A. Colin Cameron 761marginal effect is twice as large. Quite flexible modeling of heterogeneity in E[yi|xi]
and the associated marginal effects is possible using nonparametric regression of
yonx. This yields very noisy estimates for high dimensionx, leading to the use
instead of semiparametric methods such as those given in section 14.3.6.
More challenging is controlling for unobserved heterogeneity that is due to fac-
tors other than the regressors. Different individuals then have different responses
even if the individuals have the same value ofx. Failure to control for this unob-
served heterogeneity can lead to inconsistent parameter estimates and associated
marginal effects. A simple example is omitted variables bias in the linear regres-
sion model, where the omitted variables form part of the unobserved heterogeneity.
The source of the unobserved heterogeneity can also matter. In particular, in struc-
tural models of economic behavior a distinction is made as to whether or not the
unobserved (to the econometrician) heterogeneity is known to the decision maker.
Meaningful discussion of unobserved heterogeneity requires the statement of
an underlying structural relationship to be estimated in the presence of unob-
served heterogeneity. Wooldridge (2005, 2008) provides a fairly general framework.
Suppose that interest lies in a conditional meanm(x,u) =E[y|x,u], or more
formally:
m(x,u)=E[Y|X=x,U=u],
whereuis unobserved and for simplicity is a scalar. Ideallym(x,u)would be esti-
mated but, instead, analysis is restricted to what Blundell and Powell (2004) call
the average structural function (ASF):
m(x)=EU[m(x,U)],which integrates out the unobserved heterogeneity. Often, interest lies in how ASF
changes as thejth regressor, say, changes. This is the average partial effect (APE):
∂m(x)
∂xj=EU[
∂m(x,U)
∂xj]
.Unobserved heterogeneity poses a problem because the ASFm(x)in general differs
from the conditional mean E[y|x]=EU|x[m(x,U)], and hence APE differs from
∂E[y|x]/∂xj, but it is only E[y|x]that is identified from the observed data.
The simplest assumption, and one commonly made, is thatuis independent
ofx, as then E[y|x]=m(x). In a model with additive heterogeneity, analysis is
particularly straightforward. Ifm(x,u)=g(x,β)+uthen E[y|x]=g(x,β)givenu
independent ofxwith mean zero. Analysis is more complicated if unobserved het-
erogeneity enters in a nonlinear manner. For example, ifm(x,u)=g(x′β+u)then
E[y|x]=Eu[g(x′β+u)]will typically require specification of the distribution ofu
and integration over this. In some cases analytical expressions can be obtained. In
other cases numerical methods are used. Ifuis low dimensional (in many appli-
cations it is a scalar) then Gaussian quadrature methods work well. Otherwise the
simulation methods given in sections 14.3.3 and 14.3.4 can be used. Examples
include negative binomial models for counts obtained by a Poisson–gamma mix-
ture, Weibull–gamma mixtures for durations, random utility models for binary and