# Seasonal Adjustment and Time Series Issues

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```					                   Seasonal Adjustment
and Time Series Issues

United Nations Statistics Division (UNSD)

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Overview

Time series basic concepts

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Time series - Data frequency
 A time series is repeated observations over time measured
at equal space interval
 Low frequency
 Normally at annual or 5-year interval
 High frequency (Infra-annual)
 Normally at quarterly or monthly interval
 The present discussion mainly concerns time series issues
related to high frequency manufacturing statistics
 Example: IIP, new orders, sales/turnover, etc.
 one of the key tool for economic policy-making,

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Time Series - Decomposition
 The main idea of time series analysis is to decompose the series into
components with a simple interpretation
 Classical time series decomposition
 Trend – long term movements in the level of the series
 Cycle – fluctuations around the trend.
• Cycle period more than a year
• In much decomposition framework, trend and cycle
components are combined
 Seasonal – cyclical patterns that may evolve as the result of
changes associated with the seasons.
• Cycle period less than a year
 Irregular – other random or short-term unpredictable fluctuations

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Time Series - Decomposition
 Most common forms of decomposition
 Additive: Yt = Tt + St + It
 Multiplicative: Yt = Tt x St x It
where
 Yt = Observed time series at time t
 Tt = Trend-cycle component at time t
 St = Seasonal component at time t
 It = Irregular component at time t
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Time Series - Decomposition
 Then, the question is how to decompose a time series.
 Two main approaches:
 Descriptive approach
• Extract components using filter-based (smoothing) methods
• Create an approximate function that attempts to capture
important patterns and leave out the noise.
 Statistical modeling approach
• Signal and noise model
• Signal - Non-random component that evolves over time (e.g.
trend + seasonal)
• Noise - Random component, usually modelled by an ARMA
process
 The well-know seasonal adjustment tools, as will be seen later, are
mainly based on these two approaches.

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Time Series – Stocks vs flows
 Stocks
 Measure of a certain attribute at a particular point of
time
 Examples: employments, wages in the manufacturing
sector
 Flows
 A measure of activity over a period of time
 Examples: industrial production, change in inventory
 Whether a time series is a stock or flow do matter for the
seasonal adjustment process (as explained later).

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Why?
What?
 How?

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Hypothetical Case Study 1

Suppose the underlying chocolate
production remain at 100.
The effect of Easter will increase chocolate
production by 20 units in the month where
Easter lies.
In 2008, Easter fall in March.
In 2009, Easter fall in April.

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March                                   April
Without Easter        Easter              Easter            Easter
2008                100                120                   100            100
2009                100                100                   100            120
Yearly Change
(without Easter
Yearly Change
(with Easter

 Without adjusting for Easter effect, the series fall by 16.7% in March and
increase by 20% in April from 2008 – 2009.
 However, they are not the true movements of the production behaviour
underlying the series. The change is solely due to different months in the
year where Easter lies.
 This is called moving holiday effects, as will be discussed later.
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Hypothetical Case Study 2
 February 2010 has 28 days, of which 20 are working/trading days
 March 2010 has 31 days, of which 23 are working/trading days.
 Suppose a worker in the chocolate factory is paid by number of days he work.
10 workers are employed. Daily wage per workers is always \$10.
 Hence the total wage and salaries paid by the factory owner is increased by
15% .
 However, the underlying wage structure and level of employment remains
unchanged. The change is solely due to different number of working days in
each month.
 This is called working/trading effects, as will be discussed later.

Period              Wage rate        Employment           Working days   Salaries paid
February 2010 \$10                    10                   20             \$2000
March 2010          \$10              10                   23             \$2300
Monthly             0%               0%                   Increased by   Increased by
changes                                                   15%            15%
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Hypothetical Case Study 3
 Suppose the underlying sales of turkey in Sept 2010 is 100
 Every year in December the sales of turkey will increase by an addition 50 due
to Christmas
 Owing to economic crisis the sales of turkey decreased by 10. Policymaker
would like to capture this information.
 Total sales of turkey increased by 50% in Dec 2010 compared with previous
month.
 However, the underlying sales of turkey actually decreased
 Wrong signal!
Underlying sales            Total sales
Sept 2010                        100                     100
Oct 2010                          90                      90
Nov 2010                          80                      80
Dec 2010                          70                     120
Monthly Change (Nov to Dec)              -12.75%                  +50%

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Example of seasonal fluctuations

Source: Brian Monsell (2009)

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 Infra-annual manufacturing statistics are often influenced by seasonal
 mask relevant short and long-term movements of the series
 impede a clear understanding of economic phenomena.
 Main aims of seasonal adjustment:
 Filter out out variations associated with
• time of year - seasonal fluctuations
• arrangement of the calendar - calendar effects
 Reveal the ―news‖ contained in a time series,
• the seasonally adjusted results do not show ―normal‖ and
repeating events, they provide an estimate for what is ―new‖ in
the series (change in the trend, the business cycle or the
irregular component).
 Facilitates comparisons between consecutive time periods.

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 Random disruptions and unusual movements effect will not be removed.
 Examples: ―extreme‖ weather conditions, consequences of economic
policy, large scale orders or strikes, etc.
 Systematic seasonal fluctuations will be removed
 Movements recur with similar intensity in the same season each year
 On the basis of the past movements of the time series in question, can
under normal circumstances be expected to recur.
 Caused by various factors, such as weather patterns, administrative
measures, social/cultural/religious event, length of the month, etc.
 Seasonal adjustment will also remove calendar effects insofar as influences
deriving from
 Differences in the number of working/trading days
• Known as ―working/trading days effects‖
 the dates of particular days which can be statistically proven and
quantified (e.g. public holidays, weekday on the last day of the month in
the case of stock series).
• Known as ―moving holidays effects‖

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so far
In sum, the aim of seasonal adjustment is to
remove
 seasonal fluctuations
 calendar-related effects
in order to
 Aid interpretation
 Reveal the ‗news‘ of the series.
 Facilitate comparison.
This answers the WHY and WHAT questions.

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 We come to the HOW question. Basic Idea:
 Take away the seasonal component (that capture systematic
seasonal fluctuations within a year)
 By using filter-based (smoothing) or modeling approach, we can
decompose and remove the seasonal component from the series
Classical                        Model for                          Model for
Decomposition                    Original Series               Seasonally Adjusted Series
Additive:                      Yt = Tt + St + It           SA(Yt) =Tt + It
Multiplicative:                Yt = Tt x St x It           SA(Yt) =Tt x It
Log-additive:              ln(Yt) = ln(Tt + St + It) SA(Yt)= Exp(Tt + It)
Pseudo-additive:                Yt=Tt (St+It-1)            SA(Yt)= Tt x It
   Yt = Observed time series
   SA(Yt ) = Seasonal adjusted series
   Tt , St, & It : Trend-cycle, Seasonal and Irregular component respectively
( t denotes time subscript)

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Source: Brian Monsell (2009)

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Source: Brian Monsell (2009)

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 Which decomposition model to be used?
 Graphical and manual inspection
• Series contains negative number or zeros
• Seasonal component is not affected by the level of the series
• Multiplicative (pure multiplicative or log-additive)
• If seasonal component vary proportionally to the level of series
 Base on diagnostic statistics built in the programme
• information criteria such as AIC and SBIC
 In practice
 many decomposition scheme is Multiplicative (i.e. pure multiplicative or

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Source: Artur Andrysiak (2010)

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17 March 2010
Nov-00
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ECLAC, Santiago
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Serbia
for Serbia

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SA_TS_7R_noHo
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for ECLAC member states
Sep-04
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Workshop on Manufacturing Statistics
Mar-05
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Tren_TS_7R_noHo

Jul-05
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Example of additive series - IIP

Jan-07
Source: Artur Andrysiak (2010)

Mar-07
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Multiplicative Model

Source: Artur Andrysiak (2010)

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Kyrgyzstan
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SA_TS_2R_Ho
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for ECLAC member states
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Workshop on Manufacturing Statistics                                                    May-05
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Tren_TS_2R_Ho

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Example of multiplicative

Jul-06
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series – IIP for Kyrgyzstan

Nov-06
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Source: Artur Andrysiak (2010)

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Summary so far
 The aim of seasonal adjustment is to remove
 seasonal fluctuations
 calendar-related effects
 Seasonal fluctuations are captured by the seasonal component in the
time series in the decomposition scheme.
 By using appropriate decomposition scheme, we can decompose
the series based on either filter-based or modeling approach to
remove the seasonal component.
 Will be discussed later
 But how to remove the calendar-related effects?

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Calendar-related effects
 What about calendar-related effects where there is no systematic
pattern repeat every year?
 Solely due to arrangement of calendar
 ―Moving‖ holidays
• Base on different calendars other than Gregorian calendar (i.e.
what we are using now)
• Exact timing shifts every year
• Example: Easter, Rio Carnival, Chinese New Year, Ramadan
• Difference number of working days each month
 They do not fit into seasonal component in the standard classical
decomposition approach

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Monthly working days 2009-11                    Months where Easter lies
2009        2010   2011                Year        Month   Day
Jan               22        21       21                 2002 March           31
Feb               20        20       20                 2003 April           20
Mar               22        23       23                 2004 April           11
Apr               22        22       21                 2005 March           27
May               21        21       22                 2006 April           16
Jun               22        22       22                 2007 April               8
Jul               23        22       21                 2008 March           23
Aug               21        22       23
2009 April           12
Sep               22        22       22
2010 April               4
Oct               22        21       21
2011 April           24
Nov               21        22       22
Dec               23        23       22

Calendar related effects: No systematic pattern repeat each year!

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 Either treat the calendar-effects in the Irregular component (not the
most common practice)….or
 Most common practice now:
 ―Pre-adjustment‖ of the series before running seasonal
• Regression-ARIMA approach
• Create user-defined regressors ( Xt) to model calendar-
related effects

Yt   ' Xt  Zt
 Zit then can be thought as prior-adjusted series that is
 free of calendar-related effects
 assumed to follow the ARIMA process
 ‗cleansed input series‘ ready for the decomposition process.

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Example of constructing holiday
regressors
Average of Z
Number of       Proportion of                   (Mean
from the
Easter   Easter     Days belong     days belongs                    Subtracted)   Stock
same
Date     Interval   to Easter       to Easter                       Flow          Regressor
quarter over
Interval        Interval (Z)                    Regressor
500 years

Q1-2007                 8           1               0.125           0.382         -0.257        -0.257

Q2-2007    08-Apr       8           7               0.875           0.618         0.257           0

Q3-2007                 8           0                 0               0             0             0

Q4-2007                 8           0                 0               0             0             0

Q1-2008    23-Mar       8           8                 1             0.382         0.618         0.618

Q2-2008                 8           0                 0             0.618         -0.618          0

Q3-2008                 8           0                 0               0             0             0

Q4-2008                 8           0                 0               0             0             0

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Stock and flows
 The construction method of regressors applied
to the stock and flow series are different
 We can also create user-defined regressor to
model other effects (will be discussed later), such
as
 Level shifts
 Seasonal breaks
 Intervention variables
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Summary at this point
 The aim of seasonal adjustment is to remove
 seasonal fluctuations
 calendar-related effects
 Seasonal fluctuations are captured by the seasonal component in the
time series in the decomposition scheme.
 Calendar-related effects are removed by pre-adjusting the series using
Regression-ARIMA model before the decomposition process.
 Now we are going to talk about the common methods/software used,
based on either filter-based or modeling approach, to decompose the
series to remove the seasonal component.

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and softwares
Filter based methods
 The X-11 family (U. S. Census Bureau,
Model-based methods
 TRAMO/SEATS (Bank of Spain)
 STAMP (Andrew Harvey)

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Most statistical agencies and central bank
are using
 X-12-ARIMA (filtered based)
 TRAMO-SEATS (model-based)
Integrated approach/Combined platform
 X-13-ARIMA-SEATS (testing version)
 DEMETRA
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―X-11‖ Family
X-11

X-11-ARIMA

X-12-ARIMA

X-13-ARIMA-SEATS
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Features of X-12-ARIMA
 Wide variety of seasonal and trend filter options;

 Suite of modeling and seasonal adjustment diagnostics, including
 Spectral diagnostics;
 Diagnostics of the quality and stability of the seasonal adjustments;
 Out of sample forecast error model selection diagnostics.

 Extensive time series modeling and model selection capabilities
 linear regression models with ARIMA errors (regARIMA models);
 Automatic model selection options;
 User-defined regression variables.

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X-12-ARIMA
The X12-ARIMA method is best described by the following flowchart, as
presented by David Findley and by Deutsche Bundesbank respectively.

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X-12-ARIMA – Two steps
1. Cleaning data – Reg-ARIMA model
 Forecast (and Backcast) extension of series
before applying X-11 filters
 Detect and adjust for outliers and other
distorting effects to improve the forecasts and
 Detect and estimate additional components
(e.g. calendar effects)
2. Decomposition - X-11 Algorithm

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RegARIMA Model
 Yt       
log 
 D         
            t  Zt
X
   t      

transformation                                  ARIMA Process

Xt        Regressor for trading day and holiday
temporary changes, level shifts, ramps,
user-defined effects

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Understand Moving Average (MA)
Example : 3x3 MA
( 1 Yt  2 
3
1
3   Yt 1      1
3   Yt ) / 3 
( 1 Yt 1 
3
1
3   Yt        1
3   Yt 1 ) / 3 
( 1 Yt 
3
1
3   Yt 1    1
3   Yt  2 ) / 3 

1
9   Yt  2       2
9   Yt 1      1
3   Yt        2
9   Yt 1    1
9   Yt  2

t-2   t-1              t             t+1              t+2
Original              100   110              120           100              100
Series
Weight                1/9   2/9              1/3           2/9              1/9
Filtered series                              117.8

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Other example of MA
Original     3x3 MA           2x4 MA
130
Nov-09            80
120
Dec-09            90
110
Jan-10           100        100.0           100.0
100
Feb-10           110        107.8           107.5
90
Mar-10           120            111.1       110.0
80
Apr-10           110        107.8           107.5
70
May-10           100        100.0           100.0
9

0
10

11
0

0

10
0
v-0

-1

v-1
l-1
-1

Jun-10            90            92.2         92.5
n-

n-
p-
ar

ay

Ju
No

No
Ja

Ja
Se
M

M

Jul-10           80            88.9         90.0
Original       3x3 MA          2x4 MA             Aug-10            90            92.2         92.5
Sep-10           100        100.0           100.0
t-2        t-1     t          t+1       t+2               Oct-10           110        107.8           107.5
2x4 MA                                                               Nov-10           120            111.1       110.0
weights     0.13       0.25       0.25     0.25     0.13
Dec-10           110        107.8           107.5
3x3 MA
Jan-11           100
weights     0.11       0.22       0.33     0.22     0.11
Feb-11            90
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X-11 Algorithm: Basic idea

Iterative Refinement in X-11
 1. Estimate simple trend
 2. Remove the trend from the series
 3. Estimate seasonal factors
 4. Calculate seasonally adjusted series
 Repeat Steps 1 — 4
 Re-estimate trend and irregular
Complete the procedure three times (B, C, D)

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X-11 Algorithm: steps
 Apply ―Trend Moving Average ‖ to the original series (Z) to
estimate trend (T)
 Remove the trend (T) from original series Z to get the
detrended series (SI)
 Apply ―Seasonal Moving Average‖ to the detrended series
(SI) to estimate seasonal (S)
 Remove seasonal (S) from original series (Z) to estimate the
 Apply ―Trend Moving Average‖MA to the seasonal adjusted
series (SA) to estimate trend (T)
 …… repeat the whole process for 3 times

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Source: Findley, et. al. (1998)

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Source: Findley, et. al. (1998)

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Source: Findley, et. al. (1998)

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X-12: Final output

X-12-ARIMA Tables/Iterations
 A. Prior adjustments before the core X-11
procedures
 B. Preliminary estimation of Seasonal, Trend,
and extreme values
 C. Intermediate estimation of Seasonal and
Trend, final estimation of extreme values
 D. Final estimation of Seasonal, Trend,
Irregular
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D10i
B1
D10h I/S ratio D10j                               B13                        B2
D10k                                                                                           B4c
B11                                       B3                  B4b                 B4d

normalisation
B4a                       B4e
D-cycle only

B10b                                                                               & outliers

B10                                             B4              B4g                B4e2
B4f
B10a

B9                                          B5              B5g
B9f
B9e2                          B9g
B5c              B5a

B9e
normalisation
B9a
B8                          B6
& outliers                                                                        B5b
B7
B9d                                                                                            normalisation
B9b
B9c
B7c

B7d I/C ratio B7a                 Source: Gary Brown (2010)

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TRAMO/SEATS
 Consists of two linked programs
 TRAMO is a complete regARIMA modeling package, with automatic
identification of ARIMA models, outliers and other components
 SEATS takes modeling results from TRAMO and performs a model-based
signal extraction
 TRAMO
 Time Series Regression with ARIMA Noise, Missing Observations and Outliers
 SEATS
 Signal Extraction in ARIMA Time Series
 TRAMO/SEATS
 Linked programs of TRAMO and SEATS
 Developed at the Bank of Spain by Victor Gomez and Agustin Maravall (1996)
 Uses ARIMA model as basis for seasonal decomposition
 Derive model-based filters to capture frequencies associated with each
component
• Signal extraction approach based on periodogram/spectral analysis

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TRAMO

Source: Augustin Maravell (2008)

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SEATS

Source: Augustin Maravell (2008)

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Source: Augustin Maravell (2008)

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SEATS

Source: Augustin Maravell (2008)

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The adjustment filter is determined by a
model, not a finite set of moving average
filters.

In practice, SEATS sometimes gives
revisions for some irregular Census Bureau
series

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Combining X-12 & TRAMO/SEATS
X-13-ARIMA-SEATS
 X-12-ARIMA + SEATS
 Users can choose between model-based
seasonal adjustments from SEATS and non-

DEMETRA
 Developed by Eurostat
 Allow the running of X-12 and
TRAMO/SEATS in the same user-interface.
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used by NSO in Europe (used in 2006)

Source: UNECE (2006)

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used by NSO in Europe (future use of SA
methods – planned in 2006)

Source: UNECE (2006)

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Software Used by NSOs
(Europe) in 2006

Source: UNECE (2006)

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UN ESCAP Assessment - 2009
(Out of 58 countries)

TRAMO/SEATS              1

X12-ARIMA                                                                          14

X11-ARIMA                        3

X13-ARIMA/SEATS        0

Other                   2

0           2           4        6        8        10      12       14         16

Source: UNESCAP (2009)

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UN ESCAP Assessment - 2009

(Out of 58 countries)

Other                                   4

X12-ARIMA/SEATS                                                              8

TRAMO/SEATS              1

Demetra        0

0            2              4             6          8               10

Source: UNESCAP (2009)

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Source: UNESCAP (2009)
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Software
 TRAMO/SEATS
 http://www.bde.es
 X12-ARIMA
 http://www.census.gov/srd/www/x12a/
 DEMETRA
 http://circa.europa.eu/irc/dsis/eurosam/info/data/demetr
a.htm
 http://circa.europa.eu/irc/dsis/eurosam/info/data/

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Summary at this point
 The aim of seasonal adjustment is to remove
 seasonal fluctuations
 calendar-related effects
 Seasonal fluctuations are captured by the seasonal component in the
time series in the decomposition scheme.
 Calendar-related effects are removed by pre-adjusting the series using
Regression-ARIMA model before the decomposition process.
 We have reviewed the commonly used approach to for seasonal
 X-12 ARIMA (filtered-based)
 TRAMO-SEATS (model-based)

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Outliers & Level Shifts

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Outliers & Level Shifts
 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.
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Outliers & Level Shifts
 If there are underlying reasons to explain such effects
 Temporary removed before the seasonal adjustment
process using the Reg-ARIMA model
 But unlike the calendar-related effect, they will be add
back after the seasonal adjustment process . i.e. such
effect will be shown in the seasonally adjusted series.
 X-12-ARIMA and TRAMO-SEATS have automatic
procedure to account for outliers and level shifts.

 If there is no reason to explain but the data is confirmed
correct
 Don‘t make any pre-adjustment, leave the series as it is.

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Requirements before
considering SA
 High quality original timeseries
 Time series 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

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 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

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 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.

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 STEP 6 for family X – Filter choices
 It should be verified that the seasonal filters are generally in agreement
with the global moving seasonality ratio.
 STEP 7 – Monitoring of the results
 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.

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The criteria of a ―good‖ seasonal
 series which does not show the presence of seasonality
 it should not leave any residual seasonality and effects that
have been corrected (trading day, Easter effect, …) in the
 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

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Recommended practices (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
 The method and software used should be explicitly mentioned in the
 Calendar adjusted series and/or the trend-cycle estimates (in graph format)
could be also disseminated in case of user demand.

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Recommended practices (Eurostat)
 Additional information to be published
 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
 Calendar Effects
 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

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ECLAC, Santiago             for ECLAC member states
Recommended practices (Eurostat)
 Transformation Analysis
 Most popular software packages provide automatic test for log-
transformation
 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
 If the series has a decreasing level with positive values close to
zero and the series do not have negative values - multiplicative
 Time Consistency
 Time consistency of adjusted data should be maintained in case of
strong user interest, but not if the seasonality is rapidly changing
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Useful references
 Eurostat. ESS Guidelines on Seasonal Adjustment
http://epp.eurostat.ec.europa.eu/pls/portal/docs/PAGE/PGP_RESEARCH/P
GE_RESEARCH_04/ESS%20GUIDELINES%20ON%20SA.PDF
 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/7A71E7935D
23BB17CA2570B1002A31DB?OpenDocument

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