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 (λ). For instance, when λ = 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 → ∞. 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 λ = o ( n ), the ﬁlter reproduces the limit Brownian motion and eliminates the stochastic trend giving a zero ‘cyclical’ process. Simulations reveal that the λ = 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.
Phillips, Peter C.B. and Jin, Sainan, "Business Cycles, Trend Elimination, and the HP Filter" (2015). Cowles Foundation Discussion Papers. 2440.