An Introduction to Econometrics. A Self-contained Approach. Frank Westhoff. The MIT Press Cambridge, Massachusetts London, England

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1 An Introduction to Econometrics A Self-contained Approach Frank Westhoff The MIT Press Cambridge, Massachusetts London, England

2 How to Use This Book xvii 1 Descriptive Statistics 1 Chapter 1 Prep Questions Describing a Single Data Variable Introduction to Distributions Measure of the Distribution Center: Mean (Average) Measures of the Distribution Spread: Range, Variance, and Standard Deviation Histogram: Visual Illustration of a Data Variable's Distribution Describing the Relationship between Two Data Variables Scatter Diagram: Visual Illustration of How Two Data Variables Are Related Correlation of Two Variables Measures of Correlation: Covariance Independence of Two Variables Measures of Correlation: Correlation Coefficient Correlation and Causation Arithmetic of Means, Variances, and Covariances 27 Chapter 1 Review Questions 29 Chapter 1 Exercises 30 Appendix 1.1: The Arithmetic of Means, Variances, and Covariances 40 2 Essentials of Probability and Estimation Procedures 45 Chapter 2 Prep Questions Random Processes and Probability Random Process: A Process Whose Outcome Cannot Be Predicted with Certainty Random Process: A Process Whose Outcome Cannot Be Predicted with Certainty Probability: The Likelihood of a Particular Outcome of a Random Process Random Variable: A Variable That Is Associated with an Outcome of a Random Process Discrete Random Variables and Probability Distributions Probability Distribution Describes the Probability for All Possible Values of a Random Variable A Random Variable's Bad News and Good News Relative Frequency Interpretation of Probability Relative Frequency Interpretation of Probability Summary 52

3 vi 2.3 Describing a Probability Distribution of a Random Variable Center of the Probability Distribution: Mean (Expected Value) of the Random Variable Spread of the Probability Distribution: Variance of the Random Variable Continuous Random Variables and Probability Distributions Estimation Procedures: Populations and Samples Clint's Dilemma: Assessing Clint's Political Prospects Usefulness of Simulations Center of the Probability Distribution: Mean of the Random Variable Spread of the Probability Distribution: Variance of the Random Variable Mean, Variance, and Covariance: Data Variables and Random Variables 77 Chapter 2 Review Questions 77 Chapter 2 Exercises 77 3 Interval Estimates and the Central Limit Theorem 87 Chapter 3 Prep Questions Review Random Variables Relative Frequency Interpretation of Probability Populations, Samples, Estimation Procedures, and the Estimate's Probability Distribution Measure of the Probability Distribution Center: Mean of the Random Variable Measure of the Probability Distribution Spread: Variance of the Random Variable Why Is the Mean of the Estimate's Probability Distribution Important? Biased and Unbiased Estimation Procedures Why Is the Variance of the Estimate's Probability Distribution Important? Reliability of Unbiased Estimation Procedures Interval Estimates Relative Frequency Interpretation of Probability Central Limit Theorem The Normal Distribution: A Way to Estimate Probabilities Properties of the Normal Distribution: Using the Normal Distribution Table: An Example Justifying the Use of the Normal Distribution Normal Distribution's Rules of Thumb 110 Chapter 3 Review Questions 112 Chapter 3 Exercises 112 Appendix 3.1: Normal Distribution Right-Tail Probabilities Estimation Procedures, Estimates, and Hypothesis Testing 119 Chapter 4 Prep Questions Clint's Dilemma and Estimation Procedures Clint's Opinion Poll and His Dilemma Clint's Estimation Procedure: The General and the Specific Taking Stock and Our Strategy to Assess the Reliability of Clint's Poll Results: Use the General Properties of the Estimation Procedure to Assess the Reliability of the One Specific Application Importance of the Mean (Center) of the Estimate's Probability Distribution 124

4 vii Importance of the Variance (Spread) of the Estimate's Probability Distribution for an Unbiased Estimation Procedure Hypothesis Testing Motivating Hypothesis Testing: The Evidence and the Cynic Formalizing Hypothesis Testing: Five Steps Significance Levels and Standards of Proof Type I and Type II Errors: The Trade-Offs 135 Chapter 4 Review Questions 138 Chapter 4 Exercises 139 Ordinary Least Squares Estimation Procedure The Mechanics 145 Chapter 5 Prep Questions Best Fitting Line Clint's Assignment Simple Regression Model Parameters of the Model Error Term and Random Influences What Is Simple about the Simple Regression Model? Best Fitting Line Needed: A Systematic Procedure to Determine the Best Fitting Line Ordinary Least Squares (OLS) Estimation Procedure Sum of Squared Residuais Criterion Finding the Best Fitting Line Importance of the Error Term Absence of Random Influences: A "What If" Question Presence of Random Influences: Back to Reality Error Terms and Random Influences: A Closer Look Standard Ordinary Least Squares (OLS) Premises Clint's Assignment: The Two Parts 173 Chapter 5 Review Questions 174 Chapter 5 Exercises Ordinary Least Squares Estimation Procedure The Properties 181 Chapter 6 Prep Questions Clint's Assignment: Assess the Effect of Studying on Quiz Scores Review Regression Model The Error Term Ordinary Least Squares (OLS) Estimation Procedure The Estimates, b Comt and b x, Are Random Variables Strategy: General Properties and a Specific Application Review: Assessing Clint's Opinion Poll Results Preview: Assessing Professor Lord's Quiz Results Standard Ordinary Least Squares (OLS) Regression Premises New Equation for the Ordinary Least Squares (OLS) Coefficient Estimate General Properties: Describing the Coefficient Estimate's Probability Distribution 190

5 viii Mean (Center) of the Coefficient Estimate's Probability Distribution Variance (Spread) of the Coefficient Estimate's Probability Distribution Estimation Procedures and the Estimate's Probability Distribution Reliability of the Coefficient Estimate Estimate Reliability and the Variance of the Error Term's Probability Distribution Estimate Reliability and the Sample Size Estimate Reliability and the Range of x's Reliability Summary Best Linear Unbiased Estimation Procedure (BLUE) 205 Chapter 6 Review Questions 208 Chapter 6 Exercises 208 Appendix 6.1: New Equation for the OLS Coefficient Estimate 213 Appendix 6.2: Gauss-Markov Theorem Estimating the Variance of an Estimate's Probability Distribution 221 Chapter 7 Prep Questions Review Clint's Assignment General Properties of the Ordinary Least Squares (OLS) Estimation Procedure Importance of the Coefficient Estimate's Probability Distribution Strategy to Estimate the Variance of the Coefficient Estimate's Probability Distribution Step 1: Estimate the Variance of the Error Term's Probability Distribution First Attempt: Variance of the Error Term's Numerical Values Second Attempt: Variance of the Residual's Numerical Values Third Attempt: "Adjusted" Variance of the Residual's Numerical Values Step 2: Use the Estimated Variance of the Error Term's Probability Distribution to Estimate the Variance of the Coefficient Estimate's Probability Distribution Tying up a Loose End: Degrees of Freedom Reviewing our Second and Third Attempts to Estimate the Variance of the Error Term's Probability Distribution How Do We Calculate an Average? Summary: The Ordinary Least Squares (OLS) Estimation Procedure Three Important Parts Regression Results 246 Chapter 7 Review Questions 247 Chapter 7 Exercises Interval Estimates and Hypothesis Testing 251 Chapter 8 Prep Questions Clint's Assignment: Taking Stock Estimate Reliability: Interval Estimate Question Normal Distribution versus the Student f-distribution: One Last Complication Assessing the Reliability of a Coefficient Estimate Theory Assessment: Hypothesis Testing Motivating Hypothesis Testing: The Cynic Formalizing Hypothesis Testing: The Steps 268

6 ix 8.4 Summary: The Ordinary Least Squares (OLS) Estimation Procedure Regression Model and the Role of the Error Term Standard Ordinary Least Squares (OLS) Premises Ordinary Least Squares (OLS) Estimation Procedure: Three Important Estimation Procedures Properties of the Ordinary Least Squares (OLS) Estimation Procedure and the Standard Ordinary Least Squares (OLS) Premises 272 Chapter 8 Review Questions 273 Chapter 8 Exercises 273 Appendix 8.1: Student (-Distribution Table Right-Tail Critical Values 278 Appendix 8.2: Assessing the Reliability of a Coefficient Estimate Using the Student (-Distribution Table One-Tailed Tests, Two-Tailed Tests, and Logarithms 285 Chapter 9 Prep Questions A One-Tailed Hypothesis Test: The Downward Sloping Demand Curve One-Tailed versus Two-Tailed Tests A Two-Tailed Hypothesis Test: The Budget Theory of Demand Hypothesis Testing Using Clever Algebraic Manipulations Summary: One-Tailed and Two-Tailed Tests Logarithms: A Useful Econometric Tool to Eine Tuning Hypotheses The Math Interpretation of the Coefficient Estimate: Esty - b Comt + b xx Differential Approximation: Ay ~ (dy/dx)ax Derivative of a Natural Logarithm: d \og(z)/dz = Mz Dependent Variable Logarithm: y = log(z) Explanatory Variable Logarithm of z: x = log(z) Using Logarithms An Illustration: Wages and Education Linear Model: Wage t = ß Comi + ß EHSEduc t + e, Log Dependent Variable Model: LogWage t = ß Const + ßuHSEduCt + e, Log Explanatory Variable Model: Wage t = ß Const + ß ELogHSEduc t + e t Log-Log (Constant Elasticity) Model: LogWage t = ß Camt + ß ELogHSEduc t + e t Summary: Logarithms and the Interpretation of Coefficient Estimates 311 Chapter 9 Review Questions 312 Chapter 9 Exercises Multiple Regression Analysis Introduction 317 Chapter 10 Prep Questions Simple versus Multiple Regression Analysis Goal of Multiple Regression Analysis A One-Tailed Test: Downward Sloping Demand Theory A Two-Tailed Test: No Money Illusion Theory Linear Demand Model and the No Money Illusion Theory Constant Elasticity Demand Model and the No Money Illusion Theory Calculating Prob[Results IE H 0 true]: Clever Algebraic Manipulation 340 Chapter 10 Review Questions 344 Chapter 10 Exercises 345

7 X 11 Hypothesis Testing and the Wald Test 349 Chapter 11 Prep Questions No Money Illusion Theory: Taking Stock No Money Illusion Theory: Calculating Prob[Results IF H 0 True] Clever Algebraic Manipulation Wald (F-Distribution) Test Calculating Prob[Results IF H 0 true]: Let the Software Do the Work Testing the Significance of the "Entire" Model Equivalence of Two-Tailed (-Tests and Wald Tests (F-Tests) Two-Tailed t-test Wald Test Three Important Distributions 374 Chapter 11 Review Questions 375 Chapter 11 Exercises Model Specification and Development 381 Chapter 12 Prep Questions Model Specification: Ramsey REgression Specification Error Test (RESET) RESET Logic Linear Demand Model Constant Elasticity Demand Model Model Development: The Effect of Economic Conditions on Presidential Elections General Theory: "It's the economy stupid" Generate Relevant Variables Data Oddities Model Formulation and Assessment: An Iterative Process Specific Voting Models 395 Chapter 12 Review Questions 404 Chapter 12 Exercises Dummy and Interaction Variables 409 Chapter 13 Prep Questions Preliminary Mathematics: Averages and Regressions Including Only a Constant An Example: Discrimination in Academe Average Salaries Dummy Variables Models ßeware of Implicit Assumptions Interaction Variables Conclusions An Example: Internet and Television Use Similarities and Differences Interaction Variable: Economic and Political Interaction Chapter 13 Review Questions Chapter 13 Exercises 434

8 xi 14 Omitted Explanatory Variables, Multicollinearity, and Irrelevant Explanatory Variables Chapter 14 Prep Questions Review Unbiased Estimation Procedures Correlated and Independent (Uncorrelated) Variables Omitted Explanatory Variables A Puzzle: Baseball Attendance Goal of Multiple Regression Analysis Omitted Explanatory Variables and ßias Resolving the Baseball Attendance Puzzle Omitted Variable Summary Multicollinearity Perfectly Correlated Explanatory Variables Highly Correlated Explanatory Variables "Earmarks" of Multicollinearity Irrelevant Explanatory Variables 466 Chapter 14 Review Questions 469 Chapter 14 Exercises Other Regression Statistics and Pitfalls 473 Chapter 15 Prep Questions Two-Tailed Confidence Intervals Confidence Interval Approach: Which Theories Are Consistent with the Data? A Confidence Interval Example: Television Growth Rates Calculating Confidence Intervals with Statistical Software Coefficient of Determination, R-Squared (R 2 ) Pitfalls Explanatory Variable Has the Same Value for All Observations One Explanatory Variable Is a Linear Combination of Other Explanatory Variables Dependent Variable Is a Linear Combination of Explanatory Variables Outlier Observations Dummy Variable Trap 500 Chapter 15 Review Questions 506 Chapter 15 Exercises Heteroskedasticity 513 Chapter 16 Prep Questions Review Regression Model Standard Ordinary Least Squares (OLS) Premises Estimation Procedures Embedded within the Ordinary Least Squares (OLS) Estimation Procedure What Is Heteroskedasticity? Heteroskedasticity and the Ordinary Least Squares (OLS) Estimation Procedure: The Consequences The Mathematics Our Suspicions Confirming Our Suspicions 522

9 xii 16.4 Accounting for Heteroskedasticity: An Example Justifying the Generalized Least Squares (GLS) Estimation Procedure Robust Standard Errors 538 Chapter 16 Review Questions 541 Chapter 16 Exercises Autocorrelation (Serial Correlation) 545 Chapter 17 Prep Questions Review Regression Model Standard Ordinary Least Squares (OLS) Premises Estimation Procedures Embedded within the Ordinary Least Squares (OLS) Estimation Procedure Covariance and Independence What Is Autocorrelation (Serial Correlation)? Autocorrelation and the Ordinary Least Squares (OLS) Estimation Procedure: The Consequences The Mathematics Our Suspicions Confirming Our Suspicions Accounting for Autocorrelation: An Example Justifying the Generalized Least Squares (GLS) Estimation Procedure Robust Standard Errors 574 Chapter 17 Review Questions 575 Chapter 17 Exercises Explanatory Variable/Error Term Independence Premise, Consistency, and Instrumental Variables 579 Chapter 18 Prep Questions Review Regression Model Standard Ordinary Least Squares (OLS) Premises Estimation Procedures Embedded within the Ordinary Least Squares (OLS) Estimation Procedure Taking Stock and a Preview: The Ordinary Least Squares (OLS) Estimation Procedure A Closer Look at the Explanatory Variable/Error Term Independence Premise Explanatory Variable/Error Term Correlation and Bias Geometrie Motivation Confirming Our Logic Estimation Procedures: Large and Small Sample Properties Unbiased and Consistent Estimation Procedure Unbiased but Inconsistent Estimation Procedure Biased but Consistent Estimation Procedure The Ordinary Least Squares (OLS) Estimation Procedure, and Consistency Instrumental Variable (IV) Estimation Procedure: A Two Regression Procedure Motivation of the Instrumental Variables Estimation Procedure Mechanics The "Good" Instrument Conditions Justification of the Instrumental Variables Estimation Procedure 604

10 xfii Chapter 18 Review Questions 607 Chapter 18 Exercises Measurement Error and the Instrumental Variables Estimation Procedure 611 Chapter 19 Prep Questions Introduction to Measurement Error What Is Measurement Error? Modeling Measurement Error The Ordinary Least Squares (OLS) Estimation Procedure and Dependent Variable Measurement Error The Ordinary Least Squares (OLS) Estimation Procedure and Explanatory Variable Measurement Error Summary: Explanatory Variable Measurement Error Blas Explanatory Variable Measurement Error: Attenuation (Dilution) Bias Might the Ordinary Least Squares (OLS) Estimation Procedure Be Consistent? Instrumental Variable (IV) Estimation Procedure: A Two Regression Procedure Mechanics The "Good" Instrument Conditions Measurement Error Example: Annual, Permanent, and Transitory Income Definitions and Theory Might the Ordinary Least Squares (OLS) Estimation Procedure Suffer from a Serious Econometric Problem? Instrumental Variable (IV) Approach The Mechanics Comparison of the Ordinary Least Squares (OLS) and the Instrumental Variables (IV) Approaches "Good" Instrument Conditions Revisited Justifying the Instrumental Variable (IV) Estimation Procedure 631 Chapter 19 Review Questions 633 Chapter 19 Exercises Omitted Variables and the Instrumental Variable Estimation Procedure 637 Chapter 20 Prep Questions Revisit Omitted Explanatory Variable Bias Review of Our Previous Explanation of Omitted Explanatory Variable Bias Omitted Explanatory Variable Bias and the Explanatory Variable/Error Term Independence Premise The Ordinary Least Squares Estimation Procedure, Omitted Explanatory Variable Bias, and Consistency Instrumental Variable Estimation Procedure: A Two Regression Estimation Procedure Mechanics The "Good" Instrument Conditions Omitted Explanatory Variables Example: 2008 Presidential Election Instrument Variable (IV) Application: 2008 Presidential Election The Mechanics "Good" Instrument Conditions Revisited Justifying the Instrumental Variable (IV) Estimation Procedure 650 Chapter 20 Review Questions 654 Chapter 20 Exercises 654

11 xiv 21 Panel Data and Omitted Variables 657 Chapter 21 Prep Questions Taking Stock: Ordinary Least Squares (OLS) Estimation Procedure Standard Ordinary Least Squares (OLS) Premises Preview: Panel Data Examples and Strategy First Differences and Fixed Effects (Dummy Variables) Math Quiz Score Model Ordinary Least Squares (OLS) Pooled Regression First Differences Cross-sectional Fixed Effects (Dummy Variables) Pen od Fixed Effects (Dummy Variables) Chemistry Score Model Ordinary Least Squares (OLS) Pooled Regression Period Fixed Effects (Dummy Variables) Cross-sectional Random Effects Art Project Model Ordinary Least Squares (OLS) Pooled Regression Cross-sectional Random Effects Random Effects Critical Assumptions 688 Chapter 21 Review Questions 688 Chapter 21 Exercises Simultaneous Equations Models Introduction 693 Chapter 22 Prep Questions Review: Explanatory Variable/Error Term Correlation Simultaneous Equations Models: Demand and Supply Endogenous versus Exogenous Variables Single Equation versus Simultaneous Equations Models Demand Model Supply Model Summary: Endogenous Explanatory Variable Problem An Example: The Market for Beef Demand and Supply Models Ordinary Least Squares (OLS) Estimation Procedure Reduced Form (RF) Estimation Procedure: The Mechanics Comparing Ordinary Least Squares (OLS) and Reduced Form (RF) Estimates Justifying the Reduced Form (RF) Estimation Procedure Two Paradoxes Resolving the Two Paradoxes: Coefficient Interpretation Approach Review: Goal of Multiple Regression Analysis and the Interpretation of the Coefficients Paradox: Demand Model Price Coefficient Depends on the Reduced Form (RF) Feed Price Coefficients Paradox: Supply Model Price Coefficient Depends on the Reduced Form (RF) Income Coefficients The Coefficient Interpretation Approach: A Bonus 724 Chapter 22 Review Questions 725 Chapter 22 Exercises 725 Appendix 22.1: Algebraic Derivation of the Reduced Form Equations 730

12 Simultaneous Equations Models Identification 733 Chapter 23 Prep Questions Review Demand and Supply Models Ordinary Least Squares (OLS) Estimation Procedure Two-Stage Least Squares (TSLS): An Instrumental Variable (IV) Two-Step Approach A Second Way to Cope with Simultaneous Equations Models First Stage: Exogenous Explanatory Variable(s) Used to Estimate the Endogenous Explanatory Variable(s) Second Stage: In the Original Model, the Endogenous Explanatory Variable Replaced with Its Estimate Comparison of Reduced Form (RF) and Two-Stage Least Squares (TSLS) Estimates Statistical Software and Two-Stage Least Squares (TSLS) Identification of Simultaneous Equations Models: Order Condition Taking Stock Underidentification Overidentification Overidentification and Two-Stage Least Squares (TSLS) Summary of Identification Issues 762 Chapter 23 Review Questions 762 Chapter 23 Exercises 762 Binary and Truncated Dependent Variables 767 Chapter 24 Prep Questions Introduction Binary Dependent Variables Electoral College: Red and Blue States Linear Probability Model Probit Probability Model: Correcting the Linear Model's Intrinsic Problems Truncated (Censored) Dependent Variables Ordinary Least Squares (OLS) Estimation Procedure Tobit Estimation Procedure 787 Chapter 24 Review Questions 789 Chapter 24 Exercises 789 Descriptive Statistics, Probability, and Random Variables A Closer Look 793 Chapter 25 Prep Questions Descriptive Statistics: Other Measures of the Distribution Center Measure of the Distribution Center: Mode Measure of the Distribution Center: Median Relationship between the Mean and Median Event Trees: A Tool to Calculate Probabilities Calculating the Probability of a Combination of Different Outcomes Nonconditional, Conditional, and Joint Probabilities Conditional/Joint Probability Relationship The Monty Hall Problem: Mathematicians Eat "Humble Pie" 809

13 xvi 25.7 Correlation Correlated Events Correlated Random Variables and Covariance Independence Independent Events Independent Random Variables and Covariance Summary of Correlation and Independence Correlation Independence Describing Probability Distributions of Continuous Random Variables 827 Chapter 25 Review Questions 827 Chapter 25 Exercises Estimating the Mean of a Population 833 Chapter 26 Prep Questions Estimation Procedure for the Population Mean Estimated Mean's Probability Distribution Measure of the Probability Distribution's Center: Mean Measure of the Probability Distribution's Spread: Variance Taking Stock: What We Know versus What Clint Knows Estimation Procedures: Importance of the Probability Distribution's Mean (Center) and Variance (Spread) Strategy to Estimate the Variance of the Estimated Mean's Probability Distribution Step 1: Estimate the Variance of the Population First Attempt: Variance of Clint's Four Numerical Values ßased on the Actual Population Mean Second Attempt: Variance of Clint's Four Numerical Values ßased on the Estimated Population Mean Third Attempt: "Adjusted" Variance of Clint's Four Numerical Values Based on the Estimated Population Mean Step 2: Use the Estimated Variance of the Population to Estimate the Variance of the Estimated Mean's Probability Distribution Clint's Assessment of the Key West Tourist Bureau's Claim Normal Distribution and the Student (-Distribution Tying Up a Loose End: Degrees of Freedom 859 Chapter 26 Review Questions 861 Chapter 26 Exercises 861 Index 867

Econometric Analysis of Cross Section and Panel Data

Econometric Analysis of Cross Section and Panel Data Jeffrey M. Wooldridge / The MIT Press Cambridge, Massachusetts London, England Contents Preface Acknowledgments xvii xxiii I INTRODUCTION AND BACKGROUND