Business Cycles, Trend Elimination, and the HP Filter

56 Pages Posted: 25 Jun 2015

See all articles by Peter C. B. Phillips

Peter C. B. Phillips

University of Auckland Business School; Yale University - Cowles Foundation; Singapore Management University - School of Economics

Sainan Jin

Singapore Management University

Date Written: June 24, 2015

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

JEL Classification: C32

Suggested Citation

Phillips, Peter C. B. and Jin, Sainan, Business Cycles, Trend Elimination, and the HP Filter (June 24, 2015). Cowles Foundation Discussion Paper No. 2005, Available at SSRN: https://ssrn.com/abstract=2622477 or http://dx.doi.org/10.2139/ssrn.2622477

Peter C. B. Phillips (Contact Author)

University of Auckland Business School ( email )

12 Grafton Rd
Private Bag 92019
Auckland, 1010
New Zealand
+64 9 373 7599 x7596 (Phone)

Yale University - Cowles Foundation ( email )

Box 208281
New Haven, CT 06520-8281
United States
203-432-3695 (Phone)
203-432-5429 (Fax)

Singapore Management University - School of Economics

90 Stamford Road
178903
Singapore

Sainan Jin

Singapore Management University ( email )

Li Ka Shing Library
70 Stamford Road
Singapore 178901, 178899
Singapore

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