Business Cycles, Trend Elimination, and the HP Filter

Peter Phillips and Sainan Jin

No 2005, Cowles Foundation Discussion Papers from Cowles Foundation for Research in Economics, Yale University

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.

Keywords: Detrending; Graduation; Hodrick Prescott filter; Integrated process; Limit theory; Smoothing; Trend break; Whittaker filter (search for similar items in EconPapers)
JEL-codes: C32 (search for similar items in EconPapers)
Pages: 55 pages
Date: 2015-06
New Economics Papers: this item is included in nep-ecm, nep-ets, nep-mac and nep-sea
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Citations: View citations in EconPapers (37)

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Journal Article: BUSINESS CYCLES, TREND ELIMINATION, AND THE HP FILTER (2021)
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