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On the Asymptotic Behavior of Least-Squares Estimators in AR Time Series with Roots Near the Unit Circle

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  • Jeganathan, P.

Abstract

Some asymptotic properties of the least-squares estimator of the parameters of an AR model of order p, p ≥ 1, are studied when the roots of the characteristic polynomial of the given AR model are on or near the unit circle. Specifically, the convergence in distribution is established and the corresponding limiting random variables are represented in terms of functionals of suitable Brownian motions. Further, the preceding convergence in distribution is strengthened to that of convergence uniformly over all Borel subsets. It is indicated that the method employed for this purpose has the potential of being applicable in the wider context of obtaining suitable asymptotic expansions of the distributions of leastsquares estimators.

Suggested Citation

  • Jeganathan, P., 1991. "On the Asymptotic Behavior of Least-Squares Estimators in AR Time Series with Roots Near the Unit Circle," Econometric Theory, Cambridge University Press, vol. 7(3), pages 269-306, September.
    • Handle: RePEc:cup:etheor:v:7:y:1991:i:03:p:269-306_00
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    Cited by:

    1. Buchmann, Boris & Chan, Ngai Hang, 2013. "Unified asymptotic theory for nearly unstable AR(p) processes," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 952-985.
    2. Peter C. B. Phillips & Zhijie Xiao, 1998. "A Primer on Unit Root Testing," Journal of Economic Surveys, Wiley Blackwell, vol. 12(5), pages 423-470, December.
    3. Castro, Tomás del Barrio & Rodrigues, Paulo M.M. & Taylor, A.M. Robert, 2013. "The Impact Of Persistent Cycles On Zero Frequency Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 29(6), pages 1289-1313, December.
    4. Rodrigues, Paulo M.M. & Taylor, A.M. Robert, 2007. "Efficient tests of the seasonal unit root hypothesis," Journal of Econometrics, Elsevier, vol. 141(2), pages 548-573, December.
    5. Barczy, M. & Ispány, M. & Pap, G., 2011. "Asymptotic behavior of unstable INAR(p) processes," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 583-608, March.
    6. Ling, Shiqing & McAleer, Michael, 2004. "Regression quantiles for unstable autoregressive models," Journal of Multivariate Analysis, Elsevier, vol. 89(2), pages 304-328, May.
    7. Peter C.B. Phillips, 1994. "Nonstationary Time Series and Cointegration: Recent Books and Themes for the Future," Cowles Foundation Discussion Papers 1081, Cowles Foundation for Research in Economics, Yale University.
    8. Eric Ghysels & Denise R. Osborn & Paulo M. M. Rodrigues, 1999. "Seasonal Nonstationarity and Near-Nonstationarity," CIRANO Working Papers 99s-05, CIRANO.
    9. Fu, Ke-Ang & Li, Yuechao & Ng, Andrew Cheuk-Yin, 2013. "Asymptotics for the residual-based bootstrap approximation in nearly nonstationary AR(1) models with possibly heavy-tailed innovations," Statistics & Probability Letters, Elsevier, vol. 83(11), pages 2553-2562.
    10. Ploberger, Werner, 2004. "A complete class of tests when the likelihood is locally asymptotically quadratic," Journal of Econometrics, Elsevier, vol. 118(1-2), pages 67-94.
    11. Ploberger, Werner & Phillips, Peter C.B., 2012. "Optimal estimation under nonstandard conditions," Journal of Econometrics, Elsevier, vol. 169(2), pages 258-265.
    12. Niels Haldrup & Peter Lildholdt, 2005. "Local power functions of tests for double unit roots," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 59(2), pages 159-179, May.
    13. Atsushi Inoue & Lutz Kilian, 2002. "Bootstrapping Autoregressive Processes with Possible Unit Roots," Econometrica, Econometric Society, vol. 70(1), pages 377-391, January.
    14. Hwang, Kyo-Shin & Pang, Tian-Xiao, 2009. "Asymptotic inference for nearly nonstationary AR(1) processes with possibly infinite variance," Statistics & Probability Letters, Elsevier, vol. 79(22), pages 2374-2379, November.
    15. Jui-Chung Yang & Ke-Li Xu, 2013. "Estimation and Inference under Weak Identi cation and Persistence: An Application on Forecast-Based Monetary Policy Reaction Function," 2013 Papers pya307, Job Market Papers.
    16. Galtchouk, Leonid & Konev, Victor, 2010. "On asymptotic normality of sequential LS-estimate for unstable autoregressive process AR(2)," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2616-2636, November.
    17. Christis Katsouris, 2023. "Bootstrapping Nonstationary Autoregressive Processes with Predictive Regression Models," Papers 2307.14463, arXiv.org.
    18. Lacroix, R., 1999. "Testing for Zeros in the Spectrum of an Univariate Stationary Process: Part I," Working papers 70, Banque de France.
    19. Datta, Somnath, 1995. "Limit theory and bootstrap for explosive and partially explosive autoregression," Stochastic Processes and their Applications, Elsevier, vol. 57(2), pages 285-304, June.

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