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Introductory econometrics: a modern approach
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INTERNATIONAL STUDENT EDITION
I n t r o d u c t o r y
E c o n o m e t r ic s
A M o d e r n A p p r o a c h
Third Edition
THIRD EDITION
INTRODUCTORY
ECONOMETRICS
A MODERN APPROACH
JEFFREY M. WOOLDRIDGE
Michigan State University
THOMSON
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Brief Contents
Chapter I The Nature of Econometrics and Economic Data 1
PART 1: Regression Analysis with Cross-Sectional Data 23
Chapter 2 The Simple Regression Model 24
Chapter 3 Multiple Regression Analysis: Estimation 73
Chapter 4 Multiple Regression Analysis: Inference 123
Chapter 5 Multiple Regression Analysis: OLS Asymptotics 176
Chapter 6 Multiple Regression Analysis: Further Issues 192
Chapter 7 Multiple Regression Analysis with Qualitative Information:
Binary (or Dummy) Variables 230
Chapter 8 Heteroskedasticity 271
Chapter 9 More on Specification and Data Problems 304
PART 2: Regression Analysis with Time Series Data 341
Chapter 10 Basic Regression Analysis with Time Series Data 342
Chapter 11 Further Issues in Using OLS with Time Series Data 380
Chapter 12 Serial Correlation and Heteroskedasticity in Time Series Regressions 412
PART 3: Advanced Topics 447
Chapter 13 Pooling Cross Sections Across Time: Simple Panel Data Methods 448
Chapter 14 Advanced Panel Data Methods 485
Chapter 15 Instrumental Variables Estimation and Two Stage Least Squares 510
Chapter 16 Simultaneous Equations Models 552
Chapter 17 Limited Dependent Variable Models and Sample Selection Corrections 582
Chapter 18 Advanced Time Series Topics 632
Chapter 19 Carrying out an Empirical Project 678
A P PEN D IC ES_______________________________________________________________________________
Appendix A Basic Mathematical Tools 707
Appendix B Fundamentals of Probability 728
Appendix C Fundamentals of Mathematical Statistics 763
Appendix D Summary of Matrix Algebra 808
Appendix E The Linear Regression Model in Matrix Form 819
Appendix F Answers to Chapter Questions 834
Appendix G Statistical Tables 847
References 854
Glossary 859
Index 873
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Contents
C H A PT ER 1
The Nat h rc' ot Econometrics
and Economic Data 1
1.1 What Is Econom etricsI
1.2 Steps in Empirical Economic Analysis 2
1.3 The Structure ot’ Economic Data 5
Cross-Sectional Data 6
Time Series Data 8
Pooled Cross Sections 10
Panel or Longitudinal Data 10
,4 Comment on Data Structures I *
1.4 Causality and the Notion ot Ceteris Paribus
in Econometric Analysis 13
Summary 18
Key Terms 19
Problems 19
Computer Exercises 20
Regression Analysis with
Cross-Sectional Data 23
C H A PTER 2
The Simple Regression Model 24
2.1 Definition of the Simple Regression Model 24
2.2 Deriving the Ordinary Least Squares
Estimates 29
.4 Note on Terminology 38
2.3 Properties of OLS on Any Sample of Data 38
Fitted Values and Residuals 39
Algebraic Properties of OLS Statistics 40
Goodness-of-Fit 42
2.4 Units of Measurement and Functional Form 44
The Effects of Changing Units of Measurement
on OLS Statistics 44
Incorporating Xonlinearitics in SimpU
Regrcssiim 46
The Meaning oJ "Linear" Regression 4w
2.5 Expected Values and Variances of the OLS
Estimators 50
L'nbiasedness of OLS 50
Miriam es of the OI.S Estimators 56
Estimating the Error Variance 60
2.6 Regression through the Origin 63
Summary 64
Key Terms 65
Problems 66
Computer Exercises 69
Appendix 2A 71
C H A PTER 3
Multiple Regression Analysis: Estimation ” 3
3.1 Motivation for Multiple Regression 73
The Model with Two Independent Variables 73
The Model with k Independent Variables 76
3.2 Mechanics and Interpretation of Ordinary Least
Squares 78
Obtaining the OLS Estimates 7H
Interpreting the OLS Regression Equation 80
On the Meaning of "Holding Other Factors
Fixed" in Multiple Regression 82
Changing More than One Independent Variable
Simultaneously 82
OLS Fitted Values and Residuals 83
.4 “Partialling Out " Interpretation of Multiple
Regression 83
Comparison of Simple and Multiple Regression
Estimates 84
Goodness-of-Fit 85
Regression through the Origin 88
3.3 The Expected Value of the OLS Estimators 89
Including Irrelevant Variables in a Regression
Model 9 4
Omitted Variable Bias: The Simple Case 95
PART 1
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Contents v
Omitted Variable Bias: More General
Cases 98
3.4 The Variance of the OLS Estimators 99
The Components of the OLS Variances:
Multicollinearity 1 01
Variances in Misspecified Models 105
Estimating a 2: Standard Errors of the OLS
Estimators 106
3.5 Efficiency of OLS: The Gauss-Markov
Theorem 108
Summary 109
Key Terms 111
Problems 111
Computer Exercises 116
Appendix 3A 119
C H A PT ER 4
M ultiple Regression Analysis: Inference 123
4.1 Sampling Distributions
of the OLS Estimators 123
4.2 Testing Hypotheses about a Single
Population Parameter: The t Test 126
Testing against One-Sided Alternatives 129
Two-Sided Alternatives 134
Testing Other Hypotheses about (3j 136
Computing p-Values for t Tests 139
A Reminder on the Language of Classical
Hypothesis Testing 142
Economic, or Practical, versus Statistical
Significance 142
4.3 Confidence Intervals 145
4.4 Testing Hypotheses about a Single Linear
Combination of the Parameters 147
4.5 Testing Multiple Linear Restrictions:
The F Test 150
Testing Exclusion Restrictions 150
Relationship between F and t Statistics 157
The R-Squared Form of the F Statistic 158
Computing p-Values for F Tests 159
The F Statistic fo r Overall Significance
of a Regression 160
Testing General Linear Restrictions 161
4.6 Reporting Regression Results 163
Summary 165
Key Terms 167
Problems 168
Computer Exercises 173
C H A PTER 5
Multiple Regression Analysis:
O LS Asymptotics 1 76
5.1 Consistency 176
Deriving the Inconsistency in OLS 179
5.2 Asymptotic Normality and Large
Sample Inference 181
Other Large Sample Tests: The Lagrange
Multiplier Statistic 185
5.3 Asymptotic Efficiency of OLS 187
Summary 189
Key Terms 189
Problems 190
Computer Exercises 190
Appendix 5A 191
C H A PT ER 6
M ultiple Regression Analysis:
Further Issues 192
6.1 Effects of Data Scaling on OLS Statistics 192
Beta Coefficients 195
6.2 More on Functional Form 197
More on Using Logarithmic Functional
Forms 197
Models with Quadratics 200
Models with Interaction Terms 204
6.3 More on Goodness-of-Fit
and Selection of Regressors 206
Adjusted R -Squared 208
Using Adjusted R -Squared to Choose
between Nonnested Models 209
Controlling for Too Many Factors
in Regression Analysis 211
Adding Regressors to Reduce the Error
Variance 213
6.4 Prediction and Residual Analysis 214
Confidence Intervals for Predictions 214
Residual Analysis 217
Predicting y When log(y) Is the Dependent
Variable 218
Summary 221
Key Terms 222
Problems 222
Computer Exercises 224
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VI Contents
C H A PTER 7
Multiple* Regression Analysis
with Qualitative Information:
Binary (or Dum my) Variables 230
7.1 Describing Qualitative Information 230
7.2 A Single Dummy Independent Variable 232
Interpreting Coefficients on Dummy Explanatory
Variables When the Dependent Variable
Islog(y) 237
7.3 Using Dummy Variables
for Multiple Categories 239
Incorporating Ordinal Information
by Using Dummy Variables 240
7.4 Interactions Involving Dummy Variables 244
Interactions among Dummy Variables 244
Allowing for Different Slopes 245
Testing for Diff erences in Regre s sion Functions
across Groups 249
7.5 A Binary Dependent Variable:
The Linear Probability Model 252
7.6 More on Policy Analysis
and Program Evaluation 258
Summary 260
Key Terms 261
Problems 261
Computer Exercises 265
C H A PTER 8
Heteroskedasticity 271
8.1 Consequences of Heteroskedasticity
for OLS 271
8.2 Heteroskedasticity-Robust Inference
after OLS Estimation 272
Computing Heteroskedasticity-Robust
LM Tests 276
8.3 Testing for Heteroskedasticity 278
The White Test for Heteroskedasticity 282
8.4 Weighted Least Squares Estimation 284
The Heteroskedasticity Is Known
up to a Multiplicative Constant 284
The Heteroskedasticity Function Must
Be Estimated: Feasible GLS 290
8.5 The Linear Probability Model Revisited 295
Summary 297
Key Terms 298
Problems 298
Computer Exercises 300
C H A PTER 9
More on Specification
and Data Problems 304
9.1 Functional Form Misspecification 3(U
RESET as a General Test for Flint tional
Form Misspecification 308
Tests against Nonnested Altematix es 3m
9.2 Using Proxy Variables for Unobserved
Explanatory Variables 310
Using Lagged Dependent Variables
as Proxy Variables 315
A Different Slant on Multiple Regression 31'
9.3 Properties of OLS under Measurement
Error 318
Measurement Error in the Dependent
Variable 3/8
Measurement Error in tin Explanatory
Variable 321
9.4 Missing Data. Nonrandom Samples,
and Outlying Observations 325
Missing Data 325
Nonrandom Samples 326
Outliers and Influential Observ ations 328
Summary 333
Key Terms 334
Problems 334
Computer Exercises 336
m PART 2
Regression Analysis with
Time Series Data 341
C H A PT ER 10
Basic Regression Analysis
with Time Series Data 342
10.1 The Nature of Time Series Data 342
10.2 Examples of Time Series Regression
Models 344
Static Models 344
Finite Distributed Lag Models 344
A Convention about the Time Index 34
10.3 Finite Sample Properties of OLS
under Classical Assumptions 347
Unbiasedness of OLS 34 7
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Contents
The Variances of the OLS Estimators
and the Gauss-Markov Theorem 351
Inference under the Classical Linear Model
Assumptions 353
10.4 Functional Form. Dummy Variables,
and Index Numbers 355
10.5 Trends and Seasonality 363
Characterizing Trending Time Series 363
Using Trending Variables in Regression
Analysis 3 66
A Detrending Interpretation of Regre ssions
with a Time Trend 368
Computing R-Squared When the Dependent
Variable Is Trending 369
Seasonality 371
Summary 373
Key Terms 375
Problems 375
Computer Exercises 377
C H A PT ER 11
Further Issues in Using O LS
with Time Series Data 380
11.1 Stationary and Weakly Dependent Time
Series 380
Stationary and Nonstationary Time Series 381
Weakly Dependent Time Series 382
11.2 Asymptotic Properties of OLS 385
11.3 Using Highly Persistent Time Series
in Regression Analysis 392
Highly Persistent Time Series 392
Transformations on Highly Persistent Time
Series 397
Deciding Whether a Time Series Is l( I ) 398
11.4 Dynamically Complete Models
and the Absence of Serial Correlation 400
11.5 The Homoskedasticity Assumption
for Time Series Models 403
Summary 403
Key Terms 405
Problems 405
Computer Exercises 408
C H A PT ER 12
Serial Correlation and Heteroskedasticity
in Time Series Regressions 412
12.1 Properties of OLS with Serially Correlated
Errors 412
I nbiasedness and Consistency 412
Efficiency and Inference 413
Goodness-of-Fit 414
Serial Correlation in the Presence
nf Lagged Dependent Variables 415
12.2 Testing for Serial Correlation 416
A t Test for ARl /) Serial Correlation
with Strictly Exogenous Regressors 416
The Durbin-Watson Test under Classical
Assumptions 419
Testing for AR( 1) Serial Correlation
without Strictly Exogenous Regressors 420
Testing for Higher Order Serial
Correlation 422
12.3 Correcting for Serial Correlation
with Strictly Exogenous Regressors 424
Obtaining the Best Linear Unbiased Estimator
in the AR( I ) Model 424
Feasible GLS Estimation with AR( I )
Errors 425
Comparing OLS and FG LS 428
Correcting for Higher Order Serial
Correlation 430
12.4 Differencing and Serial Correlation 431
12.5 Serial Correlation-Robust
Inference after OLS 432
12.6 Heteroskedasticity in Time Series
Regressions 436
Heteroskedasticity -Robust Statistics 436
Testing for Heteroskedasticity 436
Autoregressive Conditional
He te roskedas tici ty 438
Heteroskedasticity and Serial Correlation
in Regression Models 440
Summary 441
Key Terms 442
Problems 442
Computer Exercises 443
PART 3
Advanced Topics 447
C H A PT ER 13
Pooling Cross Sections across Time:
Simple Panel Data Methods 448
13,1 Pooling Independent Cross Sections
across Time 449
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VIII Contents
The Chow Test for Structural Change
across Time 454
13.2 Policy Analysis with Pooled Cross
Sections 454
13.3 Two-Period Panel Data Analysis 460
Organizing Panel Data 467
13.4 Policy Analysis with Two-Period Panel
Data 467
13.5 Differencing with More Than
Two Time Periods 470
Potential Pitfalls in First-Differencing Panel
Data 475
Summary 476
Key Terms 476
Problems 476
Computer Exercises 478
Appendix 13A 483
C H A PTER 14
Advanced Panel Data Methods 485
14.1 Fixed Effects Estimation 485
The Dummy Variable Regression 489
Fixed Effects or First Differencing ? 491
Fixed Effects with Unbalanced Panels 492
14.2 Random Effects Models 493
Random Effects or Fixed Effects? 497
14.3 Applying Panel Data Methods
to Other Data Structures 498
Summary 500
Key Terms 501
Problems 501
Computer Exercises 503
Appendix 14A 507
C H A PT ER 15
Instrumental Variables Estimation
and Two Stage Least Squares 510
15.1 Motivation: Omitted Variables in a Simple
Regression Model 511
Statistical Inference with the IV Estimator 514
Properties of IV with a Poor Instrumental
Variable 519
Computing R -Squared after IV Estimation 520
15.2 IV Estimation of the Multiple Regression
Model 521
15.3 Two Stage Least Squares 525
A Single Endogenous Explanatory
Variable 525
Multicollinearity and 2SLS 528
Multiple Endogenous Explanatory
Variables 528
Testing Multiple Hypotheses after 2SLS
Estimation 529
15.4 IV Solutions to Errors-in-Variables
Problems 530
15.5 Testing for Endogeneity and Testing
0\eridentify ing Restrictions 532
Testing for Endogeneity 532
Testing Overidentification Restrictions 533
15.6 2SLS w ith Heteroskedasticity 535
15.7 Apply ing 2SLS to Time Series Equations 536
15.8 Apply ing 2SLS to Pooled Cross Sections
and Panel Data 538
Summary 540
Key Terms 541
Problems 541
Computer Exercises 544
Appendix 15A 549
C H A PT ER 16
Simultaneous Equations Models 552
16.1 The Nature of Simultaneous
Equations Models 552
16.2 Simultaneity Bias in OLS 557
16.3 Identifying and Estimating
a Structural Equation 559
Identification in a Two-Equation System 559
Estimation by 2SLS 564
16.4 Sy stems with More Than Two Equations 565
Identification in Systems with Three
or More Equations 566
Estimation 567
16.5 Simultaneous Equations Models
w ith Time Series 567
16.6 Simultaneous Equations Models
w ith Panel Data 571
Summary 573
Key Terms 574
Problems 574
Computer Exercises 577
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Contents
C H A PT ER 17
Limited Dependent Variable Models
and Sample Selection Corrections 582
17.1 Logit and Prohit Models for Binary
Response 583
Specifying Logit and Prohit Models 583
Maximum Likelihood Estimation of Logit and
Prohit Models 586
Testing Multiple Hypotheses 587
Interpreting the Logit and Prohit
Estimates 588
17.2 The Tobit Model
for Corner Solution Responses 595
Interpreting the Tobit Estimates 597
Specification Issues in Tobit Models 602
17.3 The Poisson Regression Model 604
17.4 Censored and Truncated Regression
Models 609
Censored Regression Models 610
Truncated Regression Models 613
17.5 Sample Selection Corrections 616
When Is OLS on the Selected Sample
Consistent? 616
Incidental Truncation 618
Summary 622
Key Terms 623
Problems 624
Computer Exercises 625
Appendix 17A 630
C H A PT ER 18
Advanced Time Series Topics 632
18.1 Infinite Distributed Lag Models 633
The Geometric (or Koyck) Distributed Lag
635
Rational Distributed Lag Models 637
18.2 Testing for Unit Roots 639
18.3 Spurious Regression 645
18.4 Cointegration and Error Correction Models 647
Cointegration 6 47
Error Correction Models 652
18.5 Forecasting 654
Types of Regression Models Used
for Forecasting 656
One-Step-Ahead Forecasting 657
Comparing One-Step-Ahead Forecasts 661
Multiple-Step-Ahead Forecasts 662
Forecasting Trending. Seasonal, and Integrated
Processes 665
Summary 670
Key Terms 671
Problems 671
Computer Exercises 674
C H A PT ER 19
Carrying Out an Empirical Project 678
19.1 Posing a Question 678
19.2 Literature Review 680
19.3 Data Collection 681
Deciding on the Appropriate Data Set 681
Entering and Storing Your Data 682
Inspecting. Cleaning, and Summarizing Your
Data 684
19.4 Econometric Analysis 685
19.5 Writing an Empirical Paper 689
Introduction 689
Conceptual (or Theoretical) Framework 689
Econometric Models and Estimation
Methods 690
The Data 692
Results 693
Conclusions 694
Sty le Hints 694
Summary 697
Key Terms 697
Sample Empirical Projects 698
List of Journals 703
Data Sources 704
APPENDICES
A PPE N D IX A
Basic Mathem atical Tools 707
A. I The Summation Operator
and Descriptive Statistics 707
A.2 Properties of Linear Functions 710
A.3 Proportions and Percentages 713
A.4 Some Special Functions and Their
Properties 714
Quadratic Functions 715
The Natural Logarithm 717
The Exponential Function 721
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