Seasonality

Thomas Url

Österreichisches Institut für Wirtschaftsforschung (WIFO)Wirtschaftsuniversität Wien

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Unobserved components model

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pt: Trend component

ct: Cyclical component

st: Seasonal component

t: Irregular component

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ttttt scpx Multiplicative model

Additive model

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A simple deterministic model

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A trigonometric seasonal

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A trigonometric seasonal

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Moving average

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Reproduces moving features of seasonality Choose weights to capture variation Ad hoc filter Start- and end-point problems

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X11 and X12 Filter

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(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

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X11 and X12 Filter

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1Example 3x3 MA:

Final Trend and irreg.:

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X11 and X12 Filter

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Spectrum of a process

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X11 gain function

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Outliers

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Spectral AMB decomposition

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Frequency domain AMB decomposition

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