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Fractional Matrix Calculus and the Distribution of Multivariate Tests

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Abstract

Fractional matrix operator methods are introduced as a new tool of distribution theory for use in multivariate analysis and econometrics. Earlier work by the author on this operational calculus is reviewed and to illustrate the use of these methods we give an exact distribution theory for a general class of tests in the multivariate linear model. This distribution theory unifies and generalizes 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^{-2} test. We also provide a simple and novel derivation of conventional asymptotic theory as a specialization of exact theory. This approach is extended to generate general formulae for higher order asymptotic expansions. Thus, the results of the paper provide a meaningful unification of conventional asymptotics, higher order asymptotic expansions and exact finite sample distribution theory in this context.

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  • 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.
    • Handle: RePEc:cwl:cwldpp:767
    Note: CFP 664.
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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d07/d0767.pdf
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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.
    2. Phillips, P C B, 1986. "The Exact Distribution of the Wald Statistic," Econometrica, Econometric Society, vol. 54(4), pages 881-895, July.
    3. Knight, John. L., 1986. "The Distribution of the Stein-Rule Estimator in a Model with Non-Normal Disturbances," Econometric Theory, Cambridge University Press, vol. 2(2), pages 202-219, August.
    4. Phillips, Peter C B, 1985. "The Exact Distribution of the SUR Estimator," Econometrica, Econometric Society, vol. 53(4), pages 745-756, July.
    5. Peter C.B. Phillips, 1983. "Finite Sample Econometrics Using ERA's," Cowles Foundation Discussion Papers 683, Cowles Foundation for Research in Economics, Yale University.
    6. Peter C.B. Phillips, 1983. "An Everywhere Convergent Series Representation of the Distribution of Hotelling's Generalized T_{0}^{2}," Cowles Foundation Discussion Papers 723R, Cowles Foundation for Research in Economics, Yale University, revised Mar 1986.
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    1. Peter C.B. Phillips, 1999. "Discrete Fourier Transforms of Fractional Processes," Cowles Foundation Discussion Papers 1243, Cowles Foundation for Research in Economics, Yale University.
    2. 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.

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