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Asymptotic Properties of the Estimators in Mildly Stable Unit Root Process

Author

Listed:
  • Xiyao Zhang

    (Nanjing University of Aeronautics and Astronautics)

  • Hui Jiang

    (Nanjing University of Aeronautics and Astronautics)

  • Weilin Xiao

    (Zhejiang University)

  • Weigang Wang

    (Zhejiang Gongshang University)

Abstract

In this paper, we study asymptotic properties of the Yule-Walker estimator and ordinary least squares (OLS) estimator in a mildly stable unit root process with Gaussian noise. By the change of measure method and asymptotic analysis technique, we establish the exponential nonuniform Berry-Esseen bounds of the two estimators. As applications, the optimal uniform Berry-Esseen bounds and optimal Cramér-type moderate deviation principles can be obtained.

Suggested Citation

  • Xiyao Zhang & Hui Jiang & Weilin Xiao & Weigang Wang, 2025. "Asymptotic Properties of the Estimators in Mildly Stable Unit Root Process," Methodology and Computing in Applied Probability, Springer, vol. 27(2), pages 1-22, June.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:2:d:10.1007_s11009-025-10175-5
    DOI: 10.1007/s11009-025-10175-5
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    References listed on IDEAS

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    1. Wang, Xiaohu & Yu, Jun, 2016. "Double asymptotics for explosive continuous time models," Journal of Econometrics, Elsevier, vol. 193(1), pages 35-53.
    2. Liudas Giraitis & Peter C. B. Phillips, 2006. "Uniform Limit Theory for Stationary Autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(1), pages 51-60, January.
    3. Phillips, Peter C B, 1988. "Regression Theory for Near-Integrated Time Series," Econometrica, Econometric Society, vol. 56(5), pages 1021-1043, September.
    4. Phillips, Peter C.B. & Magdalinos, Tassos, 2007. "Limit theory for moderate deviations from a unit root," Journal of Econometrics, Elsevier, vol. 136(1), pages 115-130, January.
    5. Phillips, Peter C.B. & Magdalinos, Tassos & Giraitis, Liudas, 2010. "Smoothing local-to-moderate unit root theory," Journal of Econometrics, Elsevier, vol. 158(2), pages 274-279, October.
    6. Phillips, Peter C B, 1977. "Approximations to Some Finite Sample Distributions Associated with a First-Order Stochastic Difference Equation," Econometrica, Econometric Society, vol. 45(2), pages 463-485, March.
    7. Yu Miao & Yanling Wang & Guangyu Yang, 2015. "Moderate Deviation Principles for Empirical Covariance in the Neighbourhood of the Unit Root," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(1), pages 234-255, March.
    8. Satchell, Stephen Ellwood, 1984. "Approximation to the Finite Sample Distribution for Nonstable First Order Stochastic Difference Equations," Econometrica, Econometric Society, vol. 52(5), pages 1271-1289, September.
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