Notice that by Lemmas 2 and 4, under the fixed effect assumption (Assumption
10), the first term of (92) is
θ2√NTGχ,τN Xxi(zzi + vi)
i
θ2√NTGx,τD-τNN X Dχ,τzi(zi + Vi)
θ2√NTGx,TD-,T {Ξχλ + op (1)} ,
where we partition
xx
zx
xz
zz
conformably to the sizes of xit and zi , and set
)
Ξx = (Ξxx , Ξxz) . Similarly, by
Lemmas 2 and 4, under the fixed effect assumption, the second term of (92) is
θT √NTGχ,τ D-T
N x Dχ,τxiz! f-N X ziz! (N χ z (u+vi)
ii
θT√NTGχ,τD-,1T {
xzΞz-z1Ξzλ+op(1) .
Therefore, the limit of (92) is
θ ' G D [ ( Ξχχ
Ξ-1 Ξ .. 0 λ + op
xz zz zx . p
Recall that it is assumed that NN → c < ∞. Also, recall that under the restric-
tions given in the theorem, Gx,T = diag (Ik22, Ik3) and Dχ,τ = diag √TTlk22, D3τ) .
Then, letting λmax (A) denote the maximum eigenvalue of matrix A, we can have
λmax (θT√NTGχ,TD-Т)
O (1) ∖∕T λmax μ √TIk22 , D3T1^ → 0.
So, under the assumptions of Part (b), the probability limit of the numerator
of (88) is
1
NT
∑∑G∙,t ⅛⅛ + op (1).
(93)
Combining (91) and (93), we can obtain Part (b). ¥
Proof of Theorem 8
Using the notation in (87) , we can express the GLS estimator γg by
^
Yg - Y
57