A. Colin Cameron, Pravin K. Trivedi9780521848053, 0521848059
Table of contents :
Half-Title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Contents……Page 9
List of Figures……Page 17
List of Tables……Page 19
Preface……Page 23
PART ONE Preliminaries……Page 25
1.1. Introduction……Page 27
1.2.1. Discreteness and Nonlinearity……Page 29
1.2.3. Greater Information Content……Page 30
1.2.4. Microeconomic Foundations……Page 31
1.2.5. Disaggregation and Heterogeneity……Page 32
1.3.1. Part 1: Preliminaries……Page 34
1.3.3. Part 3: Simulation-Based Methods……Page 36
1.3.4. Part 4: Models for Cross-Section Data……Page 37
1.4. How to Use This Book……Page 38
1.5. Software……Page 39
1.6. Notation and Conventions……Page 40
2.1. Introduction……Page 42
2.2. Structural Models……Page 44
2.3. Exogeneity……Page 46
2.4. Linear Simultaneous Equations Model……Page 47
2.4.1. The SEM Setup……Page 48
2.4.2. Causal Interpretation in SEM……Page 50
2.4.3. Extensions to Nonlinear and Latent Variable Models……Page 51
2.4.4. Interpretations of Structural Relationships……Page 52
2.5. Identification Concepts……Page 53
2.7. Potential Outcome Model……Page 55
2.7.1. The Rubin Causal Model……Page 57
Limited-Information Structural Models……Page 59
Elimination of Nuisance Parameters……Page 60
Reweighting Samples……Page 61
2.9. Bibliographic Notes……Page 62
3.1. Introduction……Page 63
3.2.1. Nature of Survey Data……Page 64
3.2.3. Multistage Surveys……Page 65
Exogenous Sampling……Page 66
Length-Biased Sampling……Page 67
3.2.5. Bias due to Sample Selection……Page 68
Problem of Survey Nonresponse……Page 69
Missing and Mismeasured Data……Page 70
3.2.7. Types of Observational Data……Page 71
3.3. Data from Social Experiments……Page 72
3.3.1. Leading Features of Social Experiments……Page 73
3.3.2. Advantages of Social Experiments……Page 74
3.3.3. Limitations of Social Experiments……Page 76
3.4. Data from Natural Experiments……Page 78
3.4.2. Differences in Differences……Page 79
3.4.3. Identification through Natural Experiments……Page 81
3.5.1. Some Sources of Microdata……Page 82
3.5.2. Handling Microdata……Page 83
3.5.4. Checking Data……Page 84
3.6. Bibliographic Notes……Page 85
PART TWO Core Methods……Page 87
4.1. Introduction……Page 89
4.2.2. Optimal Prediction……Page 90
4.2.3. Linear Prediction……Page 92
4.3. Example: Returns to Schooling……Page 93
4.4.1. Linear Regression Model……Page 94
4.4.3. Identification……Page 95
Consistency……Page 96
Limit Distribution……Page 97
4.4.5. Heteroskedasticity-Robust Standard Errors for OLS……Page 98
4.4.6. Assumptions for Cross-Section Regression……Page 99
Stratified Random Sampling……Page 100
Weakly Exogenous Regressors……Page 101
Conditionally Heteroskedastic Errors……Page 102
Small-Sample Distribution……Page 103
Heteroskedasticity-Robust Standard Errors……Page 104
4.5.1. GLS and Feasible GLS……Page 105
4.5.2. Weighted Least Squares……Page 106
4.5.3. Robust Standard Errors for LS Example……Page 108
4.6. Median and Quantile Regression……Page 109
4.6.1. Population Quantiles……Page 110
4.6.2. Sample Quantiles……Page 111
4.6.4. Quantile Regression Example……Page 112
4.7. Model Misspecification……Page 114
4.7.2. Functional Form Misspecification……Page 115
4.7.4. Omitted Variables……Page 116
4.7.6. Parameter Heterogeneity……Page 118
4.8.1. Inconsistency of OLS……Page 119
Definition of an Instrument……Page 120
Examples of an Instrument……Page 121
4.8.4. Wald Estimator……Page 122
4.8.6. IV Estimation for Multiple Regression……Page 123
4.8.7. Two-Stage Least Squares……Page 125
4.8.8. IV Example……Page 126
4.9. Instrumental Variables in Practice……Page 127
4.9.1. Weak Instruments……Page 128
Partial F-Statistics……Page 129
4.9.3. Low Precision……Page 131
4.9.4. Finite-Sample Bias……Page 132
4.9.6. IV Application……Page 134
4.11. Bibliographic Notes……Page 136
5.1. Introduction……Page 140
5.2.1. Poisson Regression Example……Page 141
5.2.2. m-Estimators……Page 142
Limit Normal Distribution……Page 143
Asymptotic Normality……Page 144
Poisson ML Example……Page 145
General Regression Function……Page 146
Finite-Difference Method……Page 147
5.3.1. Extremum Estimators……Page 148
5.3.2. Formal Consistency Theorems……Page 149
5.3.3. Asymptotic Normality……Page 151
5.3.4. Poisson ML Estimator Asymptotic Properties Example……Page 153
5.3.5. Proofs of Consistency and Asymptotic Normality……Page 154
5.3.7. Informal Approach to Consistency of an m-Estimator……Page 156
5.4. Estimating Equations……Page 157
5.4.1. Estimating Equations Estimator……Page 158
5.5. Statistical Inference……Page 159
Tests of a Single Coefficient……Page 160
Sandwich Estimate of the Variance Matrix……Page 161
Estimation of A and B……Page 162
Conditional Likelihood……Page 163
Examples……Page 164
Regularity Conditions……Page 165
5.6.4. Distribution of the ML Estimator……Page 166
5.6.5. Weibull Regression Example……Page 167
5.6.6. Variance Matrix Estimation for MLE……Page 168
5.6.7. Derivation of ML Regularity Conditions……Page 169
5.7.1. Psuedo-True Value……Page 170
5.7.3. Linear Exponential Family……Page 171
5.7.4. Generalized Linear Models……Page 173
5.8. Nonlinear Least Squares……Page 174
5.8.2. NLS Estimator……Page 175
5.8.3. Distribution of the NLS Estimator……Page 176
5.8.4. Variance Matrix Estimation for NLS……Page 178
5.8.6. Weighted NLS and FGNLS……Page 179
5.8.7. Time Series……Page 182
5.9.1. Model and Estimators……Page 183
5.9.2. Simulation and Results……Page 184
5.9.3. Comparison of Estimates and Standard Errors……Page 185
5.9.4. Coefficient Interpretation……Page 186
5.11. Bibliographic Notes……Page 187
6.1. Introduction……Page 190
6.2.1. Linear Regression……Page 191
6.2.2. Nonlinear Regression……Page 192
6.2.4. Additional Moment Restrictions……Page 193
6.2.6. Panel Data……Page 194
6.2.7. Moment Conditions from Economic Theory……Page 195
6.3.1. Method of Moments Estimator……Page 196
6.3.3. Distribution of GMM Estimator……Page 197
6.3.4. Variance Matrix Estimation……Page 198
6.3.5. Optimal Weighting Matrix……Page 199
6.3.6. Regression with Symmetric Error Example……Page 202
6.3.7. Optimal Moment Condition……Page 203
6.3.9. Derivations for the GMM Estimator……Page 206
6.4. Linear Instrumental Variables……Page 207
6.4.1. Linear GMM with Instruments……Page 208
Instrumental Variables Estimator……Page 210
Optimal GMM versus 2SLS……Page 211
GLS in a Transformed Model……Page 212
Theil’s Interpretation……Page 213
6.4.4. Alternatives to Standard IV Estimators……Page 214
Split-Sample IV……Page 215
6.5. Nonlinear Instrumental Variables……Page 216
6.5.1. Nonlinear GMM with Instruments……Page 217
6.5.2. Different Nonlinear GMM Estimators…….Page 218
6.5.3. Poisson IV Example……Page 221
6.5.4. Two-Stage Estimation in Nonlinear Models……Page 222
6.6. Sequential Two-Step m-Estimation……Page 224
6.7. Minimum Distance Estimation……Page 226
6.8. Empirical Likelihood……Page 227
6.8.1. Empirical Likelihood Estimation of Population Mean……Page 228
6.8.2. Empirical Likelihood Estimation of Regression Parameters……Page 229
6.9. Linear Systems of Equations……Page 230
6.9.1. Linear Systems of Equations……Page 231
6.9.2. Systems OLS and FGLS Estimation……Page 232
6.9.3. Seemingly Unrelated Regressions……Page 233
6.9.5. Systems IV Estimation……Page 235
6.9.6. Linear Simultaneous Equations Systems……Page 237
6.10. Nonlinear Sets of Equations……Page 238
6.10.1. Nonlinear Systems ML Estimation……Page 239
6.10.3. Nonlinear Systems Estimation……Page 240
6.10.4. Nonlinear Systems IV Estimation……Page 242
6.11. Practical Considerations……Page 243
6.12. Bibliographic Notes……Page 244
7.1. Introduction……Page 247
7.2.1. Linear Hypotheses in Linear Models……Page 248
7.2.2. Nonlinear Hypotheses……Page 249
Common Versions of the Wald Test……Page 250
7.2.4. Derivation of the Wald Statistic……Page 251
Tests of Statistical Signi.cance……Page 252
7.2.6. Tests in Misspecified Models……Page 253
7.2.7. Joint Versus Separate Tests……Page 254
Confidence Intervals……Page 255
7.2.9. Lack of Invariance of the Wald Test……Page 256
7.3.1. Wald, Likelihood Ratio, and Lagrange Multiplier (Score) Tests……Page 257
7.3.2. Poisson Regression Example……Page 260
LM or Score Test……Page 261
Invariance to Reparameterization……Page 262
Simple Interpretation of the LM Test……Page 263
Outer Product of the Gradient Versions of the LM Test……Page 264
7.4. Example: Likelihood-Based Hypothesis Tests……Page 265
7.5. Tests in Non-ML Settings……Page 267
7.5.1. Tests Based on m-Estimators……Page 268
7.5.2. Tests Based on Efficient GMM Estimators……Page 269
7.6.1. Test Size and Power……Page 270
7.6.2. Local Alternative Hypotheses……Page 271
7.6.3. Asymptotic Power of the Wald Test……Page 272
7.6.4. Derivation of Asymptotic Power……Page 273
7.7. Monte Carlo Studies……Page 274
7.7.2. Monte Carlo Details……Page 275
7.7.4. Test Size……Page 276
7.7.5. Test Power……Page 277
7.8. Bootstrap Example……Page 278
7.8.2. Bootstrap without Asymptotic Refinement……Page 279
7.9. Practical Considerations……Page 280
7.10. Bibliographic Notes……Page 281
8.1. Introduction……Page 283
8.2.1. m-Test Statistic……Page 284
Auxiliary Regressions Using the ML Estimator……Page 285
Other Auxiliary Regressions……Page 286
8.2.3. Derivations for the m-Test Statistic……Page 287
8.2.4. Conditional Moment Tests……Page 288
8.2.5. White’s Information Matrix Test……Page 289
8.2.6. Chi-Square Goodness-of-Fit Test……Page 290
8.2.8. Power and Consistency of Conditional Moment Tests……Page 291
Moments Tested……Page 293
Simulation Results……Page 294
8.3.1. Hausman Test……Page 295
Computation for Fully Efficient Estimator under the Null Hypothesis……Page 296
8.3.3. Power of the Hausman Test……Page 297
8.4.1. Tests for Omitted Variables……Page 298
8.4.3. Hausman Tests for Endogeneity……Page 299
8.4.5. RESET Test……Page 301
8.5.1. Information Criteria……Page 302
8.5.2. Cox Likelihood Ratio Test of Nonnested Models……Page 303
8.5.3. Vuong Likelihood Ratio Test of Nonnested Models……Page 304
8.5.5. Nonnested Models Example……Page 307
8.6.3. Data Mining……Page 309
8.6.4. A Practical Approach……Page 310
8.7.1. Pseudo-R2 Measures……Page 311
8.7.2. Residual Analysis……Page 313
8.7.3. Diagnostics Example……Page 314
8.8. Practical Considerations……Page 315
8.9. Bibliographic Notes……Page 316
9.1. Introduction……Page 318
9.2.1. Nonparametric Density Estimate……Page 319
9.2.2. Nonparametric Regression……Page 321
9.3.1. Histogram……Page 322
9.3.3. Kernel Functions……Page 323
9.3.4. Kernel Density Example……Page 324
Asymptotic Normality……Page 325
Mean Integrated Squared Error……Page 326
Optimal Kernel……Page 327
Cross-Validation……Page 328
9.3.9. Multivariate Kernel Density Estimate……Page 329
9.3.11. Alternative Nonparametric Density Estimates……Page 330
9.4.1. Local Weighted Averages……Page 331
9.4.2. K-Nearest Neighbors Example……Page 332
9.4.4. Statistical Inference……Page 333
9.5.1. Kernel Regression Estimator……Page 335
9.5.2. Statistical Inference……Page 336
9.5.3. Bandwidth Choice……Page 337
Cross-Validation……Page 338
Generalized Cross-Validation……Page 339
9.5.4. Confidence Intervals……Page 340
9.5.6. Conditional Moment Estimation……Page 341
9.5.7. Multivariate Kernel Regression……Page 342
9.6.1. Nearest Neighbors Estimator……Page 343
9.6.2. Local Linear Regression and Lowess……Page 344
9.6.4. Series Estimators……Page 345
9.7.1. Examples……Page 346
9.7.2. Efficiency of Semiparametric Estimators……Page 347
Robinson Difference Estimator……Page 348
Identification……Page 349
Average Derivative Estimator……Page 350
9.7.5. Generalized Additive Models……Page 351
9.7.7. Seminonparametric MLE……Page 352
9.7.8. Semiparametric Efficiency Bounds……Page 353
9.8.1. Mean and Variance of Kernel Density Estimator……Page 354
9.8.2. Distribution of Kernel Regression Estimator……Page 355
9.10. Bibliographic Notes……Page 357
10.2. General Considerations……Page 360
10.2.3. Gradient Methods……Page 361
10.2.4. Gradient Method Example……Page 362
10.2.6. Convergence Criteria……Page 363
10.2.8. Numerical and Analytical Derivatives……Page 364
10.3.1. Newton–Raphson Method……Page 365
10.3.3. BHHH Method……Page 367
10.3.5. DFP and BFGS Methods……Page 368
10.3.7. Expectation Maximization……Page 369
10.3.8. Simulated Annealing……Page 371
10.4. Practical Considerations……Page 372
10.4.1. Statistical Packages……Page 373
10.4.2. Computational Difficulties……Page 374
10.5. Bibliographic Notes……Page 376
PART THREE Simulation-Based Methods……Page 379
11.1. Introduction……Page 381
11.2.1. Bootstrap without Refinement……Page 382
11.2.3. Asymptotically Pivotal Statistic……Page 383
Bootstrap Sampling Methods……Page 384
The Number of Bootstraps……Page 385
11.2.5. Standard Error Estimation……Page 386
Tests with Asymptotic Refinement……Page 387
11.2.7. Confidence Intervals……Page 388
11.2.8. Bias Reduction……Page 389
11.3. Bootstrap Example……Page 390
11.4. Bootstrap Theory……Page 392
11.4.2. Consistency of the Bootstrap……Page 393
11.4.3. Edgeworth Expansions……Page 394
11.4.4. Asymptotic Refinement via Bootstrap……Page 395
11.4.5. Power of Bootstrapped Tests……Page 396
11.5.2. Moving Blocks Bootstrap……Page 397
11.5.5. The Jackknife……Page 398
11.6.1. Heteroskedastic Errors……Page 400
11.6.2. Panel Data and Clustered Data……Page 401
Tests with Asymptotic Refinement……Page 402
11.6.4. GMM, Minimum Distance, and Empirical Likelihood in Overidentified Models……Page 403
11.6.6. Nonsmooth Estimators……Page 404
11.7. Practical Considerations……Page 405
11.8. Bibliographic Notes……Page 406
12.1. Introduction……Page 408
12.2.1. Random Parameters Models……Page 409
12.2.2. Limited Dependent Variable Models……Page 410
12.3. Basics of Computing Integrals……Page 411
12.3.1. Deterministic Numerical Integration……Page 412
12.3.2. Integration by Direct Monte Carlo Sampling……Page 414
12.3.3. Integral Computation Example……Page 415
12.4.1. Simulators……Page 417
12.4.3. Distribution of the MSL Estimator……Page 418
12.4.4. Asymptotic Bias-Adjusted MSL……Page 420
12.4.5. Unobserved Heterogeneity Example……Page 421
12.5.1. Simulated m-Estimators……Page 422
12.5.3. Method of Simulated Moments……Page 423
12.5.4. Distribution of MSM Estimator……Page 424
12.5.5. Choosing between MSM and MSL……Page 426
12.5.6. Unobserved Heterogeneity Example……Page 427
12.6. Indirect Inference……Page 428
12.7.1. Frequency Simulator……Page 430
12.7.2. Importance Sampling……Page 431
12.7.3. Variance Reduction by Antithetic Acceleration……Page 432
12.7.4. Computation Using Quasi-Random Sequences……Page 433
12.8. Methods of Drawing Random Variates……Page 434
Inverse Transformation……Page 436
Accept–Reject Methods……Page 437
12.8.3. Multivariate Distributions……Page 439
12.9. Bibliographic Notes……Page 440
13.1. Introduction……Page 443
13.2. Bayesian Approach……Page 444
13.2.1. Bayes’ Theorem……Page 445
13.2.2. Bayes’ Theorem Example……Page 446
13.2.3. Bayesian and Non-Bayesian Approaches Compared……Page 448
13.2.4. Specification of the Prior……Page 449
Point Estimation……Page 454
Conditional Posterior Density……Page 455
13.2.6. Large-Sample Behavior of the Posterior……Page 456
13.2.7. Bayesian Decision Analysis……Page 458
13.3.1. Noninformative Priors……Page 459
13.3.2. Informative Priors……Page 461
13.3.3. Mixed Estimation……Page 463
13.3.4. Hierarchical Priors……Page 465
13.3.5. Multivariate t-and Wishart Distributions……Page 466
13.4.1. Importance Sampling……Page 467
13.5. Markov Chain Monte Carlo Simulation……Page 469
13.5.1. Markov Chains……Page 470
Gibbs Sampler……Page 472
Linear Regression Example……Page 473
13.5.3. Metropolis Algorithm……Page 474
13.5.4. The Metropolis–Hastings Algorithm……Page 475
13.6. MCMC Example: Gibbs Sampler for SUR……Page 476
13.7. Data Augmentation……Page 478
13.8. Bayesian Model Selection……Page 480
13.10. Bibliographic Notes……Page 482
PART FOUR Models for Cross-Section Data……Page 485
14.1. Introduction……Page 487
14.2. Binary Outcome Example: Fishing Mode Choice……Page 488
14.3. Logit and Probit Models……Page 489
14.3.1. General Binary Outcome Model……Page 490
14.3.3. ML Estimation……Page 491
Consistency of the MLE……Page 492
14.3.4. Logit Model……Page 493
14.3.5. Probit Model……Page 494
Theoretical Considerations……Page 495
Empirical Considerations……Page 496
Pseudo-R2……Page 497
Predicted Probabilities……Page 498
14.4.1. Index Function Models……Page 499
14.4.2. Random Utility Models……Page 500
Probit and Logit Models……Page 501
14.5. Choice-Based Samples……Page 502
14.6.1. Berkson’s Minimum Chi-Square Estimator……Page 504
14.6.2. Estimation with Aggregate Data……Page 505
14.7. Semiparametric Estimation……Page 506
14.7.2. Maximum Score Estimation……Page 507
14.7.4. Semiparametric ML Estimation……Page 509
14.8. Derivation of Logit from Type I Extreme Value……Page 510
14.10. Bibliographic Notes……Page 511
15.1. Introduction……Page 514
15.2.1. Conditional Logit: Alternative-Varying Regressors……Page 515
15.2.2. Multinomial Logit: Alternative-Invariant Regressors……Page 518
15.3. General Results……Page 519
15.3.2. ML Estimation……Page 520
15.3.4. Alternative-Varying Regressors……Page 521
15.3.5. Revealed Preference and Stated Preference Data……Page 522
15.3.6. Model Evaluation and Selection……Page 523
15.4.1. Conditional, Multinomial, and Mixed Logit Models……Page 524
Marginal Effects and Elasticities……Page 525
Comparison to Base Category……Page 526
15.4.4. Independence of Irrelevant Alternatives……Page 527
15.5.1. ARUM……Page 528
Type 1 Extreme Value Errors……Page 529
15.5.4. Welfare Analysis……Page 530
15.6. Nested Logit……Page 531
15.6.2. Nested Logit Model……Page 532
15.6.3. Estimation of Nested Logit……Page 534
15.6.4. Discussion……Page 535
15.7. Random Parameters Logit……Page 536
15.7.2. Estimation of Random Parameters Logit……Page 537
15.7.3. Generalized Random Utility Models……Page 539
15.8.1. Multinomial Probit Model……Page 540
15.8.2. Estimation of Multinomial Probit……Page 541
15.9.1. Ordered Multinomial Models……Page 543
15.9.2. Sequential Multinomial Models……Page 544
15.10.1. Bivariate Discrete Outcomes……Page 545
15.10.2. Bivariate Probit……Page 546
15.11. Semiparametric Estimation……Page 547
15.12.1. Conditional Logit……Page 548
15.12.2. Multinomial Logit……Page 549
15.12.3. Nested Logit……Page 550
15.13. Practical Considerations……Page 551
15.14. Bibliographic Notes……Page 552
16.1. Introduction……Page 553
16.2.1. Censoring and Truncation Example……Page 554
Interval Data……Page 556
Censored MLE……Page 557
Interval Data MLE……Page 558
16.2.5. Censored and Truncated Conditional Means……Page 559
16.3.1. Tobit Model……Page 560
16.3.3. Censored and Truncated Means in Linear Regression……Page 562
16.3.4. Censored and Truncated Means in the Tobit Model……Page 564
16.3.5. Marginal Effects in the Tobit Model……Page 565
NLS Estimator……Page 566
16.3.7. Specification Tests for the Tobit Model……Page 567
16.4. Two-Part Model……Page 568
16.4.2. Two-Part Model Examples……Page 569
16.5.1. Sample Selection Models……Page 570
16.5.2. A Bivariate Sample Selection Model (Type 2 Tobit)……Page 571
16.5.3. Conditional Means in the Bivariate Sample Selection Model……Page 572
16.5.4. Heckman Two-Step Estimator……Page 574
16.5.5. Identification Considerations……Page 575
16.5.7. Selection on Observables and on Unobservables……Page 576
16.6. Selection Example: Health Expenditures……Page 577
16.7.1. Roy Model……Page 579
16.7.2. Variations of the Roy Model……Page 581
16.8.1. Structural Models Based on Utility Maximization……Page 582
Endogenous Latent Variables……Page 584
Endogenous Regressors……Page 585
16.9. Semiparametric Estimation……Page 586
16.9.2. Semiparametric Estimation for Censored Models……Page 587
16.9.3. Semiparametric Estimation for Selection Models……Page 589
16.10.1. Truncated Moments of Standard Normal……Page 590
16.10.2. Asymptotic Theory for Heckman’s Two-Step Estimator in the Tobit Model……Page 591
16.11. Practical Considerations……Page 592
16.12. Bibliographic Notes……Page 593
17.1. Introduction……Page 597
17.2. Example: Duration of Strikes……Page 598
17.3.1. Survivor, Hazard, and Cumulative Hazard Functions……Page 600
17.3.2. Discrete Data……Page 601
17.4.1. Censoring Mechanisms……Page 603
17.5. Nonparametric Models……Page 604
17.5.1. Nonparametric Estimation……Page 605
17.6.1. Exponential and Weibull Distributions……Page 608
17.6.2. Some Parametric Models……Page 610
17.6.3. Maximum Likelihood Estimation……Page 611
17.6.4. Components of Likelihood……Page 612
17.6.6. Use of Model Estimates……Page 613
17.6.7. Least-Squares Estimation……Page 614
17.7.2. Accelerated Failure Time Model……Page 615
17.8. Cox PH Model……Page 616
17.8.1. Proportional Hazards Model……Page 617
17.8.2. Partial Likelihood Estimation……Page 618
17.8.3. Survivor Function for the Cox PH Model……Page 620
17.9. Time-Varying Regressors……Page 621
17.9.1. Extended Cox Model……Page 623
17.10.1. Discrete-Time Proportional Hazards……Page 624
17.10.2. Han and Hausman Approach……Page 625
17.10.3. Discrete-Time Binary Choice……Page 626
17.11. Duration Example: Unemployment Duration……Page 627
17.13. Bibliographic Notes……Page 632
18.1. Introduction……Page 635
18.2. Unobserved Heterogeneity and Dispersion……Page 636
18.2.1. Mixtures……Page 637
18.2.2. Choice of Heterogeneity Distribution……Page 638
18.2.3. Weibull–Gamma Mixture……Page 639
18.2.4. Interpreting the Mixture Hazard Function……Page 641
18.3. Identification in Mixture Models……Page 642
18.4.1. Discrete-Time PH with Gamma Heterogeneity……Page 644
18.5.1. Finite Mixture Model……Page 645
18.5.3. EM Algorithm……Page 647
18.5.4. Choosing the Number of Latent Classes……Page 648
18.6. Stock and Flow Sampling……Page 649
18.7. Specification Testing……Page 652
18.7.1. Hypothesis Tests……Page 653
18.7.2. Graphical Tools for Detecting Misspecification……Page 654
18.8. Unobserved Heterogeneity Example: Unemployment Duration……Page 656
18.10. Bibliographic Notes……Page 661
19.1. Introduction……Page 664
Competing Causes……Page 666
Independent Risks……Page 668
19.2.2. CRM with Proportional Hazards……Page 669
19.2.4. Interpretation of Regression Coefficients……Page 670
19.2.6. CRM with Dependent Competing Risks……Page 671
19.3.1. Extending Survival Concepts to a Multivariate Setting……Page 672
19.3.2. Bivariate Distributions Based on Marginals……Page 673
Properties of Copulas……Page 675
Measuring Dependence……Page 677
Examples……Page 678
19.4. Multiple Spells……Page 679
19.4.1. A Model with Two Spells……Page 680
19.4.2. A More General Model of Multiple Spells……Page 681
19.5. Competing Risks Example: Unemployment Duration……Page 682
19.5.1. Estimates under Competing Risks Framework……Page 683
19.6. Practical Considerations……Page 686
19.7. Bibliographic Notes……Page 687
20.1. Introduction……Page 689
20.2.1. Poisson Regression……Page 690
20.2.2. Poisson MLE and QMLE……Page 691
20.2.3. Interpretation of Regression Coefficients……Page 693
20.2.4. Overdispersion……Page 694
20.3. Count Example: Contacts with Medical Doctor……Page 695
20.4. Parametric Count Regression Models……Page 698
20.4.1. Negative Binomial Model……Page 699
20.4.2. Simulated Maximum Likelihood……Page 701
20.4.3. Finite Mixture Models……Page 702
20.4.4. Truncation and Censoring……Page 703
Hurdle or Two-Part Models……Page 704
With-Zeros or Zero-Inflated Model……Page 705
20.5.1. Quasi-ML Estimation……Page 706
20.5.3. Semiparametric Models……Page 708
Semiparametric Methods……Page 709
Fully Parametric Methods……Page 710
20.6.2. Count Models with Endogenous Regressors……Page 711
20.8. Practical Considerations……Page 714
20.9. Bibliographic Notes……Page 715
PART FIVE Models for Panel Data……Page 719
21.1. Introduction……Page 721
21.2.1. Panel Data Models……Page 722
Individual and Time Dummies……Page 723
Fixed Effects and Random Effects Models……Page 724
Fixed versus Random Effects Models……Page 725
Pooled OLS……Page 726
Within Estimator or Fixed Effects Estimator……Page 727
First-Differences Estimator……Page 728
21.2.3. Panel-Robust Statistical Inference……Page 729
Panel-Robust Sandwich Standard Errors……Page 730
21.3. Linear Panel Example: Hours and Wages……Page 732
21.3.1. Data Summary……Page 733
Slope Parameter Estimates……Page 734
Standard Error Estimation……Page 735
21.3.3. Graphical Analysis……Page 736
21.3.4. Residual Analysis……Page 737
21.4. Fixed Effects versus Random Effects Models……Page 739
21.4.1. Fixed Effects Example……Page 740
Computation When RE Is Fully Ef.cient……Page 741
Computation When RE Is Not Fully Ef.cient……Page 742
21.4.4. Richer Models for Random Effects……Page 743
21.5.1. Pooled OLS, FGLS, and WLS Estimators……Page 744
21.5.2. Error Variance Matrix for Short Panels……Page 745
21.5.4. The Impact of Autocorrelated Errors……Page 747
21.5.5. Hours and Wages Pooled GLS Example……Page 749
21.6.1. Within or Fixed Effects Estimator……Page 750
Asymptotic Distribution of the Within Estimator……Page 751
Derivation of the Variance of the Within Estimator……Page 752
21.6.2. First-Differences Estimator……Page 753
21.6.3. Conditional ML Estimator……Page 755
21.6.4. Least-Squares Dummy Variable Estimator……Page 756
21.6.5. Covariance Estimator……Page 757
21.7.1. GLS Estimator……Page 758
21.7.3. Other Estimators……Page 760
21.8.2. Tests for Individual-Specific Effects……Page 761
21.8.4. Two-Way Effects Models……Page 762
21.8.5. Unbalanced Panel Data……Page 763
21.10. Bibliographic Notes……Page 764
22.1. Introduction……Page 767
22.2.1. Panel GMM……Page 768
22.2.2. Panel-Robust Statistical Inference……Page 769
Two-Step GMM……Page 770
22.2.4. Selection of Instruments……Page 771
Contemporaneous Exogeneity Assumption……Page 772
Strong Exogeneity Assumption……Page 773
Redundant Instruments……Page 774
22.2.5. Computation of Panel GMM Estimators……Page 775
22.2.6. Variations on GMM Estimation……Page 776
22.2.7. Chamberlain’s Optimal Distance Estimator……Page 777
22.3. Panel GMM Example: Hours and Wages……Page 778
22.4. Random and Fixed Effects Panel GMM……Page 780
22.4.2. IV for Fixed Effects Models……Page 781
IV for the Within or Mean-Differenced Model……Page 782
22.4.3. IV for Random Effects Models……Page 783
22.4.4. IV for the Hausman–Taylor Hybrid Model……Page 784
22.4.5. SUR and Simultaneous Equations Estimation……Page 786
22.5.1. True State Dependence and Unobserved Heterogeneity……Page 787
22.5.2. Inconsistency of Standard Panel Estimators……Page 788
22.5.3. Arellano–Bond Estimator……Page 789
22.5.4. Estimation of Covariance Structures……Page 790
22.5.5. Nonstationary Panels……Page 791
22.6.1. Fixed Effects with Binary Treatment……Page 792
22.6.2. Differences in Differences……Page 793
22.7.1. Repeated Cross Sections……Page 794
22.7.2. Pseudo Panels……Page 795
22.7.3. Measurement Error Estimators for Pseudo Panels……Page 796
22.8.1. Mixed Linear Models……Page 798
22.8.2. Estimation……Page 799
22.9. Practical Considerations……Page 800
22.10. Bibliographic Notes……Page 801
23.2.1. Individual-Specific Effects Models……Page 803
The Incidental Parameters Problem……Page 805
Conditional Likelihood……Page 806
First-Differences Transformation……Page 807
Dummy Variable Model Estimation……Page 808
Parametric Models……Page 809
Finite Mixture Model……Page 810
Parametric Models……Page 811
23.2.6. Estimation and Panel-Robust Statistical Inference……Page 812
GMM Estimation……Page 813
Generalized Estimating Equations Estimation……Page 814
Fixed Effects Models……Page 815
23.3. Nonlinear Panel Example: Patents and R&D……Page 816
23.4.2. Random Effects Binary Models……Page 819
23.4.3. Fixed Effects Logit……Page 820
23.4.4. Dynamic Binary Models……Page 821
23.4.6. Derivations for Fixed Effects Logit……Page 822
23.5.1. Censored and Truncated Models……Page 824
23.6. Transition Data……Page 825
23.7.1. Individual-Specific Effects Count Models……Page 826
23.7.2. Random Effects Count Models……Page 827
23.7.3. Fixed Effects Count Models……Page 829
23.7.4. Dynamic Count Models……Page 830
23.7.5. Derivations for Random and Fixed Effects Poisson……Page 831
23.9. Practical Considerations……Page 832
23.10. Bibliographic Notes……Page 833
PART SIX Further Topics……Page 835
24.1. Introduction……Page 837
24.2.1. Current Population Survey……Page 838
Exhaustive Sampling……Page 839
Finite-Sample Correction……Page 840
24.3.1. Sample Weights……Page 841
24.3.2. Weighted Regression……Page 842
Incorrectly Specified Conditional Mean……Page 843
Should One Use Sample Weights?……Page 844
24.3.3. Prediction……Page 845
24.4.1. Stratification Schemes……Page 846
24.4.2. Endogeneity Induced by Stratification……Page 848
24.4.3. Endogenous Sampling……Page 849
24.4.4. Endogenously Stratified Samples……Page 850
24.4.5. Weighted Estimation……Page 851
Weighted ML Estimation……Page 852
24.5. Clustering……Page 853
24.5.1. Cluster-Specific Effects Models……Page 854
Notation……Page 857
OLS Standard Errors Assuming the CSRE Model……Page 858
Bias of Usual OLS Standard Errors……Page 860
24.5.3. Cluster-Specific Random Effects Estimator……Page 861
24.5.4. Cluster-Specific Fixed Effects Estimator……Page 863
Within-Clusters Estimator……Page 864
24.5.6. Clustering in Nonlinear Models……Page 865
m-Estimation with Clustering……Page 866
Nonlinear Cluster-Specific Fixed Effects……Page 867
24.6.1. Model Structure……Page 869
24.6.2. HLM for Panel Data……Page 871
24.7. Clustering Example: Vietnam Health Care Use……Page 872
24.7.1. Results and Discussion……Page 873
24.8.1. Variance Estimation in Complex Surveys……Page 877
Variance of a Linear Statistic……Page 878
Variance of Weighted Least-Squares Estimator……Page 879
Endogenous Stratification……Page 880
24.10. Bibliographic Notes……Page 881
25.1. Introduction……Page 884
25.2.1. Treatment Effects Framework……Page 886
25.2.2. Conditional Independence Assumption……Page 887
25.2.5. Propensity Scores……Page 888
25.3. Treatment Effects and Selection Bias……Page 889
25.3.1. Two Key Parameters: ATE and ATET……Page 890
25.3.2. Sampling and Selection Bias……Page 891
25.3.4. Selection on Unobservables……Page 893
25.4.1. Treatment Effect Assumptions……Page 895
25.4.2. Exact Matching……Page 896
Implementation Issues……Page 897
25.4.4. Measuring Treatment Effects……Page 898
Matching Methods……Page 899
25.4.5. Variance of ATET Based on x and p(x)……Page 901
25.5. Differences-in-Differences Estimators……Page 902
25.6.1. Discontinuous Treatment Assignment Mechanism……Page 903
25.6.2. Identification and Estimation under RD Design……Page 905
25.6.4. A Two-Stage Estimator……Page 906
25.7.1. Local ATE (LATE)……Page 907
25.7.2. Relation to Other Measures……Page 909
25.7.3. IV Estimation in a Model with Heterogeneous Treatment Effect……Page 910
25.7.4. Endogenous Treatment in Nonlinear Models……Page 912
25.8.2. Control Function Approach……Page 913
25.8.3. Differences in Differences……Page 914
25.8.4. Simple Propensity Score Estimate……Page 915
Matching Algorithms and Balancing……Page 917
ATET Estimates by Matching Methods……Page 918
25.9. Bibliographic Notes……Page 920
26.1. Introduction……Page 923
26.2. Measurement Error in Linear Regression……Page 924
26.2.1. Classical Measurement Error Model……Page 925
26.2.2. Inconsistency of OLS……Page 926
26.2.3. Measurement Error with a Scalar Regressor……Page 927
26.2.5. Measurement Error in Linear Panel Models……Page 928
26.3. Identification Strategies……Page 929
Reverse Regression……Page 930
26.3.2. Identification Using Instrumental Variables……Page 932
26.3.3. Identification via Additional Moment Restrictions……Page 933
26.3.4. Replicated Data……Page 934
26.4. Measurement Errors in Nonlinear Models……Page 935
26.4.1. Identification through Instrumental Variables……Page 936
26.4.3. Measurement Errors in Dependent Variables……Page 937
Discrete Choice Models……Page 938
26.4.4. Poisson Regression with Measurement Errors in Covariates……Page 939
Estimation of Errors-in-Variables Model……Page 940
Inconsistent and Consistent Estimators……Page 941
26.5. Attenuation Bias Simulation Examples……Page 943
26.6. Bibliographic Notes……Page 944
27.1. Introduction……Page 947
27.2. Missing Data Assumptions……Page 949
27.2.1. Missing at Random……Page 950
27.2.3. Ignorable and Nonignorable Missingness……Page 951
27.3.2. Imputation without Models……Page 952
27.4. Observed-Data Likelihood……Page 953
27.5.1. Linear Regression Example with Missing Data on a Dependent Variable……Page 954
27.6. Data Augmentation and MCMC……Page 956
27.7. Multiple Imputation……Page 958
27.8. Missing Data MCMC Imputation Example……Page 959
27.8.1. Linear Regression with Missing Data on Regressors……Page 960
27.8.2. Logit Regression with Missing Data on Regressors……Page 961
27.9. Practical Considerations……Page 963
27.10. Bibliographic Notes……Page 964
A.1. Introduction……Page 967
A.2.1. Convergence in Probability……Page 968
A.2.2. Alternative Modes of Convergence……Page 970
A.3. Laws of Large Numbers……Page 971
A.4. Convergence in Distribution……Page 972
A.5. Central Limit Theorems……Page 973
A.6.1. Multivariate Normal Limit Distributions……Page 975
A.6.3. Limit Variance Matrix……Page 976
A.6.4. Asymptotic Distribution and Variance……Page 977
A.7. Stochastic Order of Magnitude……Page 978
A.8. Other Results……Page 979
A.9. Bibliographic Notes……Page 980
APPENDIX B Making Pseudo-Random Draws……Page 981
References……Page 985
Author Index……Page 1022
Subject Index……Page 1030
cambridge.org……Page 0
Microeconometrics – Cambridge University Press……Page 1059
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