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Uniform Convergence Of Series Estimators Over Function Spaces

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  • Song, Kyungchul

Abstract

This paper considers a series estimator of E [α( Y )|λ( X ) = λ̄], (α,λ) null 𝛢 × Λ, indexed by function spaces, and establishes the estimator's uniform convergence rate over λ̄ null R , α null 𝛢, and λ null Λ, when 𝛢 and Λ have a finite integral bracketing entropy. The rate of convergence depends on the bracketing entropies of 𝛢 and Λ in general. In particular, we demonstrate that when each λ null Λ is locally uniformly null 2 -continuous in a parameter from a space of polynomial discrimination and the basis function vector p null in the series estimator keeps the smallest eigenvalue of E [ p null (λ( X )) p null (λ( X ))‼] above zero uniformly over λ null Λ, we can obtain the same convergence rate as that established by de Jong (2002, Journal of Econometrics 111, 1–9).

Suggested Citation

  • Song, Kyungchul, 2008. "Uniform Convergence Of Series Estimators Over Function Spaces," Econometric Theory, Cambridge University Press, vol. 24(06), pages 1463-1499, December.
  • Handle: RePEc:cup:etheor:v:24:y:2008:i:06:p:1463-1499_08
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    Cited by:

    1. Xiaohong Chen & Timothy M. Christensen, 2014. "Optimal Uniform Convergence Rates and Asymptotic Normality for Series Estimators under Weak Dependence and Weak Conditions," Cowles Foundation Discussion Papers 1976, Cowles Foundation for Research in Economics, Yale University.
    2. Escanciano, Juan Carlos & Jacho-Chávez, David T. & Lewbel, Arthur, 2014. "Uniform convergence of weighted sums of non and semiparametric residuals for estimation and testing," Journal of Econometrics, Elsevier, vol. 178(P3), pages 426-443.
    3. Song, Kyungchul, 2010. "Testing semiparametric conditional moment restrictions using conditional martingale transforms," Journal of Econometrics, Elsevier, vol. 154(1), pages 74-84, January.
    4. Mammen, Enno & Rothe, Christoph & Schienle, Melanie, 2016. "Semiparametric Estimation With Generated Covariates," Econometric Theory, Cambridge University Press, vol. 32(05), pages 1140-1177, October.
    5. Kyungchul Song, 2009. "Two-Step Extremum Estimation with Estimated Single-Indices," PIER Working Paper Archive 09-012, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    6. Escanciano, Juan Carlos & Song, Kyungchul, 2010. "Testing single-index restrictions with a focus on average derivatives," Journal of Econometrics, Elsevier, vol. 156(2), pages 377-391, June.
    7. Juan Carlos Escanciano & Lin Zhu, 2013. "Set inferences and sensitivity analysis in semiparametric conditionally identified models," CeMMAP working papers CWP55/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    8. Xiaohong Chen & Timothy M. Christensen, 2014. "Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions," CeMMAP working papers CWP46/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    9. Ying-Ying Lee, 2014. "Partial Mean Processes with Generated Regressors: Continuous Treatment Effects and Nonseparable Models," Economics Series Working Papers 706, University of Oxford, Department of Economics.

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