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 (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
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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