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On the Exact Statistical Distribution of Econometric Estimators and Test Statistics

Author

Listed:
  • Yong Bao

    (Purdue University)

  • Xiaotian Liu

    (Purdue University)

  • Aman Ullah

    (Department of Economics, University of California Riverside)

Abstract

Barry Arnold has made many fundamental and innovative contributions in different areas of statistics and econometrics, including estimation and inference,distribution theory, Bayesian inference, order statistics, income inequality measures, and characterization problems. His extensive work in the area of distribution theory include studies on income distributions and Lorenz curves, the exact sampling distribution theory of test statistics, and the characterization of distributions. In our paper here we consider the problem of developing exact sampling distributions of various econometric and statistical estimators and test statistics. The motivation stems from the fact that inference procedures based on the asymptotic distributions may provide misleading results if the sample size is small or moderately large. In view of this we develop a unified procedure by first observing that a large number of econometric and statistical estimators can be written as ratios of quadratic forms. Their distributions can then be straightforwardly analyzed by using Imhof’s (1961) method. We show the applications of this procedure to develop the distribution of some commonly used statistics in applied work. The exact results developed will be helpful for practitioners to conduct appropriate inference for any given size of the sample data.

Suggested Citation

  • Yong Bao & Xiaotian Liu & Aman Ullah, 2020. "On the Exact Statistical Distribution of Econometric Estimators and Test Statistics," Working Papers 202014, University of California at Riverside, Department of Economics, revised Jun 2020.
    • Handle: RePEc:ucr:wpaper:202014
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    References listed on IDEAS

    as
    1. Yong Bao & Aman Ullah & Yun Wang, 2017. "Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process," Econometric Reviews, Taylor & Francis Journals, vol. 36(6-9), pages 1039-1056, October.
    2. Phillips, P C B, 1986. "The Exact Distribution of the Wald Statistic," Econometrica, Econometric Society, vol. 54(4), pages 881-895, July.
    3. H. Kelejian, Harry & Prucha, Ingmar R., 2001. "On the asymptotic distribution of the Moran I test statistic with applications," Journal of Econometrics, Elsevier, vol. 104(2), pages 219-257, September.
    4. Filipe Marques & Carlos Coelho & Barry Arnold, 2011. "A general near-exact distribution theory for the most common likelihood ratio test statistics used in Multivariate Analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(1), pages 180-203, May.
    5. Ullah, Aman, 2004. "Finite Sample Econometrics," OUP Catalogue, Oxford University Press, number 9780198774488.
    6. Carlos A. Coelho & Barry C. Arnold, 2014. "On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 43(10-12), pages 2007-2033, May.
    7. P. C. B. Phillips, 1980. "Finite Sample Theory and the Distributions of Alternative Estimators of the Marginal Propensity to Consume," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 47(1), pages 183-224.
    8. Miller, Robert E. & Gehr, Adam K., 1978. "Sample Size Bias and Sharpe's Performance Measure: A Note," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 13(5), pages 943-946, December.
    9. Arnold, Barry C, et al, 1987. "Generating Ordered Families of Lorenz Curves by Strongly Unimodal Distributions," Journal of Business & Economic Statistics, American Statistical Association, vol. 5(2), pages 305-308, April.
    10. Barry C. Arnold & José María Sarabia, 2018. "Analytic Expressions for Multivariate Lorenz Surfaces," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(1), pages 84-111, December.
    11. Arnold, Barry C., 2012. "On the Amato inequality index," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1504-1506.
    12. Anderson, T W & Sawa, Takamitsu, 1973. "Distributions of Estimates of Coefficients of a Single Equation in a Simultaneous System and Their Asymptotic Expansions," Econometrica, Econometric Society, vol. 41(4), pages 683-714, July.
    13. Srivastava, M. S., 1987. "Asymptotic distribution of Durbin-Watson statistic," Economics Letters, Elsevier, vol. 24(2), pages 157-160.
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    More about this item

    Keywords

    Exact Distribution; Sharp Ratio; Coeficcient of Variation; Durbi-Watson Test; Moran Test; Imhof Distribution ; R-square;
    All these keywords.

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