Lesson 10: Covariances of causal ARMA Processes · Umberto Triacca Lesson 10: Covariances of causal...
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Lesson 10: Covariances of causalARMA Processes
Umberto Triacca
Dipartimento di Ingegneria e Scienze dell’Informazione e MatematicaUniversita dell’Aquila,
Umberto Triacca Lesson 10: Covariances of causal ARMA Processes
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A causal ARMA(p, q) process
Consider a causal ARMA(p, q) process
xt −φ1xt−1− ...−φpxt−p = ut + θ1ut−1 + ...+ θqut−q ∀t ∈ Z,
ut ∼ WN(0, σ2u)
φ(z) = 1− φ1z − ...− φpzp
andθ(z) = 1 + θ1z ... + θqz
q
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A causal ARMA(p, q) process
The causality assumption implies that there exists constantsψ0, ψ1, ... such that
∞∑j=0
|ψj | <∞
and
xt =∞∑j=0
ψjut−j ∀t.
The sequence {ψ0, ψ1, ...} is determined by the identity
(1− φ1z − ...− φpzp) (ψ0 + ψ1z ...) = 1 + θ1z ... + θqz
q
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A causal ARMA(p, q) process
Equating coefficients of z j , j=0,1,. . . , we obtain
ψ0 = 1
ψ1 = θ1 + ψ0φ1
ψ2 = θ2 + ψ1φ1 + ψ0φ2
and so on.
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A causal ARMA(p, q) process
We have
ψj = θj +
p∑k=1
φkψj−k , j = 0, 1...,
where θ0 = 1, θj = 0 for j > q and ψj = 0 for j < 0.
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The Autocovariance Function of a causal
ARMA(p, q) process
Since
E (xt) =∞∑j=0
ψjE (ut−j) = 0 ∀t.
We have
γ(k) = E (xtxt−k)
= E(∑∞
j=0 ψjut−j∑∞
i=0 ψiut−k−i)
=∑∞
j=0
∑∞i=0 ψjψiE (ut−jut−k−i)
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The Autocovariance Function of a causal
ARMA(p, q) process
Since ut ∼ WN(0, σ2), we have that E (ut−jut−k−i) = σ2 ifi = j − k and 0 otherwise. Therefore
γ(k) = σ2∞∑j=0
ψjψj−k k = 0, 1, ...
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The Autocovariance Function of a causal
ARMA(p, q) process
It is possible to show that the sequence {γ(k)}, is absolutelysummable, that is
∞∑k=−∞
|γ(k)| <∞.
Further, we observe that the absolute summability of thesequence {γ(k)} implies that
limk→∞
γ(k) = 0.
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The Autocovariance Function of a causal
ARMA(p, q) process
We can summarize our findings about the autocovariancefunction 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.
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Ergodicity and ARMA process
Are the causal ARMA processes ergodic?
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Ergodic Theorems
Corollary 1 (Sufficient condition for mean-ergodicity). Let xtbe a stationary process with mean µ and autocovariancefunction γx(k). If
limk→∞γx(k) = 0,
then xt is mean-ergodic.
Corollary 2. (Sufficient condition for mean-ergodicity) Let xtbe a stationary process with mean µ and and autocovariancefunction γx(k). If
∞∑k=0
|γx(k)| <∞,
then xt is mean-ergodic.
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Ergodicity under Gaussianity
Let {xt ; t ∈ Z} be a stationary process with mean µ andautocovariance function γx(k). If the process is Gaussian, then1. the condition of absolute summability of covariancefunction
∞∑k=0
|γx(k)| <∞
is sufficient to ensure ergodicity for all moments.2. the condition
limk→∞γx(k) = 0
is necessary and sufficient.
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Ergodicity and ARMA process
Thus we have that
a causal ARMA process is ergodic for the mean.
a Gaussian causal ARMA process is ergodic for allmoments.
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The Autocovariance Function of a causal
ARMA(1, 1) process
Consider a causal ARMA(1, 1) process:
xt − φxt−1 = ut + θut−1, ut ∼ WN(0, σ2)
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The Autocovariance Function of a causal
ARMA(1, 1) process
We find that
ψ0 = 1 and ψj = (φ + θ)φj−1 for j ≥ 1.
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The Autocovariance Function of a causal
ARMA(1, 1) process
Thus we have
γ(0) = σ2∑∞
j=0 ψ2j
= σ2[
1 + (φ + θ)2∑∞j=0 φ
2j]
= σ2[
1 + (φ+θ)2
1−φ2
]
γ(1) = σ2∑∞
j=0 ψjψj−1
= σ2[φ + θ + (φ + θ)2 φ
∑∞j=0 φ
2j]
= σ2[φ + θ + (φ+θ)2φ
1−φ2
]and
γ(k) = φk−1γ(1), k ≥ 2.
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The Autocovariance Function of a causal
ARMA(p, q) process
The autocovariance function, γ(k), of a causal ARMA processtends to zero, as k →∞, at an exponential rate, that is, thereexist constants, D and δ such that, as k →∞, γ(k) ≈ Dδk ,with −1 < δ < 1
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The Autocovariance Function of a causal AR(1)
process
Consider a causal AR(1) process:
xt − φxt−1 = ut , ut ∼ WN(0, σ2)
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The Autocovariance Function of a causal AR(1)
process
We haveγ(0) = σ2
(1
1−φ2
)and
γ(k) = φkγ(0), k ≥ 1.
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The Autocovariance Function of a causal AR(1)
process
Consider a causal AR(1) process:
xt − 0.7xt−1 = ut , ut ∼ WN(0, 1)
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The Autocovariance Function of a causal AR(1)
process
Consider a causal AR(1) process:
xt + 0.7xt−1 = ut , ut ∼ WN(0, 1)
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The Autocovariance Function of a causal AR(1)
process
Thus the autocovariances decay exponentially in one of twoforms.
1 If 0 < φ < 1, the autocovariances are positive;
2 If −1 < φ < 0 they oscillate between positive andnegative values.
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The Autocovariance Function of a causal AR(2)
process
Consider a causal AR(2) process:
xt = φ1xt−1 + φ2xt−2 + ut , ut ∼ WN(0, σ2u)
The autocovariance function of this process is given by thefollowing recursive formula:
γx(k) = φ1γx(k − 1) + φ2γx(k − 2) k = 2, 3, ...
with the initial conditions
γx(0) =(1− φ2)σ2
u
(1 + φ2)[
(1− φ2)2 − φ21
]and
γx(1) =φ1γx(0)
(1− φ2).
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The Autocovariance Function of a causal AR(2)
process
Consider a causal AR(2) process:
xt = 0.5xt−1 + 0.3xt−2 + ut , ut ∼ WN(0, 1)
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The Autocovariance Function of a causal AR(2)
process
In general, the autocovariance function of AR(p) processesdecays exponentially.
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The Autocovariance Function of an MA(1) process
Consider an MA(1) process:
xt = ut + θut−1, ut ∼ WN(0, σ2)
We haveγ(0) = σ2 (1 + θ2)
γ(1) = σ2θ
andγ(k) = 0, k ≥ 2.
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The Autocovariance Function of a causal MA(q)
process
In general, the autocovariance function of MA(q) models iszero for all lags greater than q.
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The Autocovariance Function of a causal
ARMA(p, q) process
Causal ARMA processes are short memory processes. Thismeans that their autocovariance function dies out quickly, andhence
∞∑k=−∞
|γ(k)| <∞
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Long memory processes
A stationary process xt with an autocovariance function γ(k)is called a long memory process, if
∞∑k=−∞
|γ(k)| =∞
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Conclusion
In this lecture, we have considered the calculation of theautocovariance function γ(·) of a causal ARMA process.We remenber that the autocorrelation function is readily foundfrom the autocovariance function on dividing by γ(0).The partial autocorrelation function is also found from thefunction γ(·).
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Conclusion
We close this lecture by introducing the lag operator L.
Definition. Lxt = xt−1
Properties.
1 Lj = xt = xt−j2 L(φxt) = φLxt = φxt−1
3 LiLjxt = Lixt−j = xt−j−i = Li+jxt4 L(xt + yt) = Lxt + Lyt = xt−1 + yt−1
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Conclusion
Finally note that by convention we define the zeroth power ofL to be the identity operator, i. e.
L0xt = xt
Thus we can consider polynomials of degree n ≥ 0 in the lagoperator L, defined as
a(L) = a0L0 + a1L + ... + anL
n
where a0, a1, ..., an are real or complex coefficients. We have
a(L)xt = a0xt + a1xt−1 + ... + anxt−n
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Conclusion
It may shown that the algebra of the polynomials of degreen ≥ 0 in the lag operator L, over the field of the real orcomplex numbers (i.e. with real or complex coefficients) isisomorphic to the algebra of polynomials in the real or complexindeterminate, say z .
We can manipulate the polynomials a(L), b(L),... like thepolynomials a(z), b(z),...
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Conclusion
ARMA(1,1) in terms of the lag operator L.
Consider an ARMA(1,1) process defined by the equation
xt − φxt−1 = ut + θut−1 ut ∼ WN(0, σ2u)
By using the lag operator this equation can be rewritten as
φ(L)xt = θ(L)ut
where φ(L) = 1− φL and θ(L) = 1 + θL
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Conclusion
ARMA(p,q) in terms of the lag operator L.
We can writeφ(L)xt = θ(L)ut
whereφ(L) = 1− φ1L− φ2L
2 − ...− φ1Lp
andφ(L) = 1 + θ1L + θ2L
2 − ... + θqLq
.
Umberto Triacca Lesson 10: Covariances of causal ARMA Processes