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Unformatted text preview: Y t = p X j =1 φ j,p Y tj + ± t . Show that the process { X t } given by X t = q X k =0 β k Y tk , where β = 1, can be written as an ARMA( p , q ) process. 1 7. HARDER QUESTION (a) Let { X t } be a real zero mean stationary process with variance σ 2 X and let X = 1 N ∑ N t =1 X t . Show that we can write var { X } = σ 2 X N 1 + 2 N N1 X i =1 N X j>i ρ ji . where { ρ τ } is the autocorrelation sequence of { X t } . (b) Now consider the AR(1) process X t = φ 1 , 1 X t1 + ± t with variance σ 2 X = σ 2 ± / (1φ 2 1 , 1 ) and autocorrelation sequence ρ τ = φ  τ  1 , 1 . Show that for this process N X j>i ρ ji = φ 1 , 1 (1φ Ni 1 , 1 ) 1φ 1 , 1 . (c) Hence show that for the AR(1) process var { X } = σ 2 ± N (1φ 2 1 , 1 ) ± 1 + 2 φ 1 , 1 N (1φ 1 , 1 ) " N(1φ N 1 , 1 ) (1φ 1 , 1 ) #! . 2...
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This note was uploaded on 10/19/2009 for the course MATH 611 taught by Professor Jsdkasj during the Spring '09 term at Kansas.
 Spring '09
 jsdkasj

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