Large-N and Large-T Properties of Panel Data Estimators and the Hausman Test

Several remarks follow. First, all of the asymptotic results given in Theorems
5-8 except for Theorem 7(b) hold as (N, T →∞), without any particular re-
striction on the convergence rates of N and T . The relative size of N and T does
not matter for the results, so long as both N and T are large. Second, one can
easily check that the convergence rates of the panel data estimates of individual
β coefficients reported in Theorems 5-8 are consistent with those from Section
2.2. Third, Theorem 6 shows that under Assumption 9 (random effects), the
between estimator of γ, b_b, is ———consistent regardless of the characteristics
of time-varying regressors. Fourth, both the between estimators of β and γ
are asymptotically biased under the sequence of local alternatives (Assumption
11). Fifth, as Theorem 7(a) indicates, the within and GLS estimators of β are
asymptotically equivalent not only under the random effects assumption, but
also under the local alternatives. Furthermore, the GLS estimator of β is asymp-
totically unbiased under the local alternatives, while the between estimator of β
is not. The asymptotic equivalence between the within and GLS estimation un-
der the random effects assumption is nothing new. Previous studies have shown
this equivalence based on a naive sequential limit method (T → ∞ followed
by — →∞) and some strong assumptions such as fixed regressors. Theorem
7(a) and (b) confirm the same equivalence result, but with more rigorous joint
limit approach as (—, T →∞) simultaneously. It is also intriguing to see that
the GLS and within estimators are equivalent even under the local alternative
hypotheses.

Sixth, somewhat surprisingly, as Theorem 7(b) indicates, even under the
fixed effects assumption (Assumption 10), the GLS estimator of β could be
asymptotically unbiased (and consistent) and equivalent to the within counter-
part, (i) if the size (—) of the cross section units does not dominate excessively
the size (T) of time series in the limit (—/T → c<∞), and (ii) if the model does
not contain trended or cross-sectionally heterogenous time-varying regressors.
This result indicates that when the two conditions are satisfied, the biases in
GLS caused by fixed effects are generally much smaller than those in between.
If at least one of these two conditions is violated, that is, if —/T →∞, or if
the other types of regressors are included, the limit of (β_g — β_w) is determined
by how fast —/T →∞and how fast the trends in the regressors increase or
decrease.¹⁹

Finally, Theorem 8(a) indicates that under the local alternative hypothe-
ses, the GLS estimator γ_g is ———consistent and asymptotically normal, but
asymptotically biased. The limiting distribution of γ_g, in this case, is equivalent
to the limiting distribution of the OLS estimator of γ in the panel model with
the known coefficients of the time-varying regressors xit (OLS on yeit — β⁰xeit =
γ⁰ei + (ui + v⅛)). Clearly, the GLS estimator γ_g is asymptotically more effi-
cient than the between estimator γ^ On the other hand, under the fixed effect
assumption, unlike the GLS estimator of β, b_g, the GLS estimator ^_g is not
consistent as (—, T → ∞). The asymptotic bias of γ_g is given in Theorem 8(b).

¹⁹ In this case, without specific assumptions on the convergence rates of N/T and the trends,
it is hard to generalize the limits of the difference of the within and the GLS estimators.

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