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Econometric Analysis of Cross Section and Panel Data

Jeffrey M. Wooldridge

The MIT Press

Cambridge, Massachusetts

London, England

Contents

Preface xvii

Acknowledgments xxiii

I INTRODUCTION AND BACKGROUND 1

1 Introduction 3

1.1 Causal Relationships and Ceteris Paribus Analysis 3

1.2 The Stochastic Setting and Asymptotic Analysis 4

1.2.1 Data Structures 4

1.2.2 Asymptotic Analysis 7

1.3 Some Examples 7

1.4 Why Not Fixed Explanatory Variables? 9

2 Conditional Expectations and Related Concepts in Econometrics 13

2.1 The Role of Conditional Expectations in Econometrics 13

2.2 Features of Conditional Expectations 14

2.2.1 De.nition and Examples 14

2.2.2 Partial Effects, Elasticities, and Semielasticities 15

2.2.3 The Error Form of Models of Conditional Expectations 18

2.2.4 Some Properties of Conditional Expectations 19

2.2.5 Average Partial Effects 22

2.3 Linear Projections 24

Problems 27

Appendix 2A 29

2.A.1 Properties of Conditional Expectations 29

2.A.2 Properties of Conditional Variances 31

2.A.3 Properties of Linear Projections 32

3 Basic Asymptotic Theory 35

3.1 Convergence of Deterministic Sequences 35

3.2 Convergence in Probability and Bounded in Probability 36

3.3 Convergence in Distribution 38

3.4 Limit Theorems for Random Samples 39

3.5 Limiting Behavior of Estimators and Test Statistics 40

3.5.1 Asymptotic Properties of Estimators 40

3.5.2 Asymptotic Properties of Test Statistics 43

Problems 45

II LINEAR MODELS 47

4 The Single-Equation Linear Model and OLS Estimation 49

4.1 Overview of the Single-Equation Linear Model 49

4.2 Asymptotic Properties of OLS 51

4.2.1 Consistency 52

4.2.2 Asymptotic Inference Using OLS 54

4.2.3 Heteroskedasticity-Robust Inference 55

4.2.4 Lagrange Multiplier (Score) Tests 58

4.3 OLS Solutions to the Omitted Variables Problem 61

4.3.1 OLS Ignoring the Omitted Variables 61

4.3.2 The Proxy Variable–OLS Solution 63

4.3.3 Models with Interactions in Unobservables 67

4.4 Properties of OLS under Measurement Error 70

4.4.1 Measurement Error in the Dependent Variable 71

4.4.2 Measurement Error in an Explanatory Variable 73

Problems 76

5 Instrumental Variables Estimation of Single-Equation Linear Models 83

5.1 Instrumental Variables and Two-Stage Least Squares 83

5.1.1 Motivation for Instrumental Variables Estimation 83

5.1.2 Multiple Instruments: Two-Stage Least Squares 90

5.2 General Treatment of 2SLS 92

5.2.1 Consistency 92

5.2.2 Asymptotic Normality of 2SLS 94

5.2.3 Asymptotic E‰ciency of 2SLS 96

5.2.4 Hypothesis Testing with 2SLS 97

5.2.5 Heteroskedasticity-Robust Inference for 2SLS 100

5.2.6 Potential Pitfalls with 2SLS 101

5.3 IV Solutions to the Omitted Variables and Measurement Error

Problems 105

5.3.1 Leaving the Omitted Factors in the Error Term 105

5.3.2 Solutions Using Indicators of the Unobservables 105

Problems 107

6 Additional Single-Equation Topics 115

6.1 Estimation with Generated Regressors and Instruments 115

Contents vi

6.1.1 OLS with Generated Regressors 115

6.1.2 2SLS with Generated Instruments 116

6.1.3 Generated Instruments and Regressors 117

6.2 Some Speci.cation Tests 118

6.2.1 Testing for Endogeneity 118

6.2.2 Testing Overidentifying Restrictions 122

6.2.3 Testing Functional Form 124

6.2.4 Testing for Heteroskedasticity 125

6.3 Single-Equation Methods under Other Sampling Schemes 128

6.3.1 Pooled Cross Sections over Time 128

6.3.2 Geographically Strati.ed Samples 132

6.3.3 Spatial Dependence 134

6.3.4 Cluster Samples 134

Problems 135

Appendix 6A 139

7 Estimating Systems of Equations by OLS and GLS 143

7.1 Introduction 143

7.2 Some Examples 143

7.3 System OLS Estimation of a Multivariate Linear System 147

7.3.1 Preliminaries 147

7.3.2 Asymptotic Properties of System OLS 148

7.3.3 Testing Multiple Hypotheses 153

7.4 Consistency and Asymptotic Normality of Generalized Least

Squares 153

7.4.1 Consistency 153

7.4.2 Asymptotic Normality 156

7.5 Feasible GLS 157

7.5.1 Asymptotic Properties 157

7.5.2 Asymptotic Variance of FGLS under a Standard

Assumption 160

7.6 Testing Using FGLS 162

7.7 Seemingly Unrelated Regressions, Revisited 163

7.7.1 Comparison between OLS and FGLS for SUR Systems 164

7.7.2 Systems with Cross Equation Restrictions 167

7.7.3 Singular Variance Matrices in SUR Systems 167

Contents vii

7.8 The Linear Panel Data Model, Revisited 169

7.8.1 Assumptions for Pooled OLS 170

7.8.2 Dynamic Completeness 173

7.8.3 A Note on Time Series Persistence 175

7.8.4 Robust Asymptotic Variance Matrix 175

7.8.5 Testing for Serial Correlation and Heteroskedasticity after

Pooled OLS 176

7.8.6 Feasible GLS Estimation under Strict Exogeneity 178

Problems 179

8 System Estimation by Instrumental Variables 183

8.1 Introduction and Examples 183

8.2 A General Linear System of Equations 186

8.3 Generalized Method of Moments Estimation 188

8.3.1 A General Weighting Matrix 188

8.3.2 The System 2SLS Estimator 191

8.3.3 The Optimal Weighting Matrix 192

8.3.4 The Three-Stage Least Squares Estimator 194

8.3.5 Comparison between GMM 3SLS and Traditional 3SLS 196

8.4 Some Considerations When Choosing an Estimator 198

8.5 Testing Using GMM 199

8.5.1 Testing Classical Hypotheses 199

8.5.2 Testing Overidenti.cation Restrictions 201

8.6 More E‰cient Estimation and Optimal Instruments 202

Problems 205

9 Simultaneous Equations Models 209

9.1 The Scope of Simultaneous Equations Models 209

9.2 Identi.cation in a Linear System 211

9.2.1 Exclusion Restrictions and Reduced Forms 211

9.2.2 General Linear Restrictions and Structural Equations 215

9.2.3 Unidenti.ed, Just Identi.ed, and Overidenti.ed Equations 220

9.3 Estimation after Identi.cation 221

9.3.1 The Robustness-E‰ciency Trade-off 221

9.3.2 When Are 2SLS and 3SLS Equivalent? 224

9.3.3 Estimating the Reduced Form Parameters 224

9.4 Additional Topics in Linear SEMs 225

Contents viii

9.4.1 Using Cross Equation Restrictions to Achieve Identi.cation 225

9.4.2 Using Covariance Restrictions to Achieve Identi.cation 227

9.4.3 Subtleties Concerning Identi.cation and E‰ciency in Linear

Systems 229

9.5 SEMs Nonlinear in Endogenous Variables 230

9.5.1 Identi.cation 230

9.5.2 Estimation 235

9.6 Different Instruments for Different Equations 237

Problems 239

10 Basic Linear Unobserved Effects Panel Data Models 247

10.1 Motivation: The Omitted Variables Problem 247

10.2 Assumptions about the Unobserved Effects and Explanatory

Variables 251

10.2.1 Random or Fixed Effects? 251

10.2.2 Strict Exogeneity Assumptions on the Explanatory

Variables 252

10.2.3 Some Examples of Unobserved Effects Panel Data Models 254

10.3 Estimating Unobserved Effects Models by Pooled OLS 256

10.4 Random Effects Methods 257

10.4.1 Estimation and Inference under the Basic Random Effects

Assumptions 257

10.4.2 Robust Variance Matrix Estimator 262

10.4.3 A General FGLS Analysis 263

10.4.4 Testing for the Presence of an Unobserved Effect 264

10.5 Fixed Effects Methods 265

10.5.1 Consistency of the Fixed Effects Estimator 265

10.5.2 Asymptotic Inference with Fixed Effects 269

10.5.3 The Dummy Variable Regression 272

10.5.4 Serial Correlation and the Robust Variance Matrix

Estimator 274

10.5.5 Fixed Effects GLS 276

10.5.6 Using Fixed Effects Estimation for Policy Analysis 278

10.6 First Differencing Methods 279

10.6.1 Inference 279

10.6.2 Robust Variance Matrix 282

Contents ix

10.6.3 Testing for Serial Correlation 282

10.6.4 Policy Analysis Using First Differencing 283

10.7 Comparison of Estimators 284

10.7.1 Fixed Effects versus First Differencing 284

10.7.2 The Relationship between the Random Effects and Fixed

Effects Estimators 286

10.7.3 The Hausman Test Comparing the RE and FE Estimators 288

Problems 291

11 More Topics in Linear Unobserved Effects Models 299

11.1 Unobserved Effects Models without the Strict Exogeneity

Assumption 299

11.1.1 Models under Sequential Moment Restrictions 299

11.1.2 Models with Strictly and Sequentially Exogenous

Explanatory Variables 305

11.1.3 Models with Contemporaneous Correlation between Some

Explanatory Variables and the Idiosyncratic Error 307

11.1.4 Summary of Models without Strictly Exogenous

Explanatory Variables 314

11.2 Models with Individual-Speci.c Slopes 315

11.2.1 A Random Trend Model 315

11.2.2 General Models with Individual-Speci.c Slopes 317

11.3 GMM Approaches to Linear Unobserved Effects Models 322

11.3.1 Equivalence between 3SLS and Standard Panel Data

Estimators 322

11.3.2 Chamberlain’s Approach to Unobserved Effects Models 323

11.4 Hausman and Taylor-Type Models 325

11.5 Applying Panel Data Methods to Matched Pairs and Cluster

Samples 328

Problems 332

III GENERAL APPROACHES TO NONLINEAR ESTIMATION 339

12 M-Estimation 341

12.1 Introduction 341

12.2 Identi.cation, Uniform Convergence, and Consistency 345

12.3 Asymptotic Normality 349

Contents x

12.4 Two-Step M-Estimators 353

12.4.1 Consistency 353

12.4.2 Asymptotic Normality 354

12.5 Estimating the Asymptotic Variance 356

12.5.1 Estimation without Nuisance Parameters 356

12.5.2 Adjustments for Two-Step Estimation 361

12.6 Hypothesis Testing 362

12.6.1 Wald Tests 362

12.6.2 Score (or Lagrange Multiplier) Tests 363

12.6.3 Tests Based on the Change in the Objective Function 369

12.6.4 Behavior of the Statistics under Alternatives 371

12.7 Optimization Methods 372

12.7.1 The Newton-Raphson Method 372

12.7.2 The Berndt, Hall, Hall, and Hausman Algorithm 374

12.7.3 The Generalized Gauss-Newton Method 375

12.7.4 Concentrating Parameters out of the Objective Function 376

12.8 Simulation and Resampling Methods 377

12.8.1 Monte Carlo Simulation 377

12.8.2 Bootstrapping 378

Problems 380

13 Maximum Likelihood Methods 385

13.1 Introduction 385

13.2 Preliminaries and Examples 386

13.3 General Framework for Conditional MLE 389

13.4 Consistency of Conditional MLE 391

13.5 Asymptotic Normality and Asymptotic Variance Estimation 392

13.5.1 Asymptotic Normality 392

13.5.2 Estimating the Asymptotic Variance 395

13.6 Hypothesis Testing 397

13.7 Speci.cation Testing 398

13.8 Partial Likelihood Methods for Panel Data and Cluster Samples 401

13.8.1 Setup for Panel Data 401

13.8.2 Asymptotic Inference 405

13.8.3 Inference with Dynamically Complete Models 408

13.8.4 Inference under Cluster Sampling 409

Contents xi

13.9 Panel Data Models with Unobserved Effects 410

13.9.1 Models with Strictly Exogenous Explanatory Variables 410

13.9.2 Models with Lagged Dependent Variables 412

13.10 Two-Step MLE 413

Problems 414

Appendix 13A 418

14 Generalized Method of Moments and Minimum Distance Estimation 421

14.1 Asymptotic Properties of GMM 421

14.2 Estimation under Orthogonality Conditions 426

14.3 Systems of Nonlinear Equations 428

14.4 Panel Data Applications 434

14.5 E‰cient Estimation 436

14.5.1 A General E‰ciency Framework 436

14.5.2 E‰ciency of MLE 438

14.5.3 E‰cient Choice of Instruments under Conditional Moment

Restrictions 439

14.6 Classical Minimum Distance Estimation 442

Problems 446

Appendix 14A 448

IV NONLINEAR MODELS AND RELATED TOPICS 451

15 Discrete Response Models 453

15.1 Introduction 453

15.2 The Linear Probability Model for Binary Response 454

15.3 Index Models for Binary Response: Probit and Logit 457

15.4 Maximum Likelihood Estimation of Binary Response Index

Models 460

15.5 Testing in Binary Response Index Models 461

15.5.1 Testing Multiple Exclusion Restrictions 461

15.5.2 Testing Nonlinear Hypotheses about b 463

15.5.3 Tests against More General Alternatives 463

15.6 Reporting the Results for Probit and Logit 465

15.7 Speci.cation Issues in Binary Response Models 470

15.7.1 Neglected Heterogeneity 470

15.7.2 Continuous Endogenous Explanatory Variables 472

Contents xii

15.7.3 A Binary Endogenous Explanatory Variable 477

15.7.4 Heteroskedasticity and Nonnormality in the Latent

Variable Model 479

15.7.5 Estimation under Weaker Assumptions 480

15.8 Binary Response Models for Panel Data and Cluster Samples 482

15.8.1 Pooled Probit and Logit 482

15.8.2 Unobserved Effects Probit Models under Strict Exogeneity 483

15.8.3 Unobserved Effects Logit Models under Strict Exogeneity 490

15.8.4 Dynamic Unobserved Effects Models 493

15.8.5 Semiparametric Approaches 495

15.8.6 Cluster Samples 496

15.9 Multinomial Response Models 497

15.9.1 Multinomial Logit 497

15.9.2 Probabilistic Choice Models 500

15.10 Ordered Response Models 504

15.10.1 Ordered Logit and Ordered Probit 504

15.10.2 Applying Ordered Probit to Interval-Coded Data 508

Problems 509

16 Corner Solution Outcomes and Censored Regression Models 517

16.1 Introduction and Motivation 517

16.2 Derivations of Expected Values 521

16.3 Inconsistency of OLS 524

16.4 Estimation and Inference with Censored Tobit 525

16.5 Reporting the Results 527

16.6 Speci.cation Issues in Tobit Models 529

16.6.1 Neglected Heterogeneity 529

16.6.2 Endogenous Explanatory Variables 530

16.6.3 Heteroskedasticity and Nonnormality in the Latent

Variable Model 533

16.6.4 Estimation under Conditional Median Restrictions 535

16.7 Some Alternatives to Censored Tobit for Corner Solution

Outcomes 536

16.8 Applying Censored Regression to Panel Data and Cluster Samples 538

16.8.1 Pooled Tobit 538

16.8.2 Unobserved Effects Tobit Models under Strict Exogeneity 540

Contents xiii

16.8.3 Dynamic Unobserved Effects Tobit Models 542

Problems 544

17 Sample Selection, Attrition, and Strati.ed Sampling 551

17.1 Introduction 551

17.2 When Can Sample Selection Be Ignored? 552

17.2.1 Linear Models: OLS and 2SLS 552

17.2.2 Nonlinear Models 556

17.3 Selection on the Basis of the Response Variable: Truncated

Regression 558

17.4 A Probit Selection Equation 560

17.4.1 Exogenous Explanatory Variables 560

17.4.2 Endogenous Explanatory Variables 567

17.4.3 Binary Response Model with Sample Selection 570

17.5 A Tobit Selection Equation 571

17.5.1 Exogenous Explanatory Variables 571

17.5.2 Endogenous Explanatory Variables 573

17.6 Estimating Structural Tobit Equations with Sample Selection 575

17.7 Sample Selection and Attrition in Linear Panel Data Models 577

17.7.1 Fixed Effects Estimation with Unbalanced Panels 578

17.7.2 Testing and Correcting for Sample Selection Bias 581

17.7.3 Attrition 585

17.8 Strati.ed Sampling 590

17.8.1 Standard Strati.ed Sampling and Variable Probability

Sampling 590

17.8.2 Weighted Estimators to Account for Strati.cation 592

17.8.3 Strati.cation Based on Exogenous Variables 596

Problems 598

18 Estimating Average Treatment Effects 603

18.1 Introduction 603

18.2 A Counterfactual Setting and the Self-Selection Problem 603

18.3 Methods Assuming Ignorability of Treatment 607

18.3.1 Regression Methods 608

18.3.2 Methods Based on the Propensity Score 614

18.4 Instrumental Variables Methods 621

18.4.1 Estimating the ATE Using IV 621

Contents xiv

18.4.2 Estimating the Local Average Treatment Effect by IV 633

18.5 Further Issues 636

18.5.1 Special Considerations for Binary and Corner Solution

Responses 636

18.5.2 Panel Data 637

18.5.3 Nonbinary Treatments 638

18.5.4 Multiple Treatments 642

Problems 642

19 Count Data and Related Models 645

19.1 Why Count Data Models? 645

19.2 Poisson Regression Models with Cross Section Data 646

19.2.1 Assumptions Used for Poisson Regression 646

19.2.2 Consistency of the Poisson QMLE 648

19.2.3 Asymptotic Normality of the Poisson QMLE 649

19.2.4 Hypothesis Testing 653

19.2.5 Speci.cation Testing 654

19.3 Other Count Data Regression Models 657

19.3.1 Negative Binomial Regression Models 657

19.3.2 Binomial Regression Models 659

19.4 Other QMLEs in the Linear Exponential Family 660

19.4.1 Exponential Regression Models 661

19.4.2 Fractional Logit Regression 661

19.5 Endogeneity and Sample Selection with an Exponential Regression

Function 663

19.5.1 Endogeneity 663

19.5.2 Sample Selection 666

19.6 Panel Data Methods 668

19.6.1 Pooled QMLE 668

19.6.2 Specifying Models of Conditional Expectations with

Unobserved Effects 670

19.6.3 Random Effects Methods 671

19.6.4 Fixed Effects Poisson Estimation 674

19.6.5 Relaxing the Strict Exogeneity Assumption 676

Problems 678

Contents xv

20 Duration Analysis 685

20.1 Introduction 685

20.2 Hazard Functions 686

20.2.1 Hazard Functions without Covariates 686

20.2.2 Hazard Functions Conditional on Time-Invariant

Covariates 690

20.2.3 Hazard Functions Conditional on Time-Varying

Covariates 691

20.3 Analysis of Single-Spell Data with Time-Invariant Covariates 693

20.3.1 Flow Sampling 694

20.3.2 Maximum Likelihood Estimation with Censored Flow

Data 695

20.3.3 Stock Sampling 700

20.3.4 Unobserved Heterogeneity 703

20.4 Analysis of Grouped Duration Data 706

20.4.1 Time-Invariant Covariates 707

20.4.2 Time-Varying Covariates 711

20.4.3 Unobserved Heterogeneity 713

20.5 Further Issues 714

20.5.1 Cox’s Partial Likelihood Method for the Proportional

Hazard Model 714

20.5.2 Multiple-Spell Data 714

20.5.3 Competing Risks Models 715

Problems 715

References 721

Index 737

Contents xvi

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