A New Method for Specifying the Tuning Parameter of ℓ1 Trend Filtering

Hiroshi Yamada

doi:10.1515/snde-2016-0073

Article

A New Method for Specifying the Tuning Parameter of ℓ₁ Trend Filtering

Hiroshi Yamada

Published/Copyright: May 1, 2018

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 22 Issue 4

Abstract

Because Hodrick–Prescott (HP) filtering and ℓ₁ trend filtering are expressed as penalized least squares problem, both of them require the specification of their tuning parameter. For HP filtering, we have accumulated knowledge for selecting the value of its tuning parameter. However, we do not have similar knowledge for ℓ₁ trend filtering. This paper presents a new method for specifying the tuning parameter of ℓ₁ trend filtering so that the sum of squared residuals of HP filtering and that of ℓ₁ trend filtering may be equivalent.

Keywords: constrained minimization; Hodrick–Prescott filtering; ℓ1 trend filtering; tuning parameter

JEL Classification: C22

Acknowledgement

The author thanks an anonymous referee for his/her valuable suggestions and comments. The usual caveat applies. This work was supported by JSPS KAKENHI Grant Number 15K13010.

Appendix

A MATLAB function for calculating x^lt and x^∗

A MATLAB function for calculating x^lt and x^∗ from y and p_c, which depends on the CVX developed by Grant and Boyd (2013), is as follows.

function [x_lt, x_ast] = l1filtering(y, p_c) n = length(y); t = (1:n)′; I_n = eye(n); D = diff(I_n, 2); psi = (2*sin(pi/p_c))^(−4) x_hp = (I_n + psi*D′*D)\y; cvx_clear cvx_begin variables x_ast(n) minimize(norm(D*x_ast, 1)) subject to sum((y − x_ast).^2) < = sum((y − x_hp).^2) cvx_end lambda = 2*norm((D*D′)\(D*(y − x_ast)), inf) cvx_clear cvx_begin variables x_lt(n) minimize(sum((y − x_lt).^2) + lambda*norm(D*x_lt, 1)) cvx_end SSR1 = sum((y − x_hp).^2) SSR2 = sum((y − x_ast).^2) SSR3 = sum((y − x_lt).^2) end

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Keywords for this article

constrained minimization; Hodrick–Prescott filtering; ℓ1 trend filtering; tuning parameter