An explicit formula for the smoother weights of the Hodrick–Prescott filter

Hiroshi Yamada; Fatima Tuj Jahra

doi:10.1515/snde-2018-0035

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An explicit formula for the smoother weights of the Hodrick–Prescott filter

Hiroshi Yamada und Fatima Tuj Jahra

Veröffentlicht/Copyright: 23. Januar 2019

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Aus der Zeitschrift Studies in Nonlinear Dynamics & Econometrics Band 23 Heft 5

Abstract

By applying the Sherman–Morrison–Woodbury (SMW) formula and a discrete cosine transformation matrix, De Jong and Sakarya [De Jong, R. M., and N. Sakarya. 2016. “The Econometrics of the Hodrick–Prescott Filter.” Review of Economics and Statistics 98 (2): 310–317] recently derived an explicit formula for the smoother weights of the Hodrick–Prescott filter. More recently, by applying the SMW formula and the spectral decomposition of a symmetric tridiagonal Toeplitz matrix, Cornea-Madeira [Cornea-Madeira, A. 2017. “The Explicit Formula for the Hodrick–Prescott Filter in Finite Sample.” Review of Economics and Statistics 99: 314–318] provided a simpler formula. This paper provides an alternative simpler formula for it and explains the reason why our approach leads to a simpler formula.

Keywords: hodrick–prescott filter; pentadiagonal toeplitz matrix; smoother weights; whittaker–henderson method of graduation

MSC 2010: 62G05

JEL Classification: C22

Funding source: Japan Society for the Promotion of Science

Award Identifier / Grant number: 16H03606

Funding statement: The Japan Society for the Promotion of Science supported this work through KAKENHI Grant Number 16H03606.

Acknowledgments

We appreciate two anonymous referees for their valuable suggestions and comments. An earlier draft entitled “An Alternative Explicit Formula for the Hodrick-Prescott Filter in Finite Sample” was presented at the 26th Annual Symposium of the Society for Nonlinear Dynamics and Econometrics held at Keio University in Japan. Our thanks to the participants for their useful comments. The usual caveat applies.

A Appendix

A.1 Proof of (5)

From (4), γ1−γ2 is

γ1−γ2=[1T2Tcos⁡((2−1)(1−0.5)πT)⋯2Tcos⁡((T−1)(1−0.5)πT)]−[1T2Tcos⁡((2−1)(2−0.5)πT)⋯2Tcos⁡((T−1)(2−0.5)πT)].

Let βj=π(j−1)/(2T) for j = 2, …, T. Then, it follows that

2Tcos⁡((j−1)(1−0.5)πT)−2Tcos⁡((j−1)(2−0.5)πT)=2T[cos⁡(βj)−cos⁡(3βj)]=32Tsin2⁡(βj)cos⁡(βj).

The last equality follows from cos⁡(βj)−cos⁡(3βj)=4sin2⁡(βj)cos⁡(βj).

A.2 Proof of (6)

From (4), γT−γT−1 is

γT−γT−1=[1T2Tcos⁡((2−1)(T−0.5)πT)⋯2Tcos⁡((T−1)(T−0.5)πT)]−[1T2Tcos⁡((2−1)(T−1−0.5)πT)⋯2Tcos⁡((T−1)(T−1−0.5)πT)]

Let βj=π(j−1)/(2T) and κj=2Tβj=π(j−1) for j = 2, …, T. Then, it follows that

2Tcos⁡((j−1)(T−0.5)πT)−2Tcos⁡((j−1)(T−1−0.5)πT)=2T[cos⁡(βj(2T−1))−cos⁡(βj(2T−3))]=2T[cos⁡(κj−βj)−cos⁡(κj−3βj)].

Here, since sin⁡(κj)=0, it follows that

cos⁡(κj−βj)−cos⁡(κj−3βj)=cos⁡(κj)cos⁡(βj)+sin⁡(κj)sin⁡(βj)−cos⁡(κj)cos⁡(3βj)−sin⁡(κj)sin⁡(3βj)=cos⁡(κj)[cos⁡(βj)−cos⁡(3βj)]+sin⁡(κj)[sin⁡(βj)−sin⁡(3βj)]=cos⁡(κj)[4sin2⁡(βj)cos⁡(βj)]+sin⁡(κj)[4sin3⁡(βj)−2sin⁡(βj)]=4sin2⁡(βj)[cos⁡(κj)cos⁡(βj)+sin⁡(κj)sin⁡(βj)]−2sin⁡(κj)sin⁡(βj)=4sin2⁡(βj)cos⁡(κj−βj),

where

κj−βj=2Tβj−βj=βj(2T−1)=π(j−1)(2T−1)2T=π(j−1)(T−0.5)T.

A.3 Application of the SMW formula to (B − αVV^⊤)⁻¹

As in Cornea-Madeira (2017), by applying the SMW formula to (B−αq1q1⊤−αqTqT⊤)−1, we obtain the following results:

(27)(B−αq1q1⊤−αqTqT⊤)−1=(B−αq1q1⊤)−1+α(B−αq1q1⊤)−1qTqT⊤(B−αq1q1⊤)−11−αqT⊤(B−αq1q1⊤)−1qT,

where

(28)(B−αq1q1⊤)−1=B−1+αB−1q1q1⊤B−11−αq1⊤B−1q1.

On the other hand, by applying the SMW formula to (B−αVV⊤)−1, we obtain

(29)(B−αVV⊤)−1=B−1−B−1V(V⊤B−1V−α−1I2)−1V⊤B−1=B−1−[B−1q1,B−1qT][q1⊤B−1q1−α−1q1⊤B−1qTqT⊤B−1q1qT⊤B−1qT−α−1]−1[q1⊤B−1qT⊤B−1].

By comparing (29) with (27) and (28), it is observable that (29) is preferable to (27), mainly because (29) is symmetric with respect to q₁ and q_T.

A.4 A MATLAB/GNU Octave function to calculate (I_T + α D^⊤D)⁻¹ in (1) based on (23)

function HP_hat_matrix = calc_HP_hat_matrix(T, alpha) % T: sample size % alpha: smoothing parameter Lam = diag( 4*(sin((1:T-2)*pi/(2*(T-1))).^2) ); G = zeros(T-2,T-2); for i = 1:T-2 for j = 1:T-2 G(i,j) = sqrt(2/(T-1))*sin(i*j*pi/(T-1)); end end invC = zeros(T-2,T-2); for i = 1:T-2 for j = 1:T-2 s = 0; for k = 1:T-2 s = s + G(i,k)*G(j,k)/( (1/alpha)+Lam(k,k)^2 ); end invC(i,j) = s; end end Xi = zeros(T,T); DG = diff([zeros(2,T-2);G;zeros(2,T-2)],2); for i = 1:T for j = 1:T s = 0; for k = 1:T-2 s = s+DG(i,k)*DG(j,k)/((1/alpha)+Lam(k,k)^2); end Xi(i,j) = s; end end Up1 = zeros(T,1); Up2 = zeros(T,1); for i=1:T s1 = 0; s2 = 0; for k = 1:T-2 s1 = s1+DG(i,k)*G(1,k)/((1/alpha)+Lam(k,k)^2); s2 = s2+DG(i,k)*G(end,k)/((1/alpha)+Lam(k,k)^2); end Up1(i) = s1; Up2(i) = s2; end c11 = invC(1,1); c22 = invC(end,end); c12 = invC(1,end); c21 = invC(end,1); den = (1+c11)*(1+c22)-c12*c21; Tau = zeros(T,T); for i=1:T for j=1:T num = (1+c22)*Up1(i)*Up1(j)-c12*Up1(i)*Up2(j)-c21*Up2(i)*Up1(j) ... +(1+c11)*Up2(i)*Up2(j); Tau(i,j) = num/den; end end Z = zeros(T,T); I = eye(T); for i=1:T for j=1:T Z(i,j) = I(i,j)-Xi(i,j)+Tau(i,j); end end HP_hat_matrix = Z; end

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Published Online: 2019-01-23

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hodrick–prescott filter; pentadiagonal toeplitz matrix; smoother weights; whittaker–henderson method of graduation

An explicit formula for the smoother weights of the Hodrick–Prescott filter

Artikel

Abstract

Acknowledgments

A Appendix

A.1 Proof of (5)

A.2 Proof of (6)

A.3 Application of the SMW formula to (B − αVV⊤)−1

A.4 A MATLAB/GNU Octave function to calculate (IT + α D⊤D)−1 in (1) based on (23)

References

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A.3 Application of the SMW formula to (B − αVV^⊤)⁻¹

A.4 A MATLAB/GNU Octave function to calculate (I_T + α D^⊤D)⁻¹ in (1) based on (23)