Filtering Persistent and Asymmetric Cycles

Luiggi Donayre

doi:10.1515/bejm-2022-0091

Artikel

Filtering Persistent and Asymmetric Cycles

Luiggi Donayre

Veröffentlicht/Copyright: 15. August 2022

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Aus der Zeitschrift The B.E. Journal of Macroeconomics Band 23 Heft 1

Abstract

This paper evaluates the ability of the trend-cycle decomposition approach of Hamilton (2018. “Why You Should Never Use the Hodrick-Prescott Filter.” The Review of Economics and Statistics 100 (5): 831–43) to adequately characterize linear and (a)symmetric nonlinear business cycles fluctuations that are known to be persistent. This ability is contrasted to that of the Hodrick–Prescott filter. By means of Monte Carlo simulations, the results indicate that neither filter is able to preserve the cyclical properties of the data-generating process nor reproduce U.S. business cycles features, although this inability is exacerbated for the decomposition of Hamilton (2018. “Why You Should Never Use the Hodrick–Prescott Filter.” The Review of Economics and Statistics 100 (5): 831–43). Based on these findings, caution is called into question when this approach is applied to linear or nonlinear processes that are thought to exhibit persistence.

Keywords: business cycles; detrending; asymmetry; persistence; Markov-switching

JEL Classification: C15; C22; E32

Corresponding author: Luiggi Donayre, Department of Economics, University of Minnesota - Duluth, 1318 Kirby Dr., Duluth, MN 55812, USA, E-mail: adonayre@d.umn.edu.

Acknowledgements

Helpful comments from two anonymous referees, Yunjong Eo, Ronald Miranda, Tara Sinclair and the participants at the 2020 MERG Annual Meeting, INFER 2020 Annual Conference, 2021 WEAI International Conference, the 2021 Asian Meeting of the Econometric Society, the 2021 Latin American Meeting of the Econometric Society, the 15th International Conference in Computational and Financial Econometrics and the University of Minnesota - Duluth are gratefully acknowledged. All errors are my own.

Appendix A: Stationarity Conditions for Hamilton (2018)’s Decomposition Approach

The main result in Hamilton (2018), that the residuals in Eq. (3) mimic the stationary component of any series y_t, hinges on the assumption that the dth difference of y_t, Δ^dy_t, is stationary for some d ≥ 1. The series Δ^dy_t = z_t is stationary if it has a fixed mean z ̄ and satisfies a functional central limit theorem. This requires that the sample mean and the mean of z_t using only Tr observations for 0 < r ≤ 1, follow a normal distribution as T → ∞. Formally,

(A-1) T − 1 / 2 ∑ s = 1 Tr z t − z ̄ → p ω ̃ W ( r )

where . is the floor function, which provides the largest integer less than or equal to the argument, ω ̃ W ( r ) is a standard Brownian motion and → p denotes weak convergence in probability. Stock (1994, p. 2749) demonstrates that a sufficient condition that implies (A-1) is given by:

(A-2) z t = z ̄ + ∑ j = 0 ∞ ψ j η j

where η_j is a martingale difference sequence with variance σ η 2 and finite fourth moment, ψ₁ ≠ 0, and ∑ j = 0 ∞ j ψ j < ∞ , in which case ω ̃ in Eq. (A-1) is given by σ_ηψ₁.

To ensure that the model (1a)–(1c) fulfills this stationarity requirement, note that any UC model with a stationary cycle admits an equivalent, univariate autoregressive integrated moving average (ARIMA) process.^[16] Substituting Eqs. (1b) and (1c) into (1a) and taking first differences yield:

(A-3) ϕ p ℓ ( L ) Δ y t = ϕ p ℓ ( 1 ) μ + ϕ p ℓ ( L ) v t + θ q ℓ ( L ) e t

where L is the lag operator, ϕ p ℓ ( L ) and θ q ℓ ( L ) are lag polynomials, μ is the drift of the permanent component, and v_t and e_t are defined in Eqs. (1a)–(1c). Because the right-hand side of Eq. (A-3) will have non-zero autocovariances through lag q ℓ * = max ( p ℓ , q ℓ + 1 ) , Granger’s lemma (Granger and Newbold 1986, p. 29) implies that the univariate representation of a stationary UC model is given by:

(A-4) ϕ p ℓ ( L ) Δ y t = μ * + θ q ℓ * ( L ) u t

where u_t is a function of v_t and e_t and μ* is a function of μ and the autoregressive parameters ϕ p ℓ ( 1 ) . It can be shown that Eq. (A-4) fulfills the sufficient condition for stationarity given in (A-2).

A.1 UC-AR(1) model

For the UC model (1a) and (1b) with a linear AR(1) process for the transitory component (1c), used in the main Monte Carlo analysis of Section 3, Eq. (A-4) becomes

(A-5) 1 − ϕ L Δ y t = μ * + v t − ϕ v t − 1 + e t − e t − 1 = μ * + u t − θ * u t − 1

where μ* = (1 − ϕ)μ, u_t = v_t + e_t with u t ∼ N 0 , σ u 2 and θ* is a function of the autoregressive coefficient ϕ. Therefore, its reduced-form is a restricted ARIMA(1,1,1) process. That is, Δ^dy_t ∼ I(0) with d = 1. From Eq. (A-5), it follows that:

(A-6) 1 − ϕ L Δ y t = μ * + u t − θ * u t − 1 Δ y t = μ * 1 − ϕ L + 1 1 − ϕ L u t − θ * 1 − ϕ L u t − 1 = μ * 1 − ϕ L + λ ( L ) 1 − θ * L u t = μ * 1 − ϕ L + ∑ j = 0 ∞ λ j 1 − θ * L u j

where the third equality follows from the stationarity condition ϕ < 1 , which implies that λ ( L ) = 1 − ϕ L − 1 = ∑ i = 0 ∞ ϕ i L i and that λ ( L ) 1 − θ * L is finite. Therefore, the coefficients λ_j are convolutions of θ* and ϕ such that lim_j→∞λ_j = 0. Since u_t is a martingale sequence, 1 − θ * L u t is a martingale difference sequence. Further, 1 − θ * L u t has a finite fourth moment since u t ∼ i.i.d N 0 , σ u 2 . Finally, λ₁ = ϕ ≠ 0, and ∑ j = 0 ∞ j λ j < ∞ . Comparing Eq. (A-2) and (A-6) reveals that Δ^dy_t satisfies the sufficient condition (A-2) and, therefore, the requirement (A-1) from Hamilton (2018).

A.2 UC-MSAR(1) model

For the case of the UC model (1a) and (1b) with a Markov-switching AR(1) process for the transitory component (1c), Eq. (A-4) involves a first-order regime-switching process:

(A-7) 1 − ϕ L Δ y t = μ S t * + v t − ϕ v t − 1 + e t − e t − 1 = μ S t * + u t − θ * u t − 1

where μ S t * = ( 1 − ϕ ) μ + ( 1 − ϕ L ) ( 1 − ϕ ) − 1 Δ π S t , with π S t = π 0 1 S t = 0 + π 1 1 S t = 1 and u t ∼ N 0 , σ u 2 as in Eq. (A-5). Therefore, the reduced-form of the UC model (1a)–(1c), with a Markov-switching AR(1) transitory component, is an ARMA(1,1) process with a regime-switching mean μ S t * .

Note that Eqs. (A-5) and (A-7) are identical, except for the terms μ* and μ S t * . It follows from Eq. (A-7) that Δ^dy_t will be integrated of order d = 1 and will satisfy the sufficient condition (A-2) if the MSAR(1) cycle is stationary, since μ S t * is a function of the switching intercept π S t . This is also consistent with the MA component in Eq. (A-7) being irrelevant for stationarity (Francq and Zakoïan 2001, p. 345), so that only μ S t * determines stationarity.

Therefore, it suffices to show that the MSAR(1) cycle is stationary. For any ARMA(1,q_ℓ) process, Francq and Zakoïan (2001, p. 345–347) show that a sufficient condition for stationarity is:

(A-8) ρ ( P ̃ ) < 1

where ρ( M ) is the spectral radius of any matrix M such that ρ ( M ) = max λ ∈ E λ , with E the set of eigenvalues of M ; and P ̃ is a m s K 2 p ℓ 2 × m s K 2 p ℓ 2 matrix of transition probabilities and autoregressive coefficients, with m_s the number of Markov states, K the number of variables and p_ℓ the number of autoregressive coefficients. In the specific case of the MSAR(1) transitory component of Section 3, m_s = 2, K = 1 and p_ℓ = 1, so P ̃ is a (2 × 2) matrix given by

(A-9) P ̃ = ϕ 2 q ( 1 − p ) ( 1 − q ) p

where p and q are the transition probabilities defined in Eqs. (2a) and (2b), and ϕ is the autoregressive coefficient. In Section 3, the two values for ϕ considered are 0.5 and 0.9. When ϕ = 0.5, and given p = 0.7 and q = 0.9, it follows that the sufficient condition of Eq. (A-8) becomes ρ ( P ̃ ) = max 0.3 , 0.5 < 1 . When ϕ = 0.9, ρ ( P ̃ ) = max 0.54 , 0.9 < 1 .

Hence, the sufficient condition (A-8) is satisfied for the values of ϕ considered, which proves that the MSAR(1) transitory components of Section 3 are stationary. This shows, subsequently, that the UC-MSAR(1) model satisfies the sufficient condition (A-2) and, consequently, the requirement (A-1) from Hamilton (2018).

Appendix B: Supplemental Material

Table 5:

Maximum likelihood estimates (MLE): asymmetric MSAR(1) transitory component.

Parameter	T		200			500
	π₀ − π₁	0.4		2.0		0.4		2.0
	ϕ	0.5	0.9	0.5	0.9	0.5	0.9	0.5	0.9
(a) HD-filtered cycle
π ₀		0.971	0.982	0.995	1.135	0.772	0.824	0.849	0.944
π ₁		−0.967	−0.971	−1.233	−1.617	−0.745	−0.816	−1.083	−1.466
ϕ		0.697	0.783	0.720	0.805	0.730	0.800	0.747	0.816
σ ²		1.265	1.341	1.464	1.813	1.353	1.408	1.540	1.860
P		0.745	0.803	0.737	0.800	0.788	0.834	0.750	0.811
Q		0.781	0.809	0.812	0.854	0.818	0.838	0.829	0.874
(b) HP-filtered cycle
π ₀		0.833	0.801	0.651	0.605	0.777	0.702	0.537	0.535
π ₁		−0.823	−0.777	−1.348	−1.296	−0.742	−0.672	−1.270	−1.246
ϕ		0.438	0.613	0.501	0.717	0.472	0.636	0.518	0.728
σ ²		0.971	0.937	1.098	1.080	1.025	0.970	1.136	1.106
P		0.648	0.702	0.573	0.662	0.637	0.710	0.565	0.657
q		0.659	0.700	0.785	0.838	0.648	0.711	0.813	0.850
(c) True cycle
π ₀		0.563	0.435	0.018	−0.032	0.485	0.340	0.004	−0.019
π ₁		−0.834	−0.823	−2.041	−2.106	−0.739	−0.689	−2.012	−2.047
ϕ		0.418	0.824	0.497	0.887	0.445	0.855	0.496	0.895
σ ²		0.863	0.887	0.977	0.979	0.907	0.936	0.992	0.992
p		0.678	0.771	0.668	0.694	0.691	0.770	0.692	0.696
q		0.712	0.802	0.889	0.896	0.727	0.833	0.895	0.897

Panels (a) and (b): Maximum likelihood (ML) estimates of the MSAR(1) cycle filtered from Eqs. (1a)–(1c) using the HD and HP filters, respectively. Panel (c): ML estimates of the true MSAR(1) cycle. In all cases, p = 0.7, q = 0.9, σ e 2 = 1 and π₀ = 0. Estimates are averaged across Monte Carlo repetitions.

Table 6:

Correlations and RSME: asymmetric MSAR(1) transitory component.

			Sample size
			200		500
	π₀ − π₁	ϕ	0.5	0.9	0.5	0.9
(a) HD approach
Correlation	0.4		0.492	0.473	0.482	0.453
	2.0		0.592	0.504	0.583	0.482
RMSE	0.4		4.414	9.860	4.654	10.067
	2.0		6.733	46.357	6.970	46.525
(b) HP filter
Correlation	0.4		0.717	0.588	0.714	0.572
	2.0		0.770	0.616	0.768	0.598
RMSE	0.4		1.012	5.126	1.016	5.070
	2.0		2.471	37.531	2.452	37.312

Correlations and RMSE between the true MSAR(1) cycle and the cycle from Eqs. (1a)–(1c) using the approach of Hamilton (2018), in panel (a), and the HP filter, in panel (b). Estimates are averaged across Monte Carlo repetitions.

Table 7:

Regression diagnostics: asymmetric MSAR(1) transitory component.

			Sample size
			200		500
	π₀ − π₁	ϕ	0.5	0.9	0.5	0.9
δ ̂	0.4		−0.116	0.019	−0.107	0.022
	2.0		−0.043	0.099	−0.043	0.098
Adjusted-R²	0.4		0.050	0.026	0.040	0.011
	2.0		0.029	0.025	0.015	0.014

Slope coefficient ( δ ̂ ) and adjusted-R² from Eq. (5). Estimates are averaged across Monte Carlo repetitions.

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Received: 2022-05-11

Revised: 2022-07-12

Accepted: 2022-07-20

Published Online: 2022-08-15

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Schlagwörter für diesen Artikel

business cycles; detrending; asymmetry; persistence; Markov-switching