T11 - 5 . , 2 1 1 = = a Z 1 2 = a σ t t a B Z B ) 3 . 1 (...

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STA4005B Time Series (2007-2008) Tutorial 11 (8/4, 9/4) (Time & Venue: T9, BMS LT; W5, MMW 704) 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 µ L + + + + = + + 2 2 1 1 ) ( ˆ t l t l t l t a a a l Z Then L + + + + = + + + + 1 2 1 1 1 ) ( ˆ t l t l t l t a a a l Z ) ( 1 2 1 1 L + + + + = + + + t l t l t l a a a Thus ) 1 ( ˆ )] ( ˆ [ ) ( ˆ 1 1 + + = + + l Z l Z Z l Z t t t l t Example 3 Suppose and t t t t a Z Z Z + + = 2 1 5 . 0 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 1
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Prediction Interval Since )]) ( [ , 0 ( ~ ) ( ) ( 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
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Unformatted text preview: 5 . , 2 1 1 = = a Z 1 2 = a σ t t a B Z B ) 3 . 1 ( ) 1 )( 2 . 1 ( − = − − 2 Revision Example: Consider the AR(1) model: t t t a Z Z + − + = − ) 2 . 10 ( 8 . 2 . 10 1 with . Suppose that we have the observation 3 . 2 = a σ 2 . 9 2006 = Z , (a) Forecast and 2007 Z 2008 Z (b) Find 95% prediction intervals for and . 2007 Z 2008 Z (c) If , update your forecast for . 1 . 9 2007 = Z 2008 Z 3...
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This note was uploaded on 11/02/2011 for the course STAT 4005 taught by Professor Wu,kaho during the Spring '08 term at CUHK.

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T11 - 5 . , 2 1 1 = = a Z 1 2 = a σ t t a B Z B ) 3 . 1 (...

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