Trend is your friend


The aim of seasonal adjustment is to estimate and remove the systematic calendar related component of a time series.

In practice, the original series that has been collected or derived will typically be comprised of at least three component parts:

  • the seasonal component, e.g. calendar related
  • the trend component, e.g. the short or long term underlying direction
  • the irregular component, e.g. one-off effects
This means that once the seasonal component has been estimated and removed from the original series, the seasonally adjusted estimates will still contain the underlying trend component, and the irregular component.

What a lot of users of time series outputs don't realise is that by definition the seasonally adjusted estimates still contain a degree of volatility which can be due to way the original data has been collected (e.g. the sampling characteristics used by the survey), the nature of the data that is being measured, or one-off events such as snow, earthquakes, strikes, riots which can distort the time series and also distort the analysis. Depending on the degree of volatility, these one-off events can make interpretation of the seasonally adjusted estimates difficult and sometimes impossible.

To help interpretation of the movements in the time series, the seasonally adjusted estimates can be filtered, or smoothed, to derive an estimate of the underlying component where the impact of the volatility has been reduced. This is commonly refered to as the trend. The trend is your friend as it helps interpretation of the underlying direction without the impact of the one-off volatility that can distort and give misleading month-on-month or quarter-on-quarter movements.

All time series components can then be considered separately:

  • the original series shows the raw data
  • the seasonally adjusted estimates show the trend and the irregular component
  • the trend estimates show the underlying direction of the series
  • the irregular component shows the impact of the one-off events
Having all the components available to analyse together helps understand how they contribute to the total and can put relative movements of the different components into perspective.

Defining a trend

There is no unique definition for the trend. This is probably the most contentious point about the use of trend estimates as it always leads to a debate about whose method is best and how trend estimates should be used. Economists can refer to business cycles as indicative of trend which may be in terms of years, whereas localised trends may be in terms of much shorter periods such as eight to twelve months. Each persons trend can be different and there is no right or wrong answer.

For producers and publishers of trend estimates there is a simple approach to get around this problem. Clearly define your trend and how you've calculated it and then let everyone know how you've done it. If you also publish seasonally adjusted estimates, then the more sophisticated users can just get on with their job and produce their own.

It is even much simpler in that as a by-product of seasonal adjustment, a trend estimate is produced. In fact, what a lot of users don't realise, is that one of the first steps in the X-12-ARIMA program is to calculate a trend. Trend estimates in the X-11 and X-12-ARIMA framework are central to the derivation of the seasonally adjusted estimates. The use of the trend estimates direct from the seasonally adjusted package would ensure a consistent set of component estimates.

However, in practice, there are other alternatives. For example, the Australian Bureau of Statistics define trend estimates by taking the published seasonally adjusted estimates and then directly applying a Henderson filter.

Where can I find out more information on trend estimates?

We believe trend estimates are an important analytical product that helps users understand the movements in a time series without having the impact of the one-off irregular events.

The references below are a useful starting point in finding out more information on trend estimates.

Reference PDF
Australian Bureau of Statistics (1987). A Guide to Smoothing Time Series - Estimates of ``Trend''. Australian Bureau of Statistics, cat. 1316.0, Canberra, Australia.
Australian Bureau of Statistics (1993). A Guide to Interpreting Time Series Monitoring ``Trend'' - an Overview. Australian Bureau of Statistics, cat. 1348.0, Canberra, Australia.
Australian Bureau of Statistics (2003). Information Paper: A Guide to Interpreting Time Series - Monitoring Trends. Australian Bureau of Statistics, cat. 1349.0, Canberra, Australia. External link
Dagum, E. B. and Luati, A. (2002), Smoothing Seasonally Adjusted Time Series, American Statistical Association, Proceedings of the Joint Statistical Meetings, p665- 670
Doherty, M. (2001) The Surrogate Henderson Filters in X11, Australia & New Zealand Journal of Statistics, Vol 43, No. 4, 385-392 External link
Gray, A. and Thomson, P. (1996). Design of moving-average trend filters using fidelity and smoothness criteria in Vol 2: Time Series Analysis in Memory of E.J. Hannan. ed. P. Robinson and M. Rosenblatt. Springer Lecture Notes in Statistics 115, 205-219.
Gray, A. and Thomson, P. (1996). Design of moving-average trend filters using fidelity, smoothness and minimum revisions criteria, Bureau of the Census, RR96/01 External link
Gray, A. and Thomson, P. (1996). On a family of moving-average trend filters for the ends of series. Proceedings of the American Statistical Association, Section on Survey Research Methods, 1996.
Henderson, R. (1916). Note on Graduation by Adjusted Average. Transactions of the American Society of Actuaries, 17, 43-48.
Kenny, P.B., and Durbin, J. (1982). Local Trend Estimation and Seasonal Adjustment of Economic and Social Time Series. Journal of the Royal Statistical Society, Series A, 145, 1-41.
Ladiray, D. and Quenneville, B. (2001). Seasonal Adjustment with the X-11 method, New York: Springer Verlan, Lecture notes in statistics, 158.
Laniel, N. (1985). Design criteria for 13 term Henderson end-weights. Technical Report Working paper TSRA-86-011, Statistics Canada, Ottawa K1A 0T6.
McLaren, C.H. (1999). "Designing Rotation Patterns and Filters for Trend Estimation in Repeated Surveys" Unpublished PhD Thesis, School of Mathematics and Applied Statistics, University of Wollongong, Australia.
McLaren, C.H. and Steel, D.G. (2001). "Rotation patterns and trend estimation for repeated surveys using rotation group estimates", Statistica Neerlandica, Vol. 55, no. 2, pages 221-238.
Shiskin, J., Young, A.H., and Musgrave, J.C. (1967). The X-11 Variant of the Census Method II Seasonal Adjustment Program. Technical Paper 15, Bureau of the Census, U.S. Department of Commerce, Washington, D.C. External link