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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) ∈ 𝛢 × Λ, indexed by function spaces, and establishes the estimator's uniform convergence rate over λ̄ ∈ R, α ∈ 𝛢, and λ ∈ 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 ∈ Λ is locally uniformly ℒ2-continuous in a parameter from a space of polynomial discrimination and the basis function vector pK in the series estimator keeps the smallest eigenvalue of E[pK(λ(X))pK(λ(X))‼] above zero uniformly over λ ∈ 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(6), pages 1463-1499, December.
    • Handle: RePEc:cup:etheor:v:24:y:2008:i:06:p:1463-1499_08
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    Citations

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    Cited by:

    1. Mammen, Enno & Rothe, Christoph & Schienle, Melanie, 2016. "Semiparametric Estimation With Generated Covariates," Econometric Theory, Cambridge University Press, vol. 32(5), pages 1140-1177, October.
    2. Ying-Ying Lee, 2018. "Partial Mean Processes with Generated Regressors: Continuous Treatment Effects and Nonseparable Models," Papers 1811.00157, arXiv.org.
    3. Javier Alejo & Antonio F. Galvao & Julián Martinez-Iriarte & Gabriel Montes-Rojas, 2023. "Unconditional Quantile Partial Effects via Conditional Quantile Regression," Working Papers 217, Red Nacional de Investigadores en Economía (RedNIE).
    4. Song, Kyungchul, 2010. "Testing semiparametric conditional moment restrictions using conditional martingale transforms," Journal of Econometrics, Elsevier, vol. 154(1), pages 74-84, January.
    5. Xiaohong Chen & Timothy M. Christensen, 2013. "Optimal uniform convergence rates for sieve nonparametric instrumental variables regression," CeMMAP working papers CWP56/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    6. repec:hum:wpaper:sfb649dp2011-064 is not listed on IDEAS
    7. Jinyong Hahn & Geert Ridder, 2013. "Asymptotic Variance of Semiparametric Estimators With Generated Regressors," Econometrica, Econometric Society, vol. 81(1), pages 315-340, January.
    8. 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.
    9. repec:hum:wpaper:sfb649dp2014-043 is not listed on IDEAS
    10. Chen, Xiaohong & Christensen, Timothy M., 2015. "Optimal uniform convergence rates and asymptotic normality for series estimators under weak dependence and weak conditions," Journal of Econometrics, Elsevier, vol. 188(2), pages 447-465.
    11. Xiaohong Chen & Timothy M. Christensen, 2013. "Optimal uniform convergence rates for sieve nonparametric instrumental variables regression," CeMMAP working papers 56/13, Institute for Fiscal Studies.
    12. 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.
    13. 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.
    14. Juan Carlos Escanciano & Lin Zhu, 2013. "Set inferences and sensitivity analysis in semiparametric conditionally identified models," CeMMAP working papers 55/13, Institute for Fiscal Studies.
    15. 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.
    16. 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 46/14, Institute for Fiscal Studies.
    17. 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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