Box Jenkins or Arima Forecasting

H:\My Documents\classes\eco346\Lectures\chap ter 7\Autoregressive Models.docH:\My Documents\classes\eco346\Lectures\chap ter 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.