Impossibility Results for Nondifferentiable Functionals

Authors

  • Keisuke Hirano,

    1. Dept. of Economics, University of Arizona, Tucson, AZ 85721, U.S.A.; hirano@u.arizona.edu
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  • Jack R. Porter

    1. Dept. of Economics, University of Wisconsin, Madison, WI 53706, U.S.A.; jrporter@ssc.wisc.edu
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    • Early versions of this paper were titled “Impossibility Results for Bounds Estimation.” We are grateful to Donald Andrews, Matias Cattaneo, Gary Chamberlain, Jinyong Hahn, Bruce Hansen, Bo Honore, Guido Imbens, Karthik Kalyanaraman, Francesca Molinari, Marcelo Moreira, Ariel Pakes, Thomas Richardson, Jamie Robins, Kevin Song, two referees, the co-editor, and numerous seminar participants for helpful comments and suggestions. We thank the National Science Foundation for research support under Grants SES-0962488 (Hirano), SES-0962422 (Porter), and SES-0438123 (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 differentiable in the parameters of the data distribution, there exist no estimator sequences that are locally asymptotically unbiased or α-quantile unbiased. This places strong limits on estimators, bias correction methods, and inference procedures, and provides motivation for considering other criteria for evaluating estimators and inference procedures, such as local asymptotic minimaxity and one-sided quantile unbiasedness.

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