MPRA

such that
⎧
⎨
E
⎩
⎧
⎨
E
⎩
{
E
∥ ∥ ⎫ ∥∥∥∥ sup 1 ∂σkt 2 (θ(k) ) ∥∥∥∥
4(1+1/δ)
⎬
< ∞, for some δ > 0, (22)
0 ) σkt 2 (θ(k) ) ∂θ (k) ⎭
θ (k) ∈V(θ (k)
∥ ∥ ⎫ ∥∥∥∥ sup 1 ∂ 2 σkt 2 (θ(k) ) ∥∥∥∥
2(1+1/δ)
⎬
< ∞, for some δ > 0, (23)
0 ) σkt 2 (θ(k) ) ∂θ (k) ∂θ (k)′ ⎭
∣ ∣∣∣∣
σkt 2 (θ(k) 0 )
s}
< ∞, for any s > 0. (24)
0 ) σkt 2 (θ(k) ) ∣
θ (k) ∈V(θ (k)
sup
θ (k) ∈V(θ (k)
Proof of Lemma 1
Iteratively using the volatility equation in (4), we obtain
{
}
∞∑
m∑
r∑
σkt(θ 2 (k) ) = ω kk + a kl a kl ′ε l,t−j−1 ε l ′ ,t−j−1 + c ks c ks ′x s,t−j−1 x s ′ ,t−j−1 .
j=0
b 2j
k
l,l ′ =1
Derive (25) with respect to θ (k) , we get
s,s ′ =1
(25)
∂σ 2 kt (θ(k) )
∂ω kk
∂σkt 2 (θ(k) )
∂a kl
∂σkt 2 (θ(k) )
∂b 2 k
∂σkt 2 (θ(k) )
∂c ks
∑
= ∞ b 2j
k = 1 ,
j=0 1 − b 2 k
∑
= 2 ∞ m∑
b 2j
k
a kl ′ε l,t−j−1 ε l ′ ,t−j−1,
j=0 l ′ =1{
∑
= ∞ ∑
ω kk +
m a kl a kl ′ε l,t−j−1 ε l ′ ,t−j−1 +
j=1
= 2 ∞ ∑
j=0
jb 2(j−1)
k
b 2j
k
l,l ′ =1
m∑
c ks ′x s,t−j−1 x s ′ ,t−j−1
s ′ =1
r ∑
s,s ′ =1
c ks c ks ′x s,t−j−1 x s ′ ,t−j−1
}
,
.
Silmilar expressions hold for the second order derivatives. Noting that
ω kk :=
inf
θ (k) ∈Θ (k) σ 2 kt > 0.
Using the moment conditions A10 and (20), we obtain (22) and (23).
The moment condition (24) will be showed even if some components of a 0 k or c0 k are
zero. Indeed, there exists a neighborhood V(θ (k)
0 ) such that for all θ (k) ∈ V(θ (k)
0 )
⎧ ⎛
⎞2
⎛
⎞ ⎫
2
⎜ m∑ a kl ⎟ ⎜ r∑ c ks ⎟
⎝ √ ε l,t−j−1 ⎠ ⎝ √ x s,t−j−1 ⎠
σkt 2 ∞ (θ(k) 0 )
σkt 2 (θ(k) ) ≤ K+K ∑ (b
⎪⎨ 0 l=1 ωkk s=1 ωkk ⎪⎬
k )2j a 0 kl≠0
b 2j ⎛
⎞2 + c 0
⎛
ks≠0
⎞2
.
j=0 k
⎜ m∑ a kl ⎟ ⎜ r∑ c ks ⎟
1 + ⎝ √ ε l,t−j−1 ⎠ 1 + ⎝ √ x s,t−j−1 ⎠
⎪⎩
ωkk ωkk ⎪⎭
l=1
a 0 kl≠0
19
s=1
c 0 ks≠0