Principles of Econometrics

Principles of Econometrics, Fifth Edition, is an introductory book for undergraduate students in economics and finance, as well as first-year graduate students in a variety of fields that include economics, finance, accounting, marketing, public policy, sociology, law, and political science. Students will gain a working knowledge of basic econometrics so they can apply modeling, estimation, inference, and forecasting techniques when working with real-world economic problems. Readers will also gain an understanding of econometrics that allows them to critically evaluate the results of others’ economic research and modeling, and that will serve as a foundation for further study of the field. This new edition of the highly-regarded econometrics text includes major revisions that both reorganize the content and present students with plentiful opportunities to practice what they have read in the form of chapter-end exercises.

Preface v List of Examples xxi 1 An Introduction to Econometrics 1 1.1 Why Study Econometrics? 1 1.2 What Is Econometrics About? 2 1.2.1 Some Examples 3 1.3 The Econometric Model 4 1.3.1 Causality and Prediction 5 1.4 How Are Data Generated? 5 1.4.1 Experimental Data 6 1.4.2 Quasi-Experimental Data 6 1.4.3 Nonexperimental Data 7 1.5 Economic Data Types 7 1.5.1 Time-Series Data 7 1.5.2 Cross-Section Data 8 1.5.3 Panel or Longitudinal Data 9 1.6 The Research Process 9 1.7 Writing an Empirical Research Paper 11 1.7.1 Writing a Research Proposal 11 1.7.2 A Format for Writing a Research Report 11 1.8 Sources of Economic Data 13 1.8.1 Links to Economic Data on the Internet 13 1.8.2 Interpreting Economic Data 14 1.8.3 Obtaining the Data 14 Probability Primer 15 P.1 Random Variables 16 P.2 Probability Distributions 17 P.3 Joint, Marginal, and Conditional Probabilities 20 P.3.1 Marginal Distributions 20 P.3.2 Conditional Probability 21 P.3.3 Statistical Independence 21 P.4 A Digression: Summation Notation 22 P.5 Properties of Probability Distributions 23 P.5.1 Expected Value of a Random Variable 24 P.5.2 Conditional Expectation 25 P.5.3 Rules for Expected Values 25 P.5.4 Variance of a Random Variable 26 P.5.5 Expected Values of Several Random Variables 27 P.5.6 Covariance Between Two Random Variables 27 P.6 Conditioning 29 P.6.1 Conditional Expectation 30 P.6.2 Conditional Variance 31 P.6.3 Iterated Expectations 32 P.6.4 Variance Decomposition 33 P.6.5 Covariance Decomposition 34 P.7 The Normal Distribution 34 P.7.1 The Bivariate Normal Distribution 37 P.8 Exercises 39 2 The Simple Linear Regression Model 46 2.1 An Economic Model 47 2.2 An Econometric Model 49 2.2.1 Data Generating Process 51 2.2.2 The Random Error and Strict Exogeneity 52 2.2.3 The Regression Function 53 2.2.4 Random Error Variation 54 2.2.5 Variation in x 56 2.2.6 Error Normality 56 2.2.7 Generalizing the Exogeneity Assumption 56 2.2.8 Error Correlation 57 2.2.9 Summarizing the Assumptions 58 2.3 Estimating the Regression Parameters 59 2.3.1 The Least Squares Principle 61 2.3.2 Other Economic Models 65 2.4 Assessing the Least Squares Estimators 66 2.4.1 The Estimator b2 67 2.4.2 The Expected Values of b1 and b2 68 2.4.3 Sampling Variation 69 2.4.4 The Variances and Covariance of b1 and b2 69 2.5 The Gauss–Markov Theorem 72 2.6 The Probability Distributions of the Least Squares Estimators 73 2.7 Estimating the Variance of the Error Term 74 2.7.1 Estimating the Variances and Covariance of the Least Squares Estimators 74 2.7.2 Interpreting the Standard Errors 76 2.8 Estimating Nonlinear Relationships 77 2.8.1 Quadratic Functions 77 2.8.2 Using a Quadratic Model 77 2.8.3 A Log-Linear Function 79 2.8.4 Using a Log-Linear Model 80 2.8.5 Choosing a Functional Form 82 2.9 Regression with Indicator Variables 82 2.10 The Independent Variable 84 2.10.1 Random and Independent x 84 2.10.2 Random and Strictly Exogenous x 86 2.10.3 Random Sampling 87 2.11 Exercises 89 2.11.1 Problems 89 2.11.2 Computer Exercises 93 Appendix 2A Derivation of the Least Squares Estimates 98 Appendix 2B Deviation from the Mean Form of b2 99 Appendix 2C b2 Is a Linear Estimator 100 Appendix 2D Derivation of Theoretical Expression for b2 100 Appendix 2E Deriving the Conditional Variance of b2 100 Appendix 2F Proof of the Gauss–Markov Theorem 102 Appendix 2G Proofs of Results Introduced in Section 2.10 103 2G.1 The Implications of Strict Exogeneity 103 2G.2 The Random and Independent x Case 103 2G.3 The Random and Strictly Exogenous x Case 105 2G.4 Random Sampling 106 Appendix 2H Monte Carlo Simulation 106 2H.1 The Regression Function 106 2H.2 The Random Error 107 2H.3 Theoretically True Values 107 2H.4 Creating a Sample of Data 108 2H.5 Monte Carlo Objectives 109 2H.6 Monte Carlo Results 109 2H.7 Random-x Monte Carlo Results 110 3 Interval Estimation and Hypothesis Testing 112 3.1 Interval Estimation 113 3.1.1 The t-Distribution 113 3.1.2 Obtaining Interval Estimates 115 3.1.3 The Sampling Context 116 3.2 Hypothesis Tests 118 3.2.1 The Null Hypothesis 118 3.2.2 The Alternative Hypothesis 118 3.2.3 The Test Statistic 119 3.2.4 The Rejection Region 119 3.2.5 A Conclusion 120 3.3 Rejection Regions for Specific Alternatives 120 3.3.1 One-Tail Tests with Alternative ‘‘Greater Than’’ (>) 120 3.3.2 One-Tail Tests with Alternative ‘‘Less Than’’ ( 2 643 15.2.4 The Least Squares Dummy Variable Model 644 15.3 Panel Data Regression Error Assumptions 646 15.3.1 OLS Estimation with Cluster-Robust Standard Errors 648 15.3.2 Fixed Effects Estimation with Cluster-Robust Standard Errors 650 15.4 The Random Effects Estimator 651 15.4.1 Testing for Random Effects 653 15.4.2 A Hausman Test for Endogeneity in the Random Effects Model 654 15.4.3 A Regression-Based Hausman Test 656 15.4.4 The Hausman–Taylor Estimator 658 15.4.5 Summarizing Panel Data Assumptions 660 15.4.6 Summarizing and Extending Panel Data Model Estimation 661 15.5 Exercises 663 15.5.1 Problems 663 15.5.2 Computer Exercises 670 Appendix 15A Cluster-Robust Standard Errors: Some Details 677 Appendix 15B Estimation of Error Components 679 16 Qualitative and Limited Dependent Variable Models 681 16.1 Introducing Models with Binary Dependent Variables 682 16.1.1 The Linear Probability Model 683 16.2 Modeling Binary Choices 685 16.2.1 The Probit Model for Binary Choice 686 16.2.2 Interpreting the Probit Model 687 16.2.3 Maximum Likelihood Estimation of the Probit Model 690 16.2.4 The Logit Model for Binary Choices 693 16.2.5 Wald Hypothesis Tests 695 16.2.6 Likelihood Ratio Hypothesis Tests 696 16.2.7 Robust Inference in Probit and Logit Models 698 16.2.8 Binary Choice Models with a Continuous Endogenous Variable 698 16.2.9 Binary Choice Models with a Binary Endogenous Variable 699 16.2.10 Binary Endogenous Explanatory Variables 700 16.2.11 Binary Choice Models and Panel Data 701 16.3 Multinomial Logit 702 16.3.1 Multinomial Logit Choice Probabilities 703 16.3.2 Maximum Likelihood Estimation 703 16.3.3 Multinomial Logit Postestimation Analysis 704 16.4 Conditional Logit 707 16.4.1 Conditional Logit Choice Probabilities 707 16.4.2 Conditional Logit Postestimation Analysis 708 16.5 Ordered Choice Models 709 16.5.1 Ordinal Probit Choice Probabilities 710 16.5.2 Ordered Probit Estimation and Interpretation 711 16.6 Models for Count Data 713 16.6.1 Maximum Likelihood Estimation of the Poisson Regression Model 713 16.6.2 Interpreting the Poisson Regression Model 714 16.7 Limited Dependent Variables 717 16.7.1 Maximum Likelihood Estimation of the Simple Linear Regression Model 717 16.7.2 Truncated Regression 718 16.7.3 Censored Samples and Regression 718 16.7.4 Tobit Model Interpretation 720 16.7.5 Sample Selection 723 16.8 Exercises 725 16.8.1 Problems 725 16.8.2 Computer Exercises 733 Appendix 16A Probit Marginal Effects: Details 739 16A.1 Standard Error of Marginal Effect at a Given Point 739 16A.2 Standard Error of Average Marginal Effect 740 Appendix 16B Random Utility Models 741 16B.1 Binary Choice Model 741 16B.2 Probit or Logit? 742 Appendix 16C Using Latent Variables 743 16C.1 Tobit (Tobit Type I) 743 16C.2 Heckit (Tobit Type II) 744 Appendix 16D A Tobit Monte Carlo Experiment 745 A Mathematical Tools 748 A.1 Some Basics 749 A.1.1 Numbers 749 A.1.2 Exponents 749 A.1.3 Scientific Notation 749 A.1.4 Logarithms and the Number e 750 A.1.5 Decimals and Percentages 751 A.1.6 Logarithms and Percentages 751 A.2 Linear Relationships 752 A.2.1 Slopes and Derivatives 753 A.2.2 Elasticity 753 A.3 Nonlinear Relationships 753 A.3.1 Rules for Derivatives 754 A.3.2 Elasticity of a Nonlinear Relationship 757 A.3.3 Second Derivatives 757 A.3.4 Maxima and Minima 758 A.3.5 Partial Derivatives 759 A.3.6 Maxima and Minima of Bivariate Functions 760 A.4 Integrals 762 A.4.1 Computing the Area Under a Curve 762 A.5 Exercises 764 B Probability Concepts 768 B.1 Discrete Random Variables 769 B.1.1 Expected Value of a Discrete Random Variable 769 B.1.2 Variance of a Discrete Random Variable 770 B.1.3 Joint, Marginal, and Conditional Distributions 771 B.1.4 Expectations Involving Several Random Variables 772 B.1.5 Covariance and Correlation 773 B.1.6 Conditional Expectations 774 B.1.7 Iterated Expectations 774 B.1.8 Variance Decomposition 774 B.1.9 Covariance Decomposition 777 B.2 Working with Continuous Random Variables 778 B.2.1 Probability Calculations 779 B.2.2 Properties of Continuous Random Variables 780 B.2.3 Joint, Marginal, and Conditional Probability Distributions 781 B.2.4 Using Iterated Expectations with Continuous Random Variables 785 B.2.5 Distributions of Functions of Random Variables 787 B.2.6 Truncated Random Variables 789 B.3 Some Important Probability Distributions 789 B.3.1 The Bernoulli Distribution 790 B.3.2 The Binomial Distribution 790 B.3.3 The Poisson Distribution 791 B.3.4 The Uniform Distribution 792 B.3.5 The Normal Distribution 793 B.3.6 The Chi-Square Distribution 794 B.3.7 The t-Distribution 796 B.3.8 The F-Distribution 797 B.3.9 The Log-Normal Distribution 799 B.4 Random Numbers 800 B.4.1 Uniform Random Numbers 805 B.5 Exercises 806 C Review of Statistical Inference 812 C.1 A Sample of Data 813 C.2 An Econometric Model 814 C.3 Estimating the Mean of a Population 815 C.3.1 The Expected Value of Y 816 C.3.2 The Variance of Y 817 C.3.3 The Sampling Distribution of Y 817 C.3.4 The Central Limit Theorem 818 C.3.5 Best Linear Unbiased Estimation 820 C.4 Estimating the Population Variance and Other Moments 820 C.4.1 Estimating the Population Variance 821 C.4.2 Estimating Higher Moments 821 C.5 Interval Estimation 822 C.5.1 Interval Estimation: σ2 Known 822 C.5.2 Interval Estimation: σ2 Unknown 825 C.6 Hypothesis Tests About a Population Mean 826 C.6.1 Components of Hypothesis Tests 826 C.6.2 One-Tail Tests with Alternative ‘‘Greater Than’’ (>) 828 C.6.3 One-Tail Tests with Alternative ‘‘Less Than’’ (