Lecture 2: ARMA(p,q) models (part 3) - unice.fr

32 Pages·2011·0.47 MB·English

by  Florian Pelgrin

Lecture 2: ARMA(p,q) models
(part 3)
Florian Pelgrin
UniversityofLausanne,E´coledesHEC
Departmentofmathematics(IMEA-Nice)
Sept. 2011 - Jan. 2012
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Introduction
Motivation
Characterize the main properties of ARMA(p,q) models.
Estimation of ARMA(p,q) models
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Introduction
Road map
1 ARMA(1,1) model
Definition and conditions
Moments
Estimation
2 ARMA(p,q) model
Definition and conditions
Moments
Estimation
3 Application
4 Appendix
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ARMA(1,1)model Definitionandconditions
1. ARMA(1,1)
1.1. Definition and conditions
Definition
A stochastic process (Xt)t∈Z is said to be a mixture autoregressive moving
average model of order 1, ARMA(1,1), if it satisfies the following
equation :
X = µ+φX +(cid:15) +θ(cid:15) ∀t
t t−1 t t−1
Φ(L)X = µ+Θ(L)(cid:15)
t t
where θ (cid:54)= 0, θ (cid:54)= 0, µ is a constant term, ((cid:15)t)t∈Z is a weak white noise
process with expectation zero and variance σ2 ((cid:15) ∼ WN(0,σ2)),
(cid:15) t (cid:15)
Φ(L) = 1−φL and Θ(L) = 1+θL.
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ARMA(1,1)model Definitionandconditions
Simulation of an ARMA(1,1) with (-0.5;-0.5) Simulation of an ARMA(1,1) with (-0.5;0.5)
4 10
2 5
0 0
-2 -5
-4 -10
0 100 200 300 0 100 200 300
Simulation of an ARMA(1,1) with (0.9;-0.5) Simulation of an ARMA(1,1) with (0.9;0.5)
10 4
5 2
0 0
-5 -2
-10 -4
0 100 200 300 0 100 200 300

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ARMA(1,1)model Definitionandconditions
The properties of an ARMA(1,1) process are a mixture of those of an
AR(1) and MA(1) processes :
The (stability) stationarity condition is the one of an AR(1) process (or
ARMA(1,0) process) :
|φ|<1.
The invertibility condition is the one of a MA(1) process (or
ARMA(0,1) process) :
|θ|<1.
The representation of an ARMA(1,1) process is fundamental or causal
if :
|φ|<1 and |θ|<1.
The representation of an ARMA(1,1) process is said to be minimal and
causal if :
|φ|<1,|θ|<1 and φ(cid:54)=θ.
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ARMA(1,1)model Definitionandconditions
If (X ) is stable and thus weakly stationary, then (X ) has an infinite
t t
moving average representation (MA(∞)) :
µ
X = +(1−φL)−1(1+θL)(cid:15)
t t
1−φ
∞
µ (cid:88)
= + a (cid:15)
k t−k
1−φ
k=0
where :
a = 1
0
a = φk +θφk−1.
k
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ARMA(1,1)model Definitionandconditions
If (X ) is invertible, then (X ) has an infinite autoregressive
t t
representation (AR(∞)) :
µ
(1−θ∗L)−1(1−φL)X = +(cid:15)
t 1−θ∗ t
i.e.
∞
µ (cid:88)
X = = + b X +(cid:15)
t 1−θ∗ k t−k t
k=1
where θ∗ = −θ, and :
b = −θ∗k −θ∗k−1φ.
k
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ARMA(1,1)model Moments
1.2. Moments
Definition
Let (X ) denote a stationary stochastic process that has a fundamental
t
ARMA(1,1) representation, X = µ+φX +(cid:15) +θ(cid:15) . Then :
t t−1 t t−1
µ
E[X ] = ≡ m
t
1−φ
1+2φθ+θ2
γ (0) ≡ V(X ) = σ2
X t 1−φ2 (cid:15)
(φ+θ)(1+φθ)
γ (1) ≡ Cov [X ,X ] = σ2
X t t−1 1−φ2 (cid:15)
γ (h) = φγ (h−1) for |h| > 1.
X X
Proof : See Appendix 1.
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ARMA(1,1)model Moments
Definition
The autocorrelation function of an ARMA(1,1) process satisfies :

1 if h = 0






ρ (h) = (φ+θ)(1+φθ) if |h| = 1
X 1+2φθ+θ2





 φρ (h−1) if |h| > 1.
X
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