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Seasonal adjustment is a statistical method for removing the seasonal component of a time series that is used when analyzing non-seasonal trends. It is normal to report seasonally adjusted data for unemployment rates to reveal the underlying trends 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] ^[3]

Time series components

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

trading days

The seasonal component, $S_{t}$ , contains the regular fluctuations associated with the time of year.
The trend component, $T_{t}$ , represents the underlying long-running movement in the series.
The cyclical component, $C_{t}$ , contains cyclical movements which take longer than a year to repeat. This component is often combined with the trend.
The error, or irregular component, $I_{t}$ , is the remainder of the original series when the seasonal, trend and cyclical components have been removed.

Time series are commonly categorised as having either an aditive or a multiplicative decomposition.

An additive time series, $Y_{t}$ , takes the form $Y_{t}$ = $T_{t}$ + $C_{t}$ + $S_{t}$ + $I_{t}$

A multiplicative time series, $Y_{t}$ , takes the form $Y_{t}$ = $T_{t}$ $C_{t}$ $S_{t}$ $I_{t}$ .

Seasonal adjustment

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 often considered to be uninteresting in their own right and to cause the interpretation of a series to be ambiguous. By removing the seasonal component, it is easier to focus on other components.^[4]

Easter

Seasonal adjustment can also be used to adjust for calendar effects including trading days and Easter. Some examples of calendar effects are:

Manufacturing typically happens on weekdays. A month with 4 weekends will see more output than the same month in another year with 5 weekends. The calendar effect makes comparing the two months directly more difficult.
Sales of chocolate eggs would be expected to be higher in an April containing Easter than and April not containing Easter.

When seasonal adjustment is not performed with monthly data, year-on-year changes are utilised in an attempt to avoid contamination with seasonality. However such comparisons should be treated with care as they will not accoutn for calendar effects.

Methods and software

Different statistical research groups have developed different methods of seasonal adjustment, for example X-12-ARIMA developed by the United States Census Bureau; TRAMO/SEATS developed by the Bank of Spain;^[5] and STAMP developed by a group led by S. J. Koopman.^{[citation needed]} 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.^{[citation needed]}

Example

One famous example is the rate of unemployment which is also presented 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. As soon as the seasonal influence is removed from this time series, the unemployment rate data can be meaningfully compared across different months. Seasonal adjustment is mostly used in the official statistics implemented by statistical software like Demetra+.

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.