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A Smoothing Method That Looks Like The Hodrick–Prescott Filter

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  • Yamada, Hiroshi

Abstract

In recent decades, in the research community of macroeconometric time series analysis, we have observed growing interest in the smoothing method known as the Hodrick–Prescott (HP) filter. This article examines the properties of an alternative smoothing method that looks like the HP filter, but is much less well known. We show that this is actually more like the exponential smoothing filter than the HP filter although it is obtainable through a slight modification of the HP filter. In addition, we also show that it is also like the low-frequency projection of Müller and Watson (2018, Econometrica 86, 775–804). We point out that these results derive from the fact that all three similar smoothing methods can be regarded as a type of graph spectral filter whose graph Fourier transform is discrete cosine transform. We then theoretically reveal the relationship between the similar smoothing methods and provide a way of specifying the smoothing parameter that is necessary for its application. An empirical examination illustrates the results.

Suggested Citation

  • Yamada, Hiroshi, 2020. "A Smoothing Method That Looks Like The Hodrick–Prescott Filter," Econometric Theory, Cambridge University Press, vol. 36(5), pages 961-981, October.
    • Handle: RePEc:cup:etheor:v:36:y:2020:i:5:p:961-981_6
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    Cited by:

    1. Sheng Zhang & Yifu Yang & Chengdi Ding & Zhongquan Miao, 2023. "The Impact of International Relations Patterns on China’s Energy Security Supply, Demand, and Sustainable Development: An Exploration of Oil Demand and Sustainability Goals," Sustainability, MDPI, vol. 15(17), pages 1-12, August.
    2. 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.
    3. Kurt Graden Lunsford, 2023. "Business Cycles and Low-Frequency Fluctuations in the US Unemployment Rate," Working Papers 23-19, Federal Reserve Bank of Cleveland.
    4. 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.
    5. Ruixue Du & Hiroshi Yamada, 2020. "Principle of Duality in Cubic Smoothing Spline," Mathematics, MDPI, vol. 8(10), pages 1-19, October.
    6. Hiroshi Yamada, 2021. "Geary’s c and Spectral Graph Theory," Mathematics, MDPI, vol. 9(19), pages 1-23, October.
    7. Hiroshi Yamada, 2023. "Quantile regression version of Hodrick–Prescott filter," Empirical Economics, Springer, vol. 64(4), pages 1631-1645, April.
    8. 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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