11_27 - STAT 420 Examples for 11/27/2007 Fall 2007 Consider...

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Unformatted text preview: STAT 420 Examples for 11/27/2007 Fall 2007 Consider the following regression (autoregressive) model: AR ( 1 ) ( Y t ) = ( Y t 1 ) + e t | | < 1 E ( e t ) = 0, Var ( e t ) = 2 & e for all t E ( e t e t' ) = 0, for t t' E ( e t Y t' ) = 0, for t' < t Define ( k ) = Cov ( Y t , Y t + k ) = E [ ( Y t ) ( Y t + k ) ] = Cov ( Y t , Y t k ) = E [ ( Y t ) ( Y t k ) ]. Then ( ) = Var ( Y t ) = E [ ( Y t ) 2 ] = E [ ( ( Y t 1 ) + e t ) 2 ] = 2 E [ ( Y t 1 ) 2 ] + 2 E [ ( Y t 1 ) e t ] + E [ e t 2 ] = 2 Var ( Y t 1 ) + Var ( e t ) = 2 ( ) + 2 & e Therefore, ( ) = Var ( Y t ) = 2 2 1 & - e . Define k = Corr ( Y t , Y t + k ) = ( k ) / ( ) . = 1. ( k ) = Cov ( Y t , Y t k ) = E [ ( Y t ) ( Y t k ) ] = E [ ( ( Y t 1 ) + e t ) ( Y t k ) ] = E [ ( Y t 1 ) ( Y t k ) ] + E [ e t ( Y t k ) ] = ( k 1 ), k 1. Then k = k 1 , k 1. Therefore, k = k = k , k 1. Sample autocorrelation coefficient: r k = ( ) ( ) ( ) & & =- = +--- N N t t k t k t t y y y y y y 1 2 1 Suppose we observe y t , t = 1, 2, , N . Then ( Y N + 1 ) = ( y N ) + e N + 1 , ( Y N + 2 ) = ( Y N + 1 ) + e N + 2 = 2 ( y N ) + e N + 1 + e N + 2 , ( Y N + 3 ) = ( Y N + 2 ) + e N + 3 = 3 ( y N ) + 2 e N + 1 + e N + 2 + e N + 3 , ( Y N + k ) = k ( y N ) + k 1 e N + 1 + + e N + k 1 + e N + k ....
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This note was uploaded on 04/29/2010 for the course STAT stat 420 taught by Professor Stepanov during the Spring '07 term at University of Illinois at Urbana–Champaign.

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11_27 - STAT 420 Examples for 11/27/2007 Fall 2007 Consider...

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