MPRA

The proofs of i), iii) and i) are very similar to the ones given in Francq and Zakoïan
(2010) for the standard GARCH without covariates and are omitted. Here I only thus
show the point ii).
Let L be the back-shift operator, i.e. L(u t ) = u t−1 . By Assumption A5, the polynomial
1 − b 2 k L is invertible for any θ(k) ∈ Θ (k) . Assume that σ 2 kt (θ(k) 0 ) = σ 2 kt (θ(k) ) a.s. We
have
( m
) 2 (
∑
m
) 2
∑
(1 − (b 0 k) 2 L) −1 a 0 klε l,t−1 − (1 − b 2 kL) −1 a kl ε l,t−1
l=1
+ (1 − (b 0 k) 2 L) −1 ( r∑
s=1
l=1
c 0 klx s,t−1
) 2
− (1 − b 2 kL) −1 ( r∑
s=1
c ks x s,t−1
) 2
=(1 − b 2 k) −1 ω kk − (1 − (b 0 k) 2 ) −1 ω 0 kk a.s.
Then
where a (k)
ill ′
∞∑
m∑
i=0 l,l ′ =1
a (k)
ill ′ ε l,t−i−1 ε l ′ ,t−i−1 +
∞∑
r∑
j=0 s,s ′ =1
c (k)
jss ′ x s,t−j−1 x s ′ ,t−j−1 = c a.s. (21)
= (b 0 k )2i a 0 kl a0 kl ′ − (b k) 2i a kl a kl ′, c (k)
jss ′ = (b 0 k )2i c 0 ks c0 ks ′ − (b k) 2i c ks c ks ′ and c = (1 −
(b 0 k )2 ) −1 ω kk − (1 − b 2 k )−1 ω 0 kk . If b0 k ≠ b k or there exists l ∗ such that a 0 kl ∗ ≠ a kl ∗ then
ε 2 l ∗ ,t−1 is a linear combination of the x s,ux s ′ ,u, s, s ′ = 1, . . . , r, u < t, the ε l,v ε l ′ ,v, l, l ′ =
1, . . . , m, v < t−1 and the ε l,t−1 ε l ′ ,t−1, (l, l ′ ) ≠ (l ∗ , l ∗ ) which is impossible by Assumption
A7. Therefore b 0 k = b k and a 0 kl = a kl for all l = 1, . . . , m and (21) becomes
∞∑
r∑
j=0 s,s ′ =1
c (k)
jss ′ x s,t−j−1 x s ′ ,t−j−1 = c
a.s.
Similarly, if there exists s ∗ such that c 0 ks ∗ ≠ c ks ∗, x2 s ∗ ,t−1 is a linear combination of
x u,t−j x u ′ ,t−j, u, u ′
= 1, . . . , r, j > 1 and x s,t−1 x s ′ ,t−1, (s, s ′ ) ≠ (s ∗ , s ∗ ) which contradicts
Assumption A8. Therefore we have c 0 ks = c ks for all s = 1, . . . , r. Hence ii) is proved. ✷
For the proof the asymptotic distribution in the Theorem 2, we need the following
lemmas
Lemma 1 Under assumptions of Theorem 2, there exists a neighborhood V(θ (k)
0 ) of θ (k)
0
18