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A Continuous Time Approximation to the Unstable First-Order Autoregressive Process: The Case without an Intercept

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  • Perron, Pierre

Abstract

We consider a first-order autoregression with i.i.d. errors and a fixed initial condition. The asymptotic distribution of the normalized least-squares estimator as the sampling interval converges to zero is shown to be the same as the exact distribution of the continuous-time estimator in an Ornstein-Uhlenbeck process. This asymptotic distribution permits explicit consideration of the effect of the initial condition. The appropriate moment-generating function is derived and used to tabulate the limiting distribution and probability density functions, the moments and some power functions. The adequacy of this asymptotic approximation is found to be excellent for values of the autoregressive parameter near one and any fixed initial condition. Copyright 1991 by The Econometric Society.

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  • Perron, Pierre, 1991. "A Continuous Time Approximation to the Unstable First-Order Autoregressive Process: The Case without an Intercept," Econometrica, Econometric Society, vol. 59(1), pages 211-236, January.
  • Handle: RePEc:ecm:emetrp:v:59:y:1991:i:1:p:211-36
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    Cited by:

    1. Abadir, Karim M. & Lucas, Andre, 2004. "A comparison of minimum MSE and maximum power for the nearly integrated non-Gaussian model," Journal of Econometrics, Elsevier, vol. 119(1), pages 45-71, March.
    2. Cosme Vodounou, 1998. "Inférence fondée sur les statistiques des rendements de long terme," CIRANO Working Papers 98s-20, CIRANO.
    3. Perron, Pierre & Vodounou, Cosme, 2004. "Tests of return predictability: an analysis of their properties based on a continuous time asymptotic framework," Journal of Empirical Finance, Elsevier, vol. 11(2), pages 203-230, March.
    4. Wang, Xiaohu & Yu, Jun, 2015. "Limit theory for an explosive autoregressive process," Economics Letters, Elsevier, vol. 126(C), pages 176-180.
    5. Pierre Perron & Cosme Vodounou, 2001. "Asymptotic approximations in the near-integrated model with a non-zero initial condition," Econometrics Journal, Royal Economic Society, vol. 4(1), pages 1-42.
    6. Paparoditis, Efstathios & Politis, Dimitris N, 2013. "The Asymptotic Size and Power of the Augmented Dickey-Fuller Test for a Unit Root," University of California at San Diego, Economics Working Paper Series qt0784p55m, Department of Economics, UC San Diego.
    7. Mukhtar Ali, 2002. "Distribution Of The Least Squares Estimator In A First-Order Autoregressive Model," Econometric Reviews, Taylor & Francis Journals, vol. 21(1), pages 89-119.
    8. Kurozumi, Eiji, 2002. "Testing for stationarity with a break," Journal of Econometrics, Elsevier, vol. 108(1), pages 63-99, May.
    9. Zhou, Qiankun & Yu, Jun, 2015. "Asymptotic theory for linear diffusions under alternative sampling schemes," Economics Letters, Elsevier, vol. 128(C), pages 1-5.
    10. Wang, Xiaohu & Yu, Jun, 2016. "Double asymptotics for explosive continuous time models," Journal of Econometrics, Elsevier, vol. 193(1), pages 35-53.
    11. Chambers, MJ & Kyriacou, M, 2016. "Jackknife Bias Reduction in the Presence of a Near-Unit Root," Economics Discussion Papers 17623, University of Essex, Department of Economics.
    12. repec:gam:jecnmx:v:6:y:2018:i:1:p:11-:d:134810 is not listed on IDEAS
    13. Mukhtar M. Ali, 1996. "Distribution of the Least Squares Estimator in a First-Order Autoregressive Model," Econometrics 9610004, EconWPA.
    14. Chambers, MJ & Kyriacou, M, 2010. "Jackknife Bias Reduction in the Presence of a Unit Root," Economics Discussion Papers 2785, University of Essex, Department of Economics.
    15. John Hatgioannides & Spiros Mesomeris, 2005. "Mean Reversion in Equity Prices: the G-7 Evidence," Money Macro and Finance (MMF) Research Group Conference 2005 64, Money Macro and Finance Research Group.

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