Palgrave Handbook of Econometrics: Applied Econometrics

William Greene 531

11.6.3 Panel data models

Hausman, Hall and Griliches (1984) (HHG) report the following conditional
density for the fixed effects negative binomial (FENB) model:

p

(

yi 1 ,yi 2 ,...,yiTi|

∑Ti

t= 1 yit

)

=

( 1 +tT=i 1 yit)(Tt=i 1 λit)

(

Ti

t= 1 yit+

Ti

t= 1 λit)

∏Ti

t= 1

(yit+λit)

( 1 +yit)(λit)

,

which is free of the fixed effects. This is the default FENB formulation used in popu-
lar software packages such as SAS, Stata and LIMDEP. Researchers accustomed to the
admonishments that fixed effects models cannot contain overall constants or time
invariant covariates are sometimes surprised to find (perhaps accidentally) that this
fixed effects model allows both. (This issue is explored at length in Allison, 2000;
Allison and Waterman, 2002.) The resolution of this apparent contradiction is that
the HHG FENB model is not obtained by shifting the conditional mean function
by the fixed effect, lnλit=x′itβ+αi, as it is in the Poisson model. Rather, the HHG
model is obtained by building the fixed effect into the model as an individual spe-
cificθiin the NB1 form. In the negative binomial models, the conditional mean
functions are:

NB1 :E[yit|xit]=θiφit=θiexp(x′itβ)=exp(x′itβ+lnθi),

NB2 :E[yit|xit]=exp(αi)φit=λit=exp(x′itβ+αi),

so, superficially, the formulations do produce the same interpretation. However,
the parameterθiin the NB1 model enters the variance function in a different
manner:

NB1 : Var[yit|xit]=θiφit[ 1 +θi],

NB2 : Var[yit|xit]=λit[ 1 +θλit].

The relationship between the mean and the variance is different for the two mod-
els. For estimation purposes, one can explain the apparent contradiction noted
earlier by observing that, in the NB1 formulation, the individual effect is iden-
tified separately from the mean in the skedastic (scaling) function. This is not
true for the FENB2 form. In order to obtain a counterpart to the HHG model, we
would replaceθwithθi(andλiwithλit). Greene (2007a) analyzes the more famil-
iar FENB2 form with the same treatment ofλit. Estimates for both models appear
below. Comparison of the suggested NB2 model to the HHG model remains for
future investigation.
Once again, theory does not provide a reason to prefer the NB1 formulation over
the more familiar NB2 model. The NB1 form does extend beyond the interpretation
of the fixed effect as carrying only the sum of all the time invariant effects in the
conditional mean function. The appearance of lnθiin the conditional mean is an
artifact of the exponential mean form;θiis a scaling parameter in this model. In
its favor, the HHG model, being conditionally independent of the fixed effects,