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Unformatted text preview: Y t = p X j =1 j,p Y tj + t . Show that the process { X t } given by X t = q X k =0 k Y tk , where = 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 2 X and let X = 1 N N t =1 X t . Show that we can write var { X } = 2 X N 1 + 2 N N1 X i =1 N X j>i ji . where { } is the autocorrelation sequence of { X t } . (b) Now consider the AR(1) process X t = 1 , 1 X t1 + t with variance 2 X = 2 / (1 2 1 , 1 ) and autocorrelation sequence =   1 , 1 . Show that for this process N X j>i ji = 1 , 1 (1 Ni 1 , 1 ) 1 1 , 1 . (c) Hence show that for the AR(1) process var { X } = 2 N (1 2 1 , 1 ) 1 + 2 1 , 1 N (1 1 , 1 ) " N(1 N 1 , 1 ) (1 1 , 1 ) #! . 2...
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 Spring '09
 jsdkasj

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