The Use of Butterworth Filters for Trend and Cycle Estimation in Economic Time Series
提出用巴特沃斯或带通滤波器替代HP滤波器，结合季节调整和ARIMA模型扩展数据，以改进经济时间序列的趋势和周期估计，并给出模型解释和误差计算。
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.