STA457H1 Chapter Notes - Chapter 5: Asparagine

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8 Apr 2014
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Chapter 5
5.1 – Preliminary Estimation
5.1.1Yule-Walker Estimation
AR(p)
Xtϕ1Xt1ϕ pXtp=Zt(¿)
ZtWN
(
0,σ2
)
Multiply
Xt1, Xt2, … , Xtp
to
(¿)
and take expectation
E
[
XtXti
]
ϕ1E
[
Xt1Xti
]
ϕ pE
[
XtpXti
]
=E
[
ZtXti
]
i=1, , p
γiϕ1γi1ϕ pγip=0
γi=ϕ1γi1+ϕpγip
¿¿
́
γp=Γṕ
ϕp¿
Γp=
[
γij
]
i,j=1
p
¿¿
¿
implies that we can use the Method of Moments and replace
́γp
and
Γp
by
̂
́γp
and
̂
Γp
and obtain the Yule-Walker estimation:
̂
́
ϕp=
̂
Γp
1
̂
́
γp
Multiply
Xt
to
(
¿
)
and take expectation
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γ
(
0
)
ϕ1γ1−ϕ pγp=E
[
ZtXt
]
=E
[
Zt
(
ϕ1Xt1++ϕpXtp+Zt
)
]
=σ2
σ2=γ
(
0
)
́
ϕp
T
̂
́
γp
Use the Method of Moments,
̂
σ2=
̂
γ
(
0
)
̂
́
ϕp
T
̂
́
γp
Theorem: Simple Yule-Walker Estimation
̂
́
ϕp=
̂
Rp
1
̂
́
ρp
̂
σ2=
̂
γ
(
0
)
(
1
̂
́
ρp
T
̂
Rp
1
̂
́
ρp
)
Here
̂
Rp
is the autocorrelation matrix
̂
Rp=
[
̂
ρij
]
i , j=1
p
̂
́ρp=
(
̂
ρ1
̂
ρp
)
Proof: HW
Proposition:
̂
σ2P
σ2as n →
Proof: not required
Theorem:
For large
n
,
n
(
̂
́
ϕṕ
ϕp
)
D
N
(
0, σ2Γp
1
)
Remark:
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For AR(p) models, the Yule-Walker estimates are as efficient as the Maximum Likelihood Estimates.
̂
́
ϕp=
̂
Γp
1
̂
́
γp
Note:
̂
γi=1
n
j=1
ni
(
Xj́
X
) (
Xj+1́
X
)
i=0, , p
Note:
E
[
Xi
]
=0
́
X 0
̂
γi1
n
j=1
ni
XjXj+1=1
n
j=1
ni
Xj
(
Xj+i1ϕ1++Xj+ipϕp+Zj+i
)
=1
n
j=1
ni
Xj
(
́
Xj+i
T́
ϕp+Zi+j
)
=1
n
j=1
ni
Xj́
Xjϕp+1
n
j=1
ni
XjZi+j
̂
́γp
̂
Γṕ
ϕp+1
n
k=p+1
ń
XkZk
́
Xi=
(
Xi1
Xip
)
̂
́
ϕṕ
ϕp
̂
Γp
11
n
k=p+1
ń
XkZk
Now 2 steps:
(1) Show
1
n
k=p+1
ń
XkZkD
N
(
0, σ2Γp
)
(2) Show
̂
ΓpP
Γp
Then Slutsky’s Theorem implies
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Document Summary

X t 1 x t 1 p x t p=zt( ) Multiply x t 1, x t 2, , xt p. E[ xt xt i] 1 e[ xt 1 x t i] p e[ x t p xt i]=e[z t x t i] and take expectation to i=1, , p. I= 1 i 1+ p i p. P=[ i j]i , j =1 p implies that we can use the method of moments and replace. (0) 1 1 p p=e[zt xt]=e[ zt ( 1 xt 1+ + p x t p+zt )]= 2. I= 1 n i n j=1 ( x j x)(x j+1 x) i=0, , p. X j x j+1= 1 n i n j=1. X j( x j+ i 1 1+ +x j+i p p+z j+i)= 1 n i n j=1. P+zi + j)= 1 n i n j=1. X j x j p+ 1 n i n j=1. P+ 1 n n k= p+1.

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