N ∑E D1τ ⅛∑(x1≠
Exι,it)
i
1 ,, „ ,,2 „
≤ N IID1TIl supE
t
T X (x1,it - Ex1,it)
t
2
→ 0,
(59)
where the last convergence result holds by Lemma 15(a). Similarly, by Lemma
15(b),
EN X D2kT T X (x2k,it - Ex2k,it)
it
∙1 sup E —= X (x2k,it — Ex2k,it)
N i,t T T t
for k = 1, 2. Finally, by Lemma 15(d), we have
11
e N∑D3kτTXC
x3k,it
EFzi x3k,it)
N suτp -TD3kτ
E -T X(x3k.it
t T t
2
→ 0,
(60)
EFzi x3k,it)
0,
(61)
for k = 1, 2, 3. The results (59), (60) and (61) imply that I3,νt →p 0.
Proof of (51): Notice that by (48) and (50),
^N X I1,i,NT^ 13,NT
^N X I2,i,NT^ i3,nt
2
N X Iι,i,NT III3,NT∣∣2 = Op (1) Op (1) ;
i
2
N X I2,i,NT ∣∣I3,NTIl2 = op (1) op (1) .
i
Thus, as (Ν,T →∞) ,
^N X I1,i,NT^ 13,NT,
^NF X I2,i,NT^ 13,NT →p 0.
We now consider the (k,l)th term of N Pi I1,i,NTI'0iNT. By the Cauchy-
Schwarz inequality,
(N X
I1,i,NT 12 ,i,NT
k,l
≤ (N X[(I1,i,NT)k]2^ (N X[(I2,i,NT)ι]2
42