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A Bootstrap Test for Positive Definiteness of Income Effect Matrices

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  • Härdle, Wolfgang
  • Hart, Jeffrey D.

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

Suggested Citation

  • Härdle, Wolfgang & Hart, Jeffrey D., 1992. "A Bootstrap Test for Positive Definiteness of Income Effect Matrices," Econometric Theory, Cambridge University Press, vol. 8(02), pages 276-292, June.
  • Handle: RePEc:cup:etheor:v:8:y:1992:i:02:p:276-292_01
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    Cited by:

    1. Joel L. Horowitz, 1996. "Bootstrap Methods in Econometrics: Theory and Numerical Performance," Econometrics 9602009, EconWPA, revised 05 Mar 1996.
    2. Manisha Chakrabarty & Anke Schmalenbach, 2002. "The Representative Agent Hypothesis: An Empirical Test," Bonn Econ Discussion Papers bgse26_2002, University of Bonn, Germany.
    3. Freyberger, Joachim & Horowitz, Joel L., 2015. "Identification and shape restrictions in nonparametric instrumental variables estimation," Journal of Econometrics, Elsevier, vol. 189(1), pages 41-53.
    4. Joachim Freyberger & Joel L. Horowitz, 2013. "Identification and shape restrictions in nonparametric instrumental variables estimation," CeMMAP working papers CWP31/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Hardle, W. & Park, B. U., 1995. "Testing increasing dispersion," Computational Statistics & Data Analysis, Elsevier, vol. 19(6), pages 641-653, June.
    6. Werner Hildenbrand & Alois Kneip, 2002. "Aggregation under structural stability: the change in consumption of a heterogeneous population," Bonn Econ Discussion Papers bgse4_2002, University of Bonn, Germany.
    7. Koebel, Bertrand M. & Falk, Martin & Laisney, François, 2000. "Imposing and testing curvature conditions on a Box-Cox function," ZEW Discussion Papers 00-70, ZEW - Zentrum für Europäische Wirtschaftsforschung / Center for European Economic Research.

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