Why does the trend extracted by the Hodrick–Prescott filtering seem to be more plausible than the linear trend?

ABSTRACT

Why does the trend extracted by the Hodrick–Prescott (HP) filtering (HP trend) seem to be more plausible than the linear trend estimated by OLS? This article provides an answer for it. Because the HP filtering is a basic econometric tool, it is necessary to have a precise understanding of the nature of it. This article concludes that the HP trend is calculated by adding the low-frequency component (the long-term periodic fluctuation) of the linearly detrended series to the linear trend, which leads to that the HP trend seems to be more plausible than the linear trend. Other than this key result, this article shows that the HP cycle, which is defined as the residuals of the HP filtering, can be interpreted as the high-frequency component (the short-term periodic fluctuation) of the linearly detrended series. An empirical illustration is also provided.

KEYWORDS:

• Hodrick–Prescott filtering
• linear trend
• low-frequency component
• lowpass filter
• Whittaker–Henderson method of graduation

JEL CLASSIFICATION:

• C22
• E32

Acknowledgements

I thank Robert King for valuable comments on an earlier version of this article. I used the data set downloaded from Pierre Perron’s website, for which I thank Pierre Perron and Tatsuma Wada. The usual caveat applies.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ See, for example, Weinert (2007).

² Yamada (2015) provides a penalized least squares (PLS) problem that estimates in (3).

³ When , is equal to 0.15828, which indicates that the corresponding periodicity, , is approximately equal to .

⁴ Yamada (2017) provides a PLS problem that estimates in (6).

⁵ The generalized filter is referred to as the Whittaker–Henderson graduation in the actuarial sciences.

Additional information

Funding

This work was supported by JSPS KAKENHI Grant Numbers 15K13010.