This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Notes 5 Models for Nonstationary 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 differenceequation 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...
View
Full
Document
This note was uploaded on 01/20/2012 for the course STA 4005 taught by Professor ? during the Spring '08 term at CUHK.
 Spring '08
 ?

Click to edit the document details