Introduction to Seasonal
Adjustment
Based on the:
Australian Bureau of Statistics’ Information Paper: An Introductory Course on Time Series
Analysis;
Hungarian Central Statistical Office: Seasonal Adjustment Methods and Practices;
Bundesbank, Robert Kirchner: X-12 ARIMA Seasonal Adjustment of Economic Data Training
Course
Artur Andrysiak
Statistics Development and Analysis Section
ESCAP
E-mail: andrysiak@un.org
Overview
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What and why
Basic concepts
Methods
Software
Recommended practices
Step by step
Issues
Useful references
Slide 2
Seasonally adjusted and original
series – Industrial Production Index
Graph 1. Original series
Graph 2. Sesonally adjusted series
135
135
125
125
115
115
105
105
95
95
85
85
75
2006M 01
75
2006M 04
Armenia
2006M 07
2006M 10
Germany
2007M 01
2007M 04
Serbia
2007M 07
2007M 10
Ukraine
2006M 01
2006M 04
Armenia
2006M 07
2006M 10
Germany
2007M 01
2007M 04
Serbia
2007M 07
2007M 10
Ukraine
Slide 3
IIP percentage change from
November 2007 to December 2007
Armenia German Serbia
y
Ukraine
Original 1.1%
-9.6%
4.4%
-0.3%
SA
1.4%
0.1%
1.4%
-2.3%
Slide 4
Why seasonally adjust?
Seasonal adjustment has three main
purposes:
 to aid in short term forecasting
 to aid in relating time series to other
series or extreme events
•

including comparison of timeseries from
different countries
to allow series to be compared from
month to month, quarter to quarter
Slide 5
Seasonal adjustment



Seasonal adjustment is an analysis
technique that estimates and then removes
from a series influences that are systematic
and calendar related.
A seasonally adjusted series can be formed
by removing the systematic calendar related
influences from the original series.
A trend series is then derived by removing
the remaining irregular influences from the
seasonally adjusted series.
Slide 6
Aim of seasonal adjustment
The aim of seasonal adjustment is to
eliminate seasonal and working day
effects. Hence there are no seasonal
and working-day effects in a perfectly
seasonally adjusted series
Source: Bundesbank
Slide 7
Aim of seasonal adjustment
In other words: seasonal adjustment
transforms the world we live in into a world
where no seasonal and working-day effects
occur. In a seasonally adjusted world the
temperature is exactly the same in winter as in
the summer, there are no holidays, Christmas
is abolished, people work every day in the
week with the same intensity (no break over
the weekend) etc.
Source: Bundesbank
Slide 8
Quarterly IPI - Mining and
Quarrying Georgia
210.00
190.00
170.00
150.00
130.00
110.00
90.00
70.00
Mining and Quarrying
Mining and Quarrying_TS_7R_011011_HO
Slide 9
Quarterly IPI - Electricity, gas
and water supply Georgia
120.00
110.00
100.00
90.00
80.00
70.00
60.00
50.00
Electricity, gas and water supply
Electricity, gas and water supply_TS_7R_011011_noHO
Slide 10
Quarterly IPI - Manufacturing
Georgia
420.00
370.00
320.00
270.00
220.00
170.00
120.00
70.00
Manufacturing
Manufacturing_TS_1R_HO
Slide 11
Jan-00
Kazakhstan
SA_TS_1R_Ho
May-07
Mar-07
Jan-07
Nov-06
Sep-06
Jul-06
May-06
Mar-06
Jan-06
Nov-05
Sep-05
Jul-05
May-05
Mar-05
Jan-05
Nov-04
Sep-04
Jul-04
May-04
Mar-04
Jan-04
Nov-03
Sep-03
Jul-03
May-03
Mar-03
Jan-03
Nov-02
Sep-02
Jul-02
May-02
Mar-02
Jan-02
Nov-01
Sep-01
Jul-01
May-01
Mar-01
Jan-01
Nov-00
Sep-00
Jul-00
May-00
Mar-00
IPI - Kazakhstan
200
180
160
140
120
100
80
Tren_TS_1R_Ho
Slide 12
Basic concepts - timeseries
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A time series is a collection of
observations of well defined data items
observed through time (measured at
equally spaced intervals).
Examples: monthly Industrial Production
Index
Data collected irregularly or only once
are not timeseries.
Slide 13
Types of timeseries

Stock series are measures of activity at a point
in time and can be thought of as stocktakes.
•

Example: the Monthly Labour Force Survey –it
takes stock of whether a person was employed in
the reference week.
Flow series are series which are a measure of
activity to a date.
•
Examples of flow series include Retail, Current
Account Deficit, Balance of Payments.
Slide 14
Basic concepts - seasonality



Seasonality can be thought of as factors that
recur one or more times per year.
A seasonal effect is reasonably stable with
respect to timing, direction and magnitude.
The seasonal component of a time series
comprises three main types of systematic
calendar related influences:
•
seasonal influences
• trading day influences
• moving holiday influences
Slide 15
Seasonal influences
Seasonal influences represent intra-year fluctuations
in the series level, that are repeated more or less
regularly year after year.
•
•
warmth in Summer and cold in Winter BUT Weather
conditions that are out of character for a particular season,
such as snow in a summer month, would appear in irregular,
not seasonal influences.
reflect traditional behaviour associated with the calendar and
the various social (Chinese New Year), business (quarterly
provisional tax payments), administrative procedures (tax
returns) and effects of Christmas and the holiday season
Slide 16
Trading day
Trading day influences refer to the impact on
the series, of the number and type of days in a
particular month. A calendar month typically
comprises four weeks (28 days) plus an extra
one, two or three days. The activity for the
month overall will be influenced by those extra
days whenever the level of activity on the days
of the week are different.
(The trading day influence is not as significant
for quarterly series)
Slide 17
Moving holidays

Moving holiday influences refer to the
impact on the series level of holidays
that occur once a year but whose exact
timing shifts systematically. Examples of
moving holidays include Easter and
Chinese New Year where the exact date
is determined by the cycles of the moon.
Slide 18
Basic concepts - trend
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The trend component is defined as the long
term movement in a series.
The trend is a reflection of the underlying level
of the series. This is typically due to influences
such as population growth, price inflation and
general economic development.
The trend component is sometimes referred to
as the trend cycle.
Slide 19
Basic concepts - irregular
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The irregular component is the remaining component of the
series after the seasonal and trend components have been
removed from the original data.
For this reason, it is also sometimes referred to as the residual
component. It attempts to capture the remaining short term
fluctuations in the series which are neither systematic nor
predictable.
The irregular component of a time series may or may not be
random. It can contain both random effects (white noise) or
artifacts of non-sampling error, which are not necessarily random.
Most time series contain some degree of volatility, causing
original and seasonally adjusted values to oscillate around the
general trend level. However, on occasions when the degree of
irregularity is unusually large, the values can deviate from the
trend by a large margin, resulting in an extreme value. Some
examples of the causes of extreme values are adverse natural
events and industrial disputes.
Slide 20
Models for decomposing a series
Components of timeseries
•
It = irregular
• St = seasonal
• Tt = trend
• Ot = original
Additive Decomposition Model
• Ot = St + Tt + It
Multiplicative Decomposition Model
•
Ot = St x Tt x It
Slide 21
Additive Decomposition Model
•
•
The additive decomposition model assumes that
the components of the series behave independently
of each other. The trend of the series fluctuates yet
the amplitude of the adjusted series (magnitude of
the seasonal spikes) remain approximately the
same, implying an additive model.
Ot = St + Tt + It
Slide 22
Additive model
Slide 23
Jan-00
Serbia
SA_TS_7R_noHo
Mar-07
Jan-07
Nov-06
Sep-06
Jul-06
May-06
Mar-06
Jan-06
Nov-05
Sep-05
Jul-05
May-05
Mar-05
Jan-05
Nov-04
Sep-04
Jul-04
May-04
Mar-04
Jan-04
Nov-03
Sep-03
Jul-03
May-03
Mar-03
Jan-03
Nov-02
Sep-02
Jul-02
May-02
Mar-02
Jan-02
Nov-01
Sep-01
Jul-01
May-01
Mar-01
Jan-01
Nov-00
Sep-00
Jul-00
May-00
Mar-00
Example of additive series - IPI for
Serbia
130
120
110
100
90
80
70
Tren_TS_7R_noHo
Slide 24
Multiplicative Decomposition Model
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As the trend of the series increases, the
magnitude of the seasonal dips also
increases, implying a multiplicative
model.
Ot = St x Tt x It
Slide 25
Multiplicative Model
Slide 26
Jan-00
Kyrgyzstan
SA_TS_2R_Ho
May-07
Mar-07
Jan-07
Nov-06
Sep-06
Jul-06
May-06
Mar-06
Jan-06
Nov-05
Sep-05
Jul-05
May-05
Mar-05
Jan-05
Nov-04
Sep-04
Jul-04
May-04
Mar-04
Jan-04
Nov-03
Sep-03
Jul-03
May-03
Mar-03
Jan-03
Nov-02
Sep-02
Jul-02
May-02
Mar-02
Jan-02
Nov-01
Sep-01
Jul-01
May-01
Mar-01
Jan-01
Nov-00
Sep-00
Jul-00
May-00
Mar-00
Example of multiplicative series – IPI
for Kyrgyzstan
170
150
130
110
90
70
50
Tren_TS_2R_Ho
Slide 27
Seasonal adjustment philosophies
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Model based method
Filter based method.
Slide 28
Model based methods
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The model based approach requires the
components of an original time series, such as
the trend, seasonal and irregular to be
modelled separately. Alternatively, the original
series could be modelled and from that model,
the trend, seasonal and irregular component
models can be derived.
Model based methods assume the irregular
component is .white noise. i.e. the irregular
has no structure, zero mean and a constant
variance.
Slide 29
Model based methods
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TRAMO/SEATS
X13-ARIMA/SEATS
STAMP
Slide 30
TRAMO/SEATS
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TRAMO (Time Series Regression with ARIMA Noise, Missing
Observations and Outliers) and SEATS (Signal Extraction in ARIMA
Time Series) are linked programs originally developed by Victor
Gómez and Agustin Maravall at Bank of Spain.
The two programs are structured to be used together, both for indepth analysis of a few series or for routine applications to a large
number of them, and can be run in an entirely automatic manner.
When used for seasonal adjustment, TRAMO preadjusts the series to
be adjusted by SEATS.
The two programs are intensively used at present by data-producing
and economic agencies, including Eurostat and the European Central
Bank.
Programs TRAMO and SEATS provide a fully model-based method
for forecasting and signal extraction in univariate time series. Due to
the model-based features, it becomes a powerful tool for a detailed
analysis of series.
Slide 31
TRAMO/SEATS
www.bde.es
Slide 32
Filter based methods

This method applies a set of fixed filters
(moving averages) to decompose the
time series into a trend, seasonal and
irregular component. Typically,
symmetric linear filters are applied to the
middle of the series, and asymmetric
linear filters are applied to the ends of
the series.
Slide 33
Filter based methods
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X11
X11-ARIMA
X12-ARIMA (uses regARIMA Models for
forecasts, backcasts and
preadjustments)
STL
SABL
SEASABS
Slide 34
X12-ARIMA
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X12-ARIMA was developed by US Census Bureau as an
extended and improved version of the X11- ARIMA method of
Statistics Canada (Dagum (1980)).
The program runs through the following steps.
• First the series is modified by any user-defined prior
adjustments.
• Then the program fits a regARIMA model to the series in
order to detect and adjust for outliers and other distorting
effects for improving forecasts and seasonal adjustment.
• The program then uses a series of moving averages to
decompose a time series into three components. In the last
step a wider range of diagnostic statistics are produced,
describing the final seasonal adjustment, and giving pointers
to possible improvements which could be made.
The X12-ARIMA method is best described by the following
flowchart, as presented by David Findley and by Deutsche
Bundesbank respectively.
Slide 35
X12-ARIMA
The X12-ARIMA method is best described by the following flowchart, as
presented by David Findley and by Deutsche Bundesbank respectively.
Slide 36
X12-ARIMA

http://www.census.gov/srd/www/x12a/
Slide 37
Slide 38
Software
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TRAMO/SEATS
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X12-ARIMA
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http://www.bde.es
http://www.census.gov/srd/www/x12a/
DEMETRA
•
•
http://circa.europa.eu/irc/dsis/eurosam/info/data/demetra.htm
http://circa.europa.eu/irc/dsis/eurosam/info/data/
Slide 39
Requirements before
considering SA QNA
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High quality original timeseries
Timeseries of minimum 4 years (16 observations)
but ideally between 5 to 7 years (minimum)
Sufficient staff and resources
Sufficient time for experimenting (1 to 2 years)
Sufficient technical know how
Established SA procedures and extensive
experience in adjusting short-term economic
statistics
Slide 40
Direct or aggregative
adjustments

It is possible to seasonally adjust an aggregate series either
directly or by seasonally adjusting a number of its components
and adding the results. The latter (aggregative) method is often
employed for most of the major aggregates in the national
accounts. Besides retaining, as far as possible, the essential
accounting relationships, the aggregative approach is needed
because many of the aggregates include components having
different seasonal and trend characteristics, and sometimes
require different methods of adjustment.
Slide 41
Suggestions for countries planning to begin
producing and publishing seasonally adjusted
statistics (UNESCAP)
 Assess your needs and the needs of your users
 Allocate sufficient resources
 Provide staff with resources to learn and develop
expertise in SA
 Evaluate the possible options and choose the most
suitable seasonal adjustment method and software
(simple and reliable)
 Allow plenty of time for experimenting and evaluation
of results
 Ensure good communication between the seasonal
adjustment team and the subject-matter area
 Do not publish any seasonally adjusted statistics until
confident with the results
Slide 42
Suggestions for countries planning to begin
producing and publishing seasonally adjusted
statistics (UNESCAP)
 Consult the main users and seek their comments
 Ensure that you have a clear seasonal adjustment
policy: covering such issues as method, software,
reanalysis, aggregation, outliers, etc.
 When publishing SA statistics ensure that users can
easily access relevant metadata
 Ensure that users can access (electronically) the
complete timeseries: original, seasonally adjusted,
(trend and working day adjusted)
 If possible keep users inform about major
events/factors affecting seasonally adjusted statistics
 Ensure subject matter areas are involved in validation
of SA statistics
Slide 43
The criteria of a “good” seasonal
adjustment process
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
series which does not show the presence of
seasonality should not be seasonally adjusted
it should not leave any residual seasonality and effects
that have been corrected (trading day, Easter effect,
…) in the seasonally adjusted data
there should not be over-smoothing
it should not lead to abnormal revisions in the
seasonal adjustment figure with respect to the
characteristics of the series
the adjustment process should prefer the
parsimonious (simpler) ARIMA models
the underlying choices should be documented
Slide 44
Recommended practices for
Seasonal Adjustment (Eurostat)

Aggregation Approach
•
Preserving relationships between data - indirect approach
• Series that have very similar seasonal components (summing up the
series together will first reinforce the seasonal pattern while allowing the
cancellation of some noise in the series) - direct adjustment

Revisions
•
•

Concurrent adjustment vs forward factors
Take into account: the revision pattern of the raw data, the main use of
the data, the stability of the seasonal component
Publication Policy
•
When seasonality is present and can be identified, series should be
made available in seasonally adjusted form.
• The method and software used should be explicitly mentioned in the
metadata accompanying the series.
• Calendar adjusted series and/or the trend-cycle estimates (in graph
format) could be also disseminated in case of user demand.
Slide 45
Recommended practices for
Seasonal Adjustment (Eurostat)

Additional information to be published
•
•
•
•
•
•
•

Calendar Effects
•
•

The decision rules for the choice of different options in the program
The aggregation policy
The outlier detection and correction methods with explanation
The decision rules for transformation
The revision policy
The description of the working/trading day adjustment
The contact address.
Proportional approach vs regression approach
model based methods - regression approach should be used
Outlier’s Detection
•
•
Expert information is especially important about outliers
Outliers should be removed before seasonal adjustment is carried out
Slide 46
Recommended practices for
Seasonal Adjustment (Eurostat)

Transformation Analysis
•
•
•
•
•

Most popular software packages provide automatic test for logtransformation
Automatic choice should be confirmed by looking at graphs of the
series
If the diagnostics are inconclusive - visually inspect the graph of the
series
If the series has zero and negative values – it must be additively
adjusted
If the series has a decreasing level with positive values close to zero
and the series do not have negative values - multiplicative
adjustment has to be used
Time Consistency
•
Time consistency of adjusted data should be maintained in case of
strong user interest, but not if the seasonality is rapidly changing
Slide 47
Forward Factors versus Concurrent
Adjustment


Forward factors rely on an annual analysis of the latest
available data to determine seasonal and trading day
factors that will be applied in the forthcoming 4
quarters or 12 months (depending if the series is
quarterly or monthly).
Concurrent adjustment uses the data available at each
reference period to re-estimate seasonal and trading
day factors. Under this method data for the current
month are used in estimating seasonal and trading
day factors for the current and previous months. This
method continually fine tunes the estimates whenever
new data becomes available.
Slide 48
Seasonal Adjustment Step by Step

STEP 0 – Length of series
•
Series has to be at least 3 year-long (36 observations) for monthly series
and 4 year-long (16 observations) for quarterly series
• For an adequate seasonal adjustment data of more than five years are
needed.
• For series under 10 years the instability of seasonally adjusted data could
arise,
• If the series is too long information regarding seasonality, many years ago
could be irrelevant today, especially if changes in concepts, definitions and
methodology occurred.

STEP 1 – Preconditions, test for seasonality
•
•
Have a look at the data and graph of the original time series
Possible outlier values should be identified
• Series with too many outliers (more than 10%) will cause estimation
problems
• The spectral graph of the original series should be examined
• If seasonality is not consistent enough for a seasonal adjustment – series
should not be seasonally adjusted.
Slide 49
Seasonal Adjustment Step by Step

STEP 2 – Transformation type
•
•

Automatic test for log-transformation is recommended
The results should be confirmed by looking at graphs of the series
STEP 3 – Calendar effect
•
It should be determined which regression effects, such as
trading/working day, leap year, moving holidays (e.g. Easter) and
national holidays, are plausible for the series
• If the effects are not plausible for the series – the regressors for the
effects should not be applied

STEP 4 – Outlier correction
•

Series with high number of outliers relative to the length of the series
should be identified - attempts can be made to re-model these series
STEP 5 – The order of the ARIMA model
•
•
Automatic procedure should be used
Not significant high-order ARIMA model coefficients should be
identified.
Slide 50
Seasonal Adjustment Step by Step

STEP 6 for family X – Filter choices
•

STEP 7 – Monitoring of the results
•
•

It should be verified that the seasonal filters are generally in
agreement with the global moving seasonality ratio.
There should not be any residual seasonal and calendar effects in
the published seasonally adjusted series or in the irregular
component.
If there is residual seasonality or calendar effect, as indicated by
the spectral peaks, the model and regressor options should be
checked in order to remove seasonality.
STEP 8 – Stability diagnostics
•
Even if no residual effects are detected, the adjustment will be
unsatisfactory if the adjusted values undergo large revisions when
they are recalculated as new data become available. In any case
instabilities should be measured and checked.
Slide 51
Forward Factors versus Concurrent
Adjustment
 Concurrent adjustment uses the data
available at each reference period to reestimate seasonal and trading day
factors. Under this method data for the
current month are used in estimating
seasonal and trading day factors for the
current and previous months. This
method continually fine tunes the
estimates whenever new data becomes
available
Slide 52
Issues that can complicate the
seasonal adjustment process



Outliers (unusual estimates):
• The focus is on unusual estimates, not unusual observations as in the
sampling sense. Outliers can cause blips in an original series, seasonally
adjusted series and trend series unless they are modified or corrected during
the seasonal adjustment process
Revisions:
• The seasonal adjustment process leads to revisions to the seasonally
adjusted and trend series. Revisions are not desirable, either for the ABS or
the users of the series. The analysis technique chosen aims to strike a
balance between revisions and quality of the seasonally adjusted and trend
series. This issue is commonly referred to as the .end point problem.
Aggregation and Disaggregation:
• Regular and irregular influences are often estimated and removed from series
at fine levels of disaggregation, such as at the State by Industry level. Higher
level seasonally adjusted series, such as at the Australia level, can be
constructed by adding up component series to a higher level (to form an
indirectly adjusted series) or by directly seasonally adjusting the higher level
series (to form a directly adjusted series). The resulting series will not be
identical. A common issue faced by time series analysts is explaining why the
two approaches do not result in the same series.
Slide 53
Outliers
Slide 54
Outliers




Outliers are data which do not fit in the tendency of the time series
observed, which fall outside the range expected on the basis of the
typical pattern of the trend and seasonal components.
Additive outlier the value of only one observation is affected. AO may
either be caused by random effects or due to an identifiable cause as
a strike, bad weather or war.
Temporary change: the value of one observation is extremely high or
low, then the size of the deviation reduces gradually (exponentially) in
the course of the subsequent observations until the time series returns
to the initial level. For example in the construction sector the
production would be higher if in a winter the weather was better than
usually (i.e. higher temperature, without snow). When the weather is
regular, the production returns to the normal level.
Level shift: starting from a given time period, the level of the time
series undergoes a permanent change. Causes could include: change
in concepts and definitions of the survey population, in the collection
method, in the economic behavior, in the legislation or in the social
traditions. For example a permanent increase in salaries.
Slide 55
Useful references



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
Eurostat. ESS Guidelines on Seasonal Adjustment
http://epp.eurostat.ec.europa.eu/pls/portal/docs/PAGE/PGP_RESEA
RCH/PGE_RESEARCH_04/ESS%20GUIDELINES%20ON%20SA.P
DF
Eurostat. Eurostat Seasonal Adjustment Project.
http://circa.europa.eu/irc/dsis/eurosam/info/data/
Hungarian Central Statistical Office (2007). Seasonal Adjustment
Methods and Practices. www.ksh.hu/hosa
US Census Bureau. The X-12-ARIMA Seasonal Adjustment
Program. http://www.census.gov/srd/www/x12a/
Bank of Spain. Statistics and Econometrics Software.
http://www.bde.es/servicio/software/econome.htm
Australian Bureau of Statistics (2005). Information Paper, An
Introduction Course on Time Series Analysis – Electronic Delivery.
1346.0.55.001.
http://www.abs.gov.au/ausstats/abs@.NSF/papersbycatalogue/7A71
E7935D23BB17CA2570B1002A31DB?OpenDocument
Slide 56
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Questions?
THANK YOU
Slide 57