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Ols Bias in a Nonstationary Autoregression

Author

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  • Abadir, Karim M.

Abstract

An analytical formula is derived to approximate the finite sample bias of the ordinary least-squares (OLS) estimator of the autoregressive parameter when the underlying process has a unit root. It is found that the bias is expressible in terms of parabolic cylinder functions which are easy to compute. Numerical evaluation of the formula reveals that the approximation is very accurate. The derived formula inspires a heuristic approximation, obtained by leastsquares fitting of the asymptotic bias. More importantly, the formula proves analytically that the bias declines at a rate which is slower than the consistency rate, thus explaining some previous simulation findings. A case where the bias increases with the sample size is also given.

Suggested Citation

  • Abadir, Karim M., 1993. "Ols Bias in a Nonstationary Autoregression," Econometric Theory, Cambridge University Press, vol. 9(01), pages 81-93, January.
  • Handle: RePEc:cup:etheor:v:9:y:1993:i:01:p:81-93_00
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    Cited by:

    1. Rolf Larsson, 1997. "On the Asymptotic Expectations of Some Unit Root Tests in a First Order Autoregressive Process in the Presence of Trend," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 49(3), pages 585-599, September.
    2. Lawford, Steve & Stamatogiannis, Michalis P., 2009. "The finite-sample effects of VAR dimensions on OLS bias, OLS variance, and minimum MSE estimators," Journal of Econometrics, Elsevier, vol. 148(2), pages 124-130, February.
    3. Abadir Karim M. & Larsson Rolf, 2012. "Biases of Correlograms and of AR Representations of Stationary Series," Journal of Time Series Econometrics, De Gruyter, vol. 4(1), pages 1-11, May.
    4. Pere, Pekka, 2000. "Adjusted estimates and Wald statistics for the AR(1) model with constant," Journal of Econometrics, Elsevier, vol. 98(2), pages 335-363, October.
    5. Hisashi Tanizaki & Shigeyuki Hamori & Yoichi Matsubayashi, 2006. "On least-squares bias in the AR(p) models: Bias correction using the bootstrap methods," Statistical Papers, Springer, vol. 47(1), pages 109-124, January.
    6. Cheung Ip, Wai & Phillips, Garry D. A., 1998. "The non-monotonicity of the bias and mean squared error of the two stage least squares estimators of exogenous variable coefficients," Economics Letters, Elsevier, vol. 60(3), pages 303-310, September.
    7. Yu, Jun, 2012. "Bias in the estimation of the mean reversion parameter in continuous time models," Journal of Econometrics, Elsevier, vol. 169(1), pages 114-122.
    8. repec:eee:touman:v:41:y:2014:i:c:p:28-42 is not listed on IDEAS
    9. 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.
    10. Hendrik Kaufmannz & Robinson Kruse, 2013. "Bias-corrected estimation in potentially mildly explosive autoregressive models," CREATES Research Papers 2013-10, Department of Economics and Business Economics, Aarhus University.
    11. Pesaran, M. Hashem & Timmermann, Allan, 2005. "Small sample properties of forecasts from autoregressive models under structural breaks," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 183-217.
    12. Tanizaki, Hisashi, 2000. "Bias correction of OLSE in the regression model with lagged dependent variables," Computational Statistics & Data Analysis, Elsevier, vol. 34(4), pages 495-511, October.
    13. Kaddour Hadri & Yao Rao, 2008. "Panel Stationarity Test with Structural Breaks," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 70(2), pages 245-269, April.
    14. Aman Ullah & Yong Bao & Ru Zhang, 2014. "Moment Approximation for Unit Root Models with Nonnormal Errors," Working Papers 201401, University of California at Riverside, Department of Economics.
    15. Jacobson, Tor & Larsson, Rolf, 1999. "Bartlett corrections in cointegration testing," Computational Statistics & Data Analysis, Elsevier, vol. 31(2), pages 203-225, August.
    16. Vougas, Dimitrios V., 2006. "Remark on the asymptotic distribution of the OLS estimator in a simple Gaussian unit-root autoregression," Statistics & Probability Letters, Elsevier, vol. 76(1), pages 27-34, January.
    17. He, Changli & Sandberg, Rickard, 2005. "Testing for Unit Roots in Nonlinear Dynamic Heterogeneous Panels," SSE/EFI Working Paper Series in Economics and Finance 582, Stockholm School of Economics.
    18. Abadir, Karim M., 1995. "Unbiased estimation as a solution to testing for random walks," Economics Letters, Elsevier, vol. 47(3-4), pages 263-268, March.
    19. Harris, Richard D. F. & Tzavalis, Elias, 1999. "Inference for unit roots in dynamic panels where the time dimension is fixed," Journal of Econometrics, Elsevier, vol. 91(2), pages 201-226, August.
    20. Wagatha, Matthias, 2007. "Integration, Kointegration und die Langzeitprognose von Kreditausfallzyklen
      [Integration, Cointegration and Long-Horizont Forecasting of Credit-Default-Cycles]
      ," MPRA Paper 8602, University Library of Munich, Germany.
    21. Nazlioglu, Saban & Karul, Cagin, 2017. "A panel stationarity test with gradual structural shifts: Re-investigate the international commodity price shocks," Economic Modelling, Elsevier, vol. 61(C), pages 181-192.

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