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

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

A simple deterministic model FUßZEILESEITE 3 with

A trigonometric seasonal FUßZEILESEITE 4

A trigonometric seasonal FUßZEILESEITE 5

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

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

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

X11 and X12 Filter FUßZEILESEITE 9

Spectrum of a process FUßZEILESEITE 10

X11 gain function FUßZEILESEITE 11

Outliers FUßZEILESEITE 12

Spectral AMB decomposition FUßZEILESEITE 13

Frequency domain AMB decomposition FUßZEILESEITE 14