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### 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

Economic Statistics Section, UNECE

Overview

- What and why
- Basic concepts
- Methods
- Software
- Recommended practices
- Step by step
- Issues
- Useful references

Seasonally adjusted and original series – Industrial Production Index

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

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.

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

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

Basic concepts - timeseries

- 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.

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.

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

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

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.

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.

Basic concepts - trend

- 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.

Basic concepts - irregular

- 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.

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 xIt

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

Multiplicative Decomposition Model

- As the trend of the series increases, the magnitude of the seasonal dips also increases, implying a multiplicative model.
- Ot = St x Tt xIt

Seasonal adjustment philosophies

- Model based method
- Filter based method.

Model based methods

- 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.

Model based methods

- TRAMO/SEATS
- X13-ARIMA/SEATS
- STAMP

TRAMO/SEATS

- 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 in-depth 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.

TRAMO/SEATS

www.bde.es

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.

Filter based methods

- X11
- X11-ARIMA
- X12-ARIMA (uses regARIMA Models for forecasts, backcasts and preadjustments)
- STL
- SABL
- SEASABS

X12-ARIMA

- 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.

X12-ARIMA

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

X12-ARIMA

- http://www.census.gov/srd/www/x12a/

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/demetra.htm
- http://circa.europa.eu/irc/dsis/eurosam/info/data/

The criteria of a “good” seasonal adjustment process

- 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

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.

Recommended practices for Seasonal Adjustment (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
- The description of the working/trading day adjustment
- The contact address.
- 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

Recommended practices for Seasonal Adjustment (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 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

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.

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.

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.

Seasonal Adjustment Step by Step

- 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.

Forward Factors versus Concurrent Adjustment

- 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

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.

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.

Useful references

- Eurostat. ESS Guidelines on Seasonal Adjustment

http://epp.eurostat.ec.europa.eu/pls/portal/docs/PAGE/PGP_RESEARCH/PGE_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/7A71E7935D23BB17CA2570B1002A31DB?OpenDocument

Questions?

- THANK YOU

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