The Use of Butterworth Filters for Trend and Cycle Estimation in Economic Time Series.

Long-term trends and business cycles are usually estimated by applying the Hodrick and Prescott (HP) filter to X-11 seasonally adjusted data. A two-stage procedure is proposed in this article to improve this methodology. The improvement is based on (a) using Butterworth or band-pass filters specifically designed for the problem at hand as an alternative to the HP filter, (b) applying the selected filter to estimated trend cycles instead of to seasonally adjusted series, and (c) using autoregressive integrated moving average models to extend the input series with forecasts and backcasts. It is shown in the article that the HP filter is a Butterworth filter and that, if a model-based method is used for seasonal adjustment, it is possible to give a fully model-based interpretation of the proposed procedure. In this case, one can compute forecasts and mean squared errors of the estimated trends and cycles. The procedure is illustrated with several examples.