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The McGraw-Hill Series
Economics
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Basic
Econometrics
Fifth 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 Bogot? Caracas Kuala Lumpur Lisbon London Madrid Mexico City
Milan Montreal New Delhi Santiago Seoul Singapore Sydney Taipei Toronto
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BASIC 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 ? 2009, 2003, 1995, 1988, 1978 by The McGraw-Hill Companies,
Inc. All rights reserved. No part of this publication may be reproduced or distributed in any form or by any
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Some ancillaries, including electronic and print components, may not be available to customers outside the
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This book is printed on acid-free paper.
1 2 3 4 5 6 7 8 9 0 VNH/VNH 0 9 8
ISBN 978-0-07-337577-9
MHID 0-07-337577-2
Publisher: Douglas Reiner
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Library of Congress Cataloging-in-Publication Data
Gujarati, Damodar N.
Basic econometrics / Damodar N. Gujarati, Dawn C. Porter. ? 5th ed.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-0-07-337577-9 (alk. paper)
ISBN-10: 0-07-337577-2 (alk. paper)
1. Econometrics. I. Porter, Dawn C. II. Title.
HB139.G84 2009
330.015195?dc22
2008035934
www.mhhe.com
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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 published 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 author of Pensions and the New York City Fiscal
Crisis (the American Enterprise Institute, 1978), Government and Business (McGraw-Hill,
1984), and Essentials of Econometrics (McGraw-Hill, 3d ed., 2006). Dr. Gujarati?s books
on econometrics 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.
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 both introductory undergraduate
and MBA statistics 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 before that was 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.
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, McGraw-Hill Irwin, 2008.
Outside of academics, Dawn has been employed as a statistical consultant for KPMG, Inc.
She has also 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.
iii
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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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Brief Contents
Preface xvi
Acknowledgments xix
PART THREE
Topics in Econometrics 523
14 Nonlinear Regression Models
525
PART ONE
15 Qualitative Response Regression
Models
541
Single-Equation Regression Models 13
16 Panel Data Regression Models
591
17 Dynamic Econometric Models:
Autoregressive and
Distributed-Lag Models
617
Introduction 1
1 The Nature of Regression Analysis
15
2 Two-Variable Regression Analysis:
Some Basic Ideas
34
3 Two-Variable Regression Model: The
Problem of Estimation
55
4 Classical Normal Linear Regression
Model (CNLRM)
97
5 Two-Variable Regression: Interval
Estimation and Hypothesis Testing
6 Extensions of the Two-Variable
Linear Regression Model
7 Multiple Regression Analysis: The
Problem of Estimation
PART FOUR
107
147
188
8 Multiple Regression Analysis: The
Problem of Inference
233
9 Dummy Variable Regression Models
277
PART TWO
Relaxing the Assumptions
of the Classical Model 315
Simultaneous-Equation Models and Time
Series Econometrics 671
18 Simultaneous-Equation Models
673
19 The Identification Problem
689
20 Simultaneous-Equation Methods
711
21 Time Series Econometrics: Some
Basic Concepts
737
22 Time Series Econometrics:
Forecasting
773
APPENDICES
A A Review of Some
Statistical Concepts
801
B Rudiments of Matrix Algebra
838
C The Matrix Approach to
Linear Regression Model
849
10 Multicollinearity: What Happens
If the Regressors Are Correlated?
320
D Statistical Tables
877
11 Heteroscedasticity: What Happens If
the Error Variance Is Nonconstant?
365
E Computer Output of EViews,
MINITAB, Excel, and STATA
894
12 Autocorrelation: What Happens If
the Error Terms Are Correlated?
412
F Economic Data on the
World Wide Web
900
13 Econometric Modeling: Model
Specification and Diagnostic Testing
467
SELECTED BIBLIOGRAPHY 902
v
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Contents
Preface xvi
Acknowledgments xix
Introduction 1
I.1
I.2
I.3
What Is Econometrics? 1
Why a Separate Discipline? 2
Methodology of Econometrics 2
1. Statement of Theory or Hypothesis 3
2. Specification of the Mathematical Model
of Consumption 3
3. Specification of the Econometric Model
of Consumption 4
4. Obtaining Data 5
5. Estimation of the Econometric Model 5
6. Hypothesis Testing 7
7. Forecasting or Prediction 8
8. Use of the Model for Control
or Policy Purposes 9
Choosing among Competing Models 9
I.4
I.5
I.6
I.7
Types of Econometrics 10
Mathematical and Statistical Prerequisites 11
The Role of the Computer 11
Suggestions for Further Reading 12
PART ONE
SINGLE-EQUATION REGRESSION
MODELS 13
CHAPTER 1
The Nature of Regression Analysis 15
1.1
1.2
Historical Origin of the Term Regression 15
The Modern Interpretation of Regression 15
1.3
Statistical versus Deterministic
Relationships 19
Regression versus Causation 19
Regression versus Correlation 20
Terminology and Notation 21
The Nature and Sources of Data for Economic
Analysis 22
Examples 16
1.4
1.5
1.6
1.7
Types of Data 22
The Sources of Data 25
The Accuracy of Data 27
A Note on the Measurement Scales
of Variables 27
vi
Summary and Conclusions 28
Exercises 29
CHAPTER 2
Two-Variable Regression Analysis: Some
Basic Ideas 34
2.1
2.2
2.3
A Hypothetical Example 34
The Concept of Population Regression
Function (PRF) 37
The Meaning of the Term Linear 38
Linearity in the Variables 38
Linearity in the Parameters 38
2.4
2.5
2.6
2.7
Stochastic Specification of PRF 39
The Significance of the Stochastic
Disturbance Term 41
The Sample Regression Function (SRF) 42
Illustrative Examples 45
Summary and Conclusions 48
Exercises 48
CHAPTER 3
Two-Variable Regression Model: The
Problem of Estimation 55
3.1
3.2
The Method of Ordinary Least Squares 55
The Classical Linear Regression Model: The
Assumptions Underlying the Method
of Least Squares 61
A Word about These Assumptions 68
3.3
Precision or Standard Errors
of Least-Squares Estimates 69
3.4
Properties of Least-Squares Estimators:
The Gauss?Markov Theorem 71
3.5
The Coefficient of Determination r2:
A Measure of ?Goodness of Fit? 73
3.6
A Numerical Example 78
3.7
Illustrative Examples 81
3.8
A Note on Monte Carlo Experiments 83
Summary and Conclusions 84
Exercises 85
Appendix 3A 92
3A.1 Derivation of Least-Squares Estimates 92
3A.2 Linearity and Unbiasedness Properties
of Least-Squares Estimators 92
3A.3 Variances and Standard Errors
of Least-Squares Estimators 93
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3A.4 Covariance Between ?^1 and ?^ 2 93
3A.5 The Least-Squares Estimator of s2 93
3A.6 Minimum-Variance Property
of Least-Squares Estimators 95
3A.7 Consistency of Least-Squares Estimators 96
The ?Zero? Null Hypothesis and the ?2-t? Rule
of Thumb 120
Forming the Null and Alternative
Hypotheses 121
Choosing a, the Level of Significance 121
The Exact Level of Significance:
The p Value 122
Statistical Significance versus Practical
Significance 123
The Choice between Confidence-Interval
and Test-of-Significance Approaches
to Hypothesis Testing 124
CHAPTER 4
Classical Normal Linear Regression
Model (CNLRM) 97
4.1
4.2
The Probability Distribution
of Disturbances ui 97
The Normality Assumption for ui 98
5.9
Regression Analysis and Analysis
of Variance 124
5.10 Application of Regression Analysis:
The Problem of Prediction 126
Why the Normality Assumption? 99
4.3
Properties of OLS Estimators under
the Normality Assumption 100
4.4
The Method of Maximum
Likelihood (ML) 102
Summary and Conclusions 102
Appendix 4A 103
4A.1 Maximum Likelihood Estimation
of Two-Variable Regression Model 103
4A.2 Maximum Likelihood Estimation
of Food Expenditure in India 105
Appendix 4A Exercises 105
Mean Prediction 127
Individual Prediction 128
5.11 Reporting the Results of Regression
Analysis 129
5.12 Evaluating the Results of Regression
Analysis 130
Normality Tests 130
Other Tests of Model Adequacy
CHAPTER 5
Two-Variable Regression: Interval
Estimation and Hypothesis Testing 107
5.1
5.2
5.3
5A.1
Statistical Prerequisites 107
Interval Estimation: Some Basic Ideas 108
Confidence Intervals for Regression
Coefficients ?1 and ?2 109
5A.2
5A.3
5A.4
Confidence Interval for ?2 109
Confidence Interval for ?1 and ?2
Simultaneously 111
5.4
5.5
5.6
Variance of Mean Prediction 145
Variance of Individual Prediction 146
Confidence Interval for s2 111
Hypothesis Testing: General Comments 113
Hypothesis Testing:
The Confidence-Interval Approach 113
5.8
Regression through the Origin 147
r2 for Regression-through-Origin Model 150
Hypothesis Testing:
The Test-of-Significance Approach 115
Testing the Significance of Regression
Coefficients: The t Test 115
Testing the Significance of s2: The ?2 Test
CHAPTER 6
Extensions of the Two-Variable Linear
Regression Model 147
6.1
Two-Sided or Two-Tail Test 113
One-Sided or One-Tail Test 115
5.7
132
Summary and Conclusions 134
Exercises 135
Appendix 5A 143
Probability Distributions Related
to the Normal Distribution 143
Derivation of Equation (5.3.2) 145
Derivation of Equation (5.9.1) 145
Derivations of Equations (5.10.2)
and (5.10.6) 145
6.2
Scaling and Units of Measurement 154
6.3
6.4
6.5
Regression on Standardized Variables 157
Functional Forms of Regression Models 159
How to Measure Elasticity: The Log-Linear
Model 159
Semilog Models: Log?Lin and Lin?Log
Models 162
A Word about Interpretation 157
118
Hypothesis Testing: Some Practical Aspects 119
The Meaning of ?Accepting? or ?Rejecting? a
Hypothesis 119
6.6
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How to Measure the Growth Rate:
The Log?Lin Model 162
The Lin?Log Model 164
6.7
Reciprocal Models 166
Log Hyperbola or Logarithmic Reciprocal
Model 172
6.8
6.9
6A.1
6A.2
6A.3
6A.4
6A.5
Choice of Functional Form 172
A Note on the Nature of the Stochastic Error
Term: Additive versus Multiplicative
Stochastic Error Term 174
Summary and Conclusions 175
Exercises 176
Appendix 6A 182
Derivation of Least-Squares Estimators
for Regression through the Origin 182
Proof that a Standardized Variable
Has Zero Mean and Unit Variance 183
Logarithms 184
Growth Rate Formulas 186
Box-Cox Regression Model 187
Allocating R2 among Regressors 206
?
The ?Game?? of Maximizing R2 206
7.9
The Cobb?Douglas Production Function:
More on Functional Form 207
7.10 Polynomial Regression Models 210
7.11 Partial Correlation Coefficients 213
Explanation of Simple and Partial
Correlation Coefficients 213
Interpretation of Simple and Partial
Correlation Coefficients 214
7A.1
7A.2
7A.3
7A.4
7A.5
CHAPTER 7
Multiple Regression Analysis:
The Problem of Estimation 188
7.1
7.2
7.3
7.4
The Three-Variable Model: Notation
and Assumptions 188
Interpretation of Multiple Regression
Equation 191
The Meaning of Partial Regression
Coefficients 191
OLS and ML Estimation of the Partial
Regression Coefficients 192
OLS Estimators 192
Variances and Standard Errors
of OLS Estimators 194
Properties of OLS Estimators 195
Maximum Likelihood Estimators 196
7.5
7.6
CHAPTER 8
Multiple Regression Analysis: The Problem
of Inference 233
8.1
8.2
8.3
8.4
The Multiple Coefficient of Determination R2
and the Multiple Coefficient
of Correlation R 196
An Illustrative Example 198
7.8
Simple Regression in the Context
of Multiple Regression: Introduction to
Specification Bias 200
R2 and the Adjusted R2 201
Comparing Two R2 Values 203
The Normality Assumption Once Again 233
Hypothesis Testing in Multiple Regression:
General Comments 234
Hypothesis Testing about Individual
Regression Coefficients 235
Testing the Overall Significance of the Sample
Regression 237
The Analysis of Variance Approach to Testing the
Overall Significance of an Observed Multiple
Regression: The F Test 238
Testing the Overall Significance of a Multiple
Regression: The F Test 240
An Important Relationship between R2 and F 241
Testing the Overall Significance of a Multiple
Regression in Terms of R2 242
The ?Incremental? or ?Marginal? Contribution
of an Explanatory Variable 243
Regression on Standardized Variables 199
Impact on the Dependent Variable of a Unit
Change in More than One Regressor 199
7.7
Summary and Conclusions 215
Exercises 216
Appendix 7A 227
Derivation of OLS Estimators
Given in Equations (7.4.3) to (7.4.5) 227
Equality between the Coefficients of PGNP
in Equations (7.3.5) and (7.6.2) 229
Derivation of Equation (7.4.19) 229
Maximum Likelihood Estimation
of the Multiple Regression Model 230
EViews Output of the Cobb?Douglas
Production Function in
Equation (7.9.4) 231
8.5
8.6
Testing the Equality of Two Regression
Coefficients 246
Restricted Least Squares: Testing Linear
Equality Restrictions 248
The t-Test Approach 249
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The F-Test Approach: Restricted Least
Squares 249
General F Testing 252
8.7
Testing for Structural or Parameter Stability
of Regression Models: The Chow Test 254
8.8
Prediction with Multiple Regression 259
8.9
The Troika of Hypothesis Tests: The
Likelihood Ratio (LR), Wald (W), and
Lagrange Multiplier (LM) Tests 259
8.10 Testing the Functional Form of Regression:
Choosing between Linear and Log?Linear
Regression Models 260
Summary and Conclusions 262
Exercises 262
Appendix 8A: Likelihood
Ratio (LR) Test 274
CHAPTER 9
Dummy Variable Regression Models 277
9.1
9.2
The Nature of Dummy Variables 277
ANOVA Models 278
Caution in the Use of Dummy Variables 281
9.3
ANOVA Models with Two Qualitative
Variables 283
9.4
Regression with a Mixture of Quantitative
and Qualitative Regressors: The ANCOVA
Models 283
9.5
The Dummy Variable Alternative
to the Chow Test 285
9.6
Interaction Effects Using Dummy
Variables 288
9.7
The Use of Dummy Variables in Seasonal
Analysis 290
9.8
Piecewise Linear Regression 295
9.9
Panel Data Regression Models 297
9.10 Some Technical Aspects of the Dummy
Variable Technique 297
The Interpretation of Dummy Variables
in Semilogarithmic Regressions 297
Dummy Variables and Heteroscedasticity 298
Dummy Variables and Autocorrelation 299
What Happens If the Dependent Variable
Is a Dummy Variable? 299
9.11 Topics for Further Study 300
9.12 A Concluding Example 300
Summary and Conclusions 304
Exercises 305
Appendix 9A: Semilogarithmic Regression
with Dummy Regressor 314
PART TWO
RELAXING THE ASSUMPTIONS OF THE
CLASSICAL MODEL 315
CHAPTER 10
Multicollinearity: What Happens
If the Regressors Are Correlated? 320
10.1 The Nature of Multicollinearity 321
10.2 Estimation in the Presence of Perfect
Multicollinearity 324
10.3 Estimation in the Presence of ?High?
but ?Imperfect? Multicollinearity 325
10.4 Multicollinearity: Much Ado about Nothing?
Theoretical Consequences
of Multicollinearity 326
10.5 Practical Consequences
of Multicollinearity 327
Large Variances and Covariances
of OLS Estimators 328
Wider Confidence Intervals 330
?Insignificant? t Ratios 330
A High R2 but Few Significant t Ratios 331
Sensitivity of OLS Estimators and Their
Standard Errors to Small Changes in Data 331
Consequences of Micronumerosity
332
10.6 An Illustrative Example 332
10.7 Detection of Multicollinearity 337
10.8 Remedial Measures 342
Do Nothing 342
Rule-of-Thumb Procedures 342
10.9 Is Multicollinearity Necessarily Bad? Maybe
Not, If the Objective Is Prediction Only 347
10.10 An Extended Example: The Longley
Data 347
Summary and Conclusions 350
Exercises 351
CHAPTER 11
Heteroscedasticity: What Happens If
the Error Variance Is Nonconstant? 365
11.1 The Nature of Heteroscedasticity 365
11.2 OLS Estimation in the Presence
of Heteroscedasticity 370
11.3 The Method of Generalized Least
Squares (GLS) 371
Difference between OLS and GLS 373
11.4 Consequences of Using OLS in the Presence
of Heteroscedasticity 374
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OLS Estimation Allowing for
Heteroscedasticity 374
OLS Estimation Disregarding
Heteroscedasticity 374
A Technical Note 376
11.5 Detection of Heteroscedasticity 376
Informal Methods 376
Formal Methods 378
11.6 Remedial Measures 389
When s 2i Is Known: The Method of Weighted
Least Squares 389
When s 2i Is Not Known 391
11.7 Concluding Examples 395
11.8 A Caution about Overreacting
to Heteroscedasticity 400
Summary and Conclusions 400
Exercises 401
Appendix 11A 409
11A.1 Proof of Equation (11.2.2) 409
11A.2 The Method of Weighted Least
Squares 409
11A.3 Proof that E(s^ 2 ) s2 in the Presence
of Heteroscedasticity 410
11A.4 White?s Robust Standard Errors 411
CHAPTER 12
Autocorrelation: What Happens If the Error
Terms Are Correlated? 412
12.1 The Nature of the Problem 413
12.2 OLS Estimation in the Presence
of Autocorrelation 418
12.3 The BLUE Estimator in the Presence
of Autocorrelation 422
12.4 Consequences of Using OLS
in the Presence of Autocorrelation 423
OLS Estimation Allowing
for Autocorrelation 423
OLS Estimation Disregarding
Autocorrelation 423
12.5 Relationship between Wages and Productivity
in the Business Sector of the United States,
1960?2005 428
12.6 Detecting Autocorrelation 429
I. Graphical Method 429
II. The Runs Test 431
III. Durbin?Watson d Test 434
IV. A General Test of Autocorrelation:
The Breusch?Godfrey (BG) Test 438
Why So Many Tests of Autocorrelation? 440
12.7 What to Do When You Find Autocorrelation:
Remedial Measures 440
12.8 Model Mis-Specification versus Pure
Autocorrelation 441
12.9 Correcting for (Pure) Autocorrelation:
The Method of Generalized Least
Squares (GLS) 442
When ? Is Known 442
When ? Is Not Known 443
12.10 The Newey?West Method of Correcting
the OLS Standard Errors 447
12.11 OLS versus FGLS and HAC 448
12.12 Additional Aspects of Autocorrelation 449
Dummy Variables and Autocorrelation 449
ARCH and GARCH Models 449
Coexistence of Autocorrelation
and Heteroscedasticity 450
12.13 A Concluding Example 450
Summary and Conclusions 452
Exercises 453
Appendix 12A 466
12A.1 Proof that the Error Term vt in
Equation (12.1.11) Is Autocorrelated 466
12A.2 Proof of Equations (12.2.3), (12.2.4),
and (12.2.5) 466
CHAPTER 13
Econometric Modeling: Model Specification
and Diagnostic Testing 467
13.1 Model Selection Criteria 468
13.2 Types of Specification Errors 468
13.3 Consequences of Model Specification
Errors 470
Underfitting a Model (Omitting a Relevant
Variable) 471
Inclusion of an Irrelevant Variable
(Overfitting a Model) 473
13.4 Tests of Specification Errors 474
Detecting the Presence of Unnecessary Variables
(Overfitting a Model) 475
Tests for Omitted Variables and Incorrect
Functional Form 477
13.5 Errors of Measurement 482
Errors of Measurement in the Dependent
Variable Y 482
Errors of Measurement in the Explanatory
Variable X 483
13.6 Incorrect Specification of the Stochastic
Error Term 486
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13.7 Nested versus Non-Nested Models 487
13.8 Tests of Non-Nested Hypotheses 488
The Discrimination Approach 488
The Discerning Approach 488
14.3 Estimating Nonlinear Regression Models:
The Trial-and-Error Method 527
14.4 Approaches to Estimating Nonlinear
Regression Models 529
13.9 Model Selection Criteria 493
The R2 Criterion 493
Adjusted R2 493
Akaike?s Information Criterion (AIC) 494
Schwarz?s Information Criterion (SIC) 494
Mallows?s Cp Criterion 494
A Word of Caution about Model
Selection Criteria 495
Forecast Chi-Square (?2) 496
13.10 Additional Topics in Econometric
Modeling 496
Outliers, Leverage, and Influence 496
Recursive Least Squares 498
Chow?s Prediction Failure Test 498
Missing Data 499
13.11 Concluding Examples 500
1. A Model of Hourly Wage Determination 500
2. Real Consumption Function for the United
States, 1947?2000 505
13.12 Non-Normal Errors and Stochastic
Regressors 509
Direct Search or Trial-and-Error
or Derivative-Free Method 529
Direct Optimization 529
Iterative Linearization Method 530
14.5 Illustrative Examples 530
Summary and Conclusions 535
Exercises 535
Appendix 14A 537
14A.1 Derivation of Equations (14.2.4)
and (14.2.5) 537
14A.2 The Linearization Method 537
14A.3 Linear Approximation of the Exponential
Function Given in Equation (14.2.2) 538
CHAPTER 15
Qualitative Response Regression Models 541
15.1 The Nature of Qualitative Response
Models 541
15.2 The Linear Probability Model (LPM) 543
Non-Normality of the Disturbances ui 544
Heteroscedastic Variances
of the Disturbances 544
Nonfulfillment of 0 = E(Yi | Xi) = 1 545
Questionable Value of R2 as a Measure
of Goodness of Fit 546
1. What Happens If the Error Term Is Not
Normally Distributed? 509
2. Stochastic Explanatory Variables 510
13.13 A Word to the Practitioner 511
Summary and Conclusions 512
Exercises 513
Appendix 13A 519
13A.1 The Proof that E(b1 2) = ?2 + ?3b3 2
[Equation (13.3.3)] 519
13A.2 The Consequences of Including an Irrelevant
Variable: The Unbiasedness Property 520
13A.3 The Proof of Equation (13.5.10) 521
13A.4 The Proof of Equation (13.6.2) 522
15.3
15.4
15.5
15.6
Applications of LPM 549
Alternatives to LPM 552
The Logit Model 553
Estimation of the Logit Model 555
Data at the Individual Level 556
Grouped or Replicated Data 556
15.7 The Grouped Logit (Glogit) Model: A
Numerical Example 558
Interpretation of the Estimated Logit
Model 558
PART THREE
TOPICS IN ECONOMETRICS 523
CHAPTER 14
Nonlinear Regression Models 525
14.1 Intrinsically Linear and Intrinsically
Nonlinear Regression Models 525
14.2 Estimation of Linear and Nonlinear
Regression Models 527
15.8 The Logit Model for Ungrouped
or Individual Data 561
15.9 The Probit Model 566
Probit Estimation with Grouped
Data: gprobit 567
The Probit Model for Ungrouped
or Individual Data 570
The Marginal Effect of a Unit Change
in the Value of a Regressor in the Various
Regression Models 571
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15.10 Logit and Probit Models 571
15.11 The Tobit Model 574
Illustration of the Tobit Model: Ray Fair?s Model
of Extramarital Affairs 575
15.12 Modeling Count Data: The Poisson
Regression Model 576
15.13 Further Topics in Qualitative Response
Regression Models 579
Ordinal Logit and Probit Models 580
Multinomial Logit and Probit Models 580
Duration Models 580
Summary and Conclusions 581
Exercises 582
Appendix 15A 589
15A.1 Maximum Likelihood Estimation of the Logit
and Probit Models for Individual (Ungrouped)
Data 589
CHAPTER 16
Panel Data Regression Models 591
16.1 Why Panel Data? 592
16.2 Panel Data: An Illustrative Example 593
16.3 Pooled OLS Regression or Constant
Coefficients Model 594
16.4 The Fixed Effect Least-Squares Dummy
Variable (LSDV) Model 596
A Caution in the Use of the Fixed Effect
LSDV Model 598
16.5 The Fixed-Effect Within-Group (WG)
Estimator 599
16.6 The Random Effects Model (REM) 602
Breusch and Pagan Lagrange
Multiplier Test 605
16.7
16.8
Properties of Various Estimators 605
Fixed Effects versus Random Effects Model:
Some Guidelines 606
16.9 Panel Data Regressions: Some Concluding
Comments 607
16.10 Some Illustrative Examples 607
Summary and Conclusions 612
Exercises 613
CHAPTER 17
Dynamic Econometric Models: Autoregressive
and Distributed-Lag Models 617
17.1 The Role of ?Time,?? or ?Lag,??
in Economics 618
17.2 The Reasons for Lags 622
17.3 Estimation of Distributed-Lag Models
623
Ad Hoc Estimation of Distributed-Lag
Models 623
17.4 The Koyck Approach to Distributed-Lag
Models 624
The Median Lag 627
The Mean Lag 627
17.5 Rationalization of the Koyck Model: The
Adaptive Expectations Model 629
17.6 Another Rationalization of the Koyck Model:
The Stock Adjustment, or Partial Adjustment,
Model 632
17.7 Combination of Adaptive Expectations
and Partial Adjustment Models 634
17.8 Estimation of Autoregressive Models 634
17.9 The Method of Instrumental
Variables (IV) 636
17.10 Detecting Autocorrelation in Autoregressive
Models: Durbin h Test 637
17.11 A Numerical Example: The Demand for
Money in Canada, 1979?I to 1988?IV 639
17.12 Illustrative Examples 642
17.13 The Almon Approach to Distributed-Lag
Models: The Almon or Polynomial Distributed
Lag (PDL) 645
17.14 Causality in Economics: The Granger
Causality Test 652
The Granger Test 653
A Note on Causality and Exogeneity 657
Summary and Conclusions 658
Exercises 659
Appendix 17A 669
17A.1 The Sargan Test for the Validity
of Instruments 669
PART FOUR
SIMULTANEOUS-EQUATION
MODELS AND TIME SERIES
ECONOMETRICS 671
CHAPTER 18
Simultaneous-Equation Models 673
18.1 The Nature of Simultaneous-Equation
Models 673
18.2 Examples of Simultaneous-Equation
Models 674
18.3 The Simultaneous-Equation Bias:
Inconsistency of OLS Estimators 679
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18.4 The Simultaneous-Equation Bias: A Numerical
Example 682
Summary and Conclusions 684
Exercises 684
CHAPTER 19
The Identification Problem 689
19.1 Notations and Definitions 689
19.2 The Identification Problem 692
Underidentification 692
Just, or Exact, Identification 694
Overidentification 697
19.3 Rules for Identification 699
The Order Condition of Identifiability 699
The Rank Condition of Identifiability 700
19.4 A Test of Simultaneity 703
Hausman Specification Test 703
19.5 Tests for Exogeneity 705
Summary and Conclusions 706
Exercises 706
CHAPTER 20
Simultaneous-Equation Methods 711
20.1 Approaches to Estimation 711
20.2 Recursive Models and Ordinary
Least Squares 712
20.3 Estimation of a Just Identified Equation: The
Method of Indirect Least Squares (ILS) 715
An Illustrative Example 715
Properties of ILS Estimators 718
20.4 Estimation of an Overidentified Equation:
The Method of Two-Stage Least Squares
(2SLS) 718
20.5 2SLS: A Numerical Example 721
20.6 Illustrative Examples 724
Summary and Conclusions 730
Exercises 730
Appendix 20A 735
20A.1 Bias in the Indirect Least-Squares
Estimators 735
20A.2 Estimation of Standard Errors of 2SLS
Estimators 736
CHAPTER 21
Time Series Econometrics:
Some Basic Concepts 737
21.1 A Look at Selected U.S. Economic Time
Series 738
21.2 Key Concepts 739
21.3 Stochastic Processes 740
Stationary Stochastic Processes 740
Nonstationary Stochastic Processes 741
21.4 Unit Root Stochastic Process 744
21.5 Trend Stationary (TS) and Difference
Stationary (DS) Stochastic Processes 745
21.6 Integrated Stochastic Processes 746
Properties of Integrated Series 747
21.7 The Phenomenon of Spurious
Regression 747
21.8 Tests of Stationarity 748
1. Graphical Analysis 749
2. Autocorrelation Function (ACF)
and Correlogram 749
Statistical Significance of Autocorrelation
Coefficients 753
21.9 The Unit Root Test 754
The Augmented Dickey?Fuller (ADF)
Test 757
Testing the Significance of More than One
Coefficient: The F Test 758
The Phi
