X-13ARIMA-SEATS

X-13ARIMA-SEATS, successor to X-12-ARIMA and X-11, is a set of statistical methods for seasonal adjustment and other descriptive analysis of time series data that are implemented in the U.S. Census Bureau's software package. These methods are or have been used by Statistics Canada, Australian Bureau of Statistics, and the statistical offices of many other countries.
X-12-ARIMA can be used together with many statistical packages, such as SAS in its econometric and time series (ETS) package, R in its (seasonal) package, Gretl or EViews which provides a graphical user interface for X-12-ARIMA, and NumXL which avails X-12-ARIMA functionality in Microsoft Excel.
Notable statistical agencies presently using X-12-ARIMA for seasonal adjustment include Statistics Canada, the U.S. Bureau of Labor Statistics and Census and Statistics Department (Hong Kong). The Brazilian Institute of Geography and Statistics uses X-13-ARIMA. 
X-12-ARIMA was the successor to X-11-ARIMA; the current version is X-13ARIMA-SEATS.
X-13-ARIMA-SEATS's source code can be found on the Census Bureau's website.
Methods.
The default method for seasonal adjustment is based on the X-11 algorithm. It is assumed that the observations in a time series, formula_1, can be decomposed additively,
or multiplicatively,
In this decomposition, formula_4 is the trend (or the "trend cycle" because it also includes cyclical movements such as business cycles) component, formula_5 is the seasonal component, and formula_6 is the irregular (or random) component. The goal is to estimate each of the three components and then remove the seasonal component from the time series, producing a seasonally adjusted time series.
The decomposition is accomplished through the iterative application of centered moving averages. For an additive decomposition of a monthly time series, for example, the algorithm follows the following pattern:
The method also includes a number of tests, diagnostics and other statistics for evaluating the quality of the seasonal adjustments.