1. Seasonality Thomas Url Österreichisches Institut für Wirtschaftsforschung (WIFO) Wirtschaftsuniversität Wien

2. Unobserved components model FUßZEILESEITE 2  p t : Trend component  c t : Cyclical component  s t : Seasonal component   t : Irregular component Multiplicative model Additive model

3. A simple deterministic model FUßZEILESEITE 3 with

4. A trigonometric seasonal FUßZEILESEITE 4

5. A trigonometric seasonal FUßZEILESEITE 5

6. Moving average FUßZEILESEITE 6  Reproduces moving features of seasonality  Choose weights to capture variation  Ad hoc filter  Start- and end-point problems

7. X11 and X12 Filter FUßZEILESEITE 7  (1) Detrend x(t) by substracting a 2x12 MA  (1) Take initial seasonal and apply MA (3x3) to get preliminary seasonal component  (1) Subtract 2x12 MA from preliminary seasonal  (1) Subtract result from x(t) to get preliminary seasonal adjusted series  (2) Subtract Henderson trend MA of this from x(t) to get refined detrended  (2) Apply (3x5) seasonal MA to result from before and subtract 2x12 MA to get final seasonal component  (3) Compute final trend and irregular components

8. X11 and X12 Filter FUßZEILESEITE 8 Example 3x3 MA: Final Trend and irreg.:

9. X11 and X12 Filter FUßZEILESEITE 9

10. Spectrum of a process FUßZEILESEITE 10

11. X11 gain function FUßZEILESEITE 11

12. Outliers FUßZEILESEITE 12

13. Spectral AMB decomposition FUßZEILESEITE 13

14. Frequency domain AMB decomposition FUßZEILESEITE 14