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# 5 - Introduction to Time Series Analysis Lecture 5 Peter...

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Introduction to Time Series Analysis. Lecture 5. Peter Bartlett www.stat.berkeley.edu/ bartlett/courses/153-fall2010 Last lecture: 1. ACF, sample ACF 2. Properties of the sample ACF 3. Convergence in mean square 1

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Introduction to Time Series Analysis. Lecture 5. Peter Bartlett www.stat.berkeley.edu/ bartlett/courses/153-fall2010 1. AR(1) as a linear process 2. Causality 3. Invertibility 4. AR(p) models 5. ARMA(p,q) models 2
AR(1) as a linear process Let { X t } be the stationary solution to X t φX t - 1 = W t , where W t WN (0 2 ) . If | φ | < 1 , X t = summationdisplay j =0 φ j W t - j is the unique solution: This infinite sum converges in mean square, since | φ | < 1 implies | φ j | < . It satisfies the AR(1) recurrence. 3

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AR(1) in terms of the back-shift operator We can write X t φX t - 1 = W t (1 φB ) bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright φ ( B ) X t = W t φ ( B ) X t = W t Recall that B is the back-shift operator: BX t = X t - 1 . 4
AR(1) in terms of the back-shift operator Also, we can write X t = summationdisplay j =0 φ j W t - j X t = summationdisplay j =0 φ j B j bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright π ( B ) W t X t = π ( B ) W t 5

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AR(1) in terms of the back-shift operator With these definitions: π ( B ) = summationdisplay j =0 φ j B j and φ ( B ) = 1 φB, we can check that π ( B ) = φ ( B ) - 1 : π ( B ) φ ( B ) = summationdisplay j =0 φ j B j (1 φB ) = summationdisplay j =0 φ j B j summationdisplay j =1 φ j B j = 1 . Thus, φ ( B ) X t = W t π ( B ) φ ( B ) X t = π ( B ) W t X t = π ( B ) W t . 6
AR(1) in terms of the back-shift operator Notice that manipulating operators like φ ( B ) , π ( B ) is like manipulating polynomials: 1 1 φz = 1 + φz + φ 2 z 2 + φ 3 z 3 + · · · , provided | φ | < 1 and | z | ≤ 1 . 7

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Introduction to Time Series Analysis. Lecture 5. 1. AR(1) as a linear process 2. Causality 3. Invertibility 4. AR(p) models 5. ARMA(p,q) models 8
AR(1) and Causality Let X t be the stationary solution to X t φX t - 1 = W t , where W t WN (0 2 ) .

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