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Some new asymptotic theory for least squares series: Pointwise and uniform results

Author

Listed:
  • Belloni, Alexandre
  • Chernozhukov, Victor
  • Chetverikov, Denis
  • Kato, Kengo

Abstract

In econometric applications it is common that the exact form of a conditional expectation is unknown and having flexible functional forms can lead to improvements over a pre-specified functional form, especially if they nest some successful parametric economically-motivated forms. Series method offers exactly that by approximating the unknown function based on k basis functions, where k is allowed to grow with the sample size n to balance the trade off between variance and bias. In this work we consider series estimators for the conditional mean in light of four new ingredients: (i) sharp LLNs for matrices derived from the non-commutative Khinchin inequalities, (ii) bounds on the Lebesgue factor that controls the ratio between the L∞ and L2-norms of approximation errors, (iii) maximal inequalities for processes whose entropy integrals diverge at some rate, and (iv) strong approximations to series-type processes.

Suggested Citation

  • Belloni, Alexandre & Chernozhukov, Victor & Chetverikov, Denis & Kato, Kengo, 2015. "Some new asymptotic theory for least squares series: Pointwise and uniform results," Journal of Econometrics, Elsevier, vol. 186(2), pages 345-366.
  • Handle: RePEc:eee:econom:v:186:y:2015:i:2:p:345-366
    DOI: 10.1016/j.jeconom.2015.02.014
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    References listed on IDEAS

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    1. Xiaohong Chen & Timothy Christensen, 2013. "Optimal Uniform Convergence Rates for Sieve Nonparametric Instrumental Variables Regression," Cowles Foundation Discussion Papers 1923, Cowles Foundation for Research in Economics, Yale University.
    2. Andrews, Donald W K, 1991. "Asymptotic Normality of Series Estimators for Nonparametric and Semiparametric Regression Models," Econometrica, Econometric Society, vol. 59(2), pages 307-345, March.
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    4. Alexandre Belloni & Victor Chernozhukov & Ivan Fernandez-Val, 2011. "Conditional quantile processes based on series or many regressors," CeMMAP working papers CWP19/11, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Newey, Whitney K., 1997. "Convergence rates and asymptotic normality for series estimators," Journal of Econometrics, Elsevier, vol. 79(1), pages 147-168, July.
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    More about this item

    Keywords

    Least squares series; Strong approximations; Uniform confidence bands;

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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