ECONOMETRIC ANALYSIS Sách về các mô hình phân tích kinh tế (Rất hay)

SEVENTH EDITION

ECONOMETRIC ANALYSIS

Q

William H. Greene
New York University

Prentice Hall


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Library of Congress Cataloging-in-Publication Data
Greene, William H., 1951–
Econometric analysis / William H. Greene.—7th ed.
p. cm.
ISBN 0-13-139538-6
1. Econometrics. I. Title.
HB139.G74 2012

330.01'5195—dc22
2010050532
10 9 8 7 6 5 4 3 2 1

ISBN 10: 0-13-139538-6
ISBN 13: 978-0-13-139538-1


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BRIEF CONTENTS

Q

Examples and Applications
Preface
xxxiii

Part I

The Linear Regression Model

Chapter 1
Chapter 2

Econometrics
1
The Linear Regression Model

Chapter 3
Chapter 4
Chapter 5

Least Squares
26
The Least Squares Estimator
51
Hypothesis Tests and Model Selection

Chapter 6
Chapter 7

Functional Form and Structural Change
149
Nonlinear, Semiparametric, and Nonparametric
Regression Models
181

Chapter 8

Endogeneity and Instrumental Variable Estimation

Part II

Generalized Regression Model and Equation Systems

Chapter 9
Chapter 10
Chapter 11

The Generalized Regression Model and Heteroscedasticity
Systems of Equations
290
Models for Panel Data
343

Part III

Estimation Methodology

Chapter 12
Chapter 13

Chapter 16

Estimation Frameworks in Econometrics
432
Minimum Distance Estimation and the Generalized
Method of Moments
455
Maximum Likelihood Estimation
509
Simulation-Based Estimation and Inference and Random
Parameter Models
603
Bayesian Estimation and Inference
655

Part IV

Cross Sections, Panel Data, and Microeconometrics

Chapter 17
Chapter 18
Chapter 19

Discrete Choice
681

Discrete Choices and Event Counts
760
Limited Dependent Variables—Truncation, Censoring,
and Sample Selection
833

Chapter 14
Chapter 15

iv

xxv

11

108

219

257


Brief Contents

Part V

Time Series and Macroeconometrics

Chapter 20
Chapter 21

Serial Correlation
Nonstationary Data

Part VI

Appendices

903
942

Appendix A Matrix Algebra
973
Appendix B Probability and Distribution Theory

1015

Appendix C Estimation and Inference
1047
Appendix D Large-Sample Distribution Theory
1066
Appendix E Computation and Optimization
1089
Appendix F Data Sets Used in Applications
References

1115

Combined Author and Subject Index

1161

1109

v


CONTENTS

Q

Examples and Applications
Preface

xxv

xxxiii

PART I The Linear Regression Model
CHAPTER 1 Econometrics
1
1.1
Introduction
1
1.2
The Paradigm of Econometrics
1.3
The Practice of Econometrics
1.4
Econometric Modeling

4
1.5
1.6

Plan of the Book
Preliminaries
9
1.6.1
1.6.2
1.6.3

1
3

7

Numerical Examples 9
Software and Replication 9
Notational Conventions 9

CHAPTER 2 The Linear Regression Model
11
2.1
Introduction
11
2.2
The Linear Regression Model
12
2.3
Assumptions of the Linear Regression Model

2.4

2.3.1
Linearity of the Regression Model 15
2.3.2
Full Rank 19
2.3.3
Regression 20
2.3.4
Spherical Disturbances 21
2.3.5
Data Generating Process for the Regressors 23
2.3.6
Normality 23
2.3.7
Independence 24
Summary and Conclusions
25

CHAPTER 3 Least Squares
26
3.1
Introduction
26
3.2
Least Squares Regression
26
3.2.1
The Least Squares Coefficient Vector

vi

15

27


Contents

3.3
3.4
3.5

3.2.2
Application: An Investment Equation 28
3.2.3
Algebraic Aspects of the Least Squares Solution 30
3.2.4
Projection 31
Partitioned Regression and Partial Regression
32
Partial Regression and Partial Correlation Coefficients
36
Goodness of Fit and the Analysis of Variance
39

3.6
3.7

3.5.1

The Adjusted R-Squared and a Measure of Fit 42
3.5.2
R-Squared and the Constant Term in the Model 44
3.5.3
Comparing Models 45
Linearly Transformed Regression
46
Summary and Conclusions
47

CHAPTER 4 The Least Squares Estimator
4.1
Introduction
51
4.2

4.3

4.4

4.5

4.6

vii

51

Motivating Least Squares
52

4.2.1
The Population Orthogonality Conditions 52
4.2.2
Minimum Mean Squared Error Predictor 53
4.2.3
Minimum Variance Linear Unbiased Estimation 54
Finite Sample Properties of Least Squares
54
4.3.1
Unbiased Estimation 55
4.3.2
Bias Caused by Omission of Relevant Variables 56
4.3.3
Inclusion of Irrelevant Variables 58
4.3.4
The Variance of the Least Squares Estimator 58
4.3.5
The Gauss–Markov Theorem 60
4.3.6
The Implications of Stochastic Regressors 60
4.3.7
Estimating the Variance of the Least Squares Estimator 61
4.3.8
The Normality Assumption 63
Large Sample Properties of the Least Squares Estimator
63
4.4.1
Consistency of the Least Squares Estimator of β 63
4.4.2
Asymptotic Normality of the Least Squares Estimator 65

4.4.3
Consistency of s2 and the Estimator of Asy. Var[b] 67
4.4.4
Asymptotic Distribution of a Function of b: The Delta
Method 68
4.4.5
Asymptotic Efficiency 69
4.4.6
Maximum Likelihood Estimation 73
Interval Estimation
75
4.5.1
Forming a Confidence Interval for a Coefficient 76
4.5.2
Confidence Intervals Based on Large Samples 78
4.5.3
Confidence Interval for a Linear Combination of Coefficients:
The Oaxaca Decomposition 79
Prediction and Forecasting
80
4.6.1
Prediction Intervals 81
4.6.2
Predicting y When the Regression Model Describes Log y 81


viii

Contents

4.6.3

4.7

4.8

Prediction Interval for y When the Regression Model Describes
Log y 83
4.6.4
Forecasting 87
Data Problems
88
4.7.1
Multicollinearity 89
4.7.2
Pretest Estimation 91
4.7.3
Principal Components 92
4.7.4
Missing Values and Data Imputation 94
4.7.5
Measurement Error 97
4.7.6
Outliers and Influential Observations 99
Summary and Conclusions
102

CHAPTER 5 Hypothesis Tests and Model Selection
5.1
Introduction

108
5.2

Hypothesis Testing Methodology

108

108

5.2.1
5.2.2
5.2.3
5.2.4
5.2.5

5.9
5.10

Restrictions and Hypotheses 109
Nested Models 110
Testing Procedures—Neyman–Pearson Methodology 111
Size, Power, and Consistency of a Test 111
A Methodological Dilemma: Bayesian versus Classical Testing
112
Two Approaches to Testing Hypotheses
112
Wald Tests Based on the Distance Measure
115
5.4.1
Testing a Hypothesis about a Coefficient 115

5.4.2
The F Statistic and the Least Squares Discrepancy 117
Testing Restrictions Using the Fit of the Regression
121
5.5.1
The Restricted Least Squares Estimator 121
5.5.2
The Loss of Fit from Restricted Least Squares 122
5.5.3
Testing the Significance of the Regression 126
5.5.4
Solving Out the Restrictions and a Caution about
Using R2 126
Nonnormal Disturbances and Large-Sample Tests
127
Testing Nonlinear Restrictions
131
Choosing between Nonnested Models
134
5.8.1
Testing Nonnested Hypotheses 134
5.8.2
An Encompassing Model 135
5.8.3
Comprehensive Approach—The J Test 136
A Specification Test
137
Model Building—A General to Simple Strategy
138

5.11

5.10.1
Model Selection Criteria 139
5.10.2
Model Selection 140
5.10.3
Classical Model Selection 140
5.10.4
Bayesian Model Averaging 141
Summary and Conclusions
143

5.3
5.4

5.5

5.6
5.7
5.8


Contents

CHAPTER 6 Functional Form and Structural Change
149
6.1
Introduction
149

6.2
Using Binary Variables
149
6.2.1
Binary Variables in Regression 149
6.2.2
Several Categories 152
6.2.3
Several Groupings 152
6.2.4
Threshold Effects and Categorical Variables 154
6.2.5
Treatment Effects and Differences in Differences
Regression 155
6.3
Nonlinearity in the Variables
158
6.3.1
Piecewise Linear Regression 158
6.3.2
Functional Forms 160
6.3.3
Interaction Effects 161
6.3.4
Identifying Nonlinearity 162
6.3.5
Intrinsically Linear Models 165
6.4
Modeling and Testing for a Structural Break
168

6.4.1
Different Parameter Vectors 168
6.4.2
Insufficient Observations 169
6.4.3
Change in a Subset of Coefficients 170
6.4.4
Tests of Structural Break with Unequal Variances 171
6.4.5
Predictive Test of Model Stability 174
6.5
Summary and Conclusions
175
CHAPTER 7
7.1
7.2

7.3

7.4
7.5
7.6

Nonlinear, Semiparametric, and Nonparametric
Regression Models
181
Introduction
181
Nonlinear Regression Models
182

7.2.1
Assumptions of the Nonlinear Regression Model 182
7.2.2
The Nonlinear Least Squares Estimator 184
7.2.3
Large Sample Properties of the Nonlinear Least Squares
Estimator 186
7.2.4
Hypothesis Testing and Parametric Restrictions 189
7.2.5
Applications 191
7.2.6
Computing the Nonlinear Least Squares Estimator 200
Median and Quantile Regression
202
7.3.1
Least Absolute Deviations Estimation 203
7.3.2
Quantile Regression Models 207
Partially Linear Regression
210
Nonparametric Regression
212
Summary and Conclusions
215

CHAPTER 8 Endogeneity and Instrumental Variable Estimation
8.1
Introduction
219

8.2
Assumptions of the Extended Model
223

219

ix


x

Contents

8.3

8.4

Estimation
224
8.3.1
Least Squares 225
8.3.2
The Instrumental Variables Estimator 225
8.3.3
Motivating the Instrumental Variables Estimator 227
8.3.4
Two-Stage Least Squares 230
Two Specification Tests
233

8.5

8.4.1
The Hausman and Wu Specification Tests
8.4.2
A Test for Overidentification 238
Measurement Error
239

8.6
8.7
8.8
8.9

8.5.1
Least Squares Attenuation 240
8.5.2
Instrumental Variables Estimation 242
8.5.3
Proxy Variables 242
Nonlinear Instrumental Variables Estimation
246
Weak Instruments
249
Natural Experiments and the Search for Causal Effects
Summary and Conclusions
254

PART II

234

251

Generalized Regression Model and Equation Systems

CHAPTER 9 The Generalized Regression Model and Heteroscedasticity
257
9.1
Introduction
257
9.2
Inefficient Estimation by Least Squares and Instrumental
Variables
258
9.2.1
Finite-Sample Properties of Ordinary Least Squares 259
9.2.2
Asymptotic Properties of Ordinary Least Squares 259
9.2.3
Robust Estimation of Asymptotic Covariance Matrices 261
9.2.4
Instrumental Variable Estimation 262
9.3
Efficient Estimation by Generalized Least Squares
264
9.3.1
Generalized Least Squares (GLS) 264
9.3.2
Feasible Generalized Least Squares (FGLS) 266

9.4
Heteroscedasticity and Weighted Least Squares
268
9.4.1
9.4.2
9.4.3
9.4.4

9.5

9.6

Ordinary Least Squares Estimation 269
Inefficiency of Ordinary Least Squares 270
The Estimated Covariance Matrix of b 270
Estimating the Appropriate Covariance Matrix for Ordinary
Least Squares 272
Testing for Heteroscedasticity
275
9.5.1
White’s General Test 275
9.5.2
The Breusch–Pagan/Godfrey LM Test 276
Weighted Least Squares
277
9.6.1
Weighted Least Squares with Known
278
9.6.2
Estimation When Contains Unknown Parameters 279

Contents

9.7

9.8

xi

Applications
280
9.7.1
Multiplicative Heteroscedasticity 280
9.7.2
Groupwise Heteroscedasticity 282
Summary and Conclusions
285

CHAPTER 10 Systems of Equations
290
10.1 Introduction
290
10.2 The Seemingly Unrelated Regressions Model

292

10.3

10.2.1

Generalized Least Squares 293
10.2.2
Seemingly Unrelated Regressions with Identical Regressors 295
10.2.3
Feasible Generalized Least Squares 296
10.2.4
Testing Hypotheses 296
10.2.5
A Specification Test for the SUR Model 297
10.2.6
The Pooled Model 299
Seemingly Unrelated Generalized Regression Models
304

10.4

Nonlinear Systems of Equations

10.5

Systems of Demand Equations: Singular Systems
307
10.5.1
Cobb–Douglas Cost Function 307
10.5.2
Flexible Functional Forms: The Translog Cost Function 310
Simultaneous Equations Models
314
10.6.1
Systems of Equations 315

10.6.2
A General Notation for Linear Simultaneous Equations
Models 318
10.6.3
The Problem of Identification 321
10.6.4
Single Equation Estimation and Inference 326
10.6.5
System Methods of Estimation 329
10.6.6
Testing in the Presence of Weak Instruments 334
Summary and Conclusions
336

10.6

10.7

305

CHAPTER 11 Models for Panel Data
343
11.1 Introduction
343
11.2 Panel Data Models
344
11.2.1
General Modeling Framework for Analyzing Panel Data
11.2.2
Model Structures 346

11.2.3
Extensions 347
11.2.4
Balanced and Unbalanced Panels 348
11.2.5
Well-Behaved Panel Data 348
11.3 The Pooled Regression Model
349
11.3.1
Least Squares Estimation of the Pooled Model 349
11.3.2
Robust Covariance Matrix Estimation 350
11.3.3
Clustering and Stratification 352
11.3.4
Robust Estimation Using Group Means 354

345


xii

Contents

11.4

11.5

11.6

11.7
11.8

11.3.5
Estimation with First Differences 355
11.3.6
The Within- and Between-Groups Estimators 357
The Fixed Effects Model
359
11.4.1
Least Squares Estimation 360
11.4.2
Small T Asymptotics 362
11.4.3
Testing the Significance of the Group Effects 363
11.4.4
Fixed Time and Group Effects 363
11.4.5
Time-Invariant Variables and Fixed Effects Vector
Decomposition 364
Random Effects
370
11.5.1
Least Squares Estimation 372
11.5.2
Generalized Least Squares 373
11.5.3
Feasible Generalized Least Squares When Is Unknown 374
11.5.4
Testing for Random Effects 376

11.5.5
Hausman’s Specification Test for the Random Effects
Model 379
11.5.6
Extending the Unobserved Effects Model: Mundlak’s
Approach 380
11.5.7
Extending the Random and Fixed Effects Models:
Chamberlain’s Approach 381
Nonspherical Disturbances and Robust Covariance Estimation
385
11.6.1
Robust Estimation of the Fixed Effects Model 385
11.6.2
Heteroscedasticity in the Random Effects Model 387
11.6.3
Autocorrelation in Panel Data Models 388
11.6.4
Cluster (and Panel) Robust Covariance Matrices for Fixed and
Random Effects Estimators 388
Spatial Autocorrelation
389

Endogeneity
394
11.8.1
Hausman and Taylor’s Instrumental Variables Estimator 394
11.8.2
Consistent Estimation of Dynamic Panel Data Models:
Anderson and Hsiao’s IV Estimator 398

11.8.3
Efficient Estimation of Dynamic Panel Data Models—The
Arellano/Bond Estimators 400
11.8.4
Nonstationary Data and Panel Data Models 410
11.9 Nonlinear Regression with Panel Data
411
11.9.1
A Robust Covariance Matrix for Nonlinear Least Squares 411
11.9.2
Fixed Effects 412
11.9.3
Random Effects 414
11.10 Systems of Equations
415
11.11 Parameter Heterogeneity
416
11.11.1 The Random Coefficients Model 417
11.11.2 A Hierarchical Linear Model 420
11.11.3 Parameter Heterogeneity and Dynamic Panel Data
Models 421
11.12 Summary and Conclusions
426


Contents

PART III

Estimation Methodology

CHAPTER 12 Estimation Frameworks in Econometrics
12.1 Introduction
432
12.2 Parametric Estimation and Inference
434

12.3

432

12.2.1
Classical Likelihood-Based Estimation 434
12.2.2
Modeling Joint Distributions with Copula Functions
Semiparametric Estimation
439
12.3.1
12.3.2
12.3.3

12.4
12.5

12.6

GMM Estimation in Econometrics 439
Maximum Empirical Likelihood Estimation 440
Least Absolute Deviations Estimation and Quantile
Regression 441

12.3.4
Kernel Density Methods 442
12.3.5
Comparing Parametric and Semiparametric Analyses
Nonparametric Estimation
444

436

443

12.4.1
Kernel Density Estimation 445
Properties of Estimators
447
12.5.1
Statistical Properties of Estimators 448
12.5.2
Extremum Estimators 449
12.5.3
Assumptions for Asymptotic Properties of Extremum
Estimators 449
12.5.4
Asymptotic Properties of Estimators 452
12.5.5
Testing Hypotheses 453
Summary and Conclusions
454

CHAPTER 13 Minimum Distance Estimation and the Generalized

Method of Moments
455
13.1 Introduction
455
13.2 Consistent Estimation: The Method of Moments
456
13.2.1
Random Sampling and Estimating the Parameters of
Distributions 457
13.2.2
Asymptotic Properties of the Method of Moments
Estimator 461
13.2.3
Summary—The Method of Moments 463
13.3 Minimum Distance Estimation
463
13.4 The Generalized Method of Moments (GMM) Estimator
468
13.4.1
Estimation Based on Orthogonality Conditions 468
13.4.2
Generalizing the Method of Moments 470
13.4.3
Properties of the GMM Estimator 474
13.5 Testing Hypotheses in the GMM Framework
479
13.5.1
Testing the Validity of the Moment Restrictions 479
13.5.2
GMM Counterparts to the WALD, LM, and LR

Tests 480

xiii


xiv

Contents

13.6

13.7

GMM Estimation of Econometric Models
482
13.6.1
Single-Equation Linear Models 482
13.6.2
Single-Equation Nonlinear Models 488
13.6.3
Seemingly Unrelated Regression Models 491
13.6.4
Simultaneous Equations Models with Heteroscedasticity 493
13.6.5
GMM Estimation of Dynamic Panel Data Models 496
Summary and Conclusions
507

CHAPTER 14 Maximum Likelihood Estimation
509

14.1 Introduction
509
14.2 The Likelihood Function and Identification of the Parameters
14.3
14.4

14.5
14.6

14.7
14.8

509

Efficient Estimation: The Principle of Maximum Likelihood
511
Properties of Maximum Likelihood Estimators
513
14.4.1
Regularity Conditions 514
14.4.2
Properties of Regular Densities 515
14.4.3
The Likelihood Equation 517
14.4.4
The Information Matrix Equality 517
14.4.5
Asymptotic Properties of the Maximum Likelihood
Estimator 517
14.4.5.a Consistency 518

14.4.5.b Asymptotic Normality 519
14.4.5.c Asymptotic Efficiency 520
14.4.5.d Invariance 521
14.4.5.e Conclusion 521
14.4.6
Estimating the Asymptotic Variance of the Maximum
Likelihood Estimator 521
Conditional Likelihoods, Econometric Models, and the GMM
Estimator
523
Hypothesis and Specification Tests and Fit Measures
524
14.6.1
The Likelihood Ratio Test 526
14.6.2
The Wald Test 527
14.6.3
The Lagrange Multiplier Test 529
14.6.4
An Application of the Likelihood-Based Test Procedures 531
14.6.5
Comparing Models and Computing Model Fit 533
14.6.6
Vuong’s Test and the Kullback–Leibler Information
Criterion 534
Two-Step Maximum Likelihood Estimation
536
Pseudo-Maximum Likelihood Estimation and Robust Asymptotic
Covariance Matrices
542

14.8.1
Maximum Likelihood and GMM Estimation 543
14.8.2
Maximum Likelihood and M Estimation 543
14.8.3
Sandwich Estimators 545
14.8.4
Cluster Estimators 546


Contents

xv

14.9

Applications of Maximum Likelihood Estimation
548
14.9.1
The Normal Linear Regression Model 548
14.9.2
The Generalized Regression Model 552
14.9.2.a Multiplicative Heteroscedasticity 554
14.9.2.b Autocorrelation 557
14.9.3
Seemingly Unrelated Regression Models 560
14.9.3.a The Pooled Model 560
14.9.3.b The SUR Model 562
14.9.3.c Exclusion Restrictions 562
14.9.4

Simultaneous Equations Models 567
14.9.5
Maximum Likelihood Estimation of Nonlinear Regression
Models 568
14.9.6
Panel Data Applications 573
14.9.6.a ML Estimation of the Linear Random Effects
Model 574
14.9.6.b Nested Random Effects 576
14.9.6.c Random Effects in Nonlinear Models: MLE Using
Quadrature 580
14.9.6.d Fixed Effects in Nonlinear Models: Full MLE 584
14.10 Latent Class and Finite Mixture Models
588
14.10.1 A Finite Mixture Model 589
14.10.2 Measured and Unmeasured Heterogeneity 591
14.10.3 Predicting Class Membership 591
14.10.4 A Conditional Latent Class Model 592
14.10.5 Determining the Number of Classes 594
14.10.6 A Panel Data Application 595
14.11 Summary and Conclusions
598

CHAPTER 15 Simulation-Based Estimation and Inference and Random Parameter
Models
603
15.1 Introduction
603
15.2

15.3
15.4
15.5

15.6

Random Number Generation
605
15.2.1
Generating Pseudo-Random Numbers 605
15.2.2
Sampling from a Standard Uniform Population 606
15.2.3
Sampling from Continuous Distributions 607
15.2.4
Sampling from a Multivariate Normal Population 608
15.2.5
Sampling from Discrete Populations 608
Simulation-Based Statistical Inference: The Method of Krinsky and
Robb
609
Bootstrapping Standard Errors and Confidence Intervals
611
Monte Carlo Studies
615
15.5.1
A Monte Carlo Study: Behavior of a Test Statistic 617
15.5.2
A Monte Carlo Study: The Incidental Parameters Problem
Simulation-Based Estimation

621
15.6.1
Random Effects in a Nonlinear Model 621

619


xvi

Contents

15.6.2

15.7

Monte Carlo Integration 623
15.6.2.a Halton Sequences and Random Draws for
Simulation-Based Integration 625
15.6.2.b Computing Multivariate Normal Probabilities Using
the GHK Simulator 627
15.6.3
Simulation-Based Estimation of Random Effects Models 629
A Random Parameters Linear Regression Model
634

15.8
15.9

Hierarchical Linear Models
639

Nonlinear Random Parameter Models

15.10 Individual Parameter Estimates
642
15.11 Mixed Models and Latent Class Models
15.12 Summary and Conclusions
653

641
650

CHAPTER 16 Bayesian Estimation and Inference
655
16.1 Introduction
655
16.2 Bayes Theorem and the Posterior Density
656
16.3 Bayesian Analysis of the Classical Regression Model
16.3.1
Analysis with a Noninformative Prior 659
16.3.2
Estimation with an Informative Prior Density
16.4 Bayesian Inference
664

658
661

16.5

16.4.1
Point Estimation 664
16.4.2
Interval Estimation 665
16.4.3
Hypothesis Testing 666
16.4.4
Large-Sample Results 668
Posterior Distributions and the Gibbs Sampler

16.6
16.7
16.8

Application: Binomial Probit Model
671
Panel Data Application: Individual Effects Models
674
Hierarchical Bayes Estimation of a Random Parameters Model

16.9

Summary and Conclusions

PART IV

668

678

Cross Sections, Panel Data, and Microeconometrics

CHAPTER 17 Discrete Choice
681
17.1 Introduction
681
17.2 Models for Binary Outcomes
683
17.2.1
Random Utility Models for Individual Choice 684
17.2.2
A Latent Regression Model 686
17.2.3
Functional Form and Regression 687
17.3 Estimation and Inference in Binary Choice Models
690
17.3.1
Robust Covariance Matrix Estimation 692
17.3.2
Marginal Effects and Average Partial Effects 693

676


Contents

17.4

17.5

17.6

17.3.2.a Average Partial Effects 696
17.3.2.b Interaction Effects 699
17.3.3
Measuring Goodness of Fit 701
17.3.4
Hypothesis Tests 703
17.3.5
Endogenous Right-Hand-Side Variables in Binary Choice
Models 706
17.3.6
Endogenous Choice-Based Sampling 710
17.3.7
Specification Analysis 711
17.3.7.a Omitted Variables 713
17.3.7.b Heteroscedasticity 714
Binary Choice Models for Panel Data
716
17.4.1
The Pooled Estimator 717
17.4.2
Random Effects Models 718
17.4.3
Fixed Effects Models 721
17.4.4
A Conditional Fixed Effects Estimator 722
17.4.5
Mundlak’s Approach, Variable Addition, and Bias
Reduction 727

17.4.6
Dynamic Binary Choice Models 729
17.4.7
A Semiparametric Model for Individual Heterogeneity 731
17.4.8
Modeling Parameter Heterogeneity 733
17.4.9
Nonresponse, Attrition, and Inverse Probability Weighting 734
Bivariate and Multivariate Probit Models
738
17.5.1
Maximum Likelihood Estimation 739
17.5.2
Testing for Zero Correlation 742
17.5.3
Partial Effects 742
17.5.4
A Panel Data Model for Bivariate Binary Response 744
17.5.5
Endogenous Binary Variable in a Recursive Bivariate Probit
Model 745
17.5.6
Endogenous Sampling in a Binary Choice Model 749
17.5.7
A Multivariate Probit Model 752
Summary and Conclusions
755

CHAPTER 18 Discrete Choices and Event Counts
18.1 Introduction

760
18.2

xvii

760

Models for Unordered Multiple Choices
761
18.2.1
Random Utility Basis of the Multinomial Logit Model
18.2.2
The Multinomial Logit Model 763
18.2.3
The Conditional Logit Model 766
18.2.4
The Independence from Irrelevant Alternatives
Assumption 767
18.2.5
Nested Logit Models 768
18.2.6
The Multinomial Probit Model 770
18.2.7
The Mixed Logit Model 771
18.2.8
A Generalized Mixed Logit Model 772

761

xviii

Contents

18.2.9

18.3

Application: Conditional Logit Model for Travel Mode
Choice 773
18.2.10 Estimating Willingness to Pay 779
18.2.11 Panel Data and Stated Choice Experiments 781
18.2.12 Aggregate Market Share Data—The BLP Random Parameters
Model 782
Random Utility Models for Ordered Choices
784
18.3.1
18.3.2
18.3.3
18.3.4

18.4

18.5

The Ordered Probit Model 787
A Specification Test for the Ordered Choice Model 791
Bivariate Ordered Probit Models 792
Panel Data Applications 794
18.3.4.a Ordered Probit Models with Fixed Effects 794

18.3.4.b Ordered Probit Models with Random Effects 795
18.3.5
Extensions of the Ordered Probit Model 798
18.3.5.a Threshold Models—Generalized Ordered Choice
Models 799
18.3.5.b Thresholds and Heterogeneity—Anchoring
Vignettes 800
Models for Counts of Events
802
18.4.1
The Poisson Regression Model 803
18.4.2
Measuring Goodness of Fit 804
18.4.3
Testing for Overdispersion 805
18.4.4
Heterogeneity and the Negative Binomial Regression
Model 806
18.4.5
Functional Forms for Count Data Models 807
18.4.6
Truncation and Censoring in Models for Counts 810
18.4.7
Panel Data Models 815
18.4.7.a Robust Covariance Matrices for Pooled
Estimators 816
18.4.7.b Fixed Effects 817
18.4.7.c Random Effects 818
18.4.8
Two-Part Models: Zero Inflation and Hurdle Models 821

18.4.9
Endogenous Variables and Endogenous Participation 826
Summary and Conclusions
829

CHAPTER 19 Limited Dependent Variables—Truncation, Censoring, and Sample
Selection
833
19.1 Introduction
833
19.2 Truncation
833
19.2.1
Truncated Distributions 834
19.2.2
Moments of Truncated Distributions 835
19.2.3
The Truncated Regression Model 837
19.2.4
The Stochastic Frontier Model 839
19.3 Censored Data
845
19.3.1
The Censored Normal Distribution 846


Contents

xix

19.3.2
19.3.3
19.3.4
19.3.5

19.4

19.5

The Censored Regression (Tobit) Model 848
Estimation 850
Two-Part Models and Corner Solutions 852
Some Issues in Specification 858
19.3.5.a Heteroscedasticity 858
19.3.5.b Nonnormality 859
19.3.6
Panel Data Applications 860
Models for Duration
861
19.4.1
Models for Duration Data 862
19.4.2
Duration Data 862
19.4.3
A Regression-Like Approach: Parametric Models of
Duration 863
19.4.3.a Theoretical Background 863
19.4.3.b Models of the Hazard Function 864
19.4.3.c Maximum Likelihood Estimation 866
19.4.3.d Exogenous Variables 867

19.4.3.e Heterogeneity 868
19.4.4
Nonparametric and Semiparametric Approaches 869
Incidental Truncation and Sample Selection
872
19.5.1
19.5.2
19.5.3
19.5.4
19.5.5

19.6

19.7

PART V

Incidental Truncation in a Bivariate Distribution 873
Regression in a Model of Selection 873
Two-Step and Maximum Likelihood Estimation 876
Sample Selection in Nonlinear Models 880
Panel Data Applications of Sample Selection Models 883
19.5.5.a Common Effects in Sample Selection Models 884
19.5.5.b Attrition 886
Evaluating Treatment Effects
888
19.6.1
Regression Analysis of Treatment Effects 890
19.6.1.a The Normality Assumption 892
19.6.1.b Estimating the Effect of Treatment on

the Treated 893
19.6.2
Propensity Score Matching 894
19.6.3
Regression Discontinuity 897
Summary and Conclusions
898

Time Series and Macroeconometrics

CHAPTER 20 Serial Correlation
903
20.1 Introduction
903
20.2 The Analysis of Time-Series Data
906
20.3 Disturbance Processes
909
20.3.1
Characteristics of Disturbance Processes
20.3.2
AR(1) Disturbances 910

909


xx

Contents

20.4

20.5

20.6
20.7

Some Asymptotic Results for Analyzing Time-Series Data
912
20.4.1
Convergence of Moments—The Ergodic Theorem 913
20.4.2
Convergence to Normality—A Central Limit Theorem 915
Least Squares Estimation
918
20.5.1
Asymptotic Properties of Least Squares 918
20.5.2
Estimating the Variance of the Least Squares Estimator 919
GMM Estimation
921
Testing for Autocorrelation
922
20.7.1
20.7.2
20.7.3
20.7.4

Lagrange Multiplier Test 922
Box and Pierce’s Test and Ljung’s Refinement 922

The Durbin–Watson Test 923
Testing in the Presence of a Lagged Dependent
Variable 923
20.7.5
Summary of Testing Procedures 924
20.8 Efficient Estimation When Is Known
924
20.9 Estimation When Is Unknown
926
20.9.1
AR(1) Disturbances 926
20.9.2
Application: Estimation of a Model with Autocorrelation 927
20.9.3
Estimation with a Lagged Dependent Variable 929
20.10 Autoregressive Conditional Heteroscedasticity
930
20.10.1 The ARCH(1) Model 931
20.10.2 ARCH(q), ARCH-in-Mean, and Generalized ARCH
Models 932
20.10.3 Maximum Likelihood Estimation of the Garch Model 934
20.10.4 Testing for Garch Effects 936
20.10.5 Pseudo–Maximum Likelihood Estimation 937
20.11 Summary and Conclusions
939
CHAPTER 21 Nonstationary Data
942
21.1 Introduction
942
21.2 Nonstationary Processes and Unit Roots

942
21.2.1
Integrated Processes and Differencing 942
21.2.2
Random Walks, Trends, and Spurious Regressions 944
21.2.3
Tests for Unit Roots in Economic Data 947
21.2.4
The Dickey–Fuller Tests 948
21.2.5
The KPSS Test of Stationarity 958
21.3 Cointegration
959
21.3.1
Common Trends 962
21.3.2
Error Correction and VAR Representations 963
21.3.3
Testing for Cointegration 965
21.3.4
Estimating Cointegration Relationships 967
21.3.5
Application: German Money Demand 967
21.3.5.a Cointegration Analysis and a Long-Run Theoretical
Model 968
21.3.5.b Testing for Model Instability 969


Contents

21.4
21.5

Nonstationary Panel Data
Summary and Conclusions

xxi

970
971

PART VI Appendices
Appendix A
Matrix Algebra
973
A.1 Terminology
973
A.2 Algebraic Manipulation of Matrices

A.3

A.4

A.5

A.6

973

A.2.1

Equality of Matrices 973
A.2.2
Transposition 974
A.2.3
Matrix Addition 974
A.2.4
Vector Multiplication 975
A.2.5
A Notation for Rows and Columns of a Matrix 975
A.2.6
Matrix Multiplication and Scalar Multiplication 975
A.2.7
Sums of Values 977
A.2.8
A Useful Idempotent Matrix 978
Geometry of Matrices
979
A.3.1
Vector Spaces 979
A.3.2
Linear Combinations of Vectors and Basis Vectors 981
A.3.3
Linear Dependence 982
A.3.4
Subspaces 983
A.3.5
Rank of a Matrix 984
A.3.6
Determinant of a Matrix 986
A.3.7

A Least Squares Problem 987
Solution of a System of Linear Equations
989
A.4.1
Systems of Linear Equations 989
A.4.2
Inverse Matrices 990
A.4.3
Nonhomogeneous Systems of Equations 992
A.4.4
Solving the Least Squares Problem 992
Partitioned Matrices
992
A.5.1
Addition and Multiplication of Partitioned Matrices 993
A.5.2
Determinants of Partitioned Matrices 993
A.5.3
Inverses of Partitioned Matrices 993
A.5.4
Deviations from Means 994
A.5.5
Kronecker Products 994
Characteristic Roots and Vectors
995
A.6.1
The Characteristic Equation 995
A.6.2
Characteristic Vectors 996
A.6.3

General Results for Characteristic Roots and Vectors 996
A.6.4
Diagonalization and Spectral Decomposition of a Matrix 997
A.6.5
Rank of a Matrix 997
A.6.6
Condition Number of a Matrix 999
A.6.7
Trace of a Matrix 999
A.6.8
Determinant of a Matrix 1000
A.6.9
Powers of a Matrix 1000
A.6.10
Idempotent Matrices 1002


xxii

Contents

A.7

A.8

A.6.11
Factoring a Matrix 1002
A.6.12
The Generalized Inverse of a Matrix 1003
Quadratic Forms and Definite Matrices

1004
A.7.1
Nonnegative Definite Matrices 1005
A.7.2
Idempotent Quadratic Forms 1006
A.7.3
Comparing Matrices 1006
Calculus and Matrix Algebra
1007
A.8.1
Differentiation and the Taylor Series 1007
A.8.2
Optimization 1010
A.8.3
Constrained Optimization 1012
A.8.4
Transformations 1014

Appendix B
Probability and Distribution Theory
1015
B.1
Introduction
1015
B.2
Random Variables
1015
B.2.1
Probability Distributions 1015
B.2.2

Cumulative Distribution Function 1016
B.3
Expectations of a Random Variable
1017
B.4
Some Specific Probability Distributions
1019
B.4.1
The Normal Distribution 1019
B.4.2
The Chi-Squared, t, and F Distributions 1021
B.4.3
Distributions with Large Degrees of Freedom 1023
B.4.4
Size Distributions: The Lognormal Distribution 1024
B.4.5
The Gamma and Exponential Distributions 1024
B.4.6
The Beta Distribution 1025
B.4.7
The Logistic Distribution 1025
B.4.8
The Wishart Distribution 1025
B.4.9
Discrete Random Variables 1026
B.5
The Distribution of a Function of a Random Variable
1026
B.6
Representations of a Probability Distribution

1028
B.7

B.8

B.9
B.10

Joint Distributions
1030
B.7.1
Marginal Distributions 1030
B.7.2
Expectations in a Joint Distribution 1031
B.7.3
Covariance and Correlation 1031
B.7.4
Distribution of a Function of Bivariate Random
Variables 1032
Conditioning in a Bivariate Distribution
1034
B.8.1
Regression: The Conditional Mean 1034
B.8.2
Conditional Variance 1035
B.8.3
Relationships Among Marginal and Conditional
Moments 1035
B.8.4
The Analysis of Variance 1037

The Bivariate Normal Distribution
1037
Multivariate Distributions
1038
B.10.1
Moments 1038


Contents

B.11

B.10.2
Sets of Linear Functions 1039
B.10.3
Nonlinear Functions 1040
The Multivariate Normal Distribution
1041
B.11.1
Marginal and Conditional Normal Distributions 1041
B.11.2
The Classical Normal Linear Regression Model 1042
B.11.3
Linear Functions of a Normal Vector 1043
B.11.4
Quadratic forms in a Standard Normal Vector 1043
B.11.5
The F Distribution 1045
B.11.6
A Full Rank Quadratic Form 1045

B.11.7
Independence of a Linear and a Quadratic Form 1046

Appendix C
Estimation and Inference
C.1
Introduction
1047
C.2
Samples and Random Sampling
C.3
Descriptive Statistics
1048

1047
1048

C.4

Statistics as Estimators—Sampling Distributions

C.5

Point Estimation of Parameters
1055
C.5.1
Estimation in a Finite Sample 1055
C.5.2
Efficient Unbiased Estimation 1058
Interval Estimation

1060
Hypothesis Testing
1062
C.7.1
Classical Testing Procedures 1062
C.7.2
Tests Based on Confidence Intervals 1065
C.7.3
Specification Tests 1066

C.6
C.7

xxiii

1051

Appendix D
Large-Sample Distribution Theory
1066
D.1 Introduction
1066
D.2 Large-Sample Distribution Theory
1067
D.2.1
Convergence in Probability 1067
D.2.2
Other forms of Convergence and Laws of Large
Numbers 1070
D.2.3

Convergence of Functions 1073
D.2.4
Convergence to a Random Variable 1074
D.2.5
Convergence in Distribution: Limiting Distributions 1076
D.2.6
Central Limit Theorems 1078
D.2.7
The Delta Method 1083
D.3 Asymptotic Distributions
1084
D.3.1
Asymptotic Distribution of a Nonlinear Function 1086
D.3.2
Asymptotic Expectations 1087
D.4 Sequences and the Order of a Sequence
1088
Appendix E
Computation and Optimization
E.1
Introduction
1089
E.2
Computation in Econometrics
1090
E.2.1
Computing Integrals 1090

1089