Class Notes (1,100,000)

CA (650,000)

Queen's (10,000)

STAT (200)

STAT 464 (20)

David Riegert (20)

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)

n−j

= lim

J→∞

J

X

j=0

ΨjX(W N)

n−j

= Ψ0X(W N)

n+

q

X

j=1

ΨjX(W N)

n−j+ lim

J→∞

J

X

j=q+1

ΨjX(W N)

n−j

= Ψ0X(W N)

n+

q

X

j=1

ΨjX(W N)

n−j+X(W N)

n−X(W N)

n+ lim

J→∞

J

X

j=q+1

ΨjX(W N)

n−j

= Ψ0X(W N)

n+X(MAq)

n−X(W N)

n+ lim

J→∞

J

X

j=q+1

ΨjX(W N)

n−j

=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 ﬁnite 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}n∈TD, 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)

n−j+X(W N)

n.

1

###### You're Reading a Preview

Unlock to view full version