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

and the correlations between x_3,it and z_i no longer play any important role in
asymptotics. Assumption 6(iii) rules out such cases.

Assumption 6 is about the asymptotic properties of means of regressors
as T →∞. We also need additional regularity assumptions on the means of
regressors that apply as N →∞.Define

H32 =

Ik32 ; H33 =

and

H32g32,i (zi)

H33g33,i (zi)
zi - Ezi

H32g32,i (zi)

H33g33,i (zi)
zi - Ezi

Γ
g32,g32,i

Γ⁰

g32, g33,i

_Γ0 _i

Γ
g32,g33,i

Γ
^g33^,g33^,i

0

With this notation, we assume the followings:

Apparently, by Assumptions 6 and 7, we assume the sequential convergence
of the means of regressors as T → ∞ followed by N →∞. However, this by
no means implies that our asymptotic analysis is a sequential one. Instead, the
uniformity conditions in Assumption 6 allow us to obtain our asymptotic results
using the joint limit approach that applies as (N, T → ∞) simultaneously.¹⁶
Joint limit results can be obtained under an alternative set of conditions that
assume uniform limits of the means of regressors sequentially as N → ∞ followed
by T →∞. Nonetheless, we adopt Assumptions 6 and 7, because they are much
more convenient to handle the trends in regressors x_1,it and x_3,it for asymptotics.

The following notation is for conditional or unconditional covariances among
time-varying regressors. Define

Γi (t, s) = [Γjl,i (t, s)]jl,

¹⁶ For the details on the relationship between the sequential and joint approaches, see Apos-
tol (1974, Theorems 8.39 and 9.16) for the cases of double indexed real number sequences,
and Phillips and Moon (1999) for the cases of random sequences.

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