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

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.