Australian Bureau of Statistics
Time Series Analysis: Issues With Seasonal Adjustment
Moving seasonality is a form of seasonality that accounts for the variability in the seasonal component of a time series from year to year. As an example, consider Retail sales in the month of January. Post Christmas specials in January have become more and more popular in recent years. This has resulted in a steady increase in sales for the month of January over the years and is reflected by the slowly evolving nature of the seasonal pattern. This is refered to as moving seasonality.
Moving holidays are holidays which occur each year, but where the exact timing shifts under the Gregorian calendar system. Examples of moving holidays include Easter and Chinese New Year. Easter generally falls in April but can also fall in late March. Its timing affects series such as Tourism because people often holiday in this period. Chinese New Year mostly occurs in February but can also occur in January. Overseas Arrivals and Departures series from some Asian countries are affected by this holiday.
The trading day effect is related to months having different numbers of each day of the week from year to year. For example, there may be more garage sales in a January with five weekends rather than four.
In each month, there are four weeks and usually an additional one, two or three days. This means that there are always at least 4 Mondays, 4 Tuesdays, 4 Wednesdays etc., but some days will occur 5 times. A 31 day month for instance, comprises four weeks (28 days) plus three extra days. The number and composition of these extra days will affect the data for the month. For example, assume that a certain store is only open during the week. Then for a 30 day month, within which the two additional days are weekdays, the level of activity is likely to be greater than if the two additional days are Saturday and Sunday.
The Trading day effect is estimated by assigning to each day of the week a weight which reflects the level of activity of that day related to others. For a population with equal levels of activity on the five weekdays and no activity on weekends, weights of 1.4 are assigned to each of the weekdays and 0.0 to each of the weekend days. If a multiplicative (additive) model is used, the daily weights sum to 7 ( 0 ) and a weight of 1.0 ( 0 ) indicates neutral activity level for the particular day.
A regression framework can be used to estimate the trading day effect from the original estimates. i.e. estimate trading day daily weights for different days. If each daily weight is assumed to be a constant (or static) over time, the data from the entire span of the time series can be used to estimate the daily weights for different days of a week. However, a constant daily weight may not reflect the true behaviour of some time series. Consumer behaviour and trading regulations change over time which means that the trading patterns can change over time. A changing trading pattern related to the day of a week is called moving trading effect.
Moving trading day is used to describe a trading day pattern that is evolving or changing slowly through time. Some changes in the trading day patterns of component series are sudden, while others are relatively gradual. Under Moving Trading Day, the trading day weights are not fixed for the entire length of the series. An example of moving trading day is evident in Australian Retail series. Sunday trading has slowly been introduced over the last ten years or so, in most states of Australia. This has resulted in a moving trading day effect in these series.
To capture the moving trading day effect, a rolling window of a segment of a time series is used to estimate the trading day effect over time.
Figure 1 shows the estimated static and moving daily weights for Australian total retail turnover series. The benchmark estimate is the best estimate of the daily weight that can be made given data available for use at a specified point in time. The graph presents the benchmark estimate (blue line), the mean of the benchmark estimate (black line) and the initial estimate (red dot) of the daily weight factors for each month. This method of presentation allows the user to see how the benchmark daily weight factors have been evolving over time. It can be observed that the daily weight factors for Sunday have shown a steady rise in later years reflecting the gradual introduction of Sunday Trading in Australia.
Figure 1: Daily weights for Australia total retail turnover
A time series will not exhibit a trading day effect if levels of activity are constant over each day of the week. However, different months have different lengths (28,29,30 and 31 days), hence monthly activity can vary purely because certain months are longer than others. This is known as the length of month effect. If a series has trading day corrections, then these adjustments will include the effect. If there is no trading day effect in a time series, then the length of month effect is accounted for in the seasonal component.
Stock series should not experience a trading day effect since they only measure the level of activity at a certain point in time and are therefore not affected by how many trading days there are in a given period of time.
Why are trading day adjustments rarely made to quarterly series?
Quarters can have 90, 91, or 92 days. Those with 91 days do not experience the trading day effect as this is a multiple of seven, hence they always contain the same number of each day. The others only lose or gain a day, and unless the activity on this day is significant, the effect is nearly impossible to quantify accurately. Trading day effects are rarely seen in ABS quarterly series.
EXTREMES or OUTLIERS
Extremes or outliers are values in a time series that are unusually large or small relative to the other data. They can distort the appearance of the underlying movement of the time series by altering the trend. For this reason, and to improve estimation of the three series components (trend, seasonal and irregular), it is necessary to detect and correct outliers.
For example, consider the Figure 2. The peak in the seasonally adjusted series in late 1994, corresponds to a significantly large irregular value. Because it has not been corrected, it is quite obviously distorting the trend around that time point.
Figure 2: Quarterly Imports of Industrial Transport Equipment
TREND AND SEASONAL BREAKS
Although a time series is a collection of consistently and rigorously defined data items, it is likely that a series will undergo structural breaks during its span. These can be the result of a change in the population they are measuring or in the way that the population is being measured.
An abrupt but sustained change in the level of a time series is known as a trend break. This is reflected in at least 6 months or 3 quarters of raised or lowered levels. If the span of anomalous values is shorter than this, they are classified as extreme values.
A trend break may be caused by:
Seasonal breaks are abrupt changes in the seasonality of a series, which do not affect the level of the series. They may be caused by changes in coverage of the survey, social traditions, administrative practices or technological innovations.
REVISIONS TO TIME SERIES
Estimates of time series components may be revised over time. This can occur for the original, seasonally adjusted and trend estimates. For example, the trend estimate for unemployment for a particular month will change from month to month as new original estimates become available. The revisions to seasonally adjusted and trend estimates will occur until there is enough original estimates available to use symmetric filters to calculate the seasonal and trend components for that month. Typically, such revisions will reduce over time and become negligible after a few months. However, this will depend on the nature of the time series.
FORWARD FACTORS VERSUS CONCURRENT ANALYSIS
There are two approaches to deriving seasonal and trading day factors:
Forward factors rely on an annual analysis of the latest available data to determine the seasonal and trading day factors that will applied to the data received during the forthcoming year.
Concurrent analysis involves re-estimating seasonal factors as each new data point becomes available. Obviously this method is more computationally intense than the forward factor method, but the seasonal factors will be more responsive to dynamic changes.
Currently, the ABS uses forward factor analysis for most time series. However, it has introduced concurrent adjustment into some series and is intending to implement this approach with the majority of ABS time series.
DIRECT (DISAGGREGATE) VERSUS INDIRECT (AGGREGATE) METHODS OF ADJUSTMENT
Sometimes we may deal with series which are related in an aggregative way. For example, we may have data relating to some activity for each individual state, but we would like to obtain a seasonally adjusted series for the Australian total. There are two ways in which we can do this.
Under an indirect (aggregate) method of adjustment, we seasonally adjust each of the lower component series individually, then sum all the values to obtain the seasonally adjusted series for the total.
Directly (disaggregatively) adjusting a series involves summing all the original series to form a total series and then seasonally adjusting the total series directly.
How do we decide which method of adjustment to use?
If the component series each have very different seasonal patterns, then indirect seasonal adjustment is preferable. However if seasonality is minor and difficult to identify in the individual series, then using direct seasonal adjustment may remove any residual seasonality from the aggregate series.
This page first published 14 November 2005, last updated 25 July 2008