HPX filter: a hybrid of Hodrick–Prescott filter and multiple regression

Hiroshi Yamada

doi:10.1515/snde-2023-0004

Abstract

This paper considers an extension of Hodrick–Prescott (HP) filter. It is a hybrid of HP filter and multiple regression. We refer to the filter as “HPX filter”. It is well known that HP filter has a unique global minimizer and the solution can be represented in matrix notation explicitly. Does HPX filter also have a unique global minimizer? Is it accomplished without any additional assumptions? Can the solution be expressed in matrix notation explicitly? In this paper, we answer these questions. In addition, this paper (i) provides an alternative perspective on the filter by representing it as a generalized ridge regression and (ii) gives an extension of it, which is a hybrid of Whittaker–Henderson method of graduation and multiple regression.

Keywords: cubic smoothing spline; generalized ridge regression; Hodrick–Prescott filter; multiple regression; Whittaker–Henderson method of graduation

JEL Classification: C22

Corresponding author: Hiroshi Yamada, Graduate School of Humanities and Social Sciences, Hiroshima University, Higashi-Hiroshima, Japan, E-mail: yamada@hiroshima-u.ac.jp

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 20K20759

Acknowledgment

I am grateful to an anonymous referee. The usual caveat applies.

Author contributions: The author has accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This study was funded by the Japan Society for the Promotion of Science (https://doi.org/10.13039/501100001691) (KAKENHI Grant Number 20K20759).
Conflict of interest statement: The author declares no conflicts of interest regarding this article.

Appendix A

A.1 Proof of Proposition 1

Given that DD ^⊤ is positive definite, it follows that ω _i > 0 for i = 1, …, T − 2. Accordingly, it follows that 0 < ρ i = λ ω i 1 + λ ω i < 1 for i = 1, …, T − 2. In addition, the spectrum of D ^⊤ D is {0, 0, ω ₁, …, ω _T−2} (Abadir and Magnus 2005, p. 167). Then, the spectrum of I T + λ D ⊤ D − 1 is 1,1 , ( 1 + λ ω 1 ) − 1 , … , ( 1 + λ ω T − 2 ) − 1 , and thus, that of Q λ = I T − I T + λ D ⊤ D − 1 is 0,0,1 − ( 1 + λ ω 1 ) − 1 , … , 1 − ( 1 + λ ω T − 2 ) − 1 . Here, it follows that 1 − ( 1 + λ ω i ) − 1 = λ ω i 1 + λ ω i = ρ i for i = 1, …, T − 2.

A.2 Proof of Lemma 1

(i) Let DX be of full column rank. Given that DΠ = 0, premultiplying Xκ ₁ + Πκ ₂ = 0 by D yields DXκ ₁ = 0. Given that DX is of full column rank, we have κ ₁ = 0. Accordingly, it follows that Πκ ₂ = 0. Given that Π is of full column rank, we also have κ ₂ = 0. Therefore, if DX is of full column rank, then [ X , Π] is of full column rank. (ii) Let [ X , Π] be of full column rank. Suppose that ζ such that DXζ = 0 is not equal to 0. Then, Xζ ≠ 0 belongs to the null space of D . Given that the null space of D is identical to the column space of Π, Xζ belongs to the column space of Π, which contradicts that [ X , Π] is of full column rank. Thus, ζ = 0. Therefore, if [ X , Π] is of full column rank, DX is of full column rank.

A.3 Another Proof of Lemma 4

(i) Let ξ ≠ 0 be an n-dimensional column vector such that Dξ ≠ 0. In other words, ξ ≠ 0 does not belong to the null space of D . Then, given that Q _X is symmetric and idempotent, it follows that ξ ⊤ Q X + λ D ⊤ D ξ = ( Q X ξ ) ⊤ ( Q X ξ ) + λ ( D ξ ) ⊤ ( D ξ ) > 0 . (ii) Let ξ ≠ 0 be an n-dimensional column vector such that Dξ = 0. In other words, ξ ≠ 0 belongs to the null space of D . Suppose that Q _X ξ = 0. Then, ξ belongs to the column space of X , and accordingly, there exists υ ≠ 0 such that ξ = Xυ . Premultiplying ξ = Xυ by D , we have Dξ = DXυ = 0, which contradicts that DX is of full column rank. Thus, Q _X ξ ≠ 0. In addition, again, Q _X is symmetric and idempotent. Then, it follows that ξ ⊤ Q X + λ D ⊤ D ξ = ( Q X ξ ) ⊤ ( Q X ξ ) > 0 . Therefore, from (i) and (ii), if DX is of full column rank, then Q _X + λ D ^⊤ D is positive definite.

A.4 Proof of Corollary 1

From (13), we have the following simultaneous equations for γ ̂ and z ̂ :

(A.1) X ⊤ X γ ̂ + X ⊤ z ̂ = X ⊤ y ,

(A.2) X γ ̂ + P λ − 1 z ̂ = y .

(i) Let us prove (17). Premultiplying (A.2) by X ^⊤ P _λ leads to X ⊤ z ̂ = − X ⊤ P λ X γ ̂ + X ⊤ P λ y . Substituting this into (A.1), we have X ⊤ Q λ X γ ̂ = X ⊤ Q λ y . From Lemma 2, X ^⊤ Q _λ X is positive definite, and thus, we obtain γ ̂ = X ⊤ Q λ X − 1 X ⊤ Q λ y . z ̂ = P λ ( y − X γ ̂ ) is immediately obtainable by premultiplying (A.2) by P _λ. (ii) Let us prove (18). Premultiplying (A.1) by X ( X ⊤ X ) − 1 , we have X γ ̂ = P X ( y − z ̂ ) . Substituting this into (A.2), we have P X ( y − z ̂ ) + z ̂ + λ D ⊤ D z ̂ = y , from which it follows that Q X + λ D ⊤ D z ̂ = Q X y . From Lemma 4, Q X + λ D ⊤ D is positive definite, and thus, we obtain z ̂ = Q X + λ D ⊤ D − 1 Q X y . Finally, γ ̂ = ( X ⊤ X ) − 1 X ⊤ ( y − z ̂ ) is immediately obtainable by premultiplying (A.1) by ( X ⊤ X ) − 1 .

A.5 MATLAB/GNU Octave user-defined functions

In this subsection, we provide MATLAB/GNU Octave user-defined functions to implement HPX filter and its extension.

A.5.1 A function to calculate the column vector defined by (16)

function [alphahat] = HPXfilter(y, X, lambda)

[T, k] = size(X); I = eye(T); D = diff(I, 2);

W = [X, I]; E = [zeros(T-2, k),D];

alphahat=(W’*W + lambda*E’*E)\(W’*y);

end

A.5.2 A function to calculate the column vector defined by (22)

function [betahat] = HPXfilter_alt(y, X, lambda)

[T, k] = size(X); Pi = [ones(T, 1),(1:T)’];

L = tril(ones(T)); Gam = cumsum(L(:, 3:T));

V = [X, Pi, Gam]; S = [zeros(T−2, k+2), eye(T−2)];

betahat=(V’*V + lambda*S’*S)\(V’*y);

end

A.5.3 A function to calculate the column vector defined by (24)

function [alphaphat] = HPXfilter_ext(y, X, lambda, p)

[T, k] = size(X); I = eye(T); Dp = diff(I, p);

W = [X, I]; Ep = [zeros(T−p, k), Dp];

alphaphat = (W’*W + lambda*Ep’*Ep)\(W’*y);

end

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Supplementary Material

This article contains supplementary material (https://doi.org/10.1515/snde-2023-0004).

Received: 2023-01-12

Accepted: 2023-06-04

Published Online: 2023-06-26

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