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Fractional Matrix Calculus and the Distribution of Multivariate Tests

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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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 767.

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Length: 23 pages
Date of creation: Sep 1985
Publication status: Published in I.B. MacNeill and G.J. Umphrey, eds., Time Series and Econometric Modelling, Reidel, 1987, pp. 219-234
Handle: RePEc:cwl:cwldpp:767
Note: CFP 664.
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Order Information: Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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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(02), 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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