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The spectral analysis of the Hodrick–Prescott filter

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  • Neslihan Sakarya
  • Robert M. de Jong

Abstract

The Hodrick–Prescott (HP) filter is a commonly used tool in macroeconomics to obtain the HP filter trend of a macroeconomic variable. In macroeconomics, the difference between the original series and this trend is called the ‘cyclical component’. In this article, we derive the autocovariance function and the spectrum of the cyclical component of a series that consists of a constant, a linear time trend, a unit root process, and a weakly stationary process. We show that the autocovariance function of the cyclical component of such a series depends on (i) the autocovariance of the innovations of the unit root process; (ii) the autocovariance of the weakly stationary process and; (iii) a component of the weights of the HP filter that is important in the middle of a large sample. The result for the spectrum of the cyclical component matches with earlier results in the literature that were obtained by using an approximate approach. Lastly, we derive the cross‐covariance function and the cross‐spectrum of the cyclical components of two cointegrated series.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jtsera:v:43:y:2022:i:3:p:479-489
    DOI: 10.1111/jtsa.12622
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    References listed on IDEAS

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    1. 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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    7. Regina Kaiser & Agustín Maravall, 1999. "Estimation of the business cycle: A modified Hodrick-Prescott filter," Spanish Economic Review, Springer;Spanish Economic Association, vol. 1(2), pages 175-206.
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    9. 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.
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