Empirical research typically involves a robustness-efficiency tradeoff. A researcher seeking to estimate a scalar parameter can invoke strong assumptions to motivate a restricted estimator that is precise but may be heavily biased, or they can relax some of these assumptions to motivate a more robust, but variable, unrestricted estimator. When a bound on the bias of the restricted estimator is available, it is optimal to shrink the unrestricted estimator towards the restricted estimator. For settings where a bound on the bias of the restricted estimator is unknown, we propose adaptive estimators that minimize the percentage increase in worst case risk relative to an oracle that knows the bound. We show that adaptive estimators solve a weighted convex minimax problem and provide lookup tables facilitating their rapid computation. Revisiting some well known empirical studies where questions of model specification arise, we examine the advantages of adapting to—rather than testing for—misspecification.
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- Acknowledgements & Disclosure
- Timothy Armstrong gratefully acknowledges support from National Science Foundation Grant SES-2049765. Liyang Sun gratefully acknowledges support from the Institute of Education Sciences, U.S. Department of Education, through Grant R305D200010, and Ayudas Juan de la Cierva Formacion. The views expressed herein are those of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.
- DOI
- https://doi.org/10.3386/w32906
- Pages
- 65
- Published in
- United States of America
Table of Contents
- Introduction 3
- An introductory example 6
- Setup 10
- B-minimax estimators 11
- Adaptation 12
- When is adaptation desirable? 14
- Main results 15
- Adaptation as minimax with scaled loss 15
- Minimax and adaptive estimators 16
- Computation, least favorable prior and lookup table 18
- Weighted average interpretation 19
- Impossibility of consistently estimating the asymptotic distribution 19
- Confidence Intervals 20
- Analytic adaptive estimators 21
- Constrained adaptation 25
- Examples 25
- Adapting to heterogeneous effects gentzkoweffect2011 26
- Adapting to endogeneity angristdoes1991 29
- Conclusion 31
- Group decision making interpretation 35
- Consensus in a single committee 36
- Consensus among committees 36
- Discussion 37
- Details and proofs 38
- Details for Theorem 4.1 and extensions 38
- Lasso interpretation of soft thresholding 41
- Additional details 45
- Constrained adaptation 45
- Details for constrained adaptation 46
- Numerical results on estimators as a function of 1-2 47
- Asymptotics as ||1 48
- Upper bounds 49
- Lower bounds 50
- Computational details 51
- Computing minimax estimators 51
- Discrete approximation to estimators and risk function 53
- Computing minimax risk in the bounded normal mean problem 53
- Computing the optimally adaptive estimator for a given 2 55
- Computing the optimally adaptive estimator based on the lookup table 56
- Computing the analytic adaptive estimators 56
- Soft thresholding 57
- Hard thresholding 58
- Adaptive ERM 58
- Pooling controls (LaLonde, 1986) 58
- Details of bivariate adaptation 62
- Pairwise adaptation 63
- Bivariate adaptation with GMM composite 64
