# Chapter 3.docx

89 views6 pages
8 Apr 2014
School
Course
Professor
Chapter 3 ARMA Models
3.1
ARMA (p,q)
Process
Definition
A weakly stationary TS
{
Xt
}
is called an
ARMA (p , q)
process
(
p ≥ 0, q ≥ 0
)
if
Xtϕ1Xt1ϕ pXtp=Zt+θ1Zt1++θqZtq(¿)
{
Zt
}
WN
(
0, σ2
)
ARMA
(
p , q
)
can be written as
Φ
(
B
)
Xt=Θ
(
B
)
Zt
Θ
(
B
)
=1+θ1B+θ2B2++θqBq
Theorem
A stationary solution of
(
¿
)
exists and is unique iff
ϕ
(
z
)
=1ϕ1zϕ pzp0zZ ,
z
=1
Example AR(1)
(
1aB
)
Xt=Zt
ϕ
(
z
)
=1az=0
z=1
a
a
1
Definition
Unlock document

This preview shows pages 1-2 of the document.
Unlock all 6 pages and 3 million more documents.

An
ARMA
(
p , q
)
process is causal if there exist constants
{
ψj
}
such that
j=0
ψj
<
Xj=
i=0
ψiZji
Theorem
Causality is equivalent to
ϕ
(
z
)
0
z
1
AR
(
1
)
causal
a
<1
Find
ψj
s!
Write
Ψ
(
B
)
=1+ψ1B+ψ2B2+
We know
Xt=Ψ
(
B
)
Zt
Meanwhile
Xt=Θ
(
B
)
Φ
(
B
)
Zt
Ψ
(
B
)
=Θ
(
B
)
Φ
(
B
)
Θ
(
B
)
=Ψ
(
B
)
Φ
(
B
)
1+θ1B++θqBq=
(
1+ψ1B+ψ2B2+
) (
1ϕ1Bϕ pBp
)
1st order coefficients:
Unlock document

This preview shows pages 1-2 of the document.
Unlock all 6 pages and 3 million more documents.

## Document Summary

{xt} is called an arma ( p,q) process ( p 0,q 0) if. X t 1 x t 1 p x t p=zt+ 1 zt 1+ + q zt q( ) ( b)=1 1 b 2 b2 p b p. ( z)=1 1 z p z p 0 z z , z =1. 1+ 1 b+ + q b q=(1+ 1 b+ 2 b2+ )(1 1 b p b p) X t=zt + 1 zt 1+ + q z t q q h j j+ h , h q. X t 1 x t 1 p x t p=zt. Ph xh+1=ph( 1 x h+ 2 x h 1+ + p xh p+1+zh +1)= 1 ph x h+ 2 p h x h 1+ + p p h x h p +1+ph zh+1= 1 x h+ 2 x h 1+ + p x h p+1.

# Get access

\$10 USD/m
Billed \$120 USD annually
Homework Help
Class Notes
Textbook Notes