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General linear process Theorem 37 The spectrum of the general linear process Y t = ∞∑ a j ε t−j , (8) j=0 where {ε t } is white noise, may be written as f(ω) = b 0 + ∞∑ b m cos(2πmω), −1/2 ≤ ω ≤ 1/2. (9) m=1 □ Any real-valued even continuous function that satisfies f(ω) = f(ω + k) for integer k can be expressed as the (implicit) Fourier series (9), so any stationary random sequence with a continuous spectrum can be represented as a general linear process—at least so far as second-order properties are concerned. □ The general linear representation (8) is only useful if it involves only a few parameters—hence the use of ARMA models, which are quite flexible linear models with finite parameters. □ The computations leading to (9) implicitly presuppose that ∑ a 2 j < ∞, which then implies that {Y t } is stationary with finite variance and covariance function. Time Series Spring 2010 – slide 136 ARMA models slide 137 Autoregressive process Definition 38 An autoregressive process of order p, AR(p), model, is of the form Y t = φ 1 Y t−1 + φ 2 Y t−2 + · · · + φ p Y t−p + ε t , (10) where {Y t } is stationary and φ 1 ,... ,φ p are constants and φ p ≠ 0. Unless otherwise mentioned, we iid assume here that ε t ∼ N(0,σ 2 ). A process with non-zero mean is obtained by replacing Y t in (10) by Y t − µ, etc. The backshift operator B can be used to write (10) in the form (1 − φ 1 B − · · · − φ p B p )Y t = φ(B)Y t = ε t , where φ(B) is the autoregressive operator, and this suggests writing Y t = φ(B) −1 ε t to get the causal representation Y t = ∑ ∞ j=0 ψ jε t−j = ψ(B)ε t , if it exists. To find the coefficients of ψ(B), we suppose that such a representation exists, and then match terms on the left and right of the equation φ(B)Y t = φ(B)ψ(B)ε t = ε t . Time Series Spring 2010 – slide 138 131
Moving average process Definition 39 A moving average model of order q, MA(q), is of the form Y t = ε t + θ 1 ε t−1 + · · · + θ q ε t−q , (11) where θ 1 ,... ,θ q are constants, θ q ≠ 0, and ε t iid ∼ N(0,σ 2 ). A process with non-zero mean is obtained by replacing Y t in (11) by Y t − µ. The backshift operator B can be used to write (11) in the form Y t = (1 + θ 1 B + · · · + θ q B q )ε t = θ(B)ε t , where θ(B) is the moving average operator. The process (11) is stationary for any values of the θ r . Example 40 Show that the MA(1) processes with parameters θ 1 and 1/θ 1 are statistically indistinguishable. Time Series Spring 2010 – slide 139 Invertibility Definition 41 A moving average process {Y t } is called invertible if it has an infinite autoregressive representation ∞∑ ε t = a j Y t−j . □ This definition is needed simply in order to ensure the identifiability of MA processes. □ In Example 40 it is easy to check which version is invertible, we write ε t = (1 + θ 1 B) −1 Y t = which is convergent iff |θ 1 | < 1. j=0 ∞∑ ∞∑ (−θ 1 B) j Y t = (−θ 1 ) j Y t−j , j=0 Time Series Spring 2010 – slide 140 j=0 132
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