Real Time Trend Extraction and Seasonal Adjustment
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Transcript Real Time Trend Extraction and Seasonal Adjustment
Real Time Trend Extraction and
Seasonal Adjustment: a Generalized
Direct Filter Approach
ISF 2011, Prague
Marc Wildi
Zurich University of Applied Sciences
[email protected]
Signalextraction vs. Forecasting
Signal
X t N oisy D ata
Filter: a set of w eights k such that
Yt
k X tk
k
is `fr ee of noise'
Yt is the S igna l
T rend, S easonally A djusted C om ponent, C yc l e
Filters: k
• Ad hoc designs: no explicit modelling of the
data
– HP-Filter, CF-Filter, BK-Filter, Henderson Filter, …
• Model-based designs
– TRAMO/SEATS, X-12-ARIMA, Stamp
• Non-parametric filters (Loess)
• Very general setting!
Real-Time Signalextraction
Time Domain
YT
k X T k `senses' the future (X T 1 , X T 2 , ...)
k
R eal-T im e Finite S am ple
YˆT
T 1
ˆ
k
X T k
k 0
M odel-B a sed A pproaches (M B A ):
k
Xˆ T k
k T
T 1
1
k
X T k
k 0
T 1
k 0
X T k
k Xˆ T k
k
1
k
k Xˆ T k
k
O ne- a nd m ulti-s tep ahead fore casts
Example
A R (1) P ro cess : X t a X t 1 t
T 1
F ilter: sym . ex p o n en tial w eig h tin g Yt c
|k |
X tk
k ( T 1)
T 1
YˆT c
1
|k |
k 0
X T k c
Xˆ T k
|k |
a
|k |
XT
k ( T 1)
T 1
c
|k |
1
|k |
k 0
|k |
k ( T 1)
T 1
c
X T k c
X T k
k 1
0
|k |
c ( a ) X T
k ( T 1)
T 1
c
k 1
|k |
X T k
c
1 a
XT
V ery cu m b erso m e w ay to d efin e a o n e-sid e d filt er!
Forecasting
YT
k X T k
k
1 1,
Forecasting:
k 0 for k -1
T his is a very particular (asym m etric) `S ignal' D efinition
M odel-B ased O ne-step ahead Forecast!
Frequency Domain
Real-Time Signalextraction
Frequency Domain
T arget: YT
k X T k
k
R eal-T im e E stim ate: YˆT
T 1
ˆ
k
X T k
k 0
T ransferfunctions
( ):=
k exp( ik ) (
if sym m etric)
k
T 1
ˆ ( ):= ˆ k exp( ik )
k 0
Example: European IPI
TRAMO/SEATS (Airline-Model in red)
Forecasting
( ):=
k exp( ik )
k
1 1,
Forecasting:
k 0 for k -1
( ) 1 * exp( i )
( ) is a very particular (allpass) Filter/T ransferfunction
R eplicates T raditional M odel-B ased O ne-s tep ahead Forecast in F-D !
Optimization Criterion: Mean-Square
Filter error: rt Yt Yˆt
C riterion: E[ rt ] m in FILT E R W E IG H T S
2
2
| ( ) ˆ ( ) | d S ( ) m in FILT E R W E IG H T S
R eal-W orld:
k
2
ˆ ) m in
( k ) ˆ ( k ) S(
k
ˆ
Choice of Spectral Estimate Sˆ ( )
• Model-based:
– TRAMO (airline-model), X-12-ARIMA, state-space
• Ad-hoc:
– implicit model (HP, CF, BK, Henderson,…)
• Non Parametric
– Periodogram
• This choice is to some extent arbitrary: it
depends on the
preference/experience/expertise of the user.
• Very general setting!
Generalized DFA: Very General Setting!
• Arbitrary signals
– Including as a special case traditional one-step ahead
forecasting
• Arbitrary finite sample Spectral Estimate
– ad hoc, model-based, non-parametric
• Generalizes
–
–
–
–
Ad hoc filters
Model-based filters
DFA (based on the periodogram)
Traditional (one-step ahead) ARIMA-modelling, statespace modelling
– Extends to multivariate filtering!
Frequency-Domain:
Timeliness-Reliability Dilemma
Control of Timeliness/Speed:
2
Cosine Law applied to ( ) ˆ ( )
ˆ ( )
( ) ˆ ( )
ˆ ( )
( ) ˆ ( )
( ) ˆ ( )
2
2
( )
2 ( ) ˆ ( ) 1 cos( ˆ ( ))
Timeliness-Criterion
T /2
2
( k ) ˆ ( k ) Sˆ ( k )
k 1
T /2
2
A ( k ) Aˆ ( k ) Sˆ ( k )
k 1
T /2
2A ( k )Aˆ ( k ) 1 cos( ˆ ( k )) Sˆ ( k )
k 1
M ean-S quare: 1
Faster Filter : > 1
S low er Filter: < 1
Emphasize Noise Rejection in Stop Band
(Reliability/Smoothness)
T /2
2
A ( k ) Aˆ ( k ) W ( k )Sˆ ( k )
k 1
T /2
2A ( k )Aˆ ( k ) 1 cos( ˆ ( k )) Sˆ ( k )
k 1
W ( k ) assigns m ore w eight to am plitude in stop band
assigns m ore w eigh t to tim e-shift in pas s band
Essence of Generalized DFA
• The new optimization criterion IS the timelinessreliability-dilemma and conversely
• `Philosophy’ may be contrasted with
– Maximum likelihood (particular parametric setting
lambda/expweight)
– Maximum entropy
• Contrast:
– Manipulate Real-Time filter characteristics explicitly
on the edge of the fundamental dilemma
– User relevant priorities (risk-aversion)
Effect of `Expweight’
Effect of Lambda
Example : European IPI
Replicate TRAMO RT-Performance:
TRAMO (red) vs. Gen. DFA (blue)
New Target: Customized Design
• Instead of optimal mean-square estimate the
user could specify a `faster’ and/or `smoother’
real-time estimate
• The new estimate is still purely model-based!
– It IS TRAMO (it could be X-12, Stamp,…)
– But it becomes faster/smoother (timelinessreliability dilemma)
Mean-Square vs. Enhanced TRAMO
• Typically, TRAMO-filter (blue) is noisy (poor noise
suppression in stop-band)
• The `customized’ filter (green) barely loses in terms of
time-shift in the pass-band. It clearly wins in terms of
noise suppression in the stop-band: better compromise
TRAMO (red) vs. Enhanced (green)
Conclusion
• As expected, the `customized’ real-time filter
(green) is as `fast’ as the MS-filter by TRAMO
(red) and it is much smoother (better noise
suppression)
SA vs. Customized RT-Trend
• Real-time customized trend filter is as fast as
traditional SA-filter and much (much)
smoother.
Conclusion
Philosophy Generalized DFA
The new criterion IS the
timeliness-reliability dilemma
Consequences
• Generalizes classical filter approaches (ad hoc,
model-based)
• Emphasizes user relevant priorities explicitly
Practicality
• Numerically (very) fast
– Closed-from approximation (I-DFA/open source)
– Fast exact optimization (Eurostat/proprietary)
• Short piece of (R-) code
– Could easily dock to any existent software/tool
Web:
•
•
•
•
SEFblog: http://blog.zhaw.ch/idp/sefblog
USRI: http://www.idp.zhaw.ch/usri
MDFA-XT: http://www.idp.zhaw.ch/MDFA-XT
SEF-page: http://www.idp.zhaw.ch/sef
Selected SEFBlog-Entries
• Forecasting the EURO-BUND-Future (6 months,
one Year)
– http://blog.zhaw.ch/idp/sefblog/index.php?/archives/
186-Forecasting-the-EURO-Bund-Future-6-monthsand-One-Year-Ahead-FirstPreliminary-Draft.html
• OECD-CLI: leading indicator for the US
– http://blog.zhaw.ch/idp/sefblog/index.php?/archives/
173-Tutorial-I-MDFA-Part-II-The-OECD-CLI-for-theUS.html
– http://blog.zhaw.ch/idp/sefblog/index.php?/archives/
175-Injecting-the-ZPC-Gene-into-I-MDFA-anApplication-to-the-OECD-CLI-for-the-US.html
SEFBlog-Entries
• Algorithmic Trading:
– http://blog.zhaw.ch/idp/sefblog/index.php?/archives/
157-A-Generalization-of-the-GARCH-in-Mean-ModelVola-in-I-MDFA-filter.html
• Tutorials Univariate Filter:
– http://blog.zhaw.ch/idp/sefblog/index.php?/archives/
159-I-DFA-Exercises-Part-I-Mean-SquareCriterion.html
– http://blog.zhaw.ch/idp/sefblog/index.php?/archives/
160-I-DFA-Exercises-Part-II-CustomizationSpeedReliability.html
SEFBlog-Entries
• Tutorials Multivariate Filter:
– http://blog.zhaw.ch/idp/sefblog/index.php?/archi
ves/172-Tutorial-I-MDFA-Part-I-Simulated-TimeSeries.html
– http://blog.zhaw.ch/idp/sefblog/index.php?/archi
ves/173-Tutorial-I-MDFA-Part-II-The-OECD-CLI-forthe-US.html