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Introductory Econometrics A Modern Approach 4e Jeffrey M. Wooldridge Michigan State University Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Introductory Econometrics, Fourth Edition © 2009, 2006 South-Western, a part of Cengage Learning Jeff rey M. Wooldridge ALL RIGHTS RESERVED. No part of this work covered by the copyright Vice President of Editorial, Business: hereon may be reproduced or used in any form or by any means—graphic, Jack W. Calhoun electronic, or mechanical, including photocopying, recording, taping, Web distribution, information storage and retrieval systems, or in any other Executive Editor: Mike Worls manner—except as may be permitted by the license terms herein. 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For your course and learning solutions, visit academic.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 1 2 3 4 5 6 7 12 11 10 09 08 Brief Contents Chapter 1 The Nature of Econometrics and Economic Data 1 PART 1: REGRESSION ANALYSIS WITH CROSS-SECTIONAL DATA 21 Chapter 2 The Simple Regression Model 22 Chapter 3 Multiple Regression Analysis: Estimation 68 Chapter 4 Multiple Regression Analysis: Inference 117 Chapter 5 Multiple Regression Analysis: OLS Asymptotics 167 Chapter 6 Multiple Regression Analysis: Further Issues 184 Chapter 7 Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables 225 Chapter 8 Heteroskedasticity 264 Chapter 9 More on Specifi cation and Data Issues 300 PART 2: REGRESSION ANALYSIS WITH TIME SERIES DATA 339 Chapter 10 Basic Regression Analysis with Time Series Data 340 Chapter 11 Further Issues in Using OLS with Time Series Data 377 Chapter 12 Serial Correlation and Heteroskedasticity in Time Series Regressions 408 PART 3: ADVANCED TOPICS 443 Chapter 13 Pooling Cross Sections across Time: Simple Panel Data Methods 444 Chapter 14 Advanced Panel Data Methods 481 Chapter 15 Instrumental Variables Estimation and Two Stage Least Squares 506 Chapter 16 Simultaneous Equations Models 546 Chapter 17 Limited Dependent Variable Models and Sample Selection Corrections 574 Chapter 18 Advanced Time Series Topics 623 Chapter 19 Carrying Out an Empirical Project 668 APPENDICES Appendix A Basic Mathematical Tools 695 Appendix B Fundamentals of Probability 714 Appendix C Fundamentals of Mathematical Statistics 747 Appendix D Summary of Matrix Algebra 788 Appendix E The Linear Regression Model in Matrix Form 799 Appendix F Answers to Chapter Questions 813 Appendix G Statistical Tables 823 References 830 Glossary 835 Index 849 iii Contents 2.5 Expected Values and Variances of the OLS CHAPTER 1 Estimators 46 The Nature of Econometrics Unbiasedness of OLS 47 and Economic Data 1 Variances of the OLS Estimators 52 Estimating the Error Variance 56 2.6 Regression through the Origin 58 1.1 What Is Econometrics? 1 Summary 59 1.2 Steps in Empirical Economic Analysis 2 Key Terms 60 1.3 The Structure of Economic Data 5 Problems 61 Cross-Sectional Data 5 Computer Exercises 64 Time Series Data 8 Appendix 2A 66 Pooled Cross Sections 9 Panel or Longitudinal Data 10 CHAPTER 3 A Comment on Data Structures 12 1.4 Causality and the Notion of Ceteris Paribus in Multiple Regression Analysis: Econometric Analysis 12 Estimation 68 Summary 17 Key Terms 17 3.1 Motivation for Multiple Regression 68 Problems 17 The Model with Two Independent Computer Exercises 18 Variables 68 The Model with k Independent Variables 71 PART 1 3.2 Mechanics and Interpretation of Ordinary Least Regression Analysis with Squares 73 Cross-Sectional Data 21 Obtaining the OLS Estimates 73 Interpreting the OLS Regression Equation 74 On the Meaning of “Holding Other Factors CHAPTER 2 Fixed” in Multiple Regression 77 The Simple Regression Model 22 Changing More Than One Independent Variable S imultaneously 77 2.1 Definition of the Simple Regression OLS Fitted Values and Residuals 77 Model 22 A “Partialling Out” Interpretation of 2.2 Deriving the Ordinary Least Squares Multiple Regression 78 Estimates 27 Comparison of Simple and Multiple Regression A Note on Terminology 35 Estimates 79 2.3 Properties of OLS on Any Sample of Data 36 Goodness-of-Fit 80 Fitted Values and Residuals 36 Regression through the Origin 83 Algebraic Properties of OLS Statistics 37 3.3 The Expected Value of the OLS Goodness-of-Fit 40 Estimators 84 2.4 Units of Measurement and Functional Including Irrelevant Variables in a Regression Form 41 Model 89 The Effects of Changing Units of Measurement Omitted Variable Bias: The Simple Case 89 on OLS Statistics 41 Omitted Variable Bias: More General Cases 93 Incorporating Nonlinearities in Simple 3.4 The Variance of the OLS Estimators 94 Regression 43 The Components of the OLS Variances: The Meaning of “Linear” Regression 46 Multicollinearity 95 iv Contents v Variances in Misspecified Models 99 5.2 Asymptotic Normality and Large Sample Estimating (cid:1)2: Standard Errors of the Inference 172 OLS Estimators 101 Other Large Sample Tests: The Lagrange 3.5 Efficiency of OLS: The Gauss-Markov Multiplier Statistic 176 Theorem 102 5.3 Asymptotic Efficiency of OLS 179 Summary 104 Summary 180 Key Terms 105 Key Terms 181 Problems 105 Problems 181 Computer Exercises 110 Computer Exercises 181 Appendix 3A 113 Appendix 5A 182 CHAPTER 4 CHAPTER 6 Multiple Regression Analysis: Multiple Regression Analysis: Inference 117 Further Issues 184 4.1 Sampling Distributions of the OLS 6.1 Effects of Data Scaling on OLS Estimators 117 Statistics 184 4.2 Testing Hypotheses about a Single Population Beta Coefficients 187 Parameter: The t Test 120 6.2 More on Functional Form 189 Testing against One-Sided Alternatives 123 More on Using Logarithmic Functional Two-Sided Alternatives 128 Forms 189 Testing Other Hypotheses about (cid:2) 130 Models with Quadratics 192 j Computing p-Values for t Tests 133 Models with Interaction Terms 197 A Reminder on the Language of Classical 6.3 More on Goodness-of-Fit and Selection Hypothesis Testing 135 of Regressors 199 Economic, or Practical, versus Statistical Adjusted R-Squared 200 Significance 135 Using Adjusted R-Squared to Choose between 4.3 Confidence Intervals 138 Nonnested Models 201 4.4 Testing Hypotheses about a Single Linear Controlling for Too Many Factors in Combination of the Parameters 140 Regression Analysis 203 4.5 Testing Multiple Linear Restrictions: Adding Regressors to Reduce the Error The F Test 143 Variance 205 Testing Exclusion Restrictions 143 6.4 Prediction and Residual Analysis 206 Relationship between F and t Statistics 149 Confidence Intervals for The R-Squared Form of the F Statistic 150 Predictions 206 Computing p-Values for F Tests 151 Residual Analysis 209 The F Statistic for Overall Significance of a Predicting y When log(y) Is the Dependent Regression 152 Variable 210 Testing General Linear Restrictions 153 Summary 215 4.6 Reporting Regression Results 154 Key Terms 215 Summary 156 Problems 216 Key Terms 158 Computer Exercises 218 Problems 159 Appendix 6A 223 Computer Exercises 163 CHAPTER 7 CHAPTER 5 Multiple Regression Analysis with Multiple Regression Analysis: Qualitative Information: Binary OLS Asymptotics 167 (or Dummy) Variables 225 5.1 Consistency 167 Deriving the Inconsistency in OLS 170 7.1 Describing Qualitative Information 225 vi Contents 7.2 A Single Dummy Independent Variable 226 RESET as a General Test for Functional Form Interpreting Coefficients on Dummy Explanatory Misspecification 303 Variables When the Dependent Variable Is Tests against Nonnested Alternatives 305 log(y) 231 9.2 Using Proxy Variables for Unobserved 7.3 Using Dummy Variables for Multiple Explanatory Variables 306 Categories 233 Using Lagged Dependent Variables as Proxy Incorporating Ordinal Informationby Using Variables 310 Dummy Variables 235 A Different Slant on Multiple 7.4 Interactions Involving Dummy Variables 238 Regression 312 Interactions among Dummy Variables 238 9.3 Models with Random Slopes 313 Allowing for Different Slopes 239 9.4 Properties of OLS under Measurement Testing for Differences in Regression Functions Error 315 across Groups 243 Measurement Error in the Dependent 7.5 A Binary Dependent Variable: The Linear Variable 316 Probability Model 246 Measurement Error in an Explanatory 7.6 More on Policy Analysis and Program Variable 318 Evaluation 251 9.5 Missing Data, Nonrandom Samples, and Summary 254 Outlying Observations 322 Key Terms 255 Missing Data 322 Problems 255 Nonrandom Samples 323 Computer Exercises 258 Outliers and Influential Observations 325 9.6 Least Absolute Deviations Estimation 330 CHAPTER 8 Summary 331 Heteroskedasticity 264 Key Terms 332 Problems 332 8.1 Consequences of Heteroskedasticity for OLS 264 Computer Exercises 334 8.2 Heteroskedasticity-Robust Inference after OLS Estimation 265 PART 2 Computing Heteroskedasticity-Robust LM Tests 269 Regression Analysis with 8.3 Testing for Heteroskedasticity 271 Time Series Data 339 The White Test for Heteroskedasticity 274 8.4 Weighted Least Squares Estimation 276 CHAPTER 10 The Heteroskedasticity Is Known up to a Multiplicative Constant 277 Basic Regression Analysis with Time The Heteroskedasticity Function Must Be Series Data 340 Estimated: Feasible GLS 282 What If the Assumed Heteroskedasticity 10.1 The Nature of Time Series Data 340 Function Is Wrong? 287 10.2 Examples of Time Series Regression Prediction and Prediction Intervals with Models 342 Heteroskedasticity 289 Static Models 342 8.5 The Linear Probability Model Revisited 290 Finite Distributed Lag Models 342 Summary 293 A Convention about the Time Index 345 Key Terms 294 10.3 Finite Sample Properties of OLS under Classical Problems 294 Assumptions 345 Computer Exercises 296 Unbiasedness of OLS 345 The Variances of the OLS Estimators and the CHAPTER 9 Gauss-Markov Theorem 349 More on Specifi cation and Inference under the Classical Linear Model Data Issues 300 Assumptions 351 10.4 Functional Form, Dummy Variables, and 9.1 Functional Form Misspecification 300 Index Numbers 353 Contents vii 10.5 Trends and Seasonality 360 Serial Correlation in the Presence of Lagged Characterizing Trending Time Series 360 Dependent Variables 411 Using Trending Variables in Regression 12.2 Testing for Serial Correlation 412 Analysis 363 A t Test for AR(1) Serial Correlation with A Detrending Interpretation of Regressions with Strictly Exogenous Regressors 412 a Time Trend 365 The Durbin-Watson Test under Classical Computing R-Squared when the Dependent Assumptions 415 Variable Is Trending 366 Testing for AR(1) Serial Correlation without Seasonality 368 Strictly Exogenous Regressors 416 Summary 370 Testing for Higher Order Serial Key Terms 371 Correlation 417 Problems 371 12.3 Correcting for Serial Correlation with Strictly Computer Exercises 373 Exogenous Regressors 419 Obtaining the Best Linear Unbiased Estimator CHAPTER 11 in the AR(1) Model 419 Feasible GLS Estimation with AR(1) Further Issues in Using OLS with Errors 421 Time Series Data 377 Comparing OLS and FGLS 423 Correcting for Higher Order Serial 11.1 Stationary and Weakly Dependent Correlation 425 Time Series 377 12.4 Differencing and Serial Correlation 426 Stationary and Nonstationary Time 12.5 Serial Correlation-Robust Inference Series 378 after OLS 428 Weakly Dependent Time Series 379 12.6 Heteroskedasticity in Time Series 11.2 Asymptotic Properties of OLS 381 Regressions 432 11.3 Using Highly Persistent Time Series in Heteroskedasticity-Robust Statistics 432 Regression Analysis 388 Testing for Heteroskedasticity 432 Highly Persistent Time Series 388 Autoregressive Conditional Transformations on Highly Persistent Heteroskedasticity 433 Time Series 393 Heteroskedasticity and Serial Correlation in Deciding Whether a Time Series Regression Models 435 Is I(1) 394 Summary 437 11.4 Dynamically Complete Models and the Absence Key Terms 437 of Serial Correlation 396 Problems 438 11.5 The Homoskedasticity Assumption for Time Computer Exercises 438 Series Models 399 Summary 400 PART 3 Key Terms 401 Problems 401 Advanced Topics 443 Computer Exercises 404 CHAPTER 13 CHAPTER 12 Pooling Cross Sections across Time: Serial Correlation and Simple Panel Data Methods 444 Heteroskedasticity in Time Series Regressions 408 13.1 Pooling Independent Cross Sections across Time 445 12.1 Properties of OLS with Serially Correlated The Chow Test for Structural Change Errors 408 across Time 449 Unbiasedness and Consistency 408 13.2 Policy Analysis with Pooled Cross Sections 450 Efficiency and Inference 409 13.3 Two-Period Panel Data Analysis 455 Goodness-of-Fit 410 Organizing Panel Data 461 viii Contents 13.4 Policy Analysis with Two-Period 15.4 IV Solutions to Errors-in-Variables Panel Data 462 Problems 525 13.5 Differencing with More Than Two 15.5 Testing for Endogeneity and Testing Time Periods 465 Overidentifying Restrictions 527 Potential Pitfalls in First Differencing Testing for Endogeneity 527 Panel Data 470 Testing Overidentification Restrictions 529 Summary 471 15.6 2SLS with Heteroskedasticity 531 Key Terms 471 15.7 Applying 2SLS to Time Series Equations 531 Problems 471 15.8 Applying 2SLS to Pooled Cross Sections and Computer Exercises 473 Panel Data 534 Appendix 13A 478 Summary 536 Key Terms 536 CHAPTER 14 Problems 536 Advanced Panel Data Methods 481 Computer Exercises 539 Appendix 15A 543 14.1 Fixed Effects Estimation 481 CHAPTER 16 The Dummy Variable Regression 485 Fixed Effects or First Differencing? 487 Simultaneous Equations Models 546 Fixed Effects with Unbalanced Panels 488 14.2 Random Effects Models 489 16.1 The Nature of Simultaneous Equations Random Effects or Fixed Effects? 493 Models 546 14.3 Applying Panel Data Methods to Other 16.2 Simultaneity Bias in OLS 550 Data Structures 494 16.3 Identifying and Estimating a Structural Summary 496 Equation 552 Key Terms 496 Identification in a Two-Equation System 552 Problems 497 Estimation by 2SLS 557 Computer Exercises 498 16.4 Systems with More Than Two Equations 559 Appendix 14A 503 Identification in Systems with Three or More Equations 559 CHAPTER 15 Estimation 560 Instrumental Variables Estimation 16.5 Simultaneous Equations Models with and Two Stage Least Squares 506 Time Series 560 16.6 Simultaneous Equations Models with Panel Data 564 15.1 Motivation: Omitted Variables in a Simple Summary 566 Regression Model 507 Key Terms 567 Statistical Inference with the Problems 567 IV Estimator 510 Computer Exercises 570 Properties of IV with a Poor Instrumental Variable 514 CHAPTER 17 Computing R-Squared after IV Estimation 516 Limited Dependent Variable 15.2 IV Estimation of the Multiple Regression Models and Sample Selection Model 517 Corrections 574 15.3 Two Stage Least Squares 521 A Single Endogenous Explanatory Variable 521 17.1 Logit and Probit Models for Binary Multicollinearity and 2SLS 523 Response 575 Multiple Endogenous Explanatory Specifying Logit and Probit Models 575 Variables 524 Maximum Likelihood Estimation of Logit and Testing Multiple Hypotheses after Probit Models 578 2SLS Estimation 525 Testing Multiple Hypotheses 579 Contents ix Interpreting the Logit and Probit 19.2 Literature Review 670 Estimates 580 19.3 Data Collection 671 17.2 The Tobit Model for Corner Solution Deciding on the Appropriate Data Set 671 Responses 587 Entering and Storing Your Data 672 Interpreting the Tobit Estimates 589 Inspecting, Cleaning, and Summarizing Specification Issues in Tobit Models 594 Your Data 673 17.3 The Poisson Regression Model 595 19.4 Econometric Analysis 675 17.4 Censored and Truncated Regression 19.5 Writing an Empirical Paper 678 Models 600 Introduction 678 Censored Regression Models 601 Conceptual (or Theoretical) Framework 679 Truncated Regression Models 604 Econometric Models and Estimation 17.5 Sample Selection Corrections 606 Methods 679 When Is OLS on the Selected Sample The Data 682 Consistent? 607 Results 682 Incidental Truncation 608 Conclusions 683 Summary 612 Style Hints 684 Key Terms 613 Summary 687 Problems 614 Key Terms 687 Computer Exercises 615 Sample Empirical Projects 687 Appendix 17A 620 List of Journals 692 Appendix 17B 621 Data Sources 693 CHAPTER 18 APPENDIX A Advanced Time Series Topics 623 Basic Mathematical Tools 695 18.1 Infinite Distributed Lag Models 624 A.1 The Summation Operator and Descriptive The Geometric (or Koyck) Distributed Lag 626 Statistics 695 Rational Distributed Lag Models 628 A.2 Properties of Linear Functions 697 18.2 Testing for Unit Roots 630 A.3 Proportions and Percentages 699 18.3 Spurious Regression 636 A.4 Some Special Functions and 18.4 Cointegration and Error Correction Models 637 Their Properties 702 Cointegration 637 Quadratic Functions 702 Error Correction Models 643 The Natural Logarithm 704 18.5 Forecasting 645 The Exponential Function 708 Types of Regression Models Used for A.5 Differential Calculus 709 Forecasting 646 Summary 711 One-Step-Ahead Forecasting 647 Key Terms 711 Comparing One-Step-Ahead Forecasts 651 Problems 711 Multiple-Step-Ahead Forecasts 652 Forecasting Trending, Seasonal, and Integrated APPENDIX B Processes 655 Fundamentals of Probability 714 Summary 660 Key Terms 661 Problems 661 B.1 Random Variables and Their Probability Computer Exercises 663 Distributions 714 Discrete Random Variables 715 CHAPTER 19 Continuous Random Variables 717 B.2 Joint Distributions, Conditional Distributions, Carrying Out an Empirical and Independence 719 Project 668 Joint Distributions and Independence 719 Conditional Distributions 721 19.1 Posing a Question 668 B.3 Features of Probability Distributions 722