On a Family of Moving-average Trend Filters for the Ends of Series

For any given central moving-average trend filter, a family of end filters is constructed using a minimum revisions criterion and a 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 are shown to be a generalisation of those developed by Musgrave (1964) for the central X-11 Henderson filters and include the BLUP end filters as a special case.