Hostname: page-component-5b777bbd6c-j65dx Total loading time: 0 Render date: 2025-06-18T19:26:29.851Z Has data issue: false hasContentIssue false
  • English
  • Français

On the edge eigenvalues of the precision matrices of nonstationary autoregressive processes

Part of: Basic linear algebra Inference from stochastic processes

Published online by Cambridge University Press:  27 May 2025

Junho Yang Open the ORCID record for Junho Yang [Opens in a new window]
Junho Yang*
Affiliation:
Academia Sinica
*
*Postal address: Institute of Statistical Science, No.128, Academia Rd., Sect. 2, Taipei 115, Taiwan. Email: junhoyang@stat.sinica.edu.tw
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper investigates structural changes in the parameters of first-order autoregressive (AR) models by analyzing the edge eigenvalues of the precision matrices. Specifically, edge eigenvalues in the precision matrix are observed if and only if there is a structural change in the AR coefficients. We show that these edge eigenvalues correspond to the zeros of a determinantal equation. Additionally, we propose a consistent estimator for detecting outliers within the panel time series framework, supported by numerical experiments.

Keywords

Autoregressive processempirical spectral distributionedge eigenvaluesstructural change model

MSC classification

Primary: 62M15: Spectral analysis
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 33
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Arbenz, P., Gander, W. and Golub, G. H. (1988). Restricted rank modification of the symmetric eigenvalue problem: Theoretical considerations. Linear Algebra Appl. 104, 75–95.10.1016/0024-3795(88)90309-6CrossRefGoogle Scholar
Aue, A. and Horváth, L. (2013). Structural breaks in time series. J. Time Series Anal. 34, 1–16.10.1111/j.1467-9892.2012.00819.xCrossRefGoogle Scholar
Bagshaw, M. and Johnson, R. A. (1977). Sequential procedures for detecting parameter changes in a time-series model. J. Amer. Statist. Assoc. 72, 593–597.10.1080/01621459.1977.10480618CrossRefGoogle Scholar
Baik, J. and Silverstein, J. W. (2006). Eigenvalues of large sample covariance matrices of spiked population models. J. Multivariate Anal. 97, 1382–1408.10.1016/j.jmva.2005.08.003CrossRefGoogle Scholar
Benaych-Georges, F. and Nadakuditi, R. R. (2011). The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Adv. Math. 227, 494–521.10.1016/j.aim.2011.02.007CrossRefGoogle Scholar
Billingsley, P. (2008). Probability and Measure. John Wiley & Sons, Hoboken, NJ.Google Scholar
Cai, T., Liu, W. and Luo, X. (2011). A constrained $\ell_1$ minimization approach to sparse precision matrix estimation. J. Amer. Statist. Assoc. 106, 594–607.10.1198/jasa.2011.tm10155CrossRefGoogle Scholar
Chang, I., Tiao, G. C. and Chen, C. (1988). Estimation of time series parameters in the presence of outliers. Technometrics 30, 193–204.10.1080/00401706.1988.10488367CrossRefGoogle Scholar
Davis, R. A., Huang, D. and Yao, Y.-C. (1995). Testing for a change in the parameter values and order of an autoregressive model. Ann. Statist. 282–304.Google Scholar
Ding, X. (2021). Spiked sample covariance matrices with possibly multiple bulk components. Random Matrices Theory Appl. 10, 2150014.10.1142/S2010326321500143CrossRefGoogle Scholar
Ding, X. and Zhou, Z. (2023). Autoregressive approximations to nonstationary time series with inference and applications. Ann. Statist. 51, 1207–1231.10.1214/23-AOS2288CrossRefGoogle Scholar
Fox, A. J. (1972). Outliers in time series. J. R. Stat. Soc. Ser. B. Stat. Methodol. 34, 350–363.10.1111/j.2517-6161.1972.tb00912.xCrossRefGoogle Scholar
Gombay, E. (2008). Change detection in autoregressive time series. J. Multivariate Anal. 99, 451–464.10.1016/j.jmva.2007.01.003CrossRefGoogle Scholar
Gombay, E. and Serban, D. (2009). Monitoring parameter change in AR(p) time series models. J. Multivariate Anal. 100, 715–725.10.1016/j.jmva.2008.08.005CrossRefGoogle Scholar
Grenander, U. and Szegö, G. (1958). Toeplitz Forms and Their Applications. University of California Press, Oakland, CA.10.1525/9780520355408CrossRefGoogle Scholar
Hillmer, S. C., William, R. and George, C. (1983). Modeling considerations in the seasonal adjustment of economic time series. Applied Time Series Analysis of Economic Data 74–100. Washington, DC: U.S. Bureau of the Census.Google Scholar
Lee, S., Ha, J., Na, O. and Na, S. (2003). The cusum test for parameter change in time series models. Scand. J. Stat. 30, 781–796.10.1111/1467-9469.00364CrossRefGoogle Scholar
Page, E. S. (1955). A test for a change in a parameter occurring at an unknown point. Biometrika 42, 523–527.10.1093/biomet/42.3-4.523CrossRefGoogle Scholar
Stroeker, R. J. (1983). Approximations of the eigenvalues of the covariance matrix of a first-order autoregressive process. J. Econometrics 22, 269–279.10.1016/0304-4076(83)90103-3CrossRefGoogle Scholar
Tsay, R. S. (1988). Outliers, level shifts, and variance changes in time series. J. Forecast. 7, 1–20.10.1002/for.3980070102CrossRefGoogle Scholar
Wu, W. B. and Pourahmadi, M. (2009). Banding sample autocovariance matrices of stationary processes. Statist. Sinica 19, 1755–1768.Google Scholar