The Use of Butterworth Filters for Trend and Cycle Estimation in Economic Time Series
Gomez, Victor, Journal of Business & Economic Statistics
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.
KEY WORDS: Business cycle; Butterworth filters; Hodrick--Prescott filter; Kalman filter; Signal extraction; Wiener--Kolmogorov filters.
The focus of this article is the presentation of stylized facts of the business cycle. This is important for economists for two reasons. First, it provides a set of "regularities" that can be used to compare actual data with time series data generated from an artificial economy before making conclusions about the validity of numerical versions of theoretical models. Second, it summarizes in some way the comovements existing among different economic series and may guide in the selection of leading indicators for economic activity.
It seems that there is no consensus in the literature as to what constitutes business fluctuations or as to what a trend is. Thus, business-cycle fluctuations are typically defined as deviations from the trend of the process, but the trend component is usually left undefined because of lack of agreement on its properties and on its relationship with the cyclical component.
As a consequence of this state of affairs, the estimation of the business-cycle is usually carried out in an ad hoc manner. The standard approach to business-cycle estimation consists of applying the Hodrick and Prescott (HP) (1997) filter to X-11 seasonally adjusted data, where X-11 should be understood as X-11 or either of its extensions, X-11-ARIMA or X-12-ARIMA.
However, some problems arise when the HP filter is used for business-cycle estimation. Several authors (e.g., Harvey and Jager 1993) have shown that the mechanical application of the HP filter may induce spurious results. Another problem with the HP filter, which is often ignored, is that the filtered series is usually subject to big revisions as new data become available. Finally, there is too much noise in the cycles estimated with the HP filter, something that may negatively affect the detection and dating of turning points.
In this article, I propose to modify the standard approach to business-cycle estimation to improve its performance. More specifically, I will show the following. First, the HP filter is a Butterworth filter. This provides useful insights as to how to select the [Lambda] parameter in the HP filter and suggests that other Butterworth or band-pass filters can, on occasion, perform better than the HP filter. This is particularly the case with business-cycle estimation, where the use of band-pass filters eliminates much high-frequency variation that would be present if the HP filter were used. Second, applying Butterworth filters to estimated trend cycles instead of to seasonally adjusted (SA) series reduces the amount of noise in the estimated cycle and, what is more important, decreases the risk of inducing spurious results. This is especially true if the estimated trend cycle has been obtained with a model-based method. Third, when a model for the input series is available, a simple algorithm can be used to apply Butterworth filters to the series extended with forecasts and backcasts obtained with that model. …