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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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Suggested Citation

  • Keisuke Hirano & Jack R. Porter, 2012. "Impossibility Results for Nondifferentiable Functionals," Econometrica, Econometric Society, vol. 80(4), pages 1769-1790, July.
  • Handle: RePEc:ecm:emetrp:v:80:y:2012:i:4:p:1769-1790 DOI: ECTA8681
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    References listed on IDEAS

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    1. Stoye, Jörg, 2009. "Minimax regret treatment choice with finite samples," Journal of Econometrics, Elsevier, pages 70-81.
    2. Keisuke Hirano & Jack R. Porter, 2009. "Asymptotics for Statistical Treatment Rules," Econometrica, Econometric Society, pages 1683-1701.
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    5. Florian Schuett, 2013. "Patent quality and incentives at the patent office," RAND Journal of Economics, RAND Corporation, pages 313-336.
    6. Charles F. Manski, 2003. "Statistical treatment rules for heterogeneous populations," CeMMAP working papers CWP03/03, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    7. Tetenov, Aleksey, 2012. "Statistical treatment choice based on asymmetric minimax regret criteria," Journal of Econometrics, Elsevier, pages 157-165.
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    9. Tetenov, Aleksey, 2012. "Statistical treatment choice based on asymmetric minimax regret criteria," Journal of Econometrics, Elsevier, pages 157-165.
    10. Charles F. Manski & John V. Pepper, 1998. "Monotone Instrumental Variables with an Application to the Returns to Schooling," NBER Technical Working Papers 0224, National Bureau of Economic Research, Inc.
    11. Jorg 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.
    12. Victor Chernozhukov & Han Hong & Elie Tamer, 2007. "Estimation and Confidence Regions for Parameter Sets in Econometric Models," Econometrica, Econometric Society, pages 1243-1284.
    13. 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.
    14. Durlauf, Steven N., 2004. "Neighborhood effects," Handbook of Regional and Urban Economics,in: J. V. Henderson & J. F. Thisse (ed.), Handbook of Regional and Urban Economics, edition 1, volume 4, chapter 50, pages 2173-2242 Elsevier.
    15. Philip A. Haile & Elie Tamer, 2003. "Inference with an Incomplete Model of English Auctions," Journal of Political Economy, University of Chicago Press, pages 1-51.
    16. Charles F. Manski & John V. Pepper, 2000. "Monotone Instrumental Variables, with an Application to the Returns to Schooling," Econometrica, Econometric Society, pages 997-1012.
    17. Hillier, Grant, 2009. "On The Conditional Likelihood Ratio Test For Several Parameters In Iv Regression," Econometric Theory, Cambridge University Press, pages 305-335.
    18. Sheetal Sekhri, 2011. "Public Provision and Protection of Natural Resources: Groundwater Irrigation in Rural India," American Economic Journal: Applied Economics, American Economic Association, pages 29-55.
    19. Fan, Yanqin & Park, Sang Soo, 2010. "Confidence sets for some partially identified parameters," MPRA Paper 37149, University Library of Munich, Germany.
    20. Charles F. Manski, 2004. "Statistical Treatment Rules for Heterogeneous Populations," Econometrica, Econometric Society, pages 1221-1246.
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    22. Canay, Ivan A., 2010. "EL inference for partially identified models: Large deviations optimality and bootstrap validity," Journal of Econometrics, Elsevier, pages 408-425.
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    Cited by:

    1. Victor Chernozhukov & Christian Hansen & Yuan Liao, 2015. "A lava attack on the recovery of sums of dense and sparse signals," CeMMAP working papers CWP56/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Lorenzo Almada & Ian McCarthy & Rusty Tchernis, 2016. "What Can We Learn about the Effects of Food Stamps on Obesity in the Presence of Misreporting?," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, pages 997-1017.
    3. Daniel Millimet & Manan Roy, 2015. "Partial identification of the long-run causal effect of food security on child health," Empirical Economics, Springer, pages 83-141.
    4. Timothy B. Armstrong, 2014. "A Note on Minimax Testing and Confidence Intervals in Moment Inequality Models," Cowles Foundation Discussion Papers 1975, Cowles Foundation for Research in Economics, Yale University.
    5. Ian M. McCarthy & Daniel L. Millimet & Manan Roy, 2014. "Bounding Treatment Effects: Stata Command for the Partial Identification of the Average Treatment Effect with Endogenous and Misreported Treatment Assignment," Emory Economics 1407, Department of Economics, Emory University (Atlanta).
    6. 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.
    7. Xu, Jiawen & Perron, Pierre, 2014. "Forecasting return volatility: Level shifts with varying jump probability and mean reversion," International Journal of Forecasting, Elsevier, pages 449-463.
    8. 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.
    9. Amin, Vikesh & Flores, Carlos A. & Flores-Lagunes, Alfonso & Parisian, Daniel J., 2016. "The effect of degree attainment on arrests: Evidence from a randomized social experiment," Economics of Education Review, Elsevier, pages 259-273.
    10. Ying Huang & Eric Laber, 2016. "Personalized Evaluation of Biomarker Value: A Cost-Benefit Perspective," Statistics in Biosciences, Springer;International Chinese Statistical Association, pages 43-65.
    11. Lukáš Lafférs, 2015. "Bounding average treatment effects using linear programming," CeMMAP working papers CWP70/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    12. Amin, Vikesh & Flores, Carlos A. & Flores-Lagunes, Alfonso & Parisian, Daniel J., 2016. "The effect of degree attainment on arrests: Evidence from a randomized social experiment," Economics of Education Review, Elsevier, pages 259-273.
    13. Song, Kyungchul, 2014. "Local asymptotic minimax estimation of nonregular parameters with translation-scale equivariant maps," Journal of Multivariate Analysis, Elsevier, pages 136-158.
    14. Chen, Xuan & Flores, Carlos A. & Flores-Lagunes, Alfonso, 2015. "Going Beyond LATE: Bounding Average Treatment Effects of Job Corps Training," IZA Discussion Papers 9511, Institute for the Study of Labor (IZA).
    15. Gerard, Francois & Rokkanen, Miikka & Rothe, Christoph, 2016. "Bounds on Treatment Effects in Regression Discontinuity Designs under Manipulation of the Running Variable, with an Application to Unemployment Insurance in Brazil," CEPR Discussion Papers 11668, C.E.P.R. Discussion Papers.
    16. repec:cup:etheor:v:33:y:2017:i:05:p:1218-1241_00 is not listed on IDEAS
    17. 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).
    18. Andrew Chesher & Adam Rosen, 2013. "Generalized instrumental variable models," CeMMAP working papers CWP43/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.

    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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