Box Jenkinsor

Arima Forecasting

• H:\My Documents\classes\eco346\Lectures\chapter 7\Autoregressive Models.doc

• All stationary time series can be modeled as AR or MA or ARMA models

• A stationary time series is one with constant mean ( ) and constant variance.

• Stationary time series are often called mean reverting series—that in the long run the mean does not change (cycles will always die out).

• If a time series is not stationary it is often possible to make it stationary by using fairly simple transformations

Nonstationary Time series

• Linear trend

• Nonlinear trend

• Multiplicative seasonality

• Heteroscedastic error terms (non constant variance)

How to make them stationary

• Linear trend– Take non-seasonal difference. What is left

over will be stationary AR, MA or ARMA

Nonlinear trend

• Exponential growth– Take logs – this makes the trend linear– Take non--seasonal difference

• Non exponential growth ?

Multiplicative seasonality

• Take logs– Multiplicative seasonality often occurs when

growth is exponential.– Take logs then a seasonal difference to

remove trend

Heteroscedsatic errors

• Take logs– Note you cannot take logs of negative

numbers

Box Jenkins Methodology

• Identification

• Estimation

• Forecasting

• Examine residuals

• Re—estimate

• Repeat until you only have noise in residuals

Identification

• What does it take to make the time series stationary?

• Is the stationary model AR, MA, ARMA– If AR(p) how big is p?– If MA(q) how big is q?– If ARMA(p,q) what are p and q?

Seasonality

• Is the seasonality AR, MA, ARMA

• What are p, q?

AR(p) models

• The ACF will show exponential decay

• The first p terms of the PACF will be significantly different from zero (outside the parallel lines)

MA(q) models

• The first q terms of the ACF will be significantly different from zero

• The PACF will decay exponentially towards zero

ARMA models

• If you can’t easily tell if the model is an AR or a MA, assume it is an ARMA model.