11_18ans - STAT 420 Examples for 11/18/2010 Fall 2010 Time...

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Unformatted text preview: STAT 420 Examples for 11/18/2010 Fall 2010 Time Series: y t , t = 1, 2, … , N . Stationary process ≈ a random process where all of its statistical properties do not vary with time. E ( Y t ) = μ . Var ( Y t ) = 2 Y σ . γ ( k ) = Cov ( 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 ) ]. γ ( ) = Var ( Y t ) = 2 Y σ . ρ k = Corr ( Y t , Y t + k ) = ( ) ( ) ( ) ( ) ( ) 2 Y σ μ μ Y Y E Y Var Y Var Y , Y Cov-- = + + + k t t k t t k t t , k = ± 1, ± 2, … . ρ k = γ ( k ) / γ ( ) . ρ = 1. Sample autocorrelation coefficient: r k = ( ) ( ) ( ) ∑ ∑ =- = +--- N N t t k t k t t y y y y y y 1 2 1 1. Calculate r 1 and r 2 for the time series 16 22 19 25 18 ( Note: In practice reliable autocorrelation estimates are only obtained from series consisting of approximately 50 observations or more. ) r k = ( ) ( ) ( ) ∑ ∑ =- = +--- N N t t k t k t t y y y y y y 1 2 1 5 18 25 19 22 16 + + + + = y = 20 . t y y y t- ( ) 2 y y t- ( ) ( ) 1 y y y y t t-- + ( ) ( ) 2 y y y y t t-- + 16 – 4 16 – 8 4 22 2 4 – 2 10 19 – 1 1 – 5 2 25 5 25 – 10 18 – 2 4 50 – 25 16 ( ) ( ) ( ) 50 25 1 2 1 1 1 1- =--- = ∑ ∑ =- = + N N t t t t t y y y y y y r = – 0.5 . ( ) ( ) ( ) 50 16 1 2 2 1 2 2 =--- = ∑ ∑ =- = + N N t t t t t y y y y y y r = 0.32 . Consider the following “regression” (autoregressive) model: AR ( 1 ) ( Y t – μ ) = φ...
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This note was uploaded on 12/17/2010 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois, Urbana Champaign.

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11_18ans - STAT 420 Examples for 11/18/2010 Fall 2010 Time...

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