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Box Jenkins or Arima Forecasting

Box Jenkins or Arima Forecasting. H:My Documentsclasseseco346Lectureschapter 7Autoregressive 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.

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Box Jenkins or Arima Forecasting

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  1. Box Jenkinsor Arima Forecasting

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

  3. 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

  4. Nonstationary Time series • Linear trend • Nonlinear trend • Multiplicative seasonality • Heteroscedastic error terms (non constant variance)

  5. How to make them stationary • Linear trend • Take non-seasonal difference. What is left over will be stationary AR, MA or ARMA

  6. Nonlinear trend • Exponential growth • Take logs – this makes the trend linear • Take non--seasonal difference • Non exponential growth ?

  7. Multiplicative seasonality • Take logs • Multiplicative seasonality often occurs when growth is exponential. • Take logs then a seasonal difference to remove trend

  8. Heteroscedsatic errors • Take logs • Note you cannot take logs of negative numbers

  9. Box Jenkins Methodology • Identification • Estimation • Forecasting • Examine residuals • Re—estimate • Repeat until you only have noise in residuals

  10. 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?

  11. Seasonality • Is the seasonality AR, MA, ARMA • What are p, q?

  12. 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)

  13. MA(q) models • The first q terms of the ACF will be significantly different from zero • The PACF will decay exponentially towards zero

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

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