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Consistency in Nonlinear Econometric Models: A Generic Uniform Law of Large Numbers

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Abstract

A basic tool of modern econometrics is a uniform law of large numbers (LLN). It is a primary ingredient used in proving consistency and asymptotic normality of parametric and nonparametric estimators in nonlinear econometric models. Thus, in a well-known review article, Burguete, Gallant, and Sousa [8, p. 162] introduce a uniform LLN with the statement: "The following theorem is the result upon which the asymptotic theory of nonlinear econometrics rests. "So pervasive is the use of uniform LLNs, that numerous authors appeal to an unspecified generic uniform LLN. Others appeal to some specific result. The purpose of this paper is to provide a generic uniform LLN that is sufficiently general to incorporate most applications of uniform LLNs in the nonlinear econometrics literature. In summary, the paper presents a result that can be used to turn state of the art pointwise LLNs into uniform LLNs over compact sets, with the addition of a single smoothness condition -- either a Lipschitz condition or a derivative condition. The latter is particularly easy to verify, and is implied by common assumptions used to prove asymptotic normality of estimators. Thus, the additional condition is not particularly restrictive. In contrast to other uniform LLNs that appear in the literature, the one given here allows the full range of heterogeneity of summands (i.e., non-identical distributions), and temporal dependence, that is available with pointwise LLNs.

Suggested Citation

  • Donald W.K. Andrews, 1986. "Consistency in Nonlinear Econometric Models: A Generic Uniform Law of Large Numbers," Cowles Foundation Discussion Papers 790, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:790
    Note: CFP 693.
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    File URL: http://cowles.yale.edu/sites/default/files/files/pub/d07/d0790.pdf
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    References listed on IDEAS

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    1. Domowitz, Ian & White, Halbert, 1982. "Misspecified models with dependent observations," Journal of Econometrics, Elsevier, vol. 20(1), pages 35-58, October.
    2. Amemiya, Takeshi, 1974. "The nonlinear two-stage least-squares estimator," Journal of Econometrics, Elsevier, vol. 2(2), pages 105-110, July.
    3. Bates, Charles & White, Halbert, 1985. "A Unified Theory of Consistent Estimation for Parametric Models," Econometric Theory, Cambridge University Press, vol. 1(02), pages 151-178, August.
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    Citations

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    Cited by:

    1. Filippo Altissimo & Giovanni L. Violante, 2001. "The non-linear dynamics of output and unemployment in the U.S," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(4), pages 461-486.
    2. Benedikt M. Potscher & Ingmar R. Prucha, 1994. "On the Formulation of Uniform Laws of Large Numbers: A Truncation Approach," NBER Technical Working Papers 0085, National Bureau of Economic Research, Inc.
    3. Donald W.K. Andrews & Ray C. Fair, 1987. "Inference in Econometric Models with Structural Change," Cowles Foundation Discussion Papers 832, Cowles Foundation for Research in Economics, Yale University.
    4. M. Hashem Pesaran & Yongcheol Shin, 2002. "Long-Run Structural Modelling," Econometric Reviews, Taylor & Francis Journals, vol. 21(1), pages 49-87.
    5. Whang, Yoon-Jae & Andrews, Donald W. K., 1993. "Tests of specification for parametric and semiparametric models," Journal of Econometrics, Elsevier, vol. 57(1-3), pages 277-318.
    6. Donald W.K. Andrews & Werner Ploberger, 1993. "Admissibility of the Likelihood Ratio Test When a Nuisance Parameter Is Present OnlyUnder the Alternative," Cowles Foundation Discussion Papers 1058, Cowles Foundation for Research in Economics, Yale University.
    7. de Jong, Robert M. & Woutersen, Tiemen, 2011. "Dynamic Time Series Binary Choice," Econometric Theory, Cambridge University Press, vol. 27(04), pages 673-702, August.
    8. Rosa L. Matzkin, 1989. "A Nonparametric Maximum Rank Correlation Estimator," Cowles Foundation Discussion Papers 918, Cowles Foundation for Research in Economics, Yale University.
    9. Benedikt M. Pötscher & Ingmar R. Prucha, 1999. "Basic Elements of Asymptotic Theory," Electronic Working Papers 99-001, University of Maryland, Department of Economics.
    10. Duffie, Darrell & Singleton, Kenneth J, 1993. "Simulated Moments Estimation of Markov Models of Asset Prices," Econometrica, Econometric Society, vol. 61(4), pages 929-952, July.
    11. Joris Pinkse, 2000. "Feasible Multivariate Nonparametric Estimation Using Weak Separability," Econometric Society World Congress 2000 Contributed Papers 1241, Econometric Society.

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