Partial identification of probability distributions

Contents......Page 10
Preface......Page 8
Introduction: Partial Identification and Credible Inference......Page 14
1.1. Anatomy of the Problem......Page 19
1.2. Means......Page 21
1.3. Parameters that Respect Stochastic Dominance......Page 24
1.4. Combining Multiple Sampling Processes......Page 26
1.5. Interval Measurement of Outcomes......Page 30
Complement 1A. Employment Probabilities......Page 31
Complement 1B. Blind-Men Bounds on an Elephant......Page 34
Endnotes......Page 36
2.1. Distributional Assumptions and Credible Inference......Page 39
2.2. Some Assumptions Using Instrumental Variables......Page 40
2.3. Outcomes Missing-at-Random......Page 42
2.4. Statistical Independence......Page 43
2.5. Mean Independence and Mean Monotonicity......Page 45
2.6. Other Assumptions Using Instrumental Variables......Page 49
Complement 2A. Estimation with Nonresponse Weights......Page 50
Endnotes......Page 51
3.1. Prediction of Outcomes Conditional on Covariates......Page 53
3.3. Jointly Missing Outcomes and Covariates......Page 54
3.4. Missing Covariates......Page 59
3.5. General Missing-Data Patterns......Page 62
3.6. Joint Inference on Conditional Distributions......Page 66
Complement 3A. Unemployment Rates......Page 68
Complement 3B. Parametric Prediction with Missing Data......Page 69
Endnotes......Page 71
4.1. The Mixture Model of Data Errors......Page 73
4.2. Outcome Distributions......Page 75
4.3. Event Probabilities......Page 76
4.4. Parameters that Respect Stochastic Dominance......Page 78
Complement 4A. Contamination Through Imputation......Page 81
Complement 4B. Identification and Robust Inference......Page 83
Endnotes......Page 85
5.1. Ecological Inference......Page 86
5.2. Anatomy of the Problem......Page 87
5.3. Long Mean Regressions......Page 89
5.4. Instrumental Variables......Page 94
Complement 5A. Structural Prediction......Page 97
Endnotes......Page 98
6.1. Reverse Regression......Page 100
6.3. The Rare-Disease Assumption......Page 102
6.4. Bounds on Relative and Attributable Risk......Page 104
6.5. Sampling from One Response Stratum......Page 107
Complement 6A. Smoking and Heart Disease......Page 110
Endnotes......Page 111
7.1. Anatomy of the Problem......Page 112
7.2. Treatment Choice in Heterogeneous Populations......Page 115
7.3. The Selection Problem and Treatment Choice......Page 118
7.4. Instrumental Variables......Page 121
Complement 7A. Identification and Ambiguity......Page 123
Complement 7B. Sentencing and Recidivism......Page 125
Complement 7C. Missing Outcome and Covariate Data......Page 127
Complement 7D. Study and Treatment Populations......Page 130
Endnotes......Page 131
8.1. Shape Restrictions......Page 133
8.2. Monotonicity......Page 136
8.3. Semi-monotonicity......Page 140
8.4. Concave Monotonicity......Page 145
Complement 8A. Downward-Sloping Demand......Page 149
Complement 8B. Econometric Response Models......Page 151
Endnotes......Page 152
9.1. Equalities and Inequalities......Page 154
9.2. Mean Monotonicity......Page 156
9.3. Mean Monotonicity and Mean Treatment Response......Page 158
Complement 9A. The Returns to Schooling......Page 162
Endnotes......Page 166
10.1. Within-Group Treatment Variation......Page 167
10.2. Known Treatment Shares......Page 170
10.3. Extrapolation from the Experiment Alone......Page 173
Complement 10A. Experiments Without Covariate Data......Page 174
Endnotes......Page 178
References......Page 180
D......Page 188
M......Page 189
R......Page 190
Z......Page 191