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)

→ p0,

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

-¹X¹XIII_{1 i}=III2 _iT → _p0; -¹X X III_{1 iτ}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 1H^limN⅛ Pi ^θ1,i^θ1,iX^τ 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= P_t Gχτ (xit - Xi) v^; and let ι ∈ R^k with ∣∣ ι∣∣ = 1. If
we can show that as (N, T →∞) ,

√N X ^ι'Qi,τ ⇒ ^N (^{0, σ}2^ι0^ψχ^ι¢ ^,
i

then, the Cramer-Wold device implies our desired result. Now let s_i²_,T =
E (ι⁰Qi_,T)² and S_N² _T = ^P_i s_i²_,T . Using similar arguments for (74) — (78) , it

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