Methodology of the Monthly Index of Services
Seasonal Adjustment
Data that are collected at regular intervals form a time series. Many of the most well known
statistics are time series, including the Index of Services and its components. Those analysing
a time series will usually be looking to see what the short term movements in the series are,
what the long term movements are, and whether any unusual occurrences, such as strikes or
bad weather, have had any major effect on the series. This type of analysis is not easy using
raw time series data because there will normally be short-term effects associated with the
time of year that will obscure other movements. For example, retail sales will go up in
December due to the effect of Christmas. The purpose of seasonal adjustment is to remove
the variations associated with the time of the year, i.e. seasonal effects; this allows
consecutive months to be compared, providing a reliable estimate of short-term change.
A series is modelled in one of two forms, either as an additive model:
Y=C+S+I
or using a multiplicative model:
Y=CxSxI
where
C is the trend-cycle
S is the seasonal component
I is the irregular component
- the medium and long term movements in the
series.
- which reflects the effect of climate and
institutional events that repeat more or less
regularly each year.
- which represents unforeseeable events of all
kinds, including errors in the data.
Seasonal adjustment is the process of identifying and removing the seasonal component from
a time series.
Other things that affect time series
There are regular effects that do not necessarily occur in the same month (or quarter) each
year. These can be identified and removed from the series, they include:
•
trading day effects, which are caused by months having differing numbers of each day of
the week from year to year, for example spending in DIY stores is likely to be higher in a
month with five, rather than four, weekends; and
•
moving holidays, which result from events that may fall in different months from year to
year, for example Easter,which can occur in either March or April.
These are known as calendar effects.
Other problems that can occur in seasonal adjustment include:
•
outliers, which are extreme values that usually have identifiable causes, such as strikes,
war or extreme weather conditions, which can distort the seasonal adjustment. They are
normally considered to be part of the irregular component;
•
trend breaks (also known as level shifts), where the trend component suddenly increases
or decreases in value. Causes of this include a change in the definition of the series that is
being measured (e.g. a reclassification of products or a change in the tax rate); and
•
seasonal breaks, where there are changes in the seasonal pattern.
Many of these problems can be solved using prior adjustments, which allow the user to
enter a set of ‘corrections’ to the data that change them before the seasonal adjustment
process takes place.
X11ARIMA
X11ARIMA is the seasonal adjustment program used in the Index of Services. It was
developed by Statistics Canada as an extended and improved version of the US Census
Bureau X-11 method. The program runs through the following steps:
•
the series is modified by any user defined prior adjustments;
•
trading day and Easter effects are identified and removed;
•
the program fits an ARIMA model (AutoRegressive Integrated Moving Average) to the
series in order to extrapolate forward (forecast) and backward (backcast) an extra year of
data;
•
the program then uses a series of moving averages to decompose a time series into the
three components, trend, seasonal and irregular. It does this in three iterations, getting
successively better estimates of the three components. During these iterations extreme
values (outliers) are identified and replaced for the purpose of identifying the seasonal
component (but not permanently replaced in the data); and
•
a range of diagnostic statistics is produced, describing the final seasonal adjustment and
giving pointers to possible improvements that could be made.
A more detailed description of X11ARIMA, including the mathematical formulation of
models, can be found in Annex D.
Prior adjustments
Prior adjustments allow the user to change the data before they are seasonally adjusted, thus
removing any features of the data that do not correspond to the general seasonal pattern, but
can be explained by economic or other factors.
If an observation does not fit the general seasonal pattern, the moving averages in
X11ARIMA will affect the seasonal adjustment of several neighbouring observations. By
changing the value of the observation (or observations) with prior adjustments a better
estimate of the general seasonal pattern can be obtained.
There are two types of prior adjustment, temporary and permanent. Both types are divided
into the original data (subtracted from, if an additive model is used) at the start of the
seasonal adjustment process. If temporary priors are used, they will be multiplied back into
(added to, if an additive model is used) the seasonally adjusted series at the end of the
process. Both temporary and permanent prior adjustments can be applied to the same series.
Temporary prior adjustments are reversed at the end of the seasonal adjustment process. They
are used when there is a 'real' effect in the data that should show up in the seasonally adjusted
series but should not affect other observations. Examples of effects that should be removed
by temporary priors include trend breaks (level shifts) caused by changes in the tax rate and
outliers caused by 'real' events such as war or extreme weather.
Permanent prior adjustments do not have their effects removed after the seasonal adjustment.
They are used for removing effects that should not appear in the seasonally adjusted series.
Examples include errors in the data, seasonal breaks, Easter effects and corrections for noncalendar data recording patterns. When adjusting for effects that are not 'errors' using
permanent priors, the increases and decreases should approximately cancel each other out,
since the change is merely a redistribution of the data between calendar months.
Seasonal adjustment of the Index of Services
In order to improve the quality and consistency of seasonal adjustment across the entire range
of National Statistics, the ONS has given responsibility for seasonal adjustment to a
specialised central unit within the Methodology Group called Time Series Analysis Branch
(TSAB). TSAB staff are highly trained in time series methods, including seasonal
adjustment. Because there are judgmental elements in the process, this helps to ensure that
seasonal adjustment is always carried out by those with a suitable level of expertise.
TSAB carries out an annual seasonal adjustment review of the IoS. In this review the data are
analysed and appropriate parameters (and prior adjustments) for X11ARIMA are set for the
following year. During the year, the prior adjustments are maintained by the IoS team.
Existing priors may be amended following revisions to the data, and new ones added if
needed to deal with some event emerging in new observations.
Trends
People are often interested in removing the irregular component as well as the seasonal in
order to produce a trend (also known as a trend-cycle or a short-term trend). There are a
number of problems associated with trend estimation; these include:
•
problems with defining a trend; how smooth it should be and whether it includes
economic and business ‘cycles’ or just long term structural effects;
•
there is conflict between wanting to identify turning points in the trend and producing a
smooth series; and
•
the irregular component is, by its nature, unpredictable, so the process is not as reliable as
removing the seasonal component. With many methods the trend can easily have different
annual totals from the original data, which can lead to public distrust of the trend.
Because of the above problems trends are not currently published for the Index of Services.