Econometric analysis [5th ed] 0130661899, 9780130661890

Citation preview

FIFTH EDITION

ECONOMETRIC ANALYSIS

Q William H. Greene New York University

Upper Saddle River, New Jersey 07458

CIP data to come

Executive Editor: Rod Banister Editor-in-Chief: P. J. Boardman Managing Editor: Gladys Soto Assistant Editor: Marie McHale Editorial Assistant: Lisa Amato Senior Media Project Manager: Victoria Anderson Executive Marketing Manager: Kathleen McLellan Marketing Assistant: Christopher Bath Managing Editor (Production): Cynthia Regan Production Editor: Michael Reynolds Production Assistant: Dianne Falcone Permissions Supervisor: Suzanne Grappi Associate Director, Manufacturing: Vinnie Scelta Cover Designer: Kiwi Design Cover Photo: Anthony Bannister/Corbis Composition: Interactive Composition Corporation Printer/Binder: Courier/Westford Cover Printer: Coral Graphics Credits and acknowledgments borrowed from other sources and reproduced, with permission, in this textbook appear on appropriate page within text (or on page XX). Copyright © 2003, 2000, 1997, 1993 by Pearson Education, Inc., Upper Saddle River, New Jersey, 07458. All rights reserved. Printed in the United States of America. This publication is protected by Copyright and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department. Pearson Education LTD. Pearson Education Australia PTY, Limited Pearson Education Singapore, Pte. Ltd Pearson Education North Asia Ltd Pearson Education, Canada, Ltd Pearson Educación de Mexico, S.A. de C.V. Pearson Education–Japan Pearson Education Malaysia, Pte. Ltd

10 9 8 7 6 5 4 3 2 1 ISBN 0-13-066189-9

For Margaret and Richard Greene

BRIEF CONTENTS

Q Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5

Introduction 1 The Classical Multiple Linear Regression Model 7 Least Squares 19 Finite-Sample Properties of the Least Squares Estimator 41 Large-Sample Properties of the Least Squares and Instrumental Variables Estimators 65

Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10

Inference and Prediction 93 Functional Form and Structural Change 116 Specification Analysis and Model Selection 148 Nonlinear Regression Models 162 Nonspherical Disturbances—The Generalized Regression Model 191

Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Appendix A Appendix B Appendix C Appendix D

Heteroscedasticity 215 Serial Correlation 250 Models for Panel Data 283 Systems of Regression Equations 339 Simultaneous-Equations Models 378 Estimation Frameworks in Econometrics 425 Maximum Likelihood Estimation 468 The Generalized Method of Moments 525 Models with Lagged Variables 558 Time-Series Models 608 Models for Discrete Choice 663 Limited Dependent Variable and Duration Models Matrix Algebra 803 Probability and Distribution Theory 845 Estimation and Inference 877 Large Sample Distribution Theory 896

756

vii

viii

Brief Contents

Appendix E Computation and Optimization

919

Appendix F Data Sets Used in Applications

946

Appendix G Statistical Tables References

959

Author Index

000

Subject Index

000

953

CONTENTS

Q CHAPTER 1 Introduction 1 1.1 Econometrics 1 1.2 Econometric Modeling 1.3 Data and Methodology 1.4 Plan of the Book 5

1 4

CHAPTER 2 The Classical Multiple Linear Regression Model 2.1 Introduction 7 2.2 The Linear Regression Model 7 2.3 Assumptions of the Classical Linear Regression Model

2.4

7

10

2.3.1 Linearity of the Regression Model 11 2.3.2 Full Rank 13 2.3.3 Regression 14 2.3.4 Spherical Disturbances 15 2.3.5 Data Generating Process for the Regressors 16 2.3.6 Normality 17 Summary and Conclusions 18

CHAPTER 3 Least Squares 19 3.1 Introduction 19 3.2 Least Squares Regression

19

3.3 3.4 3.5

3.2.1 The Least Squares Coefficient Vector 20 3.2.2 Application: An Investment Equation 21 3.2.3 Algebraic Aspects of The Least Squares Solution 24 3.2.4 Projection 24 Partitioned Regression and Partial Regression 26 Partial Regression and Partial Correlation Coefficients 28 Goodness of Fit and the Analysis of Variance 31

3.6

3.5.1 The Adjusted R-Squared and a Measure of Fit 34 3.5.2 R-Squared and the Constant Term in the Model 36 3.5.3 Comparing Models 37 Summary and Conclusions 38

ix

x

Contents

CHAPTER 4 Finite-Sample Properties of the Least Squares Estimator 4.1 Introduction 41 4.2

4.3 4.4 4.5 4.6 4.7

4.8 4.9

41

Motivating Least Squares 42 4.2.1 The Population Orthogonality Conditions 42 4.2.2 Minimum Mean Squared Error Predictor 43 4.2.3 Minimum Variance Linear Unbiased Estimation 44 Unbiased Estimation 44 The Variance of the Least Squares Estimator and the Gauss Markov Theorem 45 The Implications of Stochastic Regressors 47 Estimating the Variance of the Least Squares Estimator 48 The Normality Assumption and Basic Statistical Inference 50 4.7.1 Testing a Hypothesis About a Coefficient 50 4.7.2 Confidence Intervals for Parameters 52 4.7.3 Confidence Interval for a Linear Combination of Coefficients: The Oaxaca Decomposition 53 4.7.4 Testing the Significance of the Regression 54 4.7.5 Marginal Distributions of the Test Statistics 55 Finite-Sample Properties of Least Squares 55 Data Problems 56

4.9.1 Multicollinearity 56 4.9.2 Missing Observations 59 4.9.3 Regression Diagnostics and Influential Data Points 4.10 Summary and Conclusions 61

60

CHAPTER 5 5.1 5.2

Large-Sample Properties of the Least Squares and Instrumental Variables Estimators 65 Introduction 65 Asymptotic Properties of the Least Squares Estimator 65 5.2.1 5.2.2 5.2.3 5.2.4

5.3

5.4 5.5

Consistency of the Least Squares Estimator of β 66 Asymptotic Normality of the Least Squares Estimator 67 Consistency of s 2 and the Estimator of Asy. Var[b] 69 Asymptotic Distribution of a Function of b: The Delta Method 70 5.2.5 Asymptotic Efficiency 70 More General Cases 72 5.3.1 Heterogeneity in the Distributions of xi 72 5.3.2 Dependent Observations 73 Instrumental Variable and Two Stage Least Squares Estimation 74 Hausman’s Specification Test and an Application to Instrumental Variable Estimation 80

Contents

5.6

Measurement Error

83

5.7

5.6.1 Least Squares Attenuation 84 5.6.2 Instrumental Variables Estimation 86 5.6.3 Proxy Variables 87 5.6.4 Application: Income and Education and a Study of Twins Summary and Conclusions 90

CHAPTER 6 Inference and Prediction 93 6.1 Introduction 93 6.2 Restrictions and Nested Models 93 6.3 Two Approaches to Testing Hypotheses

6.4 6.5 6.6 6.7

95

6.3.1 The F Statistic and the Least Squares Discrepancy 95 6.3.2 The Restricted Least Squares Estimator 99 6.3.3 The Loss of Fit from Restricted Least Squares 101 Nonnormal Disturbances and Large Sample Tests 104 Testing Nonlinear Restrictions 108 Prediction 111 Summary and Conclusions 114

CHAPTER 7 Functional Form and Structural Change 7.1 Introduction 116 7.2 Using Binary Variables 116

7.3

7.4

7.5

7.6

116

7.2.1 Binary Variables in Regression 116 7.2.2 Several Categories 117 7.2.3 Several Groupings 118 7.2.4 Threshold Effects and Categorical Variables 120 7.2.5 Spline Regression 121 Nonlinearity in the Variables 122 7.3.1 Functional Forms 122 7.3.2 Identifying Nonlinearity 124 7.3.3 Intrinsic Linearity and Identification 127 Modeling and Testing for a Structural Break 130 7.4.1 Different Parameter Vectors 130 7.4.2 Insufficient Observations 131 7.4.3 Change in a Subset of Coefficients 132 7.4.4 Tests of Structural Break with Unequal Variances 133 Tests of Model Stability 134 7.5.1 Hansen’s Test 134 7.5.2 Recursive Residuals and the CUSUMS Test 135 7.5.3 Predictive Test 137 7.5.4 Unknown Timing of the Structural Break 139 Summary and Conclusions 144

88

xi

xii

Contents

CHAPTER 8 Specification Analysis and Model Selection 8.1 Introduction 148 8.2

148

8.4

Specification Analysis and Model Building 148 8.2.1 Bias Caused by Omission of Relevant Variables 148 8.2.2 Pretest Estimation 149 8.2.3 Inclusion of Irrelevant Variables 150 8.2.4 Model Building—A General to Simple Strategy 151 Choosing Between Nonnested Models 152 8.3.1 Testing Nonnested Hypotheses 153 8.3.2 An Encompassing Model 154 8.3.3 Comprehensive Approach—The J Test 154 8.3.4 The Cox Test 155 Model Selection Criteria 159

8.5

Summary and Conclusions

8.3

160

CHAPTER 9 Nonlinear Regression Models 9.1 Introduction 162 9.2

Nonlinear Regression Models

162

162

9.2.1 9.2.2 9.2.3 9.2.4

9.5

Assumptions of the Nonlinear Regression Model 163 The Orthogonality Condition and the Sum of Squares 164 The Linearized Regression 165 Large Sample Properties of the Nonlinear Least Squares Estimator 167 9.2.5 Computing the Nonlinear Least Squares Estimator 169 Applications 171 9.3.1 A Nonlinear Consumption Function 171 9.3.2 The Box–Cox Transformation 173 Hypothesis Testing and Parametric Restrictions 175 9.4.1 Significance Tests for Restrictions: F and Wald Statistics 175 9.4.2 Tests Based on the LM Statistic 177 9.4.3 A Specification Test for Nonlinear Regressions: The P E Test 178 Alternative Estimators for Nonlinear Regression Models 180

9.6

9.5.1 Nonlinear Instrumental Variables Estimation 181 9.5.2 Two-Step Nonlinear Least Squares Estimation 183 9.5.3 Two-Step Estimation of a Credit Scoring Model 186 Summary and Conclusions 189

9.3

9.4

CHAPTER 10

Nonspherical Disturbances—The Generalized Regression Model 191 10.1 Introduction 191 10.2 Least Squares and Instrumental Variables Estimation 10.2.1 10.2.2 10.2.3

192

Finite-Sample Properties of Ordinary Least Squares 193 Asymptotic Properties of Least Squares 194 Asymptotic Properties of Nonlinear Least Squares 196

Contents

xiii

10.2.4

10.3 10.4 10.5

10.6 10.7

Asymptotic Properties of the Instrumental Variables Estimator 196 Robust Estimation of Asymptotic Covariance Matrices 198 Generalized Method of Moments Estimation 201 Efficient Estimation by Generalized Least Squares 207 10.5.1 Generalized Least Squares (GLS) 207 10.5.2 Feasible Generalized Least Squares 209 Maximum Likelihood Estimation 211 Summary and Conclusions 212

CHAPTER 11 Heteroscedasticity 215 11.1 Introduction 215 11.2 Ordinary Least Squares Estimation 216 11.2.1 Inefficiency of Least Squares 217 11.2.2 The Estimated Covariance Matrix of b 217 11.2.3 Estimating the Appropriate Covariance Matrix for Ordinary Least Squares 219 11.3 GMM Estimation of the Heteroscedastic Regression Model 221 11.4 Testing for Heteroscedasticity 222 11.4.1 White’s General Test 222 11.4.2 The Goldfeld–Quandt Test 223 11.4.3 The Breusch–Pagan/Godfrey LM Test 223 11.5 Weighted Least Squares When  is Known 225 11.6 Estimation When  Contains Unknown Parameters 227 11.6.1 Two-Step Estimation 227 11.6.2 Maximum Likelihood Estimation 228 11.6.3 Model Based Tests for Heteroscedasticity 229 11.7 Applications 232 11.7.1 Multiplicative Heteroscedasticity 232 11.7.2 Groupwise Heteroscedasticity 235 11.8 Autoregressive Conditional Heteroscedasticity 238 11.8.1 The ARCH(1) Model 238 11.8.2 ARCH(q), ARCH-in-Mean and Generalized ARCH Models 240 11.8.3 Maximum Likelihood Estimation of the GARCH Model 242 11.8.4 Testing for GARCH Effects 244 11.8.5 Pseudo-Maximum Likelihood Estimation 245 11.9 Summary and Conclusions 246 CHAPTER 12 Serial Correlation 250 12.1 Introduction 250 12.2 The Analysis of Time-Series Data 12.3 Disturbance Processes 256

253

xiv

Contents

12.4

12.5

12.6 12.7

12.3.1 Characteristics of Disturbance Processes 256 12.3.2 AR(1) Disturbances 257 Some Asymptotic Results for Analyzing Time Series Data 259 12.4.1 Convergence of Moments—The Ergodic Theorem 260 12.4.2 Convergence to Normality—A Central Limit Theorem 262 Least Squares Estimation 265 12.5.1 Asymptotic Properties of Least Squares 265 12.5.2 Estimating the Variance of the Least Squares Estimator 266 GMM Estimation 268 Testing for Autocorrelation 268

12.7.1 Lagrange Multiplier Test 269 12.7.2 Box and Pierce’s Test and Ljung’s Refinement 269 12.7.3 The Durbin–Watson Test 270 12.7.4 Testing in the Presence of a Lagged Dependent Variables 270 12.7.5 Summary of Testing Procedures 271 12.8 Efficient Estimation When  Is Known 271 12.9 Estimation When  Is Unknown 273 12.9.1 AR(1) Disturbances 273 12.9.2 AR(2) Disturbances 274 12.9.3 Application: Estimation of a Model with Autocorrelation 274 12.9.4 Estimation with a Lagged Dependent Variable 277 12.10 Common Factors 278 12.11 Forecasting in the Presence of Autocorrelation 12.12 Summary and Conclusions CHAPTER 13 Models for Panel Data 13.1 Introduction 283

279

280 283

13.2

Panel Data Models

13.3

Fixed Effects 287 13.3.1 Testing the Significance of the Group Effects 289 13.3.2 The Within- and Between-Groups Estimators 289 13.3.3 Fixed Time and Group Effects 291 13.3.4 Unbalanced Panels and Fixed Effects 293 Random Effects 293

13.4

283

13.4.1 13.4.2 13.4.3 13.4.4 13.5 13.6 13.7

Generalized Least Squares 295 Feasible Generalized Least Squares When  Is Unknown 296 Testing for Random Effects 298 Hausman’s Specification Test for the Random Effects Model 301 Instrumental Variables Estimation of the Random Effects Model 303 GMM Estimation of Dynamic Panel Data Models 307 Nonspherical Disturbances and Robust Covariance Estimation 314

13.7.1

Robust Estimation of the Fixed Effects Model 314

Contents

13.8 13.9

xv

13.7.2 Heteroscedasticity in the Random Effects Model 316 13.7.3 Autocorrelation in Panel Data Models 317 Random Coefficients Models 318 Covariance Structures for Pooled Time-Series Cross-Sectional Data 320

13.9.1 Generalized Least Squares Estimation 321 13.9.2 Feasible GLS Estimation 322 13.9.3 Heteroscedasticity and the Classical Model 323 13.9.4 Specification Tests 323 13.9.5 Autocorrelation 324 13.9.6 Maximum Likelihood Estimation 326 13.9.7 Application to Grunfeld’s Investment Data 329 13.9.8 Summary 333 13.10 Summary and Conclusions 334 CHAPTER 14 Systems of Regression Equations 14.1 Introduction 339 14.2

14.3

14.4

14.5

339

The Seemingly Unrelated Regressions Model 340 14.2.1 Generalized Least Squares 341 14.2.2 Seemingly Unrelated Regressions with Identical Regressors 343 14.2.3 Feasible Generalized Least Squares 344 14.2.4 Maximum Likelihood Estimation 347 14.2.5 An Application from Financial Econometrics: The Capital Asset Pricing Model 351 14.2.6 Maximum Likelihood Estimation of the Seemingly Unrelated Regressions Model with a Block of Zeros in the Coefficient Matrix 357 14.2.7 Autocorrelation and Heteroscedasticity 360 Systems of Demand Equations: Singular Systems 362 14.3.1 Cobb–Douglas Cost Function 363 14.3.2 Flexible Functional Forms: The Translog Cost Function 366 Nonlinear Systems and GMM Estimation 369 14.4.1 GLS Estimation 370 14.4.2 Maximum Likelihood Estimation 371 14.4.3 GMM Estimation 372 Summary and Conclusions 374

CHAPTER 15 Simultaneous-Equations Models 378 15.1 Introduction 378 15.2 Fundamental Issues in Simultaneous-Equations Models 15.2.1 15.2.2 15.2.3 15.3

378

Illustrative Systems of Equations 378 Endogeneity and Causality 381 A General Notation for Linear Simultaneous Equations Models 382 The Problem of Identification 385

xvi

Contents

15.3.1 15.3.2 15.3.3 15.4 15.5

The Rank and Order Conditions for Identification 389 Identification Through Other Nonsample Information 394 Identification Through Covariance Restrictions—The Fully Recursive Model 394 Methods of Estimation 396 Single Equation: Limited Information Estimation Methods 396

15.5.1 15.5.2 15.5.3 15.5.4 15.5.5

15.6

Ordinary Least Squares 396 Estimation by Instrumental Variables 397 Two-Stage Least Squares 398 GMM Estimation 400 Limited Information Maximum Likelihood and the k Class of Estimators 401 15.5.6 Two-Stage Least Squares in Models That Are Nonlinear in Variables 403 System Methods of Estimation 404

15.7 15.8 15.9

15.6.1 Three-Stage Least Squares 405 15.6.2 Full-Information Maximum Likelihood 407 15.6.3 GMM Estimation 409 15.6.4 Recursive Systems and Exactly Identified Equations 411 Comparison of Methods—Klein’s Model I 411 Specification Tests 413 Properties of Dynamic Models 415

15.9.1 Dynamic Models and Their Multipliers 415 15.9.2 Stability 417 15.9.3 Adjustment to Equilibrium 418 15.10 Summary and Conclusions 421 CHAPTER 16 Estimation Frameworks in Econometrics 425 16.1 Introduction 425 16.2 Parametric Estimation and Inference 427 16.2.1 Classical Likelihood Based Estimation 428 16.2.2 Bayesian Estimation 429 16.2.2.a Bayesian Analysis of the Classical Regression Model 430 16.2.2.b Point Estimation 434 16.2.2.c Interval Estimation 435 16.2.2.d Estimation with an Informative Prior Density 435 16.2.2.e Hypothesis Testing 437 16.2.3 Using Bayes Theorem in a Classical Estimation Problem: The Latent Class Model 439 16.2.4 Hierarchical Bayes Estimation of a Random Parameters Model by Markov Chain Monte Carlo Simulation 444 16.3 Semiparametric Estimation 447 16.3.1 16.3.2

GMM Estimation in Econometrics 447 Least Absolute Deviations Estimation 448

Contents

16.4

16.5

16.6

xvii

16.3.3 Partially Linear Regression 450 16.3.4 Kernel Density Methods 452 Nonparametric Estimation 453 16.4.1 Kernel Density Estimation 453 16.4.2 Nonparametric Regression 457 Properties of Estimators 460 16.5.1 Statistical Properties of Estimators 460 16.5.2 Extremum Estimators 461 16.5.3 Assumptions for Asymptotic Properties of Extremum Estimators 461 16.5.4 Asymptotic Properties of Estimators 464 16.5.5 Testing Hypotheses 465 Summary and Conclusions 466

CHAPTER 17 Maximum Likelihood Estimation 468 17.1 Introduction 468 17.2 The Likelihood Function and Identification of the Parameters 468 17.3 Efficient Estimation: The Principle of Maximum Likelihood 470 17.4 Properties of Maximum Likelihood Estimators 472 17.4.1 Regularity Conditions 473 17.4.2 Properties of Regular Densities 474 17.4.3 The Likelihood Equation 476 17.4.4 The Information Matrix Equality 476 17.4.5 Asymptotic Properties of the Maximum Likelihood Estimator 476 17.4.5.a Consistency 477 17.4.5.b Asymptotic Normality 478 17.4.5.c Asymptotic Efficiency 479 17.4.5.d Invariance 480 17.4.5.e Conclusion 480 17.4.6 Estimating the Asymptotic Variance of the Maximum Likelihood Estimator 480 17.4.7 Conditional Likelihoods and Econometric Models 482 17.5 Three Asymptotically Equivalent Test Procedures 484 17.5.1 The Likelihood Ratio Test 484 17.5.2 The Wald Test 486 17.5.3 The Lagrange Multiplier Test 489 17.5.4 An Application of the Likelihood Based Test Procedures 490 17.6 Applications of Maximum Likelihood Estimation 492 17.6.1 17.6.2 17.6.3 17.6.4

The Normal Linear Regression Model 492 Maximum Likelihood Estimation of Nonlinear Regression Models 496 Nonnormal Disturbances—The Stochastic Frontier Model 501 Conditional Moment Tests of Specification 505

xviii

Contents

17.7

Two-Step Maximum Likelihood Estimation

508

17.8

Maximum Simulated Likelihood Estimation

512

17.9

Pseudo-Maximum Likelihood Estimation and Robust Asymptotic Covariance Matrices 518

17.10 Summary and Conclusions

521

CHAPTER 18 The Generalized Method of Moments 18.1 Introduction 525 18.2

18.3

18.4

18.5 18.6

Consistent Estimation: The Method of Moments 526 18.2.1 Random Sampling and Estimating the Parameters of Distributions 527 18.2.2 Asymptotic Properties of the Method of Moments Estimator 531 18.2.3 Summary—The Method of Moments 533 The Generalized Method of Moments (GMM) Estimator 533 18.3.1 Estimation Based on Orthogonality Conditions 534 18.3.2 Generalizing the Method of Moments 536 18.3.3 Properties of the GMM Estimator 540 18.3.4 GMM Estimation of Some Specific Econometric Models 544 Testing Hypotheses in the GMM Framework 548 18.4.1 Testing the Validity of the Moment Restrictions 548 18.4.2 GMM Counterparts to the Wald, LM, and LR Tests 549 Application: GMM Estimation of a Dynamic Panel Data Model of Local Government Expenditures 551 Summary and Conclusions 555

CHAPTER 19 Models with Lagged Variables 19.1 Introduction 558 19.2

19.3

19.4

19.5

525

558

Dynamic Regression Models 559 19.2.1 Lagged Effects in a Dynamic Model 560 19.2.2 The Lag and Difference Operators 562 19.2.3 Specification Search for the Lag Length 564 Simple Distributed Lag Models 565 19.3.1 Finite Distributed Lag Models 565 19.3.2 An Infinite Lag Model: The Geometric Lag Model 566 Autoregressive Distributed Lag Models 571 19.4.1 Estimation of the ARDL Model 572 19.4.2 Computation of the Lag Weights in the ARDL Model 573 19.4.3 Stability of a Dynamic Equation 573 19.4.4 Forecasting 576 Methodological Issues in the Analysis of Dynamic Models 579 19.5.1 19.5.2

An Error Correction Model 579 Autocorrelation 581

Contents

19.6

19.7

19.5.3 Specification Analysis 582 19.5.4 Common Factor Restrictions 583 Vector Autoregressions 586 19.6.1 Model Forms 587 19.6.2 Estimation 588 19.6.3 Testing Procedures 589 19.6.4 Exogeneity 590 19.6.5 Testing for Granger Causality 592 19.6.6 Impulse Response Functions 593 19.6.7 Structural VARs 595 19.6.8 Application: Policy Analysis with a VAR 19.6.9 VARs in Microeconomics 602 Summary and Conclusions 605

596

CHAPTER 20 Time-Series Models 608 20.1 Introduction 608 20.2 Stationary Stochastic Processes 609 20.2.1 20.2.2 20.2.3 20.2.4

20.3

20.4

20.5

Autoregressive Moving-Average Processes 609 Stationarity and Invertibility 611 Autocorrelations of a Stationary Stochastic Process 614 Partial Autocorrelations of a Stationary Stochastic Process 617 20.2.5 Modeling Univariate Time Series 619 20.2.6 Estimation of the Parameters of a Univariate Time Series 621 20.2.7 The Frequency Domain 624 20.2.7.a Theoretical Results 625 20.2.7.b Empirical Counterparts 627 Nonstationary Processes and Unit Roots 631 20.3.1 Integrated Processes and Differencing 631 20.3.2 Random Walks, Trends, and Spurious Regressions 632 20.3.3 Tests for Unit Roots in Economic Data 636 20.3.4 The Dickey–Fuller Tests 637 20.3.5 Long Memory Models 647 Cointegration 649 20.4.1 Common Trends 653 20.4.2 Error Correction and VAR Representations 654 20.4.3 Testing for Cointegration 655 20.4.4 Estimating Cointegration Relationships 657 20.4.5 Application: German Money Demand 657 20.4.5.a Cointegration Analysis and a Long Run Theoretical Model 659 20.4.5.b Testing for Model Instability 659 Summary and Conclusions 660

xix

xx

Contents

CHAPTER 21 Models for Discrete Choice 21.1 Introduction 663

663

21.2

Discrete Choice Models

21.3

Models for Binary Choice 665 21.3.1 The Regression Approach 665 21.3.2 Latent Regression—Index Function Models 668 21.3.3 Random Utility Models 670 Estimation and Inference in Binary Choice Models 670

21.4

663

21.4.1 21.4.2 21.4.3 21.4.4

21.5

21.6

21.7

Robust Covariance Matrix Estimation 673 Marginal Effects 674 Hypothesis Tests 676 Specification Tests for Binary Choice Models 679 21.4.4.a Omitted Variables 680 21.4.4.b Heteroscedasticity 680 21.4.4.c A Specification Test for Nonnested Models—Testing for the Distribution 682 21.4.5 Measuring Goodness of Fit 683 21.4.6 Analysis of Proportions Data 686 Extensions of the Binary Choice Model 689 21.5.1 Random and Fixed Effects Models for Panel Data 689 21.5.1.a Random Effects Models 690 21.5.1.b Fixed Effects Models 695 21.5.2 Semiparametric Analysis 700 21.5.3 The Maximum Score Estimator (MSCORE) 702 21.5.4 Semiparametric Estimation 704 21.5.5 A Kernel Estimator for a Nonparametric Regression Function 706 21.5.6 Dynamic Binary Choice Models 708 Bivariate and Multivariate Probit Models 710 21.6.1 Maximum Likelihood Estimation 710 21.6.2 Testing for Zero Correlation 712 21.6.3 Marginal Effects 712 21.6.4 Sample Selection 713 21.6.5 A Multivariate Probit Model 714 21.6.6 Application: Gender Economics Courses in Liberal Arts Colleges 715 Logit Models for Multiple Choices 719 21.7.1 21.7.2 21.7.3 21.7.4 21.7.5 21.7.6 21.7.7

The Multinomial Logit Model 720 The Conditional Logit Model 723 The Independence from Irrelevant Alternatives 724 Nested Logit Models 725 A Heteroscedastic Logit Model 727 Multinomial Models Based on the Normal Distribution A Random Parameters Model 728

727

Contents

21.7.8 21.8 21.9

Application: Conditional Logit Model for Travel Mode Choice 729 Ordered Data 736 Models for Count Data 740

21.9.1 21.9.2 21.9.3

Measuring Goodness of Fit 741 Testing for Overdispersion 743 Heterogeneity and the Negative Binomial Regression Model 744 21.9.4 Application: The Poisson Regression Model 745 21.9.5 Poisson Models for Panel Data 747 21.9.6 Hurdle and Zero-Altered Poisson Models 749 21.10 Summary and Conclusions 752 CHAPTER 22 Limited Dependent Variable and Duration Models 22.1 Introduction 756

756

22.2 Truncation 756 22.2.1 Truncated Distributions 757 22.2.2 Moments of Truncated Distributions 758 22.2.3 The Truncated Regression Model 760 22.3 Censored Data 761 22.3.1 The Censored Normal Distribution 762 22.3.2 The Censored Regression (Tobit) Model 764 22.3.3 Estimation 766 22.3.4 Some Issues in Specification 768 22.3.4.a Heteroscedasticity 768 22.3.4.b Misspecification of Prob[y* < 0] 770 22.3.4.c Nonnormality 771 22.3.4.d Conditional Moment Tests 772 22.3.5 Censoring and Truncation in Models for Counts 773 22.3.6 Application: Censoring in the Tobit and Poisson Regression Models 774 22.4 The Sample Selection Model 780 22.4.1 Incidental Truncation in a Bivariate Distribution 781 22.4.2 Regression in a Model of Selection 782 22.4.3 Estimation 784 22.4.4 Treatment Effects 787 22.4.5 The Normality Assumption 789 22.4.6 Selection in Qualitative Response Models 790 22.5 Models for Duration Data 790 22.5.1 Duration Data 791 22.5.2 A Regression-Like Approach: Parametric Models of Duration 792 22.5.2.a Theoretical Background 792 22.5.2.b Models of the Hazard Function 793 22.5.2.c Maximum Likelihood Estimation 794

xxi

xxii

Contents

22.5.2.d Exogenous Variables 796 22.5.2.e Heterogeneity 797 22.5.3 Other Approaches 798 22.6 Summary and Conclusions 801 APPENDIX A Matrix Algebra 803 A.1 Terminology 803 A.2 Algebraic Manipulation of Matrices

803

A.2.1 Equality of Matrices 803 A.2.2 Transposition 804 A.2.3 Matrix Addition 804 A.2.4 Vector Multiplication 805 A.2.5 A Notation for Rows and Columns of a Matrix 805 A.2.6 Matrix Multiplication and Scalar Multiplication 805 A.2.7 Sums of Values 807 A.2.8 A Useful Idempotent Matrix 808 A.3 Geometry of Matrices 809 A.3.1 Vector Spaces 809 A.3.2 Linear Combinations of Vectors and Basis Vectors 811 A.3.3 Linear Dependence 811 A.3.4 Subspaces 813 A.3.5 Rank of a Matrix 814 A.3.6 Determinant of a Matrix 816 A.3.7 A Least Squares Problem 817 A.4 Solution of a System of Linear Equations 819 A.4.1 Systems of Linear Equations 819 A.4.2 Inverse Matrices 820 A.4.3 Nonhomogeneous Systems of Equations 822 A.4.4 Solving the Least Squares Problem 822 A.5 Partitioned Matrices 822 A.5.1 Addition and Multiplication of Partitioned Matrices 823 A.5.2 Determinants of Partitioned Matrices 823 A.5.3 Inverses of Partitioned Matrices 823 A.5.4 Deviations from Means 824 A.5.5 Kronecker Products 824 A.6 Characteristic Roots and Vectors 825 A.6.1 A.6.2 A.6.3 A.6.4 A.6.5 A.6.6 A.6.7 A.6.8 A.6.9

The Characteristic Equation 825 Characteristic Vectors 826 General Results for Characteristic Roots and Vectors 826 Diagonalization and Spectral Decomposition of a Matrix 827 Rank of a Matrix 827 Condition Number of a Matrix 829 Trace of a Matrix 829 Determinant of a Matrix 830 Powers of a Matrix 830

Contents

xxiii

A.6.10 Idempotent Matrices 832 A.6.11 Factoring a Matrix 832 A.6.12 The Generalized Inverse of a Matrix 833 A.7 Quadratic Forms and Definite Matrices 834 A.7.1 Nonnegative Definite Matrices 835 A.7.2 Idempotent Quadratic Forms 836 A.7.3 Comparing Matrices 836 A.8 Calculus and Matrix Algebra 837 A.8.1 A.8.2 A.8.3 A.8.4

Differentiation and the Taylor Series Optimization 840 Constrained Optimization 842 Transformations 844

APPENDIX B Probability and Distribution Theory B.1 Introduction 845 B.2

837

845

Random Variables 845 B.2.1 Probability Distributions 845 B.2.2 Cumulative Distribution Function 846 B.3 Expectations of a Random Variable 847 B.4 Some Specific Probability Distributions 849 B.4.1 The Normal Distribution 849 B.4.2 The Chi-Squared, t, and F Distributions 851 B.4.3 Distributions With Large Degrees of Freedom 853 B.4.4 Size Distributions: The Lognormal Distribution 854 B.4.5 The Gamma and Exponential Distributions 855 B.4.6 The Beta Distribution 855 B.4.7 The Logistic Distribution 855 B.4.8 Discrete Random Variables 855 B.5 The Distribution of a Function of a Random Variable 856 B.6 Representations of a Probability Distribution 858 B.7 Joint Distributions 860 B.7.1 Marginal Distributions 860 B.7.2 Expectations in a Joint Distribution 861 B.7.3 Covariance and Correlation 861 B.7.4 Distribution of a Function of Bivariate Random Variables 862 B.8 Conditioning in a Bivariate Distribution 864 B.8.1 Regression: The Conditional Mean 864 B.8.2 Conditional Variance 865 B.8.3 Relationships Among Marginal and Conditional Moments 865 B.8.4 The Analysis of Variance 867 B.9 The Bivariate Normal Distribution 867 B.10 Multivariate Distributions 868 B.10.1 Moments 868

xxiv

Contents

B.10.2 Sets of Linear Functions 869 B.10.3 Nonlinear Functions 870 B.11 The Multivariate Normal Distribution 871 B.11.1 Marginal and Conditional Normal Distributions 871 B.11.2 The Classical Normal Linear Regression Model 872 B.11.3 Linear Functions of a Normal Vector 873 B.11.4 Quadratic Forms in a Standard Normal Vector 873 B.11.5 The F Distribution 875 B.11.6 A Full Rank Quadratic Form 875 B.11.7 Independence of a Linear and a Quadratic Form 876 APPENDIX C Estimation and Inference C.1 Introduction 877 C.2 Samples and Random Sampling C.3 Descriptive Statistics

877 878

878

C.4 Statistics as Estimators—Sampling Distributions

882

C.5 Point Estimation of Parameters 885 C.5.1 Estimation in a Finite Sample 885 C.5.2 Efficient Unbiased Estimation 888 C.6 Interval Estimation 890 C.7 Hypothesis Testing 892 C.7.1 C.7.2 C.7.3

Classical Testing Procedures 892 Tests Based on Confidence Intervals Specification Tests 896

APPENDIX D Large Sample Distribution Theory D.1 Introduction 896

895 896

D.2 Large-Sample Distribution Theory 897 D.2.1 Convergence in Probability 897 D.2.2 Other Forms of Convergence and Laws of Large Numbers D.2.3 Convergence of Functions 903 D.2.4 Convergence to a Random Variable 904 D.2.5 Convergence in Distribution: Limiting Distributions 906 D.2.6 Central Limit Theorems 908 D.2.7 The Delta Method 913 D.3 Asymptotic Distributions 914 D.3.1 Asymptotic Distribution of a Nonlinear Function 916 D.3.2 Asymptotic Expectations 917 D.4 Sequences and the Order of a Sequence 918 APPENDIX E Computation and Optimization E.1 Introduction 919

919

E.2 Data Input and Generation 920 E.2.1 Generating Pseudo-Random Numbers

920

900

Contents

xxv

E.2.2 Sampling from a Standard Uniform Population 921 E.2.3 Sampling from Continuous Distributions 921 E.2.4 Sampling from a Multivariate Normal Population 922 E.2.5 Sampling from a Discrete Population 922 E.2.6 The Gibbs Sampler 922 E.3 Monte Carlo Studies 923 E.4 Bootstrapping and the Jackknife

924

E.5 Computation in Econometrics 925 E.5.1 Computing Integrals 926 E.5.2 The Standard Normal Cumulative Distribution Function 926 E.5.3 The Gamma and Related Functions 927 E.5.4 Approximating Integrals by Quadrature 928 E.5.5 Monte Carlo Integration 929 E.5.6 Multivariate Normal Probabilities and Simulated Moments 931 E.5.7 Computing Derivatives 933 E.6 Optimization 933 E.6.1 Algorithms 935 E.6.2 Gradient Methods 935 E.6.3 Aspects of Maximum Likelihood Estimation 939 E.6.4 Optimization with Constraints 941 E.6.5 Some Practical Considerations 942 E.6.6 Examples 943 APPENDIX F

Data Sets Used in Applications

APPENDIX G

Statistical Tables

References

959

Author Index

000

Subject Index

000

953

946

P R E FA C E

Q 1.

THE FIFTH EDITION OF ECONOMETRIC ANALYSIS Econometric Analysis is intended for a one-year graduate course in econometrics for social scientists. The prerequisites for this course should include calculus, mathematical statistics, and an introduction to econometrics at the level of, say, Gujarati’s Basic Econometrics (McGraw-Hill, 1995) or Wooldridge’s Introductory Econometrics: A Modern Approach [South-Western (2000)]. Self-contained (for our purposes) summaries of the matrix algebra, mathematical statistics, and statistical theory used later in the book are given in Appendices A through D. Appendix E contains a description of numerical methods that will be useful to practicing econometricians. The formal presentation of econometrics begins with discussion of a fundamental pillar, the linear multiple regression model, in Chapters 2 through 8. Chapters 9 through 15 present familiar extensions of the single linear equation model, including nonlinear regression, panel data models, the generalized regression model, and systems of equations. The linear model is usually not the sole technique used in most of the contemporary literature. In view of this, the (expanding) second half of this book is devoted to topics that will extend the linear regression model in many directions. Chapters 16 through 18 present the techniques and underlying theory of estimation in econometrics, including GMM and maximum likelihood estimation methods and simulation based techniques. We end in the last four chapters, 19 through 22, with discussions of current topics in applied econometrics, including time-series analysis and the analysis of discrete choice and limited dependent variable models. This book has two objectives. The first is to introduce students to applied econometrics, including basic techniques in regression analysis and some of the rich variety of models that are used when the linear model proves inadequate or inappropriate. The second is to present students with sufficient theoretical background that they will recognize new variants of the models learned about here as merely natural extensions that fit within a common body of principles. Thus, I have spent what might seem to be a large amount of effort explaining the mechanics of GMM estimation, nonlinear least squares, and maximum likelihood estimation and GARCH models. To meet the second objective, this book also contains a fair amount of theoretical material, such as that on maximum likelihood estimation and on asymptotic results for regression models. Modern software has made complicated modeling very easy to do, and an understanding of the underlying theory is important. I had several purposes in undertaking this revision. As in the past, readers continue to send me interesting ideas for my “next edition.” It is impossible to use them all, of xxvii

xxviii

Preface

course. Because the five volumes of the Handbook of Econometrics and two of the Handbook of Applied Econometrics already run to over 4,000 pages, it is also unnecessary. Nonetheless, this revision is appropriate for several reasons. First, there are new and interesting developments in the field, particularly in the areas of microeconometrics (panel data, models for discrete choice) and, of course, in time series, which continues its rapid development. Second, I have taken the opportunity to continue fine-tuning the text as the experience and shared wisdom of my readers accumulates in my files. For this revision, that adjustment has entailed a substantial rearrangement of the material—the main purpose of that was to allow me to add the new material in a more compact and orderly way than I could have with the table of contents in the 4th edition. The literature in econometrics has continued to evolve, and my third objective is to grow with it. This purpose is inherently difficult to accomplish in a textbook. Most of the literature is written by professionals for other professionals, and this textbook is written for students who are in the early stages of their training. But I do hope to provide a bridge to that literature, both theoretical and applied. This book is a broad survey of the field of econometrics. This field grows continually, and such an effort becomes increasingly difficult. (A partial list of journals devoted at least in part, if not completely, to econometrics now includes the Journal of Applied Econometrics, Journal of Econometrics, Econometric Theory, Econometric Reviews, Journal of Business and Economic Statistics, Empirical Economics, and Econometrica.) Still, my view has always been that the serious student of the field must start somewhere, and one can successfully seek that objective in a single textbook. This text attempts to survey, at an entry level, enough of the fields in econometrics that a student can comfortably move from here to practice or more advanced study in one or more specialized areas. At the same time, I have tried to present the material in sufficient generality that the reader is also able to appreciate the important common foundation of all these fields and to use the tools that they all employ. There are now quite a few recently published texts in econometrics. Several have gathered in compact, elegant treatises, the increasingly advanced and advancing theoretical background of econometrics. Others, such as this book, focus more attention on applications of econometrics. One feature that distinguishes this work from its predecessors is its greater emphasis on nonlinear models. [Davidson and MacKinnon (1993) is a noteworthy, but more advanced, exception.] Computer software now in wide use has made estimation of nonlinear models as routine as estimation of linear ones, and the recent literature reflects that progression. My purpose is to provide a textbook treatment that is in line with current practice. The book concludes with four lengthy chapters on time-series analysis, discrete choice models and limited dependent variable models. These nonlinear models are now the staples of the applied econometrics literature. This book also contains a fair amount of material that will extend beyond many first courses in econometrics, including, perhaps, the aforementioned chapters on limited dependent variables, the section in Chapter 22 on duration models, and some of the discussions of time series and panel data models. Once again, I have included these in the hope of providing a bridge to the professional literature in these areas. I have had one overriding purpose that has motivated all five editions of this work. For the vast majority of readers of books such as this, whose ambition is to use, not develop econometrics, I believe that it is simply not sufficient to recite the theory of estimation, hypothesis testing and econometric analysis. Understanding the often subtle

Preface

xxix

background theory is extremely important. But, at the end of the day, my purpose in writing this work, and for my continuing efforts to update it in this now fifth edition, is to show readers how to do econometric analysis. I unabashedly accept the unflattering assessment of a correspondent who once likened this book to a “user’s guide to econometrics.” 2.

SOFTWARE AND DATA There are many computer programs that are widely used for the computations described in this book. All were written by econometricians or statisticians, and in general, all are regularly updated to incorporate new developments in applied econometrics. A sampling of the most widely used packages and Internet home pages where you can find information about them are: E-Views Gauss LIMDEP RATS SAS Shazam Stata TSP

www.eviews.com www.aptech.com www.limdep.com www.estima.com www.sas.com shazam.econ.ubc.ca www.stata.com www.tspintl.com

(QMS, Irvine, Calif.) (Aptech Systems, Kent, Wash.) (Econometric Software, Plainview, N.Y.) (Estima, Evanston, Ill.) (SAS, Cary, N.C.) (Ken White, UBC, Vancouver, B.C.) (Stata, College Station, Tex.) (TSP International, Stanford, Calif.)

Programs vary in size, complexity, cost, the amount of programming required of the user, and so on. Journals such as The American Statistician, The Journal of Applied Econometrics, and The Journal of Economic Surveys regularly publish reviews of individual packages and comparative surveys of packages, usually with reference to particular functionality such as panel data analysis or forecasting. With only a few exceptions, the computations described in this book can be carried out with any of these packages. We hesitate to link this text to any of them in particular. We have placed for general access a customized version of LIMDEP, which was also written by the author, on the website for this text, http://www.stern.nyu.edu/ ∼wgreene/Text/econometricanalysis.htm. LIMDEP programs used for many of the computations are posted on the sites as well. The data sets used in the examples are also on the website. Throughout the text, these data sets are referred to “TableFn.m,” for example Table F4.1. The F refers to Appendix F at the back of the text, which contains descriptions of the data sets. The actual data are posted on the website with the other supplementary materials for the text. (The data sets are also replicated in the system format of most of the commonly used econometrics computer programs, including in addition to LIMDEP, SAS, TSP, SPSS, E-Views, and Stata, so that you can easily import them into whatever program you might be using.) I should also note, there are now thousands of interesting websites containing software, data sets, papers, and commentary on econometrics. It would be hopeless to attempt any kind of a survey here. But, I do note one which is particularly agreeably structured and well targeted for readers of this book, the data archive for the

xxx

Preface

Journal of Applied Econometrics. This journal publishes many papers that are precisely at the right level for readers of this text. They have archived all the nonconfidential data sets used in their publications since 1994. This useful archive can be found at http://qed.econ.queensu.ca/jae/.

3.

ACKNOWLEDGEMENTS It is a pleasure to express my appreciation to those who have influenced this work. I am grateful to Arthur Goldberger and Arnold Zellner for their encouragement, guidance, and always interesting correspondence. Dennis Aigner and Laurits Christensen were also influential in shaping my views on econometrics. Some collaborators to the earlier editions whose contributions remain in this one include Aline Quester, David Hensher, and Donald Waldman. The number of students and colleagues whose suggestions have helped to produce what you find here is far too large to allow me to thank them all individually. I would like to acknowledge the many reviewers of my work whose careful reading has vastly improved the book: Badi Baltagi, University of Houston: Neal Beck, University of California at San Diego; Diane Belleville, Columbia University; Anil Bera, University of Illinois; John Burkett, University of Rhode Island; Leonard Carlson, Emory University; Frank Chaloupka, City University of New York; Chris Cornwell, University of Georgia; Mitali Das, Columbia University; Craig Depken II, University of Texas at Arlington; Edward Dwyer, Clemson University; Michael Ellis, Wesleyan University; Martin Evans, New York University; Ed Greenberg, Washington University at St. Louis; Miguel Herce, University of North Carolina; K. Rao Kadiyala, Purdue University; Tong Li, Indiana University; Lubomir Litov, New York University; William Lott, University of Connecticut; Edward Mathis, Villanova University; Mary McGarvey, University of Nebraska-Lincoln; Ed Melnick, New York University; Thad Mirer, State University of New York at Albany; Paul Ruud, University of California at Berkeley; Sherrie Rhine, Chicago Federal Reserve Board; Terry G. Seaks, University of North Carolina at Greensboro; Donald Snyder, California State University at Los Angeles; Steven Stern, University of Virginia; Houston Stokes, University of Illinois at Chicago; Dimitrios Thomakos, Florida International University; Paul Wachtel, New York University; Mark Watson, Harvard University; and Kenneth West, University of Wisconsin. My numerous discussions with B. D. McCullough have improved Appendix E and at the same time increased my appreciation for numerical analysis. I am especially grateful to Jan Kiviet of the University of Amsterdam, who subjected my third edition to a microscopic examination and provided literally scores of suggestions, virtually all of which appear herein. Chapters 19 and 20 have also benefited from previous reviews by Frank Diebold, B. D. McCullough, Mary McGarvey, and Nagesh Revankar. I would also like to thank Rod Banister, Gladys Soto, Cindy Regan, Mike Reynolds, Marie McHale, Lisa Amato, and Torie Anderson at Prentice Hall for their contributions to the completion of this book. As always, I owe the greatest debt to my wife, Lynne, and to my daughters, Lesley, Allison, Elizabeth, and Julianna. William H. Greene

1

INTRODUCTION

Q 1.1

ECONOMETRICS In the first issue of Econometrica, the Econometric Society stated that its main object shall be to promote studies that aim at a unification of the theoretical-quantitative and the empirical-quantitative approach to economic problems and that are penetrated by constructive and rigorous thinking similar to that which has come to dominate the natural sciences. But there are several aspects of the quantitative approach to economics, and no single one of these aspects taken by itself, should be confounded with econometrics. Thus, econometrics is by no means the same as economic statistics. Nor is it identical with what we call general economic theory, although a considerable portion of this theory has a definitely quantitative character. Nor should econometrics be taken as synonomous [sic] with the application of mathematics to economics. Experience has shown that each of these three viewpoints, that of statistics, economic theory, and mathematics, is a necessary, but not by itself a sufficient, condition for a real understanding of the quantitative relations in modern economic life. It is the unification of all three that is powerful. And it is this unification that constitutes econometrics. Frisch (1933) and his society responded to an unprecedented accumulation of statistical information. They saw a need to establish a body of principles that could organize what would otherwise become a bewildering mass of data. Neither the pillars nor the objectives of econometrics have changed in the years since this editorial appeared. Econometrics is the field of economics that concerns itself with the application of mathematical statistics and the tools of statistical inference to the empirical measurement of relationships postulated by economic theory.

1.2

ECONOMETRIC MODELING Econometric analysis will usually begin with a statement of a theoretical proposition. Consider, for example, a canonical application: Example 1.1

Keynes’s Consumption Function

From Keynes’s (1936) General Theory of Employment, Interest and Money: We shall therefore define what we shall call the propensity to consume as the functional relationship f between X , a given level of income and C, the expenditure on consumption out of the level of income, so that C = f ( X ) . The amount that the community spends on consumption depends (i) partly on the amount of its income, (ii) partly on other objective attendant circumstances, and 1

2

CHAPTER 1 ✦ Introduction

(iii) partly on the subjective needs and the psychological propensities and habits of the individuals composing it. The fundamental psychological law upon which we are entitled to depend with great confidence, both a priori from our knowledge of human nature and from the detailed facts of experience, is that men are disposed, as a rule and on the average, to increase their consumption as their income increases, but not by as much as the increase in their income.1 That is, . . . dC/dX is positive and less than unity. But, apart from short period changes in the level of income, it is also obvious that a higher absolute level of income will tend as a rule to widen the gap between income and consumption. . . . These reasons will lead, as a rule, to a greater proportion of income being saved as real income increases. The theory asserts a relationship between consumption and income, C = f ( X ) , and claims in the third paragraph that the marginal propensity to consume (MPC), dC/dX , is between 0 and 1. The final paragraph asserts that the average propensity to consume (APC), C/ X , falls as income rises, or d( C/ X ) /dX = ( MPC − APC) / X < 0. It follows that MPC < APC. The most common formulation of the consumption function is a linear relationship, C = α + β X , that satisfies Keynes’s “laws” if β lies between zero and one and if α is greater than zero. These theoretical propositions provide the basis for an econometric study. Given an appropriate data set, we could investigate whether the theory appears to be consistent with the observed “facts.” For example, we could see whether the linear specification appears to be a satisfactory description of the relationship between consumption and income, and, if so, whether α is positive and β is between zero and one. Some issues that might be studied are (1) whether this relationship is stable through time or whether the parameters of the relationship change from one generation to the next (a change in the average propensity to save, 1—APC, might represent a fundamental change in the behavior of consumers in the economy); (2) whether there are systematic differences in the relationship across different countries, and, if so, what explains these differences; and (3) whether there are other factors that would improve the ability of the model to explain the relationship between consumption and income. For example, Figure 1.1 presents aggregate consumption and personal income in constant dollars for the U.S. for the 10 years of 1970–1979. (See Appendix Table F1.1.) Apparently, at least superficially, the data (the facts) are consistent with the theory. The relationship appears to be linear, albeit only approximately, the intercept of a line that lies close to most of the points is positive and the slope is less than one, although not by much.

Economic theories such as Keynes’s are typically crisp and unambiguous. Models of demand, production, and aggregate consumption all s