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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.

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

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References

REFERENCES

1.Beran, R.Prepivoting to reduce level error of confidence sets. Biometrika 74 (1987): 457–468.Google Scholar
2.Beran, R. & Srivastava, M.S.. Bootstrap tests and confidence regions for functions of a co-variance matrix. Annals of Statistics 13 (1985): 95–115.CrossRefGoogle Scholar
3.Bickel, P.J., Gotze, F. & Zwet, W.R. van. The Edgeworth expansion for u-statistics of degree two. Annals of Statistics 14 (1986): 1463–1484.Google Scholar
4.Family Expenditure Survey, Annual Base Tapes (1968-1983) Department of Employment. Statistics Division, Her Majesty's Stationery Office, London 1968–1983. The data utilized in this hook were made available by the ESRC Data Archive at the Universtiy of Essex.Google Scholar
5.Graybill, F.A.Matrices With Applications in Statistics. Belmont, CA: Wadsworth, 1983.Google Scholar
6.Hall, P.Theoretical comparison of bootstrap confidence intervals (with discussion). Annals of Statistics 16 (1988): 927–953.Google Scholar
7.Hall, P. & Hart, J.D.. Bootstrap test for difference between means in nonparametric regression. Journal of the American Statistical Association 85 (1990): 1039–1049.Google Scholar
8.Härdle, W. Applied Nonparametric Regression. Econometric Society Monograph Series 19, Cambridge University Press, 1990.Google Scholar
9.Härdle, W. & Stoker, T.. Investigating smooth multiple regression by the method of average derivatives. Journal of the American Statistical Association 84 (1989): 986–995.Google Scholar
10.Härdle, W., Hildenbrand, W. & Jerison, M.. Empirical evidence on the law of demand. Econometrika 59(1991): 1525–1550.Google Scholar
11.Mammitzsch, V.Asymptotically optimal kernels for average derivative estimation. IMS Lecture, Davis, CA, June 1989.Google Scholar
12.Muirhead, R.C.Aspects of Multivariate Statistical Theory. New York: Wiley, 1982.CrossRefGoogle Scholar
13.Randies, R.H. & Wolfe, D.A.. Introduction to the Theory of Nonparametric Statistics. New York: Wiley, 1979.Google Scholar
14.Serfling, R.J.Approximation Theorems of Mathematical Statistics. New York: Wiley, 1980.Google Scholar
15.Stone, C.J.Additive regression and other nonparametric models. Annals of Statistics 13 (1985): 689–705.Google Scholar
16.Wilkinson, J.H.The Algebraic Eigenvalue Problem. London: Oxford University Press, 1965.Google Scholar
17.XploRe, 2.0 XploRe A Computing Environment for exploratory Regression and Data Analysis. Available from CORE (1990).Google Scholar