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On the inconsistency of the unrestricted estimator of the information matrix near a unit root

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  • Tassos Magdalinos
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    Abstract

    The unrestricted estimator of the information matrix is shown to be inconsistent for an autoregressive process with a root lying in a neighbourhood of unity with radial length proportional or smaller than 1/n, i.e. a root that takes the form rho=1+c/n^alpha, alpha>=1. In this case the information evaluated at rho-hat_n converges to a non-degenerate random variable and contributes to the asymptotic distribution of a Wald test for the null hypothesis of a random walk versus a stable AR(1) alternative. With this newly derived asymptotic distribution the above Wald test is found to improve its performance. A non local criterion of asymptotic relative efficiency based on Bahadur slopes has been employed for the first time to the problem of unit root testing. The Wald test derived in the paper is found to be as efficient as the Dickey Fuller t ratio test and to outperform the non studentised Dickey Fuller test and a Lagrange Multiplier test.

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    File URL: http://www.nottingham.ac.uk/economics/grangercentre/papers/06-05.pdf
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    Bibliographic Info

    Paper provided by University of Nottingham, Granger Centre for Time Series Econometrics in its series Discussion Papers with number 06/05.

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    Date of creation: Oct 2005
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    Handle: RePEc:not:notgts:06/05

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    Web page: http://www.nottingham.ac.uk/economics/grangercentre/
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    Keywords: Unit root distribution; neighbourhoods of unity; information matrix; inconsistency; Wald test; Bahadur slopes;

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    1. Evans, G B A & Savin, N E, 1981. "Testing for Unit Roots: 1," Econometrica, Econometric Society, vol. 49(3), pages 753-79, May.
    2. Abadir, K.M., 1992. "The Limiting Distribution of the T Ratio Under a Unit Root," Papers 1992-2, American Cairo - Economics and Political Sciences.
    3. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
    4. Liudas Giraitis & Peter C. B. Phillips, 2006. "Uniform Limit Theory for Stationary Autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(1), pages 51-60, 01.
    5. 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.
    6. Abadir, Karim M., 1993. "On the Asymptotic Power of Unit Root Tests," Econometric Theory, Cambridge University Press, vol. 9(02), pages 189-221, April.
    7. Nabeya, Seiji & Tanaka, Katsuto, 1990. "Limiting power of unit-root tests in time-series regression," Journal of Econometrics, Elsevier, vol. 46(3), pages 247-271, December.
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