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The Exact Distribution of the Wald Statistic

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Abstract

This paper derives the exact distribution of the Wald statistic for testing general linear restrictions on the coefficients in the multivariate linear model. This generalizes all previously known results including those for the standard F statistic in linear regression, for Hotelling's T^{2} test and for Hotelling's generalized T_{0}^{2} test. Conventional classical assumptions of normally distributed errors and nonrandom exogenous variables are employed.

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  • Peter C.B. Phillips, 1984. "The Exact Distribution of the Wald Statistic," Cowles Foundation Discussion Papers 722, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:722
    Note: CFP 654.
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    1. Phillips, P.C.B., 1984. "The exact distribution of the Stein-rule estimator," Journal of Econometrics, Elsevier, vol. 25(1-2), pages 123-131.
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    Cited by:

    1. 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.
    2. Attfield, C. L. F., 1997. "Estimating a cointegrating demand system," European Economic Review, Elsevier, vol. 41(1), pages 61-73, January.
    3. Phillips, Peter C B, 1994. "Some Exact Distribution Theory for Maximum Likelihood Estimators of Cointegrating Coefficients in Error Correction Models," Econometrica, Econometric Society, vol. 62(1), pages 73-93, January.
    4. Peter C.B. Phillips, 1985. "Fractional Matrix Calculus and the Distribution of Multivariate Tests," Cowles Foundation Discussion Papers 767, Cowles Foundation for Research in Economics, Yale University.
    5. Grant Hillier & Raymond Kan & Xiaolu Wang, 2008. "Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors," CeMMAP working papers CWP14/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    6. Neil R. Ericsson, 1987. "Monte Carlo methodology and the finite sample properties of statistics for testing nested and non-nested hypotheses," International Finance Discussion Papers 317, Board of Governors of the Federal Reserve System (U.S.).
    7. Giovanni Forchini, "undated". "The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables," Discussion Papers 01/12, Department of Economics, University of York.
    8. Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009. "Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications," Econometric Theory, Cambridge University Press, vol. 25(1), pages 211-242, February.
    9. Peter C. B. Phillips, 2021. "Pitfalls in Bootstrapping Spurious Regression," Journal of Quantitative Economics, Springer;The Indian Econometric Society (TIES), vol. 19(1), pages 163-217, December.

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