1. Time series analysis - lecture 3 Forecasting using ARIMA-models Step 1. Assess the stationarity of the given time series of data and form differences if necessary Step 2. Estimate auto-correlations and partial auto- correlations, and select a suitable ARMA-model Step 3. Compute forecasts according to the estimated model

2. Time series analysis - lecture 3 The general integrated auto-regressive-moving-average model ARIMA(p, q)

3. Time series analysis - lecture 3 Weekly SEK/EUR exchange rate Jan 2004 - Oct 2007

4. Time series analysis - lecture 3 Weekly SEK/EUR exchange rate Jan 2004 - Oct 2007 AR(2) model Final Estimates of Parameters Type Coef SE Coef T P AR 1 1.2170 0.0685 17.75 0.000 AR 2 -0.2767 0.0684 -4.05 0.000 Constant 0.549902 0.002726 201.69 0.000 Mean 9.21589 0.04569

5. Time series analysis - lecture 3 Consumer price index and its first order differences

6. Time series analysis - lecture 3 Consumer price index - first order differences

7. Time series analysis - lecture 3 Consumer price index – predictions using an ARI(1) model

8. Time series analysis - lecture 3 Seasonal differencing Form where S depicts the seasonal length

9. Time series analysis - lecture 3 Consumer price index and its seasonal differences

10. Time series analysis - lecture 3 Consumer price index- seasonally differenced data

11. Time series analysis - lecture 3 Consumer price index- differenced and seasonally differenced data

12. Time series analysis - lecture 3 The purely seasonal auto-regressive-moving- average model ARMA(P,Q) with period S {Y t } is said to form a seasonal ARMA(P,Q) sequence with period S if where the error terms  t are independent and N(0;  )

13. Time series analysis - lecture 3 Typical auto-correlation functions of purely seasonal ARMA(P,Q) sequences with period S Auto-correlations are non-zero only at lags S, 2S, 3S, … In addition: AR(P): Autocorrelations tail off gradually with increasing time-lags MA(Q): Auto-correlations are zero for time lags greater than q*S ARMA(P,Q): Auto-correlations tail off gradually with time-lags greater than q*S

14. Time series analysis - lecture 3 No. air passengers by week in Sweden -original series and seasonally differenced data

15. Time series analysis - lecture 3 No. air passengers by week in Sweden - seasonally differenced data

16. Time series analysis - lecture 3 No. air passengers by week in Sweden - differenced and seasonally differenced data

17. Time series analysis - lecture 3 The general seasonal auto-regressive-moving- average model ARMA(p, q, P, Q) with period S {Y t } is said to form a seasonal ARMA(p, q, P, Q) sequence with period S if where the error terms  t are independent and N(0;  ) Example: p = P = 0, q = Q = 1, S = 12.

18. Time series analysis - lecture 3 The general seasonal integrated auto-regressive-moving- average model ARMA(p, q, P, Q) with period S {Y t } is said to form a seasonal ARIMA(p, q, d, P, Q, D) sequence with period S if where the error terms  t are independent and N(0;  )

19. Time series analysis - lecture 3 Forecasting using Seasonal ARIMA-models Step 1. Assess the stationarity of the given time series of data and form differences and seasonal differences if necessary Step 2. Estimate auto-correlations and partial auto- correlations, and select a suitable ARMA-model of the short-term dependence Step 3. Estimate auto-correlations and partial auto- correlations, and select a suitable seasonal ARMA-model of the variation by season Step 4. Compute forecasts according to the estimated model

20. Time series analysis - lecture 3 Consumer price index- differenced and seasonally differenced data

21. Time series analysis - lecture 3 No. registered cars and its first order differences

22. Time series analysis - lecture 3 No. registered cars - first order differences