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Seasonality Thomas Url Österreichisches Institut für Wirtschaftsforschung (WIFO) Wirtschaftsuniversität Wien. Unobserved components model. Multiplicative model. Additive model. p t : Trend component c t : Cyclical component s t : Seasonal component  t : Irregular component.

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SeasonalityThomas UrlÖsterreichisches Institut für Wirtschaftsforschung (WIFO)Wirtschaftsuniversität Wien

unobserved components model
Unobservedcomponents model

Multiplicative model

Additive model

  • pt: Trend component
  • ct: Cyclicalcomponent
  • st: Seasonalcomponent
  • t: Irregularcomponent

Fußzeile

moving average
Movingaverage
  • Reproduces movingfeaturesofseasonality
  • Chooseweightstocapturevariation
  • Ad hoc filter
  • Start- and end-pointproblems

Fußzeile

x11 and x12 filter
X11 and X12 Filter
  • (1) Detrend x(t) bysubstracting a 2x12 MA
  • (1) Take initialseasonalandapply MA (3x3) togetpreliminaryseasonalcomponent
  • (1) Subtract 2x12 MA frompreliminaryseasonal
  • (1) Subtractresultfrom x(t) togetpreliminaryseasonaladjustedseries
  • (2) Subtract Henderson trend MA ofthisfrom x(t) togetrefineddetrended
  • (2) Apply (3x5) seasonal MA toresultfrombeforeandsubtract 2x12 MA toget final seasonalcomponent
  • (3) Compute final trendandirregularcomponents

Fußzeile

x11 and x12 filter1
X11 and X12 Filter

Example 3x3 MA:

Final Trend and irreg.:

Fußzeile

outliers
Outliers

Fußzeile