Linear trend, HP trend, and bHP trend: a further unified perspective

Estimating the trend component of an economic time series is equivalent to estimating its cycle component. This is because the cycle component can be estimated as trend residuals. Consequently, considerable attention has been devoted to estimating the trend component. Modeling trend as a linear function of time is the most primitive but still widely used trend estimation method. The Hodrick–Prescott (HP) filter is one of the most common methods for trend estimation. The HP filter provides a more flexible time trend than a linear trend. In 2021, there was a new development in the popular method. Phillips and Shi (2021) developed a new filter by applying L₂ L 2 -boosting to the HP filter and they called it the boosted HP (bHP) filter. Interest in this new trend estimation method has grown since its development. Recently, Yamada (2025a) showed that the three trends mentioned above, i.e., the linear trend, the HP trend, and the bHP trend, can be understood in a unified manner. This paper advances research in this direction. In this paper, we provide a further unified perspective on the three trends. We also present several results based on the perspective.