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Unformatted text preview: Notes 5 Models for Nonstationary 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.
 Spring '08
 WU,KaHo

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