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notes51 - Notes 5 Models for Non-stationary Time Series :...

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Unformatted text preview: Notes 5 Models for Non-stationary Time Series : In Notes 4, the time series we studied are all stationary processes. However, in prac- tice, a lot of useful time series are nonstationary. At present, we introduce a class of nonstationary time series models called the autoregressive integrated moving average models. ARIMA model Notation: Let the notation ∇ be defined as ∇ Z t = (1- B ) Z t = Z t- Z t- 1 , ∇ 2 Z t = ∇ ( ∇ Z t ) = ∇ ( Z t- Z t- 1 ) = Z t- 2 Z t- 1 + Z t- 2 , and so on . Definition: A series { Z t } is said to follow an integrated autoregressive model aver- age model if the d th difference W t = ∇ d Z t is a stationary ARMA process. If W t is ARMA(p,q), we say that Z t is ARIMA(p,d,q). In general, the ARIMA(p,d,q) model can be expressed as (1- B ) d φ ( B ) Z t = θ ( B ) a t where the stationary AR operator φ ( B ) = 1- φ 1 B- ...- φ p B p and the invertible MA operator θ ( B ) = 1- θ 1 B- ...- θ q B q share no common factors. This is a useful form for identifying models. Example: ARIMA(p,1,q) model With W t = Z t- Z t- 1 , W t = φ 1 W t- 1 + φ 2 W t- 2 + ... + φ p W t- p + a t- θ 1 a t- 1- ...- θ q a t- q . In terms of the observed series Z t- Z t- 1 = φ 1 ( Z t- 1- Z t- 2 )+ φ 2 ( Z t- 2- Z t- 3 )+ ... + φ p ( Z t- p- Z t- p- 1 )+ a t- θ 1 a t- 1- ...- θ q a t- q . Therefore Z t = (1+ φ 1 ) Z t- 1 +( φ 2- φ 1 ) Z t- 2 + ... +( φ p- φ p- 1 ) Z t- p- φ p Z t- p- 1 + a t- θ 1 a t- 1- ...- θ q a t- q . We call this the difference-equation form of the model which appears to be an ARMA(p+1,q) process. However the AR characteristic polynomial is 1- (1 + φ 1 ) x- ( φ 2- φ 1 ) x 2- ( φ 3- φ 2 ) x 3- ...- ( φ p- φ p- 1 ) x p + φ p x p +1 = (1- x )(1- φ 1 x- φ 2 x- ...- φ p x p ) 1 As a result, the ARIMA(p,1,q) model can also be written as (1- B ) φ ( B ) Z t = θ ( B ) a t where φ ( x ) = 1...
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notes51 - Notes 5 Models for Non-stationary Time Series :...

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