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notes44 - Notes 4 Models for Stationary Time Series :...

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Unformatted text preview: Notes 4 Models for Stationary Time Series : General Linear Processes A general linear process { Z t } is one that can be represented as a weighted linear combination of the present and past terms of a white noise process: Z t = a t + ψ 1 a t- 1 + ψ 2 a t- 2 + . . . (1) = ψ ( B ) a t where ψ ( B ) = 1 + ψ 1 B + ψ 2 B 2 + . . . , B is a ‘backward shift operator’ defined by BZ t = Z t- 1 , B k Z t = Z t- k . If the RHS of (1) truly an infinite series, then certain restrictions must be placed on the ψ s for the RHS to be mathematically meaningful. For our purposes, it suffices to assume that ∞ X i =1 ψ 2 i < ∞ as V ar ( Z t ) = ( ∑ ∞ i =0 ψ 2 i ) σ 2 a . We should note that since { a t } is unobservable, there is no loss of generality to assume the coefficient of a t is 1 (we put ψ = 1). Example: For ψ j = φ j , | φ | < 1, we have Z t = a t + φa t- 1 + φ 2 a t- 2 + . . . . E ( Z t ) = 0 V ar ( Z t ) = V ar ( a t + φa t- 1 + φ 2 a t- 2 + . . . ) = V ar ( a t ) + φ 2 V ar ( a t- 1 ) + φ 4 V ar ( a t- 2 ) + . . . = σ 2 a (1 + φ 2 + φ 4 + . . . ) = σ 2 a 1 1- φ 2 γ 1 = Cov ( Z t , Z t- 1 ) = Cov ( a t + φa t- 1 + φ 2 a t- 2 + . . . , a t- 1 + φa t- 2 + φ 2 a t- 3 + . . . ) = Cov ( φa t- 1 , a t- 1 ) + Cov ( φ 2 a t- 2 , φa t- 2 ) + . . . = φσ 2 a + φ 3 σ 2 a + φ 5 σ 2 a + . . . = φσ 2 a (1 + φ 2 + φ 4 + . . . ) = φσ 2 a 1- φ 2 1 Thus ρ 1 = Corr ( Z t , Z t- 1 ) = φ . Similarly, γ k = Cov ( Z t , Z t- k ) = φ k σ 2 a 1- φ 2 , k = 0 , 1 , . . . . and ρ k = Corr ( Z t , Z t- k ) = φ k , k = 0 , 1 , . . . . For a general linear process, Z t = a t + ψ 1 a t- 1 + ψ 2 a t- 2 + . . . , E ( Z t ) = 0 , γ k = σ 2 a ∞ X i =0 ψ i ψ i + k , k ≥ , ψ = 1 . A process with a nonzero mean μ may be obtained by adding μ to the RHS of (1) of the general linear process. Since the mean does not affect the covariance (or correla- tion) structure of a process, we shall assume a zero mean until we begin fitting the models to the data. Moving Average (MA) Process In the case where only a finite number of ψ s are nonzero, we have what is called a moving average process. In this case, we write Z t = a t- θ 1 a t- 1- θ 2 a t- 2- . . .- θ q a t- q . We call such a series a moving average of order q [Notation: MA( q )]. Here ψ = 1 , ψ 1 =- θ 1 , , . . . , ψ q =- θ q , θ j = 0 for all j > q . Example: MA(1). Z t = a t- θa t- 1 γ = V ar ( Z t ) = σ 2 a (1 + θ 2 ) γ 1 = Cov ( Z t , Z t- 1 ) =- θσ 2 a γ k = 0 , k ≥ 2 ρ 1 =- θ 1 + θ 2 ρ k = 0 , k ≥ 2 . Example: MA(2). Z t = a t- θ 1 a t- 1- θ 2 a t- 2 γ = V ar ( Z t ) = σ 2 a (1 + θ 2 1 + θ 2 2 ) γ 1 = Cov ( Z t , Z t- 1 ) = Cov ( a t- θ 1 a t- 1- θ 2 a t- 2 , a t- 1- θ 1 a t- 2- θ 2 a t- 3 ) = (- θ 1 + θ 1 θ 2 ) σ 2 a γ 2 = Cov ( Z t , Z t- 2 ) = Cov ( a t- θ 1 a t- 1- θ 2 a t- 2 , a t- 2- θ 1 a t- 3- θ 2 a t- 4 ) =- θ 2 σ 2 a γ k = 0 , k ≥ 3 2 ρ 1 =- θ 1 + θ 1 θ 2 1 + θ 2 1 + θ 2 2 ρ 2 =- θ 2 1 + θ 2 1...
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This note was uploaded on 01/20/2012 for the course STA 4005 taught by Professor ? during the Spring '08 term at CUHK.

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notes44 - Notes 4 Models for Stationary Time Series :...

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