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Econometric Analysis, 6th Edition
Comment from the Stata technical groupWilliam Greene’s Econometric Analysis has served as the standard reference for econometrics among economists, political scientists, and other social scientists for nearly two decades, and the newly released sixth edition is certain to carry on that tradition. The book’s abundance of examples and Greene’s emphasis on how to put econometric theory to practical use make the book valuable not just to graduate students taking their first course in econometrics but also to students and professionals who engage in empirical research. As with most econometrics texts, Greene’s Econometric Analysis begins by introducing the linear regression model. Part I of the book, consisting of seven chapters, covers the properties of the least-squares estimator, inference and prediction, and tests for functional form and specification. Part II of the book generalizes the linear regression model to allow for heteroskedasticity; then, with the generalized least-squares estimator already discussed in the context of nonspherical disturbances, presents the fixed- and random-effects panel-data models as straightforward extensions of least-squares analysis. A chapter also discusses systems of regression equations, another application of GLS. Stata’s nl and nlsur commands make fitting nonlinear models as easy as fitting linear models, and it is refreshing to see nonlinear models introduced relatively early in Econometric Analysis. Part III discusses instrumental variables and simultaneous equations. The chapter on instrumental variables has been updated to include highlights of the recent research on weak instruments, and it includes a lucid discussion of dynamic panel-data models. Part I lays out an estimation principle (least-squares analysis), and parts II and III show how that methodology can be applied to a wide variety of applications. Having a firm grasp of a general estimation framework makes learning new applications much easier, and this style of teaching continues through the rest of the book. Aside from least squares, maximum likelihood, generalized methods of moments, and, increasingly, simulation-based and Bayesian estimation frameworks represent the four most commonly used techniques in econometrics; and in part IV, a chapter is devoted to each of these. Each chapter strikes a good balance between theoretical rigor and applications illustrating their use. Many newer discrete-choice models require evaluation of multivariate normal probabilities; thus, chapter 17 includes a detailed discussion of the GHK simulator. Part V is devoted to time-series techniques, including estimation in the presence of serial correlation, models with lagged variables including vector autoregressions, stochastic processes and ARIMA models, and nonstationarity, unit roots, and cointegration. The chapters in part V frequently make use of the results obtained in part IV on estimation Econometric Analysis has long been recognized for its extensive coverage of limited dependent variable models, and part VI of the sixth edition continues in that tradition. Binomial, multinomial, and ordered outcomes for both cross-sectional and panel data are covered in one chapter. Two chapters are devoted to truncation, censoring, and sample selection, as well as count and duration models. Table of contentsExamples and Applications
Preface
Part I: The Linear Regression Model
Chapter 1: Introduction
1.1 Econometrics
1.2 Econometric Modeling 1.3 Methodology 1.4 The Practice of Econometrics 1.5 Plan of the Book
Chapter 2: The Classical Multiple Linear Regression Model
2.1 Introduction
2.2 The Linear Regression Model 2.3 Assumptions of the Classical Linear Regression Model
2.3.1 Linearity of the Regression Model
2.4 Summary and Conclusions2.3.2 Full Rank 2.3.3 Regression 2.3.4 Spherical Disturbances 2.3.5 Data Generating Process for the Regressors 2.3.6 Normality Chapter 3: Least Squares
3.1 Introduction
3.2 Least Squares Regression
3.2.1 The Least Squares Coefficient Vector
3.3 Partitioned Regression and Partial Regression3.2.2 Application: An Investment Equation 3.2.3 Algebraic Aspects of the Least Squares Solution 3.2.4 Projection 3.4 Partial Regression and Partial Correlation Coefficients 3.5 Goodness of Fit and the Analysis of Variance
3.5.1 The Adjusted R-Squared and a Measure of Fit
3.6 Summary and Conclusions
3.5.2 R-Squared and the Constant Term in the Model 3.5.3 Comparing Models Chapter 4: Statistical Properties of the Least Squares Estimator
4.1 Introduction
4.2 Motivating Least Squares
4.2.1 The Population Orthogonality Conditions
4.3 Unbiased Estimation4.2.2 Minimum Mean Squared Error Predictor 4.2.3 Minimum Variance Linear Unbiased Estimation 4.4 The Variance of the Least Squares Estimator and the Gauss–Markov Theorem 4.5 The Implications of Stochastic Regressors 4.6 Estimating the Variance of the Least Squares Estimator 4.7 The Normality Assumption and Basic Statistical Interference
4.7.1 Testing a Hypothesis about a Coefficient
4.8 Finite-Sample Properties of the Least Squares Estimator
4.7.2 Confidence Intervals for Parameters 4.7.3 Confidence Interval for a Linear Combination of Coefficients: The Oaxaca Decomposition 4.7.4 Testing the Significance of the Regression 4.7.5 Marginal Distributions of the Test Statistics
4.8.1 Multicollinearity
4.9 Large Sample Properties of the Least Squares Estimator
4.8.2 Missing Observations
4.9.1 Consistency of the Least Squares Estimator
4.10 Summary and Conclusions
4.9.2 Asymptotic Normality of the Least Squares Estimator 4.9.3 Consistency of s2 and the Estimator of Asy. Var[b] 4.9.4 Asymptotic Distribution of a Function of b: The Delta Method and the Method of Krinsky and Robb 4.9.5 Asymptotic Efficiency 4.9.6 More General Data Generating Processes
Chapter 5: Interference and Prediction
5.1 Introduction
5.2 Restrictions and Nested Models 5.3 Two Approaches to Testing Hypotheses
5.3.1 The F Statistic and the Least Squares Discrepancy
5.4 Nonnormal Disturbances and Large Sample Tests5.3.2 The Restricted Least Squares Estimator 5.3.3 The Loss of Fit from Restricted Least Squares 5.5 Testing Nonlinear Restrictions 5.6 Prediction 5.7 Summary and Conclusions
Chapter 6: Functional Form and Structural Change
6.1 Introduction
6.2 Using Binary Variables
6.2.1 Binary Variables in Regression
6.3 Nonlinearity in the Variables
6.2.2 Several Categories 6.2.3 Several Groupings 6.2.4 Threshold Effects and Categorical Variables 6.2.5 Spline Regression
6.3.1 Functional Forms
6.4 Modeling and Testing for a Structural Break
6.3.2 Identifying Nonlinearity 6.3.3 Intrinsic Linearity and Identification
6.4.1 Different Parameter Vectors
6.5 Summary and Conclusions
6.4.2 Insufficient Observations 6.4.3 Change in a Subset of Coefficients 6.4.4 Tests of a Structural Break with Unequal Variances 6.4.5 Predictive Test
Chapter 7: Specification Analysis and Model Selection
7.1 Introduction
7.2 Specification Analysis and Model Building
7.2.1 Bias Caused by Omission of Relevant Variables
7.3 Choosing Between Nonnested Models
7.2.2 Pretest Estimation 7.2.3 Inclusion of Irrelevant Variables 7.2.4 Model Building—A General to Simple Strategy
7.3.1 Testing Nonnested Hypotheses
7.4 Model Selection Criteria7.3.2 An Encompassing Model 7.3.3 Comprehensive Approach—The J Test 7.3.4 Vuon’s Test and the Kullback–Leibler Information Criterion 7.5 Model Selection
7.5.1 Classical Models Selection
7.6 Summary and Conclusions
7.5.2 Bayesian Model Averaging Part II: The Generalized Regression Model
Chapter 8: The Generalized Regression Model and Heteroscedasticity
8.1 Introduction
8.2 Least Squares Estimation
8.2.1 Finite-Sample Properties of Ordinary Least Squares
8.3 Efficient Estimation by Generalized Least Squares8.2.2 Asymptotic Properties of Least Squares 8.2.3 Robust Estimation of Asymptotic Covariance Matrices 8.4 Heteroscedasticity
8.4.1 Ordinary Least Squares Estimation
8.5 Testing for Heteroscedasticity
8.4.2 Inefficiency of Least Squares 8.4.3 The Estimated Covariance Matrix 8.4.4 Estimation the Appropriate Covariance Matrix for Ordinary Least Squares
8.5.1 White’s General Test
8.6 Weighted Least Squares When Omega is known8.5.2 The Breusch–Pagan/Godfrey LM Test 8.7 Estimation When Omega Contains Unknown Parameters 8.8 Applications
8.8.1 Multiplicative Heteroscedasticity
8.9 Summary and Conclusions
8.8.2 Group-wise Heteroscedasticity
Chapter 9: Models for Panel Data
9.1 Introduction
9.2 Panel Data Models
9.2.1 General Modeling Framework for Analyzing Panel Data
9.3 The Pooled Regression Model
9.2.2 Model Structures 9.2.3 Extensions 9.2.4 Balanced and Unbalanced Panels
9.3.1 Least Squares Estimation of the Pooled Model
9.4 The Fixed Effects Model
9.3.2 Robust Covariance Matrix Estimation 9.3.3 Clustering and Stratification 9.3.4 Robust Estimation Using Group Means 9.3.5 Estimation with First Differences 9.3.6 The Within- and Between-Groups Estimators
9.4.1 Least Squares Estimation
9.5 Random Effects
9.4.2 Small T Asymptotics 9.4.4 Testing the Significance of the Group Effects 9.4.4 Fixed Time and Group Effects
9.5.1 Generalized Least Squares
9.6 Nonspherical Disturbances and Robust Covariance Estimation
9.5.2 Feasible Generalized Least Squares When Sigma is Unknown 9.5.3 Testing for Random Effects 9.5.4 Hausman’s Specification Test for the Random Effects Model 9.5.5 Extending the Unobserved Effects Model: Mundlak’s Approach
9.6.1 Robust Estimation of the Fixed Effects Model
9.7 Extensions of the Random Effects Model
9.6.2 Heteroscedasticity in the Random Effects Model 9.6.3 Autocorrelation in Panel Data Models
9.7.1 Nested Random Effects
9.8 Parameter Heterogeneity
9.7.2 Spatial Autocorrelation
9.8.1 The Random Coefficients Model
9.9 Consistent Estimation of Dynamic Panel Data Models9.8.2 Random Parameters and Simulation-Based Estimation 9.8.3 Two-Step Estimation of Panel Data Models 9.8.4 Hierarchical Linear Models 9.8.5 Parameter Heterogeneity and Dynamic Panel Data Models 9.8.6 Nonstationary Data and Panel Data Models 9.10 Summary and Conclusions
Chapter 10: Systems of Regression Equations
10.1 Introduction
10.2 The Seemingly Unrelated Regressions Model
10.2.1 Generalized Least Squares
10.3 Panel Data Applications
10.2.2 Seemingly Unrelated Regressions with Identical Regressors 10.2.3 Feasible Generalized Least Squares 10.2.4 Testing Hypotheses 10.2.5 Heteroscedasticity 10.2.6 Autocorrelation 10.2.7 A Specification Test for the Sur Model 10.2.8 The Pooled Model
10.3.1 Random Effects Sur Models
10.4 Systems of Demand Equations: Singular Systems
10.3.2 The Random and Fixed Effects Models
10.4.1 Cobb–Douglas Cost Function (Example 6.3 Continued)
10.5 Summary and Conclusions
10.4.2 Flexible Functional Forms: The Translog Cost Function
Chapter 11: Nonlinear Regressions and Nonlinear Least Squares
11.1 Introduction
11.2 Nonlinear Regression Models
11.2.1 Assumptions of the Nonlinear Regression Model
11.3 Applications
11.2.2 The Orthogonality Condition and The Sum of Squares 11.2.3 The Linearized Regression 11.2.4 Large Sample Properties of the Nonlinear Least Squares Estimator 11.2.5 Computing the Nonlinear Least Squares Estimator
11.3.1 A Nonlinear Consumption Function
11.4 Hypothesis Testing and Parametric Restrictions
11.3.2 The Box–Cox Transformation
11.4.1 Significance Tests for Restrictions: F and Wald Statistics
11.5 Nonlinear Systems of Equations11.4.2 Tests Based on the LM Statistic 11.6 Two-Step Nonlinear Least Squares Estimation 11.7 Panel Data Applications
11.7.1 A Robust Convergence Matrix for Nonlinear Least Squares
11.8 Summary and Conclusions
11.7.2 Fixed Effects 11.7.3 Random Effects
Part III: Instrumental Variables and Simultaneous Equations Model
Chapter 12 Nonlinear Regressions and Nonlinear Least Squares
12.1 Introduction
12.2 Assumptions of the Model 12.3 Estimation
12.3.1 Ordinary Least Squares
12.4 The Hausman and Wu Specification Tests and an Application to Instrumental
Variable Estimation12.3.2 The Instrumental Variables Estimator 12.3.3 Two-Stage Least Squares 12.5 Measurement Error
12.5.1 Least Squares Attenuation
12.6 Estimation of the Generalized Regression Model12.5.2 Instrumental Variables Estimation 12.5.3 Proxy Variables 12.7 Nonlinear Instrumental Variables Estimation 12.8 Panel Data Applications
12.8.1 Instrumental Variables Estimation of the Random Effects
Model—The Hausman and Taylor Estimator
12.9 Weak Instruments12.8.2 Dynamic Panel Data Models—The Anderson/Hsiao and Arellano/Bond Estimators 12.10 Summary and Conclusions
Chapter 13: Simultaneous Equations Models
13.1 Introduction
13.2 Fundamental Issues in Simultaneous Equations Models
13.2.1 Illustrative Systems of Equations
13.3 The Problem of Identification 13.2.2 Endogeneity and Causality 13.2.3 A General Notation for Linear Simultaneous Equations Models
13.3.1 The Rank and Order Conditions for Identification
13.3 The Problem of Identification 13.3.2 Identification through Other Nonsample Information 13.4 Method Estimation 13.5 Single Equation: Limited Information Estimation Methods
13.5.1 Ordinary Least Squares
13.6 System Methods of Estimation
13.5.2 Estimation by Instrumental Variables 13.5.3 Two-Stage Least Squares 13.5.4 Limited Information Maximum Likelihood and the K Class of Estimators 13.5.5 Testing in the Presence of Weak Instruments 13.5.6 Two-Stage Least Squares in Models That Are Nonlinear in Variables
13.6.1 Three-Stage Least Squares
13.7 Comparison of Methods—Klein’s Model I 13.6.2 Full Information Maximum Likelihood 13.8 Specification Tests 13.9 Properties of Dynamic Models
13.9.1 Dynamic Models and Their Multipliers
13.10 Summary and Conclusions
13.9.2 Stability 13.9.3 Adjustment to Equilibrium Part IV Estimation Methodology
Chapter 14: Estimation Frameworks in Econometrics
14.1 Introduction
14.2 Parametric Estimation and Inference
14.2.1 Classical Likelihood-Based Estimation
14.3 Semiparametric Estimation
14.2.2 Modeling Joint Distributions with Copula Functions
14.3.1 GMM Estimation in Econometrics
14.4 Nonparametric Equations
14.3.2 Least Absolute Deviations Estimation 14.3.3 Partially Linear Regression 14.3.4 Kernel Density Methods 14.3.5 Comparing Parametric and Semiparametric Analyses
14.4.1 Kernel Density Estimation
14.5 Properties of Estimators
14.4.2 Nonparametric Regression
14.5.1 Statistical Properties of Estimators
14.6 Summary and Conclusions
14.5.2 Extremum Estimators 14.5.3 Assumptions for Asymptotic Properties of Extremum Estimators 14.5.4 Asymptotic Properties of Estimators 14.5.5 Testing Hypotheses
Chapter 15: Minimum Distance Estimation and the Generalized Method of Moments
15.1 Introduction
15.2 Consistent Estimation: The Method of Moments
15.2.1 Random Sampling and Estimating the Parameters of Distributions
15.3 Minimum Distance Estimation15.2.2 Asymptotic Properties of the Method of Moments Estimator 15.2.3 Summary–The Method of Moments 15.4 The Generalized Method of Moments (GMM) Estimator
15.4.1 Estimation Based on Orthogonality Conditions
15.5 Testing Hypotheses in the GMM Framework
15.4.2 Generalizing the Method of Moments 15.4.3 Properties of the GMM Estimator
15.5.1 Testing the Validity of the Moment Restrictions
15.6 GMM Estimation of Econometric Models
15.5.2 GMM Counterparts to the WALD, LM, and LR Tests
15.6.1 Single-Equation Linear Models
15.7 Summary and Conclusions
15.6.2 Single-Equation Nonlinear Models 15.6.3 Seemingly Unrelated Regression Models 15.6.4 Simultaneous Equations Models with Heteroscedasticity 15.6.5 GMM Estimation of Dynamic Panel Data Models
Chapter 16: Maximum Likelihood Estimation
16.1 Introduction
16.2 The Likelihood Function and Identification of the Parameters 16.3 Efficient Estimation: The Principle of Maximum Likelihood 16.4 Properties of Maximum Likelihood Estimators
16.4.1 Regularity Conditions
16.5 Conditional Likelihoods, Econometric Models, and the GMM Estimator16.4.2 Properties of Regular Densities 16.4.3 The Likelihood Equation 16.4.4 The Information Matrix Equality 16.4.5 Asymptotic Properties of the Maximum Likelihood Estimator
16.4.5.a Consistency
16.4.6 Estimating the Asymptotic Variance of the Maximum Likelihood Estimator
16.4.5.b Asymptotic Normality 16.4.5.c Asymptotic Efficiency 16.4.5.d Invariance 16.4.5.e Conclusion 16.6 Hypothesis and Specification Tests and Fit Measures
16.6.1 The Likelihood Ratio Test
16.7 Two-Step Maximum Likelihood Estimation16.6.2 The Wald Test 16.6.3 The Lagrange Multiplier Test 16.6.4 An Application of the Likelihood-Based Test Procedures 16.6.5 Comparing Models and Computing Model Fit 16.8 Pseudo-Maximum Likelihood Estimation and Robust Asymptotic Covariance Matrices
16.8.1 Maximum Likelihood and GMM Estimation
16.9 Applications of Maximum Likelihood Estimation
16.8.2 Maximum Likelihood and M Estimation 16.8.3 Sandwich Estimators 16.8.4 Cluster Estimators
16.9.1 The Normal Linear Regression Model
16.10 Summary and Conclusions
16.9.2 The Wald Test
16.9.2.a Multiplicative Heteroscedasticity
16.9.3 Seemingly Unrelated Regression Models
16.9.2.b Autocorrelation
16.9.3.a The Pooled Model
16.9.4 Simultaneous Equations Models16.9.3.b The SUR Model 16.9.3.c Exclusion Restrictions 16.9.5 Maximum Likelihood Estimation of Nonlinear Regression Models
16.9.5.a Nonnormal Disturbances—The Stochastic Frontier Model
16.9.6 Panel Data Applications
16.9.5.b ML Estimation of a Geometric Regression Model for Count Data
16.9.6.a ML Estimation of the Linear Random Effects
16.9.7 Latent Class and Finite Mixture Models
16.9.6.b Random Effects in Nonlinear Models: MLE using Quadrature 16.9.6.c Fixed Effects in Nonlinear Models: Full MLE
16.9.7.a A Finite Mixture Model
16.9.7.b Measured and Unmeasured Heterogeneity 16.9.7.c Predicting Class Membership 16.9.7.d A Conditional Latent Class Model 16.9.7.e Determining the Number of Classes 16.9.7.f A Panel Data Application
Chapter 17: Simulation-Based Estimation and Inference
17.1 Introduction
17.2 Random Number Generation
17.2.1 Generating Pseudo-Random Numbers
17.3 Monte Carlo Integration
17.2.2 Sampling from a Standard Uniform Population 17.2.3 Sampling from Continuous Distributions 17.2.4 Sampling from a Multivariate Normal Population 17.2.5 Sampling from a Discrete Population
17.3.1 Halton Sequences and Random Draws for Simulation-Based
Integration
17.4 Monte Carlo Studies
17.3.2 Importance Sampling 17.3.3 Computing Multivariate Normal Probabilities Using the GHK Estimator
17.4.1 A Monte Carlo Study: Behavior of a Test Statistic
17.5 Simulation-Based Estimation
17.4.2 A Monte Carlo Study: The Incidental Parameters Problem
17.5.1 Maximum Simulated Likelihood Estimation of Random Parameters
Models
17.6 Bootstrapping17.5.2 The Method of Simulated Moments 17.7 Summary and Conclusions
Chapter 18 Bayesian Estimation and Inference
18.1 Introduction
18.2 Bayes Theorem and the Posterior Density 18.3 Bayesian Analysis of the Classical Regression Model
18.3.1 Analysis with a Noninformative Prior
18.4 Bayesian Inference
18.3.2 Estimation with an Informative Prior Density
18.4.1 Point Estimation
18.5 Posterior Distributions and the Gibbs Sampler18.4.2 Interval Estimation 18.4.3 Hypothesis Testing 18.4.4 Large Sample Results 18.6 Application: Binomial Probit Model 18.7 Panel Data Application: Individual Effects Model 18.8 Hierarchical Bayes Estimation of a Random Parameters Model 18.9 Summary and Conclusions
Part V Time Series and Macroeconometrics
Chapter 19 Serial Correlation
19.1 Introduction
19.2 The Analysis of Time-Series Data 19.3 Disturbance Processes
19.3.1 Characteristics of Disturbance Processes
19.4 Some Asymptotic Results for Analyzing Time-Series Data
19.3.2 AR(1) Disturbances
19.4.1 Convergence of Moments—The Ergodic Theorem
19.5 Least Squares Estimation
19.4.2 Convergence to Normality—A Central Limit Theorem
19.5.1 Asymptotic Properties of Least Squares
19.6 GMM Estimation19.5.2 Estimating the Variance of the Least Squares Estimator 19.7 Testing for Autocorrelation
19.7.1 Lagrange Multiplier Test
19.8 Efficient Estimation When Omega is Known19.7.2 Box and Pierce’s Test and Ljung’s Refinement 19.7.3 The Durbin–Watson Test 19.7.4 Testing in the Presence of a Lagged Dependent Variable 19.7.5 Summary of Testing Procedures 19.9 Estimation When Omega is Unknown
19.9.1 AR(1) Disturbances
19.10 Autocorrelation in Panel Data19.9.2 Application: Estimation of a Model with Autocorrelation 19.9.3 Estimation with a Lagged Dependent Variable 19.11 Common Factors 19.12 Forecasting in the Presence of Autocorrelation 19.13 Autoregressive Conditional Heteroscedasticity
19.13.1 The ARCH(1) Model
19.14 Summary and Conclusions
19.13.2 ARCH(q), ARCH-in-Mean, and Generalized ARCH Models 19.13.3 Maximum Likelihood Estimation of the Garch Model 19.13 4 Testing for Garch Effects 19.13.5 Pseudo-Maximum Likelihood Estimation
Chapter 20 Models with Lagged Variables
20.1 Introduction
20.2 Dynamic Regression Models
20.2.1 Lagged Effects in a Dynamic Model
20.3 Simple Distributed Lag Models20.2.2 The Lag and Difference Operators 20.2.3 Specification Search for the Lag Length 20.4 Autoregressive Distributed Lag Models
20.4.1 Estimation of the ARDL Model
20.5 Methodological Issues in the Analysis of Dynamic Models
20.4.2 Computation of the Lag Weights in the ARDL Model 20.4.3 Stability of a Dynamic Equation 20.4.4 Forecasting
20.5.1 An Error Correction Model
20.6 Vector Autoregressions
20.5.2 Autocorrelation 20.5.3 Specification Analysis
20.6.1 Model Forms
20.7 Summary and Conclusions
20.6.2 Estimation 20.6.3 Testing Procedures 20.6.4 Exogeneity 20.6.5 Testing for Granger Causality 20.6.6 Impulse Response Functions 20.6.7 Structural VARS 20.6.8 Application: Policy Analysis with a VAR
20.6.8.a A VAR Model of the Macroeconomic Variables
20.6.9 VARs in Microeconometrics
20.6.8.b The Sacrifice Ratio 20.6.8.c Identification and Estimation of a Structural VAR Model 20.6.8.d Inference 20.6.8.e Empirical Results
Chapter 21 Time-Series Models
21.1 Introduction
21.2 Stationary Stochastic Processes
21.2.1 Autoregressive Moving-Average Processes
21.3 The Frequency Domain
21.2.2 Stationarity and Invertibility 21.2.3 Autocorrelations of a Stationary Stochastic Process 21.2.4 Partial Autocorrelations of a Stationary Stochastic Process 21.2.5 Modeling Univariate Time Series 21.2.6 Estimation of the Parameters of a Univariate Time Series
21.3.1 Theoretical Results
21.4 Summary and Conclusions
21.3.2 Empirical Counterparts
Chapter 22 Nonstationary Data
22.1 Introduction
22.2 Nonstationary Processes and Unit Roots
22.2.1 Integrated Processes and Differencing
22.3 Cointegration
22.2.2 Random Walks, Trends, and Spurious Regressions 22.2.3 Tests for Unit Roots in Economic Data 22.2.4 The Dicky–Fuller Tests 22.2.5 The KPSS Test of Stationarity
22.3.1 Common Trends
22.4 Nonstationary Panel Data22.3.2 Error Correlation and VAR Representations 22.3.3 Testing for Cointegration 22.3.4 Estimating Cointegration Relationships 22.3.5 Application: German Money Demand
22.3.5.a Cointegration Analysis and Long-Run Theoretical Model
22.3.5.b Testing for Model Instability 22.5 Summary and Conclusions
Part VI Cross Sections, Panel Data, and Microeconometrics
Chapter 23 Models for Discrete Choice
23.1 Introduction
23.2 Discrete Choice Models 23.3 Models for Binary Choice
23.3.1 The Regression Approach
23.4 Estimation and Inference in Binary Choice Models
23.3.2 Latent Regression—Index Function Models 23.3.3 Random Utility Models
23.4.1 Robust Covariance Matrix Estimation
23.5 Binary Choice Models for panel data
23.4.2 Marginal Effects and Average Partial Effects 23.4.3 Hypothesis Tests 23.4.4 Specification Tests for Binary Choice Models
23.4.4.a Omitted Variables
23.4.5 Measuring Goodness of FIT23.4.4.b Heteroscedasticity 23.4.6 Choice-Based Sampling 23.4.7 Dynamic Binary Choice Models
23.5.1 Random Effects Models
23.6 Semiparametric Analysis
23.5.2 Fixed Effects Models 23.5.3 Modeling Heterogeneity 23.5.4 Parameter Heterogeneity
23.6.1 Semiparametric Estimation
23.7 Endogenous Right-Hand-Side Variables in Binary Choice Models23.6.2 A Kernel Estimator for a Nonparametric Regression Function 23.8 Bivariate Probit Models
23.8.1 Maximum Likelihood Estimation
23.9 A Multivariate Probit Model23.8.2 Testing for Zero Correlation 23.8.3 Marginal Effects 23.8.4 Recursive Bivariate Probit Models 23.10 Analysis of Ordered Choices
23.10.1 The Ordered Probit Model
23.11 Models for Unordered Multiple Choices
23.10.2 Bivariate Ordered Probit Models 23.10.3 Panel Data Applications
23.10.3.a Ordered Probit Models with Fixed Effects
23.10.3.b Ordered Probit Models with Random Effects
23.11.1 The Multinomial Logit Model
23.12 Summary and Conclusions
23.11.2 The Conditional Logit Model 23.11.3 The Independence from Irrelevant Alternatives Assumption 23.11.4 Nested Logit Models 23.11.5 The Multinomial Probit Model 23.11.6 The Mixed Logit Model 23.11.7 Application: Conditional Logit Model for Travel Mode Choice 23.11.8 Panel Data and Stated Choice Experiments
Chapter 24 Truncation, Censoring, and Sample Selection
24.1 Introduction
24.2 Truncation
24.2.1 Truncated Distributions
24.3 Censored Data
24.2.2 Moments of Truncated Distributions 24.2.3 The Truncated Regression Model
24.3.1 The Censored Normal Distribution
24.4 Panel Data Applications24.3.2 The Censored Regression (Tobit) Model 24.3.3 Estimation 24.3.4 Some Issues in Specification
24.3.4.a Heteroscedasticity
24.3.4.b Misspecification of Prob[y*<0] 24.3.4.c Corner Solutions 24.3.4.d Nonnormality 24.5 Sample Selection
24.5.1 Incidental Truncation in a Bivariate Distribution
24.6 Summary and Conclusions
24.5.2 Regression in a Model of Selection 24.5.3 Estimation 24.5.4 Regression Analysis of Treatment Effects 24.5.5 The Nonnormality Assumption 24.5.6 Estimating the Effect of Treatment on the Treated 24.5.7 Sample Selection in Nonlinear Models 24.5.8 Panel Data Applications of Sample Selection Models
24.5.8.a Common Effects in Sample Selection Models
24.5.8.b Attrition
Chapter 25 Models for Event Counts and Duration
25.1 Introduction
25.2 Models for Counts of Events
25.2.1 Measuring Goodness of FIT
25.3 Panel Data Models
25.2.2 Testing for Overdispersion 25.2.3 Heterogeneity and the Negative Binomial Regression Model 25.2.4 Functional Forms for Count Data Models
25.3.1 Robust Covariance Matrices
25.4 Hurdle and Zero-Altered Poisson Models25.3.2 Fixed Effects 25.3.3 Random Effects 25.5 Censoring and Truncation in Models for Counts
25.5.1 Censoring and Truncation in the Poisson Model
25.6 Models for Duration Data
25.5.2 Application: Censoring in the Tobit and Poisson Regression Models
25.6.1 Duration Data
25.7 Summary and Conclusions
25.6.2 A Regression-Like Approach: Parametric Models of Duration
25.6.2.a Theoretical Background
25.6.3 Nonparametric and Semiparametric Approaches
25.6.2.b Models of the Hazard Approach 25.6.2.c Maximum Likelihood Estimation 25.6.2.d Exogenous Variables 25.6.2.e Heterogeneity
Part VII Appendices
Appendix A Matrix Algebra
A.1 Terminology
A.2 Algebraic Manipulation of Matrices
A.2.1 Equality of Matrices
A.3 Geometry of Matrices
A.2.2 Transposition A.2.3 Matrix Addition A.2.4 Vector Multiplication A.2.5 A Notation for Rows and Columns of a Matrix A.2.6 Matrix Multiplication and Scalar Multiplication A.2.7 Sums of Values A.2.8 A Useful Idempotent Matrix
A.3.1 Vector Spaces
A.4 Solution of a System of Linear Equations
A.3.2 Linear Combinations of Vectors and Basis Vectors A.3.3 Linear Dependence A.3.4 Subspaces A.3.5 Rank of a Matrix A.3.6 Determinant of a Matrix A.3.7 A Least Squares Problem
A.4.1 Systems of Linear Equations
A.5 Partitioned Matrices
A.4.2 Inverse Matrices A.4.3 Nonhomogeneous Systems of Equations A.4.4 Solving the Least Squares Problems
A.5.1 Addition and Multiplication of Partitioned Matrices
A.6 Characteristic Roots and Vectors
A.5.2 Determinants of Partitioned Matrices A.5.3 Inverses of Partitioned Matrices A.5.4 Deviations from Means A.5.5 Kronecker Products
A.6.1 The Characteristic Equation
A.7 Quadratic Forms and Definite Matrices
A.6.2 Characteristic Vectors A.6.3 General Results for Characteristic Roots and Vectors A.6.4 Diagonalization and Spectral Decomposition of a Matrix A.6.5 Rank of a Matrix A.6.6 Condition Number of a Matrix A.6.7 Trace of a Matrix A.6.8 Determinant of a Matrix A.6.9 Powers of a Matrix A.6.10 Idempotent Quadratic Forms A.6.11 Factoring a Matrix A.6.12 The Generalized Inverse of a Matrix
A.7.1 Nonnegative Definite Matrices
A.8 Calculus and Matrix Algebra
A.7.2 Idempotent Quadratic Forms A.7.3 Comparing Matrices
A.8.1 Differentiation and the Taylor Series
A.8.2 Optimization A.8.3 Constrained Optimizations A.8.4 Transformations
Appendix B Probability and Distribution Theory
B.1 Introduction
B.2 Random Variables
B.2.1 Probability Distributions
B.3 Expectations of a Random VariableB.2.2 Cumulative Distribution Function B.4 Some Specific Probability Distributions
B.4.1 The Normal Distribution
B.5 The Distribution of a Function of a Random VariableB.4.2 The Chi-Squared, t, and F Distributions B.4.3 Distributions with Large Degrees of Freedom B.4.4 Size Distributions: The Lognormal Distribution B.4.5 The Gamma and Exponential Distributions B.4.6 The Beta Distribution B.4.7 The Logistic Distribution B.4.8 The Wishart Distribution B.4.9 Discrete Random Variables B.6 Representations of a Probability Distribution B.7 Joint Distributions
B.7.1 Marginal Distributions
B.8 Conditioning in a Bivariate Distribution
B.7.2 Expectations in a Joint Distribution B.7.3 Covariance and Correlation B.7.4 Distribution of a Function of Bivariate Random Variables
B.8.1 Regression: The Conditional Mean
B.9 The Bivariate Normal DistributionB.8.2 Conditional Variance B.8.3 Relationships Among Marginal and Conditional Moments B.8.4 The Analysis of Variance B.10 Multivariate Distributions
B.10.1 Moments
B.11 The Multivariate Normal Distribution
B.10.2 Sets of Linear Functions B.10.3 Nonlinear Functions
B.11.1 Marginal and Conditional Normal Distributions
B.11.2 The Classical Normal Linear Regression Model B.11.3 Linear Functions of a Normal Vector B.11.4 Quadratic Forms in a Standard Normal Vector B.11.5 The F Distribution B.11.6 A Full Rank Quadratic Form B.11.7 Independence of a Linear and a Quadratic Form
Appendix C Estimation and Inference
C.1 Introduction
C.2 Samples and Random Sampling C.3 Descriptive Statistics C.4 Statistics as Estimators—Sampling Distributions C.5 Point Estimation of Parameters
C.5.1 Estimation in a Finite Sample
C.6 Interval EstimationC.5.2 Efficient Unbiased Estimation C.7 Hypotheses Testing
C.7.1 Classical Testing Procedures
C.7.2 Tests Based on Confidence Intervals C.7.3 Specification Tests
Appendix D Large-Sample Distribution Theory
D.1 Introduction
D.2 Large-Sample Distribution Theory
D.2.1 Convergence in Probability
D.3 Asymptotic Distributions
D.2.2 Other Forms of Convergence and Laws of Large Numbers D.2.3 Convergence of Functions D.2.4 Convergence to a Random Variable D.2.5 Convergence in Distribution: Limiting Distributions D.2.6 Central Limit Theorems D.2.7 The Delta Method
D.3.1 Asymptotic Distribution of a Nonlinear Function
D.4 Sequences and the Order of a Sequence
D.3.2 Asymptotic Expectations
Appendix E Computation and Optimization
E.1 Introduction
E.2 Computation in Econometrics
E.2.1 Computing Integrals
E.3 Optimization
E.2.2 The Standard Normal Cumulative Distribution Function E.2.3 The Gamma and Related Functions E.2.4 Approximating Integrals by Quadrature
E.3.1 Algorithms
E.4 Examples
E.3.2 Computing Derivatives E.3.3 Gradient Models E.3.4 Aspects of Maximum Likelihood Estimation E.3.5 Optimization with Constraints E.3.6 Some Practical Considerations E.3.7 The EM Algorithm
E.4.1 Function of One Parameter
E.4.2 Function of Two Parameters: The Gamma Distribution E.4.3 A Concentrated Log-Likelihood Function Appendix F Data Sets Used in Applications
Appendix G Statistical Tables
References
Author Index
Subject Index
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