Seasonal adjustment
Seasonal adjustment is a statistical method for removing the seasonal component of a time series that exhibits a seasonal pattern. It is usually done when wanting to analyse the trend, and cyclical deviations from trend, of a time series independently of the seasonal components. It is normal to report seasonally adjusted data for unemployment rates to reveal the underlying trends and cycles in labor markets.^{[1]} Many economic phenomena have seasonal cycles, such as agricultural production and consumer consumption, e.g. greater consumption leading up to Christmas. It is necessary to adjust for this component in order to understand what underlying trends are in the economy and so official statistics are often adjusted to remove seasonal components.^{[2]}
Contents
Time series components
The investigation of many economic time series becomes problematic due to seasonal fluctuations. Time series are made up of four components:
 S_{t}: The seasonal component
 T_{t}: The trend component
 C_{t}: The cyclical component
 E_{t}: 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.^{[3]}
The relation between decomposition of time series components
 Additive decomposition: Y_{t} = S_{t} + T_{t} + C_{t} + E_{t}, where Y_{t} is the data at time t.
 Multiplicative decomposition: Y_{t} = S_{t} * T_{t} * C_{t} * E_{t}.
 Logs turn multiplicative relationship into an additive relationship: Y_{t} = S_{t} * T_{t} * C_{t} * E_{t} => log Y_{t} = log S_{t} + log T_{t} + log C_{t} + log E_{t}:
 An additive model appropriate if the magnitude of seasonal fluctuations does not vary with level.
 If seasonal fluctuations are proportional to level of series, then a multiplicative model is appropriate. Multiplicative decomposition more prevalent with economic series.
Seasonal adjustment
This section may require cleanup to meet Wikipedia's quality standards. The specific problem is: make less terse; put in subscripting for e.g. St; what is the number 6 at start of last paragraph? (March 2017) (Learn how and when to remove this template message)

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.^{[4]}
Different statistical research groups have developed different methods of seasonal adjustment, for example X13ARIMA and X12ARIMA developed by the United States Census Bureau; TRAMO/SEATS developed by the Bank of Spain;^{[5]} MoveReg (for weekly data) developed by the United States Bureau of Labor Statistics, STAMP developed by a group led by S. J. Koopman;^{[6]} and “Seasonal and Trend decomposition using Loess” (STL) developed by Cleveland et al. (1990).^{[7]} While X12/13ARIMA can only be applied to monthly or quarterly data, STL decomposition can be used on data with any type of seasonality. Furthermore, unlike X12ARIMA, STL allows the user to control the degree of smoothness of the trend cycle and how much the seasonal component changes over time. X12ARIMA 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 backtransform after the decomposition.^{[7]}
Brief introduction to process of X12ARIMA:
For example: description assumes monthly data. Additive decomposition: Yt = St + Tt + Ct + Et: Multiplicative decomposition: Yt = St * Tt * Ct * Et
 1.Using movingaverage smoothing method to estimate the trendcycle for all periods. In the monthly data, use 12month centered moving average is appropriate to be applied to estimate the trendcycle component.
 2.Ratios of data to trend computed(called“centered ratios”)which means remove the smoothed series from Y to leave S and E.
 3.Separate 3 * 3 MA (moving average) applied to each month of centered ratios to form rough estimate of St.
 4.Divide centered ratios by estimate of St to get estimate of Et.
 5.Reduce extreme value of Et
 6.Multiply by St to get modified centered ratios.
 7.Take another 3 * 3 MA of each month of the year individually applied to modified ratios to get revised St.
 8.Original data divided by new estimate of St gives the preliminary seasonally adjusted series.
 9.Trendcycle estimated by applying a weighted Henderson MA to the preliminary seasonally adjusted values.
 10.Repeat Step 2. New ratios are obtained by dividing the original data by the new estimated trendcycle
 11.Repeat Steps 3–6 using new ratios and applying a 3 * 5 MA instead of a 3 * 3 MA.
 12.Repeat Step 7 but using a 3 * 5 MA instead of a 3 * 3 MA, that is taking 5 * 3 MA of each month of the year individually by using the modified data applied to modified ratios to get revised St.
 13.Repeat Step 8 but using the new seasonal component obtained in Step 12 to obtain seasonally adjusted values.
 14.Remainder component obtained by dividing seasonally adjusted data from Step 13 by the trendcycle obtained in Step 9.
 15.Extreme values of remainder component are reduced as in Step 5.
 16.A series of modified data is obtained by multiplying the trendcycle, seasonal component, and adjusted irregular component together.
Repeat whole process two more times with modified data. On final iteration, the 3 * 5 MA of Steps 11 and 12 is replaced by either a 3 * 3, 3 * 5, or 3 * 9 moving average, depending on the variability in the data.
6. Time series 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 X12ARIMA, 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 X12ARIMA and TRAMO/SEATS.^{[8]} R includes STL decomposition.^{[9]} The X12ARIMA method can be utilized via the R package "X12" ^{[10]}. EViews supports X12, X13, Tramo/Seats, STL and MoveReg.
Example
One wellknown 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.^{[11]} Seasonal adjustment is used in the official statistics implemented by statistical software like Demetra+.
When seasonal adjustment is not performed with monthly data, yearonyear changes are utilised in an attempt to avoid contamination with seasonality.
Moves to standardise seasonal adjustment processes
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, 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.
Use of seasonally adjusted data in regressions
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 "nonrevertible" moving average (MA) component into time series data, unit root tests (such as the Phillips–Perron test) will be biased towards nonrejection of the unit root null.^{[12]}
Shortcomings of using seasonally adjusted data
Use of seasonally adjusted time series data can be misleading. This is because the seasonally adjusted series contains both the trendcycle component and the error component. As such, the seasonally adjusted data will not be "smooth" and what appears 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, it is better to use the trendcycle component rather than the seasonally adjusted data.^{[13]}
See also
This article needs additional citations for verification. (July 2011) (Learn how and when to remove this template message)

References
 ^ http://www.bls.gov/cps/seasfaq.htm
 ^ "Retail spending rise boosts hopes UK can avoid doubledip recession". The Guardian. 17 February 2012.
 ^ https://www.otexts.org/fpp/2/1
 ^ FAQs on Seasonal Adjustment
 ^ OECD Glossary: Seasonal Adjustment
 ^ STAMP Modelling and Forecasting
 ^ ^{a} ^{b} "6.5 STL decomposition  OTexts". www.otexts.org. Retrieved 20160512.
 ^ OECD, ShortTerm Economic Statistics Expert Group (June 2002), Harmonising Seasonal Adjustment Methods in European Union and OECD Countries
 ^ Hyndman, R.J. "6.4 X12ARIMA decomposition  OTexts". www.otexts.org. Retrieved 20160515.
 ^ Kowarik, Alexander (February 20, 2015). "Xx12" (PDF). cran.rproject.org. Retrieved 20160802.
 ^ https://www.otexts.org/fpp/6/1
 ^ Maddala, G. S.; Kim, InMoo (1998). Unit Roots, Cointegration, and Structural Change. Cambridge: Cambridge University Press. pp. 364–365. ISBN 0521587824.
 ^ Hyndman, Rob J; Athanasopoulos, George. "Forecasting: principles and practice". Retrieved 20 May 2015.
Further reading
 Enders, Walter (2010). Applied Econometric Time Series (Third ed.). New York: Wiley. pp. 97–103. ISBN 9780470505397.
 Ghysels, Eric; Osborn, Denise R. (2001). The Econometric Analysis of Seasonal Time Series. New York: Cambridge University Press. pp. 93–120. ISBN 052156588X.
 Hylleberg, Svend (1986). Seasonality in Regression. Orlando: Academic Press. pp. 36–44. ISBN 0123634555.
 Jaditz, Ted (December 1994). "Seasonality: economic data and model estimation". BLS Monthly Labor Review. pp. 17–22.
External links
 Download Demetra+ from circa.europa.eu
 Seasonal adjustment at CROS portal (www.crosportal.eu)
 ESS Guidelines on Seasonal Adjustment