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

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  • Mukhtar M. Ali

    (University of Kentucky)

Abstract

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.

Suggested Citation

  • Mukhtar M. Ali, 1996. "Exact Distribution of the Least Squares Estimator in a First- Order Autoregressive Model," Econometrics 9604001, University Library of Munich, Germany.
    • 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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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Perron, Pierre, 1989. "The Calculation of the Limiting Distribution of the Least-Squares Estimator in a Near-Integrated Model," Econometric Theory, Cambridge University Press, vol. 5(2), pages 241-255, August.
    4. 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 1057-1072, June.
    5. Perron, Pierre & Phillips, Peter C. B., 1987. "Does GNP have a unit root? : A re-evaluation," Economics Letters, Elsevier, vol. 23(2), pages 139-145.
    6. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    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(1), pages 36-52, April.
    8. 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.).
    9. Evans, G B A & Savin, N E, 1981. "Testing for Unit Roots: 1," Econometrica, Econometric Society, vol. 49(3), pages 753-779, May.
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    Cited by:

    1. 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.

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    More about this item

    Keywords

    Unit Root; Saddlepoint Approximation; Asymptotic Expansion;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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