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
More intriguing information
1. MULTIMODAL SEMIOTICS OF SPIRITUAL EXPERIENCES: REPRESENTING BELIEFS, METAPHORS, AND ACTIONS2. AGRIBUSINESS EXECUTIVE EDUCATION AND KNOWLEDGE EXCHANGE: NEW MECHANISMS OF KNOWLEDGE MANAGEMENT INVOLVING THE UNIVERSITY, PRIVATE FIRM STAKEHOLDERS AND PUBLIC SECTOR
3. Strengthening civil society from the outside? Donor driven consultation and participation processes in Poverty Reduction Strategies (PRSP): the Bolivian case
4. Short Term Memory May Be the Depletion of the Readily Releasable Pool of Presynaptic Neurotransmitter Vesicles
5. The name is absent
6. The name is absent
7. The name is absent
8. THE EFFECT OF MARKETING COOPERATIVES ON COST-REDUCING PROCESS INNOVATION ACTIVITY
9. Ein pragmatisierter Kalkul des naturlichen Schlieβens nebst Metatheorie
10. The name is absent