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A Bootstrap Test for Positive Definiteness of Income Effect Matrices

Published online by Cambridge University Press:  18 October 2010

Wolfgang Härdle  and
Jeffrey D. Hart
Wolfgang Härdle
Affiliation:
CORE, Université Catholique de Louvain
Jeffrey D. Hart
Affiliation:
Texas A&M University
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Abstract

Positive definiteness of income effect matrices provides a sufficient condition for the law of demand to hold. Given cross section household expenditure data, empirical evidence for the law of demand can be obtained by estimating such matrices. Härdle, Hildenbrand, and Jerison used the bootstrap method to simulate the distribution of the smallest eigenvalue of random matrices and to test their positive definiteness. Here, theoretical aspects of this bootstrap test of positive definiteness are considered. The asymptotic distribution of the smallest eigenvalue , of the matrix estimate is obtained. This theory applies generally to symmetric, asymptotically normal random matrices. A bootstrap approximation to the distribution of is shown to converge in probability to the asymptotic distribution of . The bootstrap test is illustrated using British family expenditure survey data.

Type
Articles
Information
Econometric Theory , Volume 8 , Issue 2 , June 1992 , pp. 276 - 292
Copyright
Copyright © Cambridge University Press 1992

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