Hirano, Keisuke and Porter, Jack (2009): Impossibility Results for Nondifferentiable Functionals.
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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.
|Item Type:||MPRA Paper|
|Original Title:||Impossibility Results for Nondifferentiable Functionals|
|Keywords:||bounds analysis; moment inequality models; treatment effects; limits of experiments|
|Subjects:||C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General
C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General
C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General
|Depositing User:||Keisuke Hirano|
|Date Deposited:||02. Jul 2009 02:26|
|Last Modified:||19. Feb 2013 20:21|
Andrews, D. W. K., and P. Guggenberger (2007): “Validity of Subsampling and “Plug-in Asymptotic” Inference for Parameters Deﬁned by Moment Inequalities,” Working Paper.
Andrews, D. W. K., and G. Soares (2007): “Inference for Parameters Deﬁned by Moment Inequalities using Generalized Moment Selection,” Working Paper.
Beran, R. (1997): “Diagnosing Bootstrap Success,” Annals of the Institute of Statistical Mathematics, 49(1), 1–24.
Blumenthal, S., and A. Cohen (1968): “Estimation of the Larger of Two Normal Means,” Journal of the American Statistical Association, 63(323), 861–876.
Bugni, F. A. (2008): “Bootstrap Inference in Partially Identiﬁed Models,” Working Paper.
Canay, I. A. (2008): “EL Inference for Partially Identiﬁed Models: Large Deviations Optimality and Bootstrap Validity,” Working Paper.
Chakraborty, B., V. Strecher, and S. A. Murphy (2008): “Bias Correction and Conﬁdence Intervals for Fitted Q-Iteration,” Working Paper.
Chernozhukov, V., H. Hong, and E. Tamer (2007): “Estimation and Conﬁdence Regions for Parameter Sets in Econometric Models,” Econometrica, 75(5), 1243–1284.
Dehejia, R. H. (2005): “Program Evaluation as a Decision Problem,” Journal of Econometrics, 125, 141–173.
Doss, H., and J. Sethuraman (1989): “The Price of Bias Reduction when there is no Unbiased Estimate,” The Annals of Statistics, 17(1), 440–442.
Fan, Y., and S. S. Park (2008): “Conﬁdence Sets for Some Partially Identiﬁed Parameters,” Working Paper, University of North Carolina.
Haile, P. A., and E. Tamer (2003): “Inference with an Incomplete Model of English Auctions,” Journal of Political Economy, 111(1), 1–51.
Hirano, K., and J. R. Porter (2008): “Asymptotics for Statistical Treatment Rules,” forthcoming, Econometrica.
Imbens, G. W., and C. F. Manski (2004): “Conﬁdence Intervals for Partially Identiﬁed Parameters,” Econometrica, 72(6), 1845–1857.
Kreider, B., and J. V. Pepper (2007): “Disability and Employment: Reevaluating the Evidence in Light of Reporting Errors,” Journal of the American Statistical Association, 102(478), 432–441.
Manski, C. F. (2004): “Statistical Treatment Rules for Heterogeneous Populations,” Econometrica, 72(4), 1221–1246.
Manski, C. F., and J. V. Pepper (2000): “Monotone Instrumental Variables: with an Application to the Returns to Schooling,” Econometrica, 68(4), 997–1010.
Moodie, E. E. M., and T. S. Richardson (2007): “Bias Correction in Non-Diﬀerentiable Estimating Equations for Optimal Dynamic Regimes,” Working Paper.
Murphy, S. A. (2003): “Optimal Dynamic Treatment Regimes,” Journal of the Royal Statistical Society, Series B, 65, 331–366.
Pakes, A., J. R. Porter, K. Ho, and J. Ishii (2006): “Moment Inequalities and Their Application,” Working Paper.
Robins, J. M. (2004): “Optimal Structural Nested Models for Optimal Decisions,” in Proceedings of the Second Seattle Symposium on Biostatistics, ed. by D. Y. Lin, and P. Heagerty, New York. Springer-Verlag.
Rosen, A. M. (2008): “Conﬁdence Sets for Partially Identiﬁed Parameters that Satisfy a Finite Number of Moment Inequalities,” Journal of Econometrics, 146(1), 107–117.
Schlag, K. (2006): “Eleven – Tests Needed for a Recommendation,” EUI Working Paper ECO 2006-2.
Stoye, J. (2006): “Minimax Regret Treatment Choice with Finite Samples,” Working paper.
Stoye, J. (2008): “More on Conﬁdence Intervals for Partially Identiﬁed Parameters,” forthcoming, Econometrica.
Tetenov, A. (2007): “Statistical Treatment Choice Based on Asymmetric Minmax Regret Criteria,” Working paper.
Van der Vaart, A. W. (1991a): “An Asymptotic Representation Theorem,” International Statistical Review, 59, 99–121.
Van der Vaart, A. W. (1991b): “On Diﬀerentiable Functionals,” The Annals of Statistics, 19, 178–204.
Van der Vaart, A. W. (1998): Asymptotic Statistics. Cambridge University Press, New York.