This is a special edition of an established title widely used by colleges and universities throughout the world Pearson published this exclusive edition for the benefit of students outside the United States and Canada If you purchased this book within the United States or Canada, you should be aware that it has been imported without the approval of the Publisher or Author New to This Edition • Chapter 14 on “Big Data” and machine learning methods • Parallel treatment of prediction and causal inference using regression • Brand new General Interest Boxes, like “The Distribution of Adulthood Earnings in the United Kingdom by Childhood Socioeconomic Circumstances” and “Conditional Cash Transfers in Rural Mexico to Increase School Enrollment,” that focus on contemporary and global choice of topics • Concept Exercises in MyLab that focus on core concepts and economic interpretations Introduction to Econometrics FOURTH EDITION FOURTH EDITION • Coverage of realized volatility as well as autoregressive conditional heteroskedasticity Introduction to Econometrics Designed for a first course in undergraduate and introductory econometrics, this best-selling text reflects modern theory and practice With riveting empirical applications as well as real-world examples and data integrated into the development of the theory, the authors ensure that students grasp the relevance of econometrics by providing an effective treatment of the substantive findings of the resulting empirical analysis GLOBAL EDITION G LO B A L EDITION GLOBAL EDITION James H Stock • Mark W Watson Stock Watson Stock_04_1292264454_Final.indd 03/12/18 7:03 AM Question Help MyLab Economics homework and practice questions are correlated to the textbook, and many generate algorithmically to give students unlimited opportunity for mastery of concepts If students get stuck, Learning Aids including Help Me Solve This and eText Pages walk them through the problem and identify helpful information in the text, giving them assistance when they need it most “[MyLab Economics] provides ample practice and explanation of the concepts at hand.” — Heather Burkett, University of Nebraska at Omaha 88% eText 90% Study Plan 92% Dynamic Study Modules % of students who found learning tool helpful Dynamic Study Modules help students study chapter topics effectively on their own by continuously assessing their knowledge application and performance in real time These are available as prebuilt Prepare assignments, and are accessible on smartphones, tablets, and computers Pearson eText enhances student learning—both in and outside the classroom Worked examples, videos, and interactive tutorials bring learning to life, while algorithmic practice and self-assessment opportunities test students’ understanding of the material Accessible anytime, anywhere via MyLab or the app The MyLab Gradebook offers an easy way for students and instructors to view course performance Item Analysis allows instructors to quickly see trends by analyzing details like the number of students who answered correctly/incorrectly, time on task, and median time spend on a question by question basis And because it’s correlated with the AACSB Standards, instructors can track students’ progress toward outcomes that the organization has deemed important in preparing students to be leaders 87% of students would tell their instructor to keep using MyLab Economics For additional details visit: www.pearson.com/mylab/economics A01_STOC4455_04_GE_FM.indd 06/12/18 10:51 AM The Pearson Series in Economics Abel/Bernanke/Croushore Macroeconomics*† Acemoglu/Laibson/List Economics*† Bade/Parkin Foundations of Economics*† Berck/Helfand The Economics of the Environment Bierman/Fernandez Game Theory with Economic Applications Blair/Rush The Economics of Managerial Decisions*† Blanchard Macroeconomics*† Boyer Principles of Transportation Economics Brander/Perloff Managerial Economics and Strategy*† Branson Macroeconomic Theory and Policy Bruce Public Finance and the American Economy Carlton/Perloff Modern Industrial Organization Case/Fair/Oster Principles of Economics*† Chapman Environmental Economics: Theory, Application, and Policy Daniels/VanHoose International Monetary & Financial Economics Downs An Economic Theory of Democracy Farnham Economics for Managers Froyen Macroeconomics: Theories and Policies Fusfeld The Age of the Economist Gerber International Economics*† Gordon Macroeconomics* Greene Econometric Analysis† Gregory/Stuart Russian and Soviet Economic Performance and Structure Hartwick/Olewiler The Economics of Natural Resource Use Heilbroner/Milberg The Making of the Economic Society Heyne/Boettke/Prychitko The Economic Way of Thinking Hubbard/O’Brien Economics*† InEcon Money, Banking, and the Financial System* Hubbard/O’Brien/Rafferty Macroeconomics* Hughes/Cain American Economic History Husted/Melvin International Economics Jehle/Reny Advanced Microeconomic Theory Keat/Young/Erfle Managerial Economics Klein Mathematical Methods for Economics Krugman/Obstfeld/Melitz International Economics: Theory & Policy*† Laidler The Demand for Money Lynn Economic Development: Theory and Practice for a Divided World Miller Economics Today* Miller/Benjamin The Economics of Macro Issues Miller/Benjamin/North The Economics of Public Issues Mishkin The Economics of Money, Banking, and Financial Markets*† The Economics of Money, Banking, and Financial Markets, Business School Edition* Macroeconomics: Policy and Practice* Murray Econometrics: A Modern Introduction O’Sullivan/Sheffrin/Perez Economics: Principles, Applications and Tools*† Parkin Economics*† Perloff Microeconomics*† Microeconomics: Theory and Applications with Calculus*† Pindyck/Rubinfeld Microeconomics*† Riddell/Shackelford/Stamos/Schneider Economics: A Tool for Critically Understanding Society Roberts The Choice: A Fable of Free Trade and Protection Scherer Industry Structure, Strategy, and Public Policy Schiller The Economics of Poverty and Discrimination Sherman Market Regulation Stock/Watson Introduction to Econometrics† Studenmund A Practical Guide to Using Econometrics† Todaro/Smith Economic Development Walters/Walters/Appel/Callahan/Centanni/ Maex/O’Neill Econversations: Today’s Students Discuss Today’s Issues Williamson Macroeconomics† *denotes MyLab Economics titles Visit www.pearson.com/mylab/economics to learn more † denotes Global Edition titles A01_STOC4455_04_GE_FM.indd 18/12/18 11:25 AM Introduction to Econometrics F O U R T H E D I T I O N G L O B A L E D I T I O N James H Stock Harvard University Mark W Watson Princeton University Harlow, England • London • New York • Boston • San Francisco • Toronto • Sydney • Dubai • Singapore • Hong Kong Tokyo • Seoul • Taipei • New Delhi • Cape Town • Sao Paulo • Mexico City • Madrid • Amsterdam • Munich • Paris • Milan A01_STOC4455_04_GE_FM.indd 06/12/18 5:47 PM Vice President, Business, Economics, and UK Courseware: Donna Battista Director of Portfolio Management: Adrienne D’Ambrosio Specialist Portfolio Manager: David Alexander Editorial Assistant: Nicole Nedwidek Project Editor, Global Edition: Paromita Banerjee Project Editor, Global Edition: Punita Kaur Mann Vice President, Product Marketing: Roxanne McCarley Product Marketing Assistant: Marianela Silvestri Manager of Field Marketing, Business Publishing: Adam Goldstein Executive Field Marketing Manager: Carlie Marvel Vice President, Production and Digital Studio, Arts and Business: Etain O’Dea Director, Production and Digital Studio, Business and Economics: Ashley Santora Managing Producer, Business: Alison Kalil Content Producer: Christine Donovan Content Producer, Global Edition: Nikhil Rakshit Operations Specialist: Carol Melville Senior Manufacturing Controller, Global Edition: Kay Holman Manager, Learning Tools: Brian Surette Senior Learning Tools Strategist: Emily Biberger Managing Producer, Digital Studio and GLP: James Bateman Managing Producer, Digital Studio: Diane Lombardo Digital Studio Producer: Melissa Honig Digital Studio Producer: Alana Coles Digital Content Team Lead: Noel Lotz Digital Content Project Lead: Noel Lotz Manager, Media Production, Global Edition: Vikram Kumar Project Manager: Vikash Sharma, Cenveo Publisher Services Interior Design: Cenveo Publisher Services Cover Design: Lumina Datamatics Cover Art: GarryKillian / Shutterstock Acknowledgments of third-party content appear on the appropriate page within the text Pearson Education Limited KAO Two KAO Park Harlow CM17 9NA United Kingdom and Associated Companies throughout the world Visit us on the World Wide Web at: www.pearsonglobaleditions.com © Pearson Education Limited 2020 The rights of James H Stock and Mark W Watson, to be identified as the authors of this work, have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Authorized adaptation from the United States edition, entitled Introduction to Econometrics, 4th Edition, ISBN 978-0-13446199-1 by James H Stock and Mark W Watson, published by Pearson Education © 2020 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the prior written permission of the publisher or a license permitting restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS All trademarks used herein are the property of their respective owners The use of any trademark in this text does not vest in the author or publisher any trademark ownership rights in such trademarks, nor does the use of such trademarks imply any affiliation with or endorsement of this book by such owners For information regarding permissions, request forms, and the appropriate contacts within the Pearson Education Global Rights and Permissions department, please visit www.pearsoned.com/permissions/ This eBook is a standalone product and may or may not include all assets that were part of the print version It also does not provide access to other Pearson digital products like MyLab and Mastering The publisher reserves the right to remove any material in this eBook at any time British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN 10: 1-292-26445-4 ISBN 13: 978-1-292-26445-5 eBook ISBN 13: 978-1-292-26452-3 Typeset in Times NR MT Pro by Cenveo® Publisher Services Brief Contents PART ONE Introduction and Review Chapter Chapter Chapter Economic Questions and Data 43 Review of Probability 55 Review of Statistics 103 PART TWO Fundamentals of Regression Analysis Chapter Chapter Chapter Chapter Chapter Chapter Linear Regression with One Regressor 143 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals 178 Linear Regression with Multiple Regressors 211 Hypothesis Tests and Confidence Intervals in Multiple Regression 247 Nonlinear Regression Functions 277 Assessing Studies Based on Multiple Regression 330 PART THREE Further Topics in Regression Analysis Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Regression with Panel Data 361 Regression with a Binary Dependent Variable 392 Instrumental Variables Regression 427 Experiments and Quasi-Experiments 474 Prediction with Many Regressors and Big Data 514 PART FOUR Regression Analysis of Economic Time Series Data Chapter 15 Chapter 16 Chapter 17 Introduction to Time Series Regression and Forecasting 554 Estimation of Dynamic Causal Effects 609 Additional Topics in Time Series Regression 649 PART FIVE Regression Analysis of Economic Time Series Data Chapter 18 Chapter 19 The Theory of Linear Regression with One Regressor 687 The Theory of Multiple Regression 713 A01_STOC4455_04_GE_FM.indd 06/12/18 10:51 AM This page intentionally left blank A01_MISH4182_11_GE_FM.indd 10/06/15 11:46 am Contents Preface 27 PART ONE Introduction and Review CHAPTER Economic Questions and Data 43 Economic Questions We Examine 43 1.1 Question #1: Does Reducing Class Size Improve Elementary School Education? 43 Question #2: Is There Racial Discrimination in the Market for Home Loans? 44 Question #3: Does Healthcare Spending Improve Health Outcomes? 45 Question #4: By How Much Will U.S GDP Grow Next Year? 46 Quantitative Questions, Quantitative Answers 47 1.2 Causal Effects and Idealized Experiments 47 Estimation of Causal Effects 48 Prediction, Forecasting, and Causality 48 1.3 Data: Sources and Types 49 Experimental versus Observational Data 49 Cross-Sectional Data 50 Time Series Data 51 Panel Data 52 CHAPTER Review of Probability 55 Random Variables and Probability Distributions 56 2.1 Probabilities, the Sample Space, and Random Variables 56 Probability Distribution of a Discrete Random Variable 56 Probability Distribution of a Continuous Random Variable 58 2.2 Expected Values, Mean, and Variance 60 The Expected Value of a Random Variable 60 The Standard Deviation and Variance 61 Mean and Variance of a Linear Function of a Random Variable 62 Other Measures of the Shape of a Distribution 63 Standardized Random Variables 65 2.3 Two Random Variables 65 Joint and Marginal Distributions 65 Conditional Distributions 66 Independence 70 Covariance and Correlation 70 The Mean and Variance of Sums of Random Variables 71 A01_STOC4455_04_GE_FM.indd 20/12/18 4:23 PM 8 Contents 2.4 The Normal, Chi-Squared, Student t, and F Distributions 75 The Normal Distribution 75 The Chi-Squared Distribution 80 The Student t Distribution 80 The F Distribution 80 2.5 Random Sampling and the Distribution of the Sample Average 81 Random Sampling 81 The Sampling Distribution of the Sample Average 82 2.6 Large-Sample Approximations to Sampling Distributions 85 The Law of Large Numbers and Consistency 85 The Central Limit Theorem 86 APPENDIX 2.1 Derivation of Results in Key Concept 2.3 100 APPENDIX 2.2 The Conditional Mean as the Minimum Mean Squared Error Predictor 101 CHAPTER Review of Statistics 103 Estimation of the Population Mean 104 3.1 Estimators and Their Properties 104 Properties of Y 106 The Importance of Random Sampling 108 3.2 Hypothesis Tests Concerning the Population Mean 109 Null and Alternative Hypotheses 109 The p-Value 110 Calculating the p-Value When sY Is Known 111 The Sample Variance, Sample Standard Deviation, and Standard Error 112 Calculating the p-Value When sY Is Unknown 113 The t-Statistic 113 Hypothesis Testing with a Prespecified Significance Level 114 One-Sided Alternatives 116 3.3 Confidence Intervals for the Population Mean 117 3.4 Comparing Means from Different Populations 119 Hypothesis Tests for the Difference Between Two Means 119 Confidence Intervals for the Difference Between Two Population Means 120 3.5 Differences-of-Means Estimation of Causal Effects Using Experimental Data 121 The Causal Effect as a Difference of Conditional Expectations 121 Estimation of the Causal Effect Using Differences of Means 121 3.6 Using the t-Statistic When the Sample Size Is Small 123 The t-Statistic and the Student t Distribution 125 Use of the Student t Distribution in Practice 126 A01_STOC4455_04_GE_FM.indd 20/12/18 4:24 PM Contents 3.7 Scatterplots, the Sample Covariance, and the Sample Correlation 127 Scatterplots 127 Sample Covariance and Correlation 127 APPENDIX 3.1 The U.S Current Population Survey 141 Proofs That Y Is the Least Squares Estimator of μY 141 APPENDIX 3.3 A Proof That the Sample Variance Is Consistent 142 APPENDIX 3.2 Two PART TWO Fundamentals of Regression Analysis CHAPTER Linear Regression with One Regressor 143 4.1 The Linear Regression Model 144 4.2 Estimating the Coefficients of the Linear Regression Model 147 The Ordinary Least Squares Estimator 148 OLS Estimates of the Relationship Between Test Scores and the Student–Teacher Ratio 149 Why Use the OLS Estimator? 151 4.3 Measures of Fit and Prediction Accuracy 153 The R2 153 The Standard Error of the Regression 154 Prediction Using OLS 155 Application to the Test Score Data 155 4.4 The Least Squares Assumptions for Causal Inference 156 Assumption 1: The Conditional Distribution of ui Given Xi Has a Mean of Zero 157 Assumption 2: (Xi, Yi), i = 1, , n, Are Independently and Identically Distributed 158 Assumption 3: Large Outliers Are Unlikely 159 Use of the Least Squares Assumptions 160 4.5 The Sampling Distribution of the OLS Estimators 161 4.6 Conclusion 164 APPENDIX 4.1 The California Test Score Data Set 172 APPENDIX 4.2 Derivation of the OLS Estimators 172 APPENDIX 4.3 Sampling Distribution of the OLS Estimator 173 APPENDIX 4.4 The Least Squares Assumptions for Prediction 176 CHAPTER Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals 178 Testing Hypotheses About One of the Regression Coefficients 178 5.1 Two-Sided Hypotheses Concerning ß1 179 One-Sided Hypotheses Concerning ß1 182 Testing Hypotheses About the Intercept ß0 184 A01_STOC4455_04_GE_FM.indd 5.2 Confidence Intervals for a Regression Coefficient 184 06/12/18 10:52 AM www.freebookslides.com 786 Index Bivariate normal distribution, 77, 79 Bivariate normal p.d.f., 710 BLUE (Best Linear Unbiased Estimator), 106–107, 194–196 Bollerslev, Tim, 670 Bonferroni’s inequality, 275 Bonferroni test of joint hypothesis, 253, 274–276 Boston mortgage data, overiew of, 421 Bournoulli, Jacob, 58 Break date, 589–590 Breaks, nonstationarity, 589–596, 591t, 593f, 595f avoiding problems caused by breaks, 595–596 pseudo out-of-sample forecasts and, 594–595, 595f testing for, known date, 590 testing for, unknown date, 590–593, 591t, 593f Business cycles, defined, 672 Central limit theorem, 86–90, 87f, 88f, 90f convergence in distribution, 692–693 distribution of averages, 162 multivariate central limit theorem, 718 Chebychev’s inequality, 691–692, 711–712 Chi-squared distribution, 80, 711, 724 critical values for, A4t Chow, Gregory, 590 Chow test, 590–593, 591t, 593f Cigarette taxes cigarette consumption data set, 466 demand elasticity application, 435–437, 443–444, 450–454, 452t externalities of smoking, 451 Classical measurement error model, 337–339 Class size See Student-teacher ratio and test scores Clustered standard errors, 376 Cochrane-Orcutt estimator, 629 Coefficients C cointegrating coefficient, 664, 665–667 California Standardized Testing and confidence intervals for regression Reporting data set, 172, 488–490, coefficients, 184–186 516–523, 517t See also Studentconfidence sets for multiple coefteacher ratio and test scores ficients, 259–260, 260f, 722 Capital asset pricing model (CAPM), joint hypotheses, 252 152 Lasso estimator, 528–532, 529f, 531f Cauchy-Schwarz inequality, 696, 712, linear regression, 145–146 753 linear regression, estimating coeffiCausal effects See also Dynamic causal cients, 147–152, 147t, 148f, 151f effects; Instrumental variables logit (logistical) regression, 402–403, (IV) regression; Time series 402f regression maximum likelihood estimator average causal (treatment) effect, (MLE), 405–406 475–476 nonlinear regression, interpreting defined, 48 coefficients, 285 differences-of-means estimation, nonlinear regression, polynomial 121–123 regression models, 288 heterogeneous populations, estimates ordinary least squares (OLS) in, 498–502 estimator, 148–152, 151f idealized experiments and, 47–48 population regression line, 217–218 IV regression, variations among probit coefficients, estimation of, 401 individuals, 511–512 quadratic regression model, 279f, least squares assumptions, 156–161, 159f 280–281, 281f local average treatment effect regression with binary variables, (LATE), 500–502 186–188 potential outcomes and, 475–477 single coefficients, hypothesis tests, Causal inference 247–251 with control variables, 233–234, single restriction, multiple coefficient 245–246 tests, 258–259 defined, 143–144 Coffee, life expectancy and, 214–215 least squares assumptions, multiple Cointegrating coefficient, 664, 665–667 regression, 225–227 Cointegration, 663–667, 665t Causality, defined, 47 coefficient estimation, 665–667 c.d.f (cumulative distribution function), error correction, 663–664 57, 57ft tests for, 664–665 Censored regression models, 424 Column vector, 748–749 Census Bureau population survey, 141 Common trend, 663–667, 665t Z04_STOC4455_04_GE_IDX.indd 786 Conditional distributions, 66–67, 67t Bayes’ rule, 69 law of iterated expectations, 68–69 Conditional expectation See Conditional mean Conditional mean correlation and, 71, 157–158 defined, 67–68 law of iterated expectations, 68–69 as minimum mean squared error predictor, 70, 101–102 in multiple regression, 231–234 oracle predictor, 518 randomized controlled experiments, 157–158 Conditional mean independence, 233–234 Conditional normal distribution, 710–711 Conditional variance, 69 Confidence intervals confidence sets for multiple coefficients, 259–260, 260f defined, 117 forecast intervals and, 576 linear probability model and, 395 multiple regression, single coefficient, 249–251 for population mean, 117–118 predicted effects of changing X, 185–186 for predicted values, multiple regression, 720 for regression coefficients, 184–186 Student t distribution, 125–127 Confidence interval b1, 184–185 Confidence level, 117–118, 184 Confidence set, 117–118, 722 Consistency, 86, 105–108, 695–696 Consistent estimator, 690–691 Constant regressor, 218–219 Constant term, 218–219 Contemporaneous dynamic multiplier, 619 Continuous mapping theorem, 693–694 Continuous random variables defined, 56 expected value of, 61 probabilities and moments of, 709–710 probability distribution of, 58, 59f Control group defined, 47–48 differences-in-differences estimator, 492–494, 493f Control variable defined, 231–232 guidelines for choosing, 261–262 internal validity and, 334–336 in multiple regression, 231–234, 245–246 TSLS (two stage least squares) estimator and, 471–473 28/12/18 4:55 PM www.freebookslides.com Index 787 Convergence in distribution, 692–693 Convergence in probability, 86, 690–692 Correlation, 71 autocorrelation, 375 conditional mean and, 157–158 sample correlation coefficient, 127–130, 128f, 130f Count data, 425 Covariance, 70–71 sample covariance, 127–130, 128f, 130f Covariance matrix, 752 conditional distributions, 716 Coverage probability, 118 Cross validation, 522–523 Critical value, 115 Cross-sectional data defined, 49–50, 49t repeated cross-sectional data, 494 Cubic regression model, 287–288 Cumulative distribution, 57, 57ft Cumulative distribution function (c.d.f.), 57, 57ft Cumulative dynamic multipliers, 618–619 Cumulative probability distribution, 57, 57ft, 58, 59f Cumulative standard logistic distribution function, 401–403, 402f Currency exchange rates, 560–561, 560f Current Population Survey, U.S., 141 D Data big data, overview of, 515–516 cross-sectional data, 49–50, 49t experimental data, 49 observational data, 49 observation number, 50 panel (longitudinal) data, 51–52, 52t (See also Panel data) repeated cross-sectional data, 494 text data, 543 time series data, 50–51, 51t (See also Time series data) missing data and sample selection, 339–340 Data entry errors, 160 Degree of overidentification, 449 Degrees of freedom, 112 homoskedasticity-only t-statistic, 699 Degrees of freedom correction, 154 Demand See also Price elasticity of demand cigarette taxes, effect of, 435–437, 443–444, 450–454, 452t instrumental variables (IV) regression, 430–432 Density, 58, 59f Z04_STOC4455_04_GE_IDX.indd 787 Density function, 58, 59f See also Probability density function (p.d.f.) Dependent variable, 145–146 See also Binary dependent variables, regression with censored regression, 424 count data, 425 discrete choice data, 426 in nonlinear regression functions, 282–283 ordered response models, 425 sample selection regression models, 424–425 truncated regression, 424–425 Deterministic trends, 582–583 DFM See Dynamic factor model (DFM) Diagonal matrices, 749 Dickey-Fuller statistic, 586 EG-ADF (Engle-Granger Augmented Dickey-Fuller) test, 665, 665t unit root tests, nonnormal distributions, 661–662 Differences estimator, 476–477 Differences-in-differences estimator, 492–494, 493f Differences-of-means, 121–123 Direct forecasts, 656–658 Direct multi-period forecasts, 656–658 Discontinuity, regression designs (sharp and fuzzy), 495–496 Discrete choice data analysis, 414 Discrete random variables defined, 56 probability distribution of, 56–58, 57ft Distributed lag model with AR(1) errors, 625–627 assumptions of, 617–618 autocorrelated ut, standard errors and inference, 618 defined, 614 exogeneity and, 615–616 OLS estimation of ADL model, 627–628 Distributions See also Statistics; specific distribution names asymptotic distribution, 85 Bernoulli distribution, 58 bivariate normal distribution, 77, 79 central limit theorem, 86–90, 87f, 88f, 90f chi-squared distribution, 80 conditional distributions, 66–67, 67t conditional expectation (mean), 67–68, 70 conditional variance, 69 exact distribution, 85 F distribution, 80–81 finite-sample distribution, 85 joint probability distribution, 65–66, 66t, 67t kurtosis, 63f, 64 large-sample approximations, 85–90, 87f, 88f, 90f marginal probability distribution, 66, 66t moments of, 63–65, 63f multivariate normal distribution, 77, 79 normal distributions, 75–79, 75f, 76f of OLS estimators, 162–164, 163f sampling distribution, 83–84 skewness, 63–64, 63fig standard normal distributions, 75–79, 75f, 76f Student t distribution, 80 Dollar/pound exchange rates, 560–561, 560f DOLS (dynamic OLS) estimator, 665–667 Double-blind experiments, 480 Drift, random walk with, 584 Dummy variables, 186–188 See also Binary variables Dummy variable trap, 229–231 Dynamic causal effects See also Causal effects ADL model notation, 647–648 autocorrelated ut, standard errors and inference, 618 cumulative dynamic multipliers, 618–619 distributed lag model, 614 distributed lag model, assumptions, 617–618 distributed lag model with AR(1) errors, 625–627 distribution of OLS estimator with autocorrelated errors, 620–621 estimation with strictly exogenous regressors, 624–629 exogeneity, types of, 615–616 feasible GLS estimator, 629 generalized least squares (GLS) estimator, 628–629 HAC standard error, 621–624 infeasible GLS estimator, 628–629 OLS estimation of ADL model, 627–628 overview of, 609–610, 639 28/12/18 1:11 PM www.freebookslides.com 788 Index Dynamic factor model (DFM), 671–676, Estimation of population mean, 104–108 682 differences-of-means, 121–123 application to U.S macroeconomic Estimators See also Instrumental data, 676–680, 677t, 678t, 679f, variables (IV) regression; 680t specific estimator names Dynamic multipliers, 618–619 asymptotic distribution theory and, Dynamic OLS (DOLS) estimator, 690–692 665–667 BLUE (Best Linear Unbiased Estimator), 106–107 E Cochrane-Orcutt estimator, 629 Earnings, consistent estimator, 690–691 age and, 129–130, 130f defined, 104 education level and, 192, 193, 193f differences estimator, 476–477 gender gap, 119–120, 292–296, 294f, differences-in-differences estimator, 298–306, 301f, 305t 492–494, 493f Socioeconomic class gap, 189–190, 192 DOLS (dynamic OLS) estimator, Econometrics, definitions and uses, 665–667 43 efficient GMM estimator, 739 Economics journals, demand for, feasible GLS estimator, 629 307–309, 307f, 308t fixed effects estimator, 388–390 EG-ADF (Engle-Granger Augmented Frisch-Waugh Theorem, 243–244 Dickey-Fuller) test, 665, 665t, generalized least squares (GLS) 666–667 estimator, 628–629 Education level, earnings distributions HAC (heteroskedasticity-and and, 192, 193, 193f autocorrelation-consistent) Efficiency, 105–108 estimator, 621–624 Efficient GMM estimator, 739 heterogeneous populations, estimates Eicker-Huber-White standard errors, in, 498–502 191 See also Heteroskedasticityhomoskedasticity-only standard robust standard errors error, 191, 243 Eigenvalues, 751 infeasible GLS estimator, Eigenvectors, 751 628–629 Elasticity, 289 instrumental variable estimators, cigarette taxes, effect of, 435–437 494 demand for economics journals, Lasso, 527–532, 529f, 531f 307–309, 307f, 308t least absolute deviations (LAD) instrumental variables (IV) regression, estimator, 196 430–432 least squares estimator, 107, nonlinear regression functions, 141–142 328–329 linear conditionally unbiased Election results, sampling bias and, estimators, 726–727 108 multiple regression, OLS estimator Endogenous variables in, 219–222 defined, 428, 615 Newey-West variance estimator, TSLS in general IV regression model, 623 439–441 nonlinear least squares estimators, weak instruments and, 445 327 Engle, Robert, 669–670, 680–681 ordinary least squares (See Ordinary Engle-Granger Augmented Dickeyleast squares (OLS) estimator) Fuller (EG-ADF) test, 665, 665t, regression discontinuity estimators, 666–667 495–496 Entity and time fixed effects regression ridge regression, 524–527, 525f, model, 371–374 527f Equilibrium effects, 481 sample covariance and correlation, Error correction term, 663 127–130, 128f, 130f Errors-in-variable bias, 336–339 shrinkage estimator, 521–522 Error term, linear regression, 145–146 standard error of the regression See also Standard error of (SER), 154 regression (SER) two stage least squares (TSLS) Estimate, defined, 104 estimator, 429 Z04_STOC4455_04_GE_IDX.indd 788 weighted least squares (WLS) estimator, 195–196, 699–704 Exact distribution, 85 Exactly identified coefficients, defined, 438 Exogeneity defined, 615–616 plausibility of, 637–639 Exogeneity of instrument, 446–449 test of overidentifying restrictions, 448–449 Exogenous variables, 428 general IV regression model, 438–439 included exogenous variables, 437 instrument relevance and, 440–441 Expectation, defined, 60 See also Mean Expected value, 60–61 of Bernoulli random variable, 61 of continuous random variable, 61 Experimental data, 49 See also Data Experiments See also Quasiexperiments attrition of subjects, 479 average causal (treatment) effect, 475–476 comparison of observational and experimental estimates, 488–490 double-blind experiments, 480 Hawthorne effect, 480 heterogeneous populations, estimates in, 498–502 overview of, 474–475, 503 potential outcomes, causal effects and idealized experiments, 475–477 randomized controlled experiment, defined, 47 sample size, validity and, 481 test for random receipt of treatment, 478 treatment protocol, adherence to, 479 validity, threats to, 478–481 Explained sum of squares (ESS), 153–154 Exponential function, 289 See also Logarithms External validity, 331, 332–333 predictions and, 344–345 threats to, 481, 498 F False positive rate, 115 Fama, Eugene, 681 Fan chart, 577, 577f, 578 F distribution, 80–81, 711 critical values for, A47t–A50t Feasible GLS estimator, 629, 648 Feasible WLS estimator, 701 Final prediction error (FPE), 574, 759 Finite kurtosis, 159–160 Finite-sample distribution, 85 28/12/18 4:55 PM www.freebookslides.com Index 789 First differences, 555–558, 556f, 558t First-order autoregression, 565–567 First-stage F-statistic, 446 First-stage regression(s), 440 Fixed effects assumptions, 374–376 asymptotic distribution, fixed effects estimator, 388–390 time fixed effects, 371–374 Florida orange crop, temperature effect on data set, 610–612, 611f, 646 example analysis, price and cold weather, 630–636, 631t, 632f, 634f, 635f Forecast, defined, 48 Forecast error, 562–563 Forecasting See also Prediction fan chart, 577, 577f, 578 final prediction error (FPE), 574 forecast types and forecast errors, 562–563 forecast uncertainty and forecast intervals, 576–578 least squares assumption, multiple predictors, 571–573 mean squared forecast error (MSFE), 563–565 MSFE estimation and forecast intervals, 573–578 multi-period forecasts, 654–658 nowcasting, 676 oracle forecast, 565 overview of, 554–555, 596, 649, 682 pseudo out-of-sample forecasts, 574–576 root mean squared forecast error (RMSFE), 563–565 Forecast interval, defined, 576 FPE (final prediction error), 574, 759 Fraction correctly predicted, 406–407 Frisch-Waugh Theorem, 243–244 F-statistic defined, 253 heteroskedasticity-robust F-statistic, 254–255 homoskedasticity-only F-statistic, 254–255 multiple regression, theory of, 721–722, 725–726 OLS distribution derivation, 754–755 overall regression F-statistic, 255 weak instruments and, 446 Functional form misspecification, 336 Fuzzy regression discontinuity design, 495–496 G GARCH (generalized ARCH), 669–671 Gauss-Markov conditions, 208 Z04_STOC4455_04_GE_IDX.indd 789 Gauss-Markov conditions for multiple regression, 726–727 Gauss-Markov theorem, 191, 194–196, 726–727 proof of, 207–210, 755–756 GDP See Gross Domestic Product (GDP) Gender gap in earning, 119–120, 192 logarithm models for, 292–296, 294f nonlinear regression, variable interactions, 298–306, 301f, 305t General equilibrium effects, 481 Generalized ARCH (GARCH), 669–671 Generalized least squares (GLS) estimator, 628–629 assumptions of, 729–730 conditional mean zero assumption, 730–733 feasible GLS estimator, 629, 730 infeasible GLS estimator, 628–629, 730 multiple regression, theory of, 728–733 Generalized method of moments (GMM), 681 efficiency, proof of, 758 efficient GMM estimator, 739 GMM J-statistic, 740 time series data and, 740–741 Granger, Clive, 663, 680–681 Gross Domestic Product (GDP) autoregression, 566–567, 568 break detection, pseudo out-ofsample forecasts, 594–595, 595f defined, 46, 555 multi-period forecasts, 654–658 nonstationarity, trends, 582–589, 587t vector autoregression (VAR) modeling, 653 Growth rates time series data, 555–558, 556f, 558t H HAC See Heteroskedasticity-and autocorrelation-consistent (HAC) estimator HAC standard error, 621–624 Hansen, Lars Peter, 681 Hawthorne effect, 480 Heckman, James, 414 Heterogeneous populations, estimates in, 498–502 Heteroskedasticity, 188–192, 189f ARCH (autoregressive conditional heteroskedasticity), 669–671 GARCH (generalized ARCH), 669–671 linear probability model, 395 multiple regression model, 219 OLS estimator distribution with autocorrelated errors, 620–621 robust standard error formula, 206 weighted least squares (WLS) estimator, 195–196, 700–704 Heteroskedasticity-and autocorrelationconsistent (HAC) estimator, 621–624 direct multi-period regression, 657–658 HAC standard error, 621–624 Heteroskedasticity-and-autocorrelationrobust (HAR) standard errors, 376 Heteroskedasticity-robust F-statistic, 254–255 validity and, 343–344 Heteroskedasticity-robust J-statistic, 740 Heteroskedasticity-robust standard errors, 191–192 asymptotic distributions and, 695–696 linear probability model, 395 multiple regression, theory of, 719–720 use in linear regression with single regressor, 703–704 Heteroskedasticity-robust t-statistic, 696–697 Heteroskedasticity-robust variance estimators, 720 Homoskedasticity, 188–193, 189f, 193f multiple regression model, 219, 243 Homoskedasticity-only F-statistic, 255–258 Homoskedasticity-only standard error, 191–192 formulas for, 206–207 multiple regression, theory of, 724–725 Homoskedasticity-only t-statistic, 698–699 Homoskedastic normal regression assumptions, 196–197 Household earning, 189–190 Hypothesis tests, 109–117 acceptance region, 115 alternative hypothesis, 109 comparing means from different populations, 119–120 confidence intervals and population mean, 117–118 critical value, 115 false positive rate, 115 linear regression with single regressor, 178–184 multiple regression joint hypotheses tests, 251–258 single coefficient, 247–251 single restriction, multiple coefficient tests, 258–259 28/12/18 1:11 PM www.freebookslides.com 790 Index Hypothesis tests (continued) test of overidentifying restrictions, nonlinear regression, 287–288 448–449 null hypothesis, 109 TSLS (two stage least squares) one-sided alternative hypothesis, estimator, 429, 434–435 116–117 with control variables, 471–473 population mean, tests about, 179 derivation of formula, 466 power of the test, 115 large-sample distributions, 467–469 prespecified significance level, 114–116 weak instruments, 445–446, 469–471 p-value, 109–111, 111f Wright, Philip and Sewell, 430–432, rejection region, 115 447 significance level, 115–116 Instrument exogeneity condition, 429 size of the test, 115 Instrument relevance condition, 429 Student t distribution, 125–127 Instruments two-sided alternative hypothesis, 109 defined, 427 type I and II errors, 115 validity in quasi-experiments, 497–498 Integrated of order d, I(d), 659–662, 661f I Integrated of order one, I(1), 659–662, Idempotent matrix, 751 661f Identically distributed, 82–84 Integrated of order zero, I(0), 659–662, Impact effect, 619 661f Imperfect multicollinearity, 230–231 Interacted regressor, 298–300 Included exogenous variables, 437 Interaction regression model, 298–300 Income, distribution in U.K., 72–73, Interaction term, 298–300 72f, 73t Intercept social class, education, and, 122–123, linear regression, 145–146 122t population regression line, 217–218 Independently and identically Interest rates distributed (i.i.d.), 82–84 cointegration and, 663–667 Independent variable, 145–146 term spread, 46 Indicator variables, 186–188 See also Internal validity, 330–332 Binary variables errors-in-variable bias, 336–339 Infeasible GLS estimator, 628–629 functional form misspecification, 336 Infeasible WLS estimator, 700–701 inconsistency of OLS standard error, In-Sample prediction, 155–156 343–344 Instrumental variable estimators, 494 measurement errors, 336–339 in matrix form, 733–734 missing data and sample selection, Instrumental variables, defined, 427 339–340 Instrumental variables estimation of predictions and, 344–345 treatment effect, 479 simultaneous causality, 341–343 Instrumental variables (IV) regression threats to, overview, 331–334, 478–481 assumptions and sampling threats to, quasi-experiments, 496–498 distribution, 441–442 Iterated multi-period AR forecasts, endogenous and exogenous variables, 654–656 428 Iterated multi-period VAR forecasts, general IV regression model, 437–444 655–656 general IV regression model, IV See Instrumental variables (IV) relevance of, 440–441 regression general IV regression model, validity and, 441 J heterogeneous populations, estimates Joint hypothesis in, 500–502 Bonferroni test of, 274–276 included exogenous and control defined, 252 variables, 438–439 multiple regression, theory of, inference using TSLS estimator, 721–722 442–443 tests of, 251–258 instrument exogeneity, 446–449 Jointly stationary, 562 instrument validity, 454–459 Joint probability distribution, 65–66, IV model and assumptions, 428–429 66t, 67t overview, 427, 459 independent variables, 70 terminology, 437–438 likelihood function, 405–406 Z04_STOC4455_04_GE_IDX.indd 790 J-statistic, 449 asymptotic distribution, proof of, 756–758 GMM J-statistic, 740, 758 heteroskedasticity-robust J-statistic, 740 homoskedasticity and, 737–738 null hypothesis and, 453 K Kurtosis, 63f, 64 L Lagged value, 556 Lag operator, 606 Lag polynomial, 606, 647–648 Lags, 555–558, 556f, 558t See also Autoregressive distributed lag (ADL) model autoregressive-moving average (ARMA) model, 607 distributed lag model, 614 lag length estimation, 578–582, 580t lag length selection, 581–582 vector autoregression lag lengths, 652 Lasso (least absolute shrinkage and selection operator), 527–532, 529f, 531f LATE (Local average treatment effect), 500–502 Law of iterated expectations, 68–69 Law of large numbers, 85–86 asymptotic distribution theory and, 691–692 Least absolute deviations (LAD) estimator, 196 Least squares assumption, 157–161, 159f, 164 for causal inference, 176–177 causal interference with control variables, 233–234, 245–246 first least squares assumption for prediction, 519 forecasting with multiple predictors, 571–573 multiple regression, causal inference, 225–227 multiple regression, predictions with, 244–245 Least squares estimator, 107 See also Ordinary least squares (OLS) estimator causal inference assumption, 156–161, 159f two stage least squares (TSLS) estimator, 429 Leptokurtic, 63f, 64 Likelihood function, 405–406 Limited dependent variable, 393 See also Binary dependent variables, regression with 28/12/18 1:11 PM www.freebookslides.com Index 791 Linear conditionally unbiased estimators, 726–727 Linear deterministic time trends, 587–589, 587t Linear functions random variables, mean and variance, 62 Linear-log model, 290–291, 292f Linear probability model, 393–397, 394f, 403 Linear regression binary variables and, 186–188 causal inference and prediction, 143–144 coefficients, estimating of, 147–152, 147t, 148f, 151f confidence intervals for regression coefficients, 184–186 constant regressor, 218–219 constant term, 218–219 homoskedastic normal regression assumptions, 196–197 least absolute deviations (LAD) estimator, 196 least squares assumptions for causal inference, 156–161, 159f, 164, 176–177 measures of fit, 153–156 model for, 144–147, 146f multiple regression measures of fit, 222–225 model for, 217–219 OLS estimator in, 219–222 omitted variable bias, 211–216, 242 ordinary least squares (OLS) estimator, 148–152, 151f algebraic facts, 175 derivation of, 172–173 sampling distribution of, 161–164, 163f, 173–175 with small sample size, 196–197 terminology of, 145–146 Linear regression, single regressor, 145–146 asymptotic distribution, OLS estimator and t-statistic, 695–697 exact sampling distribution, normal error distributions, 697–699 extended least squares assumptions, 688–689 hypothesis testing, 178–184 overview of, 687 weighted least squares, 699–704 Local average treatment effect (LATE), 500–502 Logarithms, 288–296, 290f, 292f, 294f computing predicted values of Y, 295–296 Z04_STOC4455_04_GE_IDX.indd 791 elasticity of demand, 307–309, 307f, 308t linear-log model, 290–291, 292f log-linear model, 291–292 log-log model, 293–294, 294f natural logarithm, defined, 289 percentages and, 289–290 slopes and elasticities, 328–329 time series data, 555–558, 556f, 558t Logistical regression See Logit regression Logistic curve, 325–326, 326f Logit regression, 397 maximum likelihood estimator (MLE), 405–406, 423 measures of fit, 406–407 multinomial logit models, 426 nonlinear least squares estimation, 404–405 overview, 401–403, 402f Log-linear model, 291–292 Log-log model, 293–294, 294f Longitudinal data, 51–52, 52t Long-run cumulative dynamic multiplier, 619 law of iterated expectations, 68–69 linear functions of random variables, 62 sample average (mean), 82–84 sums of random variables, 71, 74 Mean squared forecast error (MSFE) estimation of, forecast intervals and, 573–578 forecast uncertainty, 576–578 overview of, 563–565 Mean squared prediction error (MSPE), 518 estimation of, m-fold cross validation, 522–523 linear regression estimated by OLS, 758–759 Mean vector, defined, 752 Measurement errors, 336–339 Measures of fit binary dependent variables, regression with, 406–407 fraction correctly predicted, 406–407 in multiple regression, 222–225 pseudo-R2, 406–407 regression R2, 153–154 M m-fold cross validation, 522–523 Machine learning, 516 MLE See Maximum likelihood Many-predictor problem, 516–523, 517t estimator (MLE) Marginal probability distribution, Moments of a distribution, 63–65, 66, 66t 63f Martingale, 583–584 Mortgage lending Massachusetts education data, 346–353, probit regression, 397–401, 398f 346t, 347f, 349t, 351t, 360, racial discrimination, questions 488–490 about, 44–45, 407–413, 408t, 410t, Matrix notation 411t addition and multiplication, 750 Mosteller, Frederick, 543 covariance matrix, 752 MSFE See Mean squared forecast error eigenvalues and eigenvectors, 751 (MSFE) idempotent matrix, 751 MSPE See Mean squared prediction matrix algebra, summary of, 748–751 error (MSPE) matrix definitions and types, 749 Multicollinearity, 226, 228–231, 716 matrix inverse, 750 Multinomial logit model, 426 positive definite and semidefinite, 751 Multinomial probit model, 426 rank, 751 Multi-period forecasts, 654–658 square root, 751 Multiple regression See also Binary trace, 751 dependent variables, regression Maximum likelihood estimator (MLE), with; Multiple regression, 405–406 theory of; Nonlinear regression for logit model, 423 functions for n i.i.d Bernoulli random variables, adjusted R2, 223–225 421–422 confidence sets for multiple for probit model, 422–423 coefficients, 259–260, 260f pseudo-R2, 423 control variables and conditional McFadden, Daniel, 414 mean, 231–234, 245–246 Mean See also Expected value dummy variable trap, 229–230 Bernoulli random variable, 62 Frisch-Waugh Theorem, 243–244 conditional expectation (mean), HAC standard error, 623–624 67–68, 70 interactions between variables, defined, 60 306 28/12/18 1:11 PM www.freebookslides.com 792 Index Multiple regression (continued) joint hypotheses, tests of, 251–258, 274–276 least squares assumption, causal inference and, 225–227 least squares assumption, predictions with, 244–245 model of, 217–219 model specification guidelines, 260–262 OLS estimator, 219–222 OLS estimator, distribution of, 227–228 perfect multicollinearity, 226–227, 228–231 R2 and adjusted R2 interpretation, 262, 263 regression R2, defined, 223 single coefficient, hypothesis tests, 247–251 single restriction, multiple coefficient tests, 258–259 standard error of regression (SER), 222–223 Multiple regression, theory of asymptotic distribution of t-statistic, 720 asymptotic normality of OLS estimator, 718–719 confidence intervals, predicted values, 720 confidence sets for multiple coefficients, 722 extended least squares assumptions, 715–716 Gauss-Markov conditions for multiple regression, 726–727 Gauss-Markov theorem, proof of, 755–756 generalized least squares, 728–733 heteroskedasticity-robust standard errors, 719–720 joint hypothesis tests, 721–722 matrix notation of multiple regression model, 714–715 multivariate central limit theorem, 718 OLS estimator, 716–717 overview, 713–714 regression statistic distributions, normal errors, 722–726 TSLS (two stage least squares) estimator asymptotic distribution, 734–735 homoskedastic errors, 735–738 matrix form, 734 Multiple regression model with control variables, 233–234, 245–246 Multi-step ahead forecasts, 562–563 Multivariate central limit theorem, 718 Z04_STOC4455_04_GE_IDX.indd 792 Multivariate distributions, 752–753 Multivariate normal distribution, 77, 79, 752 N National Statistics Socio-economic Classification (NS-SEC), 72 Natural experiments, 490 See also Quasi-experiments Natural logarithm, 289 See also Logarithms Negative exponential growth, 326, 328f Newey, Whitney, 623 Newey-West variance estimator, 623 Nonlinear least squares, 327 estimation and inference, logit and probit models, 404–405 Nonlinear least squares estimators, 327 Nonlinear regression functions changes in X and Y, 282–283 cubic regression model, 287–288 general functions with nonlinear parameters, 326–327 interactions between variables, 297 continuous and binary variable, 300–303, 301f two binary variables, 298–300 two continuous variables, 305–309 interpreting coefficients in, 285 logarithms, 288–296, 290f, 292f, 294f logistic curve, 325–326, 326f logit (logistical) regression, 397, 401–403, 402f modeling strategies, 279–286, 279f, 281f, 285–286 negative exponential growth, 326 nonlinear least squares estimation, 327 overview, 277–278, 278f polynomial regression model, 286–288 probit regression, 397–401, 398f quadratic regression model, 279f, 280–281, 281f slopes and elasticities, 328–329 standard errors of estimated effects, 284–285 Nonrandom regressors, 158–159 Nonrepresentative samples, 481 Nonsingular matrix, 750 Nonstationarity breaks, 589–596, 591t, 593f, 595f trends, 582–589, 587t unit root tests, nonnormal distributions, 661–662 Nonstationary, defined, 562 Normal distributions, 75–79, 75f, 76f multivariate normal distribution, 77, 79 Normal probability density function (p.d.f.), 710 Normal random variables linear combination and quadratic forms, 752–753 Nowcasting, 676 Null hypothesis, 109 comparing means from different populations, 119–120 false positive rate, 115 hypothesis testing about slope, 180–181 joint null hypotheses, 252–258 J-statistic and, 453 prespecified significance level, 114–116 O Observational data, 49 Observation number, defined, 50 OLS See Ordinary least squares (OLS) estimator OLS regression line, 220–222 OLS residual, 220–222 Omitted variable bias, 211–216, 242, 262, 334–336 One-sided alternative hypothesis, 116–117 One-step ahead forecasts, 562–563 Oracle forecast, 565 Oracle predictor, 518 Orange juice example analysis, price and cold weather, 630–636, 631t, 632f, 634f, 635f Florida orange crop data set, 610–612, 611f, 646 Ordered response regression models, 425 Orders of integration, 658–662, 661f Ordinary least squares (OLS) estimator, 148–152, 151f See also Instrumental variables (IV) regression adjusted R2, 223–225 algebraic facts about, 175 asymptotic distributions and, 695, 753 autocorrelated ut, standard errors and inference, 618 derivation of, 172–173 derivation of, k=1, 551 distributions of test statistics, derivations of, 754–755 DOLS (dynamic OLS) estimator, 665–667 Frisch-Waugh Theorem, 243–244 Gauss-Markov theorem for multiple regression, 726–727 heterogeneous populations, estimates in, 498–502 homoskedasticity, 190–191, 243 hypothesis tests about mean and slope, 181–182 Lasso, 528–532, 529f, 531f 28/12/18 1:11 PM www.freebookslides.com Index 793 linear probability model, 395 many-predictor problem and, 516–523, 517t MSPE for linear regression and, 758–759 multiple regression, 219–222 least squares assumptions, 225–227, 715–716 multicollinearity, 228–231 OLS distribution, 227–228 standard errors, 247–248 theory of, 716–719, 723–724 OLS regression line, 220–222 OLS residual, 220–222 predictions with, 155–156 regression R2, defined, 223 ridge regression, 524–527, 525f, 527f sampling distribution, 161–164, 163f, 173–175 shrinkage estimator and, 521–522 single regressors, extended least squares assumptions, 688–689 standard error of regression, 211–216 stochastic trends, problems caused by, 585–586 theoretical foundation, 194–196, 207–210 time series data, autocorrelated errors, 620–621 validity, inconsistency of OLS standard error, 343–344 in vector autoregression (VAR), 650–651 weighted least squares (WLS) estimator, 699–704 Ordinary least squares (OLS) regression line, 149–152, 151f Outcomes, defined, 56 Outliers kurtosis and, 63f, 64 least squares assumptions and, 159–160 Out-of-sample prediction, 155–156 computation of, 552–553 pseudo out-of-sample forecasts, 574–576 Overidentified coefficients, 438 test of overidentifying restrictions, 448–449 P Panel data before and after comparisons, 365–367, 366f asymptotic distribution, fixed effects estimator, 388–390 balanced panel, 362 defined, 51–52, 52t, 362 fixed effects regression assumptions, 374–376 Z04_STOC4455_04_GE_IDX.indd 793 regression with fixed time effects, 371–374 standard errors for fixed effect regression, 376 unbalanced panel, 362 Parameters, linear regression, 145–146 Partial compliance, 479 Partial effect, 218 Pattern recognition, 516 p.d.f (probability density function), 58, 59f Penalized sum of squared residuals, 524–527, 525f, 527f Percentages, logarithms and, 289–290 Perfect multicollinearity, 226–227, 228–231 Polynomial regression model, 286–288, 296–297, 297f Pooled standard error formula, 125–127, 197 Population mean comparing means from different populations, 119–120 confidence intervals for, 117–118 hypothesis testing, 109–117, 179 Population multiple regression model, 218–219 Population regression line (function), 145–146, 217–218 Populations See also Sampling attrition of subjects, 479 heterogeneous populations, estimates in, 498–502 simple random sampling, 81–82 Positive definite matrix, 751 Positive semidefinite matrix, 751 Potential outcomes causal effects and, 475–477 defined, 475 Power, hypothesis testing, 115 Predicted value, 149, 150, 220–222 Prediction See also Dynamic causal effects; Forecasting defined, 48 first least squares assumption for prediction, 519 internal and external validity, 344–345 Lasso, 527–532, 529f, 531f many-predictor problem and OLS, 516–523, 517t mean squared prediction error (MSPE), 518 oracle predictor, 518 with ordinary least squares (OLS) estimator, 155–156 overview of, 514–515, 542–544 principal components, 532–537, 533f, 536f, 537f ridge regression, 524–527, 525f, 527f shrinkage estimator, 521–522 sparse model, 528 standardized predictive regression model, 519–521 Price, inflation rate and, 660–661, 661f Price elasticity of demand, 45 Principal components, 532–537, 533f, 536f, 537f formulas for, 761–762 scree plot, 534–535, 536f Probability density function (p.d.f.), 58, 59f, 710 Probability distributions See also Statistics asymptotic distribution, 85 Bayes’ rule, 69 Bernoulli distribution, 58 bivariate normal distribution, 77, 79 chi-squared distribution, 80 conditional distributions, 66–67, 67t of continuous random variable, 58, 59f cumulative probability distribution, 57, 57ft defined, 56 of discrete random variable, 56–58, 57ft F distribution, 80–81 finite-sample distribution, 85 independent variables, 70 joint probability distribution, 65–66, 66t, 67t kurtosis, 63f, 64 large-sample approximations, 85–90, 87f, 88f, 90f marginal probability distribution, 66, 66t moments of a distribution, 63–65, 63f multivariate normal distribution, 77, 79 normal distributions, 75–79, 75f, 76f skewness, 63–64, 63fig standard deviation and variance, 61–62 Student t distribution, 80 Probit regression, 397–401, 398f maximum likelihood estimator (MLE), 405–406, 422–423 measures of fit, 406–407 multinomial probit models, 426 nonlinear least squares estimation, 404–405 ordered probit model, 425 Program evaluation, 474 See also Experiments; Quasi-experiments Project STAR, 482–490, 484t, 485t, 487t, 489t, 510 Pseudo out-of-sample forecasts, 574–576 28/12/18 1:11 PM www.freebookslides.com 794 Index Pseudo-R2, 406–407, 423 pth -order autoregressive [AR(p)] model, 567–568 p-value, 109–111, 111f F-statistic and, 254–255 hypothesis testing about population mean, 179 hypothesis testing about slope, 180–181 two-sided tests, 182, 182f conditional expectation (mean), 67–68, 70 conditional variance, 69 covariance and correlation, 70–71 defined, 56 expected value, 60–61 F distribution, 80–81 independent variables, 70 joint probability distribution, 65–66, 66t Q kurtosis, 63f, 64 Quadratic forms, normal random varilaw of iterated expectations, 68–69 ables, 752–753 law of large numbers, 85–86 Quadratic regression model, 279f, marginal probability distribution, 280–281, 281f 66, 66t Quandt likelihood ratio (QLR) statistic, mean and variance, linear functions, 590–593, 591t, 593f 62 Quasi-difference, 626 mean and variance, sums of variables, Quasi-experiments See also 71, 74 Experimental data moments of distribution, 63–65, defined, 490 63f differences-in-differences estimator, multivariate normal distribution, 77, 492–494, 493f 79 heterogeneous populations, estimates normal distributions, 75–79, 75f, 76f in, 498–502 skewness, 63–64, 63f instrumental variable estimators, standard deviation and variance, 494 61–62 overview of, 474–475, 503 Student t distribution, 80 potential outcomes and causal effects, Random walk, 583–584, 659 475–477 Rank of matrix, 751 regression discontinuity estimators, Realized volatility, 668–669, 669f 495–496 Reduced form, 439 repeated cross-sectional data, 494 Regression validity, external threats, 481, 498 autoregression, 565–568 validity, internal threats, 478–481, binary dependent variables and 496–498 linear probability model, 393–397, 394f R logit (logistical) regression, 397 Racial discrimination in mortgage maximum likelihood estimator lending, 44–45, 407–413, 408t, (MLE), 405–406 410t, 411t measures of fit, 406–407 Randomization, validity and, 478, nonlinear least squares estimation, 496–497 404–405 Randomization based on covariates, overview, 392–393, 413–414 477 probit regression, 397–401, 398f Randomized controlled experiment censored regression models, 424 See also Experiments; Quasicount data, 425 experiments cubic regression model, 287–288 causal and treatment effects, 121–123 discrete choice data, 426 conditional mean, 157–158 instrumental variables (See overview of, 47–48 Instrumental variables (IV) time series data and, 613 regression) Random sampling, 81–82 See also linear (See Linear regression; Linear Sampling regression, single regressor) Random variables multiple (See Multiple regression; Bernoulli random variable, 58 Multiple regression, theory of) bivariate normal distribution, 77, 79 nonlinear regression (See Nonlinear chi-squared distribution, 80 regression functions) conditional distributions, 66–67, 67t ordered response models, 425 Z04_STOC4455_04_GE_IDX.indd 794 polynomial regression model, 286–288 quadratic regression model, 279f, 280–281, 281f ridge regression, 524–527, 525f, 527f sample selection models, 424–425 spurious regression, 584–586 standardized predictive regression model, 519–521 Tobit regression, 424 truncated regression models, 424–425 vector autoregression (VAR), 649–653 Regression discontinuity, 495–496 Regression R2, 153–154 defined, 223 interpretation of, 262, 263 Regressor, 145–146 multicollinearity, 228–231 Rejection region, 115 Relevance of instrument general IV regression model, 440–441 instrumental variables (IV) regression, 444–446 Repeated cross-sectional data, 494 Residual, 149, 150 Restricted regression, 256 single restriction, multiple coefficient tests, 258–259 Restrictions, 252 Ridge regression estimator, 524–527, 525f, 527f derivation of, 759–761 precautions about, 530–531 Risk, measures of, 152 River of blood, inflation forecasts, 577, 577f, 578 RMSFE See Root mean squared forecast error (RMSFE) Roll, Richard, 636 Root mean squared forecast error (RMSFE), 563–565 forecast uncertainty, 576–578 Row vector, 748–749 rth moment, 65 S Sample average (mean), 82–84 Sample correlation, 127–130, 128f, 130f Sample correlation coefficient, 127–130, 128f, 130f Sample covariance, 127–130, 128f, 130f Sample regression function, 149–152, 151f Sample regression line, 149–152, 151f Sample selection bias, 340, 414 Sample selection regression models, 424–425 Sample space, 56 28/12/18 1:11 PM www.freebookslides.com Index 795 Sample standard deviation, 111–113 Sample variance, 111–113 consistency, 141–142 Sampling distribution, 83–84 Sargent, Thomas, 681 Scalar, defined, 749 Scatterplots, 127–130, 128f, 130f Schwartz information criterion (SIC), 579 Scree plot, 534–535, 536f, 674–675 Second difference, 659 Second-stage regression(s), 440 Serial correlation, 558–559 Sharp regression discontinuity design, 495–496 Shea, Dennis, 124 Shiller, Robert, 681 Shrinkage estimator, 521–522 Lasso, 528–532, 529f, 531f ridge regression, 524–527, 525f, 527f Significance level, hypothesis testing and, 114–116 Significance probability, 109–111, 111f Sims, Christopher, 652, 681 Simple random sampling, 81–82 Simultaneous causality, 341–343 Simultaneous equations bias, 342–343 Size of test, hypothesis testing, 115 Skewness, 63–64, 63fig Slope hypothesis testing about, 180–182 linear regression, 145–146, 149–151, 151f nonlinear regressions, 277, 278f, 328–329 (See also Nonlinear regression functions) one-sided hypothesis tests, 182–184 ordinary least squares (OLS) estimators, 149–151, 151f population regression line, 217–218 Slutsky’s theorem, 693–694 Socioeconomic class, household earnings by 189–190 Smoking See Cigarette taxes Sparse model, 528–532, 529f, 531f Spurious regression, 584–586 Square matrices, 749 Square root of matrix, 751 Standard deviation See also Statistics defined, 61 sampling distribution, estimators for, 179 Standard error clustered standard errors, 376 direct multi-period regression, 657–658 dynamic causal effects and, 618 fixed effects regression errors, 376 HAC standard error, 621–624 Z04_STOC4455_04_GE_IDX.indd 795 heteroskedasticity-and-autocorrelationrobust (HAR) standard errors, 376 heteroskedasticity-robust standard errors, 191–192, 206 homoskedasticity, 188–193, 189f, 193f, 243 homoskedasticity, error formulas, 206–207 homoskedasticity-only standard error, 191–192 linear probability model, 395 multiple regression, 222–223, 224 nonlinear regression, estimated effects, 284–285 for predicted probabilities, MLE and, 423 TSLS (two stage least squares) estimator, 442–443, 735 validity, inconsistency of OLS standard error, 343–344 Standard error of regression (SER), 154 and mean square forecast error (MSFE), 573–574 Standard error of sample average, 111–113 pooled standard error formula, 125–127 consistency, 141–142 Standardization, 65 Standardized predictive regression model, 519–521 Standardized random variables, 65 Standard normal distributions, 75–79, 75f, 76f values for, A43t–A44t Stationarity, 561–562, 572 in autoregressive model, 605–606 in autoregressive-moving average (ARMA) model, 607 Stochastic trends, 583, 584 cointegration, 663–667, 665t common trend, 663 detection and avoidance of, 586–589, 587t orders of integration and unit root tests, 658–662, 661f problems caused by, 584–586 Stock market beating the market, 563–565 capital asset pricing model (CAPM), 152 diversification and risk, 84 forecasting with macroeconomic data, 676–680, 677t, 678t, 679f, 680t performance of funds and market, 341 probability distributions, market swings, 77–79, 78f realized volatility, 668–669, 669f volatility clustering, 561, 667–668, 667f Wilshire 5000 Total Market Index, 560f, 561, 667–669, 667f, 669f Strict exogeneity, 615–616 Structural VAR modeling, 652, 681 Student t distribution, 80, 125–127, 711 critical values for, A45t small sample size and, 197 Student-teacher ratio and test scores California school testing data, 49–50, 49t, 516–523, 517t Lasso prediction model, 531–532, 531f Massachusetts data, 346–353, 346t, 347f, 349t, 351t, 360 Project STAR, Tennessee, 482–490, 484t, 485t, 487t, 489t, 510 Sum of squared residuals (SSR), 153–154 Sup-Wald statistic, 590–593, 591t, 593f Survivorship bias, 341 Symmetric matrices, 749 T Tarrifs, instrumental variables (IV) regression, 430–432 Taxes See Cigarette taxes t distribution, 80 Tennessee, Project STAR, 482–490, 484t, 485t, 487t, 489t, 510 Term spread, 47 GDP growth forecasts, 568–570, 569f, 638 vector autoregression (VAR) modeling, 653 Test for random receipt of treatment, 478 Test for the difference between two means, 119–120 Test of overidentifying restrictions, 448–449 Test power, hypothesis testing, 115 Test size, hypothesis testing, 115 Test statistic, 113–114 Text data, 516, 543 Thaler, Richard, 124 Time fixed effects regression model, 371–374 Time series data, 159 See also Dynamic causal effects; Time series regression autocorrelation (serial correlation) and autocovariance, 558–559 central limit theorem and, 693 defined, 50–51, 51t generalized method of moments (GMM), 740–741 law of large numbers and, 693 OLS estimator distribution with autocorrelated errors, 620–621 as randomized controlled experiments, 613 28/12/18 1:11 PM www.freebookslides.com 796 Index Time series regression Akaike information criterion (AIC), 579–581, 608 ARCH (autoregressive conditional heteroskedasticity), 669–671 autoregressions, 565–568 autoregressive distributed lag (ADL) model, 570–571 autoregressive-moving average (ARMA) model, 607 Bayes information criterion (BIC), 579, 580t, 581, 607–608 cointegration, 663–667, 665t dynamic factor model (DFM), 671–676 final prediction error (FPE), 574 forecast uncertainty and forecast intervals, 576–578 GARCH (generalized ARCH), 669–671 generalized method of moments (GMM) and, 740–741 lag length estimation, 578–582, 580t lag length selection, 581–582 lag operator notation, 606 lags, first differences, logarithms, and growth rates, 555–558, 556f, 558t least squares assumption, multiple predictors, 571–573 mean squared forecast error (MSFE), 563–565 MSFE estimation and forecast intervals, 573–578 multi-period forecasts, 654–658 nonstationarity, breaks, 589–596, 591t, 593f, 595f nonstationarity, trends, 582–589, 587t nowcasting, 676 orders of integration and unit root tests, 658–662, 661f overview of, 554–555, 596, 649, 682 pseudo out-of-sample forecasts, 574–576 root mean squared forecast error (RMSFE), 563–565 spurious regression, 584–586 stationarity, 561–562, 605–606 stochastic trends, 583, 584 detection and avoidance of, 586–589, 587t problems caused by, 584–586 unit root, 584 vector autoregression (VAR), 649–653 Tobin, James, 424 Tobit regression, 424 Trace of matrix, 751 Traffic deaths and alcohol taxes, 275f, 362–365 Transpose, matrices, 749 Z04_STOC4455_04_GE_IDX.indd 796 t-ratio, 113–114 Treatment effect, 121–123 instrumental variables estimation of, 479 local average treatment effect (LATE), 500–502 Treatment group defined, 47–48 repeated cross-sectional data, 494 Treatment protocol, validity and, 479, 497 Trends, 582–589, 587t cointegration, 663–667, 665t common trend, 663 deterministic trends, 582–583 orders of integration and unit root tests, 658–662, 661f random walk, 583–584 stochastic trends, 583, 584 detection and avoidance of, 586–589, 587t problems caused by, 584–586 Truncated regression models, 424–425 Truncation parameter, HAC, 622–623 TSLS See Two stage least squares (TSLS) estimator t-statistic, 113–114 asymptotic distributions and, 696–697, 720 central limit theorem and, 694 comparing means from different populations, 119–120 confidence intervals and population mean, 118 general form of, 179 homoskedasticity-only t-statistic, 698–699 hypothesis testing about population mean, 179 hypothesis testing about slope, 180–181 multiple regression, theory of, 725 with small sample size, 123, 125–127, 196–197 stochastic trends, problems caused by, 585–586 Student t distribution, 125–127 Two-sided alternative hypothesis, 109 hypothesis testing about slope, 180–181 Two stage least squares (TSLS) estimator, 429 asymptotic distribution of, 734–735 with control variables, 471–473 derivation of formula, 466–467 first- and second-stage regressions, 440 general IV regression model, 439–440 homoskedastic errors, 735–738 inference and, 442–443 instrument exogeneity and, 446–449 IV regression sampling distribution, 441–442 large-sample distribution, 467–469 local average treatment effect (LATE), 500–502 matrix form, 734 standard errors for, 735 weak instruments and, 445–446 Type I error, 115 Type II error, 115 U Unbalanced panel, 362 Unbiased estimators, 104–108 Unconfoundedness, 513 Uncorrelated variables, 71 Underidentified coefficients, 438 Unemployment rates, 560, 560f, 702 Unit root, 584 cointegration, 664–665 orders of integration and nonnormality of tests, 658–662, 661f Unrestricted regression, 256 V Validity external validity, 331, 332–333 general IV regression model, 441 Hawthorne effect, 480 instrumental variables (IV) regression, 444–449, 454–459 internal validity, 330–332 internal validity, threats to, 331–334, 478–481, 496–498 errors-in-variable bias, 336–339 functional form misspecification, 336 inconsistency of OLS standard error, 343–344 measurement errors, 336–339 missing data and sample selection, 339–340 omitted variable bias, 334–336 simultaneous causality, 341–343 predictions and, 344–345 VAR See Vector autoregression (VAR) Variables See also Statistics; specific variable names Bernoulli random variable, 58 binary variables, 186–188 constant regressor, 218–219 constant term, 218–219 continuous random variables, 56 control variable, 231–232 dependent variable, 145–146 discrete random variables, 56 dummy variables, 186–188 endogenous variables, 428 exogenous variables, 428 included exogenous variables, 437 28/12/18 1:11 PM www.freebookslides.com Index 797 independently distributed (independent) variables, 70 independent variable, 145–146 indicator variables, 186–188 standardized random variables, 65 Variance of Bernoulli random variable, 62 conditional variance, 69 defined, 61 of estimators, 104–108 homoskedasticity, 188–193, 189f, 193f, 243 linear functions of random variables, 62 sample average (mean), 82–84 sums of random variables, 71, 74 volatility clustering, 668 Vector autoregression (VAR), 682 causal analysis with, 652 inference in, 650–651 iterated multivariate forecasts, 655–656 lag length determination, 652 Z04_STOC4455_04_GE_IDX.indd 797 model of, 649–650 defined, 445 structural VAR modeling, 652 instrumental variable analysis, Vector error correction model (VECM), 469–471 663 problems with, 445 Vectors See also Matrix notation Weighted least squares (WLS) estimator, definitions and types, 748–749 195–196 eigenvectors, 751 feasible WLS, 701 multivariate distributions, 752–753 infeasible WLS, 700 Volatility linear regression, one regressor, ARCH (autoregressive conditional 699–704 heteroskedasticity), 680–681 West, Kenneth, 623 GARCH (generalized ARCH), stock Wilshire 5000 Total Market Index, 560f, market example, 670–671, 681 561, 667–669, 667f, 669f realized volatility, 668–669, 669f GARCH (generalized ARCH), volatility clustering, 561, 667–668 670–671 WLS See Weighted least squares (WLS) W estimator Wages See also Earnings, distribution Wold decomposition theorem, 607 in U.S Wright, Philip G., 430–432, 447 Wallace, David, 543 Wright, Sewell, 430–431, 447 Weak dependence, 572 Weak instruments Z checking for, 446 Zero-period dynamic multiplier, 619 28/12/18 1:11 PM www.freebookslides.com Large-Sample Critical Values for the t-statistic from the Standard Normal Distribution Significance Level 10% 5% 1% 1.64 1.96 2.58 1.28 1.64 2.33 –1.28 –1.64 –2.33 2-Sided Test ( ) Reject if |t| is greater than 1-Sided Test ( + ) Reject if t is greater than 1-Sided Test ( * ) Reject if t is less than Z04_STOC4455_04_GE_IDX.indd 798 28/12/18 1:11 PM www.freebookslides.com Large-Sample Critical Values for the F-statistic from the Fm, ∞ Distribution Reject if F + Critical Value Z04_STOC4455_04_GE_IDX.indd 799 Significance Level Degrees of Freedom (m) 10% 5% 1% 2.71 3.84 6.63 2.30 3.00 4.61 2.08 2.60 3.78 1.94 2.37 3.32 1.85 2.21 3.02 1.77 2.10 2.80 1.72 2.01 2.64 1.67 1.94 2.51 1.63 1.88 2.41 10 1.60 1.83 2.32 11 1.57 1.79 2.25 12 1.55 1.75 2.18 13 1.52 1.72 2.13 14 1.50 1.69 2.08 15 1.49 1.67 2.04 16 1.47 1.64 2.00 17 1.46 1.62 1.97 18 1.44 1.60 1.93 19 1.43 1.59 1.90 20 1.42 1.57 1.88 21 1.41 1.56 1.85 22 1.40 1.54 1.83 23 1.39 1.53 1.81 24 1.38 1.52 1.79 25 1.38 1.51 1.77 26 1.37 1.50 1.76 27 1.36 1.49 1.74 28 1.35 1.48 1.72 29 1.35 1.47 1.71 30 1.34 1.46 1.70 28/12/18 1:11 PM www.freebookslides.com This page intentionally left blank A01_MISH4182_11_GE_FM.indd 10/06/15 11:46 am ... Poverty and Discrimination Sherman Market Regulation Stock/ Watson Introduction to Econometrics? ?? Studenmund A Practical Guide to Using Econometrics? ?? Todaro/Smith Economic Development Walters/Walters/Appel/Callahan/Centanni/... Introduction to Econometrics F O U R T H E D I T I O N G L O B A L E D I T I O N James H Stock Harvard University Mark W Watson Princeton University Harlow, England • London • New York • Boston... 1988 Authorized adaptation from the United States edition, entitled Introduction to Econometrics, 4th Edition, ISBN 978-0-13446199-1 by James H Stock and Mark W Watson, published by Pearson Education