COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 767 "Fractional Matrix Calculus and the Distribution of Multivariate Tests" Peter C.B. Phillips September 1985 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. |
