asymptotically normal under the random effects assumption. The conventional
Hausman test is also well defined. The Hausman test, which is based on the
difference between the GLS and within estimators, has significant local power
to detect violations of the random effects assumption (in particular, non-zero
correlations between the time-varying regressors and unobservable individual
effects), despite the fact that the two estimators are asymptotically identical
under a sequence of local alternative hypotheses.
In this paper, we have restricted our attention to the asymptotic properties
of the existing estimators and tests when panel data contain both large numbers
of cross section and time series observations. Apparently, thus, this paper does
not provide any new estimator or test. However, this paper makes several
contributions to the literature. First, our findings have pedagogical values for
future studies. For example, we find that asymptotics as (N,T →∞) are
much more sensitive to data generating processes than asymptotics as either
N →∞or T →∞are. However, previous studies have often assumed that
data are cross-sectionally i.i.d.. Our findings suggest that future studies should
pay more attention to cross-sectional heterogeneity. Second, we consider the
cases in which the time series of time-varying regressors are not ergodic due to
their correlations with time invariant regressors. For such cases, we have shown
that the limits of averages of panel data can be derived under the assumption of
conditional α-mixing. It would also be interesting to see how this conditional
α-mixing concept can be refined and generalized to other more sophisticated
panel data models. Finally, differently from many other previous studies, we
avoid making any particular restriction on the relative sizes of N and T .We
do so using a more rigorous joint limit instead of other simple sequential limit
methods. Thus we are confident that our theoretical results apply to a broader
range of panel data.
An obvious extension of our paper is the instrumental variables estimation
of Hausman and Taylor (1981), Amemiya and MaCurdy (1986), and Breusch,
Mizon and Schmidt (1989). For an intermediate model between fixed effects and
random effects, these studies propose several instrumental variables estimators
by which both the coefficients on time-varying and time invariant regressors can
be consistently estimated. It would be interesting to investigate the large N and
large T properties of these instrumental variables estimators and the Hausman
tests based on these estimators.
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