Linear Trend, HP Trend, and bHP Trend

Abstract

The modelling of the trend component of economic time series has a long history, and the most primitive and popular model displays the trend as a linear function of time. However, the residuals of such a linear trend frequently exhibit long-period fluctuations. The Hodrick–Prescott (HP) filter is able to capture such long-period fluctuations well, resulting in a very realistic trend-cycle decomposition. It may be queried whether the HP trend residuals no longer contain useful long-period fluctuations. If such long-period fluctuations are present, then taking them into consideration could improve the HP trend. In a recent article, a new approach to address this issue, the boosted HP (bHP) filter, was proposed. The three trends mentioned above, i.e., the linear trend, the HP trend, and the bHP trend, can be treated in a unified manner. In this paper, we demonstrate the relationship in detail. We show how the bHP trend is constructed from the linear/HP trend, and long-period fluctuations remained in their trend residuals.

1. Introduction

The modelling of the trend component of economic time series has a long history (Mills, 2003 [1]). The most primitive and popular model displays the trend as a linear function of time. However, the residuals of such a linear trend frequently exhibit long-period fluctuations (see, for example, King and Rebelo (1993, Figure 1) [2] and Yamada (2018, Figure 2) [3]). The Hodrick–Prescott (HP) filter (Hodrick and Prescott, 1997 [4]) is able to capture such long-period fluctuations well, resulting in a very realistic trend-cycle decomposition. This is probably why the HP trend is often used instead of the linear trend. It removes a smooth trend as one would draw it with a free hand (Pedersen, 2001 [5]). For a historical review of the HP filter, see Weinert (2007) [6] and Phillips and Jin (2021) [7].

It may be queried whether the HP trend residuals no longer contain useful long-period fluctuations. If such long-period fluctuations are present, then taking them into consideration could improve the HP trend. In a recent article, Phillips and Shi (2021) [8] proposed a new approach to address this issue. They applied the $L_2$-boosting (Bühlmann, 2006 [9]; Tutz and Binder, 2007 [10]), a popular technique in machine learning. They called it the boosted HP (bHP) filter. For $I\left(1\right)$ unit root nonstationary time series, the HP filter cannot properly estimate the trend (Phillips and Jin, 2021, Theorem 3 [7]). However, the bHP filter is capable of doing so (Phillips and Shi, 2021, Theorem 1 [8]), making it a highly promising alternative.

Several studies concerning the bHP filter have since emerged: (i) Knight (2021) [11] established the properties of the bHP filter by deriving the penalized least-squares problems corresponding to the bHP filter; (ii) Tomal (2022) [12], Trojanek et al. (2023) [13], Xu (2023) [14], and Andrián et al. (2024) [15] conducted empirical studies using the bHP filter; (iii) Hall and Thomson (2024) [16] provided a way to use the bHP filter as a frequency-selective filter; (iv) Mei et al. (2024) [17] and Biswas et al. (2024) [18] provided further results that demonstrate the usefulness of the bHP filter; (v) Yamada (2024) [19] established the properties of the bHP filter following Knight (2021) [11]; and (vi) Jin and Yamada (2024) [20] and Bao and Yamada (2025) [21] studied the boosted version of the Whittaker–Henderson graduation (Weinert, 2007 [6]).

The three trends mentioned above, (i.e., the linear trend, the HP trend, and the bHP trend) can be treated in a unified manner. In this paper, we demonstrate the relationship in detail. We show how the bHP trend is constructed from the linear/HP trend and long-period fluctuations remained in their trend residuals.

The organization of the paper is as follows: In Section 2, we describe the three trends. In Section 3, we present a unified perspective of them. In Section 4, we add some more details to the perspective. In Section 5, we provide an empirical illustration of the obtained results. Section 6 concludes the paper. In Appendices Appendix A and Appendix B, we provide additional results and several proofs.

2. Preliminaries

In this section, we review the three trends mentioned in Section 1.

Let $y_t$ denote a macroeconomic variable y at time t for $t=1,\dots ,n$ and $\mathit{y}={[y_1,\dots ,y_n]}^{\top }$. Let $\mathsf{\Pi }$ be the $n\times 2$ matrix whose t-th row is $[1,t]$ for $t=1,\dots ,n$. We assume that $\mathit{y}$ does not belong to the column space of $\mathsf{\Pi }$. That is, $\mathit{y}$ cannot be expressed as $\mathsf{\Pi }\mathit{\alpha }$, where $\mathit{\alpha }$ is a two-dimensional column vector. The case where $\mathit{y}$ belongs to the column space of $\mathsf{\Pi }$ is discussed in Appendix A.

• Linear trend:

The linear trend estimated by ordinary least squares, denoted by $\widehat{\mathit{\tau }}$, is

$$\begin{array}{c}\hfill \widehat{\mathit{\tau }}=\mathsf{\Pi }\widehat{\mathit{\beta }},\end{array}$$

(1)

where

$$\begin{array}{cc}\hfill \widehat{\mathit{\beta }}& =arg\underset{{\beta }_1,{\beta }_2\in \mathbb{R}}{min}\sum _{t=1}^n{(y_t-{\beta }_1-{\beta }_{2}t)}^2\hfill \\ \hfill & =arg\underset{\mathit{\beta }\in {\mathbb{R}}^2}{min}{\parallel \mathit{y}-\mathsf{\Pi }\mathit{\beta }\parallel }^2\hfill \\ \hfill & ={\left({\mathsf{\Pi }}^{\top }\mathsf{\Pi }\right)}^{-1}{\mathsf{\Pi }}^{\top }\mathit{y},\hfill \end{array}$$

(2)

where $\mathit{\beta }={[{\beta }_1,{\beta }_2]}^{\top }$. Here, for a column vector $\mathit{g}$, ${\parallel \mathit{g}\parallel }^2$ denotes the squared ${\mathsf{\ell }}_2$-norm of $\mathit{g}$, and it thus identical to ${\mathit{g}}^{\top }\mathit{g}$.

• HP trend:

The HP trend, denoted by ${\widehat{\mathit{x}}}^{\mathrm{HP}}$, is

$$\begin{array}{cc}\hfill {\widehat{\mathit{x}}}^{\mathrm{HP}}& =arg\underset{x_1,\dots ,x_n\in \mathbb{R}}{min}\left\{\sum _{t=1}^n{(y_t-x_t)}^2+\lambda \sum _{t=3}^n{(x_t-2x_{t-1}+x_{t-2})}^2\right\}\hfill \\ \hfill & =arg\underset{\mathit{x}\in {\mathbb{R}}^n}{min}\left({\parallel \mathit{y}-\mathit{x}\parallel }^2+\lambda {\parallel \mathit{D}\mathit{x}\parallel }^2\right)\hfill \\ \hfill & =\mathit{S}\mathit{y},\hfill \end{array}$$

(3)

where $\lambda \in (0,\infty )$ is a smoothing parameter, $\mathit{x}={[x_1,\dots ,x_n]}^{\top }$, $\mathit{D}$ is the $(n-2)\times n$ matrix such that $\mathit{D}\mathit{x}={[x_3-2x_2+x_1,\dots ,x_n-2x_{n-1}+x_{n-2}]}^{\top }$, and

$$\begin{array}{c}\hfill \mathit{S}={({\mathit{I}}_n+\lambda {\mathit{D}}^{\top }\mathit{D})}^{-1}.\end{array}$$

(4)

Here, ${\mathit{I}}_n$ is the $n\times n$ identity matrix. Since ${\mathit{I}}_n+\lambda {\mathit{D}}^{\top }\mathit{D}$ in (4) is a positive definite matrix (Danthine and Girardin, 1989 [22]), it is nonsingular. Given that

$$\begin{array}{c}\hfill {\mathit{S}}^{\top }={\left\{{({\mathit{I}}_n+\lambda {\mathit{D}}^{\top }\mathit{D})}^{-1}\right\}}^{\top }={\left\{{({\mathit{I}}_n+\lambda {\mathit{D}}^{\top }\mathit{D})}^{\top }\right\}}^{-1}={({\mathit{I}}_n+\lambda {\mathit{D}}^{\top }\mathit{D})}^{-1}=\mathit{S},\end{array}$$

$\mathit{S}$ is a symmetric matrix.

As is widely acknowledged, there is a relationship between ${\widehat{\mathit{x}}}^{\mathrm{HP}}$ and $\widehat{\mathit{\tau }}$, as follows:

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\mathrm{HP}}\to \widehat{\mathit{\tau }},(\lambda \to \infty ).\end{array}$$

(5)

Here, (5) follows by applying the Sherman–Morrison–Woodbury formula to $\mathit{S}$. More specifically, it follows from

$$\begin{array}{c}\hfill \mathit{S}={\mathit{I}}_n-{\mathit{D}}^{\top }{\left({\displaystyle \frac{1}{\lambda }}{\mathit{I}}_{n-1}+\mathit{D}{\mathit{D}}^{\top }\right)}^{-1}\mathit{D}\to {\mathit{I}}_n-{\mathit{D}}^{\top }{\left(\mathit{D}{\mathit{D}}^{\top }\right)}^{-1}\mathit{D}={\mathit{P}}_{\mathsf{\Pi }},(\lambda \to \infty ).\end{array}$$

(6)

where ${\mathit{P}}_{\mathsf{\Pi }}=\mathsf{\Pi }{\left({\mathsf{\Pi }}^{\top }\mathsf{\Pi }\right)}^{-1}{\mathsf{\Pi }}^{\top }$. Here, $\mathit{D}{\mathit{D}}^{\top }$ is an $(n-2)\times (n-2)$ positive definite matrix. Note that the last equality in (6) holds because $[\mathsf{\Pi },{\mathit{D}}^{\top }]$ is a nonsingular matrix such that

$$\begin{array}{c}\hfill {\left({\mathit{D}}^{\top }\right)}^{\top }\mathsf{\Pi }=\mathit{D}\mathsf{\Pi }={\mathbf{0}}_{n-2,2},\end{array}$$

(7)

where ${\mathbf{0}}_{r,s}$ denotes the $r\times s$ matrix of zeros.

• bHP trend:

The bHP trend, denoted by ${\widehat{\mathit{x}}}^{\left(m\right)}$, can be considered a generalization of the HP filter. It is defined using the smoother matrix of the HP filter, $\mathit{S}$ in (4), as follows:

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\left(m\right)}={\mathit{S}}_m\mathit{y},m=1,2,\dots ,\end{array}$$

(8)

where

$$\begin{array}{c}\hfill {\mathit{S}}_m={\mathit{I}}_n-{({\mathit{I}}_n-\mathit{S})}^m.\end{array}$$

(9)

Given that ${\mathit{S}}_1={\mathit{I}}_n-({\mathit{I}}_n-\mathit{S})=\mathit{S}$, it follows that

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\left(1\right)}={\widehat{\mathit{x}}}^{\mathrm{HP}}.\end{array}$$

(10)

Therefore, as stated, the bHP trend can be considered a generalization of the HP trend.

Given that $\mathit{S}$ is a symmetric matrix, it follows that

$$\begin{array}{cc}\hfill {\mathit{S}}_m^{\top }& ={\left\{{\mathit{I}}_n-{({\mathit{I}}_n-\mathit{S})}^m\right\}}^{\top }={\mathit{I}}_n-{\left\{{({\mathit{I}}_n-\mathit{S})}^m\right\}}^{\top }={\mathit{I}}_n-{\left\{{({\mathit{I}}_n-\mathit{S})}^{\top }\right\}}^m\hfill \\ \hfill & ={\mathit{I}}_n-{({\mathit{I}}_n-\mathit{S})}^m={\mathit{S}}_m.\hfill \end{array}$$

(11)

Hence, ${\mathit{S}}_m$ is also a symmetric matrix.

3. A Unified Perspective of the Three Trends

The three trends defined in the previous section, $\widehat{\mathit{\tau }}$, ${\widehat{\mathit{x}}}^{\mathrm{HP}}$, and ${\widehat{\mathit{x}}}^{\left(m\right)}$, can be treated in a unified way. In this section, we show this.

This perspective can be regarded as an extension of the result presented in Kim et al. (2009) [23] and Yamada (2018) [3]. Thus, here, we review the result. Using (7), it follows that ${\mathit{S}}^{-1}\mathsf{\Pi }=({\mathit{I}}_n+\lambda {\mathit{D}}^{\top }\mathit{D})\mathsf{\Pi }=\mathsf{\Pi }$, which leads to

$$\begin{array}{c}\hfill \mathit{S}\mathsf{\Pi }=\mathsf{\Pi }.\end{array}$$

(12)

Therefore, it follows that

$$\begin{array}{c}\hfill \mathit{S}\widehat{\mathit{\tau }}=\mathit{S}{\mathit{P}}_{\mathsf{\Pi }}\mathit{y}={\mathit{P}}_{\mathsf{\Pi }}\mathit{y}=\widehat{\mathit{\tau }},\end{array}$$

(13)

from which we obtain

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\mathrm{HP}}=\widehat{\mathit{\tau }}+\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }}).\end{array}$$

(14)

Given that $\mathit{S}$ is a low-pass filter (Yamada, 2018 [3], Figure 1), $\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }})$ in (14) is a low-frequency component of the linear trend residuals, $\mathit{y}-\widehat{\mathit{\tau }}$. Therefore, ${\widehat{\mathit{x}}}^{\mathrm{HP}}$ is the sum of the linear trend and a low-frequency component of the linear trend residuals.

Let ${\widehat{\mathit{x}}}^{\left(0\right)}=\widehat{\mathit{\tau }}$. Then, since ${\widehat{\mathit{x}}}^{\left(1\right)}={\widehat{\mathit{x}}}^{\mathrm{HP}}$, (14) can be represented as

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\left(1\right)}={\widehat{\mathit{x}}}^{\left(0\right)}+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(0\right)}).\end{array}$$

(15)

Next, consider ${\widehat{\mathit{x}}}^{\left(2\right)}$. Given that

$$\begin{array}{c}\hfill {\mathit{S}}_2={\mathit{I}}_n-{({\mathit{I}}_n-\mathit{S})}^2=\mathit{S}+\mathit{S}({\mathit{I}}_n-\mathit{S}),\end{array}$$

(16)

it follows that

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\left(2\right)}={\widehat{\mathit{x}}}^{\left(1\right)}+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(1\right)}).\end{array}$$

(17)

Then, substituting (15) into (17) yields

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\left(2\right)}={\widehat{\mathit{x}}}^{\left(0\right)}+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(0\right)})+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(1\right)}).\end{array}$$

(18)

The next proposition generalizes (15) and (18).

Proposition 1. 

${\widehat{\mathit{x}}}^{\left(m\right)}$ in (8) can be decomposed as

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\left(m\right)}={\widehat{\mathit{x}}}^{\left(0\right)}+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(0\right)})+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(1\right)})+\dots +\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{(m-1)})\end{array}$$

(19)

for $m=1,2,\dots$, where ${\widehat{\mathit{x}}}^{\left(0\right)}=\widehat{\mathit{\tau }}$ in (1).

Proof. 

Let $\mathit{R}={\mathit{I}}_n-\mathit{S}$. Given that

$$\begin{array}{cc}\hfill {\mathit{S}}_m& ={\mathit{I}}_n-{\mathit{R}}^m=({\mathit{I}}_n-\mathit{R})({\mathit{I}}_n+\mathit{R}+\dots +{\mathit{R}}^{m-1})\hfill \\ \hfill & =\mathit{S}({\mathit{I}}_n+\mathit{R}+\dots +{\mathit{R}}^{m-1})=\mathit{S}+\mathit{S}\mathit{R}+\dots +\mathit{S}{\mathit{R}}^{m-1},\hfill \end{array}$$

(20)

the bHP trend, ${\widehat{\mathit{x}}}^{\left(m\right)}$, can be represented as

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\left(m\right)}={\mathit{S}}_m\mathit{y}=\mathit{S}\mathit{y}+\mathit{S}\mathit{R}\mathit{y}+\dots +\mathit{S}{\mathit{R}}^{m-1}\mathit{y}.\end{array}$$

(21)

Here, since ${\mathit{S}}_k={\mathit{I}}_n-{\mathit{R}}^k$, it follows that ${\mathit{R}}^k={\mathit{I}}_n-{\mathit{S}}_k$, which yields ${\mathit{R}}^k\mathit{y}=\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)}$ for $k=1,\dots ,m-1$. In addition, $\mathit{S}\mathit{y}={\widehat{\mathit{x}}}^{\left(1\right)}$. Accordingly, (21) can be rewritten using ${\widehat{\mathit{x}}}^{\left(1\right)},\dots ,{\widehat{\mathit{x}}}^{(m-1)}$ as

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\left(m\right)}={\widehat{\mathit{x}}}^{\left(1\right)}+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(1\right)})+\dots +\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{(m-1)}).\end{array}$$

(22)

Finally, by substituting (15) into (22), we obtain (19). □

Remark 1. 

(a) Again, given that $\mathit{S}$ is a low-pass filter, $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ for $k=0,\dots ,m-1$ in (19) are trend components retained in the trend residuals, $\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)}$. In this regard, see also Proposition 3 in the next section. (b) The next (23)–(26) illustrate how the linear trend, the HP trend, and the bHP trend can be treated in a unified way.

$$\begin{array}{cc}\hfill & {\widehat{\mathit{x}}}^{\left(0\right)}=\widehat{\mathit{\tau }},\left(\mathit{Linear}\mathit{trend}\right),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & {\widehat{\mathit{x}}}^{\left(1\right)}=\widehat{\mathit{\tau }}+\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }}),\left(\mathit{HP}\mathit{trend}\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & {\widehat{\mathit{x}}}^{\left(2\right)}=\widehat{\mathit{\tau }}+\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }})+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}}),(\mathit{bHP}\mathit{trend},\mathit{twicing}),\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & {\widehat{\mathit{x}}}^{\left(3\right)}=\widehat{\mathit{\tau }}+\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }})+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}})+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(2\right)}),\left(\mathit{bHP}\mathit{trend}\right).\hfill \end{array}$$

(26)

Here, the term ‘twicing’ is used in Hall and Thompson (2024) [16]. Interestingly, not only $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}})$ and $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(2\right)})$ but also $\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }})$ can be considered as the gains from boosting. (c) From Proposition 1, we immediately obtain the following recursive formula:

$$\begin{array}{c}{\widehat{\mathit{x}}}^{\left(m\right)}={\widehat{\mathit{x}}}^{(m-1)}+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{(m-1)}),\end{array}$$

(27)

$$\begin{array}{c}{\widehat{\mathit{x}}}^{\left(0\right)}=\widehat{\mathit{\tau }}\end{array}$$

(28)

for $m=1,2,\dots$.

Let ${\mathit{S}}_0={\mathit{P}}_{\mathsf{\Pi }}$. Then, it follows that ${\mathit{S}}_0\mathsf{\Pi }={\mathit{P}}_{\mathsf{\Pi }}\mathsf{\Pi }=\mathsf{\Pi }$. In addition, given that ${\mathit{S}}_1=\mathit{S}$, (12) can be represented as ${\mathit{S}}_1\mathsf{\Pi }=\mathsf{\Pi }$. Moreover, for $k=2,\dots ,m-1$, it follows that

$$\begin{array}{cc}\hfill {\mathit{S}}_k\mathsf{\Pi }& =\{{\mathit{I}}_n-{({\mathit{I}}_n-\mathit{S})}^k\}\mathsf{\Pi }=\mathsf{\Pi }-{({\mathit{I}}_n-\mathit{S})}^{k-1}({\mathit{I}}_n-\mathit{S})\mathsf{\Pi }\hfill \\ \hfill & =\mathsf{\Pi }.\hfill \end{array}$$

Combining these results, we obtain

$$\begin{array}{c}\hfill {\mathit{S}}_k\mathsf{\Pi }=\mathsf{\Pi },k=0,\dots ,m-1.\end{array}$$

(29)

This is a generalization of (12).

Given that ${\mathit{S}}_k$ is symmetric, from (29), it follows that

$$\begin{array}{c}\hfill {\mathsf{\Pi }}^{\top }{\mathit{S}}_k={\mathsf{\Pi }}^{\top }{\mathit{S}}_k^{\top }={\left({\mathit{S}}_k\mathsf{\Pi }\right)}^{\top }={\mathsf{\Pi }}^{\top },k=0,\dots ,m-1.\end{array}$$

(30)

Thereby, we obtain

$$\begin{array}{cc}\hfill {\mathsf{\Pi }}^{\top }(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})& ={\mathsf{\Pi }}^{\top }\mathit{y}-{\mathsf{\Pi }}^{\top }{\widehat{\mathit{x}}}^{\left(k\right)}={\mathsf{\Pi }}^{\top }\mathit{y}-{\mathsf{\Pi }}^{\top }{\mathit{S}}_k\mathit{y}\hfill \\ \hfill & ={\mathsf{\Pi }}^{\top }\mathit{y}-{\mathsf{\Pi }}^{\top }\mathit{y}={\mathbf{0}}_{2,1},k=0,\dots ,m-1.\hfill \end{array}$$

(31)

From (30) and (31), we have the following results.

Proposition 2. 

$\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ for $k=0,\dots ,m-1$ in (19) belong to the orthogonal complement of the column space of Π.

Proof. 

From (30) and (31), it follows that

$$\begin{array}{c}\hfill {\mathsf{\Pi }}^{\top }\left\{\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})\right\}={\mathsf{\Pi }}^{\top }(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})={\mathbf{0}}_{2,1},k=0,\dots ,m-1.\end{array}$$

(32)

Therefore, $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ for $k=0,\dots ,m-1$ belong to the orthogonal complement of the column space of $\mathsf{\Pi }$. □

Remark 2. 

(a) Proposition 2 implies that the total gains of boosting, $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(0\right)})+\dots +\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{(m-1)})$, in (19) is orthogonal to $\widehat{\mathit{\tau }}(={\widehat{\mathit{x}}}^{\left(0\right)})$. In other words, the right-hand side of (19) shows an orthogonal decomposition of ${\widehat{\mathit{x}}}^{\left(m\right)}$. (b) Denote the t-th entry of $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ by ${\zeta }_t^{\left(k\right)}$ for $k=0,\dots ,m-1$. Proposition 2 implies that

$$\begin{array}{c}\hfill \sum _{t=1}^n{\zeta }_t^{\left(k\right)}=0,\sum _{t=1}^{n}t{\zeta }_t^{\left(k\right)}=0,k=0,\dots ,m-1.\end{array}$$

(33)

4. Some More Details to the Perspective

In this section, we add some more results concerning $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ for $k=0,\dots ,m-1$ in (19).

Denote a spectral decomposition of an $n\times n$ real symmetric matrix ${\mathit{D}}^{\top }\mathit{D}$ by $\mathit{V}\mathsf{\Gamma }{\mathit{V}}^{\top }$, where $\mathsf{\Gamma }=\mathrm{diag}({\gamma }_1,\dots ,{\gamma }_n)$ and $\mathit{V}=[{\mathit{v}}_1,\dots ,{\mathit{v}}_n]$. Here, the eigenvalues, ${\gamma }_1,\dots ,{\gamma }_n$, are in ascending order. Given that ${\mathit{D}}^{\top }\mathit{D}$ is a positive semi-definite matrix whose rank is $n-2$, the eigenvalues satisfy the inequalities given by

$$\begin{array}{c}\hfill 0={\gamma }_1={\gamma }_2<{\gamma }_3\le \dots \le {\gamma }_n\le 16.\end{array}$$

(34)

Note that ${\gamma }_n\le 16$ follows from the Gershgorin circle theorem (Estrada and Knight, 2015 [24]). Figure 1 plots the eigenvalues ${\gamma }_1,\dots ,{\gamma }_n$ for $n=100$. $\mathit{V}$ is an orthogonal matrix. Given that $\mathit{D}\mathsf{\Pi }={\mathbf{0}}_{n-2,2}$, it follows that

$$\begin{array}{c}\hfill {\mathit{D}}^{\top }\mathit{D}\mathsf{\Pi }={\mathbf{0}}_{n,2}=0·\mathsf{\Pi }.\end{array}$$

Accordingly, given that $\mathsf{\Pi }$ is of full column rank, we can let

$$\begin{array}{c}\hfill {\mathit{v}}_1={\displaystyle \frac{1}{\sqrt{n}}}\mathit{\iota },{\mathit{v}}_2={\displaystyle \frac{1}{\sqrt{{\mathit{\xi }}^{\top }{\mathit{Q}}_{\iota }\mathit{\xi }}}}{\mathit{Q}}_{\iota }\mathit{\xi },\end{array}$$

(35)

where $\mathit{\iota }={[1,\dots ,1]}^{\top }\in {\mathbb{R}}^n$, $\mathit{\xi }={[1,\dots ,n]}^{\top }$, and ${\mathit{Q}}_{\iota }={\mathit{I}}_n-\mathit{\iota }{\left({\mathit{\iota }}^{\top }\mathit{\iota }\right)}^{-1}{\mathit{\iota }}^{\top }$. Note that, since $\mathsf{\Pi }=[\mathit{\iota },\mathit{\xi }]$, both ${\mathit{v}}_1$ and ${\mathit{v}}_2$ belong to the column space of $\mathsf{\Pi }$. Denote the i-th entry of ${\mathit{V}}^{\top }\mathit{y}$ by ${\eta }_i$ for $i=1,\dots ,n$, i.e., ${\mathit{V}}^{\top }\mathit{y}={[{\eta }_1,\dots ,{\eta }_n]}^{\top }$. Accordingly, ${\eta }_1={\mathit{v}}_1^{\top }\mathit{y}=\sqrt{n}\overline{y}$, where $\overline{y}={\displaystyle \frac{1}{n}}{\sum }_{t=1}^n{y}_t$.

Figure 1. Eigenvalues ${\gamma }_1,\dots ,{\gamma }_n$ for $n=100$.

The next proposition shows how the residuals, $\mathit{y}-{\widehat{\mathit{x}}}^{\left(0\right)},\dots ,\mathit{y}-{\widehat{\mathit{x}}}^{(m-1)}$, are smoothed by the smoother matrix $\mathit{S}$.

Proposition 3. 

$\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ can be represented by

$$\begin{array}{c}\hfill \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})=\sum _{i=3}^n{\displaystyle \frac{{\varphi }_i^{\left(k\right)}}{1+\lambda {\gamma }_i}}{\mathit{v}}_i,k=0,\dots ,m-1,\end{array}$$

(36)

where ${\varphi }_i^{\left(k\right)}$ denotes the i-th entry of ${\mathit{V}}^{\top }(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$. Moreover, in (36), the eigenvectors, ${\mathit{v}}_3,\dots ,{\mathit{v}}_n$, satisfy the inequalities given by

$$\begin{array}{c}\hfill 0<\parallel \mathit{D}{\mathit{v}}_3{\parallel }^2\le \dots \le {\parallel \mathit{D}{\mathit{v}}_n\parallel }^2\le 16\end{array}$$

(37)

and the coefficients on the eigenvectors satisfy the inequalities given by

$$\begin{array}{c}\hfill 1>{\displaystyle \frac{1}{1+\lambda {\gamma }_3}}\ge \dots \ge {\displaystyle \frac{1}{1+\lambda {\gamma }_n}}>0.\end{array}$$

(38)

Proof. 

See Appendix B. □

Remark 3. 

(a) The inequalities in (37) show that the degree of smoothness of ${\mathit{v}}_i$ is higher than that of ${\mathit{v}}_{i+1}$ for $i=3,\dots ,n-1$. Figure 2 illustrates the latter results by plotting ${\mathit{v}}_3$, ${\mathit{v}}_5$, ${\mathit{v}}_{10}$, and ${\mathit{v}}_{20}$ for $n=100$. (b) The inequalities in (38) are the core of smoothing. This is because only relatively short-period fluctuations are compressed. For example, ${\displaystyle \frac{1}{1+\lambda {\gamma }_n}}$ is close to zero, whereas ${\displaystyle \frac{1}{1+\lambda {\gamma }_3}}$ is close to unity. We note that ${\displaystyle \frac{1}{1+\lambda {\gamma }_1}},\dots ,{\displaystyle \frac{1}{1+\lambda {\gamma }_n}}$ are the eigenvalues of $\mathit{S}$ in descending order. See Figure 3, which plots ${\displaystyle \frac{1}{1+\lambda {\gamma }_3}},\dots ,{\displaystyle \frac{1}{1+\lambda {\gamma }_{30}}}$ for $\lambda =1600$ and $n=100$.

Figure 2. Eigenvectors ${\mathit{v}}_3$, ${\mathit{v}}_5$, ${\mathit{v}}_{10}$, ${\mathit{v}}_{20}$ for $n=100$.

Figure 3. Eigenvalues ${\displaystyle \frac{1}{1+\lambda {\gamma }_3}},\dots ,{\displaystyle \frac{1}{1+\lambda {\gamma }_{30}}}$ for $\lambda =1600$ and $n=100$.

The next proposition shows the results concerning the squared ${\mathsf{\ell }}_2$-norm of $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ for $k=0,\dots ,m-1$.

Proposition 4. 

If $\mathit{y}$ does not belong to the column space of Π, then it follows that

$$\begin{array}{c}\hfill \parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)}){\parallel }^2=\sum _{i=3}^n{\left({\displaystyle \frac{1}{1+\lambda {\gamma }_i}}\right)}^2{\left(1-{\displaystyle \frac{1}{1+\lambda {\gamma }_i}}\right)}^{2k}{\eta }_i^2>0,k=0,\dots ,m-1.\end{array}$$

(39)

Proof. 

See Appendix B. □

Remark 4. 

From Proposition 4, it follows that

$$\begin{array}{c}\hfill \parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(0\right)}){\parallel }^2=\sum _{i=3}^n{\left({\displaystyle \frac{{\eta }_i}{1+\lambda {\gamma }_i}}\right)}^2.\end{array}$$

(40)

Here, we show that this result is reasonable. Given that $\{{\mathit{v}}_1,{\mathit{v}}_2\}$ is an orthonormal basis of the column space of Π, it follows that ${\mathit{P}}_{\mathsf{\Pi }}={\mathit{V}}_1{\left({\mathit{V}}_1^{\top }{\mathit{V}}_1\right)}^{-1}{\mathit{V}}_1^{\top }={\mathit{V}}_1{\mathit{V}}_1^{\top }$, where ${\mathit{V}}_1=[{\mathit{v}}_1,{\mathit{v}}_2]$, from which we have

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\left(0\right)}=\widehat{\mathit{\tau }}={\eta }_1{\mathit{v}}_1+{\eta }_2{\mathit{v}}_2.\end{array}$$

(41)

In addition, given that $\mathit{S}{\widehat{\mathit{x}}}^{\left(0\right)}=\mathit{S}\widehat{\mathit{\tau }}=\widehat{\mathit{\tau }}$ and ${\gamma }_1={\gamma }_2=0$, we have

$$\begin{array}{cc}\hfill \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(0\right)})& =\mathit{S}\mathit{y}-\mathit{S}\widehat{\mathit{\tau }}=\mathit{V}{({\mathit{I}}_n+\lambda \mathsf{\Gamma })}^{-1}{\mathit{V}}^{\top }\mathit{y}-\widehat{\mathit{\tau }}\hfill \\ \hfill & =\sum _{i=1}^n\left({\displaystyle \frac{{\eta }_i}{1+\lambda {\gamma }_i}}\right){\mathit{v}}_i-({\eta }_1{\mathit{v}}_1+{\eta }_2{\mathit{v}}_2)=\sum _{i=3}^n\left({\displaystyle \frac{{\eta }_i}{1+\lambda {\gamma }_i}}\right){\mathit{v}}_i,\hfill \end{array}$$

(42)

which is consistent with (40).

The next proposition documents the results for the arithmetic mean and the variance of $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ for $k=0,\dots ,m-1$.

Proposition 5. 

(i) The sum of the entries of $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ is equal to zero for $k=0,\dots ,m-1$. (ii) If $\mathit{y}$ does not belong to the column space of Π, then it follows that

$$\begin{array}{c}\hfill \parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(0\right)}){\parallel }^2>\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(1\right)}){\parallel }^2>\dots >{\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{(m-1)})\parallel }^2.\end{array}$$

(43)

Proof. 

See Appendix B. □

Remark 5. 

(a) (i) shows that the arithmetic mean of $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ equals zero for $k=0,\dots ,m-1$. (b) (i) and (ii) imply that the variance of $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ strictly decreases as k increases from zero. Moreover, in addition to (43), given that $1-{\displaystyle \frac{1}{1+\lambda {\gamma }_i}}={\displaystyle \frac{\lambda {\gamma }_i}{1+\lambda {\gamma }_i}}\in (0,1)$ for $i=3,\dots ,n$, from Proposition 4, it follows that

$$\begin{array}{c}\hfill \parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)}){\parallel }^2\to 0+,(k\to \infty ).\end{array}$$

(44)

5. An Empirical Illustration

The dashed line in the top panel of Figure 4 depicts the log of US real gross domestic product (GDP) from 1947:1 to 1998:2, $\mathit{y}$. This is the same data used by Morley et al. (2003) [25] and Perron and Wada (2009) [26]. The solid line in the panel is the linear trend, $\widehat{\mathit{\tau }}$. The dashed line in the middle panel plots the linearly detrended series, $(\mathit{y}-\widehat{\mathit{\tau }})$. It clearly shows that the linear trend residuals contain long-period fluctuations. The solid line in the panel plots $\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }})$ is estimated with $\lambda =1600$. Recall that

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\mathrm{HP}}=\widehat{\mathit{\tau }}+\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }}).\end{array}$$

(45)

The two panels presented here are identical to those shown in Yamada (2018, Figure 2) [3].

Figure 4. Top panel: Log of US real GDP, $\mathit{y}$ (dashed line), and the linear trend, $\widehat{\mathit{\tau }}$ (solid line). Middle panel: The linearly detrended series, $\mathit{y}-\widehat{\mathit{\tau }}$ (dashed line), and its low-frequency component, $\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }})$, estimated with $\lambda =1600$ (solid line). Bottom panel: The HP detrended series, $\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}}$ (dashed line), and its low-frequency component, $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}})$, estimated with $\lambda =1600$ (solid line).

The dashed line in the bottom panel plots the HP detrended series, $\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}}$. The solid line in the panel plots its low-frequency component, $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}})$, which is again estimated with $\lambda =1600$. From the bottom panel, we observe that the HP detrended series, $\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}}$ still contain long-period fluctuations and they are extracted by $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}})$. Recall that

$$\begin{array}{c}\hfill {\widehat{\mathit{x}}}^{\left(2\right)}=\widehat{\mathit{\tau }}+\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }})+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}})={\widehat{\mathit{x}}}^{\mathrm{HP}}+\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}}),\end{array}$$

(46)

which is twicing. Finally, we report that the following inequalities hold:

$$\begin{array}{cc}\hfill & \parallel \mathit{S}(\mathit{y}-\widehat{\mathit{\tau }}){\parallel }^2=4,227.0>\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}}){\parallel }^2=30.9>{\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(2\right)})\parallel }^2=7.1\hfill \\ \hfill & >\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(3\right)}){\parallel }^2=3.1>\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(4\right)}){\parallel }^2=1.8>{\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(5\right)})\parallel }^2=1.3>0,\hfill \end{array}$$

(47)

which illustrate Propostions 4 and 5.

As a supplementary examination, the same analysis was conducted with $\lambda$ = 800,000. Note that both 1600 and 800,000 are the values used in Perron and Wada (2009, Figure 5) [26]. The results are illustrated in Figure 5. Even in this case, the following inequalities hold:

$$\begin{array}{cc}\hfill & \parallel \mathit{S}(\mathit{y}-\widehat{\mathit{\tau }}){\parallel }^2=2,346.9>\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}}){\parallel }^2=81.5>{\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(2\right)})\parallel }^2=5.1\hfill \\ \hfill & >\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(3\right)}){\parallel }^2=1.7>\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(4\right)}){\parallel }^2=1.2>{\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(5\right)})\parallel }^2=0.9>0.\hfill \end{array}$$

(48)

Figure 5. Top panel: Log of US real GDP, $\mathit{y}$ (dashed line), and the linear trend, $\widehat{\mathit{\tau }}$ (solid line). Middle panel: The linearly detrended series, $\mathit{y}-\widehat{\mathit{\tau }}$ (dashed line), and its low-frequency component, $\mathit{S}(\mathit{y}-\widehat{\mathit{\tau }})$, estimated with $\lambda$ = 800,000 (solid line). Bottom panel: The HP detrended series, $\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}}$ (dashed line), and its low-frequency component, $\mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\mathrm{HP}})$, estimated with $\lambda$ = 800,000 (solid line).

Regarding (47) and (48), we make a remark. If $\mathit{y}$ does not belong to the column space of $\mathsf{\Pi }$, then it follows that

$$\begin{array}{c}\hfill \parallel {({\mathit{I}}_n+{\lambda }_1{\mathit{D}}^{\top }\mathit{D})}^{-1}(\mathit{y}-\widehat{\mathit{\tau }}){\parallel }^2>{\parallel {({\mathit{I}}_n+{\lambda }_2{\mathit{D}}^{\top }\mathit{D})}^{-1}(\mathit{y}-\widehat{\mathit{\tau }})\parallel }^2,{\lambda }_1<{\lambda }_2.\end{array}$$

(49)

A proof of (49) is provided in Appendix B.

6. Concluding Remarks

Yamada (2018) [3] clarified the relationship between the linear trend in (1) and the HP trend in (3) and explained why the HP trend seems to be more plausible than the linear trend. This paper is an extension of Yamada (2018) [3]. In this paper, we treated the three trends, the linear trend, the HP trend, and the recently proposed bHP trend in (8), in a unified manner and clarified their relationship in detail. We showed how the bHP trend is constructed from the linear/HP trend, and long-period fluctuations remained in their trend residuals. The results obtained are summarized in Propositions 1–5 and illustrated in Figure 4 and Figure 5.

Finally, we make one remark. It concerns the value of m. As stated by Phillips and Shi (2021) [8], when $\lambda$ is fixed, the effective degrees of freedom of the bHP filter, defined as $\mathrm{tr}({\mathit{S}}_m$), is an increasing function of m. Thus, increasing m reduces the sum of squares of the bHP trend residuals. (As shown in Proposition 1 of Yamada (2024) [19], it tends to 0 as m goes to ∞.) This relationship is similar to that between the number of explanatory variables and the sum of squares of residuals in a linear regression model. For selecting the value of m, Phillips and Shi (2021) [8] proposed an information criterion.

Appendix B.1. Proof of Proposition 3

From the spectral decomposition of ${\mathit{D}}^{\top }\mathit{D}$, we have

$$\begin{array}{cc}\hfill \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})& ={({\mathit{I}}_n+\lambda {\mathit{D}}^{\top }\mathit{D})}^{-1}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})=\mathit{V}{({\mathit{I}}_n+\lambda \mathsf{\Gamma })}^{-1}{\mathit{V}}^{\top }(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})\hfill \\ \hfill & =\sum _{i=1}^n{\varphi }_i^{\left(k\right)}{\displaystyle \frac{1}{1+\lambda {\gamma }_i}}{\mathit{v}}_i,\hfill \end{array}$$

(A9)

where ${\varphi }_i^{\left(k\right)}={\mathit{v}}_i^{\top }(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})$ for $i=1,\dots ,n$. Here, as shown in (35), both ${\mathit{v}}_1$ and ${\mathit{v}}_2$ belong to the column space of $\mathsf{\Pi }$. Then, from (31), we have ${\varphi }_1={\varphi }_2=0$ in (A9). Next, (37) follows $0<{\gamma }_3\le \dots \le {\gamma }_n\le 16$ and

$$\begin{array}{c}\hfill \parallel \mathit{D}{\mathit{v}}_i{\parallel }^2={\mathit{v}}_i^{\top }{\mathit{D}}^{\top }\mathit{D}{\mathit{v}}_i={\gamma }_i{\mathit{v}}_i^{\top }{\mathit{v}}_i={\gamma }_i,i=3,\dots ,n.\end{array}$$

(38) follows from $\lambda \in (0,\infty )$ and $0<{\gamma }_3\le \dots \le {\gamma }_n$.

Appendix B.2. Proof of Proposition 4

Given that ${\psi }_1={\psi }_2=0$, from (A6), we have

$$\begin{array}{cc}\hfill \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})& =\mathit{V}({\mathit{I}}_n-\mathsf{\Psi }){\mathsf{\Psi }}^k{\mathit{V}}^{\top }\mathit{y}=\sum _{i=1}^n{\eta }_i(1-{\psi }_i){\psi }_i^k{\mathit{v}}_i\hfill \\ \hfill & =\sum _{i=3}^n{\eta }_i(1-{\psi }_i){\psi }_i^k{\mathit{v}}_i,k=0,\dots ,m-1.\hfill \end{array}$$

(A10)

Then, given that ${[{\eta }_3,\dots ,{\eta }_n]}^{\top }\ne {\mathbf{0}}_{n-2,1}$ by Lemma A2 and ${\psi }_i\in (0,1)$ for $i=3,\dots ,n$, from (A10), it follows that

$$\begin{array}{c}\hfill \parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)}){\parallel }^2=\sum _{i=3}^n{\eta }_i^2{(1-{\psi }_i)}^2{\psi }_i^{2k}>0,k=0,\dots ,m-1.\end{array}$$

Appendix B.3. Proof of Proposition 5

(i) It immediately follows from Proposition 2. Next, we prove (ii) by showing the inequalities given by

$$\begin{array}{c}\hfill \parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{(k-1)}){\parallel }^2>{\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})\parallel }^2,k=1,\dots ,m-1.\end{array}$$

Given that ${[{\eta }_3,\dots ,{\eta }_n]}^{\top }\ne {\mathbf{0}}_{n-2,1}$ by Lemma A2 and ${\psi }_i\in (0,1)$ for $i=3,\dots ,n$, from Proposition 4, it follows that

$$\begin{array}{cc}\hfill \parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{(k-1)}){\parallel }^2-{\parallel \mathit{S}(\mathit{y}-{\widehat{\mathit{x}}}^{\left(k\right)})\parallel }^2& =\sum _{i=3}^n{\eta }_i^2{(1-{\psi }_i)}^2{\psi }_i^{2(k-1)}-\sum _{i=3}^n{\eta }_i^2(1-{\psi }_i){\psi }_i^{2k}\hfill \\ \hfill & =\sum _{i=3}^n{\eta }_i^2{(1-{\psi }_i)}^2({\psi }_i^{2(k-1)}-{\psi }_i^{2k})\hfill \\ \hfill & =\sum _{i=3}^n{\eta }_i^2{(1-{\psi }_i)}^2{\psi }_i^{2(k-1)}(1-{\psi }_i^2)>0\hfill \end{array}$$

for $k=1,\dots ,m-1$.

Appendix B.4. Proof of (49)

From (40), we have

$$\begin{array}{cc}\hfill & \parallel {({\mathit{I}}_n+{\lambda }_1{\mathit{D}}^{\top }\mathit{D})}^{-1}(\mathit{y}-\widehat{\mathit{\tau }}){\parallel }^2-{\parallel {({\mathit{I}}_n+{\lambda }_2{\mathit{D}}^{\top }\mathit{D})}^{-1}(\mathit{y}-\widehat{\mathit{\tau }})\parallel }^2\hfill \\ \hfill & =\sum _{i=3}^n{\left({\displaystyle \frac{{\eta }_i}{1+{\lambda }_1{\gamma }_i}}\right)}^2-\sum _{i=3}^n{\left({\displaystyle \frac{{\eta }_i}{1+{\lambda }_2{\gamma }_i}}\right)}^2=\sum _{i=3}^n\left\{{\left({\displaystyle \frac{{\eta }_i}{1+{\lambda }_1{\gamma }_i}}\right)}^2-{\left({\displaystyle \frac{{\eta }_i}{1+{\lambda }_2{\gamma }_i}}\right)}^2\right\}\hfill \\ \hfill & =\sum _{i=3}^n\left({\displaystyle \frac{{\eta }_i}{1+{\lambda }_1{\gamma }_i}}+{\displaystyle \frac{{\eta }_i}{1+{\lambda }_2{\gamma }_i}}\right)\left({\displaystyle \frac{{\eta }_i}{1+{\lambda }_1{\gamma }_i}}-{\displaystyle \frac{{\eta }_i}{1+{\lambda }_2{\gamma }_i}}\right).\hfill \end{array}$$

(A11)

Here, it follows that

$$\begin{array}{c}\hfill {\displaystyle \frac{{\eta }_i}{1+{\lambda }_1{\gamma }_i}}-{\displaystyle \frac{{\eta }_i}{1+{\lambda }_2{\gamma }_i}}={\displaystyle \frac{{\eta }_i(1+{\lambda }_2{\gamma }_i-1-{\lambda }_1{\gamma }_i)}{(1+{\lambda }_1{\gamma }_i)(1+{\lambda }_2{\gamma }_i)}}={\displaystyle \frac{{\eta }_i{\gamma }_i({\lambda }_2-{\lambda }_1)}{(1+{\lambda }_1{\gamma }_i)(1+{\lambda }_2{\gamma }_i)}}.\end{array}$$

(A12)

Therefore, given that ${\lambda }_2>{\lambda }_1$, ${[{\eta }_3,\dots ,{\eta }_n]}^{\top }\ne {\mathbf{0}}_{n-2,1}$ by Lemma A2, and ${\gamma }_i>0$ for $i=3,\dots ,n$, from (A11) and (A12), we have

$$\begin{array}{c}\hfill \parallel {({\mathit{I}}_n+{\lambda }_1{\mathit{D}}^{\top }\mathit{D})}^{-1}(\mathit{y}-\widehat{\mathit{\tau }}){\parallel }^2-{\parallel {({\mathit{I}}_n+{\lambda }_2{\mathit{D}}^{\top }\mathit{D})}^{-1}(\mathit{y}-\widehat{\mathit{\tau }})\parallel }^2>0.\end{array}$$