Preface
Introduction
The Reflection Problem
The Law of Decreasing Credibility
Identification and Statistical Inference
Prediction and Decisions
Coping with Ambiguity
Organization of the Book
The Developing Literature on Partial Identification
I. Prediction with Incomplete Data
1. Conditional Prediction
1.1. Predicting Criminality
1.2. Probabilistic Prediction
Conditional Distributions
Best Predictors
Specifying a Loss Function
1.3. Estimation of Best Predictors from Random Samples
Covariates with Positive Probability
Covariates with Zero Probability but on the Support
Covariates off the Support
1.4. Extrapolation
Invariance Assumptions and Shape Restrictions
Testing and Using Theories
1.5. Predicting High School Graduation
Complement 1A. Best Predictors under Square and Absolute Loss
Square Loss
Absolute Loss
Complement 1B. Nonparametric Regression Analysis
Consistency of the Local-Average Estimate
Choosing an Estimate
Complement 1C. Word Problems
2. Missing Outcomes
2.1. Anatomy of the Problem
Identification of Event Probabilities
Identification of Quantiles
2.2. Bounding the Probability of Exiting Homelessness
Is the Cup Part Empty or Part Full?
2.3. Means of Functions of the Outcome
Bounded Random Variables
Unbounded Random Variables
2.4. Parameters That Respect Stochastic Dominance
2.5. Distributional Assumptions
Missingness at Random
Refutable and Non-refutable Assumptions
Refutability and Credibility
2.6. Wage Regressions and the Reservation-Wage Model of Labor Supply
Homogeneous Reservation Wages
Other Cases of Missingness by Choice
2.7. Statistical Inference
Sample Analogs of Identification Regions
Confidence Sets
Testing Refutable Assumptions
Complement 2A. Interval Measurement of Outcomes
Measurement Devices with Bounded Range
Complement 2B. Jointly Missing Outcomes and Covariates
Conditioning on a Subset of the Outcomes
Illustration: Bounding the Probability of Employment and the Unemployment Rate
Complement 2C. Convergence of Sets to Sets
3. Instrumental Variables
3.1. Distributional Assumptions and Credible Inference
Assumptions using Instrumental Variables
3.2. Missingness at Random
Conditioning Is Not Controlling
3.3. Statistical Independence
Binary Outcomes
Identifying Power
Combining Multiple Surveys
3.4. Equality of Means
Means Missing at Random
Mean Independence
3.5. Inequality of Means
Means Missing Monotonically
Monotone Regressions
Complement 3A. Imputations and Nonresponse Weights
Imputations
Nonresponse Weights
Complement 3B. Conditioning on the Propensity Score
Complement 3C. Word Problems
4. Parametric Prediction
4.1. The Normal-Linear Model of Market and Reservation Wages
4.2. Selection Models
A Semiparametric Model
4.3. Parametric Models for Best Predictors
Identification of the Parameters and the Best Predictor
Linear-Index Models
Statistical Inference
Complement 4A. Minimum-Distance Estimation of Partially Identified Models
5. Decomposition of Mixtures
5.1. The Inferential Problem and Some Manifestations
The Problem in Abstraction
Ecological Inference
Contaminated Sampling
The Task Ahead
5.2. Binary Mixing Covariates
Inference on One Component Distribution
Event Probabilities
Parameters That Respect Stochastic Dominance
5.3. Contamination through Imputation
Income Distribution in the United States
Corrupted Sampling
5.4. Instrumental Variables
The Identification Region
Complement 5A. Sharp Bounds on Parameters That Respect Stochastic Dominance
6. Response-Based Sampling
6.1. The Odds Ratio and Public Health
Relative and Attributable Risk
The Rare-Disease Assumption
6.2. Bounds on Relative and Attributable Risk
Relative Risk
Attributable Risk
6.3. Information on Marginal Distributions
6.4. Sampling from One Response Stratum
Using Administrative Records to Infer AFDC Transition Rates
6.5. General Binary Stratifications
Sampling from Both Strata
Sampling from One Stratum
II. Analysis of Treatment Response
7. The Selection Problem
7.1. Anatomy of the Problem
Prediction using the Empirical Evidence Alone
Comparing Treatments
Average Treatment Effects
Distributional Assumptions
7.2. Sentencing and Recidivism
7.3. Randomized Experiments
Experiments in Practice
7.4. Compliance with Treatment Assignment
Experiments without Crossover
Experiments with Crossover
Point Identification with Partial Compliance
Intention to Treat
The Effect of Treatment on Compliers
7.5. Treatment by Choice
Outcome Optimization
Parametric Selection Models
7.6. Treatment at Random in Non-Experimental Settings
Association and Causation
Sensitivity Analysis
7.7. Homogeneous Linear Response
“The” Instrumental Variables Estimator
Mean Independence and Overidentification
Complement 7A. Perspectives on Treatment Comparison
Differences in Outcome Distributions or Distributions of Outcome Differences
The Population To Be Treated or the Subpopulation of the Treated
Complement 7B. Word Problems
8. Linear Simultaneous Equations
8.1. Simultaneity in Competitive Markets
“The” Identification Problem in Econometrics
Simultaneity Is Selection
8.2. The Linear Market Model
Credibility of the Assumptions
Analysis of the Reduced Form
8.3. Equilibrium in Games
Ehrlich, the Supreme Court, and the National Research Council
8.4. The Reflection Problem
Endogenous, Contextual, and Correlated Effects
The Linear-in-Means Model
Identification of the Parameters
Inferring the Composition of Reference Groups
9. Monotone Treatment Response
9.1. Shape Restrictions
Downward-Sloping Demand
Production Analysis
9.2. Bounds on Parameters That Respect Stochastic Dominance
The General Result
Means of Increasing Functions of the Outcome
Upper Tail Probabilities
9.3. Bounds on Treatment Effects
Average Treatment Effects
9.4. Monotone Response and Selection
Interpreting the Statement “Wage Increases with Schooling”
Bounds on Mean Outcomes and Average Treatment Effects
9.5. Bounding the Returns to Schooling
Data
Statistical Considerations
Findings
10. The Mixing Problem
10.1. Extrapolation from Experiments to Rules with Treatment Variation
From Marginals to Mixtures
10.2. Extrapolation from the Perry Preschool Experiment
Prediction with the Experimental Evidence Alone
Prediction with Assumptions
10.3. Identification of Event Probabilities with the Experimental Evidence Alone
10.4. Treatment Response Assumptions
Statistically Independent Outcomes
Monotone Treatment Response
10.5. Treatment Rule Assumptions
Treatment at Random
Outcome Optimization
Known Treatment Shares
10.6. Combining Assumptions
11. Planning under Ambiguity
11.1. Studying Treatment Response to Inform Treatment Choice
Partial Identification and Ambiguity
11.2. Criteria for Choice under Ambiguity
Dominance
Bayes Rules
The Maximin Criterion
The Minimax-Regret Criterion
11.3. Treatment using Data from an Experiment with Partial Compliance
The Illinois UI Experiment
11.4. An Additive Planning Problem
The Choice Set
The Objective Function and the Optimal Treatment Rule
The Value of Covariate Information
Non-Separable Planning Problems
11.5. Planning with Partial Knowledge of Treatment Response
The Study Population and the Treatment Population
Planning under Ambiguity
11.6. Planning and the Selection Problem
Bayes Rules
The Maximin Criterion
The Minimax-Regret Rule
Sentencing Juvenile Offenders
11.7. The Ethics of Fractional Treatment Rules
Choosing Treatments for X-Pox
11.8. Decentralized Treatment Choice
The Informational Argument for Decentralization
Decentralized Treatment of X-Pox
Complement 11A. Minimax-Regret Rules for Two Treatments Are Fractional
Complement 11B. Reporting Observable Variation in Treatment Response
Complement 11C. Word Problems
12. Planning with Sample Data
12.1. Statistical Induction
12.2. Wald’s Development of Statistical Decision Theory
The Expected Welfare of a Statistical Treatment Rule
The States of Nature
Admissibility
Implementable Criteria for Treatment Choice
Unification of Identification, Statistical Inference, and Sample Design
12.3. Using a Randomized Experiment to Evaluate an Innovation
The Setting
The Admissible Treatment Rules
Some Monotone Rules
Savage on the Maximin and Minimax-Regret Criteria
III. Predicting Choice Behavior
13. Revealed Preference Analysis
13.1. Revealing the Preferences of an Individual
Observation of One Choice Setting
Observation of Multiple Choice Settings
Application to General Choice Problems
Thought Experiment or Practical Prescription for Prediction?
13.2. Random Utility Models of Population Choice Behavior
Consistency with Utility Theory
Prediction using Attributes of Alternatives and Decision Makers
Incomplete Data and Conditional Choice Probabilities
Practicality through the Conditional Logit Model
Other Distributional Assumptions
Extrapolation
13.3. College Choice in America
An Idealized Binary Choice Setting
Predicting the Enrollment Effects of Student Aid Policy
Power and Price of the Analysis
13.4. Random Expected-Utility Models
Identification of the Decision Rules of Proposers in Ultimatum Games
Rational Expectations Assumptions
How do Youth Infer the Returns to Schooling?
Complement 13A. Prediction Assuming Strict Preferences
Complement 13B. Axiomatic Decision Theory
14. Measuring Expectations
14.1. Elicitation of Expectations from Survey Respondents
Attitudinal Research
Probabilistic Expectations in Cognitive Psychology
Probabilistic Expectations in Economics
14.2. Illustrative Findings
Response Rates and Use of the Percent-Chance Scale
One-Year-Ahead Income Expectations
Social Security Expectations
14.3. Using Expectations Data to Predict Choice Behavior
Choice Expectations
Using Expectations and Choice Data to Estimate Random Expected-Utility Models
14.4. Measuring Ambiguity
Complement 14A. The Predictive Power of Intentions Data: A Best-Case Analysis
Rational Expectations Responses to Intentions Questions
Prediction of Behavior Conditional on Intentions
Prediction Not Conditioning on Intentions
Interpreting Fertility Intentions
Complement 14B. Measuring Expectations of Facts
Anchoring
15. Studying Human Decision Processes
15.1. As-If Rationality and Bounded Rationality
The As-If Argument of Friedman and Savage
Simon and Bounded Rationality
15.2. Choice Experiments
Heuristics and Biases
Widespread Irrationality or Occasional Cognitive Illusions?
15.3. Prospects for a Neuroscientific Synthesis
References
Author Index
Subject Index
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