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Three-stage semi-parametric inference: Control variables and differentiability

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  • Hahn, Jinyong
  • Ridder, Geert

Abstract

We show the usefulness of the path-derivative calculations that were introduced in econometrics by Newey (1994) for multi-step semi-parametric estimators. These estimators estimate a finite-dimensional parameter using moment conditions that depend on nonparametric regressions on observed and estimated regressors that are estimated in the second and first step of the estimation procedure, respectively. Our earlier paper showed that Newey’s calculations can be extended to three-step estimators. In the current paper we consider the control variable (CV) estimator and related statistics in semi-parametric econometric models with non-separable errors and regressors that are correlated with these errors. Non-separable econometric models with endogenous regressors are often identified by average moment restrictions that average over control variables, and these control variables are estimated in a first stage by (non)parametric regression. We study aspects of inference for such estimators where we focus on a finite-dimensional parameter vector or statistic. The asymptotic distribution and a closed-form expression for the asymptotic variance of the CV estimator were not available until now. Our path derivative calculations are much simpler than the derivation of the asymptotic distribution by a stochastic expansion that is particularly complicated for multi-step semi-parametric estimators. We also consider just- and overidentification of the parameters and we propose a diagnostic test for overidentifying restrictions in models with non-separable errors and endogenous regressors. Finally, the path-derivative calculation breaks down if the moment condition is not differentiable. In an example we show that non-differentiability is associated with irregular behavior of the estimator.

Suggested Citation

  • Hahn, Jinyong & Ridder, Geert, 2019. "Three-stage semi-parametric inference: Control variables and differentiability," Journal of Econometrics, Elsevier, vol. 211(1), pages 262-293.
  • Handle: RePEc:eee:econom:v:211:y:2019:i:1:p:262-293
    DOI: 10.1016/j.jeconom.2018.12.016
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    References listed on IDEAS

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    1. Jinyong Hahn & Geert Ridder, 2013. "Asymptotic Variance of Semiparametric Estimators With Generated Regressors," Econometrica, Econometric Society, vol. 81(1), pages 315-340, January.
    2. Guido W. Imbens & Whitney K. Newey, 2009. "Identification and Estimation of Triangular Simultaneous Equations Models Without Additivity," Econometrica, Econometric Society, vol. 77(5), pages 1481-1512, September.
    3. Keisuke Hirano & Guido W. Imbens & Geert Ridder, 2003. "Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score," Econometrica, Econometric Society, vol. 71(4), pages 1161-1189, July.
    4. 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.
    5. Jinyong Hahn & Geert Ridder, 2011. "Conditional Moment Restrictions and Triangular Simultaneous Equations," The Review of Economics and Statistics, MIT Press, vol. 93(2), pages 683-689, May.
    6. Duflo, Esther & Glennerster, Rachel & Kremer, Michael, 2008. "Using Randomization in Development Economics Research: A Toolkit," Handbook of Development Economics, in: T. Paul Schultz & John A. Strauss (ed.), Handbook of Development Economics, edition 1, volume 4, chapter 61, pages 3895-3962, Elsevier.
    7. Jinyong Hahn, 1998. "On the Role of the Propensity Score in Efficient Semiparametric Estimation of Average Treatment Effects," Econometrica, Econometric Society, vol. 66(2), pages 315-332, March.
    8. Newey, Whitney K, 1994. "The Asymptotic Variance of Semiparametric Estimators," Econometrica, Econometric Society, vol. 62(6), pages 1349-1382, November.
    9. Rivers, Douglas & Vuong, Quang H., 1988. "Limited information estimators and exogeneity tests for simultaneous probit models," Journal of Econometrics, Elsevier, vol. 39(3), pages 347-366, November.
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    11. Hausman, Jerry A., 1983. "Specification and estimation of simultaneous equation models," Handbook of Econometrics, in: Z. Griliches† & M. D. Intriligator (ed.), Handbook of Econometrics, edition 1, volume 1, chapter 7, pages 391-448, Elsevier.
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    More about this item

    Keywords

    Endogenous regressor; Non-separable errors; Control variable; Influence function; Overidentification; Differentiability;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C36 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Instrumental Variables (IV) Estimation

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