Heteroscedasticity in State Space
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Transcript of Heteroscedasticity in State Space
7/23/2019 Heteroscedasticity in State Space
http://slidepdf.com/reader/full/heteroscedasticity-in-state-space 1/12
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C 2
C 2t−1 =u
j=1
ιt− jΩt− jιt− j ∼ χ2 (u)
u
u
u
q = 50
χ2 (50)
[χ2 (.025, 50) , χ2 (.975, 50)] = [32.36, 71.42]
0.975
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0 1 2 3 4 5 6 7 8 9
−1000
0
1000
0 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
0 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
1 1 . 3
6 5 6 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
2 2 . 5
4 8 6 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
5 7 . 7
2 3 5 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
7 3 . 7
5 6 2 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
D a t a
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
0 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
0 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
1 1 . 3
6 5 6 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
2 2 . 5
4 8 6 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
5 7 . 7
2 3 5 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
7 3 . 7
5 6 2 H z
0 1 2 3 4 5 6 7 8 9
−1000
0
1000
D a t a
100
0 H z
100
0 H z
100
1 1 . 3
6 5 6 H z
100
2 2 . 5
4 8 6 H z
100
5 7 . 7
2 3 5
H z
100
7 3 . 7
5 6 2 H z
0 1 2 3 4 5 6 7 8 9
−2
0
2
I n n o v a t i o n
100
0 H z
100
0 H z
100
1 1 . 3
6 5 6 H z
100
2 2 . 5
4 8 6 H z
100
5 7 . 7
2 3 5
H z
100
7 3 . 7
5 6 2 H z
0 1 2 3 4 5 6 7 8 9
−2
0
2
I n n o v a t i o n
0 1 2 3 4 5 6 7 8 9 1010
20
30
40
50
60
70
80
90
100
0 1 2 3 4 5 6 7 8 9 1010
20
30
40
50
60
70
80
90
100
q = 50
χ2 (50)
[χ2 (.025, 50) , χ2 (.975, 50)]
= [32.36, 71.42]
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ηt = σtν t ν t ∼ N (0, 1)
σ2t = α0 +
pi=1
αiη2t−i
ηt = σtν t
ν t ∼ N (0, 1)
σ2t = α0 +
ui=1
αiη2t−i +
v j=1
β jσ2t− j
σ2t
η2t
α
β
ν t = σtηt
ηt ∼ N (0, 1)
log σ2t = α0 +
ui=1
αig
ν 2t−i
+
v j=1
β j log σ2t− j
g ν 2t−i
ν
g
ν 2t−i
= log ν 2t−i
σ
ν
σ
α1 = 1
α
β
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xt = F xt−1 + Gwt
yt = H xt + t
wt ∼ N (0, Qt)
Qt wt
Qt =
τ
(1)
t
2
0 · · · · · · · · · · · · 00 τ
(2)t
2
τ (r)t
2
τ (r+1)t
2
0
0 · · · · · · · · · · · · 0 τ (r+s)t
2
.
Q τ (k)t2
k
τ (k)t
2
log τ (k)t
2= α
(k)0 +
ui=1
α(k)i log w
(k)t−i
2+
v j=1
β (k) j log τ
(k)t− j
2.
α0
u
v
g
ν 2t− j
w
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E
Gwt− j (Gwt− j) |yt− j
= E Gwt− j|yt− jE Gwt− j|yt− j + Var Gwt− j|yt− j
= E
Gwt− j|yt− j
E
Gwt− j|yt− j
+ Var
Gwt− j|yt− j−1, ιt− j
= E
Gwt− j|yt− j
E
Gwt− j|yt− j
+
Var
Gwt− j|yt− j−1
−Cov
Gwt− j, ιt− j |yt− j−1
Var (ιt− j)−1 Cov
Gwt− j, ιt− j|yt− j−1
= K t− jιt− jι
t− jK t− j + GQt− jG − GQt− jGH Ωt− jHGQt− jG
w2
E
Gwt− j (Gwt− j) |yt− j
log τ (1)t
2
log τ (2)t
2
(2r − 1)
r
(2r + 1) log τ (r+1)t2
E
Gwt− j (Gwt− j) |yt− j
m × m
k
m × m
k < m = k + r
k × m
G−
G
Ewt− jw
t− j|yt− j
k × k
w2
Ewt− jw
t− j|yt− j
= G−K t− jιt− jι
t− jK t− jG− + Qt− j − Qt− jGH Ωt− jHGQt− j
G
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G− =
1 0
1 0
1
1
α(k)1 = α
(k)2 = · · · = α(k)
u
log τ (k)t
2= α
(k)0 + α
(k)1
ui=1
log w(k)t−i
2+
v j=1
β (k) j log τ
(k)t− j
2.
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u
w
α(k)1 = µ−1α
(k)2 = · · · = µ−uα
(k)u+1 = . . .
log τ (k)t
2= α
(k)0 + α
(k)1
t−1
i=1
µi−1 log w(k)t−i
2+
v
j=1
β (k) j log τ
(k)t− j
2.
log τ (k)t
2= α
(k)0 + α
(k)1 log w(k)
t−1
2+
v j=1
β (k) j log τ
(k)t− j
2
log w(k)t−1
2= log w
(k)t−1
2+ µ log w
(k)t−2
2
µ
µ = u−1u
u
log w(k)
t−1
2
u
log w(k)t−1
2
µ
χ2 (u)
C 20 = q
C 2t = q−1q
C 2t−1 + ιt−1Ωt−1ιt−1
C 2t−1 =
q j=1
q − 1
q j−1
ιt− jΩt− jιt− j
q
q
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ξ
σ2c
σd
σ2t = (σc + ξ tσd)2
E
Gwt− j (Gwt− j) |yt− j
w(k)t− j
2
wt− j
wt− jw
t− j
t − j
K t− j
ιt− j Ωt− j
t − j
E
Gwt− j (Gwt− j) |yt− j
=
Var
Gwt− j|yt− j
+ E
Gwt− j |yt− j
E
Gwt− j|yt− j
E
Gwt− j|yt− j
K t− jιt− j Var
Gwt− j|yt− j
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Var
Gwt− j |yt− j
= Var
Gwt− j|yt− j−1, ιt− j
= Var
Gwt− j|yt− j−1
− Cov
Gwt− j, ιt− j |yt− j−1
Var (ιt− j)−1 Cov
Gwt− j , ιt− j|yt− j−1
Cov
Gwt− j, ιt− j|yt− j−1
= E Gwt− jιt− j|yt− j−1 − E Gwt− j|yt− j−1E ιt− j |yt− j−1= E
Gwt− j
yt− j − H xt− j|t− j−1
|yt− j
= E
Gwt− j
[H (F xt− j−1 + Gwt− j) + t− j] − H
F xt− j−1|t− j−1
|yt− j
= E
Gwt− j
HF
xt− j−1 − xt− j−1|t− j−1
|yt− j
+ E
Gwt− jw
t− jGH |yt− j
+ E
Gwt− j
t− j |yt− j
= 0 + GQt−1GH + 0
= GQt−1GH
E ιt− j|yt− j−1 = 0
E
Gwt− j
HF
xt− j−1 − xt− j−1|t− j−1
|yt− j
= 0
E
Gwt− jwt− jGH |yt− j
= GQt−1GH
E
Gwt− jt− j|yt− j
= 0
E
Gwt− j (Gwt− j) |yt− j
=
K t− jιt− jιt− jK t− j + GQt− jG − GQt− jGH Ωt− jHGQt− jG
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