Non-parametric estimation of a time varying GARCH model

Proof. Let Ft−1 = σ(ǫ2 t−1,ǫ2 t−2,...). Then
�
n�
V ar (uk − u0)
k=p+1
iKh(uk − u0)(v2 k − 1)σ2 k[1,ǫ2 k−1,...,ǫ 2 k−p] ⊤
�
�
n�
= E (uk − u0)
k=p+1
2iK2 h(uk − u0)V ar �
(v2 k − 1)σ2 k[1,ǫ2 k−1,...,ǫ 2 k−p] ⊤ �
|Fk−1
�
= �
n�
E (uk − u0)
k=p+1
2iK2 h(uk − u0)V ar(v2 k) �
σ4 k[1,ǫ2 k−1,...,ǫ 2 k−p] ⊤ [1,ǫ2 k−1,...,ǫ 2 k−p] ��
= nh2i−1ν2iV ar(v2 t )Ω(1 + oP(1)), (using Lemma A.2(ii))
Proof of Theorem 4.1. Let us denote β1 = [α00,α01,...,α0d,...,αp0,...,αpd] ⊤ . Using
Taylor’s series expansion, we can write,
�
Y1 = X1 α0(u0),α (1)
0 (u0),... α(d)
+
⎡
1 ⎢
(d + 1)!
⎣
+
α (d+1)
0 (ζ0(p+1))(up+1 − u0) d+1
.
α (d+1)
0 (ζ0(n))(un − u0) d+1
⎡
p� 1 ⎢
(d + 1)!
⎣ .
j=1
0 (u0)
,α1(u0),...,αp(u0),... d! α(d)
�⊤ p (u0)
d!
⎤
⎥
⎦
α (d+1)
j (ζj(p+1))(up+1 − u0) d+1ǫ2 p+1−j
α (d+1)
j (ζj(n))(un − u0) d+1ǫ2 n−j
⎤
⎥
⎦ +σ2 ∗ (v 2 − en−p)
where σ 2 = [σ 2 p+1,σ 2 p+2,...,σ 2 n] ⊤ , v 2 = [v 2 p+1,v 2 p+2,...,v 2 n] ⊤ , ∗ denotes the component
wise product 3 of vectors and ζjk, j = 0, 1,...,p, k = p + 1,...,n are between uk and u0.
Multiplying both sides by (X ⊤ 1 W1X1) −1 X ⊤ 1 W1,
⎡
⎢
× ⎢
⎣ .
ˆβ1(u0) = β1(u0) +
⎡
⎢
× ⎢
⎣ .
α (d+1)
0 (ζ0(p+1))(up+1 − u0) d+1
α (d+1)
0 (ζ0(n))(un − u0) d+1
α (d+1)
j (ζj(p+1))(up+1 − u0) d+1ǫ2 p+1−j
α (d+1)
j (ζj(n))(un − u0) d+1ǫ2 n−j
1
(d + 1)! (X⊤ 1 W1X1) −1 X ⊤ 1 W1
⎤
⎤
⎥
⎦ +
1
(d + 1)!
Now it is not difficult to show using Lemma A.2 (i) that
⎡
⎤
X ⊤ ⎢
1 W1
⎢
⎣
α (d+1)
0 (ζ0(p+1))(up+1 − u0) d+1
.
α (d+1)
0 (ζ0(n))(un − u0) d+1
p�
(X
j=1
⊤ 1 W1X1) −1 X ⊤ 1 W1
⎥
⎦ + (X⊤ 1 W1X1) −1 X ⊤ 1 W1(σ 2 ∗ (v 2 − en−p)). (12)
3 Let x = [x1,x2,...,xp] ⊤ and y = [y1,y2,...,yp] ⊤ , then x ∗ y = [x1y1,x2y2,...,xpyp] ⊤ .
22
⎥
⎦