Eviews near singular matrix error regressors may be perfectly collinear

Chapter 10

Problems 10.10, 10.12, 10.13, and 10.17

Describe a situation in which autocorrelation might be present and which of the three methods of detecting autocorrelation you would leverage. Explain your rationale.
Describe what remedial measure you would take to address autocorrelation if it were found. Provide examples to support your response.

ESSENTIALS OF ECONOMETRICS

FOURTH EDITION

Damodar N. Gujarati Professor Emeritus of Economics, United States Military Academy, West Point

Dawn C. Porter University of Southern California

Boston Burr Ridge, IL Dubuque, IA New York San Francisco St. Louis Bangkok Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto

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ESSENTIALS OF ECONOMETRICS Published by McGraw-Hill/Irwin, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY, 10020. Copyright © 2010, 2006, 1999, 1992 by The McGraw- Hill Companies, Inc. All rights reserved. No part of this publication may be reproduced or distrib- uted in any form or by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.

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Library of Congress Cataloging-in-Publication Data

Gujarati, Damodar N. Essentials of econometrics / Damodar N. Gujarati, Dawn C. Porter.—4th ed.

p. cm. Includes index. ISBN-13: 978-0-07-337584-7 (alk. paper) ISBN-10: 0-07-337584-5 (alk. paper) 1. Econometrics. 2. Economics—Statistical methods. I. Porter, Dawn C. II. Title.

HB139.G85 2010 330.01'5195—dc22

2009010482

www.mhhe.com

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For Joan Gujarati, Diane Gujarati-Chesnut, Charles Chesnut, and my grandchildren,

“Tommy” and Laura Chesnut. DNG

For Judy, Lee, Brett, Bryan, Amy, and Autumn Porter. But especially for my adoring father, Terry.

DCP

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ABOUT THE AUTHORS

DAMODAR N. GUJARATI After teaching for more than 25 years at the City University of New York and 17 years in the Department of Social Sciences, U.S. Military Academy at West Point, New York, Dr. Gujarati is currently Professor Emeritus of economics at the Academy. Dr. Gujarati received his M.Com. degree from the University of Bombay in 1960, his M.B.A. degree from the University of Chicago in 1963, and his Ph.D. degree from the University of Chicago in 1965. Dr. Gujarati has pub- lished extensively in recognized national and international journals, such as the Review of Economics and Statistics, the Economic Journal, the Journal of Financial and Quantitative Analysis, and the Journal of Business. Dr. Gujarati was a member of the board of editors of the Journal of Quantitative Economics, the official journal of the Indian Econometric Society. Dr. Gujarati is also the au- thor of Pensions and the New York City Fiscal Crisis (the American Enterprise Institute, 1978), Government and Business (McGraw-Hill, 1984), and Basic Econometrics (McGraw-Hill, 5th ed., 2009). Dr. Gujarati’s books on economet- rics have been translated into several languages.

Dr. Gujarati was a Visiting Professor at the University of Sheffield, U.K. (1970–1971), a Visiting Fulbright Professor to India (1981–1982), a Visiting Professor in the School of Management of the National University of Singapore 1985–1986), and a Visiting Professor of Econometrics, University of New South Wales, Australia (summer of 1988). Dr. Gujarati has lectured extensively on micro- and macroeconomic topics in countries such as Australia, China, Bangladesh, Germany, India, Israel, Mauritius, and the Republic of South Korea.

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ABOUT THE AUTHORS v

DAWN C. PORTER Dawn Porter has been an assistant professor in the Information and Operations Management Department at the Marshall School of Business of the University of Southern California since the fall of 2006. She currently teaches undergraduate, M.B.A., and graduate elective statistics courses in the business school. Prior to joining the faculty at USC, from 2001–2006, Dawn was an assistant professor at the McDonough School of Business at Georgetown University and also served as a Visiting Professor in the Psychology Department at the Graduate School of Arts and Sciences at NYU. At NYU she taught a number of advanced statistical methods courses and was also an instructor at the Stern School of Business. Her Ph.D. is from the Stern School in Statistics, and her undergraduate degree is in mathematics from Cornell University.

Dawn’s areas of research interest include categorical analysis, agreement measures, multivariate modeling, and applications to the field of psychology. Her current research examines online auction models from a statistical perspective. She has presented her research at the Joint Statistical Meetings, the Decision Sciences Institute meetings, the International Conference on Information Systems, several universities including the London School of Economics and NYU, and various e-commerce and statistics seminar series. Dawn is also a co- author on Essentials of Business Statistics, 2nd edition and Basic Econometrics, 5th edition, both from McGraw-Hill.

Outside academics, Dawn has been employed as a statistical consultant for KPMG, Inc. She also has worked as a statistical consultant for many other major companies, including Ginnie Mae, Inc.; Toys R Us Corporation; IBM; Cosmaire, Inc; and New York University (NYU) Medical Center.

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CONTENTS

PREFACE xix

1 The Nature and Scope of Econometrics 1 1.1 WHAT IS ECONOMETRICS? 1 1.2 WHY STUDY ECONOMETRICS? 2 1.3 THE METHODOLOGY OF ECONOMETRICS 3

Creating a Statement of Theory or Hypothesis 3 Collecting Data 4 Specifying the Mathematical Model of Labor Force Participation 5 Specifying the Statistical, or Econometric, Model of Labor Force

Participation 7 Estimating the Parameters of the Chosen Econometric Model 9 Checking for Model Adequacy: Model Specification Testing 9 Testing the Hypothesis Derived from the Model 11 Using the Model for Prediction or Forecasting 12

1.4 THE ROAD AHEAD 12 KEY TERMS AND CONCEPTS 13 QUESTIONS 14 PROBLEMS 14 APPENDIX 1A: ECONOMIC DATA ON

THE WORLD WIDE WEB 16

PART I THE LINEAR REGRESSION MODEL 19

2 Basic Ideas of Linear Regression:The Two-Variable Model 21 2.1 THE MEANING OF REGRESSION 21 2.2 THE POPULATION REGRESSION FUNCTION (PRF):

A HYPOTHETICAL EXAMPLE 22

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2.3 STATISTICAL OR STOCHASTIC SPECIFICATION OF THE POPULATION REGRESSION FUNCTION 25

2.4 THE NATURE OF THE STOCHASTIC ERROR TERM 27 2.5 THE SAMPLE REGRESSION FUNCTION (SRF) 28 2.6 THE SPECIAL MEANING OF THE TERM “LINEAR”

REGRESSION 31 Linearity in the Variables 31 Linearity in the Parameters 32

2.7 TWO-VARIABLE VERSUS MULTIPLE LINEAR REGRESSION 33

2.8 ESTIMATION OF PARAMETERS: THE METHOD OF ORDINARY LEAST SQUARES 33

The Method of Ordinary Least Squares 34 2.9 PUTTING IT ALL TOGETHER 36

Interpretation of the Estimated Math S.A.T. Score Function 37 2.10 SOME ILLUSTRATIVE EXAMPLES 38 2.11 SUMMARY 43

KEY TERMS AND CONCEPTS 44 QUESTIONS 44 PROBLEMS 45 OPTIONAL QUESTIONS 51 APPENDIX 2A: DERIVATION OF LEAST-SQUARES

ESTIMATES 52

3 The Two-Variable Model: Hypothesis Testing 53 3.1 THE CLASSICAL LINEAR REGRESSION MODEL 54 3.2 VARIANCES AND STANDARD ERRORS OF

ORDINARY LEAST SQUARES ESTIMATORS 57 Variances and Standard Errors of the Math S.A.T. Score Example 59 Summary of the Math S.A.T. Score Function 59

3.3 WHY OLS? THE PROPERTIES OF OLS ESTIMATORS 60 Monte Carlo Experiment 61

3.4 THE SAMPLING, OR PROBABILITY, DISTRIBUTIONS OF OLS ESTIMATORS 62

3.5 HYPOTHESIS TESTING 64 Testing = 0 versus : The Confidence

Interval Approach 66 The Test of Significance Approach to Hypothesis Testing 68 Math S.A.T. Example Continued 69

3.6 HOW GOOD IS THE FITTED REGRESSION LINE: THE COEFFICIENT OF DETERMINATION, r2 71

Formulas to Compute r2 73 r2 for the Math S.A.T. Example 74 The Coefficient of Correlation, r 74

3.7 REPORTING THE RESULTS OF REGRESSION ANALYSIS 75

H1:B2 Z 0H0:B2

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CONTENTS ix

3.8 COMPUTER OUTPUT OF THE MATH S.A.T. SCORE EXAMPLE 76

3.9 NORMALITY TESTS 77 Histograms of Residuals 77 Normal Probability Plot 78 Jarque-Bera Test 78

3.10 A CONCLUDING EXAMPLE: RELATIONSHIP BETWEEN WAGES AND PRODUCTIVITY IN THE U.S. BUSINESS SECTOR, 1959–2006 79

3.11 A WORD ABOUT FORECASTING 82 3.12 SUMMARY 85

KEY TERMS AND CONCEPTS 86 QUESTIONS 86 PROBLEMS 88

4 Multiple Regression: Estimation and Hypothesis Testing 93 4.1 THE THREE-VARIABLE LINEAR REGRESSION

MODEL 94 The Meaning of Partial Regression Coefficient 95

4.2 ASSUMPTIONS OF THE MULTIPLE LINEAR REGRESSION MODEL 97

4.3 ESTIMATION OF THE PARAMETERS OF MULTIPLE REGRESSION 99

Ordinary Least Squares Estimators 99 Variance and Standard Errors of OLS Estimators 100 Properties of OLS Estimators of Multiple Regression 102

4.4 GOODNESS OF FIT OF ESTIMATED MULTIPLE REGRESSION: MULTIPLE COEFFICIENT OF DETERMINATION, R2 102

4.5 ANTIQUE CLOCK AUCTION PRICES REVISITED 103 Interpretation of the Regression Results 103

4.6 HYPOTHESIS TESTING IN A MULTIPLE REGRESSION: GENERAL COMMENTS 104

4.7 TESTING HYPOTHESES ABOUT INDIVIDUAL PARTIAL REGRESSION COEFFICIENTS 105

The Test of Significance Approach 105 The Confidence Interval Approach to Hypothesis Testing 106

4.8 TESTING THE JOINT HYPOTHESIS THAT 107

An Important Relationship between F and R2 111 4.9 TWO-VARIABLE REGRESSION IN THE CONTEXT OF

MULTIPLE REGRESSION: INTRODUCTION TO SPECIFICATION BIAS 112

4.10 COMPARING TWO R2 VALUES: THE ADJUSTED R2 113

B2 = B3 = 0 OR R 2 = 0

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4.11 WHEN TO ADD AN ADDITIONAL EXPLANATORY VARIABLE TO A MODEL 114

4.12 RESTRICTED LEAST SQUARES 116 4.13 ILLUSTRATIVE EXAMPLES 117

Discussion of Regression Results 118 4.14 SUMMARY 122

KEY TERMS AND CONCEPTS 123 QUESTIONS 123 PROBLEMS 125 APPENDIX 4A.1: DERIVATIONS OF OLS ESTIMATORS

GIVEN IN EQUATIONS (4.20) TO (4.22) 129 APPENDIX 4A.2: DERIVATION OF EQUATION (4.31) 129 APPENDIX 4A.3: DERIVATION OF EQUATION (4.50) 130 APPENDIX 4A.4: EVIEWS OUTPUT OF THE

CLOCK AUCTION PRICE EXAMPLE 131

5 Functional Forms of Regression Models 132 5.1 HOW TO MEASURE ELASTICITY: THE LOG-LINEAR

MODEL 133 Hypothesis Testing in Log-Linear Models 137

5.2 COMPARING LINEAR AND LOG-LINEAR REGRESSION MODELS 138

5.3 MULTIPLE LOG-LINEAR REGRESSION MODELS 140 5.4 HOW TO MEASURE THE GROWTH RATE: THE

SEMILOG MODEL 144 Instantaneous versus Compound Rate of Growth 147 The Linear Trend Model 148

5.5 THE LIN-LOG MODEL: WHEN THE EXPLANATORY VARIABLE IS LOGARITHMIC 149

5.6 RECIPROCAL MODELS 150 5.7 POLYNOMIAL REGRESSION MODELS 156 5.8 REGRESSION THROUGH THE ORIGIN 158 5.9 A NOTE ON SCALING AND UNITS OF MEASUREMENT 160 5.10 REGRESSION ON STANDARDIZED VARIABLES 161 5.11 SUMMARY OF FUNCTIONAL FORMS 163 5.12 SUMMARY 164

KEY TERMS AND CONCEPTS 165 QUESTIONS 166 PROBLEMS 167 APPENDIX 5A: LOGARITHMS 175

6 Dummy Variable Regression Models 178 6.1 THE NATURE OF DUMMY VARIABLES 178 6.2 ANCOVA MODELS: REGRESSION ON ONE

QUANTITATIVE VARIABLE AND ONE QUALITATIVE VARIABLE WITH TWO CATEGORIES: EXAMPLE 6.1 REVISITED 185

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6.3 REGRESSION ON ONE QUANTITATIVE VARIABLE AND ONE QUALITATIVE VARIABLE WITH MORE THAN TWO CLASSES OR CATEGORIES 187

6.4 REGRESSION ON ONE QUANTIATIVE EXPLANATORY VARIABLE AND MORE THAN ONE QUALITATIVE VARIABLE 190

Interaction Effects 191 A Generalization 192

6.5 COMPARING TWO REGESSIONS 193 6.6 THE USE OF DUMMY VARIABLES IN SEASONAL

ANALYSIS 198 6.7 WHAT HAPPENS IF THE DEPENDENT VARIABLE IS

ALSO A DUMMY VARIABLE? THE LINEAR PROBABILITY MODEL (LPM) 201

6.8 SUMMARY 204 KEY TERMS AND CONCEPTS 205 QUESTIONS 206 PROBLEMS 207

PART II REGRESSION ANALYSIS IN PRACTICE 217

7 Model Selection: Criteria and Tests 219 7.1 THE ATTRIBUTES OF A GOOD MODEL 220 7.2 TYPES OF SPECIFICATION ERRORS 221 7.3 OMISSON OF RELEVANT VARIABLE BIAS:

“UNDERFITTING” A MODEL 221 7.4 INCLUSION OF IRRELEVANT VARIABLES:

“OVERFITTING” A MODEL 225 7.5 INCORRECT FUNCTIONAL FORM 227 7.6 ERRORS OF MEASUREMENT 229

Errors of Measurement in the Dependent Variable 229 Errors of Measurement in the Explanatory Variable(s) 229

7.7 DETECTING SPECIFICATION ERRORS: TESTS OF SPECIFICATION ERRORS 230

Detecting the Presence of Unnecessary Variables 230 Tests for Omitted Variables and Incorrect Functional Forms 233 Choosing between Linear and Log-linear Regression Models:

The MWD Test 235 Regression Error Specification Test: RESET 237

7.8 SUMMARY 239 KEY TERMS AND CONCEPTS 240 QUESTIONS 240 PROBLEMS 241

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8 Multicollinearity: What Happens If Explanatory Variables are Correlated? 245 8.1 THE NATURE OF MULTICOLLINEARITY: THE

CASE OF PERFECT MULTICOLLINEARITY 246 8.2 THE CASE OF NEAR, OR IMPERFECT,

MULTICOLLINEARITY 248 8.3 THEORETICAL CONSEQUENCES OF

MULTICOLLINEARITY 250 8.4 PRACTICAL CONSEQUENCES OF MULTICOLLINEARITY 251 8.5 DETECTION OF MULTICOLLINEARITY 253 8.6 IS MULTICOLLINEARITY NECESSARILY BAD? 258 8.7 AN EXTENDED EXAMPLE: THE DEMAND FOR

CHICKENS IN THE UNITED STATES, 1960 TO 1982 259 Collinearity Diagnostics for the Demand Function for

Chickens (Equation [8.15]) 260 8.8 WHAT TO DO WITH MULTICOLLINEARITY:

REMEDIAL MEASURES 261 Dropping a Variable(s) from the Model 262 Acquiring Additional Data or a New Sample 262 Rethinking the Model 263 Prior Information about Some Parameters 264 Transformation of Variables 265 Other Remedies 266

8.9 SUMMARY 266 KEY TERMS AND CONCEPTS 267 QUESTIONS 267 PROBLEMS 268

9 Heteroscedasticity: What Happens If the Error Variance Is Nonconstant? 274 9.1 THE NATURE OF HETEROSCEDASTICITY 274 9.2 CONSEQUENCES OF HETEROSCEDASTICITY 280 9.3 DETECTION OF HETEROSCEDASTICITY: HOW DO

WE KNOW WHEN THERE IS A HETEROSCEDASTICITY PROBLEM? 282

Nature of the Problem 283 Graphical Examination of Residuals 283 Park Test 285 Glejser Test 287 White’s General Heteroscedasticity Test 289 Other Tests of Heteroscedasticity 290

9.4 WHAT TO DO IF HETEROSCEDASTICITY IS OBSERVED: REMEDIAL MEASURES 291

When �2i Is Known: The Method of Weighted Least Squares (WLS) 291 When True �2i Is Unknown 292 Respecification of the Model 297

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9.5 WHITE’S HETEROSCEDASTICITY-CORRECTED STANDARD ERRORS AND t STATISTICS 298

9.6 SOME CONCRETE EXAMPLES OF HETEROSCEDASTICITY 299

9.7 SUMMARY 302 KEY TERMS AND CONCEPTS 303 QUESTIONS 304 PROBLEMS 304

10 Autocorrelation: What Happens If Error Terms Are Correlated? 312 10.1 THE NATURE OF AUTOCORRELATION 313

Inertia 314 Model Specification Error(s) 315 The Cobweb Phenomenon 315 Data Manipulation 315

10.2 CONSEQUENCES OF AUTOCORRELATION 316 10.3 DETECTING AUTOCORRELATION 317

The Graphical Method 318 The Durbin-Watson d Test 320

10.4 REMEDIAL MEASURES 325 10.5 HOW TO ESTIMATE � 327

� � 1: The First Difference Method 327 � Estimated from Durbin-Watson d Statistic 327 � Estimated from OLS Residuals, et 328 Other Methods of Estimating � 328

10.6 A LARGE SAMPLE METHOD OF CORRECTING OLS STANDARD ERRORS: THE NEWEY-WEST (NW) METHOD 332

10.7 SUMMARY 334 KEY TERMS AND CONCEPTS 335 QUESTIONS 335 PROBLEMS 336 APPENDIX 10A: THE RUNS TEST 341 Swed-Eisenhart Critical Runs Test 342 Decision Rule 342 APPENDIX 10B: A GENERAL TEST OF

AUTOCORRELATION: THE BREUSCH-GODFREY (BG) TEST 343

PART III ADVANCED TOPICS IN ECONOMETRICS 345

11 Simultaneous Equation Models 347 11.1 THE NATURE OF SIMULTANEOUS EQUATION MODELS 348 11.2 THE SIMULTANEOUS EQUATION BIAS:

INCONSISTENCY OF OLS ESTIMATORS 350

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11.3 THE METHOD OF INDIRECT LEAST SQUARES (ILS) 352 11.4 INDIRECT LEAST SQUARES: AN ILLUSTRATIVE

EXAMPLE 353 11.5 THE IDENTIFICATION PROBLEM: A ROSE BY

ANY OTHER NAME MAY NOT BE A ROSE 355 Underidentification 356 Just or Exact Identification 357 Overidentification 359

11.6 RULES FOR IDENTIFICATION: THE ORDER CONDITION OF IDENTIFICATION 361

11.7 ESTIMATION OF AN OVERIDENTIFIED EQUATION: THE METHOD OF TWO-STAGE LEAST SQUARES 362

11.8 2SLS: A NUMERICAL EXAMPLE 364 11.9 SUMMARY 365

KEY TERMS AND CONCEPTS 366 QUESTIONS 367 PROBLEMS 367 APPENDIX 11A: INCONSISTENCY OF OLS ESTIMATORS 369

12 Selected Topics in Single Equation Regression Models 371 12.1 DYNAMIC ECONOMIC MODELS: AUTOREGRESSIVE AND

DISTRIBUTED LAG MODELS 371 Reasons for Lag 372 Estimation of Distributed Lag Models 374 The Koyck, Adaptive Expectations, and Stock Adjustment Models

Approach to Estimating Distributed Lag Models 377 12.2 THE PHENOMENON OF SPURIOUS REGRESSION:

NONSTATIONARY TIME SERIES 380 12.3 TESTS OF STATIONARITY 382 12.4 COINTEGRATED TIME SERIES 383 12.5 THE RANDOM WALK MODEL 384 12.6 THE LOGIT MODEL 386

Estimation of the Logit Model 390 12.7 SUMMARY 396

KEY TERMS AND CONCEPTS 397 QUESTIONS 397 PROBLEMS 398

INTRODUCTION TO APPENDIXES A, B, C, AND D: BASICS OF PROBABILITY AND STATISTICS 403

Appendix A: Review of Statistics: Probability and Probability Distributions 405

A.1 SOME NOTATION 405 The Summation Notation 405 Properties of the Summation Operator 406

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A.2 EXPERIMENT, SAMPLE SPACE, SAMPLE POINT, AND EVENTS 407

Experiment 407 Sample Space or Population 407 Sample Point 408 Events 408 Venn Diagrams 408

A.3 RANDOM VARIABLES 409 A.4 PROBABILITY 410

Probability of an Event: The Classical or A Priori Definition 410 Relative Frequency or Empirical Definition of Probability 411 Probability of Random Variables 417

A.5 RANDOM VARIABLES AND THEIR PROBABILITY DISTRIBUTIONS 417

Probability Distribution of a Discrete Random Variable 417 Probability Distribution of a Continuous Random Variable 419 Cumulative Distribution Function (CDF) 420

A.6 MULTIVARIATE PROBABILITY DENSITY FUNCTIONS 422 Marginal Probability Functions 424 Conditional Probability Functions 425 Statistical Independence 427

A.7 SUMMARY AND CONCLUSIONS 428 KEY TERMS AND CONCEPTS 428 REFERENCES 429 QUESTIONS 429 PROBLEMS 430

Appendix B: Characteristics of Probability Distributions 434 B.1 EXPECTED VALUE: A MEASURE OF CENTRAL

TENDENCY 434 Properties of Expected Value 436 Expected Value of Multivariate Probability Distributions 437

B.2 VARIANCE: A MEASURE OF DISPERSION 438 Properties of Variance 439 Chebyshev’s Inequality 441 Coefficient of Variation 442

B.3 COVARIANCE 443 Properties of Covariance 444

B.4 CORRELATION COEFFICIENT 445 Properties of Correlation Coefficient 445 Variances of Correlated Variables 447

B.5 CONDITIONAL EXPECTATION 447 Conditional Variance 449

B.6 SKEWNESS AND KURTOSIS 449 B.7 FROM THE POPULATION TO THE SAMPLE 452

Sample Mean 452

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Sample Variance 453 Sample Covariance 454 Sample Correlation Coefficient 455 Sample Skewness and Kurtosis 456

B.8 SUMMARY 456 KEY TERMS AND CONCEPTS 457 QUESTIONS 457 PROBLEMS 458 OPTIONAL EXERCISES 460

Appendix C: Some Important Probability Distributions 461 C.1 THE NORMAL DISTRIBUTION 462

Properties of the Normal Distribution 462 The Standard Normal Distribution 464 Random Sampling from a Normal Population 468 The Sampling or Probability Distribution of the Sample Mean X

– 468 The Central Limit Theorem (CLT) 472

C.2 THE t DISTRIBUTION 473 Properties of the t Distribution 474

C.3 THE CHI-SQUARE ( 2) PROBABILITY DISTRIBUTION 477 Properties of the Chi-square Distribution 478

C.4 THE F DISTRIBUTION 480 Properties of the F Distribution 481

C.5 SUMMARY 483 KEY TERMS AND CONCEPTS 483 QUESTIONS 484 PROBLEMS 484

Appendix D: Statistical Inference: Estimation and Hypothesis Testing 487

D.1 THE MEANING OF STATISTICAL INFERENCE 487 D.2 ESTIMATION AND HYPOTHESIS TESTING:

TWIN BRANCHES OF STATISTICAL INFERENCE 489 D.3 ESTIMATION OF PARAMETERS 490 D.4 PROPERTIES OF POINT ESTIMATORS 493

Linearity 494 Unbiasedness 494 Minimum Variance 495 Efficiency 496 Best Linear Unbiased Estimator (BLUE) 497 Consistency 497

D.5 STATISTICAL INFERENCE: HYPOTHESIS TESTING 498 The Confidence Interval Approach to Hypothesis Testing 499 Type I and Type II Errors: A Digression 500 The Test of Significance Approach to Hypothesis Testing 503

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A Word on Choosing the Level of Significance, �, and the p Value 506 The 2 and F Tests of Significance 507

D.6 SUMMARY 510 KEY TERMS AND CONCEPTS 510 QUESTIONS 511 PROBLEMS 512

Appendix E: Statistical Tables 515

Appendix F: Computer Output of EViews, MINITAB, Excel, and STATA 534

SELECTED BIBLIOGRAPHY 541

INDEXES 545 Name Index 545 Subject Index 547

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PREFACE

OBJECTIVE OF THE BOOK

As in the previous editions, the primary objective of the fourth edition of Essentials of Econometrics is to provide a user-friendly introduction to econometric theory and techniques. The intended audience is undergraduate economics ma- jors, undergraduate business administration majors, MBA students, and others in social and behavioral sciences where econometrics techniques, especially the techniques of linear regression analysis, are used. The book is designed to help students understand econometric techniques through extensive examples, care- ful explanations, and a wide variety of problem material. In each of the previous editions, I have tried to incorporate major developments in the field in an intu- itive and informative way without resorting to matrix algebra, calculus, or sta- tistics beyond the introductory level. The fourth edition continues that tradition.

Although I am in the eighth decade of my life, I have not lost my love for econometrics and I strive to keep up with the major developments in the field. To assist me in this endeavor, I am now happy to have Dr. Dawn Porter, Assistant Professor of Statistics at the Marshall School of Business at the University of Southern California in Los Angeles, as my co-author. Both of us have been deeply involved in bringing the fourth edition of Essentials of Econometrics to fruition.

MAJOR FEATURES OF THE FOURTH EDITION

Before discussing the specific changes in the various chapters, the following features of the new edition are worth noting:

1. In order to streamline topics and jump right into information about linear regression techniques, we have moved the background statistics material (formerly Chapters 2 through 5) to the appendix. This allows for easy refer- ence to more introductory material for those who need it, without disturbing the main content of the text.

2. Practically all the data used in the illustrative examples have been updated from the previous edition.

3. Several new examples have been added.

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4. In several chapters, we have included extended concluding examples that illustrate the various points made in the text.

5. Concrete computer printouts of several examples are included in the book. Most of these results are based on EViews (version 6), STATA (version 10), and MINITAB (version 15).

6. Several new diagrams and graphs are included in various chapters. 7. Several new data-based exercises are included throughout the book. 8. Small-sized data are included in the book, but large sample data are posted

on the book’s Web site, thereby minimizing the size of the text. The Web site also contains all the data used in the book.

SPECIFIC CHANGES

Some of the chapter-specific changes in the fourth edition are as follows: Chapter 1: A revised and expanded list of Web sites for economic data has been included. Chapters 2 and 3: An interesting new data example concerning the relationship between family income and student performance on the S.A.T. is utilized to introduce the two-variable regression model. Chapter 4: We have included a brief explanation of nonstochastic versus stochas- tic predictors. An additional example regarding educational expenditures among several countries that adds to the explanation of regression hypothesis testing. Chapter 5: The math S.A.T. example is revisited to illustrate various functional forms. Section 5.10 has been added to handle the topic of regression on stan- dardized variables. Also, several new data exercises have been included. Chapter 6: An example concerning acceptance rates among top business schools has been added to help illustrate the usefulness of dummy variable regression models. Several new data exercises also have been added. Chapter 8: Again, we have added several new, updated data exercises dealing with the issue of multicollinearity. Chapter 9: To illustrate the concept of heteroscedasticity, a new example relat- ing wages to education levels and years of experience has been included, as well as more real data exercises. Chapter 10: A new section concerning the Newey-West standard error correc- tion method using a data example has been added. Also, a new appendix has been included at the end of the chapter to cover the Breusch-Godfrey test of autocorrelation. Chapter 12: An expanded treatment of logistic regression has been included in this chapter with new examples to illustrate the results. Appendixes A–D: As noted above, the material in these appendixes was formerly contained in Chapters 2–5 of the main text. By placing them in the back of the book, they can more easily serve as reference sections to the main text. Data examples have been updated, and new exercises have been added.

Besides these specific changes, errors and misprints in the previous editions have been corrected. Also, our discussion of several topics in the various chap- ters has been streamlined.

xx PREFACE

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MATHEMATICAL REQUIREMENTS

In presenting the various topics, we have used very little matrix algebra or cal- culus. We firmly believe that econometrics can be taught to the beginner in an intuitive manner, without a heavy dose of matrix algebra or calculus. Also, we have not given any proofs unless they are easily understood. We do not feel that the nonspe- cialist needs to be burdened with detailed proofs. Of course, the instructor can supply the necessary proofs as the situation demands. Some of the proofs are available in our Basic Econometrics (McGraw-Hill, 5th ed., 2009).

SUPPLEMENTS AID THE PROBLEM SOLVING APPROACH

The comprehensive Web site for the fourth edition contains the following sup- plementary material to assist both instructors and students:

• Data from the text, as well as additional large set data referenced in the book. • A Solutions Manual providing answers to all of the questions and problems

throughout the text is provided for the instructors to use as they wish. • A digital image library containing all of the graphs and tables from the book.

For more information, please visit the Online Learning Center at www.mhhe .com/gujaratiess4e.

COMPUTERS AND ECONOMETRICS

It cannot be overemphasized that what has made econometrics accessible to the beginner is the availability of several user-friendly computer statistical pack- ages. The illustrative problems in this book are solved using statistical software packages, such as EViews, Excel, MINITAB, and STATA. Student versions of some of these packages are readily available. The data posted on the Web site is in Excel format and can also be read easily by many standard statistical pack- ages such as LIMDEP, RATS, SAS, and SPSS.

In Appendix E we show the outputs of EViews, Excel, MINITAB, and STATA, using a common data set. Each of these software packages has some unique features although some of the statistical routines are quite similar.

IN CLOSING

To sum up, in writing Essentials of Econometrics, our primary objective has been to introduce the wonderful world of econometrics to the beginner in a relaxed but informative style. We hope the knowledge gained from this book will prove to be of lasting value in the reader’s future academic or professional ca- reer and that the reader’s knowledge learned in this book can be further widened by reading some advanced and specialized books in econometrics. Some of these books can be found in the selected bibliography given at the end of the book.

PREFACE xxi

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ACKNOWLEDGMENTS

Our foremost thanks are to the following reviewers who made very valuable suggestions to improve the quality of the book.

Michael Allison University of Missouri, St. Louis Giles Bootheway Saint Bonaventure University Bruce Brown California State Polytechnic University, Pomona Kristin Butcher Wellesley College Juan Cabrera Queens College Tom Chen Saint John’s University Joanne Doyle James Madison University Barry Falk Iowa State University Eric Furstenberg University of Virginia, Charlottesville Steffen Habermalz Northwestern University Susan He Washington State University, Pullman Jerome Heavey Lafayette College George Jakubson Cornell University Elia Kacapyr Ithaca College Janet Kohlhase University of Houston Maria Kozhevnikova Queens College John Krieg Western Washington University William Latham University of Delaware Jinman Lee University of Illinois, Chicago Stephen LeRoy University of California, Santa Barbara Dandan Liu Bowling Green State University Fabio Milani University of California, Irvine Hillar Neumann Northern State University Jennifer Rice Eastern Michigan University Steven Stageberg University of Mary Washington Joseph Sulock University of North Carolina, Asheville Mark Tendall Stanford University Christopher Warburton John Jay College Tiemen Woutersen Johns Hopkins University

We are very grateful to Douglas Reiner, our publisher at McGraw-Hill, for help- ing us through this edition of the book. We are also grateful to Noelle Fox, edito- rial coordinator at McGraw-Hill, for working with us through all of our setbacks. We also need to acknowledge the project management provided by Manjot Singh Dodi, and the great copy editing by Ann Sass, especially since this type of text- book incorporates so many technical formulas and symbols.

Damodar N. Gujarati United States Military Academy, West Point

Dawn C. Porter University of Southern California, Los Angeles

xxii PREFACE

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CHAPTER 1 THE NATURE AND SCOPE

OF ECONOMETRICS

1

Research in economics, finance, management, marketing, and related disci- plines is becoming increasingly quantitative. Beginning students in these fields are encouraged, if not required, to take a course or two in econometrics—a field of study that has become quite popular. This chapter gives the beginner an overview of what econometrics is all about.

1.1 WHAT IS ECONOMETRICS?

Simply stated, econometrics means economic measurement. Although quan- titative measurement of economic concepts such as the gross domestic prod- uct (GDP), unemployment, inflation, imports, and exports is very important, the scope of econometrics is much broader, as can be seen from the following definitions:

Econometrics may be defined as the social science in which the tools of economic the- ory, mathematics, and statistical inference are applied to the analysis of economic phenomena.1

Econometrics, the result of a certain outlook on the role of economics, consists of the application of mathematical statistics to economic data to lend empirical support to the models constructed by mathematical economics and to obtain numerical results.2

1Arthur S. Goldberger, Econometric Theory, Wiley, New York, 1964, p. 1. 2P. A. Samuelson, T. C. Koopmans, and J. R. N. Stone, “Report of the Evaluative Committee for

Econometrica,” Econometrica, vol. 22, no. 2, April 1954, pp. 141–146.

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1.2 WHY STUDY ECONOMETRICS?

As the preceding definitions suggest, econometrics makes use of economic the- ory, mathematical economics, economic statistics (i.e., economic data), and mathematical statistics. Yet, it is a subject that deserves to be studied in its own right for the following reasons.

Economic theory makes statements or hypotheses that are mostly qualitative in nature. For example, microeconomic theory states that, other things remain- ing the same (the famous ceteris paribus clause of economics), an increase in the price of a commodity is expected to decrease the quantity demanded of that commodity. Thus, economic theory postulates a negative or inverse relation- ship between the price and quantity demanded of a commodity—this is the widely known law of downward-sloping demand or simply the law of demand. But the theory itself does not provide any numerical measure of the strength of the relationship between the two; that is, it does not tell by how much the quan- tity demanded will go up or down as a result of a certain change in the price of the commodity. It is the econometrician’s job to provide such numerical esti- mates. Econometrics gives empirical (i.e., based on observation or experiment) content to most economic theory. If we find in a study or experiment that when the price of a unit increases by a dollar the quantity demanded goes down by, say, 100 units, we have not only confirmed the law of demand, but in the process we have also provided a numerical estimate of the relationship between the two variables—price and quantity.

The main concern of mathematical economics is to express economic theory in mathematical form or equations (or models) without regard to measurability or empirical verification of the theory. Econometrics, as noted earlier, is primar- ily interested in the empirical verification of economic theory. As we will show shortly, the econometrician often uses mathematical models proposed by the mathematical economist but puts these models in forms that lend themselves to empirical testing.

Economic statistics is mainly concerned with collecting, processing, and pre- senting economic data in the form of charts, diagrams, and tables. This is the economic statistician’s job. He or she collects data on the GDP, employment, un- employment, prices, etc. These data constitute the raw data for econometric work. But the economic statistician does not go any further because he or she is not primarily concerned with using the collected data to test economic theories.

Although mathematical statistics provides many of the tools employed in the trade, the econometrician often needs special methods because of the unique nature of most economic data, namely, that the data are not usually generated as the result of a controlled experiment. The econometrician, like the meteorol- ogist, generally depends on data that cannot be controlled directly. Thus, data on consumption, income, investments, savings, prices, etc., which are collected by public and private agencies, are nonexperimental in nature. The econometri- cian takes these data as given. This creates special problems not normally dealt with in mathematical statistics. Moreover, such data are likely to contain errors of measurement, of either omission or commission, and the econometrician

2 CHAPTER ONE: THE NATURE AND SCOPE OF ECONOMETRICS

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may be called upon to develop special methods of analysis to deal with such errors of measurement.

For students majoring in economics and business there is a pragmatic reason for studying econometrics. After graduation, in their employment, they may be called upon to forecast sales, interest rates, and money supply or to estimate de- mand and supply functions or price elasticities for products. Quite often, econo- mists appear as expert witnesses before federal and state regulatory agencies on behalf of their clients or the public at large. Thus, an economist appearing before a state regulatory commission that controls prices of gas and electricity may be re- quired to assess the impact of a proposed price increase on the quantity de- manded of electricity before the commission will approve the price increase. In situations like this the economist may need to develop a demand function for electricity for this purpose. Such a demand function may enable the economist to estimate the price elasticity of demand, that is, the percentage change in the quan- tity demanded for a percentage change in the price. Knowledge of econometrics is very helpful in estimating such demand functions.

It is fair to say that econometrics has become an integral part of training in economics and business.

1.3 THE METHODOLOGY OF ECONOMETRICS

How does one actually do an econometric study? Broadly speaking, economet- ric analysis proceeds along the following lines.

1. Creating a statement of theory or hypothesis. 2. Collecting data. 3. Specifying the mathematical model of theory. 4. Specifying the statistical, or econometric, model of theory. 5. Estimating the parameters of the chosen econometric model. 6. Checking for model adequacy: Model specification testing. 7. Testing the hypothesis derived from the model. 8. Using the model for prediction or forecasting.

To illustrate the methodology, consider this question: Do economic condi- tions affect people’s decisions to enter the labor force, that is, their willingness to work? As a measure of economic conditions, suppose we use the unemploy- ment rate (UNR), and as a measure of labor force participation we use the labor force participation rate (LFPR). Data on UNR and LFPR are regularly published by the government. So to answer the question we proceed as follows.

Creating a Statement of Theory or Hypothesis

The starting point is to find out what economic theory has to say on the subject you want to study. In labor economics, there are two rival hypotheses about the effect of economic conditions on people’s willingness to work. The discouraged- worker hypothesis (effect) states that when economic conditions worsen, as

CHAPTER ONE: THE NATURE AND SCOPE OF ECONOMETRICS 3

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reflected in a higher unemployment rate, many unemployed workers give up hope of finding a job and drop out of the labor force. On the other hand, the added-worker hypothesis (effect) maintains that when economic conditions worsen, many secondary workers who are not currently in the labor market (e.g., mothers with children) may decide to join the labor force if the main breadwinner in the family loses his or her job. Even if the jobs these secondary workers get are low paying, the earnings will make up some of the loss in in- come suffered by the primary breadwinner.

Whether, on balance, the labor force participation rate will increase or decrease will depend on the relative strengths of the added-worker and discouraged- worker effects. If the added-worker effect dominates, LFPR will increase even when the unemployment rate is high. Contrarily, if the discouraged-worker effect dominates, LFPR will decrease. How do we find this out? This now becomes our empirical question.

Collecting Data

For empirical purposes, therefore, we need quantitative information on the two variables. There are three types of data that are generally available for empirical analysis.

1. Time series. 2. Cross-sectional. 3. Pooled (a combination of time series and cross-sectional).

Times series data are collected over a period of time, such as the data on GDP, employment, unemployment, money supply, or government deficits. Such data may be collected at regular intervals—daily (e.g., stock prices), weekly (e.g., money supply), monthly (e.g., the unemployment rate), quarterly (e.g., GDP), or annually (e.g., government budget). These data may be quanti- tative in nature (e.g., prices, income, money supply) or qualitative (e.g., male or female, employed or unemployed, married or unmarried, white or black). As we will show, qualitative variables, also called dummy or categorical variables, can be every bit as important as quantitative variables.

Cross-sectional data are data on one or more variables collected at one point in time, such as the census of population conducted by the U.S. Census Bureau every 10 years (the most recent was on April 1, 2000); the surveys of consumer expenditures conducted by the University of Michigan; and the opinion polls such as those conducted by Gallup, Harris, and other polling organizations.

In pooled data we have elements of both time series and cross-sectional data. For example, if we collect data on the unemployment rate for 10 countries for a period of 20 years, the data will constitute an example of pooled data—data on the unemployment rate for each country for the 20-year period will form time se- ries data, whereas data on the unemployment rate for the 10 countries for any single year will be cross-sectional data. In pooled data we will have 200 observations—20 annual observations for each of the 10 countries.

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There is a special type of pooled data, panel data, also called longitudinal or micropanel data, in which the same cross-sectional unit, say, a family or firm, is surveyed over time. For example, the U.S. Department of Commerce conducts a census of housing at periodic intervals. At each periodic survey the same household (or the people living at the same address) is interviewed to find out if there has been any change in the housing and financial conditions of that household since the last survey. The panel data that result from repeatedly in- terviewing the same household at periodic intervals provide very useful infor- mation on the dynamics of household behavior.

Sources of Data A word is in order regarding data sources. The success of any econometric study hinges on the quality, as well as the quantity, of data. Fortunately, the Internet has opened up a veritable wealth of data. In Appendix 1A we give addresses of several Web sites that have all kinds of mi- croeconomic and macroeconomic data. Students should be familiar with such sources of data, as well as how to access or download them. Of course, these data are continually updated so the reader may find the latest available data.

For our analysis, we obtained the time series data shown in Table 1-1. This table gives data on the civilian labor force participation rate (CLFPR) and the civilian unemployment rate (CUNR), defined as the number of civilians unem- ployed as a percentage of the civilian labor force, for the United States for the period 1980–2007.3

Unlike physical sciences, most data collected in economics (e.g., GDP, money supply, Dow Jones index, car sales) are nonexperimental in that the data- collecting agency (e.g., government) may not have any direct control over the data. Thus, the data on labor force participation and unemployment are based on the information provided to the government by participants in the labor market. In a sense, the government is a passive collector of these data and may not be aware of the added- or discouraged-worker hypotheses, or any other hypothesis, for that matter. Therefore, the collected data may be the result of several factors affecting the labor force participation decision made by the individual person. That is, the same data may be compatible with more than one theory.

Specifying the Mathematical Model of Labor Force Participation

To see how CLFPR behaves in relation to CUNR, the first thing we should do is plot the data for these variables in a scatter diagram, or scattergram, as shown in Figure 1-1.

The scattergram shows that CLFPR and CUNR are inversely related, perhaps suggesting that, on balance, the discouraged-worker effect is stronger than the added-worker effect.4 As a first approximation, we can draw a straight line

CHAPTER ONE: THE NATURE AND SCOPE OF ECONOMETRICS 5

3We consider here only the aggregate CLFPR and CUNR, but data are available by age, sex, and ethnic composition.

4On this, see Shelly Lundberg, “The Added Worker Effect,” Journal of Labor Economics, vol. 3, January 1985, pp. 11–37.

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6 CHAPTER ONE: THE NATURE AND SCOPE OF ECONOMETRICS

U.S. CIVILIAN LABOR FORCE PARTICIPATION RATE (CLFPR), CIVILIAN UNEMPLOYMENT RATE (CUNR), AND REAL AVERAGE HOURLY EARNINGS (AHE82)* FOR THE YEARS 1980–2007

Year CLFPR (%) CUNR (%) AHE82 ($)

1980 63.8 7.1 8.00 1981 63.9 7.6 7.89 1982 64.0 9.7 7.87 1983 64.0 9.6 7.96 1984 64.4 7.5 7.96 1985 64.8 7.2 7.92 1986 65.3 7.0 7.97 1987 65.6 6.2 7.87 1988 65.9 5.5 7.82 1989 66.5 5.3 7.75 1990 66.5 5.6 7.66 1991 66.2 6.8 7.59 1992 66.4 7.5 7.55 1993 66.3 6.9 7.54 1994 66.6 6.1 7.54 1995 66.6 5.6 7.54 1996 66.8 5.4 7.57 1997 67.1 4.9 7.69 1998 67.1 4.5 7.89 1999 67.1 4.2 8.01 2000 67.1 4.0 8.04 2001 66.8 4.7 8.12 2002 66.6 5.8 8.25 2003 66.2 6.0 8.28 2004 66.0 5.5 8.24 2005 66.0 5.1 8.18 2006 66.2 4.6 8.24 2007 66.0 4.6 8.32

*AHE82 represents average hourly earnings in 1982 dollars. Source: Economic Report of the President, 2008, CLFPR from

Table B-40, CUNR from Table B-43, and AHE82 from Table B-47.

TABLE 1-1

through the scatter points and write the relationship between CLFPR and CUNR by the following simple mathematical model:

(1.1)

Equation (1.1) states that CLFPR is linearly related to CUNR. B1 and B2 are known as the parameters of the linear function.5 B1 is also known as the intercept; it

CLFPR = B1 + B2 CUNR

5Broadly speaking, a parameter is an unknown quantity that may vary over a certain set of val- ues. In statistics a probability distribution function (PDF) of a random variable is often character- ized by its parameters, such as its mean and variance. This topic is discussed in greater detail in Appendixes A and B.

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gives the value of CLFPR when CUNR is zero.6 B2 is known as the slope. The slope measures the rate of change in CLFPR for a unit change in CUNR, or more gen- erally, the rate of change in the value of the variable on the left-hand side of the equation for a unit change in the value of the variable on the right-hand side. The slope coefficient B2 can be positive (if the added-worker effect dominates the discouraged-worker effect) or negative (if the discouraged-worker effect dominates the added-worker effect). Figure 1-1 suggests that in the present case it is negative.

Specifying the Statistical, or Econometric, Model of Labor Force Participation

The purely mathematical model of the relationship between CLFPR and CUNR given in Eq. (1.1), although of prime interest to the mathematical economist, is of limited appeal to the econometrician, for such a model assumes an exact, or deterministic, relationship between the two variables; that is, for a given CUNR, there is a unique value of CLFPR. In reality, one rarely finds such neat relation- ships between economic variables. Most often, the relationships are inexact, or statistical, in nature.

This is seen clearly from the scattergram given in Figure 1-1. Although the two variables are inversely related, the relationship between them is not perfectly or exactly linear, for if we draw a straight line through the 28 data points, not all the data points will lie exactly on that straight line. Recall that to draw a straight line we need only two points.7 Why don’t the 28 data points lie exactly on the straight

CHAPTER ONE: THE NATURE AND SCOPE OF ECONOMETRICS 7

C L

FP R

(% )

CUNR (%)

67.5

67.0

66.5

66.0

65.5

65.0

64.5

64.0

63.5 3.5 4.5 5.5 6.5

Fitted Line Plot

7.5 8.5 9.5 10.5

Regression plot for civilian labor force participationrate (%) and civilian unemployment rate (%) FIGURE 1-1

6In Chapter 2 we give a more precise interpretation of the intercept in the context of regression analysis.

7We even tried to fit a parabola to the scatter points given in Fig. 1-1, but the results were not materially different from the linear specification.

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line specified by the mathematical model, Eq. (1.1)? Remember that our data on labor force and unemployment are nonexperimentally collected. Therefore, as noted earlier, besides the added- and discouraged-worker hypotheses, there may be other forces affecting labor force participation decisions. As a result, the observed relationship between CLFPR and CUNR is likely to be imprecise.

Let us allow for the influence of all other variables affecting CLFPR in a catchall variable u and write Eq. (1.2) as follows:

(1.2)

where u represents the random error term, or simply the error term.8 We let u represent all those forces (besides CUNR) that affect CLFPR but are not explic- itly introduced in the model, as well as purely random forces. As we will see in Part II, the error term distinguishes econometrics from purely mathematical economics.

Equation (1.2) is an example of a statistical, or empirical or econometric, model. More precisely, it is an example of what is known as a linear regression model, which is a prime subject of this book. In such a model, the variable appearing on the left-hand side of the equation is called the dependent variable, and the vari- able on the right-hand side is called the independent, or explanatory, variable. In linear regression analysis our primary objective is to explain the behavior of one variable (the dependent variable) in relation to the behavior of one or more other variables (the explanatory variables), allowing for the fact that the rela- tionship between them is inexact.

Notice that the econometric model, Eq. (1.2), is derived from the mathemati- cal model, Eq. (1.1), which shows that mathematical economics and economet- rics are mutually complementary disciplines. This is clearly reflected in the definition of econometrics given at the outset.

Before proceeding further, a warning regarding causation is in order. In the regression model, Eq. (1.2), we have stated that CLFPR is the dependent vari- able and CUNR is the independent, or explanatory, variable. Does that mean that the two variables are causally related; that is, is CUNR the cause and CLFPR the effect? In other words, does regression imply causation? Not necessarily. As Kendall and Stuart note, “A statistical relationship, however strong and how- ever suggestive, can never establish causal connection: our ideas of causation must come from outside statistics, ultimately from some theory or other.”9 In our example, it is up to economic theory (e.g., the discouraged-worker hypoth- esis) to establish the cause-and-effect relationship, if any, between the depen- dent and explanatory variables. If causality cannot be established, it is better to call the relationship, Eq. (1.2), a predictive relationship: Given CUNR, can we pre- dict CLFPR?

CLFPR = B1 + B2CUNR + u

8 CHAPTER ONE: THE NATURE AND SCOPE OF ECONOMETRICS

8In statistical lingo, the random error term is known as the stochastic error term. 9M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, Charles Griffin Publishers, New

York, 1961, vol. 2, Chap. 26, p. 279.

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Estimating the Parameters of the Chosen Econometric Model

Given the data on CLFPR and CUNR, such as that in Table 1-1, how do we esti- mate the parameters of the model, Eq. (1.2), namely, B1 and B2? That is, how do we find the numerical values (i.e., estimates) of these parameters? This will be the focus of our attention in Part II, where we develop the appropriate methods of computation, especially the method of ordinary least squares (OLS). Using OLS and the data given in Table 1-1, we obtained the following results:

(1.3)

Note that we have put the symbol on CLFPR (read as “CLFPR hat”) to remind us that Eq. (1.3) is an estimate of Eq. (1.2). The estimated regression line is shown in Figure 1-1, along with the actual data points.

As Eq. (1.3) shows, the estimated value of B1 is 69.5 and that of B2 is – 0.58, where the symbol means approximately. Thus, if the unemployment rate goes up by one unit (i.e., one percentage point), ceteris paribus, CLFPR is ex- pected to decrease on the average by about 0.58 percentage points; that is, as eco- nomic conditions worsen, on average, there is a net decrease in the labor force participation rate of about 0.58 percentage points, perhaps suggesting that the discouraged-worker effect dominates. We say “on the average” because the presence of the error term u, as noted earlier, is likely to make the relationship somewhat imprecise. This is vividly seen in Figure 1-1 where the points not on the estimated regression line are the actual participation rates and the (vertical) distance between them and the points on the regression line are the estimated u’s. As we will see in Chapter 2, the estimated u’s are called residuals. In short, the estimated regression line, Eq. (1.3), gives the relationship between average CLFPR and CUNR; that is, on average how CLFPR responds to a unit change in CUNR. The value of about 69.5 suggests that the average value of CLFPR will be about 69.5 percent if the CUNR is zero; that is, about 69.5 percent of the civil- ian working-age population will participate in the labor force if there is full employment (i.e., zero