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

= τ 1(r)dr;

H32 =

Ik32 ; H33 =

τ 33(r)dr;

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

With this notation, we assume the followings:

Assumption 7 (convergence as N →∞):

^θ21,i = ^θ21,i ^- N Pi ^θ21,i; ^μg32,i = ^μg32,i ^- N Pi ^μg32,i
ɪ P∙ μ_n . As N →∞,

N iμ ^μg33,i ^,

Define Θ1,i = Θ1,i -

; and ^μg33,i

NN ∑i θι,i;
= ^μg33,i ^-

H1θ1,i H1θ1,i

(i) N∑i

θ21,i θ21,i

H³2^μg32,i H³2^μg32,i

WJ WJ

ΓΘ1,Θ1
Γ⁰Θ1,Θ21
Γ

' Θ.,μ 2

VKμ33

ΓΘ1,Θ21

^ΓΘ21^,Θ21

Γ⁰

^θ21^,μ32

Γ⁰

^θ21^,μ33

^r^θ1^,μ32
^rθ21,μ32
Γ

^μ32,^μ32

Γ⁰

μ32^,g33

^r^θ1^,μ33
^r‰^,μ33
Γ

^μ32,^μ33

^μ μ33^,μ33/

^Γg32,g32,i

(ii) NN∑ -

g32, g33, i

g33,g33,i

_Γ0 _i

(iii) The limit of ɪ Pi θɪ,iΘ1,_i

Γg32,z,i Γg32,g32

Γg33,z,i → Γ⁰g₃₂ ,g₃₃

Γzz,i Γg₃₂,z

exists.

g32^,g33

g33,g33

Γ⁰

^g33^,z

g32,z

g33,z

Γ
z,z

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.¹⁶Joint 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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