IMPOSSIBILITY RESULTS FOR NONDIFFERENTIABLE FUNCTIONALS

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 tr...

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Bibliographic Details
Published in:Econometrica Vol. 80; no. 4; pp. 1769 - 1790
Main Authors: Hirano, Keisuke, Porter, Jack R.
Format: Journal Article
Language:English
Published: Oxford, UK Econometric Society 01.07.2012
Blackwell Publishing Ltd
Wiley-Blackwell
Subjects:
Applications
Bias
bias‐correction
bounds
Differential analysis
Distribution theory
Dynamic models
Econometric models
Econometrics
Estimating techniques
Estimation
Estimation bias
Estimators
Exact sciences and technology
Functional analysis
Global analysis, analysis on manifolds
Induced substructures
Inequality
Inference
Insurance, economics, finance
Local asymptotics
Mathematical independent variables
Mathematics
Modeling
moment inequality models
Motivation
NOTES AND COMMENTS
Parameter estimation
Parametric models
Probability
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Sequences
Statistical inference
Statistical models
Statistics
Studies
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Unbiased estimators
Working papers
Statistical distribution
moment inequality models
Statistical moment
Bias
Statistical estimation
bias-correction
Functional
Statistical method
Unbiased estimation
Distribution function
bounds
Approximation theory
Econometrics
Quantile
Local asymptotics
Biased estimation
ISSN:0012-9682, 1468-0262
Online Access:Get full text
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Description
Summary: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.
Bibliography: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).
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ISSN:0012-9682
1468-0262
DOI:10.3982/ECTA8681