The purpose of seasonal adjustment is to remove seasonal and calendar effects from economic time series. It is a common procedure but also a complex one, with side effects. Seasonal adjustment has two essential stages. The first accounts for deterministic effects by means of regression and selects a general time series model. The second stage decomposes the original time series into trend-cycle, seasonal, calendar and irregular components.

Seasonal adjustment does not generally improve the quality of economic data. There is always some loss of information. Also, it is often unclear which calendar effects have been removed. And sometimes seasonal adjustment is just adding noise or fails to remove all seasonality. Moreover, seasonally adjusted data are not necessarily good trend indicators. By design, they do not remove noise and outliers. And extreme weather events or public holiday patterns are notorious sources of distortions. Estimated trends at the end of the series are subject to great uncertainty. Furthermore, seasonally adjusted time series are often revised and can be source of bias if these data are used for trading strategy backtests.

The below is a summary based on quotes from several fundamental papers and manuals. Links are provided in the breaks next to the quotes.

The post ties up with this site’s summary on quantitative methods for macro information efficiency.

### An essential ingredient of official economic data series

“The objectives of seasonal adjustment are to __identify and remove seasonal fluctuations and calendar effects, __which can mask short and long-term movements in a time series and impede a clear understanding of underlying phenomena. Seasonal adjustment is therefore a fundamental process in the interpretation of time series.” [Eurostat]

“Economic variables are influenced by __systematic and recurrent within-a-year patterns due to weather and social factors__, commonly referred to as the seasonal pattern…In a broad sense, seasonal adjustment comprises the removal of both within-a-year seasonal movements and the influence of calendar effects (such as the different number of working days or moving holidays).” [IMF]

“A common solution to deal with seasonal patterns is to look at annual rates of change: that is, compare the current quarter to the same quarter of the previous year. __Over-the-year comparisons present the disadvantage, however, of giving signals of outdated events__. Furthermore, these rates of change do not fully exclude all calendar-related effects…Easter may fall in the first or second quarter, and the number of working days of a quarter may differ between subsequent years.” [IMF]

### Pre-adjustment and decomposition of time series

“A seasonal adjustment procedure follows a two-stage approach.

- The first stage is called
**preadjustment**.__The objective of preadjustment is to select a regression model that best describes the characteristics of the original series__…__The chosen model is used to adjust the series for deterministic effects__and to extend the series with backcasts and forecasts to be used in the time series decomposition process. The preadjustment stage comprises mainly the choice of (i) how the unobserved components are related to each other (additive, multiplicative, or other mixed forms), (ii) the [parameters of the] model, (iii) calendar effects, and (iv) outliers and other intervention variables. - The second stage performs a
**decomposition**of the preadjusted series into unobserved components.__The series adjusted for deterministic effects is decomposed into three unobserved components: trend-cycle, seasonal, and irregular__…After unobserved components are estimated, the adjustment factors identified in the first stage (calendar effects, outliers, etc.) are allocated to their respective component so to end up with a full decomposition of the original series into final trend-cycle, seasonal, calendar, and irregular components.” [IMF]

“For seasonal adjustment purposes, a time series is generally assumed to be made up of four main components:

- The
**trend-cycle component**is the underlying path of the series. It includes both the long-term trend and the business-cycle movements. - The
**seasonal component**includes those seasonal fluctuations that repeat themselves with similar annual timing, direction, and magnitude. - The
**calendar component**comprises effects that are related to the different characteristics of the calendar from period to period. Calendar effects are both seasonal and nonseasonal.__Only the nonseasonal part should be included in the calendar component and treated separately__, as the seasonal one is already caught by the seasonal component. - The
**irregular component**captures all the other fluctuations that are not part of the trend-cycle, seasonal, and calendar components…Their timing, impact, and duration are unpredictable at the time of their occurrence.” [IMF]

“[A] decomposition scheme specifies how the various components – basically trend-cycle, seasonal, calendar component and irregular – combine to form the original series. __Usually, the decomposition scheme is multiplicative (either pure multiplicative or log-additive), because in most economic time series, the magnitudes of the seasonal component appear to vary proportionally__ to the level of the series.” [Eurostat]

“Currently the most commonly used seasonal adjustment methods are the signal extraction approach that starts from an ARIMA modelling of the complete series, and the semi-parametric approach based on a set of predefined moving averages.” [Eurostat]

*N.B.: ARIMA means ‘Auto-Regressive Integrated Moving Average’, which is a general model that explains a time series or its differences based on past values and past forecast errors. The model is defined by the order of autoregression (how many past values matter), the order of the moving average (how many past forecast errors matter) and the order of integration (how many times the series needs to be differenced to become stationary).*

“Broadly speaking…methods to remove seasonal patterns…can be divided into two groups: moving average methods and model-based methods.

- Methods in the first group
__derive the seasonally adjusted data by applying a sequence of moving average filters__to the original series and its transformations. These methods are all variants of the**X-11 method**, originally developed by the U.S. Census Bureau. The current version of the X-11 family is X-13ARIMA-SEATS (X-13A-S)… - Model-based methods
__derive the unobserved components in accordance with specific time series models__, primarily autoregressive integrated moving average (ARIMA) models. The most popular model-based seasonal adjustment method is**TRAMO-SEATS**developed by the Bank of Spain.” [IMF]

### Accounting not just for seasonal but also for calendar effects

“The most used calendar effects include the following:

**Trading-Day or Working-Day Effect**…The trading-day effect assumes an underlying pattern associated with each day of the week; the working-day effect postulates different behavior between the groups of weekdays and weekends.**Moving Holiday Effect**…A moving holiday is associated with events of religious or cultural significance within a country that change date from year to year (e.g., Easter or Ramadan).**Leap Year Effect**. This effect is needed to account for the extra day in February of a leap year, which may generate a four-year cycle.” [IMF]

**“**Trading day effects…are system__atic effects in monthly times series related to changes in the day-of-week composition of each month__ and, in some cases, also to changes in the length of February. For flow series (monthly accumulations of daily activity e.g. monthly sales), the increases or decreases from average day-of-week activity associated with the days that occur five times in the month in a given year are important. (If they are days of high sales volumes, the monthly value will be inflated.) For flow series, the length of February can have an impact. (More days than average should produce more sales than average for February.) For stock series, such as end-of-month inventories, the extent to which inventories tend to rise or fall on the day of measurement (e.g. the last day of the month) can have an impact that is different from year to year.” [United States Census Bureau]

**“**Moving holiday effects…are __systematic changes in the values of a time series that are associated with the timing of moving holidays__, i.e. holidays whose dates vary from year to year, such as Easter, Passover, Ramadan, Chinese New Year and U.S. Labor Day. Estimates of one or a combination of such effects define the moving holiday component of time series” [United States Census Bureau]

“Calendar effects should be removed from the series because they could affect negatively the quality of decomposition into unobserved components…__Calendar effects should be estimated and eliminated from the original series before the time series decomposition process__…The adjustment for calendar effects should be performed only for those series for which there is both statistical evidence and economic interpretation of calendar effects…All calendar effects are captured through specific deterministic effects that are meant to reproduce the changes in the calendar structure over time. These deterministic effects are called calendar regressors, as they are used as independent variables in the regARIMA.” [IMF]

“In order to take into account the national…idiosyncrasies, __different calendars are needed__. They are used to calculate calendar regressors for calendar adjustment.” [Eurostat]

### No general improvement of the quality of economic data

“Seasonal adjustment adopted by statistical agencies is sometimes seen as a potentially dangerous procedure that may compromise the intrinsic properties of the original series. In fact, __there is always some loss of information from seasonal adjustment,__ even when the seasonal adjustment process is properly conducted…Unadjusted data are useful in their own right. The non-seasonally adjusted data show the actual economic events that have occurred, while the seasonally adjusted data and the trend-cycle estimate represent an analytical elaboration of the data designed to show the underlying movements that may be hidden by the seasonal variations.” [IMF]

“__Seasonal adjustment programs may return ‘seasonally adjusted’ data even when the input data does not contain seasonal effects__. On the other hand, they may provide seasonally adjusted series that still contain residual seasonal effects…Basic diagnostics should include at a minimum tests for presence of identifiable seasonality in the original series, tests for residual seasonality in the seasonally adjusted series, significance tests of calendar effects” [IMF]

### Seasonally adjusted data can still have seasonal patterns

“Residual seasonality, like seasonality in general, is a predictable pattern of output that occurs over the year, but current methods are failing to measure it…Thus, residual seasonality is not removed in the initial seasonal adjustment process…[For example] __small seasonalities unaccounted for in the GDP components could be producing noticeable seasonal patterns in the aggregate__…In past decades, first-quarter economic growth has been substantially lower than the growth in other quarters, even after adjusting for typical seasonality.” [St. Louis Fed]

“The official seasonal adjustment [of national accounts] is [sometimes] done at the disaggregate level (the indirect approach) [as opposed to] the seasonal adjustment at the aggregate level (the direct approach)…__Seasonal adjustment at the disaggregate level might be more prone to leaving residual seasonality in aggregate data… Some of the components are not seasonally adjusted at all on the grounds that the seasonality in those components is not sufficiently pronounced__. There may be many components that each exhibit minor seasonality where that seasonality is however positively correlated across the components…Revision policies prevent seasonal adjustments being applied to historical data.” [Wright]

“__Seasonality may be gradually changing over time__. This phenomenon is called ‘moving seasonality.’” [IMF]

### Seasonally adjusted data are not generally good trend indicators

“__Seasonal adjustment is not meant for smoothing series__. A seasonally adjusted series is the sum of the trend-cycle component and the irregular component…Adjusting a series for seasonal variations removes the identifiable, regularly repeated influences on the series but not the impact of any irregular events. Consequently, if the impact of irregular events is strong, seasonally adjusted series may not represent a smooth, easily interpretable series.” [IMF]

“__The decomposition between the trend-cycle and the irregular components is subject to large uncertainty at the endpoint of the series__, where it may be difficult to distinguish and allocate the effects from new observations.” [IMF]

### Seasonal adjustment deals with outlier events in theory, but not always in practice

“__Outlier…effects manifest themselves with abrupt changes in the series,__ sometimes related to unexpected weather or socioeconomic effects (such as natural disasters, strikes, or economic and financial crises). __Such effects are not part of the underlying linear data generation process assumed for the original series. For these reasons, outlier effects are also called nonlinear effects__. In the seasonal adjustment process, outliers should be removed by means of predefined intervention variables. Three main types of outliers are often used for economic time series: i. additive outlier, which relates to only one period; ii. level shift, which changes the level of a series permanently; iii. transitory change, whose effects on a series fade out over a number of periods.” [IMF]

“Outliers are abnormal values of the series [that]…greatly affect the quality of the seasonal estimate. Various kinds of outliers (i.e. additive outliers, transitory changes, level shifts etc.) should be detected and corrected for. RegARIMA models have proved a successful method of doing this…RegARIMA models provide a possibility for modelling outliers identified by the user and an automatic procedure to detect outliers and to correct for their effects.” {Eurostat]

*N.B.: RegARIMA (linear regression model with ARIMA time series errors) models clean the series from non-linearities, such as outliers and calendar effects. For a majority of the time series this preliminary step is crucial for achieving satisfactory results from seasonal adjustment. Outliers are observations which do not fit in the tendency of the time series observed as they differ dramatically from the typical pattern of the trend and/or seasonal components.*

“__Outliers at the ends of the series present unique problems.__ A level shift at the last data point …[or] a temporary change at the last data point…cannot be distinguished from an additive outlier there.” [Eurostat]

### Seasonal adjustment is being revised and a source of bias in backtests

“__Seasonal adjustment results are severely affected by outliers and intervention variables.__ A different combination of regression effects can produce significant changes in the estimation of trend and seasonal components…Outliers are particularly difficult to detect and interpret in real-time, especially during periods of strong economic changes such as recessions.” [IMF]

“Seasonal effects may change over time. The seasonal pattern may gradually evolve as economic behavior, economic structures, and institutional and social arrangements change. The seasonal pattern may also change abruptly because of sudden institutional changes. Seasonal filters estimated using centered moving averages (like the X-11 and SEATS filters) allow the seasonal pattern of the series to change over time and allow for a gradual update of the seasonal pattern [but this requires forward averaging and hence real-time estimates of latest data often look very different]… Consequently, __new observations generally result in changes in the estimated seasonal pattern for the latest part of the series and subject seasonally adjusted data to more frequent revisions than the original non-seasonally adjusted series__.” [IMF]

“__Revisions of seasonally adjusted data can take place because of a better estimate/identification of the seasonal pattern due to new information__ provided by new unadjusted data and/or due to the characteristics of the filters and procedures removing seasonal and calendar components…In seasonal adjustment it may be the case that __just one additional observation results in revisions of the seasonally adjusted data for several years__.” [Eurostat]

### The trouble with “bridge days”, school holidays and extreme weather events

“__Bridging days are days lying between a public holiday and a weekend__. They are counted in purely calendar terms as full working days, but…they would be expected to have…a negative impact on production. Empirical investigations show clear evidence of these effects…[However] the countermovement of less leave should actually also be adjusted over the rest of the year so that no distortion of the business cycle occurs…The estimation of this countermovement is often not possible in practice.” [Eurostat]

“Country experience shows that the __estimation of bridge day and extreme weather effects is extremely uncertain__. Normal weather-related effects should be treated as part of the regular seasonal adjustment process, whereas extreme effects can be adjusted using outliers or ad hoc intervention variables.” [IMF]

“The economic __activity in a month/quarter is likely to depend on the timing of the school holidays__. Workers with school-age children take leave mainly during the school holidays, and hence interrupt their work. Empirical investigations show clear evidence of these effects. However, monthly-specific estimates of the influences of the school holidays are based in each case on only a very limited number of observations.” [Eurostat]

“Weather-induced effects do not occur repeatedly with exactly the same intensity in the same month each year. Rather, the __impairment of construction activity in the cold season depends on the intensity and, above all, on the length of the extreme weather periods__. In this sense, one may attempt to model the weather dependency of, for instance, construction output using suitable regressors in order to make it easier to draw conclusions as to economic developments. However, exceptionally severe weather-related production impairments in the cold season frequently lead to positive catch-up effects in the spring. If the winter shortfall was adjusted, the indirect knock-on effect would also have to be removed from the spring calculation in order not to unilaterally distort the business cycle picture.” [Eurostat]

### Standard software for seasonal adjustment

“United States Census Bureau …__X-11 [has been the first widely used] seasonal adjustment software. It is based on an iterative application of linear filters__…X-12-ARIMA Seasonal adjustment software…incorporates regression techniques and also ARIMA [AutoRegressive Integrated Moving Average] modelling to improve estimation of the different time series components… X-13ARIMA-SEATS (X-13A-S) seasonal adjustment software…integrates an enhanced version of X-12-ARIMA with an enhanced version of SEATS [Signal Extraction in ARIMA Time Series] __to provide both X-11 method seasonal adjustments and ARIMA model-based seasonal adjustments and diagnostics__. This is the seasonal adjustment software currently used by the Census Bureau.” [United States Census Bureau]

“**X-13ARIMA-SEATS** is a seasonal adjustment software produced, distributed, and maintained by the [U.S.] Census Bureau. Features include:

__Extensive time series modeling and model selection capabilities for linear regression models__with ARIMA errors [errors that are not necessarily independently and identically distributed but are of a more general autoregressive, integrated, and moving average form]…ARIMA models are a versatile family of models for modeling and forecasting time series data. Seasonal ARIMA models have a special form for efficiently modeling many kinds of seasonal time series and are heavily used in seasonal adjustment…[In] RegARIMA models (regression+ARIMA models) some features of the time series, such as moving holiday, trading day and outlier effects, are modeled with linear regression variables while the remaining features (those of the regression residuals, including trend, cycle and seasonal components) are modeled with a seasonal ARIMA model.- The
__capability to generate ARIMA model-based seasonal adjustment using a version of the SEATS [Signal Extraction in ARIMA Time Series] procedure__. It performs estimation based on regression models with missing observations and ARIMA errors, in the presence of several types of outlier…**Outliers**[are] data from an abrupt, untypical movement in the time series, e.g. from a hurricane, a strike, etc., that are likely to distort the estimates of seasonal, trading day or holiday effects. For seasonal adjustment, the software’s generic outlier regressors are used to estimate and temporarily (approximately) remove the outliers, in order to prevent distortion of the desired estimates. These protected estimates are removed from the original series to obtain the adjusted series. Consequently, the seasonal and perhaps trading day and holiday effect adjusted series still shows the expected outlier movements. __Diagnostics of the quality and stability__of the adjustments achieved under the options selected.” [United States Census Bureau]

“The **TRAMO-SEATS** program implements the ARIMA model-based seasonal adjustment method…It is promoted and maintained by the Bank of Spain. The SEATS decomposition is based on the signal extraction method…The pre-adjustment module (TRAMO) is similar to the one incorporated in the X-13A-S program.” [IMF]

“**JDemetra+** is __open source, platform independent, extensible software for seasonal adjustment__ (SA) and other related time series problems developed by the National Bank of Belgium…The most widely used and recommended [seasonal adjustment methods] are X-12-ARIMA2/X-13ARIMA-SEATS3 developed at the U.S. Census Bureau and TRAMO/SEATS4 developed by…the Bank of Spain. Both methods are divided into two main parts. The first part is called pre-adjustment and removes deterministic effects from the series by means of a regression model with ARIMA noise. The second part is the decomposition of the time series to estimate and remove a seasonal component. TRAMO/SEATS and X-12-ARIMA/X-13ARIMA-SEATS use a very similar approach in the first part to estimate the same model on the processing step, but they differ completely in the decomposition step. [Grudkowska]