But, by Lemma 16(b),
trE
(N Pi DTwiw
<M,
for some constant M. This implies that
N ∑i DTwi (ui - EFw ui)
2
→ 0,
as (N, T →∞). Thus, by Chebyshev’s inequality,
N Pi DTwi (ui - EFwui) →p 0, (86)
as (N, T →∞). Then, Lemma 16(a), (85) and (86) imply our desired result. ¥
Proof of Theorem 5
By Lemma 2(a) and (b). ¥
Proof of Theorem 6
By Lemma 2(c), (d), and Lemma 3. ¥
Before we prove the rest of the theorems given in Section 4, we introduce
the following notation:
Al = NN PiT Pt (xit - xi) (xit - xi)' ;
A2 = N PiT Pt (xit - xi) (vit - vi) ;
A3 = N Pi xixi; A4 = NN Pi xiui; A5 = N1 Pi xivi;
(87)
B3 = NN Pi ziz0; B4 = N1 Pi ¾⅛ B5 = N Pi Zivi;
C = nN ∑i xiZ0;
F1 =A3-CB3-1C0; F2 = A4+A5 - CB3-1 (B4+B5).
Proof of Theorem 7
Using the notation given in (87) , we can express the GLS estimator βg by
√NTG⅛ (βg - β)
= £Gx,tAιGχ,T + θTGx,t ©A3 - CB-1C0} GxT] -1
× √NTG χ,T ©A2 + θT £ (A4 + A5) - CB-1 (B4 + B5)]} , (88)
where θT = √σ2/(TσU + σV).
Part (a)
First, consider
θ2T Gx,T A3 - CB3-1C0 Gx,T
θ2 G D — 1 ½ N Pi Dχ,TXixiDx,T
θTGx,TDx,T S 1 1 Y' ∩ - ~∣∖ 1 1 γ-1 ~ ~∣∖ -1 1 1 Y' ~ ~∣ ,4 ∖
I In∑1 Dx,Txizi) Nn ∑i zizi) Nn ∑i zixiDxtT)
×Dx-, 1T Gx,T .
55
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