1. Time series analysis - lecture 1 Time series analysis Analysis of data for which the temporal order of the observations is important Two major objectives: (i) Forecasting (ii) Retrospective analysis (trend and intervention analyses)

2. Time series analysis - lecture 1 Oil and coal consumption in the US and China

3. Time series analysis - lecture 1 Consumer price index by month in Sweden

4. Time series analysis - lecture 1 No. unemployed by month in Sweden

5. Time series analysis - lecture 1 No. air passengers by week in Sweden

6. Time series analysis - lecture 1 Nitrogen concentration in the Rhine River

7. Time series analysis - lecture 1 Time series decomposition – additive model Y t = T t + S t + e t, t = 1, …, n The trend component is a smooth function of time that may have a simple parametric form The seasonal component represents a pattern that is repeated every year (or period) The error terms are supposed to be independent and identically distributed with mean zero Trend Season Error

8. Time series analysis - lecture 1 Time series decomposition – multiplicative model Y t = T t * S t * e t, t = 1, …, n The trend component is a smooth function of time that may have a simple parametric form The seasonal component represents a pattern that is repeated every year (or period) The error terms are supposed to be independent and identically distributed with mean one Trend Season Error

9. Time series analysis - lecture 1 Time series decomposition – multiplicative model

10. Time series analysis - lecture 1 Time series decomposition – multiplicative model MAPE Mean Absolute Percentage Error (MAPE) measures the accuracy of fitted time series values. It expresses accuracy as a percentage. where y t equals the actual value at time t, equals the fitted value, and n equals the number of observations.

11. Time series analysis - lecture 1 Time series decomposition – multiplicative model

12. Time series analysis - lecture 1 Time series decomposition – multiplicative model

13. Time series analysis - lecture 1 Time series decomposition – multiplicative model

14. Time series analysis - lecture 1 Time series decomposition – multiplicative model

15. Time series analysis - lecture 1 Time series decomposition – multiplicative model

16. Time series analysis - lecture 1 Time series decomposition – multiplicative model

17. Time series analysis - lecture 1 No. unemployed men by month in Sweden

18. Time series analysis - lecture 1 Moving quarterly averages of monthly data

19. Time series analysis - lecture 1 Moving 1-year averages of monthly data

20. Time series analysis - lecture 1 Moving averages of purely random components

21. Time series analysis - lecture 1 Impact of moving averages  Purely random errors are suppressed (variance reduction)  Spurious trends may appear (serial correlations are introduced)  The timing and abruptness of level shifts are influenced (abrupt changes are smoothed out and may be shifted in time)

22. Time series analysis - lecture 1 Single exponential forecast of the number of air passengers (  =0.2)

23. Time series analysis - lecture 1 Single exponential forecast of the number of air passengers (  =0.05)

24. Time series analysis - lecture 1 Scatter-chart of time-lagged observations - time lag one week

25. Time series analysis - lecture 1 Scatter-chart of time-lagged observations - time-lags one and two weeks

26. Time series analysis - lecture 1 Scatter-chart of time-lagged observations - time-lags 1, 2, 3, and 5 weeks

27. Time series analysis - lecture 1 Empirical autocorrelation function

28. Time series analysis - lecture 1 Empirical autocorrelation function

29. Time series analysis - lecture 1 Empirical autocorrelation function

30. Time series analysis - lecture 1 Information provided by empirical autocorrelation functions  Noise level (high noise implies low autocorrelations)  Long or short memory of past observations  Long-term trends  Seasonal patterns