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Fractional Matrix Calculus and the Distribution of Multivariate Tests

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Author Info
Peter C.B. Phillips () (Cowles Foundation, Yale University)

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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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Publisher Info
Paper provided by Cowles Foundation, Yale University in its series Cowles Foundation Discussion Papers with number 767.

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Length: 23 pages
Date of creation: Sep 1985
Date of revision:
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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Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
Phone: (203) 432-3702
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Web page: http://cowles.econ.yale.edu/
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Related research
Keywords: Fractional matrix calculus; multivariate tests; exact distribution theory; asymptotic expansions;

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References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Phillips, P C B, 1986. "The Exact Distribution of the Wald Statistic," Econometrica, Econometric Society, vol. 54(4), pages 881-95, July. [Downloadable!] (restricted)
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  2. 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, Yale University, revised Mar 1986. [Downloadable!]
  3. Phillips, Peter C B, 1985. "The Exact Distribution of the SUR Estimator," Econometrica, Econometric Society, vol. 53(4), pages 745-56, July. [Downloadable!] (restricted)
  4. Phillips, P.C.B., 1984. "The exact distribution of the Stein-rule estimator," Journal of Econometrics, Elsevier, vol. 25(1-2), pages 123-131. [Downloadable!] (restricted)
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  5. Peter C.B. Phillips, 1983. "Finite Sample Econometrics Using ERA's," Cowles Foundation Discussion Papers 683, Cowles Foundation, Yale University. [Downloadable!]
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Cited by:
(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Peter C.B. Phillips, 1992. "Some Exact Distribution Theory for Maximum Likelihood Estimators of Cointegrating Coefficients in Error Correction Models," Cowles Foundation Discussion Papers 1039, Cowles Foundation, Yale University. [Downloadable!]
    Other versions:
  2. Peter C.B. Phillips, 1999. "Discrete Fourier Transforms of Fractional Processes," Cowles Foundation Discussion Papers 1243, Cowles Foundation, Yale University. [Downloadable!]
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