# 5 - Notes 5 Models for Non-stationary Time Series In Notes...

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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 practice, a lot of useful time series are nonstationary. At present, we introduce a class of nonsta- tionary 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: Identify the following models as specific ARIMA models: (a) Z t = Z t- 1- . 25 Z t- 2 + a t- . 4 a t- 1 (b) Z t = 2 Z t- 1- Z t- 2- . 3 a t- 1 + a t Solution: For (a), (1- B + 0 . 25 B 2 ) Z t = (1- . 4 B ) a t (1- . 5...
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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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5 - Notes 5 Models for Non-stationary Time Series In Notes...

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