notes51

# notes51 - Notes 5 Models for Non-stationary Time Series In...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
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.

### Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online