TimeSeriesBook.pdf

First one is a method of moments procedure where the

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first one is a method of moments procedure where the theoretical moments are equated to the empirical ones. This procedure is known under the name of Yule-Walker estimator. The second procedure interprets the stochastic difference as a regression model and estimates the parameters by ordinary least-squares (OLS). These two methods work well if the underlying model is just an AR model and thus involves no MA terms. If the model comprises MA terms, a maximum likelihood (ML) approach must be pursued. 93
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94 CHAPTER 5. ESTIMATION OF ARMA MODELS 5.1 The Yule-Walker estimator We assume that the stochastic process is governed by a causal purely autore- gressive model of order p and mean zero: Φ(L) X t = Z t with Z t WN(0 , σ 2 ) . Causality with respect to { Z t } implies that there exists a sequence { ψ j } with j =0 | ψ j | < such that X t = Ψ(L) Z t . Multiplying the above differ- ence equation by X t - j , j = 0 , 1 , . . . , p and taking expectations leads to the following equation system for the parameters Φ = ( φ 1 , . . . , φ p ) 0 and σ 2 : γ (0) - φ 1 γ (1) - . . . - φ p γ ( p ) = σ 2 γ (1) - φ 1 γ (0) - . . . - φ p γ ( p - 1) = 0 . . . γ ( p ) - φ 1 γ ( p - 1) - . . . - φ p γ (0) = 0 This equation system is known as the Yule-Walker equations . It can be written compactly in matrix algebra as: γ (0) - Φ 0 γ p (1) = σ 2 , γ (0) γ (1) . . . γ ( p - 1) γ (1) γ (0) . . . γ ( p - 2) . . . . . . . . . . . . γ ( p - 1) γ ( p - 2) . . . γ (0) φ 1 φ 2 . . . φ p = γ (1) γ (2) . . . γ ( p ) , respectively γ (0) - Φ 0 γ p (1) = σ 2 , Γ p Φ = γ p (1) . The Yule-Walker estimator is obtained by replacing the theoretical mo- ments by the empirical ones and solving the resulting equation system for the unknown parameters: b Φ = b Γ - 1 p b γ p (1) = b R - 1 p b ρ p (1) b σ 2 = b γ (0) - b Φ 0 b γ p (1) = b γ (0) 1 - b ρ p (1) 0 b R - 1 p b ρ p (1) Note the recursiveness of the equation system: the estimate b Φ is obtained without knowledge of b σ 2 as the estimator b R - 1 p b ρ p (1) involves only autocor-
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5.1. THE YULE-WALKER ESTIMATOR 95 relations. The estimates b Γ p , b R p , b γ p (1), b ρ p (1), and b γ (0) are obtained in the usual way as explained in Chapter 4. 1 The construction of the Yule-Walker estimator implies that the first p val- ues of the autocovariance, respectively the autocorrelation function, implied by the estimated model exactly correspond to their estimated counterparts. It can be shown that this moment estimator always delivers coefficients b Φ which imply that { X t } is causal with respect to { Z t } . In addition, the fol- lowing Theorem establishes that the estimated coefficients are asymptotically normal. Theorem 5.1 (Asymptotic property of Yule-Walker estimator) . Let { X t } be an AR(p) process which is causal with respect to { Z t } whereby { Z t } ∼ IID(0 , σ 2 ) . Then the Yule-Walker estimator is consistent and b Φ is asymptot- ically normal with distribution given by: T b Φ - Φ d ----→ N ( 0 , σ 2 Γ - 1 p ) .
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