Econometric Analysis of Cross Section and Panel Data

Transcription

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

2 Contents Preface Acknowledgments xvii xxiii I INTRODUCTION AND BACKGROUND 1 1 Introduction Causal Relationships and Ceteris Paribus Analysis The 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 The Role of Conditional Expectations in Econometrics Features of Conditional Expectations Definition and Examples Partial Effects, Elasticities, and Semielasticities The Error Form of Models of Conditional Expectations Some Properties of Conditional Expectations Average Partial Effects 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 Convergence of Deterministic Sequences Convergence in Probability and Bounded 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 43 Problems 45

3 II LINEAR MODELS 47 4 The Single-Equation Linear Model and OLS Estimation Overview of the Single-Equation Linear Model Asymptotic Properties of OLS Consistency Asymptotic Inference Using OLS Heteroskedasticity-Robust Inference Lagrange Multiplier (Score) Tests OLS Solutions to the Omitted Variables Problem OLS Ignoring the Omitted Variables The Proxy Variable-OLS Solution Models with Interactions in Unobservables Properties of OLS under Measurement Error Measurement Error in the Dependent Variable Measurement Error in an Explanatory Variable 73 Problems 76 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 2SLS Consistency Asymptotic Normality of 2SLS Asymptotic Efficiency of 2SLS Hypothesis Testing with 2SLS Heteroskedasticity-Robust Inference for 2SLS Potential Pitfalls with 2SLS IV Solutions to the Omitted Variables and Measurement Error Problems Leaving the Omitted Factors in the Error Term Solutions Using Indicators of the Unobservables 105 Problems Additional Single-Equation Topics Estimation with Generated Regressors and Instruments 115

4 Contents Vll OLS with Generated Regressors SLS with Generated Instruments Generated Instruments and Regressors Some Specification Tests Testing for Endogeneity Testing Overidentifying Restrictions Testing Functional Form Testing for Heteroskedasticity Single-Equation Methods under Other Sampling Schemes Pooled Cross Sections over Time Geographically Stratified Samples Spatial Dependence Cluster Samples 134 Problems 135 Appendix 6A Estimating Systems of Equations by OLS and GLS Introduction Some Examples System OLS Estimation of a Multivariate Linear System Preliminaries Asymptotic Properties of System OLS Testing Multiple Hypotheses Consistency and Asymptotic Normality of Generalized Least Squares Consistency Asymptotic Normality Feasible GLS Asymptotic Properties Asymptotic Variance of FGLS under a Standard Assumption Testing Using FGLS Seemingly Unrelated Regressions, Revisited Comparison between OLS and FGLS for SUR Systems Systems with Cross Equation Restrictions Singular Variance Matrices in SUR Systems 167

5 7.8 The Linear Panel Data Model, Revisited Assumptions for Pooled OLS Dynamic Completeness A Note on Time Series Persistence Robust Asymptotic Variance Matrix Testing for Serial Correlation and Heteroskedasticity after Pooled OLS Feasible GLS Estimation under Strict Exogeneity 178 Problems System Estimation by Instrumental Variables Introduction and Examples A General Linear System of Equations Generalized Method of Moments Estimation A General Weighting Matrix The System 2SLS Estimator The Optimal Weighting Matrix The Three-Stage Least Squares Estimator Comparison between GMM 3SLS and Traditional 3SLS Some Considerations When Choosing an Estimator Testing Using GMM Testing Classical Hypotheses Testing Overidentification Restrictions More Efficient Estimation and Optimal Instruments 202 Problems Simultaneous Equations Models The 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 The Robustness-Efficiency Trade-off When Are 2SLS and 3SLS Equivalent? Estimating the Reduced Form Parameters Additional Topics in Linear SEMs 225

6 Contents ix Using Cross Equation Restrictions to Achieve Identification Using Covariance Restrictions to Achieve Identification Subtleties Concerning Identification and Efficiency in Linear Systems SEMs Nonlinear in Endogenous Variables Identification Estimation Different Instruments for Different Equations 237 Problems Basic Linear Unobserved Effects Panel Data Models Motivation: The 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 Estimating Unobserved Effects Models by Pooled OLS Random Effects Methods Estimation and Inference under the Basic Random Effects Assumptions Robust Variance Matrix Estimator A General FGLS Analysis Testing for the Presence of an Unobserved Effect Fixed Effects Methods Consistency of the Fixed Effects Estimator Asymptotic Inference with Fixed Effects The Dummy Variable Regression Serial Correlation and the Robust Variance Matrix Estimator Fixed Effects GLS Using Fixed Effects Estimation for Policy Analysis First Differencing Methods Inference Robust Variance Matrix 282

7 X Contents Testing for Serial Correlation Policy Analysis Using First Differencing Comparison of Estimators Fixed Effects versus First Differencing The Relationship between the Random Effects and Fixed Effects Estimators The Hausman Test Comparing the RE and FE Estimators 288 Problems More Topics in Linear Unobserved Effects Models Unobserved Effects Models without the Strict Exogeneity Assumption Models under Sequential Moment Restrictions Models with Strictly and Sequentially Exogenous Explanatory Variables Models with Contemporaneous Correlation between Some Explanatory Variables and the Idiosyncratic Error Summary of Models without Strictly Exogenous Explanatory Variables Models with Individual-Specific Slopes A Random Trend Model General Models with Individual-Specific Slopes GMM Approaches to Linear Unobserved Effects Models Equivalence between 3SLS and Standard Panel Data Estimators Chamberlain's Approach to Unobserved Effects Models Hausman and Taylor-Type Models Applying Panel Data Methods to Matched Pairs and Cluster Samples 328 Problems 332 Ш GENERAL APPROACHES TO NONLINEAR ESTIMATION M-Estimation Introduction Identification, Uniform Convergence, and Consistency Asymptotic Normality 349

8 Contents XI 12.4 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 The Newton-Raphson Method The Berndt, Hall, Hall, and Hausman Algorithm The Generalized Gauss-Newton Method Concentrating Parameters out of the Objective Function Simulation and Resampling Methods Monte Carlo Simulation Bootstrapping 378 Problems Maximum Likelihood Methods Introduction Preliminaries and Examples General Framework for Conditional MLE Consistency of Conditional MLE Asymptotic Normality and Asymptotic Variance Estimation Asymptotic Normality Estimating the Asymptotic Variance Hypothesis Testing Specification Testing Partial Likelihood Methods for Panel Data and Cluster Samples Setup for Panel Data Asymptotic Inference Inference with Dynamically Complete Models Inference under Cluster Sampling 409

9 Xll Contents 13.9 Panel Data Models with Unobserved Effects Models with Strictly Exogenous Explanatory Variables Models with Lagged Dependent Variables Two-Step MLE 413 Problems 414 Appendix 13A Generalized Method of Moments and Minimum Distance Estimation Asymptotic Properties of GMM Estimation under Orthogonality Conditions Systems of Nonlinear Equations Panel Data Applications Efficient Estimation A General Efficiency Framework Efficiency of MLE Efficient Choice of Instruments under Conditional Moment Restrictions Classical Minimum Distance Estimation 442 Problems 446 Appendix 14A 448 IV NONLINEAR MODELS AND RELATED TOPICS Discrete Response Models Introduction The 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 ß Tests against More General Alternatives Reporting the Results for Probit and Logit Specification Issues in Binary Response Models Neglected Heterogeneity Continuous Endogenous Explanatory Variables 472

10 Contents хш A Binary Endogenous Explanatory Variable Heteroskedasticity and Nonnormality in the Latent Variable Model Estimation under Weaker Assumptions Binary Response Models for Panel Data and Cluster Samples Pooled Probit and Logit Unobserved Effects Probit Models under Strict Exogeneity Unobserved Effects Logit Models under Strict Exogeneity Dynamic Unobserved Effects Models Semiparametric Approaches Cluster Samples Multinomial Response Models Multinomial Logit Probabilistic Choice Models Ordered Response Models Ordered Logit and Ordered Probit Applying Ordered Probit to Interval-Coded Data 508 Problems Corner Solution Outcomes and Censored Regression Models Introduction and Motivation Derivations of Expected Values Inconsistency of OLS Estimation and Inference with Censored Tobit Reporting the Results Specification Issues in Tobit Models Neglected Heterogeneity Endogenous Explanatory Variables Heteroskedasticity and Nonnormality in the Latent Variable Model Estimation under Conditional Median Restrictions Some Alternatives to Censored Tobit for Corner Solution Outcomes Applying Censored Regression to Panel Data and Cluster Samples Pooled Tobit Unobserved Effects Tobit Models under Strict Exogeneity 540

11 XIV Contents Dynamic Unobserved Effects Tobit Models 542 Problems Sample Selection, Attrition, and Stratified Sampling Introduction When Can Sample Selection Be Ignored? Linear Models: OLS and 2SLS Nonlinear Models Selection on the Basis of the Response Variable: Truncated Regression A Probit Selection Equation Exogenous Explanatory Variables Endogenous Explanatory Variables Binary Response Model with Sample Selection A Tobit Selection Equation Exogenous Explanatory Variables Endogenous Explanatory Variables Estimating Structural Tobit Equations with Sample Selection Sample Selection and Attrition in Linear Panel Data Models Fixed Effects Estimation with Unbalanced Panels Testing and Correcting for Sample Selection Bias Attrition Stratified Sampling Standard Stratified Sampling and Variable Probability Sampling Weighted Estimators to Account for Stratification Stratification Based on Exogenous Variables 596 Problems Estimating Average Treatment Effects Introduction A Counterfactual Setting and the Self-Selection Problem Methods Assuming Ignorability of Treatment Regression Methods Methods Based on the Propensity Score Instrumental Variables Methods Estimating the ATE Using IV 621

12 Contents xv Estimating the Local Average Treatment Effect by IV Further Issues Special Considerations for Binary and Comer Solution Responses Panel Data Nonbinary Treatments Multiple Treatments 642 Problems Count Data and Related Models Why Count Data Models? Poisson Regression Models with Cross Section Data Assumptions Used for Poisson Regression 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 Other QMLEs in the Linear Exponential Family Exponential Regression Models Fractional Logit Regression Endogeneity and Sample Selection with an Exponential Regression Function Endogeneity Sample Selection 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 676 Problems 678

13 XVI Contents 20 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 Further Issues Cox's Partial Likelihood Method for the Proportional Hazard Model Multiple-Spell Data Competing Risks Models 715 Problems 715 References 721 Index 737