Micro-Econometrics：Methods of Moments and Limited Dependent Variables
Second Edition



CONTENTS
Chapter 1 Methods of Moments for Single Linear Equation
Models 1
1 Least Squares Estimator (LSE) 1
1.1 LSE as aMethod ofMoment (MOM) . . . . . . . . . . . . . 1
1.1.1 Linear Model . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 LSE and Moment Conditions . . . . . . . . . . . . . . 2
1.1.3 Zero Moments and Independence . . . . . . . . . . . . 3
1.2 Asymptotic Properties of LSE . . . . . . . . . . . . . . . . . . 4
1.2.1 LLN and LSE Consistency . . . . . . . . . . . . . . . 5
1.2.2 CLT and
√
N-Consistency . . . . . . . . . . . . . . . . 6
1.2.3 LSE Asymptotic Distribution . . . . . . . . . . . . . . 7
1.3 Matrices and Linear Projection . . . . . . . . . . . . . . . . . 8
1.4 R2 and Two Examples . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Partial Regression . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Omitted Variable Bias . . . . . . . . . . . . . . . . . . . . . . 15
2 Heteroskedasticity and Homoskedasticity 17
2.1 Heteroskedasticity Sources . . . . . . . . . . . . . . . . . . . . 18
2.1.1 Forms of Heteroskedasticity . . . . . . . . . . . . . . . 18
2.1.2 Heteroskedasticity due to Aggregation . . . . . . . . . 19
2.1.3 Variance Decomposition . . . . . . . . . . . . . . . . . 20
2.1.4 Analysis of Variance (ANOVA)* . . . . . . . . . . . . 21
2.2 Weighted LSE (WLS) . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Heteroskedasticity Examples . . . . . . . . . . . . . . . . . . 24
3 Testing Linear Hypotheses 25
3.1 Wald Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Empirical Examples . . . . . . . . . . . . . . . . . . . . . . . 28
4 Instrumental Variable Estimator (IVE) 31
4.1 IVE Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1.1 IVE in Narrow Sense . . . . . . . . . . . . . . . . . . . 31
4.1.2 Instrumental Variable (IV) qualifications . . . . . . . . 32
4.1.3 Further Remarks . . . . . . . . . . . . . . . . . . . . . 34
4.2 IVE Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 IVE with More than Enough Instruments . . . . . . . . . . . 39
4.3.1 IVE in Wide Sense . . . . . . . . . . . . . . . . . . . . 39
4.3.2 Various Interpretations of IVE . . . . . . . . . . . . . 40
4.3.3 Further Remarks . . . . . . . . . . . . . . . . . . . . . 41
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xiv Contents
5 Generalized Method-of-Moment Estimator (GMM) 42
5.1 GMMBasics . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 GMMRemarks . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 GMMExamples . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 Generalized Least Squares Estimator (GLS) 48
6.1 GLS Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2 GLS Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.3 Efficiency of LSE, GLS, and GMM . . . . . . . . . . . . . . . 50
Chapter 2 Methods of Moments for Multiple Linear Equation
Systems 53
1 System LSE, IVE, and GMM 53
1.1 System LSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
1.1.1 Multiple Linear Equations . . . . . . . . . . . . . . . . 53
1.1.2 System LSE and Motivation . . . . . . . . . . . . . . . 54
1.1.3 Asymptotic Variance . . . . . . . . . . . . . . . . . . . 55
1.2 System IVE and Rank Condition . . . . . . . . . . . . . . . . 56
1.2.1 Moment Conditions . . . . . . . . . . . . . . . . . . . 56
1.2.2 System IVE and Separate IVE . . . . . . . . . . . . . 57
1.2.3 Identification Conditions . . . . . . . . . . . . . . . . . 59
1.3 System GMMand Link to Panel Data . . . . . . . . . . . . . 60
1.3.1 System GMM . . . . . . . . . . . . . . . . . . . . . . . 60
1.3.2 System GMM and Panel Data . . . . . . . . . . . . . 62
2 Simultaneous Equations and Identification 66
2.1 Relationship Between Endogenous Variables . . . . . . . . . . 66
2.2 Conventional Approach to Rank Condition . . . . . . . . . . 68
2.3 Simpler Approach to Rank Condition . . . . . . . . . . . . . 69
2.4 Avoiding Arbitrary Exclusion Restrictions* . . . . . . . . . . 71
2.4.1 Grouping and Assigning . . . . . . . . . . . . . . . . . 71
2.4.2 Patterns in Reduced-Form Ratios . . . . . . . . . . . . 72
2.4.3 Meaning of Singular Systems . . . . . . . . . . . . . . 74
3 Methods of Moments for Panel Data 75
3.1 Panel LinearModel . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.1 Typical Panel Data Layout . . . . . . . . . . . . . . . 76
3.1.2 Panel Model with a Cross-Section Look . . . . . . . . 78
3.1.3 Remarks* . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2 Panel GMMand Constructing Instruments . . . . . . . . . . 81
3.2.1 Panel IVE and GMM . . . . . . . . . . . . . . . . . . 81
3.2.2 Instrument Construction . . . . . . . . . . . . . . . . . 82
3.2.3 Specific Examples of Instruments . . . . . . . . . . . . 82
3.3 Within-Group and Between-Group Estimators . . . . . . . . 84
3.3.1 Within Group Estimator (WIT) . . . . . . . . . . . . 84
Contents xv
3.3.2 Between Group Estimator (BET) and Panel LSE
and GLS . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3.3 WIT as Fixed-Effect Estimator* . . . . . . . . . . . . 87
Chapter 3 M-Estimator and Maximum Likelihood Estimator
(MLE) 91
1 M-Estimator 91
1.1 Four Issues andMain Points . . . . . . . . . . . . . . . . . . . 91
1.2 Remarks for Asymptotic Distribution . . . . . . . . . . . . . . 92
1.3 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2 Maximum Likelihood Estimator (MLE) 96
2.1 MLE Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.2 MLE Identification . . . . . . . . . . . . . . . . . . . . . . . . 99
2.3 Asymptotic Variance Relative to M-estimator . . . . . . . . . 100
3 M-Estimator with Nuisance Parameters 102
3.1 Two-Stage M-Estimator Basics . . . . . . . . . . . . . . . . . 102
3.2 Influence Function and Correction Term . . . . . . . . . . . . 103
3.3 Various Forms of Asymptotic Variances . . . . . . . . . . . . 105
3.4 Examples of Two-Stage M-Estimators . . . . . . . . . . . . . 106
3.4.1 No First-Stage Effect . . . . . . . . . . . . . . . . . . . 106
3.4.2 First-Stage Effect . . . . . . . . . . . . . . . . . . . . . 108
4 Method-of-Moment Tests (MMT) 108
4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3 Conditional Moment Tests . . . . . . . . . . . . . . . . . . . . 112
5 Tests Comparing Two Estimators 112
5.1 Two Estimators for the Same Parameter . . . . . . . . . . . . 113
5.2 Two Estimators for the Same Variance . . . . . . . . . . . . . 115
6 Three Tests for MLE 117
6.1 Wald Test and Nonlinear Hypotheses . . . . . . . . . . . . . . 118
6.2 Likelihood Ratio (LR) Test . . . . . . . . . . . . . . . . . . . 119
6.2.1 Restricted LSE . . . . . . . . . . . . . . . . . . . . . . 119
6.2.2 Restricted MLE and LR Test . . . . . . . . . . . . . . 121
6.3 Score (LM) Test and Effective Score Test . . . . . . . . . . . 122
6.4 Further Remarks and an Empirical Example . . . . . . . . . . 124
7 Numerical Optimization and One-Step
Efficient Estimation 127
7.1 Newton–Raphson Algorithm . . . . . . . . . . . . . . . . . . . 127
7.2 Newton–Raphson Variants and Other Methods . . . . . . . . 129
7.3 One-Step Efficient Estimation . . . . . . . . . . . . . . . . . . 131
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Chapter 4 Nonlinear Models and Estimators 133
1 Nonlinear Least Squares Estimator (NLS) 133
1.1 Various NonlinearModels . . . . . . . . . . . . . . . . . . . . 134
1.1.1 Index Models . . . . . . . . . . . . . . . . . . . . . . . 134
1.1.2 Transformation-of-Variable Models . . . . . . . . . . . 135
1.1.3 Mean, Median, and More Nonlinear Models . . . . . . 136
1.2 NLS and Its Asymptotic Properties . . . . . . . . . . . . . . . 138
1.3 Three Tests for NLS . . . . . . . . . . . . . . . . . . . . . . . 141
1.4 Gauss–Newton Algorithm . . . . . . . . . . . . . . . . . . . . 143
1.5 NLS-LMTest for LinearModels* . . . . . . . . . . . . . . . . 144
2 Quantile and Mode Regression 145
2.1 Median Regression . . . . . . . . . . . . . . . . . . . . . . . . 146
2.2 Quantile Regression . . . . . . . . . . . . . . . . . . . . . . . 147
2.2.1 Asymmetric Absolute Loss and Quantile
Function . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2.2.2 Quantile Regression Estimator . . . . . . . . . . . . . 150
2.2.3 Empirical Examples . . . . . . . . . . . . . . . . . . . 151
2.3 Mode Regression . . . . . . . . . . . . . . . . . . . . . . . . . 153
2.4 Treatment Effects . . . . . . . . . . . . . . . . . . . . . . . . . 154
3 GMM for Nonlinear Models 156
3.1 GMM for Single Nonlinear Equation . . . . . . . . . . . . . . 157
3.2 Implementation and Examples . . . . . . . . . . . . . . . . . 159
3.3 Three Tests in GMM . . . . . . . . . . . . . . . . . . . . . . . 163
3.4 Efficiency of GMM . . . . . . . . . . . . . . . . . . . . . . . . 164
3.5 Weighting Matrices for Dependent Data . . . . . . . . . . . . 165
3.6 GMM for Multiple Nonlinear Equations* . . . . . . . . . . . . 166
4 Minimum Distance Estimation (MDE) 168
4.1 MDE Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.2 VariousMDE Cases . . . . . . . . . . . . . . . . . . . . . . . 171
4.3 An Empirical Example from Panel Data . . . . . . . . . . . . 174
Chapter 5 Parametric Methods for Single Equation
LDV Models 177
1 Binary Response 177
1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
1.2 Logit and Probit . . . . . . . . . . . . . . . . . . . . . . . . . 179
1.3 Marginal Effects . . . . . . . . . . . . . . . . . . . . . . . . . 183
1.4 Willingness to Pay and Treatment Effect . . . . . . . . . . . . 185
1.4.1 Willingness to Pay (WTP) . . . . . . . . . . . . . . . 185
1.4.2 Remarks for WTP Estimation . . . . . . . . . . . . . 187
1.4.3 Comparison to Treatment Effect . . . . . . . . . . . . 188
Contents xvii
2 Ordered Discrete Response 189
2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
2.2 Digression on Re-parametrization in MLE . . . . . . . . . . . 191
2.3 Ordered Probit . . . . . . . . . . . . . . . . . . . . . . . . . . 192
2.4 An Empirical Example: Contingent Valuation . . . . . . . . . 194
3 Count Response 198
3.1 Basics and PoissonMLE . . . . . . . . . . . . . . . . . . . . . 198
3.2 Poisson Over-dispersion Problem and Other Estimators . . . 200
3.2.1 Negative Binomial (NB) MLE . . . . . . . . . . . . . . 200
3.2.2 Zero-Inflated Count Responses . . . . . . . . . . . . . 202
3.2.3 Methods of Moments . . . . . . . . . . . . . . . . . . . 202
3.3 An Empirical Example: Inequality Effect on Crime . . . . . . 203
3.4 IVE for Count or Positive Responses . . . . . . . . . . . . . . 204
4 Censored Response and Related LDV Models 206
4.1 CensoredModels . . . . . . . . . . . . . . . . . . . . . . . . . 206
4.2 Censored-Model MLE . . . . . . . . . . . . . . . . . . . . . . 208
4.3 Truncated Regression and Fractional Response . . . . . . . . 210
4.4 Marginal Effects for Censored/Selection Models . . . . . . . . 211
4.5 Empirical Examples . . . . . . . . . . . . . . . . . . . . . . . 213
5 Parametric Estimators for Duration 216
5.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
5.1.1 Survival and Hazard Functions . . . . . . . . . . . . . 216
5.1.2 Log-Likelihood Functions . . . . . . . . . . . . . . . . 218
5.2 Exponential Distribution for Duration . . . . . . . . . . . . . 219
5.3 Weibull Distribution for Duration . . . . . . . . . . . . . . . . 221
5.4 Unobserved Heterogeneity and Other Parametric
Hazards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
5.5 Invariances and Extreme Value Distributions* . . . . . . . . . 225
Chapter 6 Parametric Methods for Multiple Equation LDV
Models 229
Chapter 7 Kernel Nonparametric Estimation 303
Chapter 8 Bandwidth-Free Semiparametric Methods 363
Chapter 9 Bandwidth-Dependent Semiparametric Methods 441

Appendix I: Mathematical Backgrounds and Chapter
Appendix II: Supplementary Topics 615