chapt5 - Z 2 1-h 2 1 ) ... (1 is ) ( ˆ ( and σ + + + h X...

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Stat 153 - 25 Sept 2008 D. R. Brillinger Chapter 5 - Forecasting Data x 1 ,..., x N What about x N+h , h>0 ) ( ˆ or ) , ( ˆ Forecast h x h N x N No single method universally applicable extrapolation conditional statement scenarios
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Conditional expected value, E(Y|X) 2 )) ( ( minimize to (.) X f Y E f - X can be vector-valued Y = X N+h X =(X 1 ,...,X N ) ) ,..., | ( ) ( ˆ 1 X X X E h X N h N N + =
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Multiple regression N t Z x x Y t mt m t t ,..., 1 ... 1 1 = + + + = β Fit by least squares lm() Residuals N t x x y z mt m t t t ,..., 1 ˆ ... ˆ ˆ 1 1 = - - - = Z X Y + = Z X X X Y X X X T T T T 1 1 ) ( ) ( ˆ - - + = = 2 1 ) ( ) ˆ ( σ - = X X Var T
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Linear c 's to minimize E(X N+h - c 0 X N -...-c N-1 X 1 ) 2 Conditional expected value if {X t } Gaussian/normal
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Prediction/forecast error Linear process ... ... 1 1 1 1 + + = + + = - + + + - h N h N h N t t t Z Z X Z Z X ψ The error is 1 1 1 1 ... + - - + + + + + N h h N h N Z Z Z 2
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Unformatted text preview: Z 2 1-h 2 1 ) ... (1 is ) ( ˆ ( and σ + + + h X Var N Represent series as linear process Box-Jenkins Model . arima(p,d,q) Stages of Box-Jenkins forecasting model (1) Model identification . Which p,d,q? (2) Estimation . arima() (3) Diagnostic checking . residuals (4) Consideration of alternate models . If necessary Cp. The scientific method Pertinent R functions arima() tsdiag() predict() - applied to output of arima help("predict.Arima")...
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This note was uploaded on 10/19/2009 for the course MATH 611 taught by Professor Jsdkasj during the Spring '09 term at Kansas.

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chapt5 - Z 2 1-h 2 1 ) ... (1 is ) ( ˆ ( and σ + + + h X...

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