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,

Jx-,T1Dx,TF1Dx,TJx-,T1 =Op(1),

by (102), and


θ Gi

σv


(√NTGχ,τA2) = Op τ) = 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

i

(n X Dχ,Tzi2°) (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

1N

N^Dχ,τ xiui

i=1

dx,t св-1 Vn (b4 + B5)

(N XX

i=1


Dχ,τ xiz^ (n X


0

ZiZ0


1    1N

( √N∑2ziui I + oP (1)


  Ik


Ξ Ξ-1

-Ξxz Ξzz


N Ξλ, σ2u Ξ .


As (N,T →∞) ,
fects,


θT √T
σv


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


HMNT χ2k ,


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


HMNT χ2k (η) ,


where η is the noncentral parameter.


63




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