11_13 - Y t Y t k = E Y t – μ Y t k – μ = Cov Y t Y t...

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STAT 420 Examples for 11/13/2007 Fall 2007 Time Series: y t , t = 1, 2, … , N . Stationary process a random process where all of its statistical properties do not vary with time. ( ) Y E = t ( ) 2 Y ± Y Var = t ρ k = ( ) ( ) ( ) ( )( ) 2 Y ± Y Y E Y Var Y Var Y , Y Cov - - = + + + k t t k t t k t t , k = ± 1, ± 2, … . Sample autocorrelation coefficient: r k = ( ) ( ) ( ) = - = + - - - N N t t k t k t t y y y y y y 1 2 1 Consider the following “regression” (autoregressive) model: ( Y t μ ) = φ ( Y t – 1 μ ) + e t 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 γ ( k ) = Cov
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Unformatted text preview: ( Y t , Y t + k ) = E [ ( Y t – μ ) ( Y t + k – μ ) ] = Cov ( Y t , Y t – k ) = E [ ( Y t – μ ) ( Y t – k – μ ) ]. Then γ ( ) = Var ( Y t ) = E [ ( Y t – μ ) 2 ] = E [ ( ( Y t – 1 – μ ) + e t ) 2 ] = 2 E [ ( Y t – 1 – μ ) 2 ] + 2 E [ ( Y t – 1 – μ ) e t ] + E [ e t 2 ] = 2 Var ( Y t – 1 ) + Var ( e t ) = 2 γ ( ) + 2 ± e Therefore, γ ( ) = Var ( Y t ) = 2 2 1 ±-e , | | < 1....
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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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