1. What is a time series?
A time series is sequence of measures of a given phenomenon
taken at regular time intervals such as hourly, daily, weekly, monthly,
quarterly, annually, or every so many years.
An example of a time series is shown below. This time series happens to be
Retail Sales from Shoe Stores.
2. What is seasonal adjustment?
Seasonal adjustment is the process of estimating and removing the
seasonal effects from a
time series, and by seasonal, we mean an effect that happens
at the same time and with the same magnitude and direction every year.
The basic goal of seasonal adjustment is to decompose a time series into several
different components including a seasonal component and an irregular component.
Because the seasonal effects are an unwanted feature of the time series,
seasonal adjustment can be thought of as focused noise reduction.
For more information on why seasonal effects are unwanted, please see the paper
(in PDF)
"Why Seasonal Adjustment" by James Ashley.
3. What kind of data do we need for seasonal adjustment?
We need a time series for seasonal adjustment. Because seasonal effects are annual effects,
the data must be collected at a frequency less than annually, usually monthly or quarterly.
For the data to be useful for time series analysis, the data should be
comparable over time. That means
- the measurements should be taken over discrete (nonoverlapping) consecutive periods, i.e.,
every month or every quarter, and
- the definition of the concept and the way it is measured
should be consistent over time.
4. Why seasonally adjust data?
Economists, policy makers, and consumers use time series we publish to
make decisions. They try to identify important features of economic
series such as direction, turning points, and consistency between other
economic indicators. Sometimes seasonal movements can make these
features difficult to see, so we publish economic series with the seasonal
movements removed for those who prefer to view data without seasonal
movements.
There is some information lost when a series is seasonally adjusted,
therefore it is also useful for many people to have both the original series
and the seasonally adjusted series available for analysis.
For more information, please see the paper (in PDF)
"Why Seasonal Adjustment" by James Ashley.
5. How much data do I need to get a reasonable adjustment?
X-12-ARIMA will work with four years of data. To get good regARIMA models (see
question 8 for more
information on regARIMA models), it is better
to have at least seven years of data. Studies have shown that TRAMO/SEATS
works best with at least six years of data.
Keep in mind that longer series aren't necessarily better. If the series has changed
the way the data is measured or defined, it might be better to cut off the early part
of the series to keep the series as homogeneous as possible. The best way to decide if
your series needs to be shortened is to investigate the data collection methods
and the economic factors associated with your series and choose a length that gives
you the most homegeneous series possible.
7. What effects are removed during seasonal adjustment?
During seasonal adjustment, we remove seasonal effects from the original series.
If present, we also remove trading day and moving holiday effects.
The seasonally adjusted series is therefore a combination of the trend
and irregular components. (See
question 5 above for definitions of these components.)
One common misconception is that seasonal adjustment will also hide
any outliers present. This is not the case. If there is some kind of unusual
event, we need that information for analysis, and outliers are included
in the seasonally adjusted series.
8. What is a regARIMA model?
A regARIMA model is a regression model
with ARIMA errors. ARIMA stands for
AutoRegressive Integrated Moving Average.
When we use regression models to estimate some of the components in a time series,
the errors from the regression model are correlated, and we use ARIMA models
to model the correlation in the errors.
ARIMA models are one way to describe the relationships
between points in a time series.
Besides using regARIMA models to estimate regression effects (such as
outliers, trading day, and moving holidays), we also use ARIMA models
to forecast the series. Research has shown that using forecasted values
gives smaller revisions at the end of the series.
9. How do I generate a seasonally adjusted series?
When generating seasonally adjusted series, it is important to use software
specifically designed for the purpose of seasonal adjustment.
Seasonal adjustment is more complicated than it might seem on the surface.
For most series, seasonal adjustment can NOT be done properly by hand or with
a spreadsheet program like Excel.
A good seasonal adjustment program has at least the following features:
- unbiased
- robust against outliers
- moving holiday and working/trading day estimation
- diagnostics
Seasonal adjustment is usually done with an off-the-shelf program.
Three popular programs are
- X-12-ARIMA (Census Bureau),
- X-11-ARIMA (Statistics Canada), and
- TRAMO/SEATS (Bank of Spain).
X-12-ARIMA is used at the U.S. Census Bureau for all
official seasonal adjustments in the publications. For more information on
X-12-ARIMA, please see the questions in the FAQ related to
X-12-ARIMA. TRAMO/SEATS is used by
some statistical offices and banks, primarily in Europe. For more
information on TRAMO/SEATS,
please see the questions in the FAQ related to
TRAMO/SEATS.
10. What are seasonal filters?
A filter is a weighted average
where the weights sum to 1. Seasonal filters
are the filters used to estimate the seasonal component. Ideally, seasonal filters
are computed using values from the same month or quarter, for example, an estimate for
January would come from a weighted average of the surrounding Januaries.
The seasonal filters available in X-12-ARIMA consist of seasonal moving averages
of consecutive values within a given month or quarter.
An n x m moving average is an m-term simple average
taken over n consecutive sequential spans.
An example of a 3x3 filter for January 2003 (or Quarter 1, 2003) is:
2001.1 + 2002.1 + 2003.1 +
2002.1 + 2003.1 + 2004.1 +
2003.1 + 2004.1 + 2005.1
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An example of a 3x5 filter for January 2003 (or Quarter 1, 2003) is:
2000.1 + 2001.1 + 2002.1 + 2003.1 + 2004.1 +
2001.1 + 2002.1 + 2003.1 + 2004.1 + 2005.1 +
2002.1 + 2003.1 + 2004.1 + 2005.1 + 2006.1
_____________________________________________________________
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11. What are trend filters?
Trend filters are weighted
averages of consecutive months or quarters used to estimate the trend component.
An example of a 2x4 filter for First Quarter 2005:
2004.3 + 2004.4 + 2005.1 + 2005.2
2004.4 + 2005.1 + 2005.2 + 2005.3
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8
Notice that we are using the closest points, not just the closest points within
the First Quarter like with the seasonal filters above.
Notice also that every quarter has a weight of 1/4, though
the Third Quarter uses values in both 2004 and 2005.
12. Why do the seasonal factors change
when new data are added?
Keep in mind that the data used in the seasonal and the trend filters
can go back several years.
Let's look at an example using X-12-ARIMA's seasonal moving average filters.
For example, if the last point in your series is
January 2006, and you're using 3x5 seasonal filters, the value at January 2006
will effect the estimates for Januaries in 2003, 2004, and 2005. You can see
the value for January 2003 in the equations below.
The 3x5 filter for January 2003:
2000.1+2001.1+2002.1+2003.1+2004.1
2001.1+2002.1+2003.1+2004.1+2005.1
2002.1+2003.1+2004.1+2005.1+2006.1
________________________________________________
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The 3x5 filter for January 2004:
2001.1+2002.1+2003.1+2004.1+2005.1
2002.1+2003.1+2004.1+2005.1+2006.1
2003.1+2004.1+2005.1+2006.1+2007.1
________________________________________________
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The 3x5 filter for January 2005:
2002.1+2003.1+2004.1+2005.1+2006.1
2003.1+2004.1+2005.1+2006.1+2007.1
2004.1+2005.1+2006.1+2007.1+2008.1
________________________________________________
15
You can see in the above equations that the new point at January 2006 will
affect the estimates for the other Januaries.
There is a similar effect for the trend filters, with new data effecting estimates
for half the filter length.
13. How do I choose between
an additive or multiplicative model?
There are two basic seasonal adjustment decomposition models.
Additive Model: Y = T + S + I or Y = T + (S + TD + H) + I
Multiplicative Model: Y = T * S * I or Y = T * (S * TD * H) * I
Under the additive decomposition, the seasonally adjusted series (A)
is obtained by subtracting
the estimated seasonal component from the original series:
A = Y - S' = T + I where S' may also include other calendar effects,
S' = S + TD + H.
Under the multiplicative decomposition, the seasonally adjusted series (A)
is obtained by dividing
the original series by the estimated seasonal component:
A = Y/S' = T * I where S' may also include other calendar effects,
S' = S * TD * H.
An additive model/decomposition is appropriate if the magnitude of the seasonal fluctuations does not vary with the level of the series
and the series does not contain any zero or negative values.
The multiplicative model/decomposition is usually appropriate for series of
positive values where the size of the seasonal oscillations increases
with the level of the series.
For a multiplicative decomposition, we generally take logarithms
of the series. Logarithms turn multiplicative relationships into
additive relationships.
For example, the multiplicative model: Y = T * S * I is equivalent to
log(Y) = log(T) + log(S) + log(I).
Logarithms also have a variance stabilizing effect on the series as seen in the
two graphs below. Notice that without a log transformation for the Retail Sales
from Shoe Stores series, the seasonal variations in the series increase as the
level of the series increases.
14. What is an indirect (or a direct) adjustment? Why would I need one?
A series that is made up of several smaller series is called a
composite series. The smaller series are called
component series.
For example, the total housing starts for the United States consists of
housing starts information for four regions of the United States.
The component series are housing starts for each region, and the composite or
aggregate series is the total for the US.
If the component series are seasonally adjusted first and then combined, the adjustment
for the total is called the indirect adjustment. An
indirect adjustment is generally of better quality if the component series have quite distinct
seasonal patterns and have adjustments of good quality.
If the component series are combined first and then we adjust the total, it is called
direct adjustment. If
the component series have similar patterns, then some of the noise
in the series may cancel out and the direct adjustment may be of
better quality than the indirect adjustment.
Example: United States Total Housing Starts-
Indirect Adjustment = SA(Northeast) + SA(Midwest) + SA(South) + SA(West)
Direct Adjustment = SA(Northeast + Midwest + South + West)
