AR(p) + MA(q) = ARMA(p, q)
§3.4: ARMA(p, q) Model Homework 3a
Outline
1 §3.4: ARMA(p, q) Model
2 Homework 3a
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 2/ 12
§3.4: ARMA(p, q) Model Homework 3a
ARMA(p, q) Model
Definition (ARMA(p, q) Model)A time series is ARMA(p, q) if it is stationary and satisfies
Zt = α+ φ1Zt−1 + · · ·+ φpZt−p︸ ︷︷ ︸AR part
+ θ1at−1 + · · ·+ θqat−q︸ ︷︷ ︸MA part
+at (?)
Using the AR and MA operators, we can rewrite (?) as
φ(B)Zt = θ(B)at
The textbook defines the ARMA process with a slightly differentparametrization given by
Zt = α+ φ1Zt−1 + · · ·+ φpZt−p − θ1at−1 − · · · − θqat−q + at (?)
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 3/ 12
§3.4: ARMA(p, q) Model Homework 3a
ARMA(p, q) Model
Definition (ARMA(p, q) Model)A time series is ARMA(p, q) if it is stationary and satisfies
Zt = α+ φ1Zt−1 + · · ·+ φpZt−p︸ ︷︷ ︸AR part
+ θ1at−1 + · · ·+ θqat−q︸ ︷︷ ︸MA part
+at (?)
Using the AR and MA operators, we can rewrite (?) as
φ(B)Zt = θ(B)at
The textbook defines the ARMA process with a slightly differentparametrization given by
Zt = α+ φ1Zt−1 + · · ·+ φpZt−p − θ1at−1 − · · · − θqat−q + at (?)
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 3/ 12
§3.4: ARMA(p, q) Model Homework 3a
ARMA(p, q) Model
Definition (ARMA(p, q) Model)A time series is ARMA(p, q) if it is stationary and satisfies
Zt = α+ φ1Zt−1 + · · ·+ φpZt−p︸ ︷︷ ︸AR part
+ θ1at−1 + · · ·+ θqat−q︸ ︷︷ ︸MA part
+at (?)
Using the AR and MA operators, we can rewrite (?) as
φ(B)Zt = θ(B)at
The textbook defines the ARMA process with a slightly differentparametrization given by
Zt = α+ φ1Zt−1 + · · ·+ φpZt−p − θ1at−1 − · · · − θqat−q + at (?)
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 3/ 12
§3.4: ARMA(p, q) Model Homework 3a
Causality and Invertibility of ARMA(p, q)
Facts:The ARMA(p, q) model given by φ(B)Zt = θ(B)at is causal if and only if
|φ(z)| 6= 0 when |z| ≤ 1
i.e. all roots including complex roots of φ(z) lie outside the unit disk.
The ARMA(p, q) model given by φ(B)Zt = θ(B)at is invertible if and onlyif
|θ(z)| 6= 0 when |θ| ≤ 1
i.e. all roots including complex roots of θ(z) lie outside the unit disk.
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 4/ 12
§3.4: ARMA(p, q) Model Homework 3a
Causality and Invertibility of ARMA(p, q)
Facts:The ARMA(p, q) model given by φ(B)Zt = θ(B)at is causal if and only if
|φ(z)| 6= 0 when |z| ≤ 1
i.e. all roots including complex roots of φ(z) lie outside the unit disk.
The ARMA(p, q) model given by φ(B)Zt = θ(B)at is invertible if and onlyif
|θ(z)| 6= 0 when |θ| ≤ 1
i.e. all roots including complex roots of θ(z) lie outside the unit disk.
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 4/ 12
§3.4: ARMA(p, q) Model Homework 3a
Causality and Invertibility of ARMA(p, q)
Facts:The ARMA(p, q) model given by φ(B)Zt = θ(B)at is causal if and only if
|φ(z)| 6= 0 when |z| ≤ 1
i.e. all roots including complex roots of φ(z) lie outside the unit disk.
The ARMA(p, q) model given by φ(B)Zt = θ(B)at is invertible if and onlyif
|θ(z)| 6= 0 when |θ| ≤ 1
i.e. all roots including complex roots of θ(z) lie outside the unit disk.
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 4/ 12
§3.4: ARMA(p, q) Model Homework 3a
Parameter Redundancy
The processZt = at (?)
is equivalent to.5Zt−1 = .5at−1 (??)
which is also equivalent to (?)− (??), i.e.
Zt − .5Zt−1 = at − .5at−1
Zt in the last representation is still white noise, but has an ARMA(1,1)representation.
We remove redundancies in an ARMA model φ(B)Zt = θ(B)at bycanceling the common factors in θ(B) and φ(B).
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 5/ 12
§3.4: ARMA(p, q) Model Homework 3a
Parameter Redundancy
The processZt = at (?)
is equivalent to.5Zt−1 = .5at−1 (??)
which is also equivalent to (?)− (??), i.e.
Zt − .5Zt−1 = at − .5at−1
Zt in the last representation is still white noise, but has an ARMA(1,1)representation.
We remove redundancies in an ARMA model φ(B)Zt = θ(B)at bycanceling the common factors in θ(B) and φ(B).
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 5/ 12
§3.4: ARMA(p, q) Model Homework 3a
Parameter Redundancy
The processZt = at (?)
is equivalent to.5Zt−1 = .5at−1 (??)
which is also equivalent to (?)− (??), i.e.
Zt − .5Zt−1 = at − .5at−1
Zt in the last representation is still white noise, but has an ARMA(1,1)representation.
We remove redundancies in an ARMA model φ(B)Zt = θ(B)at bycanceling the common factors in θ(B) and φ(B).
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 5/ 12
§3.4: ARMA(p, q) Model Homework 3a
Parameter Redundancy
The processZt = at (?)
is equivalent to.5Zt−1 = .5at−1 (??)
which is also equivalent to (?)− (??), i.e.
Zt − .5Zt−1 = at − .5at−1
Zt in the last representation is still white noise, but has an ARMA(1,1)representation.
We remove redundancies in an ARMA model φ(B)Zt = θ(B)at bycanceling the common factors in θ(B) and φ(B).
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 5/ 12
§3.4: ARMA(p, q) Model Homework 3a
Parameter Redundancy
The processZt = at (?)
is equivalent to.5Zt−1 = .5at−1 (??)
which is also equivalent to (?)− (??), i.e.
Zt − .5Zt−1 = at − .5at−1
Zt in the last representation is still white noise, but has an ARMA(1,1)representation.
We remove redundancies in an ARMA model φ(B)Zt = θ(B)at bycanceling the common factors in θ(B) and φ(B).
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 5/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – Removing Redundancy
ExampleConsider the process
Zt = .4Zt−1 + .45Zt−2 + at + at−1 + .25at−2
which is equivalent to
(1− .4B− .45B2)︸ ︷︷ ︸φ(B)
Zt = (1 + B + .25B2)︸ ︷︷ ︸θ(B)
at
Is this process really ARMA(2,2)?
No!
Factoring φ(z) and θ(z) gives
φ(z) = 1− .4z− .45z2 = (1 + .5z)(1− .9z)
θ(z) = (1 + z + .25z2) = (1 + .5z)2
Removing the common term (1 + .5z) gives the reduced model
Zt = .9Zt−1 + .5at−1 + at
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – Removing Redundancy
ExampleConsider the process
Zt = .4Zt−1 + .45Zt−2 + at + at−1 + .25at−2
which is equivalent to
(1− .4B− .45B2)︸ ︷︷ ︸φ(B)
Zt = (1 + B + .25B2)︸ ︷︷ ︸θ(B)
at
Is this process really ARMA(2,2)?
No!
Factoring φ(z) and θ(z) gives
φ(z) = 1− .4z− .45z2 = (1 + .5z)(1− .9z)
θ(z) = (1 + z + .25z2) = (1 + .5z)2
Removing the common term (1 + .5z) gives the reduced model
Zt = .9Zt−1 + .5at−1 + at
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – Removing Redundancy
ExampleConsider the process
Zt = .4Zt−1 + .45Zt−2 + at + at−1 + .25at−2
which is equivalent to
(1− .4B− .45B2)︸ ︷︷ ︸φ(B)
Zt = (1 + B + .25B2)︸ ︷︷ ︸θ(B)
at
Is this process really ARMA(2,2)?
No!
Factoring φ(z) and θ(z) gives
φ(z) = 1− .4z− .45z2 = (1 + .5z)(1− .9z)
θ(z) = (1 + z + .25z2) = (1 + .5z)2
Removing the common term (1 + .5z) gives the reduced model
Zt = .9Zt−1 + .5at−1 + at
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – Removing Redundancy
ExampleConsider the process
Zt = .4Zt−1 + .45Zt−2 + at + at−1 + .25at−2
which is equivalent to
(1− .4B− .45B2)︸ ︷︷ ︸φ(B)
Zt = (1 + B + .25B2)︸ ︷︷ ︸θ(B)
at
Is this process really ARMA(2,2)? No! Factoring φ(z) and θ(z) gives
φ(z) = 1− .4z− .45z2 = (1 + .5z)(1− .9z)
θ(z) = (1 + z + .25z2) = (1 + .5z)2
Removing the common term (1 + .5z) gives the reduced model
Zt = .9Zt−1 + .5at−1 + at
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – Removing Redundancy
ExampleConsider the process
Zt = .4Zt−1 + .45Zt−2 + at + at−1 + .25at−2
which is equivalent to
(1− .4B− .45B2)︸ ︷︷ ︸φ(B)
Zt = (1 + B + .25B2)︸ ︷︷ ︸θ(B)
at
Is this process really ARMA(2,2)? No! Factoring φ(z) and θ(z) gives
φ(z) = 1− .4z− .45z2 = (1 + .5z)(1− .9z)
θ(z) = (1 + z + .25z2) = (1 + .5z)2
Removing the common term (1 + .5z) gives the reduced model
Zt = .9Zt−1 + .5at−1 + at
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – Removing Redundancy
ExampleConsider the process
Zt = .4Zt−1 + .45Zt−2 + at + at−1 + .25at−2
which is equivalent to
(1− .4B− .45B2)︸ ︷︷ ︸φ(B)
Zt = (1 + B + .25B2)︸ ︷︷ ︸θ(B)
at
Is this process really ARMA(2,2)? No! Factoring φ(z) and θ(z) gives
φ(z) = 1− .4z− .45z2 = (1 + .5z)(1− .9z)
θ(z) = (1 + z + .25z2) = (1 + .5z)2
Removing the common term (1 + .5z) gives the reduced model
Zt = .9Zt−1 + .5at−1 + at
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 6/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – Causality and Invertibility
Example (cont.)
The reduced model is φ(B)Zt = θ(B)at where
φ(z) = 1− .9z
θ(z) = 1 + .5z
There is only one root of φ(z) which is z = 10/9, and |10/9| lies outsidethe unit circle so this ARMA model is causal.
There is only one root of θ(z) which is z = −2, and | − 2| lies outside theunit circle so this ARMA model is invertible.
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 7/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – Causality and Invertibility
Example (cont.)
The reduced model is φ(B)Zt = θ(B)at where
φ(z) = 1− .9z
θ(z) = 1 + .5z
There is only one root of φ(z) which is z = 10/9, and |10/9| lies outsidethe unit circle so this ARMA model is causal.
There is only one root of θ(z) which is z = −2, and | − 2| lies outside theunit circle so this ARMA model is invertible.
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 7/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – Causality and Invertibility
Example (cont.)
The reduced model is φ(B)Zt = θ(B)at where
φ(z) = 1− .9z
θ(z) = 1 + .5z
There is only one root of φ(z) which is z = 10/9, and |10/9| lies outsidethe unit circle so this ARMA model is causal.
There is only one root of θ(z) which is z = −2, and | − 2| lies outside theunit circle so this ARMA model is invertible.
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 7/ 12
§3.4: ARMA(p, q) Model Homework 3a
ARMA Mathematics
Start with the ARMA equation
φ(B)Zt = θ(B)at
To solve for Zt, we simply divide! That is
Zt =θ(B)φ(B)︸ ︷︷ ︸ψ(B)
at “MA(∞) representation”
Similarly, solving for at gives
at =φ(B)θ(B)︸ ︷︷ ︸π(B)
Zt “AR(∞) representation”
The key is to figuring out the ψj and πj in
ψ(B) = 1 +∞∑
j=1
ψjBj ψ(B) = 1 +∞∑
j=1
ψjBj
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 8/ 12
§3.4: ARMA(p, q) Model Homework 3a
ARMA Mathematics
Start with the ARMA equation
φ(B)Zt = θ(B)at
To solve for Zt, we simply divide! That is
Zt =θ(B)φ(B)︸ ︷︷ ︸ψ(B)
at “MA(∞) representation”
Similarly, solving for at gives
at =φ(B)θ(B)︸ ︷︷ ︸π(B)
Zt “AR(∞) representation”
The key is to figuring out the ψj and πj in
ψ(B) = 1 +∞∑
j=1
ψjBj ψ(B) = 1 +∞∑
j=1
ψjBj
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 8/ 12
§3.4: ARMA(p, q) Model Homework 3a
ARMA Mathematics
Start with the ARMA equation
φ(B)Zt = θ(B)at
To solve for Zt, we simply divide! That is
Zt =θ(B)φ(B)︸ ︷︷ ︸ψ(B)
at “MA(∞) representation”
Similarly, solving for at gives
at =φ(B)θ(B)︸ ︷︷ ︸π(B)
Zt “AR(∞) representation”
The key is to figuring out the ψj and πj in
ψ(B) = 1 +∞∑
j=1
ψjBj ψ(B) = 1 +∞∑
j=1
ψjBj
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 8/ 12
§3.4: ARMA(p, q) Model Homework 3a
ARMA Mathematics
Start with the ARMA equation
φ(B)Zt = θ(B)at
To solve for Zt, we simply divide! That is
Zt =θ(B)φ(B)︸ ︷︷ ︸ψ(B)
at “MA(∞) representation”
Similarly, solving for at gives
at =φ(B)θ(B)︸ ︷︷ ︸π(B)
Zt “AR(∞) representation”
The key is to figuring out the ψj and πj in
ψ(B) = 1 +∞∑
j=1
ψjBj ψ(B) = 1 +∞∑
j=1
ψjBj
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 8/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – MA(∞) RepresentationExample (cont.)Note that
ψ(z) =θ(z)φ(z)
=1 + .5z1− .9z
= (1 + .5z)(1 + .9z + .92z2 + .93z3 + · · · ) for |z| ≤ 1
= 1 + (.5 + .9)z + (.5(.9) + .92)z2 + (.5(.92) + .93)z3 + · · · for |z| ≤ 1
= 1 + (.5 + .9)z + (.5 + .9)(.9)z2 + (.5 + .9)(.92)z3 + · · · for |z| ≤ 1
= 1 + (.5 + .9)︸ ︷︷ ︸1.4
∞∑j=1
.9j−1zj for |z| ≤ 1
This gives the following MA(∞) representation:
Zt =θ(B)φ(B)
at =
1 + 1.4∞∑
j=1
.9j−1Bj
at = at + 1.4∞∑
j=1
.9j−1at−j
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 9/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – MA(∞) RepresentationExample (cont.)Note that
ψ(z) =θ(z)φ(z)
=1 + .5z1− .9z
= (1 + .5z)(1 + .9z + .92z2 + .93z3 + · · · ) for |z| ≤ 1
= 1 + (.5 + .9)z + (.5(.9) + .92)z2 + (.5(.92) + .93)z3 + · · · for |z| ≤ 1
= 1 + (.5 + .9)z + (.5 + .9)(.9)z2 + (.5 + .9)(.92)z3 + · · · for |z| ≤ 1
= 1 + (.5 + .9)︸ ︷︷ ︸1.4
∞∑j=1
.9j−1zj for |z| ≤ 1
This gives the following MA(∞) representation:
Zt =θ(B)φ(B)
at =
1 + 1.4∞∑
j=1
.9j−1Bj
at = at + 1.4∞∑
j=1
.9j−1at−j
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 9/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – AR(∞) Representation
Example (cont.)Similarly, one can show
π(z) =φ(z)θ(z)
=1− .9z1 + .5z
...
= 1− 1.4∞∑
j=1
(−.5)j−1zj for |z| ≤ 1
This gives the following AR(∞) representation:
Zt = 1.4∞∑
j=1
(−.5)j−1at−j + at
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 10/ 12
§3.4: ARMA(p, q) Model Homework 3a
Example – AR(∞) Representation
Example (cont.)Similarly, one can show
π(z) =φ(z)θ(z)
=1− .9z1 + .5z
...
= 1− 1.4∞∑
j=1
(−.5)j−1zj for |z| ≤ 1
This gives the following AR(∞) representation:
Zt = 1.4∞∑
j=1
(−.5)j−1at−j + at
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 10/ 12
§3.4: ARMA(p, q) Model Homework 3a
Outline
1 §3.4: ARMA(p, q) Model
2 Homework 3a
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 11/ 12
§3.4: ARMA(p, q) Model Homework 3a
Homework 3a
Read §4.1 and §4.2.
Derive the equation for π(z) on slide 10.Do exercise #3.14(a) only for models (i), (ii), and (iv)Do exercise #3.14(b,c) only for model (ii)
Arthur Berg AR(p) + MA(q) = ARMA(p, q) 12/ 12