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Boosting the HP Filter for Trending Time Series with Long Range Dependence

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This paper extends recent asymptotic theory developed for the Hodrick Prescott (HP) filter and boosted HP (bHP) filter to long range dependent time series that have fractional Brownian motion (fBM) limit processes after suitable standardization. Under general conditions it is shown that the asymptotic form of the HP filter is a smooth curve, analogous to the finding in Phillips and Jin (2021) for integrated time series and series with deterministic drifts. Boosting the filter using the iterative procedure suggested in Phillips and Shi (2021) leads under well defined rate conditions to a consistent estimate of the fBM limit process or the fBM limit process with an accompanying deterministic drift when that is present. A stopping criterion is used to automate the boosting algorithm, giving a data-determined method for practical implementation. The theory is illustrated in simulations and two real data examples that highlight the differences between simple HP filtering and the use of boosting. The analysis is assisted by employing a uniformly and almost surely convergent trigonometric series representation of fBM.

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  • Eva Biswas & Farzad Sabzikar & Peter C. B. Phillips, 2022. "Boosting the HP Filter for Trending Time Series with Long Range Dependence," Cowles Foundation Discussion Papers 2347, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:2347
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    File URL: https://cowles.yale.edu/sites/default/files/2022-12/d2347.pdf
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    1. Hodrick, Robert J & Prescott, Edward C, 1997. "Postwar U.S. Business Cycles: An Empirical Investigation," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 29(1), pages 1-16, February.
    2. Tucker McElroy, 2008. "Exact formulas for the Hodrick-Prescott filter," Econometrics Journal, Royal Economic Society, vol. 11(1), pages 209-217, March.
    3. Davidson, James & Hashimzade, Nigar, 2009. "Type I and type II fractional Brownian motions: A reconsideration," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2089-2106, April.
    4. James D. Hamilton, 2018. "Why You Should Never Use the Hodrick-Prescott Filter," The Review of Economics and Statistics, MIT Press, vol. 100(5), pages 831-843, December.
    5. Szabados, Tamás, 2001. "Strong approximation of fractional Brownian motion by moving averages of simple random walks," Stochastic Processes and their Applications, Elsevier, vol. 92(1), pages 31-60, March.
    6. 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.
    7. Qiying Wang & Yan-Xia Lin & Chandra M. Gulati, 2003. "Strong Approximation for Long Memory Processes with Applications," Journal of Theoretical Probability, Springer, vol. 16(2), pages 377-389, April.
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