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# notes44 - Notes 4 Models for Stationary Time Series General...

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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 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 0 = 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

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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 i =0 ψ i ψ i + k , k 0 , ψ 0 = 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 0 = 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 0 = 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 0 = 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 + θ 2 2 ρ k = 0 , k 3 . For MA( q ), ρ k = - θ k + θ 1 θ k +1 + θ 2 θ k +2 + ... + θ q - k θ q 1+ θ 2 1 + θ 2 2 + ... + θ 2 q k = 1 , 2 , . . . , q ; 0 k q + 1 . Process Variance γ 0 = (1 + θ 2 1 + . . . + θ 2 q ) σ 2 a . Autoregressive (AR) Process A p th order autoregressive process { Z t } satisfies the equation Z t = φ 1 Z t - 1 + φ 2 Z t - 2 + . . . + φ p Z t - p + a t Notation: AR( p ).

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

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