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Brief Contents Preface Part I The Linear Regression Model Chapter 1 Introduction Chapter 2 The Classical Multiple Linear Regression Model Chapter 3 Least Squares Chapter 4 Properties of the Least Squares Estimator Chapter 5 Inference and Prediction Chapter 6 Functional Form and Structural Change Chapter 7 Specification Analysis and Model Selection Part II The Generalized Regression Model Chapter 8 The Generalized Regression Model and Heteroscedasticity Chapter 9 Models for Panel and Stratified Data Chapter 10 Systems of Regression Equations Chapter 11 Nonlinear Regressions and Nonlinear Least Squares Part III Instrumental Variables and Simultaneous Equations Models Chapter 12 Instrumental Variables Estimation Chapter 13 Simultaneous-Equations Models Part IV Estimation Methodology Chapter 14 Estimation Frameworks in Econometrics Chapter 15 Minimum Distance Estimation and The Generalized Method of Moments Chapter 16 Maximum Likelihood Chapter 17 Simulation Based Estimation and the Bootstrap Chapter 18 Bayesian Inference in Econometrics Part V Time Series and Macroeconometrics Chapter 19 Serial Correlation Chapter 20 Models with Lagged Variables Chapter 21 Time Series Models Chapter 22 Nonstationary Data Part VI Cross Sections, Panel Data and Microeconometrics Chapter 23 Models for Discrete Choice Chapter 24 Limited Dependent Variable Models Chapter 25 Sample Selection and Treatment Effects Chapter 26 Even Counts and Duration Models Part VII Appendices Appendix A Matrix Algebra Appendix B Probability and Distribution Theory Appendix C Estimation and Inference Appendix D Large Sample Distribution Theory Appendix E Computation and Optimization Appendix F Data Sets Used in Applications Appendix G Statistical Tables References Author Index Subject Index Contents 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 2.3.2 Full Rank 2.3.3 Regression 2.3.4 Spherical Disturbances 2.3.5 Data Generating Process for the Regressors 2.3.6 Normality 2.4 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 Projection 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 Summary and Conclusions CHAPTER 4 Statistical Properties of the Least Squares Estimator 4.1 Introduction 4.2 Motivating Least Squares 4.2.1 The Population Orthogonality Conditions 4.2.2 Minimum Mean Squared Error Projection 4.2.3 Minimum Variance Linear Unbiased Estimation 4.3 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 Inference 4.7.1 Testing a Hypothesis About a Coefficient 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 Finite-Sample Properties of the Least Squares Estimator 4.8.1 Multicollinearity 4.8.2 Missing Observations 4.9 Large Sample Properties of the Least Squares Estimator 4.9.1 Consistency of the Least Squares Estimator of _ 4.9.2 Asymptotic Normality of the Least Squares Estimator 4.9.3 Consistency of s_ and the Estimator of Asy.Var[_] 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 4.10 Summary and Conclusions CHAPTER 5 Inference 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 5.3.2 The Restricted Least Squares Estimator 5.3.3 The Loss of Fit From Restricted Least Squares 5.4 Nonnormal Disturbances and Large Sample Tests 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 6.2.2 Several Categories 6.2.3 Several Groupings 6.2.4 Threshold Effects and Categorical Variables 6.2.5 Spline Regression 6.3 Nonlinearity in the Variables 6.3.1 Functional Forms 6.3.2 Identifying Nonlinearity 6.3.3 Intrinsic Linearity and Identification 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 6.5 Summary and Conclusions 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 7.2.2 Pretest Estimation 7.2.3 Inclusion of Irrelevant Variables 7.2.4 Model Building - A General to Simple Strategy 7.3 Choosing Between Nonnested Models 7.3.1 Testing Nonnested Hypotheses 7.3.2 An Encompassing Model 7.3.3 Comprehensive Approach - The J Test 7.3.4 Vuong's Test and Kullback-Leibler Information Criterion 7.4 Model Selection Criteria 7.5 Model Selection 7.5.1 Classical Model Selection 7.5.2 Bayesian Model Averaging 7.6 Summary and Conclusions 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 Least Squares 8.2.2 Asymptotic Properties of Least Squares 8.2.3 Robust Estimation of Asymptotic Covariance Matrices 8.3 Efficient Estimation by Generalized Least Squares 8.3.1 Generalized Least Squares (GLS) 8.3.2 Feasible Generalized Least Squares (FGLS) 8.4 Heteroscedasticity 8.4.1 Ordinary Least Squares Estimation 8.4.2 Inefficiency of Least Squares 8.4.3 The Estimated Covariance Matrix of b 8.4.4 Estimating the Appropriate Covariance Matrix for Ordinary Least Squares 8.5 Testing for Heteroscedasticity 8.5.1 White's General Test 8.5.2 The Breusch-Pagan-Godfrey LM Test 8.6 Weighted Least Squares When _ is Known 8.7 Estimation When _ Contains Unknown Parameters 8.8 Applications 8.8.1 Multiplicative Heteroscedasticity 8.8.2 Groupwise Heteroscedasticity 8.9 Summary and Conclusions CHAPTER 9 Models for Panel Data 9.1 Introduction 9.2 Panel Data Models 9.2.1 General Modeling Structures for Analyzing Panel Data 9.2.2 Model Structures 9.2.3 Extensions 9.2.4 Balanced and Unbalanced Panels 9.3 The Pooled Regression Model 9.3.1 Least Squares Estimation of the Pooled 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 The Fixed Effects Model 9.4.1 Least Squares Estimation 9.4.2 Small T Asymptotics 9.4.3 Testing the Significance of the Group Effects 9.4.4 Fixed Time and Group Effects 9.5 Random Effects 9.5.1 Generalized Least Squares 9.5.2 Feasible Generalized Least Squares when _ 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 Nonspherical Disturbances and Robust Covariance Matrix Estimation 9.6.1 Robust Estimation of the Fixed Effects Model 9.6.2 Heteroscedasticity in the Random Effects Model 9.6.3 Autocorrelation in Panel Data Models 9.7 Extensions of the Random Effects Model 9.7.1 Nested Random Effects 9.7.2 Spatial Autocorrelation 9.8 Parameter Heterogeneity 9.8.1 The Random Coefficients Model 9.8.2 Random Parameters and Simulation Based Estimation 9.8.3 Two Step estimation of Panel Data Models 9.8.4 Hierarchical Models 9.8.5 Parameter Heterogeneity and Dynamic Panel Data Models 9.8.6 Nonstationary Data and Panel Data Models 9.9 Consistent Estimation of Dynamic Panel Data Models 9.10 Summary and Conclusions CHAPTER 10 System of Regression Equations 10.1 Introduction 10.2 The Seemingly Unrelated Regressions Model 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 Heteroscedasticity 10.2.6 Autocorrelation 10.2.7 A Specification Test for the SUR Model 10.2.8 The Pooled Model 10.3 Panel Data Applications 10.3.1 Random Effects SUR Models 10.3.2 The Random and Fixed Effects Models 10.4 Systems of Demand Equations: Singular Systems 10.4.1 Cobb-Douglas Cost Function 10.4.2 Flexible Functional Forms: The Translog Cost Function 10.5 Summary and Conclusions 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 11.2.2 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 Applications 11.3.1 A Nonlinear Consumption Function 11.3.2 The Box-Cox Transformation 11.4 Hypothesis Testing and Parametric Restrictions 11.4.1 Significance Tests for Restrictions: F and Wald Tests 11.4.2 Tests Based on the LM Statistic 11.5 Nonlinear Systems of Equations 11.6 Two-Step Nonlinear Least Squares Estimation 11.7 Panel Data Applications 11.7.1 A Robust Covariance Matrix for Nonlinear Least Squares 11.7.2 Fixed Effects 11.7.3 Random Effects 11.8 Summary and Conclusions Part III Instrumental Variables and Simultaneous Equations Models CHAPTER 12 Instrumental Variables Estimation 12.1 Introduction 12.2 Assumptions of the Model 12.3 Estimation 12.3.1 Ordinary Least Squares 12.3.2 The Instrumental Variables Estimator 12.3.3 Two Stage Least Squares 12.4 The Hausman and Wu Specification Tests and an Application to Instrumental Variable Estimation 12.5 Measurement Error 12.5.1 Least Squares Attenuation 12.5.2 Instrumental Variables Estimation 12.5.3 Proxy Variables 12.6 Estimation of the Generalized Regression Model 12.7 Nonlinear Instrumental Variables Estimation 12.8 Panel Data Applications 12.8.1 Instrumental Variables Estimation of the Random Effects Model 12.8.2 Dynamic Panel Data Models the Anderson/Hsiao and Arellano/Bond Estimator 12.9 Weak Instruments 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 13.2.2 Endogeneity and Causality 13.2.3 A General Notation for Linear Simultaneous Equations Models 13.3 The Problem of Identification 13.3.1 The Rank and Order Conditions for Identification 13.3.2 Identification Through Other Nonsample Information 13.4 Methods of Estimation 13.5 Single Equation: Limited Information Estimation Methods 13.5.1 Ordinary Least Squares 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 System Methods of Estimation 13.6.1 Three Stage Least Squares 13.6.2 Full Information Maximum Likelihood 13.7 Comparison of Methods - Klein's Model I 13.8 Specification Tests 13.9 Properties of Dynamic Models 13.9.1 Dynamic Models and Their Multipliers 13.9.2 Stability 13.9.3 Adjustment to Equilibrium 13.10 Summary and Conclusions 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 14.2.2 Modeling Joint Distributions with Copula Functions 14.3 Semiparametric Estimation 14.3.1 GMM Estimation in Econometrics 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 Nonparametric Estimation 14.4.1 Kernel Density Estimation 14.4.2 Nonparametric Regression 14.5 Properties of Estimators 14.5.1 Statistical Properties of Estimators 14.5.2 Extremum Estimators 14.5.3 Assumptions for Asymptotic Properties of Extremum Estimators 14.5.4 Properties of Extremum Estimators 14.5.5 Testing Hypotheses 14.6 Summary and Conclusions 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 15.2.2 Asymptotic Properties of the Method of Moments Estimator 15.2.3 Summary - The Method of Moments 15.3 Minimum Distance Estimation 15.4 The Generalized Method of Moments Estimator 15.4.1 Estimation Based on Orthogonality Conditions 15.4.2 Generalizing the Method of Moments 15.4.3 Properties of the GMM Estimator 15.5 Testing Hypothesis in the GMM Framework 15.5.1 Testing the Validity of the Moment Restrictions 15.5.2 GMM Counterparts to the Wald, LM and LR Tests 15.6 GMM Estimation of Econometric Models 15.6.1 Single Equation Linear Models 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 15.7 Summary and Conclusions CHAPTER 16 Maximum Likelihood 16.1 Introduction 16.2 The Likelihood Function and Identification of Parameters 16.3 Efficient Estimation: The Principle of Maximum Likelihood 16.4 Properties of Maximum Likelihood Estimators 16.4.1 Regularity Conditions 16.4.2 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 16.4.5.b Asymptotic Normality 16.4.5.c Asymptotic Efficiency 16.4.5.d Invariance 16.4.5.e Conclusion 16.4.6 Estimating the Asymptotic Variance of the Maximum Likelihood Estimator 16.5 Conditional Likelihoods, Econometric models and the GMM Estimator 16.6 Hypothesis and Specification Tests and Fit Measures 16.6.1 The Likelihood Ratio Test 16.6.2 The Wald Test 16.6.3 The Lagrange Multiplier Test 16.6.4 An Application of the Likelihood Based Procedures 16.6.5 Comparing Models and Computing Model Fit 16.7 Two Step Maximum Likelihood Estimation 16.8 Pseudo-Maximum Likelihood Estimation and Robust Asymptotic Covariance Matrices 16.8.1 Maximum Likelihood and GMM Estimation 16.8.2 Maximum Likelihood Estimation and M Estimation 16.8.3 Sandwich Estimators 16.8.4 Cluster Estimators 16.9 Applications of Maximum Likelihood Estimation 16.9.1 The Normal Linear Regression Model 16.9.2 The Generalized Regression Model 16.9.2.a Multiplicative Heteroscedasticity 16.9.2.b Autocorrelation 16.9.3 Seemingly Unrelated Regression Models 16.9.3.a The Pooled Model 16.9.3.b The SUR Model 16.9.3.c Exclusion Restrictions 16.9.4 Simultaneous Equations Models 16.9.5 Maximum Likelihood Estimation of Nonlinear Regression Models 16.9.5.a Nonnormal Disturbances - The Stochastic Frontier Model 16.9.5.b ML Estimation of a Geometric Regression Model for Count Data 16.9.6 Panel Data Applications 16.9.6.a ML Estimation of the Linear Random Effects Model 16.9.6.b Random Effects in Nonlinear Models: MLE Using rature 16.9.6.c Fixed Effects in Nonlinear Models: Full MLE 16.9.7 Latent Class and Finite Mixture Models 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 16.9.8 Summary and Conclusions CHAPTER 17 Simulation Based Estimation and Inference 17.1 Introduction 17.2 Random Number Generation 17.2.1 Generating Pseudo-Random Numbers 17.2.2 Sampling from a Standard Uniform Population 17.2.3 Sampling from a Continuous Distributions 17.2.4 Sampling from a Multivaraite Normal Population 17.2.5 Sampling from a Discrete Population 17.3 Monte Carlo Integration 17.3.1 Halton Sequences and Random Draws for Simulation Based Integration 17.3.2 Importance Sampling 17.3.3 Computing Multivvariate Normal Probabilities Using the GHK Simulator 17.4 Monte Carlo Studies 17.4.1 A Monte Carlo Study: Behavior of a Test Statistic 17.4.2 A Monte Carlo Study: The Incidental Parameters Problem 17.5 Simulation Based Estimation 17.5.1 Maximum Simulated Likelihood Estimation of Random Parameters Models 17.5.2 The Method of Simulated Moments 17.6 Bootstrapping 17.7 Summary and Conclusion CHAPTER 18 Bayesian Inference in Econometrics 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 18.3.2 Estimation with an Informative Prior Density 18.4 Bayesian Inference 18.4.1 Point Estimation 18.4.2 Interval Estimation 18.4.3 Hypothesis Testing 18.4.4 Large Sample Results 18.5 Posterior Distributions and the Gibbs Sampler 18.6 Application: Binomial Probit Model 18.7 Panel Data Application: Individual Effects Models 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 19.3.2 AR(1) Disturbances 19.4 Some Asymptotic Results for Analyzing Time Series Data 19.4.1 Convergence of Moments - The Ergodic Theorem 19.4.2 Convergence to Normality - A Central Limit Theorem 19.5 Least Squares Estimation 19.5.1 Asymptotic Properties of Least Squares 19.5.2 Estimating the Variance of the Least Squares Estimator 19.6 GMM Estimation 19.7 Test for Autocorrelation 19.7.1 Lagrange Multiplier Test 19.7.2 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 19.8 Efficient Estimation when _ is Known 19.9 Estimation when _ is Unknown 19.9.1 AR(1) Disturbances 19.9.2 Application: Estimation of a Model with Autocorrelation 19.9.3 Estimation with a Lagged Dependent Variable 19.10 Autocorrelation in Panel Data 19.11 Common Factors 19.12 Forecasting in the Presence of Autocorrelation 19.13 Autoregressive Conditional Heteroscedasticity 19.13.1 The ARCH(1) Model 19.13.2 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 19.14 Summary and Conclusions CHAPTER 20 Models with Lagged Variables 20.1 Introduction 20.2 Dynamic Regression Models 20.2.1 Lagged Effects in a Dynamic Model 20.2.2 The Lag and Difference Operators 20.2.3 Specification Search for the Lag Length 20.3 Simple Distributed Lag Models 20.4 Autoregressive Distributed Lag Models 20.4.1 Estimation of the ARDL Model 20.4.2 Computation of the Lag Weights in the ARDL Model 20.4.3 Stability of a Dynamic Equation 20.4.4 Forecasts 20.5 Methodological Issues in the Analysis of Dynamic Models 20.5.1 An Error Correction Model Autocorrelation 20.5.2 Specification Analysis 20.6 Vector Autoregressions 20.6.1 Model Forms 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 for Macroeconomic Variables 20.6.8.b The Sacrifice Ration 20.6.8.c Identification and Estimation of a Structural VAR Model 20.6.8.d Inference 20.6.8.e Empirical Results 20.6.9 VARs in Microeconomics 20.7 Summary and Conclusions CHAPTER 21 Time-Series Models 21.1 Introduction 21.2 Stationary Stochastic Processes 21.2.1 Autoregressive Moving Average Models 21.2.2 Stationarity and Invertibility 21.2.3 Autocorrelations of a Stationary Stochastic Process 21.2.4 Parial 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 The Frequency Domain 21.3.1 Theoretical Results 21.3.2 Empirical Counterparts 21.4 Summary and Conclusions CHAPTER 22 Nonstationary Data 22.1 Introduction 22.2 Nonstationary Processes and Unit Roots 22.2.1 Integrated Processes and Differences 22.2.2 Random Walks, Trends and Spurious Regressions 22.2.3 Tests for Unit Roots in Economic Data 22.2.4 The Dickey-Fuller Tests 22.2.5 The KPSS Test of Stationarity 22.3 Cointegration 22.3.1 Common Trends 22.3.2 Error Correction and VAR Representations 22.3.3 Testing for Cointegration 22.3.4 Estimating Cointegrating Relationships 22.3.5 Application: German Money Demand 22.3.5.a Cointegration Analysis and a Long Run Theoretical Model 22.3.5.b Testing for Model Instability 22.4 Nonstationary Panel Data 22.5 Summary and Conclusions Part VI Cross Sections, Panel Data and Microeconometrics CHAPTER 23 Models for Discrete Choice 23.1 Models for Discrete Choice 23.2 Discrete Choice Models 23.3 Models for Binary Choice 23.3.1 The Regression Approach 23.3.2 Latent Regression - Index Function Models 23.3.3 Random Utility Models 23.4 Estimation and Inference in Binary Choice Models 23.4.1 Robust covariance Matrix Estimation 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 23.4.4.b Heteroscedasticity 23.4.5 Measuring Goodness of Fit 23.4.6 Choice Based Sampling 23.4.7 Dynamic Binary Choice Models 23.5 Binary Choice Models for Panel Data 23.5.1 Random Effects Models 23.5.2 Fixed Effects Models 23.5.3 Modeling Heterogeneity 23.5.4 Parameter Heterogeneity 23.6 Semiparametric Analysis 23.6.1 Semiparametric Estimation 23.6.2 A Kernel Estimator for a Nonparametric Regression Function 23.7 Endogenous Right Hand Side Variables in Binary Choice Models 23.8 Bivariate Probit Models 23.8.1 Maximum Likelihood Estimation 23.8.2 Testing for Zero Correlation 23.8.3 Marginal Effects 23.8.4 Recursive Bivariate Probit Models 23.9 A Multivariate Probit Model 23.10 Analysis of Ordered Choice 23.10.1 The Ordered Probit Model 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 Models for Unordered Multiple Choices 23.11.1 The Multinomial Logit Model 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 23.12 Summary and Conclusions CHAPTER 24 Truncation, Censoring and Sample Selection 24.1 Introduction 24.2 Truncation 24.2.1 Truncated Distributions 24.2.2 Moments of Truncated Distributuions 24.2.3 The Truncated Regression Model 24.3 Censored Data 24.3.1 The Censored Normal Distribution 24.3.2 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 24.3.4.e Conditional Moment Tests 24.4 Panel Data Applications 24.5 Sample Selection 24.5.1 Incidental Truncation in a Bivariate Distribution 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 Normality 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 24.6 Summary and Conclusion 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 25.2.2 Testing for Overdispersion 25.2.3 Heterogeneity and the Negative Binomial Regression Model 25.2.4 Functional Forms for Count Data 25.3 Panel Data Models 25.3.1 Robust Covariance Matrices 25.3.2 Fixed Effects 25.3.3 Random Effects 25.4 Hurdle and Zero Altered Poisson Models 25.5 Censoring and Truncation in Models for Counts 25.5.1 Censoring and Truncation in the Poisson Model 25.5.2 Application: Censoring in the Tobit and Poisson Regression Models 25.6 Models for Duration Data 25.6.1 Duration Data 25.6.2 A Regression-Like Approach: Parametric Models of Duration 25.6.2.a Theoretical Background 25.6.2.b Models of the Hazard Function 25.6.2.c Maximum Likelihood Estimation 25.6.2.d Exogenous Variables 25.6.2.e Heterogeneity 25.6.3 Nonparametric and Semiparametric Approaches 25.7 Summary and Conclusions Part VII 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 Solution of a System of Linear Equations A.4.2 Inverse Matrices A.4.3 Nonhomogeneous Systems of Equation A.4.4 Solving the Least Squares Problem 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 ratic Forms and Definite Matrices A.7.1 Nonnegative Definite Matrices A.7.2 Idempotent ratic 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 Distribution 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 Distributions B.11.2 The Classical Normal Linear Regression Model B.11.3 Linear Functions of a Normal Vector B.11.4 ratic Forms in a Standard Normal Vector B.11.5 The F Distribution B.11.6 A Full Rank ratic Form B.11.7 Independence of a Linear and a ratic 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 Internal 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 rature 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 Application APPENDIX G Statistical Tables REFERENCES AUTHOR INDEX SUBJECT INDEX Examples and Applications CHAPTER 1 Introduction Example 1.1 Behavioral Models and the Nobel Laureates Example 1.2 Keynes's Consumption Function CHAPTER 2 The Classical Multiple Linear Regression Model Example 2.1 Keynes's Consumption Function Example 2.2 Earnings and Education Example 2.3 The U.S. Gasoline Market Example 2.4 The Translog Model Example 2.5 Short Rank CHAPTER 3 Least Squares Example 3.1 Partial Correlations Example 3.2 Fit of a Consumption Function Example 3.3 Analysis of Variance for an Investment Equation CHAPTER 4 Statistical Properties of the Least Squares Estimator Example 4.1 The Sampling Distribution of a Least Squares Estimator Example 4.2 Sampling Variance in the Two-Variable Regression Model Example 4.3 Earning Estimation Example 4.4 Confidence Interval for the Income Elasticity of Demand for Gasoline Example 4.5 F Test for the Earnings Equation Example 4.6 Multicollinearity in the Longley Data Example 4.7 Nonlinear Functions of Parameters: The Delta Method Example 4.8 Nonlinear Functions of Parameters: The Krinsky and Robb Method Example 4.9 The Gamma Regression Model CHAPTER 5 Inference and Prediction Example 5.1 Restricted Investment Equation Example 5.2 Production Function Example 5.3 A Long-Run Marginal Propensity to Consume Example 5.4 Prediction for Investment CHAPTER 6 Functional Form and Structural Change Example 6.1 Dummy Variables in an Earnings Function Example 6.2 Analysis of Covariance Example 6.3 Functional Form for a Nonlinear Cost Function Example 6.4 Intrinsically Linear Regression Example 6.5 CES Production Function Example 6.6 The World Health Report Example 6.7 Structural Break in the Gasoline Market CHAPTER 7 Specification Analysis and Model Selection Example 7.1 Omitted Variable Example 7.2 J Test for a Consumption Function Example 7.3 Vuong Test for a Consumption Function Example 7.4 Bayesian Averaging of Classical Estimates CHAPTER 8 The Generalized Regression Model and Heteroscedasticity Example 8.1 Heteroscedastic Regression Example 8.2 The White Estimator Example 8.3 Testing for Heteroscedasticity Example 8.4 Multiplicative Heteroscedasticity Example 8.5 Groupwise Heteroscedasticity CHAPTER 9 Models for Panel Data Example 9.1 Wage Equation Example 9.2 Robust Estimators of the Wage Equation Example 9.3 Analysis of Covariance and the World Health Organization Data Example 9.4 Fixed Effects Wage Equation Example 9.5 Testing for Random Effects Example 9.6 Estimates of the Random Effects Model Example 9.7 Hausman Test for Fixed vs. Random Effects Example 9.8 Variable Addition Test for Fixed vs. Random Effects Example 9.9 Staewide Productivity Example 9.10 Spatial Autocorrelation in Real Estate Sales Example 9.11 Spatial Lags in Health Expenditures Example 9.12 Random Coefficients Model Example 9.13 Minimum Sum of Squares Estimates of the Production Function Example 9.14 Two Step Estimation of Cornwell and Rupert's Wage Equation Example 9.15 Fammie Mae's Pass Through Example 9.16 Hierarchical Linear Model of Home Sales Example 9.17 Mixed Linear Model for Wages Example 9.18 Dynamic Panel Data Models Example 9.19 A Mixed Fixed Growth Model for Developing Countries CHAPTER 10 Systems of Regression Equations Example 10.1 Munnell's Statewide Production Data Example 10.2 Estimated SUR Model for Regional Output Example 10.3 Hypothesis Tests in the SUR Model Example 10.4 Testing for Cross Equation Correlation Example 10.5 Demand for Electricity and Gas Example 10.6 Hospital Costs Example 10.7 Stone's Expenditure System Example 10.8 A Cost Function for U.S. Manufacturing CHAPTER 11 Nonlinear Regressions and Nonlinear Least Squares Example 11.1 CES Production Function Example 11.2 Translog Demand System Example 11.3 First Order Conditions for a Nonlinear Model Example 11.4 Linearized Regression Example 11.5 Analysis of a Nonlinear Consumption Function Example 11.6 Multicollinearity in Nonlinear Regression Example 11.7 Flexible Cost Function Example 11.8 Hypothesis Tests in a Nonlinear Regression Model Example 11.9 Two-Step Estimation of a Credit Scoring Model Example 11.10 Health Care Utilization Example 11.11 Exponential Model with Fixed Effects CHAPTER 12 Instrumental Variables Estimation Example 12.1 Models in Which Least Squares is Inconsistent Example 12.2 Streams as Instruments Example 12.3 Labor Supply Market Example 12.4 Hausman Test for a Consumption Function Example 12.5 Income and Education in a Study of Twins Example 12.6 Instrumental Variable Estimation of the Consumption Function Example 12.7 The Returns to Schooling Example 12.8 Dynamic Labor Supply Equation CHAPTER 13 Simultaneous-Equations Models Example 13.1 A Small Macroeconomic Model Example 13.2 Klein's Model I Example 13.3 Structure and Reduced Form Example 13.4 Observational Equivalence Example 13.5 Identification Example 13.6 Identification of Klein's Model I Example 13.7 Testing Overidentifying Restrictions Example 13.8 Dynamic Model CHAPTER 14 Estimation Frameworks in Econometrics Example 14.1 The Linear Regression Model Example 14.2 The Stochastic Frontier Model Example 14.3 Joint Modeling of a Pair of Event Counts Example 14.4 LAD Estimation of a Cobb-Douglas Function Example 14.5 Partially Linear Translog Cost Function Example 14.6 Semiparametric Estimation of Binary Choice Models Example 14.7 A Model of Vacation Expenditures CHAPTER 15 Minimum Distance Estimation and The Generalized Method of Moments Example 15.1 Euler Equations and Life Cycle Consumption Example 15.2 Method of Moments Estimator for N[_,__] Example 15.3 Inverse Gaussian (Wald) Distribution Example 15.4 Mixtures of Normal Distributions Example 15.5 Gamma Distribution Example 15.6 Minimum Distance Estimation of a Hospital Cost Function Example 15.7 GMM Estimation of the Parameters of a Gamma Distribution Example 15.8 Empirical Moment Equation for Instrumental Variables Example 15.9 Overidentifying Restrictions Example 15.10 GMM Estimation of a Dynamic Model of Local Government Expenditures CHAPTER 16 Maximum Likelihood Example 16.1 Identification of Parameters Example 16.2 Log Likelihood Function and Likelihood Equations for the Normal Distribution Example 16.3 Information Matrix for the Normal Distribution Example 16.4 Variance Estimators for an MLE Example 16.5 Two Step ML Estimation Example 16.6 Heteroscedastic Regression Example 16.7 Autocorrelation in a Money Demand Equation Example 16.8 ML Estimates of a Seemingly Unrelated Regressions Model Example 16.9 Stochastic Frontier Model Example 16.10 Geometric Model for Doctor visits Example 16.11 Maximum Likelihood and FGLS Estimates of a Wage Equation Example 16.12 Random Effects Geometric Regression Model Example 16.13 Fixed and Random Effects Geometric Regression Model Example 16.14 Latent Class Model for Grade Point Averages Example 16.15 Latent Class Regression Model for Grade Point Averages Example 16.16 Latent Class Model for Health Care Utilization CHAPTER 17 Simulation Based Estimation and Inference Example 17.1 Fractional Moments of the Truncated Normal Distribution Example 17.2 Estimating the Lognormal Mean Example 17.3 Mean of a Lognormal Distribution Example 17.4 Monte Carlo Study of the Mean versus the Median 17.4.1 A Monte Carlo Study: Behavior of a Test Statistic 17.4.2 A Monte Carlo Study: The Incidental Parameters Problem Example 17.5 Random Effects Geometric Regression Example 17.6 Maximum Simulated Likelihood Estimation of a Binary Choice Model Example 17.7 Bootstrapping the Variance of the Median CHAPTER 18 Bayesian Inference in Econometrics Example 18.1 Bayesian Estimation of a Proportion Example 18.2 Estimation with a Conjugate Prior Example 18.3 Bayesian Estimate of the Marginal Propensity to Consume Example 18.4 Posterior Odds for the Classical Regression Model Example 18.5 Gibbs Sampling from the Normal Distribution Example 18.6 Application: Binomial Probit Model Example 18.7 Gibbs Sampler for a Probit Model CHAPTER 19 Serial Correlation Example 19.1 Money Demand Equation Example 19.2 Autocorrelation Induced by Misspecification of the Model Example 19.3 Negative Autocorrelation in the Phillips Curve Example 19.4 Autocorrelation Consistent Covariance Matrix Example 19.5 Panel Data Models with Autocorrelation Example 19.6 Test for Common Factors Example 19.7 Stochastic Volatility CHAPTER 20 Models with Lagged Variables Example 20.1 A Structural Model of the Demand for Gasoline Example 20.2 Expectations Augmented Phillips Curve Example 20.3 Price and Income Elasticities of Demand for Gasoline Example 20.4 A Rational Lag Model Example 20.5 An Error Correction Model for Consumption Example 20.6 Granger Causality 20.6.8 Application: Policy Analysis with a VAR Example 20.7 VAR for Municipal Expenditures CHAPTER 21 Time-Series Models Example 21.1 ACF and PACF for a Series of Bond Yields Example 21.2 Spectral Density Function for an AR(1) Process Example 21.3 Spectral Analysis of the Growth Rate of Real GNP CHAPTER 22 Nonstationary Data Example 22.1 A Nonstationary Series Example 22.2 Tests for Unit Roots Example 22.3 Augmented Dicky-Fuller Test for a Unit Root in GNP Example 22.4 Is There a Unit Root in GNP? 22.4.5 Application: German Money Demand Example 22.5 Cointegration in Consumption and Output Example 22.6 Several Cointegrated Series Example 22.7 Multiple Cointegrating Vectors Example 22.8 Cointegration in Consumption and Output CHAPTER 23 Models for Discrete Choice Example 23.1 Labor Force Participation Model Example 23.2 Structural Equations for a Probit Model Example 23.3 Probability Models Example 23.4 Average Partial Effects Example 23.5 Specification Tests in a Labor Force Participation Model Example 23.6 Prediction with a Probit Model Example 23.7 An Intertemporal Labor Force Participation Equation Example 23.8 Binary Choice Models for Panel Data Example 23.9 Fixed Effects Logit Models: Magazine Prices Revisited Example 23.10 Semiparametric Models of Heterogeneity Example 23.11 Parameter Heterogeneity in a Binary Choice Model 23.11.1 Application: Conditional Logit Model for Travel Mode Choice Example 23.12 A Comparison of Binary Choice Estimators Example 23.13 Labor Supply Model Example 23.14 Tetrachoric Correlation Example 23.15 Bivariate Probit Model for Health Care Utilization Example 23.16 A Multivariate Probit Model for Product Innovations Example 23.17 Rating Assignments Example 23.18 Calculus and Intermediate Economics Courses Example 23.19 Health Satisfaction CHAPTER 24 Truncation, Censoring and Sample Selection Example 24.1 Truncated Uniform Distribution Example 24.2 A Truncated Lognormal Income Distribution Example 24.3 Censored Random Variable Example 24.4 Estimated Tobit Equations for Hours Worked Example 24.5 Multiplicative Heteroscedasticity in the Tobit Model Example 24.6 Incidental Truncation Example 24.7 A Model of Labor Supply Example 24.8 Female Labor Supply Example 24.9 A Mover Stayer Model for Migration Example 24.10 Treatment Effects on Earnings Example 24.11 Doctor Visits and Insurance CHAPTER 25 Models for Event Counts and Duration Example 25.1 Count Data Models for Doctor Visits Example 25.2 Panel Datra Models for Doctor Visits Example 25.3 A Split Population Model for Major Derogatory Reports Example 25.4 Survival Models for Strike Duration 25.5.2 Application: Censoring in the Tobit and Poisson Regression Models APPENDIX C Models for Event Counts and Duration Example C.1 Descriptive Statistics for Random Sample Example C.2 Kernel Density for the Income Data Example C.3 Sampling Distribution of a Sample Mean Example C.4 Sampling Distribution of the Sample Minimum Example C.5 Mean-Squared Error of the Sample Variance Example C.6 Likelihood Functions for Exponential and Normal Distributions Example C.7 Variance Bound for the Poisson Distribution Example C.8 Confidence Intervals for the Normal Mean Example C.9 Confidence Intervals for a Normal Mean and Variance Example C.10 Testing a Hypothesis About a Mean Example C.11 Consistent Test About a Mean Example C.12 Testing a Hypothesis About a Mean with a Confidence Interval Example C.13 One-Sided Test About a Mean APPENDIX D Large Sample Distribution Theory Example D.1 Mean Square Convergence of the Sample Minimum in Exponential Sampling Example D.2 Estimating a Function of the Mean Example D.3 Probability Limit of a Function of x and s_ Example D.4 Limiting Distribution of tn-1 Example D.5 The F Distribution Example D.6 The Lindeberg-Levy Central Limit Theorem Example D.7 Asymptotic Distribution of the Mean of an Exponential Sample Example D.8 Asymptotic Inefficiency of the Median in Normal Sampling Example D.9 Asymptotic Moments of the Sample Variance APPENDIX E Computation and Optimization Example E.1 Function of One Parameter Example E.2 Function of Two Parameters: The Gamma Distribution Example E.3 A Concentrated Log-Likelihood Function INDEX
Library of Congress Subject Headings for this publication:
Econometrics.