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Business Cycles, Trend Elimination, And The Hp Filter

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  • Peter C. B. Phillips
  • Sainan Jin

Abstract

Trend elimination and business cycle estimation are analyzed by finite sample and asymptotic methods. An overview history is provided, operator theory is developed, limit theory as the sample size n→∞ is derived, and filtered series properties are studied relative to smoothing parameter (λ) behavior. Simulations reveal that limit theory with λ=O(n4) delivers excellent approximations to the HP filter for common sample sizes but fails to remove stochastic trends, contrary to standard thinking in macroeconomics and thereby explaining “spurious cycle” effects of the HP filter. The findings are related to the long run effects of the global financial crisis.

Suggested Citation

  • 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.
    • Handle: RePEc:wly:iecrev:v:62:y:2021:i:2:p:469-520
    DOI: 10.1111/iere.12494
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    References listed on IDEAS

    as
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    2. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2014. "Specification Sensitivity in Right-Tailed Unit Root Testing for Explosive Behaviour," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 76(3), pages 315-333, June.
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    5. Blix, Mårten, 1999. "Forecasting Swedish Inflation With a Markov Switching VAR," Working Paper Series 76, Sveriges Riksbank (Central Bank of Sweden).
    6. Frederick R. Macaulay, 1931. "The Smoothing of Time Series," NBER Books, National Bureau of Economic Research, Inc, number maca31-1, March.
    7. Peter C.B. Phillips, 1987. "Multiple Regression with Integrated Time Series," Cowles Foundation Discussion Papers 852, Cowles Foundation for Research in Economics, Yale University.
    8. Peter C. B. Phillips & Hyungsik R. Moon, 1999. "Linear Regression Limit Theory for Nonstationary Panel Data," Econometrica, Econometric Society, vol. 67(5), pages 1057-1112, September.
    9. 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.
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    13. Stock, James H. & Watson, Mark W., 1999. "Forecasting inflation," Journal of Monetary Economics, Elsevier, vol. 44(2), pages 293-335, October.
    14. Peter C. B. Phillips, 2003. "Laws and Limits of Econometrics," Economic Journal, Royal Economic Society, vol. 113(486), pages 26-52, March.
    15. Adriana Cornea-Madeira, 2017. "The Explicit Formula for the Hodrick-Prescott Filter in a Finite Sample," The Review of Economics and Statistics, MIT Press, vol. 99(2), pages 314-318, May.
    16. White Halbert & Granger Clive W.J., 2011. "Consideration of Trends in Time Series," Journal of Time Series Econometrics, De Gruyter, vol. 3(1), pages 1-40, February.
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    18. Shiller, Robert J, 1973. "A Distributed Lag Estimator Derived from Smoothness Priors," Econometrica, Econometric Society, vol. 41(4), pages 775-788, July.
    19. 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.
    20. Peter C. B. Phillips, 1998. "New Tools for Understanding Spurious Regressions," Econometrica, Econometric Society, vol. 66(6), pages 1299-1326, November.
    21. Phillips, Peter C.B., 2014. "Optimal estimation of cointegrated systems with irrelevant instruments," Journal of Econometrics, Elsevier, vol. 178(P2), pages 210-224.
    22. 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.
    23. 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.
    24. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    25. Cogley, Timothy & Nason, James M., 1995. "Effects of the Hodrick-Prescott filter on trend and difference stationary time series Implications for business cycle research," Journal of Economic Dynamics and Control, Elsevier, vol. 19(1-2), pages 253-278.
    26. Morten O. Ravn & Harald Uhlig, 2002. "On adjusting the Hodrick-Prescott filter for the frequency of observations," The Review of Economics and Statistics, MIT Press, vol. 84(2), pages 371-375.
    27. Peter C. B. Phillips & Shu-Ping Shi & Jun Yu, 2011. "Specification Sensitivity in Right-Tailed Unit Root Testing for Explosive Behavior," Working Papers 15-2011, Singapore Management University, School of Economics.
    28. Osborn, Denise R, 1995. "Moving Average Detrending and the Analysis of Business Cycles," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 57(4), pages 547-558, November.
    29. Sun, Yixiao, 2014. "Let’s fix it: Fixed-b asymptotics versus small-b asymptotics in heteroskedasticity and autocorrelation robust inference," Journal of Econometrics, Elsevier, vol. 178(P3), pages 659-677.
    30. Phillips, Peter C.B., 2007. "Unit root log periodogram regression," Journal of Econometrics, Elsevier, vol. 138(1), pages 104-124, May.
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    JEL classification:

    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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