National accounts, non-financial sector accountsQ1 2020


About seasonal adjustment

General information on seasonal adjustment

What is seasonal adjustment?

Monthly and quarterly time series are often characterised by considerable seasonal variations that might complicate their inter-period comparability. Such time series are therefore subjected to a process of seasonal adjustment in order to remove the effects of these seasonal fluctuations. Once data have been adjusted for seasonal effects by X-13-ARIMA or some other seasonal adjustment tool, a clearer picture of the time series emerges.

For more information on seasonal adjustment:

Why seasonally adjust these statistics?

Why do we seasonally adjust QSA ?

The quarterly institutional sector accounts (QSA) consist of a set of incomes and expenditures from which the items net savings and net lending/net borrowing are derived.

The balancing items, savings and net lending/net borrowing are shown in the accounts for incomes and expenditures for one sector. The saving ratio (the ratio between saving and disposable income) indicates how the sector has been financed. If savings are denoted by a negative sign, it means that a sector has used more resources than it had and had to finance its expenses through borrowing.

Some of the most relevant series in the QSAs show clear evidence of seasonality, for example gross value added and for total consumption. Most of the other series show seasonality in quite different ways. It seems to be that series for income and expenditure do not have identical seasonality.

Therefore, in order to evaluate the current saving rate in an appropriate way, the seasonality must be removed from its components. Series for the QSAs are now extensive enough to identify their seasonality.

Series are long enough to run X-13-ARIMA but remain quite short and therefore some problems of instability can arise. For different reasons as mentioned above, some of the aggregate series show non-identical seasonal patterns before and after 2006.

Seasonally adjusted series

All series included in the QSA have been seasonally adjusted. The indirect approach has been used for the main aggregates. This means that consistency is maintained for aggregation and definitions in the released tables for the seasonally-adjusted figures. We get better results for the main components when adjusting in this way (indirectly) in place to adjust directly. To arrive to such conclusion we use the results as well as other analysis using figures and output from X-13-ARIMA..
It seems to be that the indirect adjustment of the main components leads to more stable series (lower irregular components) and therefore less revisions in the future. This is particularly relevant for the saving ratio and net lending/borrowing.
The main elements for the seasonal adjustment of the QSA’s serer are given by:
• We use X13-ARIMA without stipulate that annual figures for unadjusted and seasonally adjusted are identical.

• We are using X-13_ARIMA for adjust each individual time series.. Seasonally adjusted aggregates emerge as a result by aggregating the seasonally adjusted components.

• The series that do not have season (there are many in QSA) are not adjusted. Mao: adjusted and unadjusted are identical
• In a few cases, we use trend in place for seasonally adjusted in order to avoid negative numbers.

• Some series are not long enough but they clearly shows seasonal pattern. In that case series are extrapolated to estimate seasonal factors
• Series corresponding with QNA was adjusted by identical factors than the QNA

• Only the last three years are reviewed when figures for new quarter are included.

Relationship and consistency demands careful checking via established graphs when new figures for seasonal adjustment are treated.

The following table shows the method used to adjust the individual time series:

Income Resident sectors, total Non-financial corporations Financial corporations General government Households NPISH Rest of the world
Imports of goods              QNA
Imports of services              QNA
Value added, gross  QNA  Residual QNA  QNA  J QNA  S
Compensation of employees  Sum        J    J
Operating surplus  Sum  S  S  S  S  S  S
Subsidies  Sum  Residual  J  N  A  N  
Property income  Sum  J  T  T  J  J  T

Financial Intermediation Services Indirectly Measured

 Sum  A  A  N  A  A  T
Balance of primary income  Sum  S  S  S  S  S  S
Contributions to social security  Sum      J      
Current taxes on income and wealth  Sum      J      M
Pensions and social benefits from public administr.  Sum        Residual    J
Unfunded and privately funded social benefits  Sum  N  Residual  N  J  A  A
Other current transfers  Sum  J  J  J  J  J  J
Disposable income  Sum  S  S  S  S  S  S
Adjustment, household pension funds  Sum        J    
Saving  Sum  S  S  S  S  S  S
Capital transfers  Sum  Residual  N  A  N    A
Net lending  Sum  S  S  S  S  S  S













Expenditures Resident sectors, total Non-financial Corporations Financial corporations General government Households NPISH Rest of the world
Exports of goods              QNA
Exports of services              QNA
Consumption of fixed capital  Sum  Residual N    
Compensation of employees  Sum  Residual  J  N  J  N  J
Taxes on production and imports  Sum  Residual  J  N A  N  
Property expenses  Sum  Residual  T  J J J
Financial Intermediation Services Indirectly Measured  Sum  J  Residual  N

Contributions to social security

 Sum        J    
Current taxes on income and wealth  Sum  Residual  J    M  J  T
Pensions and social benefits from public administration  Sum      J      A
Unfunded and privately funded social benefits  Sum  N  Residual  N  J  A  A
Other current transfers  Sum  Residual  J  J  J  J  A
Consumption  QNA      QNA  QNA  QNA  
Adjustment, household pension funds  Sum    J        
Capital transfers  Sum  N  N  J  N    N
Gross fixed capital formation  QNA  Residual  QNA  QNA  J  QNA  
Changes in stocks and statistical discrepancies  QNA  Residual          
Net aqcuisition of non-produced non-financial assets  Sum  Residual      N   N












Description of the table: J: Direct seasonal adjustment, A: Additive adjustment due to negative values, N: No clear seasonal pattern, M: Manual adjustment due to short time series, S: Sum of components, T: Using trend series, QNA: Series which are based on pre-existing databases are denoted as QNA, from Quarterly national accounts.

Setting up a full set of QSA data set implies that certain accounting relations between institutional sectors are still preserved after the seasonal adjustment. For each transaction total uses must equal total resources and thereby preserving the same horizontal consistency as QSA. This is achieved for every transaction by residualizing a carefully selected series. The choice of residual sector cannot be a series from an already existing data set like QNA. The data series must be rather large so that any erratic pattern is diluted in the time series. And the QSA raw data series cannot be estimated with a linear trend, for example where we have annual series divided by 4 quarters.

Seasonally adjusted series for the balancing items such as gross operating surplus, gross disposable income, saving etc. are derived by indirect calculation. The balancing items are thus calculated taking into account all the upstream transactions (vertical consistency).

For more information on the properties of the series please refer to "Analyzing the series of Quarterly Sector Accounts.doc“


Pre-treatment routines/schemes

Pre-treatment is an adjustment for variations caused by calendar effects and outliers

•Running an automatic pre-treatment of the raw data based on standard options in the seasonal adjustment tools. The method used to calculate the raw data for some of the series allows to remove both seasonal and outlier effects. This is especially true for the compensation of employees and for taxes and subsidies on production.

Calendar adjustment

Calendar adjustment involves adjusting for the effects of working days/trading days and for moving holidays. Working days/trading days are adjustment for both the number of working days/trading days and for that the composition of days can vary from one month to anoth

•The series tested on the number of working days in a quarter can affect the results. For quarterly series, these effects are rarely identified. This is the case for the Norwegian QSAs. None of the series have been pre-treated for trading-day effects.

•More relevant is the Easter effect. Four series have been corrected for Easter effects: production, compensation of employees, benefits from pension funds and consumption.

Methods for trading/working day adjustment

•RegARIMA correction – in this case, the effect of trading days is estimated in a RegArima framework. The effect of trading days can be estimated by using a correction for the length of the month or leap year, regressing the series on the number of working days, etc. In this case, the residuals will have an ARIMA structure.

Correction for moving holidays

•Use of standard options X-12-ARIMA, RegARIMA modeling, to identify and remove Easter effects.

•Consumption: Correction based on an estimation of the duration of the moving holidays effects, specifically adjusted to Norwegian circumstances.

National and EU/euro area calendars

•Use of default calendars. The default in X-12-ARIMA is the US calendar.

Comments : Final consumption expenditure for households uses the Norwegian calendar.

Treatment of outliers

Outliers or extreme value, are abnormal values of the series

•Outliers are detected automatically by the seasonal adjustment tool. The outliers are removed before seasonal adjustment is carried out, and then reintroduced into the seasonally adjusted data.

Model selection

Pre-treatment requires choosing an ARIMA model, as well as deciding whether the data should be log-transformed or not.

•X-12-ARIMA automatically identifies one model if the average of the forecast errors is less than a previously established value. If all the models are refused one of them is manually selected (i.e. a default option). Under these premises the automatic selection must be synonymous with better quality of the results.

•The model (0, 1, 1) (0, 1, 1), often referred to as the airline model, is generally the best one. This model has only 2 parameters and is easy to interpret.

Decomposition scheme

The decomposition scheme specifies how the various components – basically trend-cycle, seasonal and irregular – combine to form the original series. Most frequently used decomposition schemes are the multiplicative, additive or log additive.

•It has been taken manual selection after graphical inspection of time series.

•For stationary series (constant mean and variance) additive decomposition has been used.

Seasonal adjustment

Choice of seasonal adjustment approach


Consistency between raw and seasonally adjusted data

•The seasonal adjusted quarterly data will not sum up to the annual unadjusted data. This is in line with international recommendations. For the annual figures the seasonally adjusted and the raw series will be alike. This means that the sum of the seasonally adjusted figures for the 4 quarters in a year do not normally add up to the seasonally adjusted annual numbers.

Consistency between aggregate/definition of seasonally adjusted data

•The chosen method imposes the equality between aggregated series and the component series.

•Definitions and relationships also apply for seasonally adjusted figures.This is the main reason for choosing the indirect method for component series.

Direct versus indirect approach

Direct seasonal adjustment is performed if all time series, including aggregates, are seasonally adjusted on an individual basis. Indirect seasonal adjustment is performed if the seasonally adjusted estimate for a time series is derived by combining the estimates for two or more directly adjusted series.

•Indirect approach has been used. The seasonal adjustment of components occurs using the same approach and software, and then totals are derived by aggregation of the seasonally adjusted components.

Horizon for estimating the model and the correction factors

When performing seasonal adjustment of a time series, it is possible to choose the period to be used in estimating the model and the correction factors. Correction factors are the factors used in the pre-treatment and seasonal adjustment of the series.

•The whole time series is used to estimate the model and the correction factors.

Audit procedures

Seasonally adjusted data may change due to a revision of the unadjusted (raw) data or the addition of new data. Such changes are called revisions, and there are several ways to deal with the problem of revisions when publishing the seasonally adjusted statistics.

•Both raw and seasonally adjusted data are revised between two consecutive official releases of the release calendar

General revision policy

Seasonally adjusted data are revised in accordance with a well-defined and publicly available revision policy and release calendar.

Concurrent versus current adjustment

•Partial concurrent adjustment: the model is identified and estimated yearly, while filters, outliers and regression parameters are re-identified and estimated continuously as new or revised data become available.

Horizon for published revisions

•The revision period for the seasonally adjusted results is limited to 3-4 years (preferably 4) prior to the revision period of the unadjusted data, while older data are frozen.

Quality of seasonal adjustment

Evaluation of seasonally adjustment data

Continuous/periodical evaluation using standard measures proposed by different seasonal adjustment tools.

Quality measures for seasonal adjustment

•For most of the series, a selected set of diagnostics and graphical facilities for bulk treatment of data is used.

X-13-ARIMA chooses automatically the most appropriate model for the individual series.

ANOVA shows that the rates of change for the original series are primarily due to seasonal effects.

Special cases

Seasonal adjustment of short time series

•All series are sufficiently long to perform an optimal seasonal adjustment.

Treatment of problematic series

•Problematic series are treated in a special way only when they are relevant. The remaining series are treated according to normal procedures.

Posting procedures

Data availability


•Raw and seasonally adjusted data are available.

•All metadata information associated with an individual time series is available.

•Historical data are available to enable revision analysis.

Press releases

•In addition to raw data, at least one of the following series is released: pre-treated, seasonally adjusted, seasonally plus working day adjusted, trend-cycle series.

•Only levels or different forms of growth rates are presented.

•For each series, some quality measures of the seasonal adjustment are presented.


ESS Guidelines on Seasonal Adjustment

Eurostat. (2008). TF-QSA-MAY08-07B: Seasonal Adjustment of key indicators and components – Quarterly (non-financial) sector accounts of the euro area and the European Union

•Hungarian Central Statistical Office (2007): Seasonal adjustment methods and practices (European Commission Grant 10300.2005.021- 2005.709)

U.S. Census Bureau (2002). X-12-ARIMA Reference Manual