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 iτIII0 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