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Impossibility Results for Nondifferentiable Functionals

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  • Keisuke Hirano
  • Jack R. Porter

Abstract

We examine challenges to estimation and inference when the objects of interest are nondifferentiable functionals of the underlying data distribution. This situation arises in a number of applications of bounds analysis and moment inequality models, and in recent work on estimating optimal dynamic treatment regimes. Drawing on earlier work relating differentiability to the existence of unbiased and regular estimators, we show that if the target object is not continuously differentiable in the parameters of the data distribution, there exist no locally asymptotically unbiased estimators and no regular estimators. This places strong limits on estimators, bias correction methods, and inference procedures.

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Bibliographic Info

Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 80 (2012)
Issue (Month): 4 (07)
Pages: 1769-1790

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Handle: RePEc:ecm:emetrp:v:80:y:2012:i:4:p:1769-1790

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References

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  1. Adam Rosen, 2006. "Confidence sets for partially identified parameters that satisfy a finite number of moment inequalities," CeMMAP working papers CWP25/06, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  2. Dehejia, Rajeev H., 2005. "Program evaluation as a decision problem," Journal of Econometrics, Elsevier, vol. 125(1-2), pages 141-173.
  3. Haile,P.A. & Tamer,E.T., 2000. "Inference with an incomplete model of English auctions," Working papers 18, Wisconsin Madison - Social Systems.
  4. Tetenov, Aleksey, 2012. "Statistical treatment choice based on asymmetric minimax regret criteria," Journal of Econometrics, Elsevier, vol. 166(1), pages 157-165.
  5. Kreider, Brent & Pepper, John V., 2007. "Disability and Employment: Reevaluating the Evidence in Light of Reporting Errors," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 432-441, June.
  6. Keisuke Hirano & Jack R. Porter, 2009. "Asymptotics for Statistical Treatment Rules," Econometrica, Econometric Society, vol. 77(5), pages 1683-1701, 09.
  7. Fan, Yanqin & Park, Sang Soo, 2010. "Confidence sets for some partially identified parameters," MPRA Paper 37149, University Library of Munich, Germany.
  8. Charles F. Manski & John V. Pepper, 2000. "Monotone Instrumental Variables, with an Application to the Returns to Schooling," Econometrica, Econometric Society, vol. 68(4), pages 997-1012, July.
  9. Karl Schlag, 2006. "ELEVEN - Tests needed for a Recommendation," Economics Working Papers ECO2006/2, European University Institute.
  10. Stoye, Jörg, 2009. "Minimax regret treatment choice with finite samples," Journal of Econometrics, Elsevier, vol. 151(1), pages 70-81, July.
  11. Victor Chernozhukov & Han Hong & Elie Tamer, 2007. "Estimation and Confidence Regions for Parameter Sets in Econometric Models," Econometrica, Econometric Society, vol. 75(5), pages 1243-1284, 09.
  12. 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.
  13. S. A. Murphy, 2003. "Optimal dynamic treatment regimes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 331-355.
  14. Charles F. Manski, 2004. "Statistical Treatment Rules for Heterogeneous Populations," Econometrica, Econometric Society, vol. 72(4), pages 1221-1246, 07.
  15. Canay, Ivan A., 2010. "EL inference for partially identified models: Large deviations optimality and bootstrap validity," Journal of Econometrics, Elsevier, vol. 156(2), pages 408-425, June.
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Cited by:
  1. Ham, John C. & Woutersen, Tiemen, 2011. "Calculating Confidence Intervals for Continuous and Discontinuous Functions of Estimated Parameters," IZA Discussion Papers 5816, Institute for the Study of Labor (IZA).
  2. Hiroaki Kaido, 2014. "Asymptotically efficient estimation of weighted average derivatives with an interval censored variable," CeMMAP working papers CWP03/14, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  3. Donald W.K. Andrews, 2011. "Similar-on-the-Boundary Tests for Moment Inequalities Exist, But Have Poor Power," Cowles Foundation Discussion Papers 1815R, Cowles Foundation for Research in Economics, Yale University, revised Mar 2012.
  4. Song, Kyungchul, 2014. "Local asymptotic minimax estimation of nonregular parameters with translation-scale equivariant maps," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 136-158.
  5. Kyungchul Song, 2009. "Point Decisions for Interval-Identified Parameters," PIER Working Paper Archive 09-036, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.

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