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Moment Approximation for Unit Root Models with Nonnormal Errors

Author

Listed:
  • Aman Ullah

    (Department of Economics, University of California Riverside)

  • Yong Bao

    (Purdue University)

  • Ru Zhang

    (University of California, Riverside)

Abstract

Phillips (1977a, 1977b) made seminal contributions to time series finite-sample theory, and then, he was among the first to develop the distributions of estimators and forecasts in stationary time series models, see Phillips (1978, 1979), among others. From the mid-eighties Phillips (1987a, 1987b), through his fundamental papers, opened the path of asymptotic (large-sample) theory for the unit root type non-stationary models. This has certainly created a large literature of important papers, including many of Phillips’’own papers. However, not much is known about the analytical finite-sample properties of estimators under the unit root, although see Kiviet and Phillips (2005) for the case when the errors are normally distributed. An objective of this paper is to analyze the …finite-sample behavior of the estimator in the first-order autoregressive model with unit root and nonnormal errors. In particular, we derive analytical approximations for the first two moments in terms of model parameters and the distribution parameters. Through Monte Carlo simulations, we find that our approximate formula perform quite well across different distribution specifications in small samples. However, when the noise to signal ratio is huge, and bias distortion can be quite substantial, and our approximations do not fare well.

Suggested Citation

  • 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.
    • Handle: RePEc:ucr:wpaper:201401
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    References listed on IDEAS

    as
    1. Ullah, Aman, 2004. "Finite Sample Econometrics," OUP Catalogue, Oxford University Press, number 9780198774488.
    2. 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.
    3. 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.
    4. Bao, Yong & Ullah, Aman, 2007. "The second-order bias and mean squared error of estimators in time-series models," Journal of Econometrics, Elsevier, vol. 140(2), pages 650-669, October.
    5. Peter C. B. Phillips, 2012. "Folklore Theorems, Implicit Maps, and Indirect Inference," Econometrica, Econometric Society, vol. 80(1), pages 425-454, January.
    6. Grubb, David & Symons, James, 1987. "Bias in Regressions With a Lagged Dependent Variable," Econometric Theory, Cambridge University Press, vol. 3(3), pages 371-386, June.
    7. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    8. Phillips, Peter C.B. & Lee, Ji Hyung, 2013. "Predictive regression under various degrees of persistence and robust long-horizon regression," Journal of Econometrics, Elsevier, vol. 177(2), pages 250-264.
    9. Kiviet, Jan F. & Phillips, Garry D.A., 1993. "Alternative Bias Approximations in Regressions with a Lagged-Dependent Variable," Econometric Theory, Cambridge University Press, vol. 9(1), pages 62-80, January.
    10. Phillips, Peter C. B., 1979. "The sampling distribution of forecasts from a first-order autoregression," Journal of Econometrics, Elsevier, vol. 9(3), pages 241-261, February.
    11. Tsui, Albert K. & Ali, Mukhtar M., 1994. "Exact distributions, density functions and moments of the last squares estimator in a first-order autoregressive model," Computational Statistics & Data Analysis, Elsevier, vol. 17(4), pages 433-454, May.
    12. Bao, Yong, 2007. "The Approximate Moments Of The Least Squares Estimator For The Stationary Autoregressive Model Under A General Error Distribution," Econometric Theory, Cambridge University Press, vol. 23(5), pages 1013-1021, October.
    13. Sawa, Takamitsu, 1978. "The exact moments of the least squares estimator for the autoregressive model," Journal of Econometrics, Elsevier, vol. 8(2), pages 159-172, October.
    14. Abadir, Karim M., 1993. "Ols Bias in a Nonstationary Autoregression," Econometric Theory, Cambridge University Press, vol. 9(1), pages 81-93, January.
    15. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    16. Phillips, Peter C B, 1977. "A General Theorem in the Theory of Asymptotic Expansions as Approximations to the Finite Sample Distributions of Econometric Estimators," Econometrica, Econometric Society, vol. 45(6), pages 1517-1534, September.
    17. Evans, G B A & Savin, N E, 1981. "Testing for Unit Roots: 1," Econometrica, Econometric Society, vol. 49(3), pages 753-779, May.
    18. Jan F. Kiviet & Garry D. A. Phillips, 2005. "Moment approximation for least-squares estimators in dynamic regression models with a unit root *," Econometrics Journal, Royal Economic Society, vol. 8(2), pages 115-142, July.
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    Cited by:

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    Keywords

    unit root; nonnormal; moment approximation.;
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