h ¼ hsðwÞ ¼ hdðwÞ
The variables in zð1Þ shift the labor supply curve, and zð2Þ contains labor demand
shifters. By defining y1 ¼ h and y2 ¼ logðwÞ we can write the equations in equilib-
rium as a linear simultaneous equations model:
y1 ¼ g1y2 þ zð1Þdð1Þ þ u1 ð9:4Þ
y1 ¼ g2y2 þ zð2Þdð2Þ þ u2 ð9:5Þ
Nothing about the general system (9.1) rules out having the same variable on the left-
hand side of more than one equation.
What is needed to identify the parameters in, say, the supply curve? Intuitively,
since we observe only the equilibrium quantities of hours and wages, we cannot dis-
tinguish the supply function from the demand function if zð1Þ and zð2Þ contain exactly
the same elements. If, however, zð2Þ contains an element not in zð1Þ—that is, if there is
some factor that exogenously shifts the demand curve but not the supply curve—then
we can hope to estimate the parameters of the supply curve. To identify the demand
curve, we need at least one element in zð1Þ that is not also in zð2Þ.
To formally study identification, assume that g10 g2; this assumption just means
that the supply and demand curves have di¤erent slopes. Subtracting equation (9.5)
from equation (9.4), dividing by g2  g1, and rearranging gives
y2 ¼ zð1Þp21 þ zð2Þp22 þ v2 ð9:6Þ
where p211 dð1Þ=ðg2  g1Þ, p22 ¼ dð2Þ=ðg2  g1Þ, and v21 ðu1  u2Þ=ðg2  g1Þ. This
is the reduced form for y2 because it expresses y2 as a linear function of all of the
exogenous variables and an error v2 which, by assumption (9.2), is orthogonal to all
exogenous variables: Eðz 0v2Þ ¼ 0. Importantly, the reduced form for y2 is obtained
from the two structural equations (9.4) and (9.5).
Given equation (9.4) and the reduced form (9.6), we can now use the identification
condition from Chapter 5 for a linear model with a single right-hand-side endogenous
variable. This condition is easy to state: the reduced form for y2 must contain at least
one exogenous variable not also in equation (9.4). This means there must be at least
one element of zð2Þ not in zð1Þ with coe‰cient in equation (9.6) di¤erent from zero.
Now we use the structural equations. Because p22 is proportional to dð2Þ, the condi-
tion is easily restated in terms of the structural parameters: in equation (9.5) at least
one element of zð2Þ not in zð1Þ must have nonzero coe‰cient. In the supply and de-
mand example, identification of the supply function requires at least one exogenous
variable appearing in the demand function that does not also appear in the supply
function; this conclusion corresponds exactly with our earlier intuition.
Simultaneous Equations Models 213

15.7 Specification Issues in Binary Response Models
We now turn to several issues that can arise in applying binary response models to
economic data. All of these topics are relevant for general index models, but features
of the normal distribution allow us to obtain concrete results in the context of probit
models. Therefore, our primary focus is on probit models.
15.7.1 Neglected Heterogeneity
We begin by studying the consequences of omitting variables when those omitted
variables are independent of the included explanatory variables. This is also called the
neglected heterogeneity problem. The (structural) model of interest is
Pðy ¼ 1 j x; cÞ ¼ Fðxb þ gcÞ ð15:34Þ
where x is 1  K with x11 1 and c is a scalar. We are interested in the partial e¤ects
of the xj on the probability of success, holding c (and the other elements of x) fixed.
We can write equation (15.34) in latent variable form as y ¼ xb þ gc þ e, where
y ¼ 1½y > 0 and e j x; c@Normalð0; 1Þ. Because x1 ¼ 1, EðcÞ ¼ 0 without loss of
generality.
Now suppose that c is independent of x and c@Normalð0; t2Þ. [Remember, this
assumption is much stronger than Covðx; cÞ ¼ 0 or even Eðc j xÞ ¼ 0: under indepen-
dence, the distribution of c given x does not depend on x.] Given these assumptions,
the composite term, gc þ e, is independent of x and has a Normalð0; g2t2 þ 1Þ dis-
tribution. Therefore,
Pðy ¼ 1 j xÞ ¼ Pðgc þ e >
xb j xÞ ¼ Fðxb=sÞ ð15:35Þ
where s21 g2t2 þ 1. It follows immediately from equation (15.35) that probit of y
on x consistently estimates b=s. In other words, if b^ is the estimator from a probit of
y on x, then plim b^j ¼ bj=s. Because s ¼ ðg2t2 þ 1Þ
1=2 > 1 (unless g ¼ 0 or t2 ¼ 0Þ,
jbj=sj < jbjj.
The attenuation bias in estimating bj in the presence of neglected heterogeneity has
prompted statements of the following kind: ‘‘In probit analysis, neglected heteroge-
neity is a much more serious problem than in linear models because, even if the
omitted heterogeneity is independent of x, the probit coe‰cients are inconsistent.’’
We just derived that probit of y on x consistently estimates b=s rather than b, so
the statement is technically correct. However, we should remember that, in nonlinear
models, we usually want to estimate partial e¤ects and not just parameters. For the
purposes of obtaining the directions of the e¤ects or the relative e¤ects of the ex-
planatory variables, estimating b=s is just as good as estimating b.
Chapter 15470