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On the Existence of Moments of Ratios ofQuadratic Forms

Published online by Cambridge University Press:  11 February 2009

Leigh A. Roberts
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Abstract

We obtain simple and generally applicable conditionsfor the existence of mixed momentsE([X′AX]″/[X′BX]U)of the ratio of quadratic forms T =X′AX/X′BXwhere A and B aren × n symmetricmatrices and X is a randomn-vector. Our principal theoremis easily stated when X has anelliptically symmetric distribution, which classincludes the multivariate normal andt distributions, whetherdegenerate or not. The result applies to the ratioof multivariate quadratic polynomials and can beexpected to remain valid in most situations in whichX is subject to linearconstraints.

If u ≤ v, the precisedistribution of X, and inparticular the existence of moments ofX, is virtually irrelevant to theexistence of the mixed moments ofT; if u >v, a prerequisite for existenceof the (u, v)thmixed moment is the existence of the2(u − v)thmoment of X WhenXis not degenerate, the principalfurther requirement for the existence of the mixedmoment is that B has rank exceeding2v.

Information

Type
Research Article
Information
Econometric Theory , Volume 11 , Issue 4 , August 1995 , pp. 750 - 774
Copyright
Copyright © Cambridge University Press 1995

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