Forecasting_solution_2008 - Practice question 2: Consider...

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Practice question 2: Consider the ARIMA(1,1,0) model ) , 0 ( ~ , ) 9 . 0 1 )( 1 ( 2 σ NID a a X B B t t t = + - . The most recent 8 observations for 1989 to 1996 were ) 3 . 6 , 2 . 7 , 9 . 4 , 3 . 4 , 2 . 2 , 5 . 1 , 1 . 0 , 0 ( ) , , ( 96 89 - - - - - - - = X X . (a) Write out the recursive formula for the t X values. l t l t l t l t t t t t t t t t a X X X a X X X a X B B a X B B + + - + - + - - + + = + + = = - - = + - 2 1 2 1 2 9 . 0 1 . 0 9 . 0 1 . 0 ) 9 . 0 1 . 0 1 ( ) 9 . 0 1 )( 1 ( 3 ), 2 ( ˆ 9 . 0 ) 1 ( ˆ 1 . 0 ) ( ˆ ) 1 ( ˆ 9 . 0 ) 2 ( ˆ 1 . 0 ) 3 ( ˆ 9 . 0 ) 1 ( ˆ 1 . 0 ) 2 ( ˆ 9 . 0 1 . 0 ) 1 ( ˆ 1 - + - = + = + = + = - l l X l X l X X X X X X X X X X t t t t t t t t t t t t (b) Derive the formulas for predictions for 1997 to 1999 in terms of previously observed values. (These may be expressed in terms of other predictions, as long as you describe how to calculate each term before you use it another formula.) 0371 . 7 ) 3 ( ˆ ] [ 381 . 6 ) 2 ( ˆ ] [ 11 . 7 ) 1 ( ˆ ] [ 96 99 96 96 98 96 96 97 96 - = = - = = - = = X X E X X E X X E (c) Let X be the mean value of this series in the 90’s (i.e., from 1990 to 1999). Use the values above and your formulas to calculate the estimate X ˆ of X . (You should give this estimate both as a formula and numerically.) Assume that all earlier values of the series are zero if you need them in your predictions. 1
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This note was uploaded on 09/27/2009 for the course STA STA457 taught by Professor Lin during the Winter '08 term at University of Toronto- Toronto.

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Forecasting_solution_2008 - Practice question 2: Consider...

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