Several least-squares problems related to the Hodrick–Prescott filtering

ABSTRACT

The Hodrick–Prescott (HP) filtering is frequently used in macroeconometrics to decompose time series, such as real gross domestic product, into their trend and cyclical components. Because the HP filtering is a basic econometric tool, it is necessary to have a precise understanding of the nature of it. This article contributes to the literature by listing several (penalized) least-squares problems that are related to the HP filtering, three of which are newly introduced in the article, and showing their properties. We also remark on their generalization.

KEYWORDS:

• Hodrick–Prescott filtering
• Penalized least squares
• Whittaker–Henderson method of graduation

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G05

JEL CLASSIFICATION:

• C22

Notes

1 For example, (i) the Organisation for Economic Co-operation and Development (OECD) began to use the HP filtering to calculate the composite leading indicators (CLIs) in December 2008 (OECD 2012) and (ii) Ball and Mankiw (2002) used it to estimate time-varying non accelerating inflation rate of unemployment (NAIRU).

2 See Weinert (2007) for a historical review of this method. See also Phillips (2010).

3 The transformation from (2(2) ) to (3(3) ) may be regarded as a special case of that of Yanagihara (2012).

4 (6(6) ) is referred to as the pure HP filtering in Yamada (2015), which is a companion paper to this article.

5 Of course, (19(19) ) is also obtainable by applying the matrix inversion lemma to (I_n + λD′D)^{− 1}.

Additional information

Funding

The author thanks Hirokazu Yanagihara and an anonymous referee for their valuable comments. The usual caveat applies. This work was partly supported by JSPS KAKENHI Grant Number 15K13010.