IDEAS home Printed from https://ideas.repec.org/a/spr/indpam/v56y2025i2d10.1007_s13226-023-00506-y.html
   My bibliography  Save this article

Limit theory for an AR(1) model with intercept and a possible infinite variance

Author

Listed:
  • Qing Liu

    (Jiangxi University of Finance and Economics
    Jiangxi University of Finance and Economics
    Jiangxi University of Finance and Economics)

  • Chiyu Xia

    (Jiangxi University of Finance and Economics
    Jiangxi University of Finance and Economics)

  • Xiaohui Liu

    (Jiangxi University of Finance and Economics
    Jiangxi University of Finance and Economics)

Abstract

In this paper, we derive the limit distribution of the least squares estimator for an AR(1) model with a non-zero intercept and a possible infinite variance. It turns out that the estimator has a quite different limit for the cases of $$|\rho | 1$$ | ρ | > 1 , and $$\rho = 1 + \frac{c}{n^\alpha }$$ ρ = 1 + c n α for some constant $$c \in R$$ c ∈ R and $$\alpha \in (0, 1]$$ α ∈ ( 0 , 1 ] , and whether or not the variance of the model errors is infinite also has a great impact on both the convergence rate and the limit distribution of the estimator.

Suggested Citation

  • Qing Liu & Chiyu Xia & Xiaohui Liu, 2025. "Limit theory for an AR(1) model with intercept and a possible infinite variance," Indian Journal of Pure and Applied Mathematics, Springer, vol. 56(2), pages 615-630, June.
    • Handle: RePEc:spr:indpam:v:56:y:2025:i:2:d:10.1007_s13226-023-00506-y
    DOI: 10.1007/s13226-023-00506-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13226-023-00506-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s13226-023-00506-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Wang, Xiaohu & Yu, Jun, 2015. "Limit theory for an explosive autoregressive process," Economics Letters, Elsevier, vol. 126(C), pages 176-180.
    2. Phillips, P.C.B., 1990. "Time Series Regression With a Unit Root and Infinite-Variance Errors," Econometric Theory, Cambridge University Press, vol. 6(1), pages 44-62, March.
    3. 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.
    4. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2009. "Heteroskedastic Time Series With A Unit Root," Econometric Theory, Cambridge University Press, vol. 25(5), pages 1228-1276, October.
    5. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-1072, June.
    6. Donald W. K. Andrews & Patrik Guggenberger, 2014. "A Conditional-Heteroskedasticity-Robust Confidence Interval for the Autoregressive Parameter," The Review of Economics and Statistics, MIT Press, vol. 96(2), pages 376-381, May.
    7. Sai-Hua Huang & Tian-Xiao Pang & Chengguo Weng, 2014. "Limit Theory for Moderate Deviations from a Unit Root Under Innovations with a Possibly Infinite Variance," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 187-206, March.
    8. So, Beong Soo & Shin, Dong Wan, 1999. "Cauchy Estimators For Autoregressive Processes With Applications To Unit Root Tests And Confidence Intervals," Econometric Theory, Cambridge University Press, vol. 15(2), pages 165-176, April.
    9. Donald W. K. Andrews & Patrik Guggenberger, 2009. "Hybrid and Size-Corrected Subsampling Methods," Econometrica, Econometric Society, vol. 77(3), pages 721-762, May.
    10. Anna Mikusheva, 2007. "Uniform Inference in Autoregressive Models," Econometrica, Econometric Society, vol. 75(5), pages 1411-1452, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Andrews, Donald W.K. & Cheng, Xu & Guggenberger, Patrik, 2020. "Generic results for establishing the asymptotic size of confidence sets and tests," Journal of Econometrics, Elsevier, vol. 218(2), pages 496-531.
    2. Christis Katsouris, 2022. "Asymptotic Theory for Unit Root Moderate Deviations in Quantile Autoregressions and Predictive Regressions," Papers 2204.02073, arXiv.org, revised Aug 2023.
    3. Donald W. K. Andrews & Patrik Guggenberger, 2014. "A Conditional-Heteroskedasticity-Robust Confidence Interval for the Autoregressive Parameter," The Review of Economics and Statistics, MIT Press, vol. 96(2), pages 376-381, May.
    4. Skrobotov Anton, 2023. "Testing for explosive bubbles: a review," Dependence Modeling, De Gruyter, vol. 11(1), pages 1-26, January.
    5. Guo, Gangzheng & Wang, Shaoping & Sun, Yixiao, 2018. "Testing for Moderate Explosiveness in the Presence of Drift," University of California at San Diego, Economics Working Paper Series qt2k26h10n, Department of Economics, UC San Diego.
    6. Andrews, Donald W.K. & Guggenberger, Patrik, 2012. "Asymptotics for LS, GLS, and feasible GLS statistics in an AR(1) model with conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 169(2), pages 196-210.
    7. Lin, Yingqian & Tu, Yundong, 2020. "Robust inference for spurious regressions and cointegrations involving processes moderately deviated from a unit root," Journal of Econometrics, Elsevier, vol. 219(1), pages 52-65.
    8. Kruse, Robinson & Kaufmann, Hendrik & Wegener, Christoph, 2018. "Bias-corrected estimation for speculative bubbles in stock prices," Economic Modelling, Elsevier, vol. 73(C), pages 354-364.
    9. Xiaohui Liu & Yuzi Liu & Yao Rao & Fucai Lu, 2021. "A Unified test for the Intercept of a Predictive Regression Model," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 83(2), pages 571-588, April.
    10. Westerlund, Joakim, 2014. "On the choice of test for a unit root when the errors are conditionally heteroskedastic," Computational Statistics & Data Analysis, Elsevier, vol. 69(C), pages 40-53.
    11. Bykhovskaya, Anna & Phillips, Peter C.B., 2020. "Point optimal testing with roots that are functionally local to unity," Journal of Econometrics, Elsevier, vol. 219(2), pages 231-259.
    12. Mohitosh Kejriwal & Xuewen Yu & Pierre Perron, 2020. "Bootstrap procedures for detecting multiple persistence shifts in heteroskedastic time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 41(5), pages 676-690, September.
    13. Jingjie Xiang & Gangzheng Guo & Qing Zhao, 2022. "Testing for a Moderately Explosive Process with Structural Change in Drift," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 84(2), pages 300-333, April.
    14. Lui, Yiu Lim & Phillips, Peter C.B. & Yu, Jun, 2024. "Robust testing for explosive behavior with strongly dependent errors," Journal of Econometrics, Elsevier, vol. 238(2).
    15. Kim, Jihyun & Meddahi, Nour, 2020. "Volatility regressions with fat tails," Journal of Econometrics, Elsevier, vol. 218(2), pages 690-713.
    16. Fei, Yijie, 2018. "Limit theory for mildly integrated process with intercept," Economics Letters, Elsevier, vol. 163(C), pages 98-101.
    17. Zhishui Hu & Ioannis Kasparis & Qiying Wang, 2020. "Locally trimmed least squares: conventional inference in possibly nonstationary models," Papers 2006.12595, arXiv.org.
    18. Andrews, Donald W.K. & Guggenberger, Patrik, 2009. "Incorrect asymptotic size of subsampling procedures based on post-consistent model selection estimators," Journal of Econometrics, Elsevier, vol. 152(1), pages 19-27, September.
    19. Breitung, Jörg & Demetrescu, Matei, 2015. "Instrumental variable and variable addition based inference in predictive regressions," Journal of Econometrics, Elsevier, vol. 187(1), pages 358-375.
    20. Plakandaras, Vasilios & Gupta, Rangan & Balcilar, Mehmet & Ji, Qiang, 2022. "Evolving United States stock market volatility: The role of conventional and unconventional monetary policies," The North American Journal of Economics and Finance, Elsevier, vol. 60(C).

    More about this item

    Keywords

    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:indpam:v:56:y:2025:i:2:d:10.1007_s13226-023-00506-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.