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Distribution Of The Least Squares Estimator In A First-Order Autoregressive Model


  • Mukhtar Ali


This paper investigates the finite sample distribution of the least squares estimator of the autoregressive parameter in a first-order autoregressive model. A uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained. Accuracy of the 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. In the literature, there exist a number of approximations to this distribution which are specifically designed to apply in some special cases of this model. The present approximation compares favorably with those approximations and in fact, its accuracy is, with almost no exception, as good as or better than these other approximations. Convenience of numerical computations seems also to favor the present approximations over the others. An application of the finding is illustrated with examples.

Suggested Citation

  • 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.
  • Handle: RePEc:taf:emetrv:v:21:y:2002:i:1:p:89-119 DOI: 10.1081/ETC-120008725

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    References listed on IDEAS

    1. Larsson, Rolf, 1995. "The Asymptotic Distributions Of Some Test Statistics in Near-Integrated AR Processes," Econometric Theory, Cambridge University Press, vol. 11(02), pages 306-330, February.
    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. 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.
    4. 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.
    5. 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.
    6. Perron, Pierre, 1991. "A Continuous Time Approximation to the Unstable First-Order Autoregressive Process: The Case without an Intercept," Econometrica, Econometric Society, vol. 59(1), pages 211-236, January.
    7. Perron, Pierre, 1991. "A Continuous Time Approximation to the Stationary First-Order Autoregressive Model," Econometric Theory, Cambridge University Press, vol. 7(02), pages 236-252, June.
    8. 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.
    9. Phillips, Peter C B, 1988. "Regression Theory for Near-Integrated Time Series," Econometrica, Econometric Society, vol. 56(5), pages 1021-1043, September.
    10. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    11. Knight, J.L. & Satchell, S.E., 1993. "Asymptotic Expansions for Random Walks with Normal Errors," Econometric Theory, Cambridge University Press, vol. 9(03), pages 363-376, June.
    12. Schwert, G. William, 1987. "Effects of model specification on tests for unit roots in macroeconomic data," Journal of Monetary Economics, Elsevier, vol. 20(1), pages 73-103, July.
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    Cited by:

    1. Aman Ullah & Yong Bao & Yun Wang, 2014. "Exact Distribution of the Mean Reversion Estimator in the Ornstein-Uhlenbeck Process," Working Papers 201413, University of California at Riverside, Department of Economics.

    More about this item


    Unit root; Saddlepoint approximation; Asymptotic expansion; JEL Classification: C13; C22;

    JEL classification:

    • 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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