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07_ts_chap5 - Outline MSE prediction Box-Jenkins procedure...

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Outline MSE prediction Box-Jenkins procedure Forecasting and Box-Jenkins Methodology Haipeng Xing, AMS316, Stony Brook University Forecasting and Box-Jenkins Methodology Outline MSE prediction Box-Jenkins procedure Outline 1 Mean square error prediction 2 Box-Jenkins procedure Haipeng Xing, AMS316, Stony Brook University Forecasting and Box-Jenkins Methodology
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Outline MSE prediction Box-Jenkins procedure Mean square error prediction Suppose that the observations X 1 , . . . , X n follow some time series model. Denote F n = ( X 1 , . . . , X n ) . We want to find an estimate f ( X 1 , . . . , X n ) for X n + k ( k 1) , i.e., a k -step ahead forecast at forecast origin n . A common used forecast criterion is to consider minimizing the mean square error (MSE) of your forecast, i.e., min f E X n + k - f ( X 1 , . . . , X n ) 2 |F n = Var ( X n + k |F n ) + min f E E ( X n + k |F n ) - f ( X 1 , . . . , X n ) 2 |F n . The above MSE is minimized by choosing f = E ( X n + k |F n ) . Therefore, we refer to E ( X n + k |F n ) as the k -step ahead forecast .
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