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T4 - STA4005B Time Series(2007-2008 Tutorial 4(5/2...

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Unformatted text preview: STA4005B Time Series (2007-2008) Tutorial 4 (5/2, 13/2) (Time & Venue: T9, BMS LT; W5, MMW 704) Table for Stationary Conditions and Invertible Conditions { } t Z is the following model Is { } t Z stationary? Is invertible? { } t Z AR model MA model ARMA model ARIMA model (“?” means that we don’t know whether { is stationary/invertible or not. We should check the model with the below theorem.) } t Z Stationary Conditions and Invertible Conditions for the General Mixed ARMA(p,q) Process Theorem Suppose { follows an ARMA(p,q) process such that } t Z , ) ( ) ( t t a B Z B θ φ = where ) 1 ( ) 1 )( 1 ( 1 ) ( 2 1 1 B B B B B B p p p δ δ δ φ φ φ − − − = − − − = L L and ) 1 ( ) 1 )( 1 ( 1 ) ( 2 1 1 B B B B B B q q q µ µ µ θ θ θ − − − = − − − = L L Remarks: (i) ) ( B φ and ) ( B θ are relatively prime. (ii) p and are non-negative integer. q Then, (i) { is stationary iff the absolute value of zeros of characteristic equation } t Z ) ( = B φ all exceed 1. all exceed 1....
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T4 - STA4005B Time Series(2007-2008 Tutorial 4(5/2...

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