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X-13ARIMA-SEATS Seasonal Adjustment Program

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Design of Moving-Average Trend Filters Using Fidelity, Smoothness and Minimum Revisions Criteria

Alistair Gray (1) and Peter Thomson (2)


Many seasonal adjustment procedures decompose time series into trend, seasonal, irregular and other components using simple non-seasonal finnite moving-average trend filters. This report considers the design of such filters, both in the body and at the ends of series, based on specified criteria and simple dynamic models operating locally within the span of the filter.

In the body of the series a flexible family of finite moving-average trend filters is developed from specified smoothness and fidelity criteria. These filters are based on local dynamic models and generalise the standard Macaulay and Henderson filters used in practice. The properties of these central filters are determined and evaluated both in theory and in practice.

At the ends of the series the central moving-average trend filter used in the body needs to be extended to handle missing observations. A family of end filters is constructed using a minimum revisions criterion and based on the local dynamic model operating within the span of the central filter. These end filters are equivalent to evaluating the central filter with unknown observations replaced by constrained optimal linear predictors. Two prediction methods are considered; best linear unbiased prediction (BLUP) and best linear biased prediction where the bias is time invariant (BLIP). The BLIP end filters generalise those developed by Musgrave for the central X-11 Henderson filters and include the BLUP end filters as a special case.

The properties of these end filters are determined both in theory and practice. In particular, they are compared to the Musgrave end filters used by X-11 and to the case where the central filter is evaluated with unknown observations predicted by global ARIMA models. The latter parallels the forecast extension method used in X-11-ARIMA.


Moving-average filters; local trend estimation; dynamic models; fidelity; smoothness; minimum revisions; best linear unbiased prediction; best linear biased prediction; X-11; seasonal time series; seasonal adjustment.

(1) Alistair Gray is currently an associate with Statistics Research Associates, Ltd of New Zealand. email : allistair@statsresearch.co.nz

(2) Peter Thomson is Professor at Victoria University in Wellington, New Zealand. email : peter@isor.vuw.ac.nz

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Source: U.S. Census Bureau | Center for Statistical Research and Methodology | (301) 763-1649 (or x12@census.gov) |  Last Revised: November 19, 2012