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



In addition, we have

N X T X III3,iTIII3,iT
it

1      1         D1T (x1,it - Ex1,it)

N Σ T Σ I (x2,it - Ex2,it)

i t      (x3,it - Ex3,it)

ɪ X 1 X E

N 2 iT,t


ID1T (x1,it - Ex1,it)

(x2,it - Ex2,it)

(x3,it - Ex3,it)


p0,

(78)


as (N, T →∞), as shown in (75) and (76). Thus,

N X τ X πi3,iτ 1113 ,iτ p 0.
it

From the Cauchy-Schwarz inequality, (74) , (77) and (78) , we have

-1X1XIII1 i=III2 iT p0; -1X X III1 III0 iT 0;

N T        1,iT    2,iT       p ; N T        1,iT 3,iT p ;

NX TX III2,iTiii3,iT P0,
it

as (N, T →∞), Combining all of these, we have

N X TT X GxT (xit - Xi) (xit - Xi)0 GxT

[ R0 (τ 1 - R τ 1HlimN⅛ Pi θ1,iθ1,iXτ 1 - R τ 1 ¢0 dr 00 ʌ

p 0                                               Φ22 Φ23

0                                                       Φ023 Φ33

Ψx.

as (N,T →∞).

Part (b)

First, let Qi,τ = ^1= Pt Gχτ (xit - Xi) v^; and let ι Rk with ∣∣ ι∣∣ = 1. If
we can show that as (
N, T →∞) ,

N X ι'Qi,τ N (0, σ2ι0ψχι¢ ,
i

(79)


then, the Cramer-Wold device implies our desired result. Now let si2,T =
E (ι0Qi,T)2 and SN2 T = Pi si2,T . Using similar arguments for (74) — (78) , it

51



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