Seasonal adjustment

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

• ${\displaystyle S_{t}}$: The seasonal component
• ${\displaystyle T_{t}}$: The trend component
• ${\displaystyle C_{t}}$: The cyclical component
• ${\displaystyle E_{t}}$: 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.[4]

The relation between decomposition of time series components

• Additive decomposition: ${\displaystyle Y_{t}=S_{t}+T_{t}+C_{t}+E_{t}}$, where ${\displaystyle Y_{t}}$ is the data at time ${\displaystyle t}$.
• Multiplicative decomposition: ${\displaystyle Y_{t}=S_{t}\cdot T_{t}\cdot C_{t}\cdot E_{t}}$.
• Logs turn multiplicative relationship into an additive relationship: ${\displaystyle Y_{t}=S_{t}\cdot T_{t}\cdot C_{t}\cdot E_{t}\Rightarrow \log Y_{t}=\log S_{t}+\log T_{t}+\log C_{t}+\log E_{t}}$:
• 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.

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, STAMP developed by a group led by S. J. Koopman;[7] and “Seasonal and Trend decomposition using Loess” (STL) developed by Cleveland et al. (1990).[8] 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.[8]

Brief introduction to process of X-12-ARIMA:

For example: description assumes monthly data. Additive decomposition: ${\displaystyle Y_{t}=S_{t}+T_{t}+C_{t}+E_{t}}$: Multiplicative decomposition: ${\displaystyle Y_{t}=S_{t}\cdot T_{t}\cdot C_{t}\cdot E_{t}}$

• 1. Using moving-average smoothing method to estimate the trend-cycle for all periods. In the monthly data, use 12-month centered moving average is appropriate to be applied to estimate the trend-cycle component.
• 2. Ratios of data to trend computed(called“centered ratios”)---which means remove the smoothed series from ${\displaystyle Y}$ to leave ${\displaystyle S}$ and ${\displaystyle E}$.
• 3. Separate 3 * 3 MA (moving average) applied to each month of centered ratios to form rough estimate of ${\displaystyle S_{t}}$.
• 4. Divide centered ratios by estimate of ${\displaystyle S_{t}}$ to get estimate of ${\displaystyle E_{t}}$.
• 5. Reduce extreme value of ${\displaystyle E_{t}}$
• 6. Multiply by ${\displaystyle S_{t}}$ 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 ${\displaystyle S_{t}}$.
• 8. Original data divided by new estimate of ${\displaystyle S_{t}}$ gives the preliminary seasonally adjusted series.
• 9. Trend-cycle 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 trend-cycle
• 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 ${\displaystyle S_{t}}$.
• 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 trend-cycle 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 trend-cycle, 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 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.[9] R includes STL decomposition.[10] The X-12-ARIMA method can be utilized via the R package "X12".[11] 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.[12][13][14] 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, 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.[15]

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]

• Ergograph
• List of statistical packages
• Seasonally adjusted annual rate
• Seasonal year

• 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