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Publication date: 07/30/2020

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 13.1 User’s Guide (search for “Historical Development of X-11”).

The multiplicative adjustment fits the following model:


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:

The adjusted additive trend is Ot - St.

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