# STA457H1 Chapter Notes - Chapter 6: Autoregressive Integrated Moving Average

Chapter 6 – Non-stationary (ARIMA)

Time Series Models

6.1 – ARIMA models for non-stationary TS

Definition

If d is a non-negative integer, then

{

Xt

}

is an ARIMA(p,d,q) model if

Yt=

(

1−B

)

dXt

is a causal

ARMA(p,q) process

In mathematical terms

Φ

(

B

) (

1−B

)

dXt=Θ

(

B

)

Zt

ZtWN

(

0,σ2

)

Φ

(

B

)

has roots all outside of the unit circle

Properties

- ACF of ARIMA(p,d,q) is very slowly decaying

Suppose we have ARIMA(p,d,q) model

If we fit an AR(1) model to it, then the fitted AR coefficient will be close to 1

Note there are no ACF or PACF for ARIMA(p,d,q) for d>=1

However, sample ACF and PACF are still well defined

6.2 – Identification Techniques

(a) Preliminary transformation

- Trend removal (moving window, harmonic regression)

- Differentiation (ARIMA(