Preface

Acknowledgments

I INTRODUCTION AND BACKGROUND

1 Introduction

1.1 Causal Relationships and Ceteris Paribus Analysis

1.2 Stochastic Setting and Asymptotic Analysis

1.2.1 Data Structures

1.2.2 Asymptotic Analysis

1.3 Some Examples

1.4 Why Not Fixed Explanatory Variables?

2 Conditional Expectations and Related Concepts in Econometrics

2.1 Role of Conditional Expectations in Econometrics

2.2 Features of Conditional Expectations

2.2.1 Definition and Examples

2.2.2 Partial Effects, Elasticities, and Semielasticities

2.2.3 Error Form of Models of Conditional Expectations

2.2.4 Some Properties of Conditional Expectations

2.2.5 Average Partial Effects

2.3 Linear Projections

Problems

Appendix 2A

2.A.1 Properties of Conditional Expectations

2.A.2 Properties of Conditional Variances and Covariances

2.A.3 Properties of Linear Projections

3 Basic Asymptotic Theory

3.1 Convergence of Deterministic Sequences

3.2 Convergence in Probability and Boundedness in Probability

3.3 Convergence in Distribution

3.4 Limit Theorems for Random Samples

3.5 Limiting Behavior of Estimators and Test Statistics

3.5.1 Asymptotic Properties of Estimators

3.5.2 Asymptotic Properties of Test Statistics

Problems

II LINEAR MODELS

4 Single-Equation Linear Model and Ordinary Least Squares Estimation

4.1 Overview of the Single-Equation Linear Model

4.2 Asymptotic Properties of Ordinary Least Squares

4.2.1 Consistency

4.2.2 Asymptotic Inference Using Ordinary Least Squares

4.2.3 Heteroskedasticity-Robust Inference

4.2.4 Lagrange Multiplier (Score) Tests

4.3 Ordinary Least Squares Solutions to the Omitted Variables Problem

4.3.1 Ordinary Least Squares Ignoring the Omitted Variables

4.3.2 Proxy Variable–Ordinary Least Squares Solution

4.3.3 Models with Interactions in Unobservables: Random Coefficient Models

4.4 Properties of Ordinary Least Squares under Measurement Error

4.4.1 Measurement Error in the Dependent Variable

4.4.2 Measurement Error in an Explanatory Variable

Problems

5 Instrumental Variables Estimation of Single-Equation Linear Models

5.1 Instrumental Variables and Two-Stage Least Squares

5.1.1 Motivation for Instrumental Variables Estimation

5.1.2 Multiple Instruments: Two-Stage Least Squares

5.2 General Treatment of Two-Stage Least Squares

5.2.1 Consistency

5.2.2 Asymptotic Normality of Two-Stage Least Squares

5.2.3 Asymptotic Efficiency of Two-Stage Least Squares

5.2.4 Hypothesis Testing with Two-Stage Least Squares

5.2.5 Heteroskedasticity-Robust Inference for Two-Stage Least Squares

5.2.6 Potential Pitfalls with Two-Stage Least Squares

5.3 IV Solutions to the Omitted Variables and Measurement Error Problems

5.3.1 Leaving the Omitted Factors in the Error Term

5.3.2 Solutions Using Indicators of the Unobservables

Problems

6 Additional Single-Equation Topics

6.1 Estimation with Generated Regressors and Instruments

6.1.1 Ordinary Least Squares with Generated Regressors

6.1.2 Two-Stage Least Squares with Generated Instruments

6.1.3 Generated Instruments and Regressors

6.2 Control Function Approach to Endogeneity

6.3 Some Specification Tests

6.3.1 Testing for Endogeneity

6.3.2 Testing Overidentifying Restrictions

6.3.3 Testing Functional Form

6.3.4 Testing for Heteroskedasticity

6.4 Correlated Random Coefficient Models

6.4.1 When Is the Usual IV Estimator Consistent?

6.4.2 Control Function Approach

6.5 Pooled Cross Sections and Difference-in-Differences Estimation

6.5.1 Pooled Cross Sections over Time

6.5.2 Policy Analysis and Difference-in-Differences Estimation
Problems

Appendix 6A

7 Estimating Systems of Equations by Ordinary Least
Squares and Generalized Least Squares

7.1 Introduction

7.2 Some Examples

7.3 System Ordinary Least Squares Estimation of a Multivariate Linear System

7.3.1 Preliminaries

7.3.2 Asymptotic Properties of System Ordinary Least Squares

7.3.3 Testing Multiple Hypotheses

7.4 Consistency and Asymptotic Normality of Generalized Least Squares

7.4.1 Consistency

7.4.2 Asymptotic Normality

7.5 Feasible Generalized Least Squares

7.5.1 Asymptotic Properties

7.5.2 Asymptotic Variance of Feasible Generalized Least Squares under a Standard Assumption

7.5.3 Properties of Feasible Generalized Least Squares with (Possibly Incorrect) Restrictions
on the Unconditional Variance Matrix

7.6 Testing the Use of Feasible Generalized Least Squares

7.7 Seemingly Unrelated Regressions, Revisited

7.7.1 Comparison between Ordinary Least Squares and Feasible Generalized Least Squares for Seemingly Unrelated Regressions Systems

7.7.2 Systems with Cross Equation Restrictions

7.7.3 Singular Variance Matrices in Seemingly Unrelated Regressions Systems

7.8 The Linear Panel Data Model, Revisited

7.8.1 Assumptions for Pooled Ordinary Least Squares

7.8.2 Dynamic Completeness

7.8.3 Note on Time Series Persistence

7.8.4 Robust Asymptotic Variance Matrix

7.8.5 Testing for Serial Correlation and Heteroskedasticity after Pooled Ordinary Least Squares

7.8.6 Feasible Generalized Least Squares Estimation under Strict Exogeneity

Problems

8 System Estimation by Instrumental Variables

8.1 Introduction and Examples

8.2 General Linear System of Equations

8.3 Generalized Method of Moments Estimation

8.3.1 General Weighting Matrix

8.3.2 System Two-Stage Least Squares Estimator

8.3.3 Optimal Weighting Matrix

8.3.4 The Generalized Method of Moments Three-Stage Least Squares Estimator

8.4 Generalized Instrumental Variables Estimator

8.4.1 Derivation of the Generalized Instrumental Variables Estimator and Its Asymptotic Properties

8.4.2 Comparison of Generalized Method of Moment, Generalized Instrumental Variables,
and the Traditional Three-Stage Least Squares Estimator

8.5 Testing Using Generalized Method of Moments

8.5.1 Testing Classical Hypotheses

8.5.2 Testing Overidentification Restrictions

8.6 More Efficient Estimation and Optimal Instruments

8.7 Summary Comments on Choosing an Estimator

Problems

9 Simultaneous Equations Models

9.1 Scope of Simultaneous Equations Models

9.2 Identification in a Linear System

9.2.1 Exclusion Restrictions and Reduced Forms

9.2.2 General Linear Restrictions and Structural Equations

9.2.3 Unidentified, Just Identified, and Overidentified Equations

9.3 Estimation after Identification

9.3.1 Robustness-Efficiency Trade-off

9.3.2 When Are 2SLS and 3SLS Equivalent?

9.3.3 Estimating the Reduced Form Parameters

9.4 Additional Topics in Linear Simultaneous Equations Methods

9.4.1 Using Cross Equation Restrictions to Achieve Identification

9.4.2 Using Covariance Restrictions to Achieve Identification

9.4.3 Subtleties Concerning Identification and Efficiency in Linear Systems

9.5 Simultaneous Equations Models Nonlinear in Endogenous Variables

9.5.1 Identification

9.5.2 Estimation

9.5.3 Control Function Estimation for Triangular Systems

9.6 Different Instruments for Different Equations

Problems

10 Basic Linear Unobserved Effects Panel Data Models

10.1 Motivation: Omitted Variables Problem

10.2 Assumptions about the Unobserved Effects and Explanatory Variables

10.2.1 Random or Fixed Effects?

10.2.2 Strict Exogeneity Assumptions on the Explanatory Variables

10.2.3 Some Examples of Unobserved Effects Panel Data Models

10.3 Estimating Unobserved Effects Models by Pooled Ordinary Least Squares

10.4 Random Effects Methods

10.4.1 Estimation and Inference under the Basic Random Effects Assumptions

10.4.2 Robust Variance Matrix Estimator

10.4.3 General Feasible Generalized Least Squares Analysis

10.4.4 Testing for the Presence of an Unobserved Effect

10.5 Fixed Effects Methods

10.5.1 Consistency of the Fixed Effects Estimator

10.5.2 Asymptotic Inference with Fixed Effects

10.5.3 Dummy Variable Regression

10.5.4 Serial Correlation and the Robust Variance Matrix Estimator

10.5.5 Fixed Effects Generalized Least Squares

10.5.6 Using Fixed Effects Estimation for Policy Analysis

10.6 First Differencing Methods

10.6.1 Inference

10.6.2 Robust Variance Matrix

10.6.3 Testing for Serial Correlation

10.6.4 Policy Analysis Using First Differencing

10.7 Comparison of Estimators

10.7.1 Fixed Effects versus First Differencing

10.7.2 The Relationship between the Random Effects and Fixed Effect Estimators

10.7.3 The Hausman Test Comparing Random Effects and Fixed Effects Estimators

Problems

11 More Topics in Linear Unobserved Effects Models

11.1 Generalized Method of Moments Approaches to the Standard Linear Unobserved Effects Model

11.1.1 Equivalence between GMM 3SLS and Standard Estimators

11.1.2 Chamberlain’s Approach to Unobserved Effects Models

11.2 Random and Fixed Effects Instrumental Variables Methods

11.3 Hausman and Taylor–Type Models

11.4 First Differencing Instrumental Variables Methods

11.5 Unobserved Effects Models with Measurement Error

11.6 Estimation under Sequential Exogeneity

11.6.1 General Framework

11.6.2 Models with Lagged Dependent Variables

11.7 Models with Individual-Specific Slopes

11.7.1 Random Trend Model

11.7.2 General Models with Individual-Specific Slopes

11.7.3 Robustness of Standard Fixed Effects Methods

11.7.4 Testing for Correlated Random Slopes

Problems

III GENERAL APPROACHES TO NONLINEAR ESTIMATION

12 M-Estimation, Nonlinear Regression, and Quantile Regression

12.1 Introduction

12.2 Identification, Uniform Convergence, and Consistency

12.3 Asymptotic Normality

12.4 Two-Step M-Estimators

12.4.1 Consistency

12.4.2 Asymptotic Normality

12.5 Estimating the Asymptotic Variance

12.5.1 Estimation without Nuisance Parameters

12.5.2 Adjustments for Two-Step Estimation

12.6 Hypothesis Testing

12.6.1 Wald Tests

12.6.2 Score (or Lagrange Multiplier) Tests

12.6.3 Tests Based on the Change in the Objective Function

12.6.4 Behavior of the Statistics under Alternatives

12.7 Optimization Methods

12.7.1 Newton-Raphson Method

12.7.2 Berndt, Hall, Hall, and Hausman Algorithm

12.7.3 Generalized Gauss-Newton Method

12.7.4 Concentrating Parameters out of the Objective Function

12.8 Simulation and Resampling Methods

12.8.1 Monte Carlo Simulation

12.8.2 Bootstrapping

12.9 Multivariate Nonlinear Regression Methods

12.9.1 Multivariate Nonlinear Least Squares

12.9.2 Weighted Multivariate Nonlinear Least Squares

12.10 Quantile Estimation

12.10.1 Quantiles, the Estimation Problem, and Consistency

12.10.2 Asymptotic Inference

12.10.3 Quantile Regression for Panel Data

Problems

13 Maximum Likelihood Methods

13.1 Introduction

13.2 Preliminaries and Examples

13.3 General Framework for Conditional Maximum Likelihood Estimation

13.4 Consistency of Conditional Maximum Likelihood Estimation

13.5 Asymptotic Normality and Asymptotic Variance Estimation

13.5.1 Asymptotic Normality

13.5.2 Estimating the Asymptotic Variance

13.6 Hypothesis Testing

13.7 Specification Testing

13.8 Partial (or Pooled) Likelihood Methods for Panel Data

13.8.1 Setup for Panel Data

13.8.2 Asymptotic Inference

13.8.3 Inference with Dynamically Complete Models

13.9 Panel Data Models with Unobserved Effects

13.9.1 Models with Strictly Exogenous Explanatory Variables

13.9.2 Models with Lagged Dependent Variables

13.10 Two-Step Estimators Involving Maximum Likelihood

13.10.1 Second-Step Estimator Is Maximum Likelihood Estimator

13.10.2 Surprising Efficiency Result When the First-Step Estimator
Is Conditional Maximum Likelihood Estimator

13.11 Quasi-Maximum Likelihood Estimation

13.11.1 General Misspecification

13.11.2 Model Selection Tests

13.11.3 Quasi-Maximum Likelihood Estimation in the Linear Exponential Family

13.11.4 Generalized Estimating Equations for Panel Data

Problems

Appendix 13A

14 Generalized Method of Moments and Minimum Distance Estimation

14.1 Asymptotic Properties of Generalized Method of Moments

14.2 Estimation under Orthogonality Conditions

14.3 Systems of Nonlinear Equations

14.4 Efficient Estimation

14.4.1 General Efficiency Framework

14.4.2 Efficiency of Maximum Likelihood Estimator

14.4.3 Efficient Choice of Instruments under Conditional Moment Restrictions

14.5 Classical Minimum Distance Estimation

14.6 Panel Data Applications

14.6.1 Nonlinear Dynamic Models

14.6.2 Minimum Distance Approach to the Unobserved Effects Model

14.6.3 Models with Time-Varying Coefficients on the Unobserved Effects

Problems

Appendix 14A

IV NONLINEAR MODELS AND RELATED TOPICS

15 Binary Response Models

15.1 Introduction

15.2 The Linear Probability Model for Binary Response

15.3 Index Models for Binary Response: Probit and Logit

15.4 Maximum Likelihood Estimation of Binary Response Index Models

15.5 Testing in Binary Response Index Models

15.5.1 Testing Multiple Exclusion Restrictions

15.5.2 Testing Nonlinear Hypotheses about β

15.5.3 Tests against More General Alternatives

15.6 Reporting the Results for Probit and Logit

15.7 Specification Issues in Binary Response Models

15.7.1 Neglected Heterogeneity

15.7.2 Continuous Endogenous Explanatory Variables

15.7.3 Binary Endogenous Explanatory Variable

15.7.4 Heteroskedasticity and Nonnormality in the Latent Variable Model

15.7.5 Estimation under Weaker Assumptions

15.8 Binary Response Models for Panel Data

15.8.1 Pooled Probit and Logit

15.8.2 Unobserved Effects Probit Models under Strict Exogeneity

15.8.3 Unobserved Effects Logit Models under Strict Exogeneity

15.8.4 Dynamic Unobserved Effects Models

15.8.5 Probit Models with Heterogeneity and Endogenous Explanatory Variables

15.8.6 Semiparametric Approaches

Problems

16 Multinomial and Ordered Response Model

16.1 Introduction

16.2 Multinomial Response Models

16.2.1 Multinomial Logit

16.2.2 Probabilistic Choice Models

16.2.3 Endogenous Explanatory Variables

16.2.4 Panel Data Methods

16.3 Ordered Response Models

16.3.1 Ordered Logit and Ordered Probit

16.3.2 Specification Issues in Ordered Models

16.3.3 Endogenous Explanatory Variables

16.3.4 Panel Data Methods

Problems

17 Corner Solution Responses

17.1 Motivation and Examples

17.2 Useful Expressions for Type I Tobit

17.3 Estimation and Inference with the Type I Tobit Model

17.4 Reporting the Results

17.5 Specification Issues in Tobit Models

17.5.1 Neglected Heterogeneity

17.5.2 Endogenous Explanatory Models

17.5.3 Heteroskedasticity and Nonnormality in the Latent Variable Model

17.5.4 Estimating Parameters with Weaker Assumptions

17.6 Two-Part Models and Type II Tobit Model

17.6.1 Truncated Normal Hurdle Model

17.6.2 Lognormal Hurdle Model and Exponential Conditional Mean

17.6.3 Exponential Type II Tobit Model

17.7 Two-Limit Tobit Model

17.8 Panel Data Methods

17.8.1 Pooled Methods

17.8.2 Unobserved Effects Models under Strict Exogeneity

17.8.3 Dynamic Unobserved Effects Tobit Models

Problems

18. Count, Fractional, and Other Nonnegative Responses

18.1 Introduction

18.2 Poisson Regression

18.2.1 Assumptions Used for Poisson Regression and Quantities of Interest

18.2.2 Consistency of the Poisson QMLE

18.2.3 Asymptotic Normality of the Poisson QMLE

18.2.4 Hypothesis Testing

18.2.5 Specification Testing

18.3 Other Count Data Regression Models

18.3.1 Negative Binomial Regression Models

18.3.2 Binomial Regression Models

18.4 Gamma (Exponential) Regression Model

18.5 Endogeneity with an Exponential Regression Function

18.6 Fractional Responses

18.6.1 Exogenous Explanatory Variables

18.6.2 Endogenous Explanatory Variables

18.7 Panel Data Models

18.7.1 Pooled QMLE

18.7.2 Specifying Models of Conditional Expectations with Unobserved Effects

18.7.3 Random Effects Methods

18.7.4 Fixed Effects Poisson Estimation

18.7.5 Relaxing the Strict Exogeneity Assumption

18.7.6 Fractional Response Models for Panel Data

Problems

19. Censored Data, Sample Selection, and Attrition

19.1 Introduction

19.2 Data Censoring

19.2.1 Binary Censoring

19.2.2 Interval Coding

19.2.3 Censoring from Above and Below

19.3 Overview of Sample Selection

19.4 When Can Sample Selection Be Ignored?

19.4.1 Linear Models: Estimation by OLS and 2SLS

19.4.2 Nonlinear Models

19.5 Selection on the Basis of the Response Variable: Truncated Regression

19.6 Incidental Truncation: A Probit Selection Equation

19.6.1 Exogenous Explanatory Variables

19.6.2 Endogenous Explanatory Variables

19.6.3 Binary Response Model with Sample Selection

19.6.4 An Exponential Response Function

19.7 Incidental Truncation: A Tobit Selection Equation

19.7.1 Exogenous Explanatory Variables

19.7.2 Endogenous Explanatory Variables

19.7.3 Estimating Structural Tobit Equations with Sample Selection

19.8 Inverse Probability Weighting for Missing Data

19.9 Sample Selection and Attrition in Linear Panel Data Models

19.9.1 Fixed and Random Effects Estimation with Unbalanced Panels

19.9.2 Testing and Correcting for Sample Selection Bias

19.9.3 Attrition

Problems

20 Stratified Sampling and Cluster Sampling

20.1 Introduction

20.2 Stratified Sampling

20.2.1 Standard Stratified Sampling and Variable Probability Sampling

20.2.2 Weighted Estimators to Account for Stratification

20.2.3 Stratification Based on Exogenous Variables

20.3 Cluster Sampling

20.3.1 Inference with a Large Number of Clusters and Small Cluster Sizes

20.3.2 Cluster Samples with Unit-Specific Panel Data

20.3.3 Should We Apply Cluster-Robust Inference with Large Group Sizes?

20.3.4 Inference When the Number of Clusters is Small

20.4 Complex Survey Sampling

Problems

21 Estimating Average Treatment Effects

21.1 Introduction

21.2 A Counterfactual Setting and the Self-Selection Problem

21.3 Methods Assuming Ignorability (or Unconfoundedness) of Treatment

21.3.1 Identification

21.3.2 Regression Adjustment

21.3.3 Propensity Score Analysis

21.3.4 Combining Regression Adjustment and Propensity Score Weighting

21.3.5 Matching Methods

21.4 Instrumental Variables Methods

21.4.1 Estimating the Average Treatment Effect Using IV

21.4.2 Correction and Control Function Approaches

21.4.3 Estimating the Local Average Treatment Effect by IV

21.5 Regression Discontinuity Designs

21.5.1 The Sharp Regression Discontinuity Design

21.5.2 The Fuzzy Regression Discontinuity Design

21.5.3 Unconfoundedness versus the Fuzzy Regression Discontinuity

21.6 Further Issues

21.6.1 Special Considerations for Responses with Discreteness or Limited Range

21.6.2 Multivalued Treatments

21.6.3 Multiple Treatments

21.6.4 Panel Data

Problems

22 Duration Analysis

22.1 Introduction

22.2 Hazard Functions

22.2.1 Hazard Functions without Covariates

22.2.2 Hazard Functions Conditional on Time-Invariant Covariates

22.2.3 Hazard Functions Conditional on Time-Varying Covariates

22.3 Analysis of Single-Spell Data with Time-Invariant Covariates

22.3.1 Flow Sampling

22.3.2 Maximum Likelihood Estimation with Censored Flow Data

22.3.3 Stock Sampling

22.3.4 Unobserved Heterogeneity

22.4 Analysis of Grouped Duration Data

22.4.1 Time-Invariant Covariates

22.4.2 Time-Varying Covariates

22.4.3 Unobserved Heterogeneity

22.5 Further Issues

22.5.1 Cox’s Partial Likelihood Method for the Proportional Hazard Model

22.5.2 Multiple-Spell Data

22.5.3 Competing Risks Models

Problems

References

Index