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The Maximum Number of Parameters for the Hausman Test When the Estimators are from Different Sets of Equations

Author

Listed:
  • Kazumitsu Nawata

    (Graduate School of Engineering, University of Tokyo)

  • Michael McAleer

    (Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam and Tinbergen Institute, The Netherlands, Department of Quantitative Economics, Complutense University of Madrid, and Institute of Economic Research, Kyoto University.)

Abstract

Hausman (1978) developed a widely-used model specification test that has passed the test of time. The test is based on two estimators, one being consistent under the null hypothesis but inconsistent under the alternative, and the other being consistent under both the null and alternative hypotheses. In this paper, we show that the asymptotic variance of the difference of the two estimators can be a singular matrix. Moreover, in calculating the Hausman test there is a maximum number of parameters which is the number of different equations that are used to obtain the two estimators. Three illustrative examples are used, namely an exogeneity test for the linear regression model, a test for the Box-Cox transformation, and a test for sample selection bias.

Suggested Citation

  • Kazumitsu Nawata & Michael McAleer, 2013. "The Maximum Number of Parameters for the Hausman Test When the Estimators are from Different Sets of Equations," Documentos de Trabajo del ICAE 2013-39, Universidad Complutense de Madrid, Facultad de Ciencias Económicas y Empresariales, Instituto Complutense de Análisis Económico.
    • Handle: RePEc:ucm:doicae:1339
    Note: This paper was supported by a Grant-in-Aid for Scientific Research “Analyses of the Large Scale Medical Survey Data and the Policy Evaluations in Japan (Grant Number: 24330067)” of the Japan Society of Science for the first author, and Australian Research Council and the National Science Council, Taiwan for the second author.
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    References listed on IDEAS

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    1. Wu, De-Min, 1973. "Alternative Tests of Independence Between Stochastic Regressors and Disturbances," Econometrica, Econometric Society, vol. 41(4), pages 733-750, July.
    2. Smith, Richard, 1983. "On the classical nature of the Wu-Hausman statistics for the independence of stochastic regressors and disturbance," Economics Letters, Elsevier, vol. 11(4), pages 357-364.
    3. James J. Heckman, 1976. "The Common Structure of Statistical Models of Truncation, Sample Selection and Limited Dependent Variables and a Simple Estimator for Such Models," NBER Chapters, in: Annals of Economic and Social Measurement, Volume 5, number 4, pages 475-492, National Bureau of Economic Research, Inc.
    4. Heckman, James, 2013. "Sample selection bias as a specification error," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 31(3), pages 129-137.
    5. Kazumitsu Nawata, 2013. "A new estimator of the Box-Cox transformation model using moment conditions," Economics Bulletin, AccessEcon, vol. 33(3), pages 2287-2297.
    6. Barnett,William A. & Powell,James & Tauchen,George E. (ed.), 1991. "Nonparametric and Semiparametric Methods in Econometrics and Statistics," Cambridge Books, Cambridge University Press, number 9780521424318, October.
    7. Hausman, Jerry, 2015. "Specification tests in econometrics," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 38(2), pages 112-134.
    8. Holly, Alberto, 1982. "A Remark on Hausman's Specification Test," Econometrica, Econometric Society, vol. 50(3), pages 749-759, May.
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    10. Smith, Richard J, 1984. "A Note on Likelihood Ratio Tests for the Independence between a Subset of Stochastic Regressors and Disturbances," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 25(1), pages 263-269, February.
    11. Smith, Richard J., 1985. "Wald tests for the independence of stochastic variables and disturbance of a single linear stochastic simultaneous equation," Economics Letters, Elsevier, vol. 17(1-2), pages 87-90.
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    Cited by:

    1. Kazumitsu Nawata, 2015. "Robust estimation based on the third-moment restriction of the error terms for the Box-Cox transformation model: An estimator consistent under heteroscedasticity," Economics Bulletin, AccessEcon, vol. 35(2), pages 1056-1064.

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    More about this item

    Keywords

    : Hausman test; specification test; number of parameters; instrumental variable (IV) model; Box-Cox model; Sample selection bias.;
    All these keywords.

    JEL classification:

    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling
    • I18 - Health, Education, and Welfare - - Health - - - Government Policy; Regulation; Public Health

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