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
