BUREAU OF THE CENSUS STATISTICAL RESEARCH DIVISION Statistical Research Report Series No. RR96/01 Design of Moving-Average Trend Filters using Fidelity, Smoothness and Minimum Revisions Criteria

Many seasonal adjustment procedures decompose time series into trend, seasonal, irregular and other components using simple non-seasonal nite moving-average trend lters. This report considers the design of such lters, both in the body and at the ends of series, based on speci ed criteria and simple dynamic models operating locally within the span of the lter. In the body of the series a exible family of nite moving-average trend lters is developed from speci ed smoothness and delity criteria. These lters are based on local dynamic models and generalise the standard Macaulay and Henderson lters used in practice. The properties of these central lters are determined and evaluated both in theory and in practice. At the ends of the series the central moving-average trend lter used in the body needs to be extended to handle missing observations. A family of end lters is constructed using a minimum revisions criterion and based on the local dynamic model operating within the span of the central lter. These end lters are equivalent to evaluating the central lter 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 lters generalise those developed by Musgrave for the central X-11 Henderson lters and include the BLUP end lters as a special case. The properties of these end lters are determined both in theory and practice. In particular, they are compared to the Musgrave end lters used by X-11 and to the case where the central lter is evaluated with unknown observations predicted by global ARIMA models. The latter parallels the forecast extension method used in X-11-ARIMA.