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Empirical Economics

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19-12-2017

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

Author: Hiroshi Yamada

Published in: Empirical Economics | Issue 4/2018

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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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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).

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).

${{\varvec{D}}}{{\varvec{D}}}'$ 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.

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

Dow M (2003) Explicit inverses of Toeplitz and associated matrices. ANZIAM J 44:185–215CrossRef

Grant M. Boyd S. (2013): CVX: Matlab software for disciplined convex programming, version 2.0 beta. (http://cvxr.com/cvx)

Han C (2007) Determinants of covariance matrices of differenced AR(1) processes. Econom Theory 23:1248–1253CrossRef

Harchaoui Z, Lévy-Leduc C (2010) Multiple change-point estimation with a total variation penalty. J Am Stat Assoc 105(492):1480–1493CrossRef

Hatanaka M, Yamada H (2003) Co-trending: a statistical system analysis of economic trends. Springer, TokyoCrossRef

Hodrick RJ, Prescott EC (1997) Postwar U.S. business cycles: an empirical investigation. J Money Credit Bank 29(1):1–16CrossRef

Kim S, Koh K, Boyd S, Gorinevsky D (2009) $\ell _{1}$ trend filtering. SIAM Rev 51(2):339–360CrossRef

Osborne MR, Presnell B, Turlach BA (2000) On the LASSO and its dual. J Comput Gr Stat 9(2):319–337

Perron P (1989) The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57(6):1361–1401CrossRef

Rappoport P, Reichlin L (1989) Segmented trends and non-stationary time series. Econ J 99:168–177CrossRef

Sato K (2002) From fast to last: the Japanese economy in the 1990s. J Asian Econ 13(2):213–235CrossRef

Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc B 58(1):267–288

Tibshirani RJ, Taylor J (2011) The solution path of the generalized lasso. Ann Stat 39(3):1335–1371CrossRef

Tibshirani RJ (2014) Adaptive piecewise polynomial estimation via trend filtering. Ann Stat 42(1):285–323CrossRef

Yamada H (2015) Ridge regression representations of the generalized Hodrick–Prescott filter. J Jpn Stat Soc 45:121–128CrossRef

Yamada H (2017) Estimating the trend in US real GDP using the $\ell _{1}$ trend filtering. Appl Econ Lett 24(10):713–716CrossRef

Yamada H (2018): Why does the trend extracted by the Hodrick–Prescott filtering seem to be more plausible than the linear trend?. Appl Econ Lett 25(2):102–105CrossRef

Yamada H, Jin L (2013) Japan’s output gap estimation and $\ell _{1}$ trend filtering. Empir Econ 45(1):81–88CrossRef

Yamada H, Yoon G (2014) When Grilli and Yang meet Prebisch and Singer: piecewise linear trends in primary commodity prices. J Int Money Finance 42:193–207CrossRef

Yamada H, Yoon G (2016a) Measuring the US NAIRU as a step function. Empir Econ 51(4):1679–1688CrossRef

Yamada H, Yoon G (2016b) Selecting the tuning parameter of the $\ell _{1}$ trend filter. Stud Nonlinear Dyn Econom 20(1):97–105

Title: A trend filtering method closely related to trend filtering
Author: Hiroshi Yamada
Publication date: 19-12-2017
Publisher: Springer Berlin Heidelberg
Published in: Empirical Economics / Issue 4/2018
Print ISSN: 0377-7332
Electronic ISSN: 1435-8921
DOI: https://doi.org/10.1007/s00181-017-1349-8

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