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Optimal Decision Rules Under Partial Identification

Kohei Yata

TL;DR

This paper develops a finite-sample decision framework for choosing between two policies when the welfare contrast is potentially only partially identified. It leverages the modulus of continuity to reduce the minimax-regret problem to a hardest one-dimensional subproblem, yielding a simple, data-driven decision rule that is either nonrandomized or randomized depending on identification strength and variance. The key theoretical contributions include a complete characterization of the minimax regret rule, its interpretation as a plug-in estimator with a bias-variance tradeoff, and a close connection to optimal estimation under restricted parameter spaces. The framework is illustrated with two empirical contexts: an eligibility-cutoff choice in a regression discontinuity setting and a BRIGHT school-construction program in Burkina Faso, demonstrating practical computation and policy implications for extrapolation and policy choice under partial identification.

Abstract

I consider a class of statistical decision problems in which the policymaker must decide between two policies to maximize social welfare (e.g., the population mean of an outcome) based on a finite sample. The framework introduced in this paper allows for various types of restrictions on the structural parameter (e.g., the smoothness of a conditional mean potential outcome function) and accommodates settings with partial identification of social welfare. As the main theoretical result, I derive a finite-sample optimal decision rule under the minimax regret criterion. This rule has a simple form, yet achieves optimality among all decision rules; no ad hoc restrictions are imposed on the class of decision rules. I apply my results to the problem of whether to change an eligibility cutoff in a regression discontinuity setup, and illustrate them in an empirical application to a school construction program in Burkina Faso.

Optimal Decision Rules Under Partial Identification

TL;DR

This paper develops a finite-sample decision framework for choosing between two policies when the welfare contrast is potentially only partially identified. It leverages the modulus of continuity to reduce the minimax-regret problem to a hardest one-dimensional subproblem, yielding a simple, data-driven decision rule that is either nonrandomized or randomized depending on identification strength and variance. The key theoretical contributions include a complete characterization of the minimax regret rule, its interpretation as a plug-in estimator with a bias-variance tradeoff, and a close connection to optimal estimation under restricted parameter spaces. The framework is illustrated with two empirical contexts: an eligibility-cutoff choice in a regression discontinuity setting and a BRIGHT school-construction program in Burkina Faso, demonstrating practical computation and policy implications for extrapolation and policy choice under partial identification.

Abstract

I consider a class of statistical decision problems in which the policymaker must decide between two policies to maximize social welfare (e.g., the population mean of an outcome) based on a finite sample. The framework introduced in this paper allows for various types of restrictions on the structural parameter (e.g., the smoothness of a conditional mean potential outcome function) and accommodates settings with partial identification of social welfare. As the main theoretical result, I derive a finite-sample optimal decision rule under the minimax regret criterion. This rule has a simple form, yet achieves optimality among all decision rules; no ad hoc restrictions are imposed on the class of decision rules. I apply my results to the problem of whether to change an eligibility cutoff in a regression discontinuity setup, and illustrate them in an empirical application to a school construction program in Burkina Faso.

Paper Structure

This paper contains 60 sections, 17 theorems, 99 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Literature.
  3. Problem Setup and Examples
  4. Data-generating Model
  5. Policy Choice Problem
  6. Optimality Criterion
  7. Relation to Existing Frameworks
  8. Examples
  9. Minimax Regret Rules in General Setup
  10. Minimax Regret Rules for One-dimensional Subproblems
  11. Case (i): $L(\bar{\theta})>0$ and $\boldsymbol{m}(\bar{\theta})\neq \boldsymbol{0}$.
  12. Case (ii): $L(\bar{\theta})>0$ and $\boldsymbol{m}(\bar{\theta})= \boldsymbol{0}$.
  13. Hardest One-dimensional Subproblem
  14. Main Result: Minimax Regret Rule for the Full Problem
  15. The Role of Randomization
  16. ...and 45 more sections

Key Result

Lemma 1

Suppose Assumption assumption:problem holds, and consider a one-dimensional subproblem for $[-\bar{\theta},\bar{\theta}]$, where $\bar{\theta}\in\Theta$ and $L(\bar{\theta})\ge 0$. Then, the following holds.

Figures (7)

  • Figure 1: Illustration of Eligibility Cutoff Choice
  • Figure 2: Optimal Decisions: Probability of Choosing the New Policy
  • Figure 3: Weight to Each Village Attached by the Minimax Regret Rule
  • Figure 4: Estimated Effects of the New Policy on the Enrollment Rate
  • Figure 5: Maximum Regret of the Minimax Regret Rule and Plug-in MSE Rules
  • ...and 2 more figures

Theorems & Definitions (40)

  • Example 1: Evidence Aggregation ishihara2021meta
  • Example 2: Choice of Treatment Assignment Policy under Unconfoundedness
  • Lemma 1: Minimax Regret Rules for One-dimensional Subproblems
  • proof
  • Lemma 2
  • proof
  • Lemma 3: Hardest One-dimensional Subproblem
  • proof
  • Remark 1: Role of the Centrosymmetry of $\Theta$
  • Example 1: Continued
  • ...and 30 more