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Some Results on ℓ 1 Polynomial Trend Filtering

Author

Listed:
  • Hiroshi Yamada

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

  • Ruixue Du

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

Abstract

ℓ 1 polynomial trend filtering, which is a filtering method described as an ℓ 1 -norm penalized least-squares problem, is promising because it enables the estimation of a piecewise polynomial trend in a univariate economic time series without prespecifying the number and location of knots. This paper shows some theoretical results on the filtering, one of which is that a small modification of the filtering provides not only identical trend estimates as the filtering but also extrapolations of the trend beyond both sample limits.

Suggested Citation

  • Hiroshi Yamada & Ruixue Du, 2018. "Some Results on ℓ 1 Polynomial Trend Filtering," Econometrics, MDPI, vol. 6(3), pages 1-10, July.
    • Handle: RePEc:gam:jecnmx:v:6:y:2018:i:3:p:33-:d:157210
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    References listed on IDEAS

    as
    1. Mohr, Matthias, 2005. "A trend-cycle(-season) filter," Working Paper Series 499, European Central Bank.
    2. 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.
    3. Peter Phillips, 2010. "Two New Zealand pioneer econometricians," New Zealand Economic Papers, Taylor & Francis Journals, vol. 44(1), pages 1-26.
    4. Yamada Hiroshi, 2018. "A New Method for Specifying the Tuning Parameter of ℓ1 Trend Filtering," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 22(4), pages 1-8, September.
    5. Matthias Mohr, 2005. "A Trend-Cycle(-Season) Filter," Econometrics 0508004, University Library of Munich, Germany.
    6. Winkelried, Diego, 2016. "Piecewise linear trends and cycles in primary commodity prices," Journal of International Money and Finance, Elsevier, vol. 64(C), pages 196-213.
    7. Yamada, Hiroshi & Yoon, Gawon, 2014. "When Grilli and Yang meet Prebisch and Singer: Piecewise linear trends in primary commodity prices," Journal of International Money and Finance, Elsevier, vol. 42(C), pages 193-207.
    8. Harchaoui, Z. & Lévy-Leduc, C., 2010. "Multiple Change-Point Estimation With a Total Variation Penalty," Journal of the American Statistical Association, American Statistical Association, vol. 105(492), pages 1480-1493.
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    10. Yamada Hiroshi & Yoon Gawon, 2016. "Selecting the tuning parameter of the ℓ1 trend filter," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 20(1), pages 97-105, February.
    11. Yamada Hiroshi, 2018. "A New Method for Specifying the Tuning Parameter of ℓ1 Trend Filtering," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 22(4), pages 1-8, September.
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