Standard signal extraction results for both stationary and nonstationary time series are expressed as linear filters applied to the observed series. Computation of the filter weights, and of the corresponding filter transfer function, is relevant for studying properties of the filter and of the resulting signal extraction estimates. Methods for doing such computations for symmetric, doubly infinite filters are well established. This paper develops an algorithm for computing filter weights for asymmetric, semi-infinite signal extraction filters, including the important case of the concurrent filter (for signal extraction at the current time point). The setting is where the time series components being estimated follow ARIMA (autoregressive-integrated-moving average) models. The algorithm provides expressions for the asymmetric signal extraction filters as rational polynomial functions of the backshift operator. The filter weights are then readily generated by simple expansion of these expressions, and the filter transfer function may be directly evaluated. Recursive expressions are also developed that relate the weights for filters that use successively increasing amounts of data. The results for the filter weights are then used to develop methods for computing mean squared error results for the asymmetric signal extraction estimates.
Source: U.S. Census Bureau, Statistical Research Division