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A trend filtering method closely related to \(\ell _{1}\) trend filtering

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Abstract

The filtering method developed by Kim et al. (SIAM Rev 51:339–360, 2009), \(\ell _{1}\) trend filtering, is attractive because it enables us to estimate a continuous piecewise linear trend. This paper introduces a new filtering method closely related to \(\ell _{1}\) trend filtering in order to contribute to the accumulation of knowledge on \(\ell _{1}\) trend filtering. We show that the piecewise linearity, which is the key feature of \(\ell _{1}\) trend filtering, is derived from the new filtering. For this reason, we refer to the filtering as ‘pure’ \(\ell _{1}\) trend filtering. We also demonstrate some other miscellaneous results concerning the new filtering.

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Notes

  1. Section 3 contains an empirical result to illustrate them. Also, see empirical applications of the filterings in Yamada and Jin (2013), Yamada and Yoon (2014, 2016a), and Yamada (2017).

  2. Similarly to (6), we may consider the following filtering:

    $$\begin{aligned} \widehat{{{\varvec{z}}}}_{\text {HP}}={{\varvec{D}}}'({{\varvec{D}}}{{\varvec{D}}}')^{-1}\widehat{{{\varvec{\zeta }}}}, \end{aligned}$$

    where

    $$\begin{aligned} \widehat{{{\varvec{\zeta }}}}=\mathop {{\text {arg min}}}\limits _{{{\varvec{\zeta }}}\in \mathbb {R}^{(T-2)}}\,\left( \Vert {{\varvec{y}}}-{{\varvec{D}}}'({{\varvec{D}}}{{\varvec{D}}}')^{-1}{{\varvec{\zeta }}}\Vert _{2}^{2}+\phi \Vert {{\varvec{\zeta }}}\Vert _{2}^{2}\right) . \end{aligned}$$

    As shown in Yamada (2015), \(\widehat{{{\varvec{z}}}}_{\text {HP}}\), which is referred to as pure HP trend, satisfies \(\widehat{{{\varvec{x}}}}_{\text {HP}}=\widehat{{{\varvec{\tau }}}}+\widehat{{{\varvec{z}}}}_{\text {HP}}\). See also Yamada (2018).

  3. \({{\varvec{D}}}{{\varvec{D}}}'\) is a banded Toeplitz matrix of which the determinant is \(T^{2}(T^{2}-1)/12\). See Dow (2003). See also Han (2007), which provides a more general result.

  4. The author is indebted to Kazuhiko Hayakawa for deriving this formula. In addition, Dow (2003) provides the exact expression of \(({{\varvec{D}}}{{\varvec{D}}}')^{-1}\).

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Authors and Affiliations

  1. Department of Economics, Hiroshima University, Higashi-Hiroshima, 739-8525, Japan

    Hiroshi Yamada

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  1. Hiroshi Yamada
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Correspondence to Hiroshi Yamada.

Additional information

The author would like to thank Peter C. B. Phillips, Kazuhiko Hayakawa, Chirok Han, Hiro Y. Toda, Hiroshi Kurata, and Hirofumi Wakaki for their useful suggestions and comments. He also appreciates an anonymous referee and an associate editor for their valuable comments. The usual caveat applies. The Japan Society for the Promotion of Science supported this work through KAKENHI Grant Number 15K13010.

Appendix A: MATLAB function for calculating \(\widehat{{{\varvec{z}}}}\) in (5)

Appendix A: MATLAB function for calculating \(\widehat{{{\varvec{z}}}}\) in (5)

The MATLAB function for calculating \(\widehat{{{\varvec{z}}}}\) in (5), which depends on the CVX software tool developed by Grant and Boyd (2013), is as follows:

figure a

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Yamada, H. A trend filtering method closely related to \(\ell _{1}\) trend filtering. Empir Econ 55, 1413–1423 (2018). https://doi.org/10.1007/s00181-017-1349-8

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