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Econometric Analysis, Seventh Edition

William H. Greene
Publisher: Prentice Hall
Copyright: 2012
ISBN-13: 978-0-13-139538-1
Pages: 1,188; hardcover
Price: $179.00

Comment from the Stata technical group

William Greene’s Econometric Analysis has served as the standard reference for econometrics among economists, political scientists, and other social scientists for two decades. The newly released seventh 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 only 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, the book begins by introducing the linear regression model. Part I of the book, consisting of eight chapters, begins with properties of the least-squares estimator; inference and prediction; and tests for functional form and specification. Chapter 7 covers nonlinear models, including a new discussion of interaction effects. Part I ends with a revised Chapter 8 that covers instrumental variables and endogeneity.

Part II of the book generalizes the linear regression model to allow for heteroskedasticity. Then, with the generalized least-squares (GLS) estimator already discussed in the context of nonspherical disturbances, Greene presents fixed- and random-effects panel-data models as straightforward extensions of least squares. Another chapter applies GLS to systems of regression equations.

Part III devotes one chapter to each of four popular estimation methods: the generalized method of moments, maximum likelihood, simulation, and Bayesian inference. Each chapter strikes a good balance between theoretical rigor and applications. Many newer discrete-choice models require evaluation of multivariate normal probabilities; thus, Chapter 15 includes a detailed discussion of the GHK simulator. New in Chapter 15 is an expanded treatment of the bootstrap.

Part IV covers advanced techniques for microeconometrics. Chapter 17 details binary choice models for both cross-sectional and panel data; a new subsection shows how to account for attrition by using inverse probability weighting. Also included in Part IV are bivariate and multivariate probit models; models for count, multinomial, and ordered outcomes; and models for truncated data, duration data, and sample selection. Part IV ends with a section on treatment effects, propensity-score matching, and regression discontinuity.

Part V covers advanced techniques for macroeconometrics. Chapter 20 on stationary time series describes estimation in the presence of serial correlation, tests for autocorrelation, lagged dependent variables, and ARCH models. Chapter 21 on nonstationary series covers unit roots and cointegration. The chapters in Part V frequently use the results obtained in Part III on estimation. The book concludes with appendices on matrix algebra, probability, distribution theory, and optimization.

Table of contents

Examples and Applications
Part I The Linear Regression Model
Chapter 1 Econometrics
1.1 Introduction
1.2 The Paradigm of Econometrics
1.3 The Practice of Econometrics
1.4 Econometric Modeling
1.5 Plan of the Book
1.6 Preliminaries
1.6.1 Numerical Examples
1.6.2 Software and Replication
1.6.3 Notational Conventions
Chapter 2 The Linear Regression Model
2.1 Introduction
2.2 The Linear Regression Model
2.3 Assumptions of the Linear Regression Model
2.3.1 Linearity of the Regression Model
2.3.2 Full Rank
2.3.3 Regression
2.3.4 Spherical Disturbances
2.3.5 Data Generating Process for the Regressors
2.3.6 Normality
2.3.7 Independence
2.4 Summary and Conclusions
Chapter 3 Least Squares
3.1 Introduction
3.2 Least Squares Regression
3.2.1 The Least Squares Coefficient Vector
3.2.2 Application: An Investment Equation
3.2.3 Algebraic Aspects of the Least Squares Solution
3.2.4 Projection
3.3 Partitioned Regression and Partial Regression
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
3.5.2 R-Squared and the Constant Term in the Model
3.5.3 Comparing Models
3.6 Linearly Transformed Regression
3.7 Summary and Conclusions
Chapter 4 The Least Squares Estimator
4.1 Introduction
4.2 Motivating Least Squares
4.2.1 The Population Orthogonality Conditions
4.2.2 Minimum Mean Squared Error Predictor
4.2.3 Minimum Variance Linear Unbiased Estimation
4.3 Finite Sample Properties of Least Squares
4.3.1 Unbiased Estimation
4.3.2 Bias Caused by Omission of Relevant Variables
4.3.3 Inclusion of Irrelevant Variables
4.3.4 The Variance of the Least Squares Estimator
4.3.5 The Gauss–Markov Theorem
4.3.6 The Implications of Stochastic Regressors
4.3.7 Estimating the Variance of the Least Squares Estimator
4.3.8 The Normality Assumption
4.4 Large Sample Properties of the Least Squares Estimator
4.4.1 Consistency of the Least Squares Estimator Β
4.4.2 Asymptotic Normality of the Least Squares Estimator
4.4.3 Consistency of s2 and the Estimator of Asy. Var[b]
4.4.4 Asymptotic Distribution of a Function of b: The Delta Method
4.4.5 Asymptotic Efficiency
4.4.6 Maximum Likelihood Estimation
4.5 Interval Estimation
4.5.1 Forming a Confidence Interval for a Coefficient
4.5.2 Confidence Intervals Based on Large Samples
4.5.3 Confidence Interval for a Linear Combination of Coefficients: The Oaxaca Decomposition
4.6 Prediction and Forecasting
4.6.1 Prediction Intervals
4.6.2 Predicting y When the Regression Model Describes Log y
4.6.3 Prediction Interval for y When the Regression Model Describes Log y
4.6.4 Forecasting
4.7 Data Problems
4.7.1 Multicollinearity
4.7.2 Pretest Estimation
4.7.3 Principal Components
4.7.4 Missing Values and Data Imputation
4.7.5 Measurement Error
4.7.6 Outliers and Influential Observations
4.8 Summary and Conclusions
Chapter 5 Hypothesis Tests and Model Selection
5.1 Introduction
5.2 Hypothesis Testing Methodology
5.2.1 Restrictions and Hypotheses
5.2.2 Nested Models
5.2.3 Testing Procedures—Neyman–Pearson Methodology
5.2.4 Size, Power, and Consistency of a Test
5.2.5 A Methodological Dilemma: Bayesian Versus Classical Testing
5.3 Two Approaches to Testing Hypotheses
5.4 Wald Tests Based on the Distance Measure
5.4.1 Testing of a Hypothesis about a Coefficient
5.4.2 The F Statistic and the Least Squares Discrepancy
5.5 Testing Restrictions Using the Fit of the Regression
5.5.1 The Restricted Least Squares Estimator
5.5.2 The Loss of Fit from Restricted Least Squares
5.5.3 Testing the Significance of the Regression
5.5.4 Solving Out the Restrictions and a Caution about Using R2
5.6 Nonnormal Disturbances and Large-Sample Tests
5.7 Testing Nonlinear Restrictions
5.8 Choosing between Nonnested Models
5.8.1 Testing Nonnested Hypotheses
5.8.2 An Encompassing Model
5.8.3 Comprehensive Approach—The J Test
5.9 A Specification Test
5.10 Model Building—A General Guide to Simple Strategy
5.10.1 Model Selection Criteria
5.10.2 Model Selection
5.10.3 Classical Model Selection
5.10.4 Bayesian Model Averaging
5.11 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
6.2.2 Several Categories
6.2.3 Several Groupings
6.2.4 Threshold Effects and Categorical Variables
6.2.5 Treatment Effects and Differences in Differences Regression
6.3 Nonlinearity in the Variables
6.3.1 Piecewise Linear Regression
6.3.2 Functional Forms
6.3.3 Interaction Effects
6.3.4 Identifying Nonlinearity
6.3.5 Intrinsically Linear Models
6.4 Modeling and Testing for a Structural Break
6.4.1 Different Parameter Vectors
6.4.2 Insufficient Observations
6.4.3 Change in a Subset of Coefficients
6.4.4 Tests of Structural Break with Unequal Variances
6.4.5 Predictive Test of Model Stability
6.5 Summary and Conclusions
Chapter 7 Nonlinear, Semiparametric, and Nonparametric Regression Models
7.1 Introduction
7.2 Nonlinear Regression Models
7.2.1 Assumptions of the Nonlinear Regression Model
7.2.2 The Nonlinear Least Squares Estimator
7.2.3 Large Sample Properties of the Nonlinear Least Squares Estimator
7.2.4 Hypothesis Testing and Parametric Restrictions
7.2.5 Applications
7.2.6 Computing the Nonlinear Least Squares Estimator
7.3 Median and Quantile Regression
7.3.1 Least Absolute Deviations Estimation
7.3.2 Quantile Regression Models
7.4 Partially Linear Regression
7.5 Nonparametric Regression
7.6 Summary and Conclusions
Chapter 8 Endogeneity and Instrumental Variable Estimation
8.1 Introduction
8.2 Assumptions of the Extended Model
8.3 Estimation
8.3.1 Least Squares
8.3.2 The Instrumental Variables Estimator
8.3.3 Motivating the Instrumental Variables Estimator
8.3.4 Two-Stage Least Squares
8.4 Two Specification Tests
8.4.1 The Hausman and Wu Specification Tests
8.4.2 A Test for Overidentification
8.5 Measurement Error
8.5.1 Least Squares Attenuation
8.5.2 Instrumental Variables Estimation
8.5.3 Proxy Variables
8.6 Nonlinear Instrumental Variables Estimation
8.7 Weak Instruments
8.8 Natural Experiments and the Search for Causal Effects
8.9 Summary and Conclusions
Part II Generalized Regression Model and Equation Systems
Chapter 9 The Generalized Regression Model and Heteroscedasticity
9.1 Introduction
9.2 Inefficient Estimation by Least Squares and Instrumental Variables
9.2.1 Finite-Sample Properties of Ordinary Least Squares
9.2.2 Asymptotic Properties of Ordinary Least Squares
9.2.3 Robust Estimation of Asymptotic Covariance Matrices
9.2.4 Instrumental Variable Estimation
9.3 Efficient Estimation by Generalized Least Squares
9.3.1 Generalized Least Squares (GLS)
9.3.2 Feasible Generalized Least Squares (FGLS)
9.4 Heteroscedasticity and Weighted Least Squares
9.4.1 Ordinary Least Squares Estimation
9.4.2 Inefficiency of Ordinary Least Squares
9.4.3 The Estimated Covariance Matrix of b
9.4.4 Estimating the Appropriate Covariance Matrix for Ordinary Least Squares
9.5 Testing for Heteroscedasticity
9.5.1 White’s General Test
9.5.2 The Breusch–Pagan/Godfrey LM Test
9.6 Weighted Least Squares
9.6.1 Weighted Least Squares with Known Ω
9.6.2 Estimation when Ω Contains Unknown Parameters
9.7 Applications
9.7.1 Multiplicative Heteroscedasticity
9.7.2 Groupwise Heteroscedasticity
9.8 Summary and Conclusions
Chapter 10 Systems of Equations
10.1 Introduction
10.2 The Seemingly Unrelated Regression Models
10.2.1 Generalized Least Squares
10.2.2 Seemingly Unrelated Regressions with Identical Regressors
10.2.3 Feasible Generalized Least Squares
10.2.4 Testing Hypotheses
10.2.5 A Specification Test for the SUR Model
10.2.6 The Pooled Model
10.3 Seemingly Unrelated Generalized Regression Models
10.4 Nonlinear Systems of Equations
10.5 Systems of Demand Equations: Singular Systems
10.5.1 Cobb–Douglas Cost Function
10.5.2 Flexible Functional Forms: The Translog Cost Function
10.6 Simultaneous Equations Models
10.6.1 Systems of Equations
10.6.2 A General Notation for Linear Simultaneous Equations Models
10.6.3 The Problem of Identification
10.6.4 Single Equation Estimation and Inference
10.6.5 System Methods of Estimation
10.6.6 Testing in the Presence of Weak Instruments
10.7 Summary and Conclusions
Chapter 11 Models for Panel Data
11.1 Introduction
11.2 Panel Data Models
11.2.1 General Modeling Framework for Analyzing Panel Data
11.2.2 Model Structures
11.2.3 Extensions
11.2.4 Balanced and Unbalanced Panels
11.2.5 Well-Behaved Panel Data
11.3 The Pooled Regression Model
11.3.1 Least Squares Estimation of the Pooled Model
11.3.2 Robust Covariance Matrix Estimation
11.3.3 Clustering and Stratification
11.3.4 Robust Estimation Using Group Means
11.3.5 Estimation with First Differences
11.3.6 The Within- and Between-Groups Estimators
11.4 The Fixed Effects Model
11.4.1 Least Squares Estimation
11.4.2 Small T Asymptotics
11.4.3 Testing the Significance of the Group Effects
11.4.4 Fixed Time and Group Effects
11.4.5 Time-Invariant Variables and Fixed Effects Vector Decomposition
11.5 Random Effects
11.5.1 Least Squares Estimation
11.5.2 Generalized Least Squares
11.5.3 Feasible Generalized Least Squares when Σ Is Unknown
11.5.4 Testing for Random Effects
11.5.5 Hausman’s Specification Test for the Random Effects Model
11.5.6 Extending the Unobserved Effects Model: Mundlak’s Approach
11.5.7 Extending the Random and Fixed Effects Models: Chamberlain’s Approach
11.6 Nonspherical Disturbances and Robust Covariance Estimation
11.6.1 Robust Estimation of the Fixed Effects Model
11.6.2 Heteroscedasticity in the Random Effects Model
11.6.3 Autocorrelation in Panel Data Models
11.6.4 Cluster (and Panel) Robust Covariance Matrices for Fixed and Random Effects Estimators
11.7 Spatial Autocorrelation
11.8 Endogeneity
11.8.1 Hausman and Taylor’s Instrumental Variables Estimator
11.8.2 Consistent Estimation of Dynamic Panel Models: Anderson and Hsiao’s IV Estimator
11.8.3 Efficient Estimation of Dynamic Panel Data Models—The Arellano/Bond Estimators
11.8.4 Nonstationary Data and Panel Data Models
11.9 Nonlinear Regression with Panel Data
11.9.1 A Robust Covariance Matrix for Nonlinear Least Squares
11.9.2 Fixed Effects
11.9.3 Random Effects
11.10 Systems of Equations
11.11 Parameter Heterogeneity
11.11.1 The Random Coefficients Model
11.11.2 A Hierarchical Linear Model
11.11.3 Parameter Heterogeneity and Dynamic Panel Data Models
11.12 Summary and Conclusions
Part III Estimation Methodology
Chapter 12 Estimation Frameworks in Econometrics
12.1 Introduction
12.2 Parametric Estimation and Inference
12.2.1 Classical Likelihood-Based Estimation
12.2.2 Modeling Joint Distributions with Copula Functions
12.3 Semiparametric Estimation
12.3.1 GMM Estimation in Econometrics
12.3.2 Maximum Empirical Likelihood Estimation
12.3.3 Least Absolute Deviations Estimation and Quantile Regression
12.3.4 Kernel Density Methods
12.3.5 Comparing Parametric and Semiparametric Analyses
12.4 Nonparametric Estimation
12.4.1 Kernel Density Estimation
12.5 Properties of Estimators
12.5.1 Statistical Properties of Estimators
12.5.2 Extremum Estimators
12.5.3 Assumptions for Asymptotic Properties of Extremum Estimators
12.5.4 Asymptotic Properties of Estimators
12.5.5 Testing Hypotheses
12.6 Summary and Conclusions
Chapter 13 Minimum Distance Estimation and the Generalized Method of Moments
13.1 Introduction
13.2 Consistent Estimation: The Method of Moments
13.2.1 Random Sampling and Estimating the Parameters of Distributions
13.2.2 Asymptotic Properties of the Method of Moments Estimator
13.2.3 Summary—The Method of Moments
13.3 Minimum Distance Estimation
13.4 The Generalized Method of Moments (GMM) Estimator
13.4.1 Estimation Based on Orthogonality Conditions
13.4.2 Generalizing the Method of Moments
13.4.3 Properties of the GMM Estimator
13.5 Testing Hypotheses in the GMM Framework
13.5.1 Testing the Validity of the Moment Restrictions
13.5.2 GMM Counterparts in the WALD, LM, and LR Tests
13.6 GMM Estimation of Econometric Models
13.6.1 Single-Equation Linear Models
13.6.2 Single-Equation Nonlinear Models
13.6.3 Seemingly Unrelated Regression Models
13.6.4 Simultaneous Equations Models with Heteroscedasticity
13.6.5 GMM Estimation of Dynamic Panel Data Models
13.7 Summary and Conclusions
Chapter 14 Maximum Likelihood Estimation
14.1 Introduction
14.2 The Likelihood Function and Identification of the Parameters
14.3 Efficient Estimation: The Principle of Maximum Likelihood
14.4 Properties of the Maximum Likelihood Estimators
14.4.1 Regularity Conditions
14.4.2 Properties of Regular Densities
14.4.3 The Likelihood Equation
14.4.4 The Information Matrix Equality
14.4.5 Asymptotic Properties of the Maximum Likelihood Estimator
14.4.5a Consistency
14.4.5b Asymptotic Normality
14.4.5c Asymptotic Efficiency
14.4.5d Invariance
14.4.5e Conclusion
14.4.6 Estimating the Asymptotic Variance of the Maximum Likelihood Estimator
14.5 Conditional Likelihoods, Econometric Models, and the GMM Estimator
14.6 Hypothesis and Specification Tests and Fit Measures
14.6.1 The Likelihood Ratio Test
14.6.2 The Wald Test
14.6.3 The Lagrange Multiplier Test
14.6.4 An Application of the Likelihood-Based Test Procedures
14.6.5 Comparing Models and Computing Model Fit
14.6.6 Vuong’s Test and the Kullback–Leibler Information Criterion
14.7 Two-Step Maximum Likelihood Estimation
14.8 Pseudo-Maximum Likelihood Estimation and Robust Asymptotic Covariance Matrices
14.8.1 Maximum Likelihood and GMM Estimation
14.8.2 Maximum Likelihood and M Estimation
14.8.3 Sandwich Estimators
14.8.4 Cluster Estimators
14.9 Applications of Maximum Likelihood Estimation
14.9.1 The Normal Linear Regression Model
14.9.2 The Generalized Regression Model
14.9.2.a Multiplicative Heteroscedasticity
14.9.2.b Autocorrelation
14.9.3 Seemingly Unrelated Regression Models
14.9.3.a The Pooled Model
14.9.3.b The SUR Model
14.9.3.c Exclusion Restrictions
14.9.4 Simultaneous Equation Models
14.9.5 Maximum Likelihood Estimation of Nonlinear Regression Models
14.9.6 Panel Data Applications
14.9.6.a ML Estimation of the Linear Random Effects Model
14.9.6.b Nested Random Effects
14.9.6.c Random Effects in Nonlinear Models: MLE Using Quadrature
14.9.6.d Fixed Effects in Nonlinear Models: Full MLE
14.10 Latent Class and Finite Mixture Models
14.10.1 A Finite Mixture Model
14.10.2 Measured and Unmeasured Heterogeneity
14.10.3 Predicting Class Membership
14.10.4 A Conditional Latent Class Model
14.10.5 Determining the Number of Classes
14.10.6 A Panel Data Application
14.11 Summary and Conclusions
Chapter 15 Simulation-Based Estimation and Inference and Random Parameter Models
15.1 Introduction
15.2 Random Number Generation
15.2.1 Generating Pseudo-Random Numbers
15.2.2 Sampling from a Standard Uniform Population
15.2.3 Sampling from Continuous Distributions
15.2.4 Sampling from a Multivariate Normal Population
15.2.5 Sampling from Discrete Populations
15.3 Simulation-Based Statistical Inference: The Method of Krinsky and Robb
15.4 Bootstrapping Standard Errors and Confidence Intervals
15.5 Monte Carlo Studies
15.5.1 A Monte Carlo Study: Behavior of a Test Statistic
15.5.2 A Monte Carlo Study: The Incidental Parameters Problem
15.6 Simulation-Based Estimation
15.6.1 Random Effects in a Nonlinear Model
15.6.2 Monte Carlo Integration
15.6.2.a Halton Sequences and Random Draws for Simulation-Based Integration
15.6.2.b Computing Multivariate Normal Probabilities Using the GHK Simulator
15.6.3 Simulation-Based Estimation of Random Effects Models
15.7 A Random Parameters Linear Regression Model
15.8 Hierarchical Linear Models
15.9 Nonlinear Random Parameter Models
15.10 Individual Parameter Estimates
15.11 Mixed Models and Latent Class Models
15.12 Summary and Conclusions
Chapter 16 Bayesian Estimation and Inference
16.1 Introduction
16.2 Bayes Theorem and the Posterior Density
16.3 Bayesian Analysis of the Classical Regression Model
16.3.1 Analysis with a Noninformative Prior
16.3.2 Estimation with an Informative Prior Density
16.4 Bayesian Inference
16.4.1 Point Estimation
16.4.2 Interval Estimation
16.4.3 Hypothesis Testing
16.4.4 Large-Sample Results
16.5 Posterior Distributions and the Gibbs Sampler
16.6 Application: Binomial Probit Model
16.7 Panel Data Application: Individual Effects Models
16.8 Hierarchical Bayes Estimation of a Random Parameters Model
16.9 Summary and Conclusions
Part IV Cross Sections, Panel Data, and Microeconometrics
Chapter 17 Discrete Choice
17.1 Introduction
17.2 Models for Binary Outcomes
17.2.1 Random Utility Models for Individual Choice
17.2.2 A Latent Regression Model
17.2.3 Functional Form and Regression
17.3 Estimation and Inference in Binary Choice Models
17.3.1 Robust Covariance Matrix Estimation
17.3.2 Marginal Effects and Average Partial Effects
17.3.2.a Average Partial Effects
17.3.2.b Interaction Effects
17.3.3 Measuring Goodness of Fit
17.3.4 Hypothesis Tests
17.3.5 Endogenous Right-Hand-Side Variables in Binary Choice Models
17.3.6 Endogenous Choice-Based Sampling
17.3.7 Specification Analysis
17.3.7.a Omitted Variables
17.3.7.b Heteroscedasticity
17.4 Binary Choice Models for Panel Data
17.4.1 The Pooled Estimator
17.4.2 Random Effects Models
17.4.3 Fixed Effects Models
17.4.4 A Conditional Fixed Effects Estimator
17.4.5 Mundlak’s Approach, Variable Addition, and Bias Reduction
17.4.6 Dynamic Binary Choice Models
17.4.7 A Semiparametric Model for Individual Heterogeneity
17.4.8 Modeling Parameter Heterogeneity
17.4.9 Nonresponse, Attrition, and Inverse Probability Weighting
17.5 Bivariate and Multivariate Probit Models
17.5.1 Maximum Likelihood Estimation
17.5.2 Testing for Zero Correlation
17.5.3 Partial Effects
17.5.4 A Panel Data Model for Bivariate Binary Response
17.5.5 Endogenous Binary Variable in a Recursive Bivariate Probit Model
17.5.6 Endogenous Sampling in a Binary Choice Model
17.5.7 A Multivariate Probit Model
17.6 Summary and Conclusions
Chapter 18 Discrete Choices and Event Counts
18.1 Introduction
18.2 Models for Unordered Multiple Choices
18.2.1 Random Utility Basis of the Multinomial Logit Model
18.2.2 The Multinomial Logit Model
18.2.3 The Conditional Logit Model
18.2.4 The Independence from Irrelevant Alternatives Assumption
18.2.5 Nested Logit Models
18.2.6 The Multinomial Probit Model
18.2.7 The Mixed Logit Model
18.2.8 A Generalized Mixed Logit Model
18.2.9 Application: Conditional Logit Model for Travel Mode Choice
18.2.10 Estimating Willingness to Pay
18.2.11 Panel Data and Stated Choice Experiments
18.2.12 Aggregate Market Share Data—The BLP Random Parameters Model
18.3 Random Utility Models for Ordered Choices
18.3.1 The Ordered Probit Model
18.3.2 A Specification Test for the Ordered Choice Model
18.3.3 Bivariate Ordered Probit Models
18.3.4 Panel Data Applications
18.3.4.a Ordered Probit Models with Fixed Effects
18.3.4.b Ordered Probit Models with Random Effects
18.3.5 Extensions of the Ordered Probit Model
18.3.5.a Threshold Models—Generalized Ordered Choice Models
18.3.5.b Thresholds and Heterogeneity—Anchoring Vignettes
18.4 Models for Counts of Events
18.4.1 The Poisson Regression Model
18.4.2 Measuring Goodness of Fit
18.4.3 Testing for Overdispersion
18.4.4 Heterogeneity and the Negative Binomial Regression Model
18.4.5 Functional Forms for Count Data Models
18.4.6 Truncation and Censoring in Models for Counts
18.4.7 Panel Data Models
18.4.7.a Robust Covariance Matrices for Pooled Estimators
18.4.7.b Fixed Effects
18.4.7.c Random Effects
18.4.8 Two-Part Models: Zero-Inflation and Hurdle Models
18.4.9 Endogenous Variables and Endogenous Participation
18.5 Summary and Conclusions
Chapter 19 Limited Dependent Variables—Truncation, Censoring, and Sample Selection
19.1 Introduction
19.2 Truncation
19.2.1 Truncated Distributions
19.2.2 Moments of Truncated Distributions
19.2.3 The Truncated Regression Model
19.2.4 The Stochastic Frontier Model
19.3 Censored Data
19.3.1 The Censored Normal Distribution
19.3.2 The Censored Regression (Tobit) Model
19.3.3 Estimation
19.3.4 Two-Part Models and Corner Solutions
19.3.5 Some Issues in Specification
19.3.5.a Heteroscedasticity
19.3.5.b Nonnormality
19.3.6 Panel Data Applications
19.4 Models for Duration
19.4.1 Models for Duration Data
19.4.2 Duration Data
19.4.3 A Regression-Like Approach: Parametric Models of Duration
19.4.3.a Theoretical Background
19.4.3.b Models of the Hazard Function
19.4.3.c Maximum Likelihood Estimation
19.4.3.d Exogenous Variables
19.4.3.e Heterogeneity
19.4.4 Nonparametric and Semiparametric Approaches
19.5 Incidental Truncation and Sample Selection
19.5.1 Incidental Truncation in a Bivariate Distribution
19.5.2 Regression in a Model of Selection
19.5.3 Two-Step and Maximum Likelihood Estimation
19.5.4 Sample Selection in Nonlinear Models
19.5.5 Panel Data Applications of Sample Selection Models
19.5.5.a Common Effects in Sample Selection Models
19.5.5.b Attrition
19.6 Evaluating Treatment Effects
19.6.1 Regression Analysis of Treatment Effects
19.6.1.a The Normality Assumption
19.6.1.b Estimating the Effect of Treatment on the Treated
19.6.2 Propensity Score Matching
19.6.3 Regression Discontinuity
19.7 Summary and Conclusions
Part V Time Series and Macroeconomics
Chapter 20 Serial Correlation
20.1 Introduction
20.2 The Analysis of Time-Series Data
20.3 Disturbance Processes
20.3.1 Characteristics of Disturbance Processes
20.3.2 AR(1) Disturbances
20.4 Some Asymptotic Results for Analyzing Time-Series Data
20.4.1 Convergence of Moments—The Ergodic Theorem
20.4.2 Convergence of Normality—A Central Limit Theorem
20.5 Least Squares Estimation
20.5.1 Asymptotic Properties of Least Squares
20.5.2 Estimating the Variance of the Least Squares Estimator
20.6 GMM Estimation
20.7 Testing for Autocorrelation
20.7.1 Lagrange Multiplier Test
20.7.2 Box and Pierce’s Test and Ljung’s Refinement
20.7.3 The Durbin–Watson Test
20.7.4 Testing in the Presence of a Lagged Dependent Variable
20.7.5 Summary of Testing Procedures
20.8 Efficient Estimation When Ω Is Known
20.9 Estimation When Ω Is Unknown
20.9.1 AR(1) Disturbances
20.9.2 Application: Estimation of a Model with Autocorrelation
20.9.3 Estimation with a Lagged Dependent Variable
20.10 Autoregressive Conditional Heteroscedasticity
20.10.1 The ARCH(1) Model
20.10.2 ARCH(q), ARCH-in-Mean, and Generalized ARCH Models
20.10.3 Maximum Likelihood Estimation of the Garch Model
20.10.4 Testing for Garch Effects
20.10.5 Pseudo-Maximum Likelihood Estimation
20.11 Summary and Conclusions
Chapter 21 Nonstationary Data
21.1 Introduction
21.2 Nonstationary Processes and Unit Roots
21.2.1 Integrated Processes and Differencing
21.2.2 Random Walks, Trends, and Spurious Regressions
21.2.3 Tests for Unit Roots in Economic Data
21.2.4 The Dickey–Fuller Tests
21.2.5 The KPSS Test of Stationarity
21.3 Cointegration
21.3.1 Common Trends
21.3.2 Error Correction and VAR Representations
21.3.3 Testing for Cointegration
21.3.4 Estimating Cointegration Relationships
21.3.5 Application: German Money Demand
21.3.5.a Cointegration Analysis and a Long-Run Theoretical Model
21.3.5.b Testing for Model Instability
21.4 Nonstationary Panel Data
21.5 Summary and Conclusions
Part VI Appendices
Appendix A Matrix Algebra
A.1 Terminology
A.2 Algebraic Manipulation of Matrices
A.2.1 Equality 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 Geometry of Matrices
A.3.1 Vector Spaces
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 Solution of a System of Linear Equations
A.4.1 Systems of Linear Equations
A.4.2 Inverse Matrices
A.4.3 Nonhomogeneous Systems of Equations
A.4.4 Solving the Least Squares Problems
A.5 Partitioned Matrices
A.5.1 Addition and Multiplication of Partitioned Matrices
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 Characteristic Roots and Vectors
A.6.1 The Characteristic Equation
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 Matrices
A.6.11 Factoring a Matrix
A.6.12 The Generalized Inverse of a Matrix
A.7 Quadratic Forms and Definite Matrices
A.7.1 Nonnegative Definite Matrices
A.7.2 Idempotent Quadratic Forms
A.7.3 Comparing Matrices
A.8 Calculus and Matrix Algebra
A.8.1 Differentiation and the Taylor Series
A.8.2 Optimization
A.8.3 Constrained Optimization
A.8.4 Transformations
Appendix B Probability and Distribution Theory
B.1 Introduction
B.2 Random Variables
B.2.1 Probability Distributions
B.2.2 Cumulative Distributions Function
B.3 Expectations of a Random Variable
B.4 Some Specific Probability Distributions
B.4.1 The Normal Distribution
B.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.5 The Distribution of a Function of a Random Variable
B.6 Representations of a Probability Distribution
B.7 Joint Distributions
B.7.1 Marginal Distributions
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 Conditioning in a Bivariate Distribution
B.8.1 Regression: The Conditional Mean
B.8.2 Conditional Variance
B.8.3 Relationships Among Marginal and Conditional Moments
B.8.4 The Analysis of Variance
B.9 The Bivariate Normal Distribution
B.10 Multivariate Distributions
B.10.1 Moments
B.10.2 Sets of Linear Functions
B.10.3 Nonlinear Functions
B.11 The Multivariate Normal Distribution
B.11.1 Marginal and Conditional Normal Distribution
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.5.2 Efficient Unbiased Estimation
C.6 Interval Estimation
C.7 Hypothesis 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.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 Asymptotic Distributions
D.3.1 Asymptotic Distribution of a Nonlinear Function
D.3.2 Asymptotic Expectations
D.4 Sequences and the Order of a Sequence
Appendix E Computation and Optimization
E.1 Introduction
E.2 Computation in Econometrics
E.2.1 Computing Integrals
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 Optimization
E.3.1 Algorithms
E.3.2 Computing Derivatives
E.3.3 Gradient Methods
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 Examples
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
Combined Author and Subject Index
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