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Lecture 14

# STAT 464 Lecture Notes - Lecture 14: Nth MetalExam

Department
Statistics
Course Code
STAT 464
Professor
David Riegert
Lecture
14

This preview shows half of the first page. to view the full 1 pages of the document. AR(p) vs. MA(q)
Saturday, February 2nd, 2019
A causal AR(p) model:
X(ARp)
n=
X
j=0
ΨjX(W N)
nj
= lim
J→∞
J
X
j=0
ΨjX(W N)
nj
= Ψ0X(W N)
n+
q
X
j=1
ΨjX(W N)
nj+ lim
J→∞
J
X
j=q+1
ΨjX(W N)
nj
= Ψ0X(W N)
n+
q
X
j=1
ΨjX(W N)
nj+X(W N)
nX(W N)
n+ lim
J→∞
J
X
j=q+1
ΨjX(W N)
nj
= Ψ0X(W N)
n+X(MAq)
nX(W N)
n+ lim
J→∞
J
X
j=q+1
ΨjX(W N)
nj
=X(MAq)
n+X(ARp,res)
n.
where:
the limiting summations are each deﬁned in terms of the mean-squared error and
both converge to a nite value with 100% certainty;
Ψj=θjfor each j {1,· · · , q};
X(ARp,res)
nis the n’th element of X(ARp,res)={X(ARp,res)
n}nTD, a residual-
component, discrete-time process.
The textbook calls the AR(p)-process a special example of the causal MA() pro-
cesses, of the form,
X(MA)
n=
X
j=1
ΨjX(W N)
nj+X(W N)
n.
1