By the Cauchy-Schwarz inequality,
kQiT k2 ≤
≤
T∑
t
(Dit (xι,it — Exι,it)
x2,it - Ex2,it
x3,it - Ex3,it
kD1T (x1,it
22
- Ex1,it)k4 + kx2,it
+ kx3,it - Ex3,it k
- Ex2,it k4
Notice that by Assumptions 2(i), 3(iii),
supE kx1,it - Ex1,it k4, supE kx2,it -Ex2,itk4 < ∞,
i,t i,t
(75)
and by Assumption 4(iii) and Assumption 6(iii),
sup E kx3,it - Ex3,itk4
i,t
≤ M1 sup EFzi°x3,it - EFzi x3,it°4 +M1E°EFzix3,it
- Ex3,it °
< ∞.
(76)
Thus, we have
sup E kQiT k2 < ∞.
i,T
From this,
supE ∣∣QiτII 1 {∣∣Qiτ∣∣ > M} ≤ supi,τ E∣lQiTk → 0,
i,T M
as M →∞. By Corollary 14 and Assumption 8(ii), then,
NN Σ T Σ IIIι,iτ II10,iτ
it
1 1 ( 0 0 0 ʌ
= N Σ QiT →P^1T N Σ EQiT = I 0 φ22 φ23 ) .
i , i 0 Φ023 Φ33
Next, by Assumption 6(i) and (ii), Lemma 11, and Lemma 12, we have
N Σ T ΣIII2,iTIII2,iT
it
1 1 DDT (Exι,it - Exι,i)∖DDit (Exι,it - Exι,i)∖
N∑T∑ EX2,it — EX2,i EX2,it — EX2,i
i t ∖Exi,it — EX3,i J∖Ex3,it — EX3,i )
[ Ro1 (τ 1 — R τ il i∣imN 1 Pi θ1,iθ1,i∏τ 1 — R τ 1/dr 0 0 ʌ
.(77)
000
000
50