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

But, by Lemma 16(b),

for some constant M. This implies that

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;

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 β_g by

√NTG⅛ (βg - β)

=  £Gx,tAιGχ,T + θTGx,t ©A3 - CB-¹C⁰} G_xT] -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 D^χ,T^XixiD^x,^T

^θT^Gx,^TDx,T S     1 1 Y'  ∩   - ~∣∖ 1 1 γ-₁ ~ ~∣∖ ^-1 1 1 Y' ~ ~∣ ,4   ∖

I    In∑1 Dx,T^xizi) Nn ∑i zizi) Nn ∑i ^zixiDx_tT)

×Dx^-, ¹T Gx,T .

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