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# 6 - Introduction to Time Series Analysis Lecture 6 Peter...

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Introduction to Time Series Analysis. Lecture 6. Peter Bartlett www.stat.berkeley.edu/ bartlett/courses/153-fall2010 Last lecture: 1. Causality 2. Invertibility 3. AR(p) models 4. ARMA(p,q) models 1

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Introduction to Time Series Analysis. Lecture 6. Peter Bartlett www.stat.berkeley.edu/ bartlett/courses/153-fall2010 1. ARMA(p,q) models 2. Stationarity, causality and invertibility 3. The linear process representation of ARMA processes: ψ . 4. Autocovariance of an ARMA process. 5. Homogeneous linear difference equations. 2
Review: Causality A linear process { X t } is causal (strictly, a causal function of { W t } ) if there is a ψ ( B ) = ψ 0 + ψ 1 B + ψ 2 B 2 + · · · with summationdisplay j =0 | ψ j | < and X t = ψ ( B ) W t . 3

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Review: Invertibility A linear process { X t } is invertible (strictly, an invertible function of { W t } ) if there is a π ( B ) = π 0 + π 1 B + π 2 B 2 + · · · with summationdisplay j =0 | π j | < and W t = π ( B ) X t . 4
Review: AR(p), Autoregressive models of order p An AR(p) process { X t } is a stationary process that satisfies X t φ 1 X t - 1 − · · · − φ p X t - p = W t , where { W t } ∼ WN (0 2 ) . Equivalently, φ ( B ) X t = W t , where φ ( B ) = 1 φ 1 B − · · · − φ p B p . 5

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Review: AR(p), Autoregressive models of order p Theorem: A (unique) stationary solution to φ ( B ) X t = W t exists iff the roots of φ ( z ) avoid the unit circle: | z | = 1 φ ( z ) = 1 φ 1 z − · · · − φ p z p negationslash = 0 . This AR(p) process is causal iff the roots of φ ( z ) are outside the unit circle: | z | ≤ 1 φ ( z ) = 1 φ 1 z − · · · − φ p z p negationslash = 0 . 6
Reminder: Polynomials of a complex variable Every degree p polynomial a ( z ) can be factorized as a ( z ) = a 0 + a 1 z + · · · + a p z p = a p ( z z 1 )( z z 2 ) · · · ( z z p ) , where z 1 ,...,z p C are the roots of a ( z ) . If the coefficients a 0 ,a 1 ,...,a p are all real, then the roots are all either real or come in complex conjugate pairs, z i = ¯ z j . Example: z + z 3 = z (1 + z 2 ) = ( z 0)( z i )( z + i ) , that is, z 1 = 0 , z 2 = i , z 3 = i . So z 1 R ; z 2 ,z 3 negationslash∈ R ; z 2 = ¯ z 3 . Recall notation: A complex number z = a + ib has Re( z ) = a , Im( z ) = b , ¯ z = a ib , | z | = a 2 + b 2 , arg( z ) = tan - 1 ( b/a ) ( π,π ] . 7

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Review: Calculating ψ for an AR(p): general case φ ( B ) X t = W t , X t = ψ ( B ) W t so 1 = ψ ( B ) φ ( B ) 1 = ( ψ 0 + ψ 1 B + · · · )(1 φ 1 B − · · · − φ p B p ) 1 = ψ 0 , 0 = ψ j ( j< 0) , 0 = φ ( B ) ψ j ( j> 0) . We can solve these linear difference equations in several ways: numerically, or by guessing the form of a solution and using an inductive proof, or by using the theory of linear difference equations. 8
Introduction to Time Series Analysis. Lecture 6. 1. Review: Causality, invertibility, AR(p) models 2. ARMA(p,q) models 3. Stationarity, causality and invertibility 4. The linear process representation of ARMA processes: ψ . 5. Autocovariance of an ARMA process.

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