Time Series Analysis: Seasonal Adjustment Methods


Filter based methods of seasonal adjustment are often known as X11 style methods. These are based on the ‘ratio to moving average’ procedure described in 1931 by Fredrick R. Macaulay, of the National Bureau of Economic Research in the US. The procedure consists of the following steps:

1) Estimate the trend by a moving average
2) Remove the trend leaving the seasonal and irregular components
3) Estimate the seasonal component using moving averages to smooth out the irregulars.

Seasonality generally cannot be identified until the trend is known, however a good estimate of the trend cannot be made until the series has been seasonally adjusted. Therefore X11 uses an iterative approach to estimate the components of a time series. As a default, it assumes a multiplicative model.

To illustrate the basic steps involved in X11, consider the decomposition of a monthly time series under a multiplicative model.

Step 1: Initial estimate of the trend

A symmetric 13 term (2x12) moving average is applied to an original monthly time series, Ot, to produce an initial estimate of the trend Tt. The trend is then removed from the original series, to give an estimate of the seasonal and irregular components.

Equation - the initial trend estimate is removed from the original series to produce first seasonal and irregular estimates

Six values at each end of the series are lost as a result of the end point problem - only symmetric filters are used.

Step 2: Preliminary estimate of the seasonal component

A preliminary estimate of the seasonal component can then be found by applying a weighted 5 term moving average (S3x3) to the St.It series for each month separately. Although this filter is the default within X11, the ABS uses 7 term moving averages (S3x5) instead. The seasonal components are adjusted to add to 12 approximately over a 12 month period, so that they average to 1 in order to ensure that the seasonal component does not change the level of the series (does not affect the trend). The missing values at the ends of the seasonal component are replaced by repeating the value from the previous year.

Step 3: Preliminary estimate of the adjusted data

An approximation of the seasonally adjusted series is found by dividing the estimate of the seasonal from the previous step into the original series:
Equation - the seasonal estimate is removed from the original data to produce estimate of seasonally adjusted series

Step 4: A better estimate of the trend

A 9, 13 or 23 term Henderson moving average is applied to the seasonally adjusted values, depending on the volatility of the series (a more volatile series requires a longer moving average), to produce an improved estimate of the trend. The resulting trend series is divided into the original series to give a second estimate of the seasonal and irregular components.
Equation - the second trend estimate is removed from the original series to produce improved seasonal and irregular estimates

Asymmetric filters are used at the ends of the series, hence there are no missing values like in step 1.

Step 5: Final estimate of the seasonal component

Step two is repeated to obtain a final estimate of the seasonal component.

Step 6: Final estimate of the adjusted data

A final seasonally adjusted series is found by dividing the second estimate of the seasonal from the previous step into the original series:

Equation - the second seasonal estimate is removed from the original data to produce a final estimate of seasonally adjusted series

Step 7: Final estimate of the trend

A 9, 13 or 23 term Henderson moving average is applied to the final estimate of the seasonally adjusted series, which has been corrected for extreme values. This gives an improved and final estimate of the trend. In more advanced versions of X11 (such as X12ARIMA and SEASABS), any odd length Henderson moving average can be used.

Step 8: Final estimate of the irregular component

The irregulars can then be estimated by dividing the trend estimates into the seasonally adjusted data.

Equation - the final trend estimate is removed from the seasonally adjusted data to estimate irregular component

Obviously these steps will depend on which model (multiplicative, additive and pseudo-additive) is chosen within X11. There are also small differences in the steps in X11 between various versions.

An additional step in estimating the seasonal factors, is to improve the robustness of the averaging process, by modification of the SI values for extremes. For more information on the major steps involved, refer to section 7.2 of the Information paper: An Introductory Course on Time Series Analysis - Electronic Delivery .



The most commonly used seasonal adjustment packages are those in the X11 family. X11 was developed by the U.S Bureau of Census and began operation in the United States in 1965. It was soon adopted by many statistical agencies around the world, including the ABS. It has been integrated into a number of commercially available software packages such as SAS and STATISTICA. It uses filters to seasonally adjust data and estimate the components of a time series.


The X11 method involves applying symmetric moving averages to a time series in order to estimate the trend, seasonal and irregular components. However at the end of the series, there is insufficient data available to use symmetric weights – the ‘end-point’ problem. Consequently, either asymmetric weights are be used, or the series must be extrapolated.

The X11ARIMA method, developed by Statistics Canada in 1980 and updated in 1988 to X11ARIMA88, uses Box Jenkins AutoRegressive Integrated Moving Average (ARIMA) models to extend a time series. Essentially, the use of ARIMA modelling on the original series helps reduce revisions in the seasonally adjusted series so that the effect of the end-point problem is reduced.

X11ARIMA88 also differs from the original X11 method in its treatment of extreme values. It can be obtained by contacting Statistics Canada.


In the late 1990’s,the U.S. Census Bureau released X12ARIMA. It uses regARIMA models (regression models with ARIMA errors) to allow the user to extend the series with forecasts and preadjust the series for outlier and calendar effects before seasonal adjustment takes place. X12ARIMA can be obtained from the Bureau.


Developed by Victor Gomez and Augustín Maravall, SEATS (Signal Extraction in ARIMA Time Series) is a program which estimates and forecasts the trend, seasonal and irregular components of a time series using signal extraction techniques applied to ARIMA models. TRAMO (Time Series Regression with ARIMA Noise, Missing Observations and Outliers) is a companion program for estimation and forecasting of regression models with ARIMA errors and missing values. It is used to preadjust a series, which will then be seasonally adjusted by SEATS. To freely download the two programs from the internet, contact the Bank of Spain: www.bde.es/homee.htm


Eurostat has focuses on two seasonal adjustment methods: TRAMO/SEATS and X12ARIMA. Versions of these programs have been implemented in a single interface, called "DEMETRA". This facilitates the application of these techniques to large scale sets of time series. DEMETRA contains two main modules: seasonal adjustment and trend estimation with an automated procedure (e.g. for inexperienced users or for large-scale sets of time series), and with a user-friendly procedure for detailed analysis of single time series.


The main tool used in the Australian Bureau of Statistics is SEASABS (SEASonal analysis, ABS standards). SEASABS is a seasonal adjustment software package with a core processing system based on X11 and X12ARIMA. SEASABS is a knowledge based system which can aid time series analysts in making appropriate and correct judgements in the analysis of a time series. SEASABS is one part of the ABS seasonal adjustment system. Other components include the ABSDB (ABS information warehouse) and FAME (Forecasting, Analysis and Modelling Environment, used to store and manipulate time series data).


SEASABS performs four major functions:
  • Data review
  • Seasonal reanalysis of time series
  • Investigation of time series
  • Maintenance of time series knowledge

SEASABS allows both expert and client use of the X11 method (which has been enhanced significantly by the ABS). This means that a user does not need detailed knowledge of the X11 package to appropriately seasonally adjust a time series. An intelligent interface guides users through the seasonal analysis process, making suitable choices of parameters and adjustment methods with little or no guidance necessary on the users part.

The basic iteration process involved in SEASABS is:

1) Test for and correct seasonal breaks.
2) Test for and remove large spikes in the data.
3) Test for and correct trend breaks.
4) Test for and correct extreme values for seasonal adjustment purposes.
5) Estimate any trading day effect present.
6) Insert or change moving holiday corrections.
7) Check moving averages (trend moving averages, and then seasonal moving averages).
8) Run X11.
9) Finalise the adjustment.

SEASABS keeps records of the previous analysis of a series so it can compare X11 diagnostics over time and 'knows' what parameters led to the acceptable adjustment at the last analysis. It identifies and corrects trend and seasonal breaks as well as extreme values, inserts trading day factors if necessary, and allows for moving holiday corrections.

        • Statistics New Zealand uses X-13ARIMA-SEATS
        • Office of National Statistics, UK uses X-12-ARIMA
        • Statistics Canada is phasing out X-11ARIMA (as of November 2015) and replacing it with X-12-ARIMA
        • U.S Census Bureau uses X-13ARIMA-SEATS
        • Eurostat uses Demetra