Correction to: ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell_{1}$$\end{document} Common Trend Filtering

The ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}$$\end{document} trend filtering enables us to estimate a continuous piecewise linear trend of univariate time series. This filter and its variants have subsequently been applied in various fields, including astronomy, climatology, economics, electronics, environmental science, finance, and geophysics. Although the ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}$$\end{document} trend filtering can estimate a continuous piecewise linear trend of univariate time series, it cannot estimate a common continuous piecewise linear trend of multiple time series. This paper develops a statistical procedure that enables us to estimate it, which is a multivariate extension of the ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}$$\end{document} trend filtering. We provide an algorithm for estimating it and a clue to specify the tuning parameter of the procedure, both required for its application. We also (i) numerically illustrate how well the algorithm works, (ii) provide an empirical illustration, and (iii) introduce a generalization of our novel method.

The ' 1 trend filtering is defined by replacing the squared ' 2 -norm penalty of the Hodrick-Prescott (HP) (1997) filtering with the ' 1 -norm penalty. 2 It is notable that, even though the modification seems to be somewhat minor, the ' 1 trend filtering provides a continuous piecewise linear trend, whereas the HP filtering provides a smooth trend. In econometrics, such a continuous piecewise linear trend was dealt with by Perron (1989) and Rappoport and Reichlin (1989) and it reflects the idea that 'economic events that have large permanent effects are relatively rare' (Hamilton 1994). Thus, it is possible to say that the ' 1 trend filtering is a method to obtain the trend considered by Perron (1989) and Rappoport and Reichlin (1989).
Although the ' 1 trend filtering can estimate a continuous piecewise linear trend of univariate time series, it cannot estimate a common continuous piecewise linear trend of multiple time series. In this paper, we develop a statistical procedure that enables us to estimate it, which is a multivariate extension of the ' 1 trend filtering. To explain more precisely, let y i;t be an observation of a univariate time series i at t, where i ¼ 1; . . .; n and t ¼ 1; . . .; T, and suppose that it has a continuous piecewise linear trend x i;t . As stated, the ' 1 trend filtering can be applied for estimating x i;t from y i;t . In this paper, we consider the situation such that x i;t can be expressed as x i;t ¼ a i x t ; i ¼ 1; . . .; n; t ¼ 1; . . .; T; where x t is a continuous piecewise linear trend and a i is a loading coefficient. Given that (1) can be represented as 1 '' 1 trend filtering' is the terminology used by Kim et al. (2009). The approach is a form of ' 1 -norm penalized least squares and may also be regarded as a type of generalized lasso regression (Tibshirani 1996;Kim et al. 2009;Tibshirani and Taylor 2011) and as a generalization of one-dimensional total variation denoising (Rudin et al. 1992;Steidl et al. 2006;Guntuboyina et al. 2020 Phillips and Shi (2020), Sakarya and de Jong (2020), and Yamada (2020). HP filtering has been used to extract the cyclical component of a univariate time series. For other such methods, see, e.g., Pollock (2016) and Michaelides et al. (2018). Also, it is a type of the Whittaker-Henderson (WH) method of graduation. For a historical survey of the WH method of graduation, see, e.g., Weinert (2007) and Phillips (2010). even though y 1;t ; . . .; y n;t commonly have x t , their linear combination b 1 y 1;t þ Á Á Á þ b n y n;t no longer has x t if ½b 1 ; . . .; b n 0 is a vector that belongs to the orthogonal complement of the space spanned by ½a 1 ; . . .; a n 0 . Hatanaka and Yamada (2003) referred to it as 'co-trending.' 3 In this paper, by extending the ' 1 trend filtering, we develop a novel method to estimate x t and a i from y i;t . Recall that i ¼ 1; . . .; n and t ¼ 1; . . .; T, where n (resp. T) represents the number of univariate time series (resp. observations). We refer to the novel filtering method as '' 1 common trend filtering.' We provide an algorithm for estimating this and a clue to specify the tuning parameter of the procedure, both of which are required for its application. We also (i) numerically illustrate how well the algorithm works, (ii) provide an empirical illustration, and (iii) introduce a generalization of our novel method.
Here, we remark that (2) is not an unlikely model of trends in macroeconomic time series but has strong relevance. To explain more precisely, let y 1;t ; . . .; y n;t be macroeconomic time series in natural logarithms and e 1;t ; . . .; e n;t be such that It is a similar concept to cointegration (Engle and Granger 1987). In the case of cointegration, variables have common stochastic trends. For more details, see, e.g., Hatanaka (1996).
Let g i;t ¼ Dy i;t ð¼ y i;t À y i;tÀ1 Þ. Accordingly, g i;t for i ¼ 1; . . .; n represent the growth rates of the original time series. Then, (2) and (3) are equivalent to with initial conditions such as y i;1 ¼ a i x 1 þ e i;1 and Dy i;2 ¼ a i Dx 2 þ De i;2 , where b t ¼ Dx t À Dx tÀ1 and v i;t ¼ De i;t À De i;tÀ1 . Recall that Dg i;t in (4) denotes the difference of growth rates of variable i at t. Given that x t in (2) is a continuous piecewise linear trend, only a few of b 3 ; . . .; b T are not equal to 0. We may regard such nonzero b t s in (4) as occasional permanent shocks that shift growth rates of multiple time series simultaneously and a i for i ¼ 1; . . .; n in (4) represent individual reaction coefficients of the time series. A typical example of such occasional permanent shocks is the oil price shock in 1973. It is natural to consider that, at the time, the growth rates of macroeconomic time series changed simultaneously with their own reaction rates. This paper is organized as follows. Section 2 introduces the novel filtering method and provides its reduced-rank-regression (RRR) representations. Section 3 discusses a numerical computation method for x t and a i in (1). Section 4 provides a clue to specify the tuning parameter of the procedure required for its application. Section 5 numerically illustrates how well the novel statistical procedure works. Section 6 includes an empirical illustration. Section 7 mentions a generalization of our method. Section 8 concludes the paper.

' 1 Trend Filtering
The ' 1 trend filtering is defined by where w [ 0 is a tuning parameter. In matrix notation, it is expressed as

' 1 Common Trend Filtering
In this paper, we extend the ' 1 trend filtering so that we may estimate a common continuous piecewise linear trend of multiple time series, y 1;t ; . . .; y n;t . The filtering method we introduce in this paper is: where k [ 0 is a tuning parameter. We refer to the filtering method described by (8) as ' 1 common trend filtering. In matrix notation, the filtering is expressed as Furthermore, given that P n i¼1 ky i À a i xk 2 2 ¼ kY À xa 0 k 2 F and P n i¼1 a 2 i ¼ kak 2 2 , (9) can be represented by min This is an ' 1 -norm penalized RRR.

Another Representation
Let b 2 R T be a column vector such as x ¼ Xb. Given that (i) X is nonsingular and (ii) DX ¼ J (Paige and Trindade 2010), we obtain another RRR representation of (10): We remark that when Jb is sparse, x ¼ Xb represents a continuous piecewise linear trend. 4 In addition, interestingly, (11) is similar to Eq. (8) in Chen and Huang (2012). 5

Two Key Results
Given kak 2 2 ¼ 1, the objective function in (11) can be represented as Suppose that b 2 R T is given. Then, Xbð¼ xÞ 2 R T is a known column vector. Because both trðY 0 YÞ and trðXbb 0 X 0 Þ in (12) do not depend on a, when b 2 R T is given, (11) reduces to We remark that, given x ¼ Xb, it follows that tr Y 0 ðXba 0 Þ f g¼ P n i¼1 y 0 i ða i xÞ, which is quite reasonable as an objective function for estimating a 1 ; . . .; a n . Moreover, letting / ¼ Y 0 Xb 2 R n , it follows that tr Y 0 ðXba 0 Þ f g¼/ 0 a. Therefore, instead of (13), we may consider the following constrained maximization problem: Given kak 2 ¼ 1, by the Cauchy-Schwarz inequality, we obtain 3.1.2 The Case Where a 2 R n is Given Next, suppose that a 2 R n such that kak 2 2 ¼ 1 is given. Then, (11) reduces to Let A ? 2 R nÂðnÀ1Þ be a matrix of an orthonormal basis of the orthogonal complement of the space spanned by a. Then, given kak 2 2 ¼ 1, ½a; A ? 2 R nÂn is an orthogonal matrix. Thus, given that ðY À Xba 0 Þ½a; We remark here that (17) can be represented as See, e.g., Eq. (9) in Kim et al. (2009). 6 As (17) is a problem whose objective function is coercive and strictly convex over R T , it has a unique global minimizer. Thus, denoting the solution by b b, we have the following result. Given a 2 R n such that kak 2 2 ¼ 1, we have the following inequality:

A Numerical Algorithm
Based on the above two inequalities, (15) and (19), we introduce a numerical algorithm.
Then, we have the following result. 6 The dual problem of (18) and its implication are given in Section A.3 in the Appendix.
Proof (i) From (15) and (20), we have f ðb a iþ1 ; b b i Þ f ða; b b i Þ for any a 2 R n and we (19) and (21), Given Lemma 3.1, we have the following result.
has a finite limit.
Proof From Lemma 3.1, ðf i Þ i2N is a nonincreasing sequence. In addition, as f i ! 0 for i 2 N, it is bounded below. Consequently, it has a finite limit. h Proposition 3.2 implies that the objective function in (11) converges by alternatively minimizing it over a and b. Denote a and b such that the objective function in (11) is converged by b a and b b. A Matlab user-defined function for estimating b a and b b from Y and k, l1_common_trend_filter, is provided in the supplementary material.

A Clue for Specifying the Tuning Parameter
Applying the ' 1 common trend filtering requires the specification of its tuning parameter. In this section, we provide a clue for specifying it.
Consider the following convex problem: where c [ 0 and a 2 R n is a given vector. Given that X is nonsingular and hðbÞ ! 0, if b ¼ X À1 Ya is feasible, i.e., kDYak 1 c, then b ¼ X À1 Ya is the solution of the above convex problem. Here, recall JX À1 ¼ D. In the case, hðX À1 YaÞ ¼ 0. If it is not feasible, i.e., kDYak 1 [ c, then b ¼ X À1 Ya cannot be the solution. In the case, the solution, denoted by e b, locates at the boundary. Thus, we have kJ e bk 1 ¼ c. More precisely, concerning e b, we have the following results.
and (ii) kJ e bk 1 ¼ c holds.
Proof See Section A.4 in the Appendix. h Proposition 4.1 implies that we can obtain b a and b b by specifying c in (22) instead of k in (17). A Matlab user-defined function for calculating b a and b b from Y and c, l1_common_trend_filter_c, is provided in the supplementary material. Here, we point out that specifying c is much easier than specifying k. We do not have any useful information for specifying k, whereas we have such an information for specifying c. As stated in Proposition 4.1, we may estimate b b such that We may utilize this relation for specifying c. See, e.g., (A.1). In the case, Thus, we may specify rough range of c from the plots of first differences of multiple time series. 7 Finally, given Xb ¼ x and Jb ¼ Dx, we remark that the convex problem (22) may be replaced with the following convex problem: where c [ 0.

Numerical Illustrations
In this section, we numerically illustrate how well the algorithm described in the last section works. Figure 1 plots generated Xb, where T ¼ 100, b 1 ¼ 10, b 2 ¼ 0:5, and P T t¼3 jb t j ¼ 2:7243. Ten bullets in the figure depict the kink points and accordingly, 10 entries of Jb ¼ ½b 3 ; . . .; b T 0 are not equal to 0. Using Xb shown in Fig. 1, we generated y 1 , y 2 , and y 3 by where a ¼ ½1; 0:6; 0:2 0 and vecðEÞ $ Nð0; r 2 I 3T Þ with r ¼ 1. and b b such that kJ b bk 1 ¼ 3, the latter of which is consistent with Proposition 4.1(ii). The value of k for obtaining b b from Yb a is 10.3730. Figure 3 illustrates the results. The solid line in Fig. 3 plots the estimated b a 1 X b b. The dashed line in the figure plots a 1 Xb. Note that, given a 1 ¼ 1, it is identical to the solid line depicted in Fig. 1. From the figure, we can see that b a 1 X b b looks very much like a 1 Xb. Figure 4 also illustrates the results. The solid lines in Fig. 4 are identical to those in Fig. 2. The dashed lines on y 1 , y 2 and y 3 respectively plot b Again, the figure shows that our novel procedure works well.
As a supplementary examination, we generated an additional data set, repeated the same analysis, and revealed similar results. For example, we obtained  Fig. 2 y 1 , y 2 , and y 3 respectively denote y 1 , y 2 , and y 3 generated by (27). The dashed lines on y 1 , y 2 , and y 3 plot Xb, 0:6Xb, and 0:2Xb, respectively. Here, Xb is identical to the solid line depicted in Fig. 1 b From the figure, we can observe that these two time series seem to contain a common piecewise linear trend such that a major kink point is located at around 1991, which corresponds to the peak of the Japanese asset price bubble. Actually, the statistical procedure developed by Hatanaka and Yamada (2003) detected a common piecewise linear trend. [See Section 8.5 of Hatanaka and Yamada (2003).] Figure 6 depicts the corresponding demeaned series. We estimated a common piecewise linear trend of these demeaned data. Denote it by X b b. For the estimation, we used l1_common_trend_filter_c with c ¼ 0:018. We specified the value of c by reference to the plots of time series in first differences shown in Fig. 7. The . . .; b b T , where T ¼ 87). From the panels, we may observe that (i) a major kink point is located at around 1991 and (ii) P T t¼3 j b b t j equals the value of c. The solid lines in Fig. 9 are identical to those plotted in Fig. 6. The dashed line in the upper (resp. lower) panel plots b Finally, the dashed lines in Fig. 10 are the mean-restored estimated piecewise linear trends. The solid lines in the figure are identical to those in Fig. 5.

A Generalization
In this section, we mention a generalization of our method briefly. Let D p 2 R ðTÀpÞÂT be the p-th order difference matrix such that D p x i ¼ ½D p x i;pþ1 ; . . .; D p x i;T 0 . Explicitly, D p is a Toeplitz matrix as follows: where a k ¼ ðÀ1Þ pÀk p k for k ¼ 0; . . .; p. Accordingly, D 2 equals D. Then, without any difficulty, we may extend our procedure to min x 2 R T a 1 ; . . .; a n 2 R X n i¼1 ky i À a i xk 2 2 þ k p kD p xk 1 ; s.t.
where p 2 N and k p [ 0 is a tuning parameter. We refer to (31) as '' 1 common polynomial trend filtering.' The solution of the problem represents a continuous piecewise ðp À 1Þ-th order polynomial trend. Note that the corresponding X p ¼ ½P p ; W p such that D p X p ¼ ½D p P p ; D p W p ¼ ½0; I TÀp 2 R ðTÀpÞÂT is given in Yamada (2015).  Hatanaka and Yamada (2003) 8 Concluding Remarks In this paper, we developed an extension of the ' 1 trend filtering. The ' 1 trend filtering can estimate a continuous piecewise linear trend of univariate time series. However, it cannot estimate a common continuous piecewise linear trend of multiple time series, which the novel statistical procedure developed in this paper enables. We provided an algorithm and a clue to specify the tuning parameter of the procedure, both of which are required for its application. We also numerically illustrated how well the algorithm works, provided an empirical illustration, and introduced a generalization of our novel method. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/. Moreover, these three trends are connected at t ¼ 3 and t ¼ 5 as Thus, x 1 ; . . .; x 7 are on the following continuous piecewise linear function of t: where ðfÞ þ ¼ f if f [ 0 and ðfÞ þ ¼ 0 if f 0. In this function, ð3; að3ÞÞ and ð6; að6ÞÞ are kink points. Accordingly, fx 1 ; x 2 ; x 3 g are on the same straight line whose slope is b 2 , fx 3 ; x 4 ; x 5 g are on the same straight line whose slope is b 2 þ b 4 , and fx 5 ; x 6 ; x 7 g are on the same straight line whose slope is b 2 þ b 4 þ b 6 . We point out a similarity between the ' 1 -norm penalized RRR (11) and Eq. (8) in Chen and Huang (2012): where Z 2 R TÂp is a matrix of full column rank, B ðiÞ denotes the ith row of B, and k i [ 0 (i ¼ 1; . . .; p) are tuning parameters. To clarify the similarity, let r ¼ 1, p ¼ T, and k i ¼ k for i ¼ 1; . . .; T in (A.3). Then, A 2 R n , B 2 R T , and Z 2 R TÂT , and we denote them by a, b, and X, respectively. Given kB ðiÞ k 2 ¼ jb i j in this setting, where b ¼ ½b 1 ; . . .; b T 0 , (A.3) finally becomes Fig. 9 The solid lines are identical to those plotted in Fig. 6. The dashed lines are estimated piecewise linear trend from the common piecewise linear trend shown in Fig. 8. More precisely, denoting the estimated common piecewise linear trend by X b b, the dashed line in the upper (resp. lower) panel plots which is not identical to (11), but they are similar.
Proof See Section A.5.2. h Now, we are ready to give a proof of Proposition 4.1. From Lemma A.2, if c\kDYak 1 , then ðYa À X e bÞ in the stationarity condition in Lemma A.1 is not equal to 0. In addition, X 0 is of full column rank. Hence, if c\kDYak 1 , then we have lJ 0 e v ¼ 2X 0 ðYa À X e bÞ 6 ¼ 0, which leads to l 6 ¼ 0. Thus, given l ! 0, it follows that l [ 0. From the complementary slackness condition in Lemma A.1, it follows that kJ e bk 1 ¼ c [ 0. Given that ðJ e bÞ 0 e v ¼ P T t¼3 j e b t j ¼ kJ e bk 1 , premultiplying the stationarity condition by e b 0 yields À2ðX e bÞ 0 ðYa À X e bÞ þ lkJ e bk 1 ¼ 0. Then, given kJ e bk 1 ¼ c [ 0, we obtain (23). In addition, given l [ 0, from the stationarity condition, it follows that e b equals b b estimated with k ¼ l.