7 - Notes 7 Parameter estimation In general for AR(p MA(q...

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Notes 7 Parameter estimation : In general, for AR(p), MA(q) and ARMA(p,q) models, what we want to estimate are the series mean μ , the AR parameters φ ’s, the MA parameters θ ’s and the error variance σ 2 a . Estimation methods: 1. Method of moment 2. Method based on Least squares estimation 3. Method based on Unconditional least squares estimation 4. Maximum likelihood estimation 1. Method of moment The method consists of equating sample moments to theoretical moments and solving the resultant equations for obtaining estimates of the unknown parameters. (a) For time series mean μ , the estimator is the sample mean ¯ Z . (b) For AR(p), from Yule Walker equations ρ 1 ρ 2 . . ρ p = ρ 0 ρ 1 . . ρ p - 1 ρ 1 ρ 0 ρ 1 . ρ p - 2 . . . . . . . . . . ρ p - 1 ρ p - 1 . . ρ 0 φ 1 φ 2 . . φ p . Replace ρ k ’s by r k ’s and then solve for ˆ φ 1 , ˆ φ 2 ,... , ˆ φ p in terms of r 1 ,r 2 ,...,r p . Example. For an AR(1) process. Since ρ 1 = φ 1 , we have ˆ φ 1 = r 1 . Example. For an AR(2) process. Since ρ 1 = φ 1 + ρ 1 φ 2 and ρ 2 = ρ 1 φ 1 + φ 2 . 1
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ˆ φ 1 = r 1 (1 - r 2 ) 1 - r 2 1 and ˆ φ 2 = r 2 - r 2 1 1 - r 2 1 . (c) MA(q) - the method of moments may cause problem. For example, let us consider a MA(1) process, recall that ρ 1 = - θ 1 + θ 2 ρ 1 θ 2 + θ + ρ 1 = 0 θ = - 1 ± q 1 - 4 ρ 2 1 2 ρ 1 . Therefore ˆ θ = - 1 ± q 1 - 4 r 2 1 2 r 1 . If | r 1 | < 0 . 5, two roots, say θ 1 , θ 2 exist but only one of the roots satisfies the invertibility condition since θ 1 θ 2 = 1. If | r 1 | = 0 . 5, θ = ± 1, neither is invertible. If | r 1 | > 0 . 5, no real roots exists, and the method of moments fails to yield an estimate of θ . Clearly, for higher MA(q) model, the method of moment are more complicated. d) ARMA(p,q) model. Example. For an ARMA(1,1) process Recall that ρ k = ( φ - θ )(1 - φθ ) (1 - 2 φθ + θ 2 ) φ k - 1 , k 1 ρ 2 = ( φ - θ )(1 - φθ ) (1 - 2 φθ + θ 2 ) φ ρ 1 = ( φ - θ )(1 - φθ ) (1 - 2 φθ + θ 2 ) . Therefore
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This note was uploaded on 11/02/2011 for the course STAT 4005 taught by Professor Wu,kaho during the Spring '08 term at CUHK.

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7 - Notes 7 Parameter estimation In general for AR(p MA(q...

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