Lesson 10: Covariances of causal ARMA Processes

35 Pages·2014·0.22 MB·English

by  Umberto Triacca

Lesson 10: Covariances of causal
ARMA Processes
Umberto Triacca
DipartimentodiIngegneriaeScienzedell’InformazioneeMatematica
Universit`adell’Aquila,
[email protected]
UmbertoTriacca Lesson10:CovariancesofcausalARMAProcesses
A causal ARMA(p,q) process
Consider a causal ARMA(p,q) process
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
u ∼ WN(0,σ2)
t u
φ(z) = 1−φ z −...−φ zp
1 p
and
θ(z) = 1+θ z...+θ zq
1 q
UmbertoTriacca Lesson10:CovariancesofcausalARMAProcesses
A causal ARMA(p,q) process
The causality assumption implies that there exists constants
ψ ,ψ ,... such that
0 1
∞
(cid:88)
|ψ | < ∞
j
j=0
and
∞
(cid:88)
x = ψ u ∀t.
t j t−j
j=0
The sequence {ψ ,ψ ,...} is determined by the identity
0 1
(1−φ z −...−φ zp)(ψ +ψ z...) = 1+θ z...+θ zq
1 p 0 1 1 q
UmbertoTriacca Lesson10:CovariancesofcausalARMAProcesses
A causal ARMA(p,q) process
Equating coefficients of zj, j=0,1,..., we obtain
ψ = 1
0
ψ = θ +ψ φ
1 1 0 1
ψ = θ +ψ φ +ψ φ
2 2 1 1 0 2
and so on.
UmbertoTriacca Lesson10:CovariancesofcausalARMAProcesses
A causal ARMA(p,q) process
We have
p
(cid:88)
ψ = θ + φ ψ , j = 0,1...,
j j k j−k
k=1
where θ = 1, θ = 0 for j > q and ψ = 0 for j < 0.
0 j j
UmbertoTriacca Lesson10:CovariancesofcausalARMAProcesses
The Autocovariance Function of a causal
ARMA(p,q) process
Since
∞
(cid:88)
E (x ) = ψ E (u ) = 0 ∀t.
t j t−j
j=0
We have
γ(k) = E (x x )
t t−k
(cid:16) (cid:17)
= E (cid:80)∞ ψ u (cid:80)∞ ψ u
j=0 j t−j i=0 i t−k−i
= (cid:80)∞ (cid:80)∞ ψ ψ E (u u )
j=0 i=0 j i t−j t−k−i
UmbertoTriacca Lesson10:CovariancesofcausalARMAProcesses
The Autocovariance Function of a causal
ARMA(p,q) process
Since u ∼ WN(0,σ2), we have that E (u u ) = σ2 if
t t−j t−k−i
i = j −k and 0 otherwise. Therefore
∞
(cid:88)
γ(k) = σ2 ψ ψ k = 0,1,...
j j−k
j=0
UmbertoTriacca Lesson10:CovariancesofcausalARMAProcesses
The Autocovariance Function of a causal
ARMA(p,q) process
It is possible to show that the sequence {γ(k)}, is absolutely
summable, that is
∞
(cid:88)
|γ(k)| < ∞.
k=−∞
Further, we observe that the absolute summability of the
sequence {γ(k)} implies that
lim γ(k) = 0.
k→∞
UmbertoTriacca Lesson10:CovariancesofcausalARMAProcesses
The Autocovariance Function of a causal
ARMA(p,q) process
We can summarize our findings about the autocovariance
function of a causal ARMA(p,q) process as follows:
The autocovariance function of a causal ARMA(p,q)
process is absolutely summable;
The autocovariance function of a causal ARMA(p,q)
process vanishes when the lag tends to infinity.
UmbertoTriacca Lesson10:CovariancesofcausalARMAProcesses
Ergodicity and ARMA process
Are the causal ARMA processes ergodic?
UmbertoTriacca Lesson10:CovariancesofcausalARMAProcesses