# sol3 - Solution of Assignment 3 Problem 2.15 By the...

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Problem 2.15 By the properties on the best linear prediction (Brockwell and Davis (2002), page 68), we have for n > p P n X n +1 = P n ( φ 1 X n + ... + φ p X n - p +1 ) + P n ( Z n +1 ) = P n ( φ 1 X n ) + ... + P n ( φ p X n - p +1 ) + P n ( Z n +1 ) = φ 1 P n ( X n ) + ... + φ p P n ( X n - p +1 ) + E [ Z n +1 ] = φ 1 X n + ... + φ p X n - p +1 The mean squared erroris E [( X n +1 - P n X n +1 ) 2 ] = E [ Z 2 n +1 ] = σ 2 . Problem 2.22 Let X 1 ,X 2 ,X 3 ,X 4 ,X 5 be observations from AR(1) model. The results follow from equations (2.5.12)-(2.5.14) in BD pages 66-67. a. The best linear predictor of X 3 given X 1 ,X 2 is P ( X 3 /X 1 ,X 2 ) = a 1 X 1 + a 2 X 2 where ( a 1 ,a 2 ) are such that ± 1 φ φ 1 a 1 a 2 ! = ± φ 2 φ ! . Then, a 1 = 0 and a 2 = φ . The best linear predictor of X 3 given X 1 ,X 2 is P ( X 3 /X 1 ,X 2 ) = φX 2 . b. The best linear predictor of X 3 given X 4 ,X 5 is P ( X 3 /X 4 ,X 5 ) = a 1 X 4 + a 2 X 5 where ( a 1 ,a 2 ) are such that ± 1 φ φ 1 a

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## This note was uploaded on 03/01/2012 for the course MATH 310 taught by Professor Smith during the Spring '12 term at Georgia Tech.

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sol3 - Solution of Assignment 3 Problem 2.15 By the...

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