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# X-12-ARIMA

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## Introduction

X-12-ARIMA is the well-known, reliable and widely used seasonal adjustment software developed, distributed, and maintained by the United States Census Bureau. The method has been developed on an empirical basis, without an explicitly defined statistical decomposition model.

## The foundations of X-12-ARIMA

The X-12-ARIMA program originates from X-11 Variant of the Census Method II seasonal adjustment program and includes a wide variety of enhancements to overcome the X-11 deficiencies.

X-11 method can be regarded as a filtering operation resulting from successive application of seasonal and non-seasonal moving averages and Henderson trend moving averages. It includes trading day adjustments as well as detection and correction for extreme observations (outliers). Although X-11 method is based on symmetric moving averages, at both ends of the series the asymmetric filters are used. It is because the symmetric filter cannot be applied for the first (last) few observations due to the lack of preceding (following) observations.

As a result the current estimates are revised once the new observations become available. As a result, the estimators for observations in the ends of the time series are less reliable than estimators for central observations. These drawbacks adversely affect the X-11 output and stimulate the development of this method.

## X-12-ARIMA basic structure

X-12-ARIMA is the last United States Census Bureau program. It includes all the capabilities of X-11 and a pre-processing program RegARIMA. Therefore, X-12-ARIMA algorithm is divided in two parts:

- In a first part, RegARIMA (linear regression model with ARIMA time series errors) models are used to clean the series from non-linearities, such as outliers and calendar effects. This model is also used for calculating and extending time series by backcasts and forecasts. The description of the model can be found here.
- In a second part, an enhanced version of the X11 algorithm is used to decompose the extended, adjusted for non-linearites time series to the trend-cycle, the seasonal fluctuations and the irregular component.

The relations between those two parts are presented in the following simple operation-flow diagram:

Flow Diagram for Seasonal Adjustment with X12; source: Findley D. et al, (1998).