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Publication date: 11/10/2021

Statistical Details for X-11 Decomposition

This method adjusts the original time series using either a multiplicative or an additive decomposition. The model is fit using an iterative process to estimate the three X-11 components: trend cycle, seasonal, and irregular. The trend cycle component contains both the long-term trend and the long-term cyclical effects. The irregular component contains the effects of variation unexplained by the trend and seasonal components. For a historical overview of the development of the X-11 method, see SAS/ETS 15.2 User’s Guide (search for “Historical Development of X-11”).

The multiplicative adjustment fits the following model:

Equation shown here


Ot is the original time series

Ct is the trend cycle component

St is the seasonal component

It is the irregular component

The adjusted multiplicative trend is Ot /St.

The additive adjustment fits the following model:

Equation shown here

The adjusted additive trend is Ot - St.

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