AR and MA representation of partial autocorrelation functions, with applications

Inoue, Akihiko

doi:10.1007/s00440-007-0074-1

Cited by 22 publications

(27 citation statements)

References 42 publications

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“…From this perspective it appears more plausible to characterize time series in terms of their PACF rather than the ACF. This opinion has been also advocated by Inoue (2008). In fact, nothing is lost by such reparameterization since there exists a one-to-one correspondence between the finite sequences ðq i Þ n i¼1 and ða i Þ n i¼1 .…”

Section: Introductionmentioning

confidence: 87%

On processes with hyperbolically decaying autocorrelations

Dębowski

2011

Journal of Time Series Analysis

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We discuss some relations between autocorrelations (ACFs) and partial autocorrelations (PACFs) of weakly stationary processes. First, we construct an extension of a process ARIMA(0,d,0) for d 2 ()¥, 0), which enjoys non-summable partial autocorrelations and autocorrelations decaying as rapidly as q n G n )1+2d . Such a situation is impossible if the absolute sum of autocorrelations is sufficiently small. We show that then the PACF is less than the ACF up to a multiplicative constant. Our second result complements a similar result of Baxter (1962).

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Section: Introductionmentioning

confidence: 87%

On processes with hyperbolically decaying autocorrelations

Dębowski

2011

Journal of Time Series Analysis

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“…The proof of statement (i) is based on Baxter's Theorem (Baxter (1962), see also (Bingham, 2012), p. 302), extended by Inoue (2008), according to which the stated assumptions imply the summability the partial autocorrelations φ kk , j |φ jj | < ∞, and therefore j |π jj | < ∞; morever, the AR coefficients φ j are also absolutely summable by Theorem 3.8.4 in Brillinger (1981).…”

Section: Proof Of Lemmamentioning

confidence: 99%

A Durbin-Levinson Regularized Estimator of High Dimensional Autocovariance Matrices

Proietti

Giovannelli

2017

SSRN Journal

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We consider the problem of estimating the high-dimensional autocovariance matrix of a stationary random process, with the purpose of out of sample prediction and feature extraction. This problem has received several solutions. In the nonparametric framework, the literature has concentrated on banding and tapering the sample autocovariance matrix. This paper proposes and evaluates an alternative approach, based on regularizing the sample partial autocorrelation function, via a modified Durbin-Levinson algorithm that receives as input the banded and tapered partial autocorrelations and returns a sample autocovariance sequence which is positive definite. We show that the regularized estimator of the autocovariance matrix is consistent and its convergence rates is established. We then focus on constructing the optimal linear predictor and we assess its properties. The computational complexity of the estimator is of the order of the square of the banding parameter, which renders our method scalable for high-dimensional time series. The performance of the autocovariance estimator and the corresponding linear predictor is evaluated by simulation and empirical applications.

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“…See Inoue [In1,In2,In3], Inoue-Kasahara [IK1,IK2] and Bingham et al [BIK] for relevant results with application in prediction theory. Theorem 4.1 (ii) can be derived from (i) with the aid of Geronimus' theorem and the next parameterization for the one-step extensions of a Nehari sequence γ = (γ 1 , γ 2 , .…”

Section: Verblunsky Coefficients and Nehari Sequencesmentioning

confidence: 99%

Verblunsky coefficients and Nehari sequences

Kasahara

Bingham

2013

Trans. Amer. Math. Soc.

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We are concerned with a rather unfamiliar condition in the theory of the orthogonal polynomials on the unit circle. In general, the Szegö function is determined by its modulus, while the condition in question is that it is also determined by its argument, or in terms of the function theory, that the square of the Szegö function is rigid. In prediction theory, this is known as a spectral characterization of complete nondeterminacy for stationary processes, studied by Bloomfield, Jewel and Hayashi (1983) going back to a small but important result in the work of Levinson and McKean (1964). It is also related to the cerebrated result of Adamyan, Arov and Krein (1968) for the Nehari problem, and there is a one-to-one correspondence between the Verblunsky coefficients and the negatively indexed Fourier coefficients of the phase factor of the Szegö function, which we call a Nehari sequence. We present some fundamental results on the correspondence, including extensions of the strong Szegö and Baxter's theorems.

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AR and MA representation of partial autocorrelation functions, with applications

Cited by 22 publications

References 42 publications

On processes with hyperbolically decaying autocorrelations

On processes with hyperbolically decaying autocorrelations

A Durbin-Levinson Regularized Estimator of High Dimensional Autocovariance Matrices

Verblunsky coefficients and Nehari sequences

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