$E}_{t$Brief introduction to process of X-12-ARIMA:

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

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.

## Example

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.

## Indirect seasonal adjustment

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*.

## 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 "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]}

## Shortcomings of using seasonally adjusted data

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]}

## See also

## References