chap5 - ARMA Modeling and Forecasting(Chap 5 5.1...

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1 1 ARMA Modeling and Forecasting (Chap 5) 5.1 Preliminary Estimation Useful for order identification (requires the fitting of a number of competing models). initial parameter estimates for likelihood optimization. ARMA(p,q) Model: Based on observations x 1 ,..., x n , from the model φ( B) X t = θ( B) Z t , { Z t } ~ WN(0, σ 2 ), want to estimate φ = φ =(φ 1 , . . ., φ p ) and θ = θ =(θ 1 , . . ., θ p ) , where the orders p and q are assumed known (for the moment). 2 AR(p) Processes: Yule-Walker Estimation (moment estimates) Burg Estimation ARMA(p,q) Processes: Innovations Algorithm Hannan-Rissanen Estimates
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2 3 Recall from Chapter 3, that γ( 0 )−φ 1 γ (1) −− φ p γ (p) = σ 2 γ( k )−φ 1 γ (k-1) φ p γ (k-p) = 0, k=1, . . .,p or in matrix form, Γ p φ = φ = γ p where Γ p is the covariance matrix of X 1 , . . ., X p , and φ = φ = (φ 1 , . . ., φ p )’, γ p = (γ(1) , . . ., γ (p))’. The Y-W estimates are then found by replacing the ACVF γ( ), by its estimated value γ( ) . Yule-Walker Estimates (for AR(p) processes): L L 4 Remarks: The fitted model is causal. The sample ACVF and fitted model ACVF agree at lags h=0,1,. . . ,p. Estimates are asymptotically efficient (i.e. have the same limit behavior as MLE). φ Ν(φ φ φ , n -1 σ 2 Γ p -1 ). Sample Yule-Walker Equations: φ = φ = Γ p −1 γ p , σ 2 = γ(0) − γ p Γ p −1 γ p . (These estimates are computed in ITSM . )
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3 5 Usually there is no true AR model. Goal is to find an AR model which represents the data in some sense. Two Techniques. If { X t } is an AR(p) process with { Z t } ~ IID(0, σ 2 ), then φ mm = α( m ) ∼ Ν(0, n -1 ) for all m > p. ( α( m ) is the sample PACF). Estimate p as the smallest value m such that | α( k ) | < 1.96 n -.5 for k > m. Estimate p by minimizing the AICC statistic AICC = 2ln L( φ p ) + 2(p+1)n/(n p −2 ) where L( ) denotes the Gaussian likelihood. Order Selection for AR Models 6 Ex 5.1.1 (Dow-Jones Utilities Index; DOWJ.DAT ). Plot of D 1 , . . ., D 78 0 1 02 03 04 05 06 07 08 0 110 115 120 Lag ACF 0 -0.4 0.0 0.2 0.4 0.6 0.8 1.0
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4 7 Lag ACF 0 5 10 15 20 25 30 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Fig 5.1 ACF of differenced series Y t = D t D t-1 Lag PACF 0 5 10 15 20 25 30 Fig 5.2 8 Preliminary Model for Dow-Jones: Using the Preliminary Estimation option of ITSM, the fitted model is X t = .4219 X t-1 + Z t , { Z t } ~ WN(0, .1479) (.094) where X t = Y t .1336, Y t = D t D t-1 , Remark: We can also arrive at this model using the automatic AICC minimization option in Preliminary Estimation.
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5 9 The Yule-Walker estimates ( φ 1 , . . . , φ p ) satisfy P p X p+1 = φ 1 X p + + φ p X 1 where P p is the prediction operator relative to the fitted Y-W model.
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This note was uploaded on 06/16/2009 for the course STAT 525 taught by Professor Brockwell during the Spring '09 term at Colorado State.

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chap5 - ARMA Modeling and Forecasting(Chap 5 5.1...

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