Introductory econometrics A Modern ApproAch SIXTH EdITIon Jeffrey M. Wooldridge Michigan State University Australia • Brazil • Mexico • Singapore • United Kingdom • United States Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. This is an electronic version of the print textbook. Due to electronic rights restrictions, some third party content may be suppressed. Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. 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Brief contents Chapter 1 The Nature of Econometrics and Economic Data 1 Part 1: Regression Analysis with Cross-Sectional Data 19 Chapter 2 The Simple Regression Model 20 Chapter 3 Multiple Regression Analysis: Estimation 60 Chapter 4 Multiple Regression Analysis: Inference 105 Chapter 5 Multiple Regression Analysis: OLS Asymptotics 149 Chapter 6 Multiple Regression Analysis: Further Issues 166 Chapter 7 Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables 205 Chapter 8 Heteroskedasticity 243 Chapter 9 More on Specification and Data Issues 274 Part 2: Regression Analysis with Time Series Data 311 Chapter 10 Basic Regression Analysis with Time Series Data 312 Chapter 11 Further Issues in Using OLS with Time Series Data 344 Chapter 12 Serial Correlation and Heteroskedasticity in Time Series Regressions 372 Part 3: Advanced Topics 401 Chapter 13 Pooling Cross Sections Across Time: Simple Panel Data Methods 402 Chapter 14 Advanced Panel Data Methods 434 Chapter 15 Instrumental Variables Estimation and Two Stage Least Squares 461 Chapter 16 Simultaneous Equations Models 499 Chapter 17 Limited Dependent Variable Models and Sample Selection Corrections 524 Chapter 18 Advanced Time Series Topics 568 Chapter 19 Carrying Out an Empirical Project 605 aPPendices Appendix A Basic Mathematical Tools 628 Appendix B Fundamentals of Probability 645 Appendix C Fundamentals of Mathematical Statistics 674 Appendix D Summary of Matrix Algebra 709 Appendix E The Linear Regression Model in Matrix Form 720 Appendix F Answers to Chapter Questions 734 Appendix G Statistical Tables 743 References 750 Glossary 756 Index 771 iii Copyright 2016 Cengage Learning. 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Contents Preface xii 2-4 Units of Measurement and Functional Form 36 About the Author xxi 2-4a The Effects of Changing Units of Measurement on OLS Statistics 36 chapter 1 The Nature of Econometrics 2-4b Incorporating Nonlinearities in Simple Regression 37 and Economic Data 1 2-4c The Meaning of “Linear” Regression 40 2-5 Expected Values and Variances of the OLS 1-1 What Is Econometrics? 1 Estimators 40 1-2 Steps in Empirical Economic Analysis 2 2-5a Unbiasedness of OLS 40 1-3 The Structure of Economic Data 5 2-5b Variances of the OLS Estimators 45 1-3a Cross-Sectional Data 5 2-5c Estimating the Error Variance 48 1-3b Time Series Data 7 2-6 Regression through the Origin and Regression 1-3c Pooled Cross Sections 8 on a Constant 50 1-3d Panel or Longitudinal Data 9 Summary 51 1-3e A Comment on Data Structures 10 Key Terms 52 1-4 Causality and the Notion of Ceteris Paribus Problems 53 in Econometric Analysis 10 Computer Exercises 56 Summary 14 Appendix 2A 59 Key Terms 14 Problems 15 chapter 3 Multiple Regression Analysis: Computer Exercises 15 Estimation 60 Part 1 3-1 Motivation for Multiple Regression 61 3-1a The Model with Two Independent Variables 61 Regression Analysis with 3-1b The Model with k Independent Variables 63 Cross-Sectional Data 19 3-2 Mechanics and Interpretation of Ordinary Least Squares 64 chapter 2 The Simple Regression Model 20 3-2a Obtaining the OLS Estimates 64 3-2b Interpreting the OLS Regression Equation 65 2-1 Definition of the Simple Regression Model 20 3-2c On the Meaning of “Holding Other Factors Fixed” 2-2 Deriving the Ordinary Least Squares Estimates 24 in Multiple Regression 67 2-2a A Note on Terminology 31 3-2d Changing More Than One Independent Variable Simultaneously 68 2-3 Properties of OLS on Any Sample of Data 32 3-2e OLS Fitted Values and Residuals 68 2-3a Fitted Values and Residuals 32 3-2f A “Partialling Out” Interpretation of Multiple 2-3b Algebraic Properties of OLS Statistics 32 Regression 69 2-3c Goodness-of-Fit 35 iv Copyright 2016 Cengage Learning. 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Contents v 3-2g Comparison of Simple and Multiple Regression 4-6 Reporting Regression Results 137 Estimates 69 Summary 139 3-2h Goodness-of-Fit 70 Key Terms 140 3-2i Regression through the Origin 73 Problems 141 3-3 The Expected Value of the oLS Estimators 73 Computer Exercises 146 3-3a Including Irrelevant Variables in a Regression Model 77 chapter 5 Multiple Regression Analysis: 3-3b Omitted Variable Bias: The Simple Case 78 OLS Asymptotics 149 3-3c Omitted Variable Bias: More General Cases 81 3-4 The Variance of the oLS Estimators 81 5-1 Consistency 150 3-4a The Components of the OLS Variances. 5-1a Deriving the Inconsistency in OLS 153 Multicollinearity 83 3-4b Variances in Misspecified Models 86 5-2 Asymptotic normality and Large Sample 3-4c Estimating s2 Standard Errors of the OLS Inference 154 Estimators 87 5-2a Other Large Sample Tests: The Lagrange Multiplier Statistic 158 3-5 Efficiency of oLS: The Gauss-Markov Theorem 89 5-3 Asymptotic Efficiency of oLS 161 3-6 Some Comments on the Language of Multiple Regression Analysis 90 Summary 162 Summary 91 Key Terms 162 Key Terms 93 Problems 162 Problems 93 Computer Exercises 163 Computer Exercises 97 Appendix 5A 165 Appendix 3A 101 chapter 6 Multiple Regression Analysis: chapter 4 Multiple Regression Analysis: Further Issues 166 Inference 105 6-1 Effects of data Scaling on oLS Statistics 166 6-1a Beta Coefficients 169 4-1 Sampling distributions of the oLS Estimators 105 6-2 More on Functional Form 171 4-2 Testing Hypotheses about a Single Population Parameter: The t Test 108 6-2a More on Using Logarithmic Functional Forms 171 4-2a Testing against One-Sided Alternatives 110 6-2b Models with Quadratics 173 4-2b Two-Sided Alternatives 114 6-2c Models with Interaction Terms 177 4-2c Testing Other Hypotheses about b 116 6-2d Computing Average Partial Effects 179 j 4-2d Computing p-Values for t Tests 118 6-3 More on Goodness-of-Fit and Selection 4-2e A Reminder on the Language of Classical of Regressors 180 Hypothesis Testing 120 6-3a Adjusted R-Squared 181 4-2f Economic, or Practical, versus Statistical 6-3b Using Adjusted R-Squared to Choose between Significance 120 Nonnested Models 182 4-3 Confidence Intervals 122 6-3c Controlling for Too Many Factors in Regression Analysis 184 4-4 Testing Hypotheses about a Single Linear 6-3d Adding Regressors to Reduce the Error b Combination of the Parameters 124 j Variance 185 4-5 Testing Multiple Linear Restrictions: The F Test 127 6-4 Prediction and Residual Analysis 186 4-5a Testing Exclusion Restrictions 127 6.4a Confidence Intervals for Predictions 186 4-5b Relationship between F and t Statistics 132 6-4b Residual Analysis 190 4-5c The R-Squared Form of the F Statistic 133 6-4c Predicting y When log(y) Is the Dependent 4-5d Computing p-Values for F Tests 134 Variable 190 4-5e The F Statistic for Overall Significance of a 6-4d Predicting y When the Dependent Variable Regression 135 Is log(y): 192 4-5f Testing General Linear Restrictions 136 Copyright 2016 Cengage Learning. 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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. vi Contents Summary 194 8-4c What If the Assumed Heteroskedasticity Function Is Key Terms 196 Wrong? 262 8-4d Prediction and Prediction Intervals with Problems 196 Heteroskedasticity 264 Computer Exercises 199 8-5 The Linear Probability Model Revisited 265 Appendix 6A 203 Summary 267 Key Terms 268 chapter 7 Multiple Regression Analysis with Problems 268 Qualitative Information: Binary (or Dummy) Computer Exercises 270 Variables 205 chapter 9 More on Specification and Data 7-1 describing Qualitative Information 205 Issues 274 7-2 A Single dummy Independent Variable 206 7-2a Interpreting Coefficients on Dummy Explanatory Variables When the Dependent Variable Is 9-1 Functional Form Misspecification 275 log(y) 211 9-1a RESET as a General Test for Functional Form Misspecification 277 7-3 Using dummy Variables for Multiple Categories 212 9-1b Tests against Nonnested Alternatives 278 7-3a Incorporating Ordinal Information by Using 9-2 Using Proxy Variables for Unobserved Explanatory Dummy Variables 214 Variables 279 7-4 Interactions Involving dummy Variables 217 9-2a Using Lagged Dependent Variables as Proxy Variables 283 7-4a Interactions among Dummy Variables 217 9-2b A Different Slant on Multiple Regression 284 7-4b Allowing for Different Slopes 218 7-4c Testing for Differences in Regression Functions 9-3 Models with Random Slopes 285 across Groups 221 9-4 Properties of oLS under Measurement Error 287 7-5 A Binary dependent Variable: The Linear Probability 9-4a Measurement Error in the Dependent Variable 287 Model 224 9-4b Measurement Error in an Explanatory Variable 289 7-6 More on Policy Analysis and Program 9-5 Missing data, nonrandom Samples, and outlying Evaluation 229 observations 293 7-7 Interpreting Regression Results with discrete 9-5a Missing Data 293 dependent Variables 231 9-5b Nonrandom Samples 294 Summary 232 9-5c Outliers and Influential Observations 296 Key Terms 233 9-6 Least Absolute deviations Estimation 300 Problems 233 Summary 302 Computer Exercises 237 Key Terms 303 Problems 303 chapter 8 Heteroskedasticity 243 Computer Exercises 307 8-1 Consequences of Heteroskedasticity for oLS 243 Pa rt 2 8-2 Heteroskedasticity-Robust Inference after oLS Estimation 244 Regression Analysis with Time 8-2a Computing Heteroskedasticity-Robust LM Tests 248 Series Data 311 8-3 Testing for Heteroskedasticity 250 8-3a The White Test for Heteroskedasticity 252 chapter 10 Basic Regression Analysis with 8-4 Weighted Least Squares Estimation 254 Time Series Data 312 8-4a The Heteroskedasticity Is Known up to a Multiplicative Constant 254 10-1 The nature of Time Series data 312 8-4b The Heteroskedasticity Function Must Be Estimated: Feasible GLS 259 10-2 Examples of Time Series Regression Models 313 Copyright 2016 Cengage Learning. 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Contents vii 10-2a Static Models 314 Problems 365 10-2b Finite Distributed Lag Models 314 Computer Exercises 368 10-2c A Convention about the Time Index 316 chapter 12 Serial Correlation and 10-3 Finite Sample Properties of oLS under Classical Heteroskedasticity in Time Series Assumptions 317 Regressions 372 10-3a Unbiasedness of OLS 317 10-3b The Variances of the OLS Estimators and the 12-1 Properties of oLS with Serially Correlated Gauss-Markov Theorem 320 Errors 373 10-3c Inference under the Classical Linear Model 12-1a Unbiasedness and Consistency 373 Assumptions 322 12-1b Efficiency and Inference 373 10-4 Functional Form, dummy Variables, and Index 12-1c Goodness of Fit 374 numbers 323 12-1d Serial Correlation in the Presence 10-5 Trends and Seasonality 329 of Lagged Dependent Variables 374 10-5a Characterizing Trending Time 12-2 Testing for Serial Correlation 376 Series 329 12-2a A t Test for AR(1) Serial Correlation with Strictly 10-5b Using Trending Variables in Regression Exogenous Regressors 376 Analysis 332 12-2b The Durbin-Watson Test under Classical 10-5c A Detrending Interpretation of Regressions Assumptions 378 with a Time Trend 334 12-2c Testing for AR(1) Serial Correlation without 10-5d Computing R-Squared When the Dependent Strictly Exogenous Regressors 379 Variable Is Trending 335 12-2d Testing for Higher Order Serial 10-5e Seasonality 336 Correlation 380 Summary 338 12-3 Correcting for Serial Correlation with Strictly Key Terms 339 Exogenous Regressors 381 Problems 339 12-3a Obtaining the Best Linear Unbiased Estimator in Computer Exercises 341 the AR(1) Model 382 12-3b Feasible GLS Estimation with AR(1) chapter 11 Further Issues in Using OLS Errors 383 12-3c Comparing OLS and FGLS 385 with Time Series Data 344 12-3d Correcting for Higher Order Serial Correlation 386 11-1 Stationary and Weakly dependent Time Series 345 12-4 differencing and Serial Correlation 387 11-1a Stationary and Nonstationary Time 12-5 Serial Correlation–Robust Inference Series 345 after oLS 388 11-1b Weakly Dependent Time Series 346 12-6 Heteroskedasticity in Time Series 11-2 Asymptotic Properties of oLS 348 Regressions 391 12-6a Heteroskedasticity-Robust Statistics 392 11-3 Using Highly Persistent Time Series in Regression Analysis 354 12-6b Testing for Heteroskedasticity 392 11-3a Highly Persistent Time Series 354 12-6c Autoregressive Conditional Heteroskedasticity 393 11-3b Transformations on Highly Persistent Time Series 358 12-6d Heteroskedasticity and Serial Correlation in Regression Models 395 11-3c Deciding Whether a Time Series Is I(1) 359 Summary 396 11-4 dynamically Complete Models and the Absence of Serial Correlation 360 Key Terms 396 11-5 The Homoskedasticity Assumption for Time Series Problems 396 Models 363 Computer Exercises 397 Summary 364 Key Terms 365 Copyright 2016 Cengage Learning. 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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. viii Contents Part 3 chapter 15 Instrumental Variables Estimation and Two Stage Least Squares 461 Advanced Topics 401 15-1 Motivation: omitted Variables in a Simple chapter 13 Pooling Cross Sections across Regression Model 462 Time: Simple Panel Data Methods 402 15-1a Statistical Inference with the IV Estimator 466 15-1b Properties of IV with a Poor Instrumental 13-1 Pooling Independent Cross Sections across Variable 469 Time 403 15-1c Computing R-Squared after IV Estimation 471 13-1a The Chow Test for Structural Change 15-2 IV Estimation of the Multiple Regression across Time 407 Model 471 13-2 Policy Analysis with Pooled Cross 15-3 Two Stage Least Squares 475 Sections 407 15-3a A Single Endogenous Explanatory Variable 475 13-3 Two-Period Panel data Analysis 412 15-3b Multicollinearity and 2SLS 477 13-3a Organizing Panel Data 417 15-3c Detecting Weak Instruments 478 13-4 Policy Analysis with Two-Period Panel 15-3d Multiple Endogenous Explanatory Variables 478 data 417 15-3e Testing Multiple Hypotheses after 2SLS 13-5 differencing with More Than Two Time Estimation 479 Periods 420 15-4 IV Solutions to Errors-in-Variables Problems 479 13-5a Potential Pitfalls in First Differencing Panel 15-5 Testing for Endogeneity and Testing Data 424 overidentifying Restrictions 481 Summary 424 15-5a Testing for Endogeneity 481 Key Terms 425 15-5b Testing Overidentification Restrictions 482 Problems 425 15-6 2SLS with Heteroskedasticity 484 Computer Exercises 426 15-7 Applying 2SLS to Time Series Equations 485 Appendix 13A 432 15-8 Applying 2SLS to Pooled Cross Sections and Panel data 487 chapter 14 Advanced Panel Data Summary 488 Methods 434 Key Terms 489 Problems 489 14-1 Fixed Effects Estimation 435 Computer Exercises 492 14-1a The Dummy Variable Regression 438 Appendix 15A 496 14-1b Fixed Effects or First Differencing? 439 14-1c Fixed Effects with Unbalanced chapter 16 Simultaneous Equations Panels 440 Models 499 14-2 Random Effects Models 441 14-2a Random Effects or Fixed Effects? 444 16-1 The nature of Simultaneous Equations 14-3 The Correlated Random Effects Models 500 Approach 445 16-2 Simultaneity Bias in oLS 503 14-3a Unbalanced Panels 447 16-3 Identifying and Estimating a Structural 14-4 Applying Panel data Methods to other data Equation 504 Structures 448 16-3a Identification in a Two-Equation System 505 Summary 450 16-3b Estimation by 2SLS 508 Key Terms 451 16-4 Systems with More Than Two Equations 510 Problems 451 16-4a Identification in Systems with Three or More Computer Exercises 453 Equations 510 Appendix 14A 457 16-4b Estimation 511 Copyright 2016 Cengage Learning. 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