# Econometric Analysis of Cross Section and Panel Data Second Edition. Jeffrey M. Wooldridge. The MIT Press Cambridge, Massachusetts London, England

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1 Econometric Analysis of Cross Section and Panel Data Second Edition Jeffrey M. Wooldridge The MIT Press Cambridge, Massachusetts London, England

2 Preface Acknowledgments xxi xxix I INTRODUCTION AND BACKGROUND 1 1 Introduction Causal Relationships and Ceteris Paribus Analysis Stochastic Setting and Asymptotic Analysis Data Structures Asymptotic Analysis Some Examples Why Not Fixed Explanatory Variables? 9 2 Conditional Expectations and Related Concepts in Econometrics Role of Conditional Expectations in Econometrics Features of Conditional Expectations Definition and Examples Partial Effects, Elasticities, and Semielasticities Error Form of Models of Conditional Expectations Some Properties of Conditional Expectations Average Partial Effects Linear Projections 25 Problems 27 Appendix 2A 30 2.A.I Properties of Conditional Expectations 30 2.A.2 Properties of Conditional Variances and Covariances 32 2.A.3 Properties of Linear Projections 34 3 Basic Asymptotic Theory Convergence of Deterministic Sequences Convergence in Probability and Boundedness in Probability Convergence in Distribution Limit Theorems for Random Samples Limiting Behavior of Estimators and Test Statistics Asymptotic Properties of Estimators Asymptotic Properties of Test Statistics 45 Problems ' 47

3 II LINEAR MODELS 51 4 Single-Equation Linear Model and Ordinary Least Squares Estimation Overview of the Single-Equation Linear Model Asymptotic Properties of Ordinary Least Squares Consistency Asymptotic Inference Using Ordinary Least Squares Heteroskedasticity-Robust Inference Lagrange Multiplier (Score) Tests Ordinary Least Squares Solutions to the Omitted Variables Problem Ordinary Least Squares Ignoring the Omitted Variables Proxy Variable-Ordinary Least Squares Solution Models with Interactions in Unobservables: Random Coefficient Models Properties of Ordinary Least Squares under Measurement Error Measurement Error in the Dependent Variable Measurement Error in an Explanatory Variable 78 Problems 82 5 Instrumental Variables Estimation of Single-Equation Linear Models Instrumental Variables and Two-Stage Least Squares Motivation for Instrumental Variables Estimation Multiple Instruments: Two-Stage Least Squares General Treatment of Two-Stage Least Squares Consistency Asymptotic Normality of Two-Stage Least Squares Asymptotic Efficiency of Two-Stage Least Squares Hypothesis Testing with Two-Stage Least Squares Heteroskedasticity-Robust Inference for Two-Stage Least Squares Potential Pitfalls with Two-Stage Least Squares IV Solutions to the Omitted Variables and Measurement Error Problems Leaving the Omitted Factors in the Error Term Solutions Using Indicators of the Unobservables 112 Problems 115

4 vii 6 Additional Single-Equation Topics Estimation with Generated Regressors and Instruments Ordinary Least Squares with Generated Regressors Two-Stage Least Squares with Generated Instruments Generated Instruments and Regressors Control Function Approach to Endogeneity Some Specification Tests Testing for Endogeneity Testing Overidentifying Restrictions Testing Functional Form Testing for Heteroskedasticity Correlated Random Coefficient Models When Is the Usual IV Estimator Consistent? Control Function Approach Pooled Cross Sections and Difference-in-Differences Estimation Pooled Cross Sections over Time Policy Analysis and Difference-in-Differences Estimation 147 Problems 152 Appendix 6A Estimating Systems of Equations by Ordinary Least Squares and Generalized Least Squares Introduction Some Examples System Ordinary Least Squares Estimation of a Multivariate Linear System Preliminaries Asymptotic Properties of System Ordinary Least Squares Testing Multiple Hypotheses Consistency and Asymptotic Normality of Generalized Least Squares Consistency Asymptotic Normality Feasible Generalized Least Squares Asymptotic Properties Asymptotic Variance of Feasible Generalized Least Squares under a Standard Assumption 180

5 viii Contents Properties of Feasible Generalized Least Squares with (Possibly Incorrect) Restrictions on the Unconditional Variance Matrix Testing the Use of Feasible Generalized Least Squares Seemingly Unrelated Regressions, Revisited Comparison between Ordinary Least Squares and Feasible Generalized Least Squares for Seemingly Unrelated Regressions Systems Systems with Cross Equation Restrictions Singular Variance Matrices in Seemingly Unrelated Regressions Systems Linear Panel Data Model, Revisited Assumptions for Pooled Ordinary Least Squares Dynamic Completeness Note on Time Series Persistence Robust Asymptotic Variance Matrix Testing for Serial Correlation and Heteroskedasticity after Pooled Ordinary Least Squares Feasible Generalized Least Squares Estimation under Strict Exogeneity 200 Problems System Estimation by Instrumental Variables Introduction and Examples General Linear System of Equations Generalized Method of Moments Estimation General Weighting Matrix System Two-Stage Least Squares Estimator Optimal Weighting Matrix The Generalized Method of Moments Three-Stage Least Squares Estimator Generalized Instrumental Variables Estimator Derivation of the Generalized Instrumental Variables Estimator and Its Asymptotic Properties Comparison of Generalized Method of Moment, Generalized Instrumental Variables, and the Traditional Three-Stage Least Squares Estimator 224

6 8.5 Testing Using Generalized Method of Moments Testing Classical Hypotheses Testing Overidentification Restrictions More Efficient Estimation and Optimal Instruments Summary Comments on Choosing an Estimator 232 Problems Simultaneous Equations Models Scope of Simultaneous Equations Models Identification in a Linear System Exclusion Restrictions and Reduced Forms General Linear Restrictions and Structural Equations Unidentified, Just Identified, and Overidentified Equations Estimation after Identification Robustness-Efficiency Trade-off When Are 2SLS and 3SLS Equivalent? Estimating the Reduced Form Parameters Additional Topics in Linear Simultaneous Equations Methods Using Cross Equation Restrictions to Achieve Identification Using Covariance Restrictions to Achieve Identification Subtleties Concerning Identification and Efficiency in Linear Systems Simultaneous Equations Models Nonlinear in Endogenous Variables Identification Estimation Control Function Estimation for Triangular Systems Different Instruments for Different Equations 271 Problems Basic Linear Unobserved Effects Panel Data Models Motivation: Omitted Variables Problem Assumptions about the Unobserved Effects and Explanatory Variables Random or Fixed Effects? Strict Exogeneity Assumptions on the Explanatory Variables Some Examples of Unobserved Effects Panel Data Models 289

7 10.3 Estimating Unobserved Effects Models by Pooled Ordinary Least Squares Random Effects Methods Estimation and Inference under the Basic Random Effects Assumptions Robust Variance Matrix Estimator General Feasible Generalized Least Squares Analysis Testing for the Presence of an Unobserved Effect Fixed Effects Methods Consistency of the Fixed Effects Estimator Asymptotic Inference with Fixed Effects Dummy Variable Regression Serial Correlation and the Robust Variance Matrix Estimator Fixed Effects Generalized Least Squares Using Fixed Effects Estimation for Policy Analysis First Differencing Methods Inference Robust Variance Matrix Testing for Serial Correlation Policy Analysis Using First Differencing Comparison of Estimators Fixed Effects versus First Differencing Relationship between the Random Effects and Fixed Effects Estimators Hausman Test Comparing Random Effects and Fixed Effects Estimators 328 Problems More Topics in Linear Unobserved Effects Models Generalized Method of Moments Approaches to the Standard Linear Unobserved Effects Model Equivalance between GMM 3SLS and Standard Estimators Chamberlain's Approach to Unobserved Effects Models Random and Fixed Effects Instrumental Variables Methods Hausman and Taylor-Type Models First Differencing Instrumental Variables Methods 361

8 xi 11.5 Unobserved Effects Models with Measurement Error Estimation under Sequential Exogeneity General Framework Models with Lagged Dependent Variables Models with Individual-Specific Slopes Random Trend Model General Models with Individual-Specific Slopes Robustness of Standard Fixed Effects Methods Testing for Correlated Random Slopes 384 Problems 387 III GENERAL APPROACHES TO NONLINEAR ESTIMATION M-Estimation, Nonlinear Regression, and Quantile Regression Introduction Identification, Uniform Convergence, and Consistency Asymptotic Normality Two-Step M-Estimators Consistency Asymptotic Normality Estimating the Asymptotic Variance Estimation without Nuisance Parameters Adjustments for Two-Step Estimation Hypothesis Testing Wald Tests Score (or Lagrange Multiplier) Tests Tests Based on the Change in the Objective Function Behavior of the Statistics under Alternatives Optimization Methods Newton-Raphson Method Berndt, Hall, Hall, and Hausman Algorithm Generalized Gauss-Newton Method Concentrating Parameters out of the Objective Function Simulation and Resampling Methods Monte Carlo Simulation Bootstrapping 438

9 12.9 Multivariate Nonlinear Regression Methods Multivariate Nonlinear Least Squares Weighted Multivariate Nonlinear Least Squares Quantile Estimation Quantiles, the Estimation Problem, and Consistency Asymptotic Inference Quantile Regression for Panel Data 459 Problems Maximum Likelihood Methods Introduction Preliminaries and Examples General Framework for Conditional Maximum Likelihood Estimation Consistency of Conditional Maximum Likelihood Estimation Asymptotic Normality and Asymptotic Variance Estimation Asymptotic Normality Estimating the Asymptotic Variance Hypothesis Testing Specification Testing Partial (or Pooled) Likelihood Methods for Panel Data Setup for Panel Data Asymptotic Inference Inference with Dynamically Complete Models Panel Data Models with Unobserved Effects Models with Strictly Exogenous Explanatory Variables Models with Lagged Dependent Variables Two-Step Estimators Involving Maximum Likelihood Second-Step Estimator Is Maximum Likelihood Estimator Surprising Efficiency Result When the First-Step Estimator Is Conditional Maximum Likelihood Estimator Quasi-Maximum Likelihood Estimation General Misspecification Model Selection Tests Quasi-Maximum Likelihood Estimation in the Linear Exponential Family 509

10 Generalized Estimating Equations for Panel Data 514 Problems 517 Appendix 13A Generalized Method of Moments and Minimum Distance Estimation Asymptotic Properties of Generalized Method of Moments Estimation under Orthogonality Conditions Systems of Nonlinear Equations Efficient Estimation General Efficiency Framework Efficiency of Maximum Likelihood Estimator Efficient Choice of Instruments under Conditional Moment Restrictions Classical Minimum Distance Estimation Panel Data Applications Nonlinear Dynamic Models Minimum Distance Approach to the Unobserved Effects Model Models with Time-Varying Coefficients on the Unobserved Effects 551 Problems 555 Appendix 14A 558 IV NONLINEAR MODELS AND RELATED TOPICS Binary Response Models Introduction Linear Probability Model for Binary Response Index Models for Binary Response: Probit and Logit Maximum Likelihood Estimation of Binary Response Index Models Testing in Binary Response Index Models Testing Multiple Exclusion Restrictions Testing Nonlinear Hypotheses about fi Tests against More General Alternatives 571

11 15.6 Reporting the Results for Probit and Logit Specification Issues in Binary Response Models Neglected Heterogeneity Continuous Endogenous Explanatory Variables Binary Endogenous Explanatory Variable Heteroskedasticity and Nonnormality in the Latent Variable Model Estimation under Weaker Assumptions Binary Response Models for Panel Data Pooled Probit and Logit Unobserved Effects Probit Models under Strict Exogeneity Unobserved Effects Logit Models under Strict Exogeneity Dynamic Unobserved Effects Models Probit Models with Heterogeneity and Endogenous Explanatory Variables Semiparametric Approaches 632 Problems Multinomial and Ordered Response Models Introduction Multinomial Response Models Multinomial Logit Probabilistic Choice Models Endogenous Explanatory Variables Panel Data Methods Ordered Response Models Ordered Logit and Ordered Probit Specification Issues in Ordered Models Endogenous Explanatory Variables Panel Data Methods 662 Problems Corner Solution Responses Motivation and Examples Useful Expressions for Type I Tobit 671

12 xv 17.3 Estimation and Inference with the Type I Tobit Model Reporting the Results Specification Issues in Tobit Models Neglected Heterogeneity Endogenous Explanatory Models Heteroskedasticity and Nonnormality in the Latent Variable Model Estimating Parameters with Weaker Assumptions Two-Part Models and Type II Tobit for Corner Solutions Truncated Normal Hurdle Model Lognormal Hurdle Model and Exponential Conditional Mean Exponential Type II Tobit Model Two-Limit Tobit Model Panel Data Methods Pooled Methods Unobserved Effects Models under Strict Exogeneity Dynamic Unobserved Effects Tobit Models 713 Problems Count, Fractional, and Other Nonnegative Responses Introduction Poisson Regression Assumptions Used for Poisson Regression and Quantities of Interest Consistency of the Poisson QMLE Asymptotic Normality of the Poisson QMLE Hypothesis Testing Specification Testing Other Count Data Regression Models Negative Binomial Regression Models Binomial Regression Models Gamma (Exponential) Regression Model Endogeneity with an Exponential Regression Function Fractional Responses 748

13 xvi Contents Exogenous Explanatory Variables Endogenous Explanatory Variables Panel Data Methods Pooled QMLE Specifying Models of Conditional Expectations with Unobserved Effects Random Effects Methods Fixed Effects Poisson Estimation Relaxing the Strict Exogeneity Assumption Fractional Response Models for Panel Data 766 Problems Censored Data, Sample Selection, and Attrition Introduction Data Censoring Binary Censoring Interval Coding Censoring from Above and Below Overview of Sample Selection When Can Sample Selection Be Ignored? Linear Models: Estimation by OLS and 2SLS Nonlinear Models Selection on the Basis of the Response Variable: Truncated Regression Incidental Truncation: A Probit Selection Equation Exogenous Explanatory Variables Endogenous Explanatory Variables Binary Response Model with Sample Selection An Exponential Response Function Incidental Truncation: A Tobit Selection Equation Exogenous Explanatory Variables Endogenous Explanatory Variables Estimating Structural Tobit Equations with Sample Selection Inverse Probability Weighting for Missing Data 821

14 xvii 19.9 Sample Selection and Attrition in Linear Panel Data Models Fixed and Random Effects Estimation with Unbalanced Panels Testing and Correcting for Sample Selection Bias Attrition 837 Problems Stratified Sampling and Cluster Sampling Introduction Stratified Sampling Standard Stratified Sampling and Variable Probability Sampling Weighted Estimators to Account for Stratification Stratification Based on Exogenous Variables Cluster Sampling Inference with a Large Number of Clusters and Small Cluster Sizes Cluster Samples with Unit-Specific Panel Data Should We Apply Cluster-Robust Inference with Large Group Sizes? Inference When the Number of Clusters Is Small Complex Survey Sampling 894 Problems Estimating Average Treatment Effects Introduction A Counterfactual Setting and the Self-Selection Problem Methods Assuming Ignorability (or Unconfoundedness) of Treatment Identification Regression Adjustment Propensity Score Methods Combining Regression Adjustment and Propensity Score Weighting Matching Methods 934

15 21.4 Instrumental Variables Methods Estimating the Average Treatment Effect Using IV Correction and Control Function Approaches Estimating the Local Average Treatment Effect by IV Regression Discontinuity Designs The Sharp Regression Discontinuity Design The Fuzzy Regression Discontinuity Design Unconfoundedness versus the Fuzzy Regression Discontinuity Further Issues Special Considerations for Responses with Discreteness or Limited Range Multivalued Treatments Multiple Treatments Panel Data 968 Problems Duration Analysis Introduction Hazard Functions Hazard Functions without Covariates Hazard Functions Conditional on Time-Invariant Covariates Hazard Functions Conditional on Time-Varying Covariates Analysis of Single-Spell Data with Time-Invariant Covariates Flow Sampling Maximum Likelihood Estimation with Censored Flow Data Stock Sampling Unobserved Heterogeneity Analysis of Grouped Duration Data Time-Invariant Covariates Time-Varying Covariates Unobserved Heterogeneity 1017

16 xix 22.5 Further Issues Cox's Partial Likelihood Method for the Proportional Hazard Model Multiple-Spell Data Competing Risks Models Problems References Index

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