Friday, August 7, 2015

The H-P Filter and Unit Roots

The Hodrick-Prescott (H-P) filter is widely used for trend removal in economic time-series, and as a basis for business cycle analysis, etc. I've posted about the H-P filter before (e.g., here).

There's a widespread belief that application of the H-P filter will not only isolate the deterministic trend in a series, but it will also remove stochastic trends - i.e., unit roots. For instance, you'll often hear that if the H-P filter is applied to quarterly data, the filtered series will be stationary, even if the original series is integrated of order up to 4.

Is this really the case?

Let's take a look at two classic papers relating to this topic, and a very recent one that provides a bit of an upset.

King and Rebelo (1993) discuss the H-P filter in some detail, from the perspectives of both the time domain and the frequency domain. Because the filter involves a centered fourth-difference, they argue that for large sample sizes this ".......renders stationary time series that are 'difference-stationary' and, indeed, integrated of higher order."

Cogley and Nason (1995) show that when the H-P filter is applied to a series that is I(1), it operates like a two-step linear filter - the first step being first-differencing, and the second being an asymmetric moving-average filter. Importantly, Cogley and Nason showec that when the H-P filter is applied to an integrated time-series, it can generate business cycle characteristics even when none are present in the original data.

Now, fast-forward to June of this year.

Phillips and Jin (2015) have recently developed the asymptotic distribution theory for the H-P filter when it is applied to a variety of different types of time-series data. Among other things, they show that their results are also applicable in the case of sample sizes that are typical of those used in empirical macroeconomics.

One implication is that when the H-P filter is used to remove deterministic trends, it doesn't remove stochastic trends (unit roots)! This runs contrary to the accepted wisdom, and provides a formal, mathematical, explanation for the folklore (and the evidence provided by Cogley and Nason) that the H-P filter can generate "spurious cycles" in the filtered data.

As Phillips and Jin note, their results are important to the debate about the long-run effects of the global financial crisis.


Cogley, T. and J. M. Nason, 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, 19, 253-278.

King, R. G. and S. T. Rebelo, 1993. Low-frequency filtering and real business cycles. Journal of Economic Dynamics and Control, 17, 207-231.

Phillips, P. C. B. and S. Jin, 2015. Business cycles, trend elimination, and the HP filter. Cowles Discussion Paper No. 2005, Yale University.

© 2015, David E. Giles


  1. Hello, Prof.Giles. My name is Bow. Now I'm graduate student in Thailand. My thesis is about natural interest rate or equilibrium real interest rate in so I used HP filter with short interest rate data to make natural interest rate proxy.The trend from data means Natural interest rate but after I used HP filter with raw data and then I take the unit root test with the data (after HP filter) why EViews4 responds with error message" Near singular metrix". My addition, I use monthly data from June 2000 to December 2014 (about 175observations), choose 24 lags in the unit root test process. If you know why message "Near singular metrix" please tell me because it's vary useful for my thesis. I'm appreciate to hearing the answer from you. Thank you.

    1. Don't use 24 lags - let EViews select the optimal number of lags.


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