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X-13ARIMA-SEATS Seasonal Adjustment Program

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Bayesian Seasonal Adjustment of Long Memory Time Series

Scott H. Holan (1) and Tucker S. McElroy (2)


Existing approaches to the seasonal adjustment of economic time series are typically either nonparametric or model-based. In both cases, the goal is to remove seasonal variation from the time series. In each of the two paradigms, both the seasonally adjusted series and the seasonal component are latent processes. As such, seasonal adjustment can be viewed as an unobserved components (UC) problem and specifically that of UC estimation. Though the nonparametric approach has a rich history going back to the development of X-11 and X-11 ARIMA (Dagum, 1980; Shiskin et al., 1967), our focus centers on model-based methodology.

Within the model-based framework, two directions have emerged. The first direction, and the direction pursued here, directly specifies models for the components and is known as the structural time series approach (Harvey, 1990). Alternatively, one could start with a model for the observed time series and derive appropriate models for each component (Hillmer and Tiao, 1982). This latter approach is often referred to as "canonical decomposition." In the seasonal adjustment of economic time series it is common to "preadjust" the series. This preadjustment often includes interpolation of missing values, outlier adjustment and adjustment for trading day and holiday effects. In addition to the customary preadjustments, many model-based approaches require that the observed series be differenced (and/or seasonally differenced) to handle nonstationarity. One question that naturally arises when implementing such an approach is whether or not the correct number of differencing operations have been imposed. In practice, typically, only integer orders of integration are considered. Nevertheless, it is possible that differencing the data once results in a series that still exhibits nonstationary behavior, whereas, imposing a second difference may result in a series that is "over-differenced" and thus noninvertible. In these cases, a natural alternative is to difference the observed series and then model the residual series as a fractionally differenced process.

The models and signal extraction methodology we propose are applied to nonstationary data. However, the approach we develop assumes that, after suitable differencing, the residual series is stationary but allows for long-range dependence (in the seasonal and/or trend component) or anti-persistance (sometimes referred to as intermediate or negative memory).

(1) Scott Holan is a professor in the Department of Statistics, University of Missouri-Columbia.

(2) Tucker S. McElroy is Mathematical Statistican, Center for Statistical Research and Methodology U. S. Census Bureau, 4600 Silver Hill Road, Washington, DC 20233. email :

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Source: U.S. Census Bureau | Statistical Research Division | (301) 763-1649 (or |  Last Revised: November 19, 2012