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Seasonal adjustment and trend-cycle estimation

Scope and purpose

Seasonal adjustment consists of estimating seasonal factors and applying them to a time series to remove the seasonal variations. These variations represent the composite effect of climatic and institutional factors that repeat with a certain regularity within the year. The seasonally adjusted series consists of the trend-cycle and the irregular components. The trend is the underlying long-term movement lasting many years. The cycle, usually called the business-cycle, is a quasi-periodic oscillation lasting from three to four years. The irregular component represents those random variations that are unforeseeable movements related to events of all kinds.

Statistical agencies publish many of their series in seasonally adjusted form to reveal the underlying trend-cyclical movements and to help data analysis. Seasonally adjusted series comprise not only the trend-cycle but also the irregular component; consequently, they only give an approximate idea of the underlying trend-cyclical movements. Smoothing the seasonally adjusted series further is often desirable to eliminate the irregular component and to publish trend-cycle estimates as a complement to the seasonally adjusted series.

This section is a transcript of the Agency’s Guidelines for Seasonal Adjustment and Trend-Cycle Estimation (Statistics Canada, 2000b).

Principles

Seasonally adjust a time series only when there is evidence that the series is influenced by seasonal forces, and when the series contains identifiable seasonality. Identifiable seasonality is defined as a seasonal pattern that is not obscured by a high degree of irregular fluctuations and thus can be identified reliably (Lothian and Morry, 1978).

A good seasonal adjustment procedure does not leave any residual seasonality in the series, and the resulting seasonally adjusted series is much smoother than the original.

The revisions to the seasonally adjusted estimates should be minimal as more data points are added to the end of the series. The X11ARIMA (Dagum, 1980) and the X11ARIMA/88 (Dagum, 1988) seasonal adjustment methods were adopted by Statistics Canada with the exact purpose of reducing the size of revisions (Dagum, 1975; Kuiper, 1976; Dagum, 1982).

The frequency of revisions should be minimized. Although revisions arise with each new data point added, implement revisions only when they bring about improvement in the estimates, that is, when the revised estimate moves appreciably closer to the final estimate.

Wherever seasonally adjusted figures pertaining to the same economic activity are published, coordinate the seasonal adjustment options applied by the areas involved, and make every effort to treat related series in a consistent manner.

When trend-cycle estimates are produced as a complement to the seasonally adjusted series it is important to keep in mind that they are used for providing a reading of the stage of the business cycle, and therefore their accuracy is important with regard to the direction of movement, the amplitude of the cycle and especially the timing of turning points.

The trend-cycle estimates should be consistent with the published seasonally adjusted estimates. If the latter estimates are frozen in the database after three months, apply the trend-cycle estimator to the seasonally adjusted estimates as they appear on the base and in the publication.

Guidelines

  • Before seasonally adjusting a series for the first time, and after that every few years, conduct a thorough seasonal analysis to assess if seasonality is identifiable and publish the series in seasonally adjusted form only if it is identifiable.

  • During seasonal adjustment it is recommended that ARIMA extrapolations be used in the calculations of the seasonal factors to reduce the size of the revisions. Use the automatic ARIMA extrapolation subroutine of the X11ARIMA program whenever possible. If none of the built-in models is selected, it is recommended that the user supply an ARIMA model.

  • For the seasonal adjustment of recent observations, use a concurrent seasonal factor (Dagum, 1987). This is a factor obtained using all the available data points. An exception to this guideline may apply when the most recent observations have been subjected to historically large revisions. In this case year-ahead (forecast) seasonal factors may be more appropriate (Morry, 1992). These seasonal factors are based on data that ended at the end of the previous year.

  • When a concurrent seasonal factor is used, it is not necessary to revise the seasonally adjusted estimates more than one period back when the next observation becomes available. An exception to this guideline applies when preliminary observations are used: it is recommended to revise the seasonal factors whenever the original figures are revised significantly. On an annual basis, revise the seasonally adjusted values for the last three years when the first month (quarter) of the next year becomes available (Dagum, 1987). When seasonally adjusted values are obtained with year-ahead (forecast) seasonal factors, the annual revision applies to the last four years.

  • For series with trading-day variations, use the daily weights that are automatically estimated by the X11ARIMA program. During the current year, keep them fixed by supplying them as prior daily weights. They will be modified at the next annual revision. Exceptions to this guideline may occur when a-priori daily weights can be provided by subject matter experts based on better knowledge of the series in question.

  • For series with Easter variations, use the Easter effect factors calculated automatically by the X11ARIMA program.

  • For aggregate series resulting from the combination of component series, seasonally adjust only those component series that contain identifiable seasonality, and leave the others unadjusted. Seasonally adjust the aggregate series by the indirect or direct method. In the indirect method, combine the seasonally adjusted components and the unadjusted ones to obtain the seasonally adjusted aggregate. In the direct method, seasonally adjust the aggregate, and restore additivity (if required) by raking the components, if possible without modifying the unadjusted ones. When choosing between the indirect and the direct approach the following factors need to be considered: the aggregate should not contain residual seasonality and it should be relatively smooth (Lothian and Morry, 1977). Choose the raked direct approach only if the raking does not distort too much the month-to-month (quarter-to-quarter) movements of the component series.

  • Wherever seasonally adjusted figures pertaining to the same economic activity are published, coordinate the seasonal adjustment options applied by the areas involved. For example, when possible, make consistent choices between direct and indirect adjustment of composite series, and ensure that extreme occurrences in the time series brought about by events such as plant closures, strikes, natural disasters, etc. are treated in a consistent fashion by the different areas.

  • Use the Henderson moving averages, available in the X11ARIMA program, to produce the trend-cycle estimates. To ensure that the trend-line lies within the scatter plot of the seasonally adjusted series, apply the Henderson moving averages to the published seasonally adjusted series.

  • Before applying the trend-cycle estimator, extend the seasonally adjusted series with one year of forecasted values from an ARIMA model fitted to the seasonally adjusted series.

  • Apply the Henderson moving averages to the extended seasonally adjusted series from which the extremes have been previously corrected.
  • Use the Henderson moving average automatically selected by the X11ARIMA program: the selection is based on the value of the I/C (irregular to trend-cycle) ratio, which measures the relative importance of the irregular variations in the seasonally adjusted series (Shiskin, et al., 1967).

  • Inform the users that the last few trend-cycle estimates (and especially the very last estimate) are subject to large revisions, and often to a reversal of movement when one more data point is added to the series (Dagum and Laniel, 1987). This high uncertainty associated with the estimates around the end can be indicated, for example, by a dashed line on the trend graph or by a written caveat to users.

References

Dagum, E.B. (1975). Seasonal factor forecasts from ARIMA models. Proceedings of the International Institute of Statistics, 40th Session, Contributed Papers, 3, Warsaw, 206-219.

Dagum, E.B. (1980). The X11ARIMA seasonal adjustment method. Catalogue No. 12-564E, Statistics Canada.

Dagum, E.B. (1982). The effects of asymmetric filters on seasonal factor revisions. Journal of the American Statistical Association, 77, 732-738.

Dagum, E.B. (1987). Current issues on seasonal adjustment. Survey Methodology, 13, 63-74.

Dagum, E.B. (1988). The X11ARIMA/88 seasonal adjustment method - foundations and user’s manual. Time Series Research and Analysis Division, Statistics Canada technical report.

Dagum, E.B. and Laniel, N. (1987). Revisions of trend-cycle estimators of moving average seasonal adjustment methods. Journal of Business and Economic Statistics, 5, 177-189.

Kuiper, J. (1976). A survey and comparative analysis of various methods of seasonal adjustment. Proceedings of the NBER/Bureau of the Census Conference on Seasonal Analysis of Economic Time Series, Arnold Zellner (ed.), Washington, D.C. 59-76.

Lothian, J. and Morry, M. (1977). The problem of aggregation: direct or indirect seasonal adjustment. Research paper 77-08-001E, Seasonal Adjustment and Time Series Staff, Statistics Canada.

Lothian, J. and Morry, M. (1978). A test for the presence of identifiable seasonality when using the X-11 program. Research paper 78-10-002E, Seasonal Adjustment and Time Series Staff, Statistics Canada.

Morry, M. (1992). Comparison of revisions in seasonally adjusted external trade series obtained through the use of concurrent versus forecast factors. Statistics Canada Working Paper, TSRA-92-004E.

Shiskin, J., Young, A.H. and Musgrave, J.C. (1967). The X-11 variant of the census method II seasonal adjustment. Technical Paper No. 15, Bureau of the Census, U.S. Department of Commerce.

Statistics Canada (2000b). Guidelines for Seasonal Adjustment and Trend-Cycle Estimation. Issued by the Methods and Standards Committee, March, 2000.



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