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Linear Trend, HP Trend, and bHP Trend

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  • Hiroshi Yamada

    (Graduate School of Humanities and Social Sciences, Hiroshima University, 1-2-1 Kagamiyama, Higashihiroshima 739-8525, Japan)

Abstract

The modelling of the trend component of economic time series has a long history, and the most primitive and popular model displays the trend as a linear function of time. However, the residuals of such a linear trend frequently exhibit long-period fluctuations. The Hodrick–Prescott (HP) filter is able to capture such long-period fluctuations well, resulting in a very realistic trend-cycle decomposition. It may be queried whether the HP trend residuals no longer contain useful long-period fluctuations. If such long-period fluctuations are present, then taking them into consideration could improve the HP trend. In a recent article, a new approach to address this issue, the boosted HP (bHP) filter, was proposed. The three trends mentioned above, i.e., the linear trend, the HP trend, and the bHP trend, can be treated in a unified manner. In this paper, we demonstrate the relationship in detail. We show how the bHP trend is constructed from the linear/HP trend, and long-period fluctuations remained in their trend residuals.

Suggested Citation

  • Hiroshi Yamada, 2025. "Linear Trend, HP Trend, and bHP Trend," Mathematics, MDPI, vol. 13(11), pages 1-15, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:11:p:1893-:d:1672652
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    References listed on IDEAS

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    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.
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