Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates

Manabu Asai; Shelton Peiris; Michael McAleer; David E. Allen

doi:10.1515/jtse-2018-0024

Published by De Gruyter March 7, 2020

Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates

Manabu Asai , Shelton Peiris , Michael McAleer and David E. Allen

From the journal Journal of Time Series Econometrics

https://doi.org/10.1515/jtse-2018-0024

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Abstract

Recent developments in econometric methods enable estimation and testing of general long memory processes, which include the general Gegenbauer process. This paper considers the error correction model for a vector general long memory process, which encompasses the vector autoregressive fractionally integrated moving average and general Gegenbauer processes. We modify the tests for unit roots and cointegration, based on the concept of heterogeneous autoregression. The Monte Carlo simulations show that the finite sample properties of the modified tests for unit roots are satisfactory, while the conventional tests suffer from size distortion. The experiments also indicate that the modified tests for cointegration improve the problem of finding too many cointegration relationships which arises for fractionally integrated series. Empirical results for interest rates series for the USA and Australia indicate that: (1) the modified unit root test detected unit roots for all series; (2) after differencing, all series favour the general Gegenbauer (GG) process; (3) the modified test for cointegration found only two cointegrating vectors; and (4) the zero interest rate policy in the USA had no effect on the cointegrating vectors for the two countries.

Keywords: long memory processes; Gegenbauer process; Dickey–Fuller Tests; cointegration; differencing; interest rates

JEL Classification: C22; C32; C51

Acknowledgements

Initial draft of this article was written while the first author was a Visiting Scholar at the University of Sydney. The authors are most grateful to Yoshi Baba, Karen Lewis, the Editor and two anonymous reviewers for very helpful comments and suggestions. The first author acknowledges the financial support of the Japan Ministry of Education, Culture, Sports, Science and Technology, the Japan Society for the Promotion of Science (Grant Number: JSPS KAKENHI JP16K03603), the School of Mathematics and Statistics at The University of Sydney, the Zengin Foundation for Studies on Economics and Finance, and the Australian Academy of Science. The second author acknowledges the support from the Faculty of Economics at Soka University. The third author is most grateful for the financial support of the Australian Research Council, Ministry of Science and Technology (MOST), Taiwan, and the Japan Society for the Promotion of Science. The fourth author acknowledges the Australian Research Council.

A Appendix

A.1 Identification and Estimation of Gegenbauer Frequencies

We discuss the identification of k and give a short explanation of semiparametric estimation of the location frequency parameters. First, we explain the semiparametric technique of Hidalgo and Soulier (2004) for estimating the location frequency parameters of (ω1,…,ωk) for the univariate GG process. We assume that k is known until we discuss the identification of parameters.

Let IT(z,λ) be the periodogram defined by:

IT(z,λ)=(2πT)−1∑t=1Tzteitλ2,−π<λ≤π,

for z1,…,zT. Define n˜=⌊(T−1)/2⌋, and let λ_j = 2jπ/T(−n˜≤j≤n˜) be the Fourier frequencies.

For purposes of introducing the approach of Hidalgo and Soulier (2004), we consider a simple case of a univariate process which produces I_T(λ), under the assumptions d = 0, ω₁≠ 0, ω₂≠ 0, d1≥d2, and k = 2. Then we can estimate ω₁ consistently as:

ωˆ1=2πTargmax1≤j≤nIT(λj),

where zT=Texp(−ln(T)), and n is an integer between 1 and ⌊(T−1)/2⌋, satisfying at least:

1n+nT→0as T→∞.

After we estimate ω₁, it is possible to estimate consistently the second location parameter, ω₂, which has sufficient distance from the first location, as:

ωˆ2=2πTargmax1≤j≤n|λj−ωˆ1|≥zT/TIT(λj).

For general k, we can estimate (ω1,…,ωk) sequentially, by applying the above procedure.

Hidalgo and Soulier (2004) modified the GPH estimator of Geweke and Porter-Hudak (1983), which was originally suggested to estimate the long memory parameter, d, using a log-periodogram regression, in order to estimate d_l at the Gegenbauer frequency ω_l. In order to identify the number of location frequencies, k, we follow the approach of Hidalgo and Soulier (2004), based on their modified GPH estimator for d1,…,dk, which is defined by:

(16)dˆl=∑1≤|j|≤n0<ωˆl+λj≤πξklnIT(ωˆl+λj),

where ξk=sn−2(ζ(λj)−ζˉn), ζ(λ)=−ln(|1−eiλ|), ζˉn=n−1∑j=1nζ(λj), and sn2=∑j=1n(ζ(λj)−ζˉn)2. Hidalgo and Soulier (2004) show that n1/2(dˆl−dl) converges weakly to N(0, π²/12), under the assumption of a Gaussian process. For the case ω_l = 0, π, Hidalgo and Soulier (2004) also show that the limiting distribution is N (0, π²/6).

The procedure of Hidalgo and Soulier (2004) consists of the following steps: (i) Find the largest periodogram ordinate; (ii) if the corresponding estimate of d_l is significant, add the respective Gegenbauer filter to the model, otherwise terminate the procedure; (iii) exclude the neighborhood of the last pole from the periodogram, and repeat the procedure from (i) onward. In the empirical analysis, we apply the method just described for identifying k and estimating (ω1,…,ωk) for the GG model.

A.2 Generation of GG Processes

We generate the GG process by the modified Durbin’s algorithm based on the theoretical covariance function for the whole sample, applying the discussion in Doornik and Ooms (2003). In the following, we first explain the calculation of the coefficients of the MA(∞) representation of the general Gegenbauer process in eq. (2) in order to show the calculation of the autocovariance functions.

Even for the simple Gegenbauer process with ARMA parameters, it is not easy to obtain explicit formulae for the coefficients for the MA(∞) representation, and the autocovariances, that are valid for all lags. Recently, McElroy and Holan (2012, 2016) developed a computationally efficient method for calculating these values. The spectral density of the general Gegenbauer process is given by (5). For convenience, we define κ(z) so that g(λ)=|κ(e−iλ)|2. Then, κ(z) takes the form κ(z)=∏l(1−ζlz)pl for (possibly complex) reciprocal roots, ζ_l, of the moving average and autoregressive polynomials, where p_l is one if l corresponds to a moving average root, and minus one if l corresponds to an autoregressive root. We set α = max{d_l} for notational convenience.

Define:

gj=2∑lplζljj,βj=2jd0+2∑l=1kdlcos(ωlj)+gj,ψ˜j=12j∑m=1lmβmψ˜j−m,ψ˜0=1.

McElroy and Holan (2012) showed that the MA(∞) representation of eq. (2) is given by:

yt=μ+∑j=0∞ψ˜jηt−j,

and the autocovariances of h_t for l ≥ 0 are given by:

γl=σ2∑j=0J−1ψ˜jψ˜j+l+RJ(l),

where

RJ(l)=σ2J−1+2αF(1−α,1−2α;2−2α;−l/J)KlΓ2(α)(1−2α){1+o(1)},

and F(a, b; c; z) is the hypergeometric function evaluated at z. In the above, K_l is a component depending on the structure of (5), and our GG processes have

Kl=2cos(lw1)21−cos(2ω1)−d121−cos(ω1−ω2)−d2×21−cos(ω1+ω2)−d21+ϕ2−2ϕcos(ω1)−1,

with k = 2, d₀ = 0, d1≥d2. Note that γ−l=γl. McElroy and Holan (2012) recommend using the cutoff value J ≥ 2000, and we set J = 20000 in this paper.

A.3 Estimation of Cointegrating Vectors

For the modified ECM in (8), we obtain the estimate of α as αˆ=[v1⋯vr], where vi is the eigenvector corresponding to the eigenvalue λ_i (i = 1, …, r), which satisfies the normalization V^'S₁₁V = I, with V=[v1⋯vm]. Simultaneously, we obtain γˆ=S01αˆ. In the decomposition (9), we need to estimate γ_⊥ and α_⊥. Gonzalo and Granger (1995) proved that the ML estimator is obtained by the following procedure. First, solve the equation:

|λS00−S01S11−1S10|=0,

which yields eigenvalues λ1≥λ2≥⋯≥λm and eigenvectors M=[m1⋯mm], normalized such that M^'S₀₀M = I. Using these eigenvalues and eigenvectors, we obtain the ML estimator, γˆ⊥=[mr+1⋯mm], and αˆ⊥=S10γˆ⊥.

Based on eq. (8), we can modify the test on the cointegrating vector as suggested by Johansen (1991). The null hypothesis of the test is:

H0:α=Gφ,

where G is an m × s matrix, and φ is an s × r matrix (r ≤ s ≤ m). The alternative hypothesis is that α consists of r cointegrating vectors without any restrictions. We first solve:

|λG′S11G−G′S10S00−1S01G|=0,

to obtain λg,1≥⋯≥λg,s. Under the null hypothesis α = Gφ, the test statistic:

(17)Qg=T∑i=1rlog(1−λg,i)/(1−λi),

is expected to have an asymptotic χ² distribution with degrees of freedom given by (m – s)r.

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Published Online: 2020-03-07

Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates

Abstract

Acknowledgements

A Appendix

A.1 Identification and Estimation of Gegenbauer Frequencies

A.2 Generation of GG Processes

A.3 Estimation of Cointegrating Vectors

References

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