TS_P3 - Forecasting Time Series Let Yt = a +bt + Xt where...

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1 Forecasting Time Series Let Y t = a +bt + X t where X t = SARIMA(p,d,q)(P,D,Q)s Given X 1 , …., X n , we want to predict the next h values X n+1 , … , X n+h,, [ h = forecast horizon, lead time ] Def: An h-step ahead minimum mean square forecast of X n+h is defined by = E ( X n+h |X n , X n-1 , … X 1 ) E.g Suppose {X t } satisfies X t = 1 X t-1 + Z t Then = 1 X n [1- step ahead forecast] = 1 Exercise : Suppose X t = 1 X t-1 + Z t, Express in terms of X n ) ( ˆ h X n ) 1 ( ˆ n X ) ( ˆ h X n ) 1 ( ˆ - h X n ) ( ˆ h X n
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Examples: Forecasting an AR(p) : E.g. Suppose {X 1 , …X 100 } satisfies X t = c + 1 X t-1 + Z t (a) Find c in terms of 1 and ± = E(X t ) . (b) The recurrence equation for h-step ahead forecasts c) ± =10.2 and 1 = 0.7 and X 100 =10.6 Find the 1-step ahead forecast Solution: (a) c = (1- 1 ) ± (b) = c + 1 c ) c = (1- 0.7)10.2 = 3.06 Hence = 3.06 + 0.7(10.6) = 10.48 ) 1 ( ˆ -
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TS_P3 - Forecasting Time Series Let Yt = a +bt + Xt where...

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