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Exact Distribution of the Least Squares Estimator in a First- Order Autoregressive Model

  • Mukhtar M. Ali

    (University of Kentucky)

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    This paper investigates the finite sample distribution of the least squares estimator of the autoregressive parameter in a first-order autoregressive model. Uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained. Accuracy of approximation to the distribution by a first few terms of this expansion is then investigated. It is found that the leading term of this expansion approximates well the distribution. The approximation is, in almost all cases, accurate to the second decimal place throughout the distribution. Only rarely the accuracy improves by including further term beyond the first term of this expansion in the approximation. As a matter of fact, often the accuracy of such an approximation with additional term(s) deteriorates. An application of the finding is illustrated with examples.

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    Paper provided by EconWPA in its series Econometrics with number 9604001.

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    Length: 22 pages
    Date of creation: 03 Apr 1996
    Date of revision:
    Handle: RePEc:wpa:wuwpem:9604001
    Note: Type of Document - Binary WordPerfect (V5.1) Document; prepared on IBM PC - Compatible; to print on HP LaserJet II; pages: 22. Contains many special characters and equations created with WordPerfect's Equation Editor. Conversion to other formats may cause problems.
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    1. Francis X. Diebold & Marc Nerlove, 1988. "Unit roots in economic time series: a selective survey," Finance and Economics Discussion Series 49, Board of Governors of the Federal Reserve System (U.S.).
    2. Perron, P., 1987. "The Calculation of the Limiting Distribution of the Least Squares Estimator in Near-Integrated Model," Cahiers de recherche 8748, Universite de Montreal, Departement de sciences economiques.
    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 463-85, March.
    4. Peter C.B. Phillips & Pierre Perron, 1986. "Testing for a Unit Root in Time Series Regression," Cowles Foundation Discussion Papers 795R, Cowles Foundation for Research in Economics, Yale University, revised Sep 1987.
    5. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-72, June.
    6. Perron, P. & Phillips, P.C.B., 1986. "Does Gnp Have a Unit Root? a Reevaluation," Cahiers de recherche 8640, Universite de Montreal, Departement de sciences economiques.
    7. Cryer, Jonathan D. & Nankervis, John C. & Savin, N.E., 1989. "Mirror-Image and Invariant Distributions in ARMA Models," Econometric Theory, Cambridge University Press, vol. 5(01), pages 36-52, April.
    8. Evans, G B A & Savin, N E, 1981. "Testing for Unit Roots: 1," Econometrica, Econometric Society, vol. 49(3), pages 753-79, May.
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