ARMA(p,q) models

40 Pages·2011·0.72 MB·English

by  Florian Pelgrin

Lecture 2: ARMA(p,q) models
(part 2)
Florian Pelgrin
UniversityofLausanne,E´coledesHEC
Departmentofmathematics(IMEA-Nice)
Sept. 2011 - Jan. 2012
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Introduction
Motivation
Characterize the main properties of MA(q) models.
Estimation of MA(q) models
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Introduction
Road map
1 Introduction
2 MA(1) model
3 Application of a ”counterfactual” MA(1)
4 Moving average model of order q, MA(q)
5 Application of a MA(q) model
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MA(1)model
Moving average models
2.1. Moving average model of order 1, MA(1)
Definition
A stochastic process (Xt)t∈Z is said to be a moving average model of
order 1 if it satisfies the following equation :
X = µ+(cid:15) −θ(cid:15) ∀t
t t t−1
where 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)
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MA(1)model
Remarks :
1. In lag notation, one has :
X = µ+Θ(L)(cid:15) ≡ µ+(1−θL)(cid:15)
t t t
2. The previous process can be written in mean-deviation as follows :
X˜ = (cid:15) −θ(cid:15)
t t t−1
where
X˜ = X −µ.
t t
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MA(1)model
Remarks (cont’d) :
3. The properties of (X ) only depend on those of the weak white noise
t
process ((cid:15) ). To some extent, the behavior of (X ) is more noisy
t t
relative to an AR(1) process...
4. Iterating on the past infinite (and with some regularity conditions),
the infinite autoregressive representation writes :
∞
µ (cid:88)
X = + θkX +(cid:15) .
t t−k t
1−θ
k=1
5. The infinite autoregressive representation illustrates the fact that a
certain form of persistence is captured by a moving average model,
especially when θ is close to one.
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MA(1)model
Simulation of a moving average process of order 1 (θ = 0.9)

4
3
2
1
0
-1
-2
-3
-4
-5
50 100 150 200 250 300 350 400 450 500

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MA(1)model
Scatter plots of a moving average process of order 1: Left panel (Xt-1 versus Xt) and right panel (Xt-2 versus Xt)

4 4
3 3
2 2
1 1
X_(t-1) -10 X_(t-2) -10
-2 -2
-3 -3
-4 -4
-5 -5
-5 -4 -3 -2 -1 0 1 2 3 4 -5 -4 -3 -2 -1 0 1 2 3 4
X_t X_t

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Figure: Scatter plots
MA(1)model
Stationarity and invertibility conditions
Since ((cid:15) ) is a weak noise process, (X ) is weakly stationary (by
t t
definition).
The invertibility condition is the counterpart of the stability
(stationary) condition of an AR(1) process :
1 If |θ|<1, then (Xt) is invertible.
2 If |θ|=1, then (Xt) is non invertible.
3 If |θ|>1, there exists a non-causal invertible representation of (Xt)
that we rule out.
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MA(1)model
Alternatively, if |θ| < 1, then :
∞
(cid:88)
(1−θL)−1 = θkLk
k=0
and
(cid:15) = (1−θL)−1(X −µ)
t t
i.e.
∞
µ (cid:88)
X = + θkX +(cid:15)
t t−k t
1−θ
k=1
1 This is the infinite autoregressive representation of a MA(1) process.
2 The MA(1) representation is then called the fundamental or causal
representation.
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