Non-parametric estimation of a time varying GARCH model

and
X ⊤ ⎡
α
⎢
2 W2
⎢
⎣
(d+1) (ξ12))(u2 − u0) d+1ǫ2 1
.
α (d+1) (ξ1n)(un − u0) d+1ǫ2 ⎤
⎥
⎦
n−1
= nh d+1
2 α (d+1) (u0)[w2,w4,λ2] ⊤ (1 + oP(1)) ⊗ D2,
X ⊤ ⎡
β
⎢
2 W2
⎢
⎣
(d+1) (ξ22))(u2 − u0) d+1ˆσ 2 1
.
β (d+1) (ξ2n)(un − u0) d+1ˆσ 2 ⎤
⎥
⎦
n−1
= nh d+1
2 β (d+1) (u0)[λ1,λ2,λ3] ⊤ (1 + oP(1)) ⊗ D2
X ⊤ ⎡
⎢ β(u2)(b0(u1) +
⎢
2 W2
⎢
⎣
p �
bj(u1)ǫ
j=1
2 1−j)
.
β(un)(b0(un−1) + p �
bj(un−1)ǫ
j=1
2 ⎤
⎥
⎦
n−1−j)
Using Lemma A.6,
Therefore,
Bias( ˆ β2(u0))
= β(u0)[λ1b,λ2b,λ3b)(1 + oP(1)] ⊤ ⊗ D ∗ .
(X ⊤ 2 W2X2) −1 = (1/n)S −1
2 (1 + oP(1)) ⊗ A −1
2 .
= hd+1
2
(d+1)! (S−1 2 (1 + oP(1)) ⊗ A −1
2 ) ��
ω (d+1) (u0)[1,w2,λ1] ⊤
+ α (d+1) (u0)[w2,w4,λ2] ⊤ + β (d+1) (u0)[λ1,λ2,λ3] ⊤�
− β(u0)S −1
2 [λ1b,λ2b,λ3b] ⊤ ⊗ A −1
2 D ∗ + oP(h d+1
2 )
= hd+1
2
(d+1)! (S−1 2 ⊗ A −1
2 ) �
(1 + oP(1)) ⊗ A −1
2 D �
(S2[ω (d+1) (u0),α (d+1) (u0),β (d+1) (u0)] ⊤ ) ⊗ D2
− β(u0)S −1
2 [λ1b,λ2b,λ3b] ⊤ ⊗ A −1
2 D ∗ + oP(h d+1
2 )
= hd+1
2
(d+1)! [ω(d+1) (u0),α (d+1) (u0),β (d+1) (u0)] ⊤ ⊗ A −1
2 D2
− β(u0)S −1
2 [λ1b,λ2b,λ3b] ⊤ ⊗ A −1
2 D∗ + oP(h d+1
2 ).
The bias expressions can be obtained after some simplification by using
Bias(ˆω(u0)) = e ⊤ 1,3(d+1) Bias(ˆ β2(u0)), Bias(ˆα(u0)) = e ⊤ d+1,3(d+1) Bias(ˆ β2(u0))
and Bias( ˆ β(u0)) = e ⊤ 2d+3,3(d+1) Bias(ˆ β2(u0)).
Now using Lemma A.7
V ar( ˆ β2(u0)) = (1/n)S −1
2 (1 + oP(1)) ⊗ A −1
2 V ar(X ⊤ 2 W2(σ 2 ∗ (v 2 − en−p)))
× (1/n)S −1
2 (1 + oP(1)) ⊗ A −1
2
= 1
nh2 V ar(v2 t )(S −1
2 ⊗ A −1
2 )(Ω2 ⊗ B2)(S −1
2 ⊗ A −1
2 )(1 + oP(1)).
26
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