# T10 - − = − − 2 1 5 1 1 5 11 9 2 1 = = Z Z and 10 3 =...

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STA4005B Time Series (2007-2008) Tutorial 10 (1/4, 2/4) (Time & Venue: T9, BMS LT; W5, MMW 704) Forecasting Def: Given a time series the MMSE (Minimum Mean Squares Error) forecast at time t with l is , , , , 2 1 t Z Z Z K ) , , , | ( ) ( ˆ 1 1 Z Z Z Z E l Z t t l t t K + = and the l -step ahead forecast error is given by . ) ( ˆ ) ( l Z Z l e t l t t = + Example 1 Random Walk with Drift Given and 1 1 a Z = t t t a Z Z + + = 1 0 θ K , 4 , 3 , 2 , = t Find , ) ( ˆ 1 l Z 1 l 1

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Forecasting with General Linear Process For any ARMA model, we can be rewritten the series in the following form: L L + + + + + + = + + + + t l l t l t l t l t a a a a Z ψ ψ ψ µ 2 2 1 1 as 0 ) , , , | ( 1 1 = + Z Z Z a E t t i t K for all , so 0 > i L + + + + = + + 2 2 1 1 ) ( ˆ t l t l t l t a a a l Z ψ ψ ψ µ and 1 1 2 2 1 1 ) ( + + + + + + + + = t l l t l t l t t a a a a l e ψ ψ ψ L Example 2 Given an AR(1) model: t t t a Z Z + = 1 φ , find the value of . ) ( l e t 2
Updating the ARIMA Forecasts If new information arrives, we need to re-calculate the forecasting L + + + + = + + + + 2 3 1 2 1 ) 1 ( ˆ t l t l t l t a a a l Z ψ ψ ψ µ L + + + + = + + 2 2 1 1 ) ( ˆ t l t l t l t a a a l Z ψ ψ ψ µ Then L + + + + = + + + + 1 2 1 1 1 ) ( ˆ t l t l t l t a a a l Z ψ ψ ψ µ ) ( 1 2 1 1 L + + + + = + + + t l t l t l a a a ψ ψ µ ψ Thus ) 1 ( ˆ )] ( ˆ [ ) ( ˆ 1 1 + + =

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Unformatted text preview: − + = − − 2 1 5 . 1 . 1 5 11 , 9 2 1 = = Z Z and 10 3 = Z . Find and . ) 1 ( ˆ 3 Z ) 2 ( ˆ 3 Z If what is ? , 12 4 = Z ) 1 ( ˆ 4 Z 3 Prediction Interval Since )]) ( [ , ( ~ ) ( ) ( ˆ 1 l e Var N l e l Z Z t t t t = − + The )% 1 ( 100 α − prediction interval for is given by l t Z + ) )) ( ( ) ( ˆ , )) ( ( ) ( ˆ ( 2 2 l e Var z l Z l e Var z l Z t t t t + − Example 4 Given and . Find the 95% prediction interval for 2-step ahead forecast at t =1 for the ARMA(1, 1) model 5 . , 2 1 1 = = a Z 1 2 = a σ t t a B Z B ) 3 . 1 ( ) 1 )( 2 . 1 ( − = − − 4...
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