Economic time series at monthly and quarterly level are often influenced by phenomena that occur at the same time each year, what we in more technical terms describes as seasonal variations.

For instance, in the external trade in goods statistics fish exports are affected by the fishing season for various types of fish. The main  fishing season for skrei is in February and March and  salmon export peaks in the autumn.

Other examples of seasonal variation in more detailed data are higher imports of alpine and cross country skis in October and November. Furthermore, during  spring months imports of bicycles increases.

Exports of crude oil and natural gas in February, which only has 28 days (if it is not a leap year), are lower than in January and March. This is due to calendar effects that affect the seasonal pattern. Production and thus exports are higher in a month with many working days. Furthermore, the demand for natural gas is also affected by seasonal variations, and the export is higher during winter when the demand from Europe is higher.  The seasonally adjusted export value of natural gas and fish and the import value of bicycles in April 2022 should be a figure that can be compared directly with the seasonally adjusted figures for this trade in the other months of 2022.

To arrive at such seasonally adjusted figures, the time series must first be precorrected and then seasonally adjusted. Seasonally adjusted figures are cleansed of the effects of calendar effects and seasonal variations, and give a better picture of the development than the unadjusted figures.


The purpose of pre-treatment is to remove the variations in unadjusted figures that are caused by  calendar effects and extreme values.

The number of working days can vary from month to month, and have a major impact on economic time series. This can, if we do not take it into account, lead to incorrect conclusions about the figures of imports and exports in the external trade in goods.

During the pre-treatment, unadjusted figures are also corrected for extreme values ​​(or outliers). There are many occurrences of extreme values ​​in external trade in goods , especially level shifts and additive outliers.

Level shifts are phenomena that affect unadjusted figures so that they remain permanently at a higher or lower level. In external trade data there wasa level shift with the start of oil exports from the Johan Sverdrup field in November 2019. Then we saw a significant increase in the number of exported oil barrels.

Additive outliers are observations that occur at one point in time, in a month or quarter,  and disappear the next month or quarter. One example of this is found in March 2013, when the production- and storage ship Skarv and the frigate KNM Thor Heyerdahl were imported. Another additive outlier was registered in April 2015 when the oil platform Goliat was imported.

Technically speaking, the correction for calendar effects and extreme observations are made in the same way, but there is still a difference that is important to be aware of. The correction for the extreme values ​​is temporary, to identify the seasonal component. The fugures are included in the seasonally adjusted figures.  On the other side, the correction of the calendar effects  is permanent and these effects are also removed in the final seasonally adjusted series.

The pre-treatment is done  using a regression model - regARIMA.

After the time series has been precorrected, the actual seasonal adjustment takes place.

A general model in time series analysis, for an observerable time series (Yt ), can be split into three components:

         • The trend (Tt), - that shows the long-term tendency, which increases or decreases over time

         • The seasonal component (St) - that shows the variation in a time series during a year

          • The irregular component (It) - which is the part of the observed time series that does not                    include the trend or the seasonal component. It is often referred to as random variation, or                  unexplained variation. In the end of  the seasonal adjustment process extreme values ​​are                    added to this component (except for level shifts).

The decomposition can be additive, and is written in the form:

Yt = Tt + St + It

Or it can be multiplicative, and is written in the form:

Yt = Tt • St • It.

It is only Yt that can be observed, while the other components must be estimated. The multiplicative model is a better choice if the effects of the seasonal variations are greater the higher the level of the seasonally adjusted series. This model is most often used in economical time series.