Autoregressive-Moving Average (ARMA) models

34 Pages·2014·0.62 MB·English

by  Umberto Triacca

Lesson 9: Autoregressive-Moving
Average (ARMA) models
Umberto Triacca
DipartimentodiIngegneriaeScienzedell’InformazioneeMatematica
Universit`adell’Aquila,
[email protected]
UmbertoTriacca Lesson9:Autoregressive-MovingAverage(ARMA)models
Introduction
We have seen that in the class of stationary, zero mean,
Gaussian processes the probabilistic structure of a stochastic
process is completly characterized by the autocovariance
function.
UmbertoTriacca Lesson9:Autoregressive-MovingAverage(ARMA)models
Autocovariance function
Stationary, zero mean, Gaussian process
(cid:39)(cid:36)
DGP γ (k)
x
(cid:38)(cid:37)
(cid:19)(cid:55)
(cid:19)
(cid:19)
(cid:19)
(cid:19)
(cid:19)
(cid:19)
(cid:19)
(cid:19)
(cid:19)
(cid:27) (cid:63)(cid:19) (cid:24)
x ,...,x
1 T
(cid:26) (cid:25)
UmbertoTriacca Lesson9:Autoregressive-MovingAverage(ARMA)models
Introduction
However, in general, to know the autocovariance function
means to know a sequence composed by an infinite number of
elements.
We have to estimate a infinite number of parameters
γ (0),γ (1),γ (2),...,
x x x
from observed data.
This mission is impossible
UmbertoTriacca Lesson9:Autoregressive-MovingAverage(ARMA)models
Introduction
We introduce a very important class of stochastic processes,
which autocovariance functions depend on a finite number of
unknown parameters:
the class of the AutoregRessive Moving Average (ARMA)
processes.
UmbertoTriacca Lesson9:Autoregressive-MovingAverage(ARMA)models
Autoregressive-Moving Average (ARMA) models
Definition. The process {x ;t ∈ Z} is an autoregressive
t
moving average process of order (p,q), denoted with
x ∼ ARMA(p,q),
t
if
x −φ x −...−φ x = u +θ u +...+θ u ∀t ∈ Z,
t 1 t−1 p t−p t 1 t−1 q t−q
where u ∼ WN(0,σ2), and φ ,...,φ ,θ ,...,θ are p +q
t u 1 p 1 q
constants and the polynomials
φ(z) = 1−φ z −...−φ zp
1 p
and
θ(z) = 1+θ z...+θ zq
1 q
have no common factors.
UmbertoTriacca Lesson9:Autoregressive-MovingAverage(ARMA)models
Autoregressive-Moving Average (ARMA) models
For q = 0 the process reduces to an autoregressive process of
order p, denoted with x ∼ AR(p),
t
x −φ x −...−φ x = u ∀t ∈ Z,
t 1 t−1 p t−p t
For p = 0 to a moving average process of order q, denoted
with x ∼ MA(q)
t
x = u +θ u +...+θ u ∀t ∈ Z,
t t 1 t−1 q t−q
UmbertoTriacca Lesson9:Autoregressive-MovingAverage(ARMA)models
An example of Autoregressive-Moving Average
(ARMA) process
The process {x ;t ∈ Z} defined by
t
x = 0.3x +u +0.7u ∀t ∈ Z,
t t−1 t t−1
where u ∼ WN(0,σ2), is an ARMA(1,1) process.
t u
Here
φ(z) = 1−0.3z
and
θ(z) = 1+0.7z.
UmbertoTriacca Lesson9:Autoregressive-MovingAverage(ARMA)models
An example of Autoregressive-Moving Average
(ARMA) process
A realizzation of the ARMA(1,1) process
x = 0.3x +u +0.7u is presented in the following figure.
t t−1 t t−1
UmbertoTriacca Lesson9:Autoregressive-MovingAverage(ARMA)models
An example of Autoregressive (AR) process
The process {x ;t ∈ Z} defined by
t
x = 0.7x −0.5x +u ∀t ∈ Z,
t t−1 t−1 t
where u ∼ WN(0,σ2), is an AR(2) process.
t u
Here
φ(z) = 1−0.7z +0.5z2
UmbertoTriacca Lesson9:Autoregressive-MovingAverage(ARMA)models