IDEAS home Printed from
   My bibliography  Save this paper

Filtering permanent cycles with complex unit roots


  • Donald S. Allen


Separating cyclical movement from trend growth at seasonal and business cycle frequencies is important to macroeconomic research. At business cycle frequencies, time trends, first differences and the more recent Hodrick-Prescott (HP) filter are used to separate trends from cycles. At seasonal frequencies, ad-hoc methods like the Census Bureau's X-11 seasonal filter are applied. This paper reviews the criteria for permanent cycles in systems characterized by difference equations and looks at the effect of filtering data which exhibit permanent cyclicality. Second order moving averages with complex unit roots at appropriate frequencies are used to filter data at seasonal and business cycle frequencies; and spectral analysis of the filtered data is used to illustrate the effect. The X-11 seasonal filter and the HP filter are also discussed in this framework. As with any filter that is applied to data where the data generating process is unknown, filtering for specific frequencies can induce cycles at harmonics of the fundamental frequency.

Suggested Citation

  • Donald S. Allen, 1997. "Filtering permanent cycles with complex unit roots," Working Papers 1997-001, Federal Reserve Bank of St. Louis.
  • Handle: RePEc:fip:fedlwp:1997-001

    Download full text from publisher

    File URL:
    Download Restriction: no

    File URL:
    Download Restriction: no

    References listed on IDEAS

    1. Ghysels, Eric & Perron, Pierre, 1993. "The effect of seasonal adjustment filters on tests for a unit root," Journal of Econometrics, Elsevier, vol. 55(1-2), pages 57-98.
    2. Harvey, A C & Jaeger, A, 1993. "Detrending, Stylized Facts and the Business Cycle," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 8(3), pages 231-247, July-Sept.
    3. Cogley, Timothy & Nason, James M., 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, Elsevier, vol. 19(1-2), pages 253-278.
    Full references (including those not matched with items on IDEAS)


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:fip:fedlwp:1997-001. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Kathy Cosgrove). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.