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# hwk6 - by f XY ω = σ 2 Z(0 84 0 8 i sin ω/π Evaluate...

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Time Series Analysis Problem Sheet 6 1. Show that the cross-covariance function of the discrete bivariate process { X t , Y t } where X t = Z 1 ,t + β 11 Z 1 ,t - 1 + β 12 Z 2 ,t - 1 Y t = Z 2 ,t + β 21 Z 1 ,t - 1 + β 22 Z 2 ,t - 1 and { Z 1 ,t } , { Z 2 ,t } are independent purely random processes with zero mean and variance σ 2 Z , is given by γ XY ( k ) = σ 2 Z ( β 11 β 21 + β 12 β 22 ) k = 0 β 21 σ 2 Z k = 1 β 12 σ 2 Z k = - 1 0 otherwise. Hence evaluate the cross-spectrum. 2. Define the cross-correlation function, ρ XY ( τ ) , of a bivariate stationary pro- cess and show that | ρ XY ( τ ) | ≤ 1 for all τ . Two MA processes X t = Z t + 0 . 4 Z t - 1 Y t = Z t - 0 . 4 Z t - 1 are formed from a purely random process, { Z t } , which has zero mean and variance σ 2 Z . Find the cross-covariance and cross-correlation functions of the bivariate process { X t , Y t } and hence show that the cross-spectrum is given
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Unformatted text preview: by f XY ( ω ) = σ 2 Z (0 . 84 + 0 . 8 i sin ω ) /π. Evaluate the co-, quadrature, cross-amplitude, phase spectra. 3. Find the cross-covariance and cross-correlation functions of the bivariate processes: X t = 1 2 X t-1 + Z 1 ,t Y t = 1 2 X t-2 + Z 2 ,t where { Z 1 ,t } , { Z 2 ,t } are independent, purely random processes with zero mean and variance σ 2 Z . Hence show that the cross-spectrum is given by f XY ( ω ) = 2 σ 2 Z e-2 iω 4 π (1-cos ω + 1 4 ) . (Part answer: γ XY ( k ) = 2 3 ( 1 2 ) | k-2 | σ 2 Z for all k ). 1...
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