where Ak,l and ak denote the (k, l)th and the kth elements of matrix A and vector
a, respectively. In view of (48) and (49) (the limits of N Pi I1,i,NTI1 i nt and
N ∑i I2,i,NTI'2,i,Nτ), we can see that all of the elements in N ∑i I1,i,NTI'2,i,Nτ
converge to zero, except for
N X D32τ (χ32,i - EFzi χ32,i) (EFzi χ32,i - EFzχ32) D32T.
i
Thus, we can complete the proof by showing that this term converges in prob-
ability to a conformable zero matrix.
Let
Qι,iτ = D32τ (χ32,i — EFzi χ32,i) ;
Q2,iT = D32T (EFzi x32,i - EFz x32~) ';
Q2,i = H32 ^g32,i (zi) - N X g32,i (zi) + μg32,i
N∑μg32j
i

By the Cauchy-Schwarz inequality and Lemma 12,
N X Q1,iT (Q2,iT - Q2,i)'
i
(N X kQι>iτ∣∣2) (N X kQ2,iτ
^Ν X kQ1,iTliɔ sup kQ2,iT - Q2,ik2
Op(1)o(1)=op(1),
where the last line holds since
E(⅛ X kQ1,iTll2) <M
by Lemma 15(d) and by Assumption 6. Thus,
11
N Q1,iTQ2,iT = N Q1,iTQ2,i + oP (1) .
(62)
Next, notice that EQ1,iTQ'2,i =0. Then, by the Cauchy-Schwarz inequality,
(N X vec (Q1,iTQ2,i))
^N X (Q2,i ° Q1,iT))
2
= N2 X E (Q2,iQ2,iQ1,iT Q1,iT )
i
N2 XhE kQ2,ik4 E kQ1,iTk4i 1/2 .
i
43
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