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

Here, the last line holds because under the local alternative hypotheses,

J_x^-_,T¹Dx,TF1Dx,TJ_x^-_,T¹ =Op(1),

by (102), and

θ Gi

σv

(√NTGχ,τA2) = Op (θτ) = op (1),

by Lemma 2(a), (b) and Assumption 12.

Finally, by Lemma 2(c), as (N,T →∞) ,

Dx,TF1Dx,T

Dx,TA3Dx,T - Dx,TCB3^-1C⁰Dx,T

N X Dx,TXiX0D

(n X Dχ^,T^zi²°) (n X ziz⁰)

-1

(⅛ X Dx,Txiz⁰

Ξ - Ξ Ξ^-1Ξ > 0

p xx ^- xz zz zx > .

Also, Lemmas 2(d) and 3 imply that under the local alternative hypotheses, as
(N,T →∞),

√NDχ.τ F2

√NDχ,τ A4 + √NDχ,τ A5

1^N

√N^Dχ,τ xiui

i=1

d_x,t св-¹ Vn (b₄ + B5)

(N XX

i=1

Dχ^,τ xiz^ (n X

ZiZ⁰

¹ 1^N

( √N∑2ziui I ⁺ oP ⁽¹⁾

⇒ Ik

Ξ Ξ^-1

-Ξ_xz Ξ_zz

N Ξλ, σ²_u Ξ .

As (N,T →∞) ,
fects,

^θT √T →
σ_v

ɪ. Therefore, under the hypothesis of random ef-
σu

HMNT ⇒ χ²_k ,

a χ² distribution with the degrees of freedom equal to k. In contrast, under the
local alternative hypotheses,

HMNT ⇒ χ²_k (η) ,

where η is the noncentral parameter.