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Business Cycles, Trend Elimination, and the HP Filter

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Abstract

We analyze trend elimination methods and business cycle estimation by data filtering of the type introduced by Whittaker (1923) and popularized in economics in a particular form by Hodrick and Prescott (1980/1997; HP). A limit theory is developed for the HP filter for various classes of stochastic trend, trend break, and trend stationary data. Properties of the filtered series are shown to depend closely on the choice of the smoothing parameter (lambda). For instance, when lambda = O(n^4) where n is the sample size, and the HP filter is applied to an I(1) process, the filter does not remove the stochastic trend in the limit as n approaches infinity. Instead, the filter produces a smoothed Gaussian limit process that is differentiable to the 4'th order. The residual 'cyclical' process has the random wandering non-differentiable characteristics of Brownian motion, thereby explaining the frequently observed 'spurious cycle' effect of the HP filter. On the other hand, when lambda = o(n), the filter reproduces the limit Brownian motion and eliminates the stochastic trend giving a zero 'cyclical' process. Simulations reveal that the lambda = O(n^4) limit theory provides a good approximation to the actual HP filter for sample sizes common in practical work. When it is used as a trend removal device, the HP filter therefore typically fails to eliminate stochastic trends, contrary to what is now standard belief in applied macroeconomics. The findings are related to recent public debates about the long run effects of the global financial crisis.

Suggested Citation

  • Peter C. B. Phillips & Sainan Jin, 2015. "Business Cycles, Trend Elimination, and the HP Filter," Cowles Foundation Discussion Papers 2005, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:2005
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    References listed on IDEAS

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    16. 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.
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    26. 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.
    27. 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.
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    More about this item

    Keywords

    Detrending; Graduation; Hodrick Prescott filter; Integrated process; Limit theory; Smoothing; Trend break; Whittaker filter;
    All these keywords.

    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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