The Holt-Winters filter and the one-sided HP filter: A close correspondence

Summary

Focus

The Hodrick-Prescott (HP) filter is heavily used in economics and by policymakers. In many cases, the two-sided, full sample version is applied to determine trends. But the one-sided filter that is estimated recursively also has important applications. Most notably, it is embedded in the Basel III framework for the countercyclical capital buffer. In this paper, we show that the trend of the one-sided HP filter has a very close correspondence with the trend of the Holt-Winters (HW) filter.

Contribution

We draw on the literature to show that the trend of the one-sided HP filter can be approximated by the HW filter. This implies an elegant, moving average representation. And it makes the computation of trends so simple that it can be implemented by any spreadsheet user. We also provide results how to apply the HW filter in practice.

Findings

In addition to showing that the one-sided HP filter can be very closely approximated by the HW filter, we highlight three further results. First, we derive a mapping between the smoothing parameter of the HP filter (typically called lambda) and the key parameters of the HW filter. Second, we provide an exact formula for the weights of the HW filter. Third, we discuss different ways to set the initial values for the HW filter.

We illustrate our approach by generating credit-to-GDP gaps using the different filters. We find that the HW filter closely approximates trends generated by the one-sided HP filter, independent of the starting point chosen for the HW filter.

Abstract

We show that the trend of the one-sided HP filter can be asymptotically approx-imated by the Holt-Winters (HW) filter. The later is an elegant, moving average representation and facilitates the computation of trends tremendously. We confirm the accuracy of this approximation empirically by comparing the one-sided HP filter with the HW filter for generating credit-to-GDP gaps. We find negligible differences, most of them concentrated at the beginning of the sample.