Large-N and Large-T Properties of Panel Data Estimators and the Hausman Test



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

√NTG⅛g - 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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