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A PROPERTY OF THE HODRICK–PRESCOTT FILTER AND ITS APPLICATION

Published online by Cambridge University Press:  23 March 2020

Neslihan Sakarya  and
Robert M. de Jong
Neslihan Sakarya*
Affiliation:
University of Essex
Robert M. de Jong
Affiliation:
Ohio State University
*
Address correspondence to Neslihan Sakarya, Department of Economics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK; ns17003@essex.ac.uk.
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Abstract

This article explores a simple property of the Hodrick–Prescott (HP) filter: when the HP filter is applied to a series, the cyclical component is equal to the HP-filtered trend of the fourth difference of the series, except for the first and last two observations, for which different formulas are needed. We use this result to derive small sample results and asymptotic results for a fixed smoothing parameter. We first apply this property to analyze the consequences of a deterministic break. We find that the effect of a deterministic break on the cyclical component is asymptotically negligible for the points that are away from the break point, while for the points in the neighborhood of the break point, the effect is not negligible even asymptotically. Second, we apply this property to show that the cyclical component of the HP filter when applied to series that are integrated up to order 2 is weakly dependent, while the situation for series that are integrated up to order 3 or 4 is more subtle. Third, we characterize the behavior of the HP filter when applied to deterministic polynomial trends and show that in the middle of the sample, the cyclical component reduces the order of the polynomial by 4, while the end point behavior is different. Finally, we give a characterization of the HP filter when applied to an exponential deterministic trend, and this characterization shows that the filter is effectively incapable of dealing with a trend that increases this fast. Our results are compared with those of Phillips and Jin (2015, Business cycles, trend elimination, and the HP filter).

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 5 , October 2020 , pp. 840 - 870
Copyright
© Cambridge University Press 2020

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