11_13 - ( Y t , Y t + k ) = E [ ( Y t ) ( Y t + k ) ] = Cov...

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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 ) ( 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 , | | &lt; 1....
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