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A Property Of The Hodrick–Prescott Filter And Its Application

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  • Sakarya, Neslihan
  • de Jong, Robert M.

Abstract

This article explores a simple property of the Hodrick–Prescott (HP) filter: when the HP filter is applied to a series, the cyclical component is equal to the HP-filtered trend of the fourth difference of the series, except for the first and last two observations, for which different formulas are needed. We use this result to derive small sample results and asymptotic results for a fixed smoothing parameter. We first apply this property to analyze the consequences of a deterministic break. We find that the effect of a deterministic break on the cyclical component is asymptotically negligible for the points that are away from the break point, while for the points in the neighborhood of the break point, the effect is not negligible even asymptotically. Second, we apply this property to show that the cyclical component of the HP filter when applied to series that are integrated up to order 2 is weakly dependent, while the situation for series that are integrated up to order 3 or 4 is more subtle. Third, we characterize the behavior of the HP filter when applied to deterministic polynomial trends and show that in the middle of the sample, the cyclical component reduces the order of the polynomial by 4, while the end point behavior is different. Finally, we give a characterization of the HP filter when applied to an exponential deterministic trend, and this characterization shows that the filter is effectively incapable of dealing with a trend that increases this fast. Our results are compared with those of Phillips and Jin (2015, Business cycles, trend elimination, and the HP filter).

Suggested Citation

  • Sakarya, Neslihan & de Jong, Robert M., 2020. "A Property Of The Hodrick–Prescott Filter And Its Application," Econometric Theory, Cambridge University Press, vol. 36(5), pages 840-870, October.
    • Handle: RePEc:cup:etheor:v:36:y:2020:i:5:p:840-870_3
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    Citations

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    Cited by:

    1. Ye Lu & Adrian Pagan, 2023. "To Boost or Not to Boost? That is the Question," CAMA Working Papers 2023-12, Centre for Applied Macroeconomic Analysis, Crawford School of Public Policy, The Australian National University.
    2. Peter C. B. Phillips & Zhentao Shi, 2021. "Boosting: Why You Can Use The Hp Filter," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 62(2), pages 521-570, May.
    3. Peter C. B. Phillips & Sainan Jin, 2021. "Business Cycles, Trend Elimination, And The Hp Filter," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 62(2), pages 469-520, May.
    4. Germaschewski, Yin & Horvath, Jaroslav & Rubini, Loris, 2021. "Property rights, expropriations, and business cycles in China," Journal of Economic Dynamics and Control, Elsevier, vol. 125(C).
    5. Germaschewski, Yin & Wang, Shu-Ling, 2022. "Fiscal stabilization in high-debt economies without monetary independence," Journal of Macroeconomics, Elsevier, vol. 72(C).
    6. Neslihan Sakarya & Robert M. de Jong, 2022. "The spectral analysis of the Hodrick–Prescott filter," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(3), pages 479-489, May.
    7. Nadav Ben Zeev, 2019. "Asymmetric Business Cycles In Emerging Market Economies," Working Papers 1909, Ben-Gurion University of the Negev, Department of Economics.
    8. Ziwei Mei & Peter C. B. Phillips & Zhentao Shi, 2022. "The boosted HP filter is more general than you might think," Papers 2209.09810, arXiv.org, revised Apr 2024.
    9. Yin Germaschewski, 2023. "House price volatility in China: Demand versus supply," Economic Inquiry, Western Economic Association International, vol. 61(1), pages 199-220, January.
    10. Hiroshi Yamada, 2023. "Quantile regression version of Hodrick–Prescott filter," Empirical Economics, Springer, vol. 64(4), pages 1631-1645, April.
    11. Hiroshi Yamada & Ruoyi Bao, 2022. "$$\ell _{1}$$ ℓ 1 Common Trend Filtering," Computational Economics, Springer;Society for Computational Economics, vol. 59(3), pages 1005-1025, March.

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