On Adjusting the One-Sided Hodrick–Prescott Filter

Abstract

We show that one should not use the one-sided Hodrick–Prescott (HP-1s) filter as the real-time version of the two-sided HP (HP-2s) filter: First, in terms of the extracted cyclical component, HP-1s fails to remove low-frequency fluctuations to the same extent as HP-2s. Second, HP-1s dampens fluctuations at all frequencies—even those it is meant to extract. As a remedy, we propose two small adjustments to HP-1s, aligning its properties closely with those of HP-2s: (i) a lower value for the smoothing parameter and (ii) a multiplicative rescaling of the extracted cyclical component. For example, for HP-2s with $\lambda =1,600$ (value of smoothing parameter), the adjusted one-sided HP filter uses ${\lambda }^{\ast }=650$ and rescales the extracted cyclical component by a factor of 1.1513. Using simulated and empirical data, we illustrate the relevance of these adjustments. For instance, financial cycles may appear to be 72% more volatile than business cycles, where, in fact, volatilities differ only marginally.