Introductory Econometrics Copyright 2012 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 Copyright 2012 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 Introductory Econometrics A Modern Approach Fifth Edition Jeffrey M Wooldridge Michigan State University Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Copyright 2012 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 The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.cengage.com/highered to search by ISBN#, author, title, or keyword for materials in your areas of interest Copyright 2012 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 Introductory Econometrics: A Modern Approach, Fifth Edition Jeffrey M Wooldridge Senior Vice President, LRS/Acquisitions & Solutions Planning: Jack W Calhoun Editorial Director, Business & Economics: Erin Joyner © 2013, 2009 South-Western, Cengage Learning ALL RIGHTS RESERVED No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher Editor-in-Chief: Joe Sabatino Executive Editor: Michael Worls Associate Developmental Editor: Julie Warwick Editorial Assistant: Libby Beiting-Lipps Brand Management Director: Jason Sakos For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions Further permissions questions can be emailed to permissionrequest@cengage.com Market Development Director: Lisa Lysne Senior Brand Manager: Robin LeFevre Library of Congress Control Number: 2012945120 Senior Market Development Manager: John Carey ISBN-13: 978-1-111-53104-1 Content Production Manager: Jean Buttrom Rights Acquisition Director: Audrey Pettengill Rights Acquisition Specialist, Text/Image: John Hill ISBN-10: 1-111-53104-8 South-Western 5191 Natorp Boulevard Mason, OH 45040 USA Media Editor: Anita Verma Senior Manufacturing Planner: Kevin Kluck Senior Art Director: Michelle Kunkler Production Management and Composition: PreMediaGlobal Internal Designer: PreMediaGlobal Cover Designer: Rokusek Design Cengage Learning products are represented in Canada by Nelson Education, Ltd For your course and learning solutions, visit www.cengage.com Purchase any of our products at your local college store or at our p referred online store www.cengagebrain.com Cover Image: © Elena R/Shutterstock.com; Milosz Aniol/Shutterstock.com Printed in the United States of America 16 15 14 13 12 Copyright 2012 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 Brief Contents Chapter The Nature of Econometrics and Economic Data PART 1: Regression Analysis with Cross-Sectional Data Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter The Simple Regression Model Multiple Regression Analysis: Estimation Multiple Regression Analysis: Inference Multiple Regression Analysis: OLS Asymptotics Multiple Regression Analysis: Further Issues Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables Heteroskedasticity More on Specification and Data Issues 21 22 68 118 168 186 227 268 303 PART 2: Regression Analysis with Time Series Data 343 Chapter 10 Basic Regression Analysis with Time Series Data Chapter 11 Further Issues in Using OLS with Time Series Data Chapter 12 Serial Correlation and Heteroskedasticity in Time Series Regressions 344 380 412 PART 3: Advanced Topics 447 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 448 484 512 554 583 632 676 Pooling Cross Sections Across Time: Simple Panel Data Methods Advanced Panel Data Methods Instrumental Variables Estimation and Two Stage Least Squares Simultaneous Equations Models Limited Dependent Variable Models and Sample Selection Corrections Advanced Time Series Topics Carrying Out an Empirical Project Appendices Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F Appendix G References Glossary Index Basic Mathematical Tools Fundamentals of Probability Fundamentals of Mathematical Statistics Summary of Matrix Algebra The Linear Regression Model in Matrix Form Answers to Chapter Questions Statistical Tables 703 722 755 796 807 821 831 838 844 862 v Copyright 2012 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 Contents Preface xv About the Author xxv Chapter 1 The Nature of Econometrics and Economic Data 1.1 What Is Econometrics? 1.2 Steps in Empirical Economic Analysis 1.3 The Structure of Economic Data Cross-Sectional Data Time Series Data Pooled Cross Sections Panel or Longitudinal Data 10 A Comment on Data Structures 11 1.4 Causality and the Notion of Ceteris Paribus in Econometric Analysis 12 Summary 16 2.5 Expected Values and Variances of the OLS Estimators 45 Unbiasedness of OLS 45 Variances of the OLS Estimators 50 Estimating the Error Variance 54 2.6 Regression through the Origin and Regression on a Constant 57 Summary 58 Key Terms 59 Computer Exercises 63 Problems 17 Computer Exercises 17 PART Regression Analysis with Cross-Sectional Data 21 Model 22 2.4 Units of Measurement and Functional Form 39 The Effects of Changing Units of Measurement on OLS Statistics 40 Incorporating Nonlinearities in Simple Regression 41 The Meaning of “Linear” Regression 44 Problems 60 Key Terms 17 Chapter 2 The 2.3 Properties of OLS on Any Sample of Data 35 Fitted Values and Residuals 35 Algebraic Properties of OLS Statistics 36 Goodness-of-Fit 38 Simple Regression 2.1 Definition of the Simple Regression Model 22 2.2 Deriving the Ordinary Least Squares Estimates 27 A Note on Terminology 34 Appendix 2A 66 Chapter 3 Multiple Regression Analysis: Estimation 68 3.1 Motivation for Multiple Regression 69 The Model with Two Independent Variables 69 The Model with k Independent Variables 71 3.2 Mechanics and Interpretation of Ordinary Least Squares 72 Obtaining the OLS Estimates 72 Interpreting the OLS Regression Equation 74 On the Meaning of “Holding Other Factors Fixed” in Multiple Regression 76 Changing More Than One Independent Variable Simultaneously 77 vi Copyright 2012 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 Contents OLS Fitted Values and Residuals 77 A “Partialling Out” Interpretation of Multiple Regression 78 Comparison of Simple and Multiple Regression Estimates 78 Goodness-of-Fit 80 Regression through the Origin 81 3.3 The Expected Value of the OLS Estimators 83 Including Irrelevant Variables in a Regression Model 88 Omitted Variable Bias: The Simple Case 88 Omitted Variable Bias: More General Cases 91 3.4 The Variance of the OLS Estimators 93 The Components of the OLS Variances: Multicollinearity 94 Variances in Misspecified Models 98 Estimating s 2: Standard Errors of the OLS Estimators 99 3.5 Efficiency of OLS: The Gauss-Markov Theorem 101 3.6 Some Comments on the Language of Multiple Regression Analysis 103 4.5 Testing Multiple Linear Restrictions: The F Test 143 Testing Exclusion Restrictions 143 Relationship between F and t Statistics 149 The R-Squared Form of the F Statistic 150 Computing p-Values for F Tests 151 The F Statistic for Overall Significance of a Regression 152 Testing General Linear Restrictions 153 4.6 Reporting Regression Results 154 Summary 157 Key Terms 159 Problems 159 Computer Exercises 164 chapter 5 Multiple Regression Analysis: OLS Asymptotics 168 5.1 Consistency 169 Deriving the Inconsistency in OLS 172 Key Terms 105 5.2 Asymptotic Normality and Large Sample Inference 173 Other Large Sample Tests: The Lagrange Multiplier Statistic 178 Problems 106 5.3 Asymptotic Efficiency of OLS 181 Computer Exercises 110 Summary 182 Appendix 3A 113 Key Terms 183 Summary 104 vii Problems 183 Chapter 4 Multiple Regression Analysis: Inference 118 Computer Exercises 183 4.1 Sampling Distributions of the OLS Estimators 118 chapter 6 Multiple 4.2 Testing Hypotheses about a Single Population Parameter: The t Test 121 Testing against One-Sided Alternatives 123 Two-Sided Alternatives 128 Testing Other Hypotheses about bj 130 Computing p-Values for t Tests 133 A Reminder on the Language of Classical Hypothesis Testing 135 Economic, or Practical, versus Statistical Significance 135 4.3 Confidence Intervals 138 4.4 Testing Hypotheses about a Single Linear Combination of the Parameters 140 Appendix 5A 185 Regression Analysis: Further Issues 186 6.1 Effects of Data Scaling on OLS Statistics 186 Beta Coefficients 189 6.2 More on Functional Form 191 More on Using Logarithmic Functional Forms 191 Models with Quadratics 194 Models with Interaction Terms 198 6.3 More on Goodness-of-Fit and Selection of Regressors 200 Adjusted R-Squared 202 Using Adjusted R-Squared to Choose between Nonnested Models 203 Copyright 2012 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 viii Contents Controlling for Too Many Factors in Regression Analysis 205 Adding Regressors to Reduce the Error Variance 206 6.4 Prediction and Residual Analysis 207 Confidence Intervals for Predictions 207 Residual Analysis 211 Predicting y When log(y) Is the Dependent Variable 212 Summary 216 Key Terms 217 Problems 218 Computer Exercises 220 Appendix 6A 225 chapter 7 Multiple Regression Analysis with Qualitative Information: Binary (or Dummy) Variables 227 chapter 8 Heteroskedasticity 8.1 Consequences of Heteroskedasticity for OLS 268 8.2 Heteroskedasticity-Robust Inference after OLS Estimation 269 Computing Heteroskedasticity-Robust LM Tests 274 8.3 Testing for Heteroskedasticity 275 The White Test for Heteroskedasticity 279 8.4 Weighted Least Squares Estimation 280 The Heteroskedasticity Is Known up to a Multiplicative Constant 281 The Heteroskedasticity Function Must Be Estimated: Feasible GLS 286 What If the Assumed Heteroskedasticity Function Is Wrong? 290 Prediction and Prediction Intervals with Heteroskedasticity 292 8.5 The Linear Probability Model Revisited 294 Summary 296 7.1 Describing Qualitative Information 227 Key Terms 297 7.2 A Single Dummy Independent Variable 228 Interpreting Coefficients on Dummy Explanatory Variables When the Dependent Variable Is log(y) 233 Problems 297 7.3 Using Dummy Variables for Multiple Categories 235 Incorporating Ordinal Information by Using Dummy Variables 237 7.4 Interactions Involving Dummy Variables 240 Interactions among Dummy Variables 240 Allowing for Different Slopes 241 Testing for Differences in Regression Functions across Groups 245 7.5 A Binary Dependent Variable: The Linear Probability Model 248 7.6 More on Policy Analysis and Program Evaluation 253 7.7 Interpreting Regression Results with Discrete Dependent Variables 256 Summary 257 Key Terms 258 Problems 258 Computer Exercises 262 268 Computer Exercises 299 chapter 9 More on Specification and Data Issues 303 9.1 Functional Form Misspecification 304 RESET as a General Test for Functional Form Misspecification 306 Tests against Nonnested Alternatives 307 9.2 Using Proxy Variables for Unobserved Explanatory Variables 308 Using Lagged Dependent Variables as Proxy Variables 313 A Different Slant on Multiple Regression 314 9.3 Models with Random Slopes 315 9.4 Properties of OLS under Measurement Error 317 Measurement Error in the Dependent Variable 318 Measurement Error in an Explanatory Variable 320 9.5 Missing Data, Nonrandom Samples, and Outlying Observations 324 Copyright 2012 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 868 Index firm sales See sales first differencing assumptions for, 481–483 defined, 461 fixed effects vs., 489–491 I(1) time series and, 396 panel data, pitfalls in, 473–474 first order autocorrelation, 397 first order conditions, 30, 73–74, 719, 808 first-differenced equations, 461 fitted values See also OLS (ordinary least squares) in multiple regressions, 77–78 in simple regressions, 30, 35–36 fixed effects assumptions for, 509–511 defined, 460 dummy variable regression, 488–489 estimation, 484–492 first differencing vs., 489–491 random effects vs., 495–496 with unbalanced panels, 491–492 forecast intervals, 655–656 forecasting multiple-step-ahead, 660–662 one-step-ahead, 655–659 overview and definitions, 652–654 trending, seasonal, and integrated processes, 662–667 types of models used for, 654–655 free throw shooting, 728, 730 freeway width and commute time, 788–789 frequency, data, frequency distributions, 401(k) plans, 174 functional forms in multiple regressions with interaction terms, 198–200 logarithmic, 191–194 misspecification, 304–308 quadratic, 194–198 in simple regressions, 39–44 in time series regressions, 356–357 G Gauss-Markov assumptions for multiple linear regressions, 83–88, 93 for simple linear regressions, 45–50, 51–54 Gauss-Markov Theorem for multiple linear regressions, 101–102, 116–117 for OLS in matrix form, 812 for time series regressions, 352–354 GDL (geometric distributed lag), 635–637 GDP See gross domestic product (GDP) gender as binary variable See qualitative information oversampling, 326 wage gap, 451–453 generalized least squares (GLS) estimators for AR(1) models, 424–428 with heteroskedasticity and AR(1) serial correlations, 439 when heteroskedasticity function must be estimated, 286–290 when heteroskedasticity is known up to a multiplicative constant, 282–283 geometric distributed lag (GDL), 635–637 GLS estimators See generalized least squares (GLS) estimators Goldberger, Arthur, 96 goodness-of-fit See also predictions; R-squareds change in unit of measurement and, 41 in multiple regressions, 80–81 overemphasizing, 205–206 percent correctly predicted, 251, 590 in simple regressions, 38–39 in time series regressions, 414 Google Scholar, 677 government policies economic growth and, housing prices and, 9–10 GPA See college GPA Granger, Clive W J., 169 Granger causality, 657 gross domestic product (GDP) data frequency for, government policies and, high persistence, 393–394 in real terms, 360 seasonal adjustment of, 372 unit root test, 643 growth rate, 366, 396 gun control laws, 255 H HAC standard errors, 432 Hartford School District, 211–212 Hausman test, 290, 496 Head Start participation, 255 Heckit method, 618–619 heterogeneity bias, 460 Copyright 2012 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 heteroskedasticity See also weighted least squares estimation 2SLS with, 538 consequences of, for OLS, 268–269 defined, 51 HAC standard errors, 432 heteroskedasticity-robust procedures, 269–275 linear probability model and, 294–296 testing for, 275–280 in time series regressions, 434–439 in wage equation, 52 high school and college GPAs See college GPA highly persistent time series deciding whether I(0) or I(1), 396–399 description of, 391–395 transformations on, 395–396 histogram, 401(k) plan participation, 174 homoskedasticity for 2SLS, 552 for IV estimation, 517 for multiple linear regressions, 93–94, 101 for OLS in matrix form, 811 for simple linear regressions, 51–54 for time series regressions, 352–353, 387–388, 402 hourly wages See wages housing prices and expenditures air pollution and See air pollution and housing prices general linear restrictions, 153–154 heteroskedasticity BP test, 278 White test, 280 incinerators and inconsistency in OLS, 173 pooled cross sections, 454–457 income and, 706 inflation, 637–639 investment and computing R-squared, 370–371 spurious relationship, 367 over controlling, 206 property taxes and, 9–10 with qualitative information, 234 RESET, 307 residual analysis, 211 rooms and See rooms and housing prices savings and, 557–558 hypotheses See also hypothesis testing about single linear combination of parameters, 140–143 about single population parameter See t tests Index 869 after 2SLS estimation, 532 expectations, 16 language of classical testing, 135 in logit and probit models, 588–589 multiple linear restrictions See F tests stating, in empirical analysis, hypothesis testing about mean in normal population, 780–783 asymptotic tests for nonnormal populations, 783–784 computing and using p-values, 784–787 confidence intervals and, 787–788 in matrix form, Wald statistics for, 818 overview and fundamentals, 777–780 practical vs statistical significance, 788–789 I I(0) and I(1) processes, 396–399 idempotent matrices, 802–803 identification defined, 516 in systems with three or more equations, 567–568 in systems with two equations, 560–565 identity matrices, 797 idiosyncratic error, 460 IDL (infinite distributed lag models), 633–639 IIP (index of industrial production), 359–360 impact propensity/multiplier, 347 incidental truncation, 615, 617–621 incinerators and housing prices inconsistency in OLS, 173 pooled cross sections, 454–457 including irrelevant variables, 88 income See also wages family See family income housing expenditure and, 706 PIH, 570–571 savings and See under savings inconsistency in OLS, deriving, 172–173 inconsistent estimators, 764 independence, joint distributions and, 727–729 independent variables See also regression analysis; specific event studies changing simultaneously, 77 defined, 23 maximum likelihood estimation with, 630 measurement error in, 320–323 in misspecified models, 88–92 random, 728 simple vs multiple regression, 69–72 Copyright 2012 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 870 Index independently pooled cross sections See also pooled cross sections across time, 449–453 defined, 448 index numbers, 359–362 industrial production, index of (IIP), 359–360 infant mortality rates, outliers, 330–331 inference in multiple regressions confidence intervals, 138–140 hypotheses See hypotheses statistical, with IV estimator, 517–521 in time series regressions, 355–356, 413–414 infinite distributed lag models, 633–639 inflation from 1948 to 2003, 345 interest rates and See under interest rates openness and, 564–565, 566 random walk model for, 392 unemployment and expectations augmented Phillips curve, 390–391 forecasting, 656 static Phillips curve, 346, 355–356 unit root test, 642 influential observations, 326–331 information set, 653 in-sample criteria, 658–659 instrumental variables See also two stage least squares computing R-squared after estimation, 523 in multiple regressions, 524–527 overview and definitions, 513, 514, 516 properties, with poor instrumental variable, 521–523 in simple regressions, 513–523 solutions to errors-in-variables problems, 532–534 statistical inference, 517–521 integrated of order zero/one processes, 396–399 integrated processes, forecasting, 662–667 interaction effect, 198–200 interaction terms, 240–241 intercept shifts, 229–230 intercepts See also OLS estimators; regression analysis change in unit of measurement and, 40–41 defined, 23, 705 in regressions on a constant, 58 in regressions through origin, 57–58 interest rates inflation, deficits, and differencing, 430 inference under CLM assumptions, 356 T-bill See T-bill rates interval estimation, 755, 770–772 See also confidence intervals inverse Mills ratio, 598 inverse of matrix, 801 IQ ability and, 309–312, 314–315 nonrandom sampling, 325 irrelevant variables, including, 88 IV See instrumental variables J JEL (Journal of Economic Literature), 677 job training See also training grants scrap rates and See scrap rates and job training as self-selection problem, 255 worker productivity and program evaluation, 254 sample model, joint distributions features of, 730–737 independence and, 727–729 joint hypotheses tests, 144 jointly statistically significant/insignificant, 148 Journal of Economic Literature (JEL), 677 journals listing, 700–701 junior colleges vs universities, 140–143 just identified equations, 568 K Koyck distributed lag, 635–637 kurtosis, 737 L labor economists, 676, 678 labor force See employment and unemployment; women in labor force labor supply and demand, 555–556 labor supply function, 715 LAD (least absolute deviations) estimation, 331–334 lag distribution, 347 lagged dependent variables as proxy variables, 313–314 serial correlation and, 415–416 lagged endogenous variables, 659 lagged explanatory variables, 349 Copyright 2012 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 Lagrange multiplier (LM) statistics heteroskedasticity-robust, 274–275 See also heteroskedasticity main discussion, 178–181 land quality and fertilizers, 25 large sample properties, 763–767 See also asymptotic entries; OLS asymptotics latent variable models, 585 law enforcement city crime levels and (causality), 14–15 murder rates and (SEM), 557 law of iterated expectations, 743–744 law of large numbers, 765 law school rankings as dummy variables, 239–240 residual analysis, 211 LDV See limited dependent variables leads and lags estimators, 650 least absolute deviations (LAD) estimation, 331–334 least squares estimator, 770 likelihood ratio statistic, 589 limited dependent variables asymptotic standard errors in, 630–631 binary response See logit and probit models censored and truncated regression models, 609–615 corner solution response See Tobit model count response, Poisson regression for, 604–609 overview, 583–584 sample selection corrections, 615–621 linear functions, 705–707 linear in parameters assumption for 2SLS, 551–552 for multiple linear regressions, 83 for OLS in matrix form, 809–810 for simple linear regressions, 45, 49 for time series regressions, 349–350 linear independence, 801 linear probability model (LPM) See also limited dependent variables heteroskedasticity and, 294–296 main discussion, 248–253 linear regression model, 44, 72 See also multiple regression analysis; simple regression model linear relationship among independent variables, 95–96 linear time trends, 364–365 linearity and weak dependence assumption, 384–385 literature review, 678–679 LM statistics See Lagrange multiplier (LM) statistics loan approval rates F and t statistics, 150 Index 871 multicollinearity, 97 program evaluation, 254–255 logarithms in multiple regressions, 191–194 natural, overview, 712–715 predicting y when log(y) is dependent, 212–215 qualitative information and, 233–235 real dollars and, 361 in simple regressions, 41–43 in time series regressions, 356–357 logit and probit models interpreting estimates, 589–596 maximum likelihood estimation of, 587–588 specifying, 584–587 testing multiple hypotheses, 588–589 log-likelihood functions, 588 longitudinal data See also panel data long-run elasticity, 357 long-run propensity (LRP), 348 loss functions, 653 LPM See linear probability model (LPM) LRP (long-run propensity), 348 lunch program and math performance, 50 M macroeconomists, 677 MAE (mean absolute error), 659 marginal effect, 705 marital status See qualitative information martingale difference sequence, 639 martingale functions, 653 matched pair samples, 500 math performance and lunch program, 50 mathematical statistics See statistics matrices See also OLS in matrix form basic definitions, 796–797 differentiation of linear and quadratic forms, 803 idempotent, 802–803 linear independence and rank of, 801 moments and distributions of random vectors, 803–805 operations, 797–801 quadratic forms and positive definite, 802 matrix notation, 808 maximum likelihood estimation, 587–588, 630, 769–770 mean, using summation operator, 704–705 mean absolute error (MAE), 659 mean independence, 25 mean squared error (MSE), 763 Copyright 2012 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 872 Index measurement error IV solutions to, 532–534 properties of OLS under, 317–323 measures of association, 737–739 measures of central tendency, 730–734 measures of variability, 734–737 median, 704, 733–734 men, return to education, 518–519 method of moments approach, 28, 768–769 micronumerosity, 96 military personnel survey, oversampling in, 326 minimum variance unbiased estimators, 119, 769, 815 minimum wages employment/unemployment and AR(1) serial correlation, testing for, 420–421 causality, 15 detrending, 369–370 logarithmic form, 356–357 SC-robust standard error, 434 in Puerto Rico, effects of, 8–9 minorities and loans See loan approval rates missing data, 324 misspecification in empirical projects, 684–685 functional form, 304–308 unbiasedness and, 88–92 variances, 98–99 motherhood, teenage, 499–500 moving average process of order one, 383 MSE (mean squared error), 763 multicollinearity 2SLS and, 530–531 among explanatory variables, 324 main discussion, 94–98 multiple hypotheses tests, 144 multiple regression analysis See also data issues; estimation and estimators; heteroskedasticity; hypotheses; OLS (ordinary least squares); predictions; R-squareds adding regressors to reduce error variance, 206–207 advantages over simple regression, 68–72 confidence intervals, 138–140 functional forms See under functional forms over controlling, 205–206 with qualitative information See under qualitative information multiple restrictions, 144 multiple-step-ahead forecasts, 660–662 multiplicative measurement error, 319 multivariate normal distribution, 804 municipal bond interest rates, 237–238 murder rates SEM, 557 static Phillips curve, 346 N natural experiments, 457, 521 natural logarithms, 712–715 See also logarithms netting out, 78 no perfect collinearity assumption for multiple linear regressions, 84–86, 87 for OLS in matrix form, 810 for time series regressions, 350, 385 no serial correlation assumption See also serial correlation for 2SLS, 553 for OLS in matrix form, 811 for time series regressions, 353–354, 387–388 nominal dollars, 360 nonexperimental data, nonlinear functions, 710–716 nonlinearities, incorporating in simple regressions, 41–44 nonnested models choosing between, 203–205 functional form misspecification and, 307–308 nonrandom samples, 324–326, 615 nonstationary time series processes, 381–382 normal distribution, 745–749 normal sampling distributions for multiple linear regressions, 120–121 for time series regressions, 355–356 normality assumption for multiple linear regressions, 118–121 for time series regressions, 355 normality of errors assumption, 813 normality of estimators in general, asymptotic, 766–767 normality of OLS, asymptotic in multiple regressions, 173–178, 185 in time series regressions, 387–391 n-R-squared statistic, 178–181 null hypothesis, 122–123, 778 See also hypotheses O observational data, OLS (ordinary least squares) See also heteroskedasticity; panel data; predictions; other OLS entries Copyright 2012 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 cointegration and, 649–650 comparison of simple and multiple regression estimates, 78–80 consistency See consistency of OLS logit and probit vs., 593–595 in multiple regressions algebraic properties, 72–83 computational properties, 72–83 effects of data scaling, 186–191 fitted values and residuals, 77–78 goodness-of-fit, 80–81 interpreting equations, 74–76 measurement error and, 317–323 obtaining estimates, 72–74 partialling out, 78 regression through origin, 81–83 statistical properties, 83–92 Poisson vs., 606, 607–608 in simple regressions algebraic properties, 35–39 defined, 30 deriving estimates, 27–35 statistical properties, 45–56 units of measurement, changing, 40–41 simultaneity bias in, 558–560 in time series regressions correcting for serial correlation, 425–428 FGLS vs., 427–428 finite sample properties, 349–356 SC-robust standard errors, 431–434 with serially correlated errors, properties of, 412–416 Tobit vs., 601–603 OLS asymptotics in matrix form, 815–818 in multiple regressions consistency, 169–173 efficiency, 181–182 Lagrange multiplier (LM) statistic, 178–181 normality, 173–178, 185 overview, 168 in time series regressions consistency, 384–387 normality, 387–391 OLS estimators See also heteroskedasticity defined, 45 in multiple regressions asymptotics See OLS asymptotics efficiency of, 101–102 expected value of, 83–92 Index 873 sampling distributions of, 118–121 unbiasedness of, 87–88 variances of, 93–101 in simple regressions unbiasedness of, 45–50 variances of, 50–56 in time series regressions sampling distributions of, 355–356 unbiasedness of, 349–352 variances of, 352–354 OLS in matrix form asymptotic analysis, 815–818 finite sample properties, 809–813 overview, 807–809 statistical inference, 813–815 Wald statistics for testing multiple hypotheses, 818 OLS intercept estimates, defined, 74 OLS regression line See also OLS (ordinary least squares) defined, 31 in multiple regressions, 74 OLS slope estimates, defined, 74 omitted variable bias See also instrumental variables general discussions, 88–92, 115–116 using proxy variables, 308–314 one-sided alternatives, 780–782 one-step-ahead forecasts, 655–659 one-tailed tests, 124, 781 See also t tests online databases, 680 online search services, 678–679 order condition, 531, 563 ordinal variables, 237–240 ordinary least squares See OLS (ordinary least squares) outliers guarding against, 331–334 main discussion, 326–331 out-of-sample criteria, 658–659 over controlling, 205–206 overall significance of regressions, 152–153 overdispersion, 607 overidentified equations, 568 overidentifying restrictions, testing, 535–538 overspecifying the model, 88 P pairwise uncorrelated random variables, 740–741 panel data applying 2SLS to, 540–542 Copyright 2012 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 874 Index panel data (continued) applying methods to other structures, 499–501 correlated random effects, 497–499 differencing with more than two periods, 468–473 fixed effects, 484–492 independently pooled cross sections vs., 448–449 organizing, 465 overview, 10–11 pitfalls in first differencing, 473–474 random effects, 492–496 simultaneous equations models with, 572–574 two-period, analysis, 459–465 two-period, policy analysis with, 465–468 unbalanced, 491–492 Panel Study of Income Dynamics, 680 parameters defined, 5, 755 estimation, general approach to, 768–770 partial derivatives, 717–718 partial effect, 74, 76–77 See also regression analysis partial effect at the average (PEA), 591–592, 600 partialling out, 78 partitioned matrix multiplication, 800 pdf (probability density functions), 724–725 percent correctly predicted, 251, 590 percentages, 707–709 perfect collinearity, 84–86 permanent income hypothesis, 570–571 per-student spending See standardized test scores pesticide usage, over controlling, 206 physical attractiveness and wages, 238–239 pizzas, expected revenue, 732 plug-in solution, 309 point estimates, 755 point forecasts, 655 Poisson regression model, 604–609 policy analysis with pooled cross sections, 454–459 with qualitative information, 232, 253–256 with two-period panel data, 465–468 pollution See air pollution and housing prices pooled cross sections See also independently pooled cross sections applying 2SLS to, 540–542 overview, 9–10 policy analysis with, 454–459 population, defined, 755 See also confidence intervals; hypothesis testing population model, defined, 83 population regression function (PRF), 25–26 population R-squareds, 202 positive definite and semi-definite matrices, defined, 802 poverty rate in absence of suitable proxies, 315 excluding from model, 91 power of a test, 779–780 practical vs statistical significance, 135–138, 788–789 Prais-Winsten (PW) estimation, 425–426, 428, 433 predetermined variables, 659 predicted variables, 23 See also dependent variables predictions confidence intervals for, 207–211 with heteroskedasticity, 292–294 residual analysis, 211–212 for y when log(y) is dependent, 212–215 predictor variables, 23 See also independent variables price index, 360–361 prisons population and crime rates, 573–574 recidivism, 611–612 probability See also conditional distributions; joint distributions features of distributions, 730–737 independence, 727–729 normal and related distributions, 745–752 overview, 722 random variables and their distributions, 722–727 probability density functions (pdf), 724–725 probability limits, 764–766 probit model See logit and probit models productivity See worker productivity program evaluation, 232, 253–256 projects See empirical analysis property taxes and housing prices, 9–10 proportions, 707–709 proxy variables, 308–314 pseudo R-squareds, 590–591 public finance study researchers, 676 Puerto Rico, employment in detrending, 369–370 logarithmic form, 356–357 time series data, 8–9 p-values computing and using, 784–787 for F tests, 151–152 for t tests, 133–135 Copyright 2012 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 Q quadratic form for matrices, 802, 803 quadratic functions, 194–198, 710–712 quadratic time trends, 366 qualitative information See also linear probability model (LPM) in multiple regressions allowing for different slopes, 241–245 binary dependent variable, 248–253 describing, 227–228 discrete dependent variables, 256–257 interactions among dummy variables, 240–241 with log(y) dependent variable, 233–235 multiple dummy independent variables, 235–240 ordinal variables, 237–240 overview, 227 policy analysis and program evaluation, 253–256 proxy variables, 312–313 single dummy independent variable, 228–235 testing for differences in regression functions across groups, 245–248 in time series regressions main discussion, 357–363 seasonal, 372–373 quantile regression, 334 quasi- (natural) experiments, 457, 521 quasi-demeaned data, 493 quasi-differenced data, 424 quasi-likelihood ratio statistic, 607 quasi-maximum likelihood estimation (QMLE), 606, 815 R R&D and sales confidence intervals, 139–140 nonnested models, 203–204 outliers, 327–328, 329–330 R2j, 95–96 race arrests and, 253 baseball player salaries and, 244–245 discrimination in hiring asymptotic confidence interval, 776–777 hypothesis testing, 784 p-value, 787 loans and See loan approval rates rational distributed lag models, 637–639 Index 875 random coefficient model, 315–317 random effects assumptions for, 509–511 correlated, 497–499 fixed effects vs., 495–496 main discussion, 492–495 random sampling assumption for 2SLS, 552 for multiple linear regressions, 84 for simple linear regressions, 45–46, 47, 49 cross-sectional data and, 6–7 defined, 756 random slope model, 315–317 random variables, 722–727 random vectors, 803 random walks, 391–395 rank condition, 531, 552, 562–563 rank of matrix, 801 RDL (rational distributed lag models), 637–639 real dollars, 360–361 recidivism, duration analysis, 611–612 reduced form equations, 525, 559 regressands, 23 See also dependent variables regression analysis, 57–58 See also multiple regression analysis; simple regression model; time series data regression specification error test (REST), 306–307 regressors, 23, 206–207 See also independent variables rejection region, 780 rejection rule, 124 See also t tests relative change, 708–709 relative efficiency, 762–763 relevant variables, excluding, 88–92 reporting multiple regression results, 154–156 resampling methods, defined, 225 rescaling, 186–189 residual analysis, 211–212 residual sum of squares (SSR) See sum of squared residuals residuals See also OLS (ordinary least squares) in multiple regressions, 77–78, 328–329 in simple regressions, 30, 35–36, 55 studentized, 328–329 response probability, 249, 584 response variables, 23 See also dependent variables REST (regression specification error test), 306–307 restricted model, 145–146 See also F tests retrospective data, Copyright 2012 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 876 Index returns on equity and CEO salaries fitted values and residuals, 35–36 goodness-of-fit, 39 OLS Estimates, 32–33 RMSE (root mean squared error), 56, 100, 659 robust regression, 334 rooms and housing prices beta coefficients, 190–191 interaction effect, 198–199 quadratic functions, 196–198 residual analysis, 211 root mean squared error (RMSE), 56, 100, 659 row vectors, 797 R-squareds See also predictions adjusted, 202–205, 414 after IV estimation, 523 change in unit of measurement and, 41 for F statistic, 150–151 in fixed effects estimation, 487, 488–489 in multiple regressions, main discussion, 80–83 for probit and logit models, 590–591 for PW estimation, 426 in regressions through origin, 57–58, 81–83 in simple regressions, 38–39, 57–58 size of, 200–201 in time series regressions, 414 trending dependent variables and, 370–371 uncentered, 237 S salaries See CEO salaries; income; wages sales CEO salaries and constant elasticity model, 43 nonnested models, 204–205 motivation for multiple regression, 71–72 R&D and See R&D and sales sales tax increase, 709 sample average, 757 sample correlation coefficient, 769 sample covariance, 768 sample regression function (SRF), 31, 74 sample selection corrections, 615–621 sample standard deviation, 765 sample variation in the explanatory variable assumption, 46, 49 sampling, nonrandom, 324–326 sampling distributions defined, 758 of OLS estimators, 118–121 sampling standard deviation, 777 sampling variances of estimators in general, 760–762 of OLS estimators for multiple linear regressions, 94, 116 for simple linear regressions, 53–54 savings housing expenditures and, 557–558 income and heteroskedasticity, 281–282 scatterplot, 27 measurement error in, 318 with nonrandom sample, 325 scalar multiplication, 798 scalar variance-covariance matrices, 811 scatterplots R&D and sales, 328 savings and income, 27 wage and education, 29 school lunch program and math performance, 50 school size and student performance, 127–128 score statistic, 178–181 scrap rates and job training 2SLS, 541–542 confidence interval, 774–775 confidence interval and hypothesis testing, 788 fixed effects estimation, 486–487 measurement error in, 319 program evaluation, 254 p-value, 786–787 statistical vs practical significance, 137 two-period panel data, 466 unbalanced panel data, 492 seasonality forecasting, 662–667 serial correlation and, 422–423 of time series, 371–373 selected samples, 616 self-selection problems, 255–256 SEM See simultaneous equations models semi-elasticity, 44, 715 sensitivity analysis, 685 sequential exogeneity, 401 serial correlation correcting for, 423–429 differencing and, 429–430 dynamic completeness and, 399–401 heteroskedasticity and, 438–439 lagged dependent variables and, 415–416 Copyright 2012 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 no serial correlation assumption, 353–354, 387–388 properties of OLS with, 412–416 testing for, 416–423 serial correlation-robust standard errors, 431–434 short-run elasticity, 357 significance level, 123 See also t tests simple regression model See also OLS (ordinary least squares) defined, 22–26 incorporating nonlinearities in, 41–44 IV estimation, 513–523 multiple regression vs., 68–71 regression on a constant, 58 regression through origin, 57–58 simultaneous equations models bias in OLS, 558–560 identifying and estimating structural equations, 560–566 overview and nature of, 554–558 with panel data, 572–574 systems with more than two equations, 567–568 with time series, 568–572 skewness, 737 sleeping vs working, 463–465 slopes See also OLS estimators; regression analysis change in unit of measurement and, 40–41, 43 defined, 23, 705 qualitative information and, 241–245 random, 315–317 in regressions on a constant, 58 in regressions through origin, 57–58 smearing estimates, 213 smoking birth weight and asymptotic standard error, 178 data scaling, 186–189 IV estimation, 522–523 cigarette taxes and consumption, 459 demand for cigarettes, 288–289 measurement error, 323 Social Sciences Citation Index, 677 soybean yields and fertilizers causality, 13, 14 simple equation, 23–24 spreadsheets, 681 spurious regression, 366–367, 644–646 square matrices, 796–797 SRF (sample regression function), 31, 74 SSE (explained sum of squares), 37–38, 80–81 Index 877 SSR (residual sum of squares) See sum of squared residuals SST (total sum of squares), 37–38, 80–81 SSTj (total sample variation in xj), 94–95 stable AR(1) processes, 383–384 standard deviation of bˆj, 101 defined, 51, 736 estimating, 56 properties of, 736 standard error of the regression (SER), 56, 100 standard errors asymptotic, 177–178 of bˆ 1, 56 of bˆj, 101 heteroskedasticity-robust, 271–273 of OLS estimators, 99–101 serial correlation-robust, 431–434 standard normal distribution, 746–748, 831–832 standardized coefficients, 189–191 standardized random variables, 736 standardized test scores beta coefficients, 189 collinearity, 84–85 interaction effect, 199–200 motivation for multiple regression, 69, 70 omitted variable bias, 91 omitting unobservables, 315 residual analysis, 211–212 static models, 346, 386 static Phillips curve, 346, 355–356, 418, 428 stationary time series processes, 381–382 statistical inference with IV estimator, 517–521 for OLS in matrix form, 813–815 statistical significance defined, 129 economic/practical significance vs., 135–138, 788–789 joint, 148 statistical tables, 831–837 statistics See also hypothesis testing asymptotic properties of estimators, 763–767 finite sample properties of estimators, 756–763 interval estimation and confidence intervals, 770–777 notation, 789–790 overview and definitions, 755–756 parameter estimation, general approaches to, 768–770 Copyright 2012 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 878 Index stepwise regression, 686 stochastic processes, 345, 381 See also time series data stock prices and trucking regulations, 359 stock returns, 438 See also efficient markets hypothesis (EMH) stratified sampling, 325–326 strict exogeneity assumption, 461, 468, 634 strict stationarity, 382 strictly exogenous variables serial correlation correcting for, 423–429 testing for, 416–423 in time series regressions, 351–352 strongly dependent time series See highly persistent time series structural equations definitions, 524, 555, 556, 559 identifying and estimating, 560–566 student enrollment, t test, 131–132 student performance See also college GPA; final exam scores; standardized test scores in math, lunch program and, 50 school expenditures and, 96–97 school size and, 127–128 studentized residuals, 328–329 style hints for empirical papers, 692–694 sum of squared residuals See also OLS (ordinary least squares) minimizing, 66–67 in multiple regressions, 80–81 in simple regressions, 37–38 summation operator, 703–705 supply shock, 390 Survey of Consumer Finances, 679 symmetric matrices, 800 system estimation methods, 568 systematic part, defined, 26 T t distribution critical values table, 833 discussions, 121–122, 749–750, 751, 805 t statistics See also t tests asymptotic, 177 defined, 122, 781 F statistic and, 149–150 heteroskedasticity-robust, 271–273 See also heteroskedasticity t tests See also t statistics for AR(1) serial correlation, 416–418 null hypothesis, 122–123 one-sided alternatives, 123–128 other hypotheses about bj, 130–133 overview, 121–123 p-values for, 133–135 two-sided alternatives, 128–130 tables, statistical, 831–837 tax exemption See under fertility rate T-bill rates cointegration, 646–647, 650 error correction model, 652 inflation, deficits, and See under interest rates random walk characterization of, 393, 394 unit root test, 641 teachers, salary-pension tradeoff, 155–156 teenage motherhood, 499–500 tenure See also wages interpreting equations, 76 motivation for multiple regression, 71–72 test scores, as indicators of ability, 534 See also college GPA; final exam scores; standardized test scores test statistics, 780 text files and editors, 680–681 theorems for 2SLS, 551–553 asymptotic efficiency of OLS, 182 asymptotic normality of OLS for multiple linear regressions, 175–178 for time series regressions, 387–391 consistency of OLS for multiple linear regressions, 169–171 for time series regressions, 384–387 Gauss-Markov for multiple linear regressions, 101–102, 116–117 for time series regressions, 352–354 normal sampling distributions, 120–121 for OLS in matrix form Gauss-Markov, 812 statistical inference, 814–815 unbiasedness, 813 variance-covariance matrix of OLS estimator, 811 sampling variances of OLS estimators for multiple linear regressions, 94, 116 for simple linear regressions, 53–54 for time series regressions, 352–354 t distribution for standardized estimators, 121–122 Copyright 2012 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 unbiased estimation of s2 for multiple linear regressions, 100–101 for simple linear regressions, 56 for time series regressions, 354 unbiasedness of OLS for multiple linear regressions, 87–88, 114–115 for simple linear regressions, 48–49 for time series regressions, 349–352 theoretical framework, 687 three stage least squares, 568 time series data See also forecasting; panel data; pooled cross sections; serial correlation; trends absence of serial correlation, 399–401 applying 2SLS to, 538–540 cointegration, 646–651 dynamically complete models, 399–401 error correction models, 651–652 examples of models, 345–349 functional forms, 356–357 heteroskedasticity in, 434–439 highly persistent See highly persistent time series homoskedasticity assumption for, 402 infinite distributed lag models, 633–639 nature of, 344–345 OLS See under OLS (ordinary least squares); OLS estimators overview, 8–9 in panel data, 10–11 in pooled cross sections, 9–10 with qualitative information See under qualitative information seasonality, 371–373 simultaneous equations models with, 568–572 spurious regression, 644–646 stationary and nonstationary, 381–382 unit roots, testing for, 639–644 weakly dependent, 382–384 time trends See trends time-demeaned data, 485 time-varying error, 460 Tobit model interpreting estimates, 598–603 overview, 596–598 specification issues in, 603–604 top coding, 610 total sample variation in xj, 94–95 total sum of squares (SST), 37–38, 80–81 trace of matrix, 800 traffic fatalities beer taxes and, 205 Index 879 drunk driving laws and, 467–468 training grants See also job training program evaluation, 254 single dummy variable, 233 transpose of matrix, 799–800 treatment group, 232 trends characterizing trending time series, 363–366 detrending, 368–370 forecasting, 662–667 high persistence vs., 394 R-squared and trending dependent variable, 370–371 seasonality and, 373 using trending variables, 366–368 trend-stationary processes, 384 trucking regulations and stock prices, 359 true model, defined, 83 truncated regression models, 609, 613–615 two stage least squares applied to pooled cross sections and panel data, 540–542 applied to time series data, 538–540 assumptions and theorems for, 551–553 with heteroskedasticity, 538 multicollinearity and, 530–531 multiple endogenous explanatory variables, 531 for SEM, 565–566, 568 single endogenous explanatory variable, 528–530 tesing multiple hypotheses after estimation, 532 testing for endogeneity, 534–535 testing overidentifying restrictions, 535–538 two-period panel data analysis, 459–465 policy analysis with, 465–468 two-sided alternatives, 780–783 two-tailed tests, 128, 782 See also t tests two-variable linear regression model See simple regression model Type I/II error, 779 U u (“unobserved” term) See also regression analysis CEV assumption and, 323 foregoing specifying models with, 314–315 general discussions, 4–5, 23–25 in time series regressions, 351 using proxy variables for, 308–314 unanticipated inflation, 390 Copyright 2012 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 880 Index unbalanced panels, 491–492 unbiased estimation of s2 for multiple linear regressions, 100–101 for simple linear regressions, 56 for time series regressions, 354 unbiasedness in general, 758–760 of OLS in matrix form, 810 in multiple regressions, 87–88, 114–115 in simple regressions, 45–50 in time series regressions, 349–352, 412–413 of sˆ 2, 813 uncentered R-squareds, 237 unconditional forecasts, 654 uncorrelated random variables, 739 underspecifying the model, 88–92 unemployment See employment and unemployment unidentified equations, 568 unit roots forecasting processes with, 665–666 process, 393, 395 testing for, 639–644 units of measurement, effects of changing, 40–41, 186–188 universities vs junior colleges, 140–143 unobserved effects/heterogeneity, 460, 485 See also fixed effects “unobserved” terms See u (“unobserved” term) unrestricted model, 145–146 See also F tests unsystematic part, defined, 26 upward bias, 91 utility maximization, 2–3 V VAR model, 657, 666–667 variables See also dependent variables; independent variables; specific types dummy, 227 See also qualitative information in multiple regressions, 69–72 in simple regressions, 22–23 variance inflation factor (VIF), 98 variance of prediction error, 210 variance-covariance matrices, 803–804, 811 variances conditional, 744–745 of OLS estimators in multiple regressions, 93–101 in simple regressions, 50–56 in time series regressions, 352–354 overview and properties of, 734–735, 740–741 vector autoregressive model, 657, 666–667 vectors, defined, 797 veterans, earnings of, 521 voting outcomes campaign expenditures and deriving OLS estimate, 34 perfect collinearity, 85–86 economic performance and, 362–363 W wages See also CEO salaries; income; minimum wages; women in labor force ability and See ability and wage education and See also subheading multiple regressions 2SLS, 542 causality, 13–14 conditional expectation, 741–742 heteroskedasticity, 52–53 independent cross sections, 451–453 logarithmic equation, 715 nonlinear relationship, 41–43 OLS estimates, 33–34 partial effect, 718 return to education, over time, 451–453 rounded averages, 37 scatterplot, 29 simple equation, 24 experience and See under experience gender gap independent cross sections, 451–453 panel data, 451–453 with heteroskedasticity-robust standard errors, 272 labor supply and demand, 555–556 labor supply function, 715 multiple regressions See also subheading with qualitative information beta coefficients, 189 homoskedasticity, 93 hypotheses with more than one parameter, 140–143 interpreting equations, 76 misspecified functional forms, 304 motivation for multiple regression, 69, 70 nonrandom sampling, 325, 326 normality assumption and, 120 null hypothesis, 122 Copyright 2012 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 omitted variable bias, 89, 90–91, 92 quadratic functions, 194–196 random slope model, 316 reporting results, 155–156 t test, 125 with unobservables, general approach, 314–315 with unobservables, using proxy, 308–312 nominal vs real, 360 productivity and, 398 quadratic function, 711–712 with qualitative information of baseball players, race and, 244–245 computer usage and, 241 with different slopes, 241–245 education and, 241–244 gender and, 228–231, 234–236, 241–244 with interacting terms, 241 law school rankings and, 239–240 with log(y) dependent variable, 234–235 marital status and, 235–236 with multiple dummy variables, 235–236 with ordinal variables, 238–240 physical attractiveness and, 238–239 random effects model, 494–495 working individuals in 1976, 6–7 Wald test/statistics, 588–589, 598, 818 weak instruments, 523 weakly dependent time series, 382–384 wealth See financial wealth weighted least squares estimation linear probability model, 294–296 overview, 280–281 prediction and prediction intervals, 292–294 for time series regressions, 433, 437 when assumed heteroskedasticity function is wrong, 290–292 when heteroskedasticity function must be estimated, 286–290 when heteroskedasticity is known up to a multiplicative constant, 281–286 White test for heteroskedasticity, 279–280 within estimators, 485 See also fixed effects women in labor force Index 881 binary dependent variable, 249–251 heteroskedasticity, 294–296 LPM, logit, and probit estimates, 593–595 OLS and Tobit estimates, 601–603 return to education 2SLS, 530 IV estimation, 518–519 testing for endogeneity, 535 testing overidentifying restrictions, 537 sample selection correction, 619–620 SEM, 563–566 women’s fertility See fertility rate worker compensation laws and weeks out of work, 458 worker productivity See also scrap rates and job training job training and program evaluation, 254 sample model, in U.S., trend in, 364 wages and, 398 working vs sleeping, 463–465 working women See women in labor force writing empirical papers, 686–694 Y year dummy variables in fixed effects model, 486–488 pooling independent cross sections across time, 449–453 in random effects model, 494–495 Z zero conditional mean assumption homoskedasticity vs., 51 for multiple linear regressions, 70–71, 86–87 for OLS in matrix form, 810 for simple linear regressions, 25–26, 47, 49 for time series regressions, 350–351, 385 zero mean and zero correlation assumption, 171 zero-one variables, 227 See also qualitative information Copyright 2012 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 Copyright 2012 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 ... Measure of Central Tendency: The Median 733 Measures of Variability: Variance and Standard Deviation 734 Variance 734 Standard Deviation 736 Standardizing a Random Variable 736 Skewness and... using random samples An important feature of a modern approach is that the explanatory variables—along with the dependent variable—are treated as outcomes of random variables For the social sciences,... work plays an integral role Updated Data Sets Handbook An extensive data description manual is also available online This manual contains a list of data sources along with suggestions for ways to