A. Colin Cameron 759To relate this to the treatment effects literature,xitmay include a binary treat-
mentditthat is correlated with the error termαi+εit(selection on unobservables),
but only with the componentαiof the error term that is time invariant. For exam-
ple, an individual may self-select into a training program due to unobserved high
ability, but this high ability is assumed to be time invariant.
Pooled OLS regression ofyitonxitwill lead to inconsistent estimation ofβ, due
to correlation of regressors with the error. The random effects estimator ofβ, the
feasible GLS estimator of (14.48) under the assumption that bothαiandεitare
i.i.d., is also inconsistent if, in fact,αiis correlated withxit. For this reason many
microeconometric studies shy away from random effects models that are widely
used in other fields.
Estimation of transformed models that eliminateαican lead to consistent esti-
mation. The fixed-effects or within estimator is obtained by OLS estimation of the
within-model:
(yit−y ̄i)=(xit−x ̄i)′β+(uit−u ̄i). (14.49)A standard procedure is to use cluster-robust standard errors from this regression,
assumingTis small andN→∞. Hansen (2007a) presents asymptotic theory
that additionally allowsT→∞and Hansen (2007b) considers more efficient GLS
estimation. The first differences estimator is obtained by OLS estimation of the first
differences model:
yit=x′itβ+uit. (14.50)Note that in both cases only the coefficients of time-varying regressors can be
identified.
Extension to nonlinear models is possible only for some specific models, as there
is an incidental parameters problem. The asymptotics rely onN→∞, so the
number of parameters (kregression coefficients plusNfixed effectsαi) is going
to infinity with the sample size. Some models permit transformations that elimi-
nateαi, while others do not. For nonlinear models with additive error the within
and first differences transformations can again be used. For binary outcomes fixed
effects estimation is possible for the logit model (see Chamberlain, 1980), but not
the probit model. For count data, Hausman, Hall and Griliches (1984) presented
fixed effects estimation for the Poisson model and a particular parameterization of
the negative binomial model. The Poisson fixed effects estimator does not require
that the data be Poisson distributed, as it is consistent provided the conditional
mean is correctly specified. An active area of research is developing methods for
general nonlinear fixed effects panel models that, while inconsistent due to the
incidental parameters problem, are less inconsistent than existing methods (see,
for example, Arellano and Hahn, 2007).
Panel data also provide the opportunity to model individual-level dynamic
behavior, since the individual is observed at more than one point in time. A simple
dynamic linear fixed effects model includes a lagged dependent variable, so that:
yit=ρyi,t− 1 +x′itβ+αi+εit,i=1,...,N,t=1,...,T. (14.51)