# SFB 303 Discussion Paper No. A - 199

**Author**: Härdle, W., and J. D. Hart

**Title**: A Bootstrap Test for Positive Definiteness of Income Effect Matrices

**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 (1988) 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, lambda hat, of the matrix estimate is obtained. This theory applies generally to
symmetric, asymptotically normal random matrices. A bootstrap approximation to the distribution of lambda hat
is shown to converge in probability to the asymptotic distribution of lambda hat. The bootstrap test is illustrated
using British family expenditure survey data.

**Keywords**: Bootstrap, central limit theorem, average derivative estimation (ADE), kernel estimator, U -statistics.

**JEL-Classification-Number**:

**Creation-Date**: March 1989

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