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Calculating confidence intervals for continuous and discontinuous functions of parameters

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  • Tiemen Woutersen

    (Institute for Fiscal Studies and John Hopkins University)

  • John Ham

    (Institute for Fiscal Studies and University of Maryland)

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    Abstract

    Applied researchers often need to estimate confidence intervals for functions of parameters, such as the effects of counterfactual policy changes. If the function is continuously differentiable and has non-zero and bounded derivatives, then they can use the delta method. However, if the function is nondifferentiable (as in the case of simulating functions with zero-one outcomes), has zero derivatives, or unbounded derivatives, then researchers usually use the nonparametric bootstrap or sample from the asymptotic distribution of the estimated parameter vector. Researchers also use these bootstrap approaches when the function is well-behaved but complicated. Indeed, these approaches are advocated by two very influential published articles. We first show that both of these bootstrap procedures can produce confidence intervals whose asymptotic coverage is less than advertised, i.e. confidence intervals that are too small. We then propose two procedures that provide correct coverage. In applications, we find that the bootstrap approaches mentioned above produce confidence intervals that are significantly smaller than their consistent counterparts, suggesting that previous empirical work is likely to have been overly optimistic in terms of the precision of estimated counterfactual effects.

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    File URL: http://www.cemmap.ac.uk/wps/cwp231313.pdf
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    Bibliographic Info

    Paper provided by Centre for Microdata Methods and Practice, Institute for Fiscal Studies in its series CeMMAP working papers with number CWP23/13.

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    Date of creation: May 2013
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    Handle: RePEc:ifs:cemmap:23/13

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    Related research

    Keywords: confidence intervals; simulation; structural models; policy effects;

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    References

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    1. Stephen Ryan, 2006. "The Costs of Environmental Regulation in a Concentrated Industry," 2006 Meeting Papers 9, Society for Economic Dynamics.
    2. Gaure, Simen & Røed, Knut & Westlie, Lars, 2012. "Job search incentives and job match quality," Labour Economics, Elsevier, vol. 19(3), pages 438-450.
    3. Jörg Stoye, 2008. "More on confidence intervals for partially identified parameters," CeMMAP working papers CWP11/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Gunter J. Hitsch & Ali Horta�su & Dan Ariely, 2010. "Matching and Sorting in Online Dating," American Economic Review, American Economic Association, vol. 100(1), pages 130-63, March.
    5. Andrews, Donald W.K. & Guggenberger, Patrik, 2010. "ASYMPTOTIC SIZE AND A PROBLEM WITH SUBSAMPLING AND WITH THE m OUT OF n BOOTSTRAP," Econometric Theory, Cambridge University Press, vol. 26(02), pages 426-468, April.
    6. Donald W. K. Andrews & Xu Cheng, 2012. "Estimation and Inference With Weak, Semi‐Strong, and Strong Identification," Econometrica, Econometric Society, vol. 80(5), pages 2153-2211, 09.
    7. Hoderlein, Stefan & Mihaleva, Sonya, 2008. "Increasing the price variation in a repeated cross section," Journal of Econometrics, Elsevier, vol. 147(2), pages 316-325, December.
    8. Andrews, Donald W.K. & Cheng, Xu, 2013. "Maximum likelihood estimation and uniform inference with sporadic identification failure," Journal of Econometrics, Elsevier, vol. 173(1), pages 36-56.
    9. Donald W. K. Andrews, 2000. "Inconsistency of the Bootstrap when a Parameter Is on the Boundary of the Parameter Space," Econometrica, Econometric Society, vol. 68(2), pages 399-406, March.
    10. Fitzenberger, Bernd & Osikominu, Aderonke & Paul, Marie, 2010. "The heterogeneous effects of training incidence and duration on labor market transitions," ZEW Discussion Papers 10-077, ZEW - Zentrum für Europäische Wirtschaftsforschung / Center for European Economic Research.
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