

Cov [F(x',fi+u),F(x'^fi+u)]

EF(x^fi+u)

which can be shown to be nonnegative. The joint probability of {yit), t = 1, 2,. . . , T, is given by

Р(Уп, Уа,- ■ ■ >Уіт) (9.7.4)

= Ещ {ft 1 ~ F«P + И/)],_л} •

The likelihood function of the model is the product of (9.7.4) over all individ­uals i=l,2,. . . , n. It is assumed that (ut) are i. i.d. over individuals.

The expectation occurring in (9.7.4) is computationally cumbersome in practice, except for the case where F = Ф, the standard normal distribution function, and и, is also normal. In this case, Butler and Moffitt (1982) pointed out that using the method ofGaussian quadrature, an evaluation of(9.7.4) for a model with, say, n = 1500 and t = 10 is within the capacity of most re­searchers having access to a high-speed computer. We shall come back to a generalization of model (9.7.4) in Section 9.7.2. But here we shall discuss a computationally simpler model in which (9.7.4) is expressed as a product and ratio of gamma functions.

Heckman and Willis proposed what they called a beta-logistic model, de­fined as24

where Uj is distributed according to the beta density

/,(«,) = «Г1 (1 - K,)6'-1, (9.7.6)

Г(а,)Г(^)

Oiu^l, a,>0, bt> 0,

where Г(а) = JoXa~'e~x dx. It is assumed that (ytt) are serially independent conditional on ut. (Independence over individuals is always assumed.)

The beta density has mean (a + b)~la and variance a(a + b)~2(a + b + 1 )ab and can take various shapes: a bell shape if a, b> 1; monotonically increasing if a > 1 and b < 1; monotonically decreasing if a < 1 and b> 1; uniform if a = b = 1; and U shaped if a, b< 1.

We have (suppressing subscript i)

>Eu = P{yt=),

where the inequality follows from Vu > 0.

Heckman and Willis postulated a, = exp (x('a) and bt — exp (x-/l), where x,- is a vector of time-independent characteristics of the zth individual. Thus, from (9.7.5), we have

P(yit= 1) = Л[х'(а-/1)]. (9.7.8)

Equation (9.7.8) shows that if we consider only marginal probabilities we have a logit model; in this sense a beta-logistic model is a generalization of a logit
model. By maximizing П"_іЩ_іЛ[хї(а — P)] we can obtain a consistent esti­mate of a — fi. However, we can estimate each of о and P consistently and also more efficiently by maximizing the full likelihood function. If the ith person

worked st periods out of T, і = 1, 2........... n, the likelihood function of the

beta-logistic model is given by

L = fE[u{ (9.7.9)

" Г(ъ + Ь,) naj + SiWibi+T-Sj)

1Па,)ЦЬ,) ЦО' + Ь+Т) ■

An undesirable feature of the beta-logistic model (9.7.9) is that the indepen­dent variables x( are time independent. Thus, in their empirical study of 1583 women observed over a five-year period, Heckman and Willis were forced to use the values of the independent variables at the first period as x, even though some of the values changed with t.