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Seasonality
Thomas Url
Österreichisches Institut für Wirtschaftsforschung (WIFO)Wirtschaftsuniversität Wien
FUẞZEILE
Unobserved components model
SEITE 2
pt: Trend component
ct: Cyclical component
st: Seasonal component
t: Irregular component
componentsunobserved
ttttt scpx
componentsunobserved
ttttt scpx Multiplicative model
Additive model
FUẞZEILE
A simple deterministic model
SEITE 3
S
jjtjt
t
dcs
btap
1
ttttt cspx
with
FUẞZEILE
A trigonometric seasonal
SEITE 4
.2,,2,1,
2sin
2
,cos1
12,,2,1,
2cos
2
2sin
2cos
1
12
1´
2/
1
Sj
S
jkm
S
kmS
Sj
S
jkm
S
S
jt
S
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S
ssj
S
ssS
S
ssj
S
jjjt
tttst cos2sin2cos 211
FUẞZEILE
A trigonometric seasonal
SEITE 5
FUẞZEILE
Moving average
SEITE 6
Reproduces moving features of seasonality Choose weights to capture variation Ad hoc filter Start- and end-point problems
111 11011 ktktttktkt xcxcxcxcxcy
FUẞZEILE
X11 and X12 Filter
SEITE 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
FUẞZEILE
X11 and X12 Filter
SEITE 8
ttt
tt
ttt
pN
NTrendHendersonp
sxN
ˆˆˆ
ˆˆ
ˆˆ
tj
jS
j
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1
1
1
115
1Example 3x3 MA:
Final Trend and irreg.:
FUẞZEILE
X11 and X12 Filter
SEITE 9
FUẞZEILE
Spectrum of a process
SEITE 10
FUẞZEILE
X11 gain function
SEITE 11
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Outliers
SEITE 12
FUẞZEILE
Spectral AMB decomposition
SEITE 13
FUẞZEILE
Frequency domain AMB decomposition
SEITE 14
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