MA, AR and ARMA

65 Pages·2008·0.72 MB·English

by  Matthieu Stigler

Stationary models
MA, AR and ARMA
Matthieu Stigler
November 14, 2008
Version 1.1
ThisdocumentisreleasedundertheCreativeCommonsAttribution-Noncommercial2.5India
license.
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Lectures list
1 Stationarity
2 ARMA models for stationary variables
3 Seasonality
4 Non-stationarity
5 Non-linearities
6 Multivariate models
7 Structural VAR models
8 Cointegration the Engle and Granger approach
9 Cointegration 2: The Johansen Methodology
10 Multivariate Nonlinearities in VAR models
11 Multivariate Nonlinearities in VECM models
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Outline
1 Last Lecture
2 AR(p) models
Autocorrelation of AR(1)
Stationarity Conditions
Estimation
3 MA models
ARMA(p,q)
The Box-Jenkins approach
4 Forecasting
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Recall: auto-covariance
Definition (autocovariance)
Cov(X ,X ) ≡ γ (t) ≡ E[(X −µ)(X −µ)]
t t−k k t t−k
Definition (Autocorrelation)
Corr(X ,X ) ≡ ρ (t) ≡ Cov(Xt,Xt−k)
t t−k k Var(Xt)
Proposition
Corr(X ,X ) = Var(X )
t t−0 t
Corr(X ,X ) = φj depend on the lage: plot its values at each lag.
t t−j
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Recall: stationarity
The stationarity is an essential property to define a time series process:
Definition
A process is said to be covariance-stationary, or weakly stationary, if
its first and second moments are time invariant.
E(Y ) = E[Y ] = µ ∀t
t t−1
Var(Y ) = γ < ∞ ∀t
t 0
Cov(Y ,Y ) = γ ∀t, ∀k
t t−k k
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Recall: The AR(1)
The AR(1): Y = c +ϕY +ε ε ∼ iid(0,σ2)
t t−1 t t
with |ϕ| < 1, it can be can be written as:
t−1
c (cid:88)
Y = + ϕiε
t t−i
1−ϕ
i=0
Its ’moments’ do not depend on the time: :
E(X ) = c
t 1−ϕ
Var(X ) = σ2
t 1−ϕ2
Cov(X ,X ) = ϕj σ2
t t−j 1−ϕ2
Corr(X ,X ) = φj
t t−j
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Outline
1 Last Lecture
2 AR(p) models
Autocorrelation of AR(1)
Stationarity Conditions
Estimation
3 MA models
ARMA(p,q)
The Box-Jenkins approach
4 Forecasting
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Outline
1 Last Lecture
2 AR(p) models
Autocorrelation of AR(1)
Stationarity Conditions
Estimation
3 MA models
ARMA(p,q)
The Box-Jenkins approach
4 Forecasting
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Autocorrelation function
A usefull plot to understand the dynamic of a process is the
autocorrelation function:
Plot the autocorrelation value for different lags.
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ff ==0 ff ==0.5
8 8
0. 0.
k k
rr 0.4 rr 0.4
0 0
0. 0.
2 4 6 8 10 12 14 2 4 6 8 10 12 14
Lag k Lag k
ff ==0.9 ff ==1
8 8
0. 0.
k k
rr 0.4 rr 0.4
0 0
0. 0.
2 4 6 8 10 12 14 2 4 6 8 10 12 14
Lag k Lag k
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