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**Seasonal adjustment** or **deseasonalization** is a statistical method for removing the seasonal component of a time series. It is usually done when wanting to analyse the trend, and cyclical deviations from trend, of a time series independently of the seasonal components. Many economic phenomena have seasonal cycles, such as agricultural production, (crop yields fluctuate with the seasons) and consumer consumption (increased personal spending leading up to Christmas). It is necessary to adjust for this component in order to understand underlying trends in the economy, so official statistics are often adjusted to remove seasonal components.^{ [1] } Typically, seasonally adjusted data is reported for unemployment rates to reveal the underlying trends and cycles in labor markets.^{ [2] }^{ [3] }

The investigation of many economic time series becomes problematic due to seasonal fluctuations. Time series are made up of four components:

- : The seasonal component
- : The trend component
- : The cyclical component
- : The error, or irregular component.

The difference between seasonal and cyclic patterns:

- Seasonal patterns have a fixed and known length, while cyclic patterns have variable and unknown length.
- Cyclic pattern exists when data exhibit rises and falls that are not of fixed period (duration usually of at least 2 years).
- The average length of a cycle is usually longer than that of seasonality.
- The magnitude of cyclic variation is usually more variable than that of seasonal variation.
^{ [4] }

The relation between decomposition of time series components

- Additive decomposition: , where is the data at time .
- Multiplicative decomposition: .
- Logs turn multiplicative relationship into an additive relationship: :
- An additive model is appropriate if the magnitude of seasonal fluctuations does not vary with level.
- If seasonal fluctuations are proportional to the level of the series, then a multiplicative model is appropriate. Multiplicative decomposition is more prevalent with economic series.

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Unlike the trend and cyclical components, seasonal components, theoretically, happen with similar magnitude during the same time period each year. The seasonal components of a series are sometimes considered to be uninteresting and to hinder the interpretation of a series. Removing the seasonal component directs focus on other components and will allow better analysis.^{ [5] }

Different statistical research groups have developed different methods of seasonal adjustment, for example X-13-ARIMA and X-12-ARIMA developed by the United States Census Bureau; TRAMO/SEATS developed by the Bank of Spain;^{ [6] }MoveReg (for weekly data) developed by the United States Bureau of Labor Statistics;^{ [7] } STAMP developed by a group led by S. J. Koopman;^{ [8] } and “Seasonal and Trend decomposition using Loess” (STL) developed by Cleveland et al. (1990).^{ [9] } While X-12/13-ARIMA can only be applied to monthly or quarterly data, STL decomposition can be used on data with any type of seasonality. Furthermore, unlike X-12-ARIMA, STL allows the user to control the degree of smoothness of the trend cycle and how much the seasonal component changes over time. X-12-ARIMA can handle both additive and multiplicative decomposition whereas STL can only be used for additive decomposition. In order to achieve a multiplicative decomposition using STL, the user can take the log of the data before decomposing, and then back-transform after the decomposition.^{ [9] }

Each group provides software supporting their methods. Some versions are also included as parts of larger products, and some are commercially available. For example, SAS includes X-12-ARIMA, while Oxmetrics includes STAMP. A recent move by public organisations to harmonise seasonal adjustment practices has resulted in the development of Demetra+ by Eurostat and National Bank of Belgium which currently includes both X-12-ARIMA and TRAMO/SEATS.^{ [10] } R includes STL decomposition.^{ [11] } The X-12-ARIMA method can be utilized via the R package "X12".^{ [12] } EViews supports X-12, X-13, Tramo/Seats, STL and MoveReg.

One well-known example is the rate of unemployment, which is represented by a time series. This rate depends particularly on seasonal influences, which is why it is important to free the unemployment rate of its seasonal component. Such seasonal influences can be due to school graduates or dropouts looking to enter into the workforce and regular fluctuations during holiday periods. Once the seasonal influence is removed from this time series, the unemployment rate data can be meaningfully compared across different months and predictions for the future can be made.^{ [3] }

When seasonal adjustment is not performed with monthly data, year-on-year changes are utilised in an attempt to avoid contamination with seasonality.

When time series data has seasonality removed from it, it is said to be *directly seasonally adjusted*. If it is made up of a sum or index aggregation of time series which have been seasonally adjusted, it is said to have been *indirectly seasonally adjusted*. Indirect seasonal adjustment is used for large components of GDP which are made up of many industries, which may have different seasonal patterns and which are therefore analyzed and seasonally adjusted separately. Indirect seasonal adjustment also has the advantage that the aggregate series is the exact sum of the component series.^{ [13] }^{ [14] }^{ [15] } Seasonality can appear in an indirectly adjusted series; this is sometimes called *residual seasonality*.

Due to the various seasonal adjustment practices by different institutions, a group was created by Eurostat and the European Central Bank to promote standard processes. In 2009 a small group composed of experts from European Union statistical institutions and central banks produced the ESS Guidelines on Seasonal Adjustment,^{ [16] } which is being implemented in all the European Union statistical institutions. It is also being adopted voluntarily by other public statistical institutions outside the European Union.

By the Frisch–Waugh–Lovell theorem it does not matter whether dummy variables for all but one of the seasons are introduced into the regression equation, or if the independent variable is first seasonally adjusted (by the same dummy variable method), and the regression then run.

Since seasonal adjustment introduces a "non-revertible" moving average (MA) component into time series data, unit root tests (such as the Phillips–Perron test) will be biased towards non-rejection of the unit root null.^{ [17] }

Use of seasonally adjusted time series data can be misleading because a seasonally adjusted series contains both the trend-cycle component and the error component. As such, what appear to be "downturns" or "upturns" may actually be randomness in the data. For this reason, if the purpose is finding turning points in a series, using the trend-cycle component is recommended rather than the seasonally adjusted data.^{ [3] }

In mathematics, an **abelian group**, also called a **commutative group**, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

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**Agustín Maravall Herrero** is a Spanish economist known for his contributions in statistics and econometrics time series analysis, in particular seasonal adjustment and in the estimation of signals in economic time series. He has completed a methodology and several computer programs that are used throughout the world by analysts, researchers, and data producers. An important use is official production of series adjusted for seasonality and (perhaps) other undesirable effects such as noise, outliers, or missing observations. Maravall has received several awards and distinctions and retired in December 2014 from the Bank of Spain.

- ↑ "Retail spending rise boosts hopes UK can avoid double-dip recession".
*The Guardian*. 17 February 2012. Archived from the original on 8 March 2017. - ↑ "What is seasonal adjustment?".
*www.bls.gov*. Archived from the original on 2011-12-20. - 1 2 3 Hyndman, Rob J; Athanasopoulos, George.
*Forecasting: principles and practice*. pp. Chapter 6.1. Archived from the original on 12 May 2018. - ↑
*2.1 Graphics - OTexts*.*www.otexts.org*. Archived from the original on 2018-01-17. - ↑ "MCD - Seasonal Adjustment Frequently Asked Questions".
*www.census.gov*. Archived from the original on 2017-01-13. - ↑ Directorate, OECD Statistics. "OECD Glossary of Statistical Terms - Seasonal adjustment Definition".
*stats.oecd.org*. Archived from the original on 2014-04-26. - ↑ MoveReg
- ↑ "STAMP".
*www.stamp-software.com*. Archived from the original on 2015-05-09. - 1 2
*6.5 STL decomposition | OTexts*.*www.otexts.org*. Archived from the original on 2018-05-12. Retrieved 2016-05-12. - ↑ OECD, Short-Term Economic Statistics Expert Group (June 2002),
*Harmonising Seasonal Adjustment Methods in European Union and OECD Countries* - ↑ Hyndman, R.J.
*6.4 X-12-ARIMA decomposition | OTexts*.*www.otexts.org*. Archived from the original on 2018-01-17. Retrieved 2016-05-15. - ↑ Kowarik, Alexander (February 20, 2015). "Xx12" (PDF).
*cran.r-project.org*. Archived (PDF) from the original on December 6, 2016. Retrieved 2016-08-02. - ↑ Hungarian Central Statistical Office.Seasonal adjustment methods and practices, Budapest, July 2007
- ↑ Thomas D. Evans. Direct vs. Indirect Seasonal Adjustment for CPS National Labor Force Series, Proceedings of the Joint Statistical Meetings, 2009, Business and Economic Statistics Section
- ↑ Marcus Scheiblecker, 2014. "Direct Versus Indirect Approach in Seasonal Adjustment," WIFO Working Papers 460, WIFO. Abstract at IDEAS/REPEC
- ↑ http://ec.europa.eu/eurostat/documents/3859598/5910549/KS-RA-09-006-EN.PDF
- ↑ Maddala, G. S.; Kim, In-Moo (1998).
*Unit Roots, Cointegration, and Structural Change*. Cambridge: Cambridge University Press. pp. 364–365. ISBN 0-521-58782-4.

- Enders, Walter (2010).
*Applied Econometric Time Series*(Third ed.). New York: Wiley. pp. 97–103. ISBN 978-0-470-50539-7. - Ghysels, Eric; Osborn, Denise R. (2001).
*The Econometric Analysis of Seasonal Time Series*. New York: Cambridge University Press. pp. 93–120. ISBN 0-521-56588-X. - Hylleberg, Svend (1986).
*Seasonality in Regression*. Orlando: Academic Press. pp. 36–44. ISBN 0-12-363455-5. - Jaditz, Ted (December 1994). "Seasonality: economic data and model estimation". BLS Monthly Labor Review. pp. 17–22.

- Download Demetra+ from circa.europa.eu
- Seasonal adjustment at CROS portal (www.cros-portal.eu)
- ESS Guidelines on Seasonal Adjustment

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