Introductory Econometrics: A Modern Approach, 7th Edition
By Jeffrey M. Wooldridge
Content:
Preface xii
About the Author xxii
chapter 1 The Nature of Econometrics
and Economic Data 1
1-1 What Is Econometrics? 1
1-2 Steps in Empirical Economic Analysis 2
1-3 The Structure of Economic Data 5
1-3a Cross-Sectional Data 5
1-3b Time Series Data 7
1-3c Pooled Cross Sections 8
1-3d Panel or Longitudinal Data 9
1-3e A Comment on Data Structures 10
1-4 Causality, Ceteris Paribus, and Counterfactual
Reasoning 10
Summary 14
Key Terms 15
Problems 15
Computer Exercises 15
Part 1
Regression Analysis with
Cross- Sectional Data 19
chapter 2 The Simple Regression Model 20
2-1 Definition of the Simple Regression Model 20
2-2 Deriving the Ordinary Least Squares Estimates 24
2-2a A Note on Terminology 31
2-3 Properties of OLS on Any Sample of Data 32
2-3a Fitted Values and Residuals 32
2-3b Algebraic Properties of OLS Statistics 32
2-3c Goodness-of-Fit 35
2-4 Units of Measurement and Functional Form 36
2-4a The Effects of Changing Units of Measurement
on OLS Statistics 36
2-4b Incorporating Nonlinearities in Simple
Regression 37
2-4c The Meaning of “Linear” Regression 40
2-5 Expected Values and Variances of the OLS
Estimators 40
2-5a Unbiasedness of OLS 40
2-5b Variances of the OLS Estimators 45
2-5c Estimating the Error Variance 48
2-6 Regression through the Origin and Regression on a Constant 50
2-7 Regression on a Binary Explanatory Variable 51
2-7a Counterfactual Outcomes, Causality, and Policy
Analysis 53
Summary 56
Key Terms 57
Problems 58
Computer Exercises 62
chapter 3 Multiple Regression Analysis: Estimation 66
3-1 Motivation for Multiple Regression 67
3-1a The Model with Two Independent Variables 67
3-1b The Model with k Independent Variables 69
3-2 Mechanics and Interpretation of Ordinary Least
Squares 70
3-2a Obtaining the OLS Estimates 70
3-2b Interpreting the OLS Regression Equation 71
3-2c On the Meaning of “Holding Other Factors Fixed”
in Multiple Regression 73
3-2d Changing More Than One Independent Variable
Simultaneously 74
3-2e OLS Fitted Values and Residuals 74
3-2f A “Partialling Out” Interpretation of Multiple
Regression 75
3-2g Comparison of Simple and Multiple Regression
Estimates 75
3-2h Goodness-of-Fit 76
3-2i Regression through the Origin 79
3-3 The Expected Value of the OLS Estimators 79
3-3a Including Irrelevant Variables in a Regression
Model 83
3-3b Omitted Variable Bias: The Simple Case 84
3-3c Omitted Variable Bias: More General Cases 87
3-4 The Variance of the OLS Estimators 87
3-4a The Components of the OLS Variances:
Multicollinearity 89
3-4b Variances in Misspecified Models 92
3-4c Estimating s2: Standard Errors of the OLS
Estimators 93
3-5 Efficiency of OLS: The Gauss-Markov
Theorem 95
3-6 Some Comments on the Language of Multiple
Regression Analysis 96
3-7 Several Scenarios for Applying Multiple
Regression 97
3-7a Prediction 98
3-7b Efficient Markets 98
3-7c Measuring the Tradeoff between Two
Variables 99
3-7d Testing for Ceteris Paribus Group
Differences 99
3-7e Potential Outcomes, Treatment Effects, and Policy
Analysis 100
Summary 102
Key Terms 104
Problems 104
Computer Exercises 109
chapter 4 Multiple Regression Analysis:
Inference 117
4-1 Sampling Distributions of the OLS
Estimators 117
4-2 Testing Hypotheses about a Single Population
Parameter: The t Test 120
4-2a Testing against One-Sided Alternatives 122
4-2b Two-Sided Alternatives 126
4-2c Testing Other Hypotheses about bj 128
4-2d Computing p-Values for t Tests 130
4-2e A Reminder on the Language of Classical
Hypothesis Testing 132
4-2f Economic, or Practical, versus Statistical Significance 132
4-3 Confidence Intervals 134
4-4 Testing Hypotheses about a Single Linear
Combination of the Parameters 136
4-5 Testing Multiple Linear Restrictions: The F Test 139
4-5a Testing Exclusion Restrictions 139
4-5b Relationship between F and t Statistics 144
4-5c The R-Squared Form of the F Statistic 145
4-5d Computing p-values for F Tests 146
4-5e The F Statistic for Overall Significance of a
Regression 147
4-5f Testing General Linear Restrictions 148
4-6 Reporting Regression Results 149
4-7 Revisiting Causal Effects and Policy
Analysis 151
Summary 152
Key Terms 154
Problems 154
Computer Exercises 159
chapter 5 Multiple Regression Analysis: OLS
Asymptotics 163
5-1 Consistency 164
5-1a Deriving the Inconsistency in OLS 167
5-2 Asymptotic Normality and Large Sample
Inference 168
5-2a Other Large Sample Tests: The Lagrange
Multiplier Statistic 172
5-3 Asymptotic Efficiency of OLS 175
Summary 176
Key Terms 176
Problems 176
Computer Exercises 178
chapter 6 Multiple Regression Analysis:
Further Issues 181
6-1 Effects of Data Scaling on OLS Statistics 181
6-1a Beta Coefficients 184
6-2 More on Functional Form 186
6-2a More on Using Logarithmic Functional
Forms 186
6-2b Models with Quadratics 188
6-2c Models with Interaction Terms 192
6-2d Computing Average Partial Effects 194
6-3 More on Goodness-of-Fit and Selection of
Regressors 195
6-3a Adjusted R-Squared 196
6-3b Using Adjusted R-Squared to Choose between
Nonnested Models 197
6-3c Controlling for Too Many Factors in Regression
Analysis 199
6-3d Adding Regressors to Reduce the Error
Variance 200
6-4 Prediction and Residual Analysis 201
6.4a Confidence Intervals for Predictions 201
6-4b Residual Analysis 205
6-4c Predicting y When log(y) Is the Dependent
Variable 205
6-4d Predicting y When the Dependent Variable Is
log(y) 207
Summary 209
Key Terms 211
Problems 211
Computer Exercises 214
chapter 7 Multiple Regression Analysis with
Qualitative Information 220
7-1 Describing Qualitative Information 221
7-2 A Single Dummy Independent
Variable 222
7-2a Interpreting Coefficients on Dummy
Explanatory Variables When the Dependent
Variable Is log(y) 226
7-3 Using Dummy Variables for Multiple
Categories 228
7-3a Incorporating Ordinal Information by Using
Dummy Variables 230
7-4 Interactions Involving Dummy Variables 232
7-4a Interactions among Dummy Variables 232
7-4b Allowing for Different Slopes 233
7-4c Testing for Differences in Regression Functions
across Groups 237
7-5 A Binary Dependent Variable: The Linear
Probability Model 239
7-6 More on Policy Analysis and Program
Evaluation 244
7-6a Program Evaluation and Unrestricted Regression
Adjustment 245
7-7 Interpreting Regression Results with Discrete
Dependent Variables 249
Summary 250
Key Terms 251
Problems 251
Computer Exercises 256
chapter 8 Heteroskedasticity 262
8-1 Consequences of Heteroskedasticity for OLS 262
8-2 Heteroskedasticity-Robust Inference after OLS
Estimation 263
8-2a Computing Heteroskedasticity-Robust LM
Tests 267
8-3 Testing for Heteroskedasticity 269
8-3a The White Test for Heteroskedasticity 271
8-4 Weighted Least Squares Estimation 273
8-4a The Heteroskedasticity Is Known up to a
Multiplicative Constant 273
8-4b The Heteroskedasticity Function
Must Be Estimated: Feasible GLS 278
8-4c What If the Assumed Heteroskedasticity Function Is
Wrong? 281
8-4d Prediction and Prediction Intervals with
Heteroskedasticity 283
8-5 The Linear Probability Model Revisited 284
Summary 286
Key Terms 287
Problems 287
Computer Exercises 290
chapter 9 More on Specification and
Data Issues 294
9-1 Functional Form Misspecification 295
9-1a RESET as a General Test for Functional
Form Misspecification 297
9-1b Tests against Nonnested Alternatives 298
9-2 Using Proxy Variables for Unobserved Explanatory
Variables 299
9-2a Using Lagged Dependent Variables as Proxy
Variables 303
9-2b A Different Slant on Multiple Regression 304
9-2c Potential Outcomes and Proxy Variables 305
9-3 Models with Random Slopes 306
9-4 Properties of OLS under Measurement Error 308
9-4a Measurement Error in the Dependent Variable 308
9-4b Measurement Error in an Explanatory
Variable 310
9-5 Missing Data, Nonrandom Samples, and Outlying
Observations 313
9-5a Missing Data 313
9-5b Nonrandom Samples 315
9-5c Outliers and Influential Observations 317
9-6 Least Absolute Deviations Estimation 321
Summary 323
Key Terms 324
Problems 324
Computer Exercises 328
Pa r t 2
Regression Analysis with Time
Series Data 333
chapter 10 Basic Regression Analysis with
Time Series Data 334
10-1 The Nature of Time Series Data 334
10-2 Examples of Time Series Regression
Models 335
10-2a Static Models 336
10-2b Finite Distributed Lag Models 336
10-2c A Convention about the Time Index 338
10-3 Finite Sample Properties of OLS under Classical
Assumptions 339
10-3a Unbiasedness of OLS 339
10-3b The Variances of the OLS Estimators and the
Gauss-Markov Theorem 342
10-3c Inference under the Classical Linear Model
Assumptions 344
10-4 Functional Form, Dummy Variables, and Index
Numbers 345
10-5 Trends and Seasonality 351
10-5a Characterizing Trending Time Series 351
10-5b Using Trending Variables in Regression
Analysis 354
10-5c A Detrending Interpretation of Regressions
with a Time Trend 356
10-5d Computing R-Squared When the Dependent
Variable Is Trending 357
10-5e Seasonality 358
Summary 360
Key Terms 361
Problems 361
Computer Exercises 363
chapter 11 Further Issues in Using OLS with
Time Series Data 366
11-1 Stationary and Weakly Dependent Time
Series 367
11-1a Stationary and Nonstationary Time Series 367
11-1b Weakly Dependent Time Series 368
11-2 Asymptotic Properties of OLS 370
11-3 Using Highly Persistent Time Series in Regression
Analysis 376
11-3a Highly Persistent Time Series 376
11-3b Transformations on Highly Persistent Time
Series 380
11-3c Deciding Whether a Time Series Is I(1) 381
11-4 Dynamically Complete Models and the Absence of
Serial Correlation 382
11-5 The Homoskedasticity Assumption for Time
Series Models 385
Summary 386
Key Terms 387
Problems 387
Computer Exercises 390
chapter 12 Serial Correlation and
Heteroskedasticity in Time Series
Regressions 394
12-1 Properties of OLS with Serially Correlated
Errors 395
12-1a Unbiasedness and Consistency 395
12-1b Efficiency and Inference 395
12-1c Goodness-of-Fit 396
12-1d Serial Correlation in the Presence
of Lagged Dependent Variables 396
12-2 Serial Correlation–Robust Inference
after OLS 398
12-3 Testing for Serial Correlation 401
12-3a A t Test for AR(1) Serial Correlation with
Strictly Exogenous Regressors 402
12-3b The Durbin-Watson Test under Classical
Assumptions 403
12-3c Testing for AR(1) Serial Correlation without
Strictly Exogenous Regressors 404
12-3d Testing for Higher-Order Serial Correlation 406
12-4 Correcting for Serial Correlation with Strictly
Exogenous Regressors 407
12-4a Obtaining the Best Linear Unbiased
Estimator in the AR(1) Model 408
12-4b Feasible GLS Estimation with AR(1)
Errors 409
12-4c Comparing OLS and FGLS 411
12-4d Correcting for Higher-Order Serial
Correlation 413
12-4e What if the Serial Correlation Model Is
Wrong? 413
12-5 Differencing and Serial Correlation 414
12-6 Heteroskedasticity in Time Series
Regressions 415
12-6a Heteroskedasticity-Robust Statistics 416
12-6b Testing for Heteroskedasticity 416
12-6c Autoregressive Conditional
Heteroskedasticity 417
12-6d Heteroskedasticity and Serial Correlation in
Regression Models 418
Summary 419
Key Terms 420
Problems 420
Computer Exercises 421
Pa r t 3
Advanced Topics 425
chapter 13 Pooling Cross Sections across
Time: Simple Panel Data Methods 426
13-1 Pooling Independent Cross Sections across
Time 427
13-1a The Chow Test for Structural Change across
Time 431
13-2 Policy Analysis with Pooled Cross Sections 431
13-2a Adding an Additional Control Group 436
13-2b A General Framework for Policy Analysis
with Pooled Cross Sections 437
13-3 Two-Period Panel Data Analysis 439
13-3a Organizing Panel Data 444
13-4 Policy Analysis with Two-Period Panel
Data 444
13-5 Differencing with More Than Two Time
Periods 447
13-5a Potential Pitfalls in First Differencing Panel
Data 451
Summary 451
Key Terms 452
Problems 452
Computer Exercises 453
chapter 14 Advanced Panel
Data Methods 462
14-1 Fixed Effects Estimation 463
14-1a The Dummy Variable Regression 466
14-1b Fixed Effects or First Differencing? 467
14-1c Fixed Effects with Unbalanced Panels 468
14-2 Random Effects Models 469
14-2a Random Effects or Pooled OLS? 473
14-2b Random Effects or Fixed Effects? 473
14-3 The Correlated Random Effects Approach 474
14-3a Unbalanced Panels 476
14-4 General Policy Analysis with Panel Data 477
14-4a Advanced Considerations with Policy
Analysis 478
14-5 Applying Panel Data Methods to Other Data
Structures 480
Summary 483
Key Terms 484
Problems 484
Computer Exercises 486
chapter 15 Instrumental Variables Estimation
and Two-Stage Least Squares 495
15-1 Motivation: Omitted Variables in a Simple
Regression Model 496
15-1a Statistical Inference with the IV
Estimator 500
15-1b Properties of IV with a Poor Instrumental
Variable 503
15-1c Computing R-Squared after IV Estimation 505
15-2 IV Estimation of the Multiple Regression
Model 505
15-3 Two-Stage Least Squares 509
15-3a A Single Endogenous Explanatory
Variable 509
15-3b Multicollinearity and 2SLS 511
15-3c Detecting Weak Instruments 512
15-3d Multiple Endogenous Explanatory
Variables 513
15-3e Testing Multiple Hypotheses after 2SLS
Estimation 513
15-4 IV Solutions to Errors-in-Variables Problems 514
15-5 Testing for Endogeneity and Testing Overidentifying
Restrictions 515
15-5a Testing for Endogeneity 515
15-5b Testing Overidentification Restrictions 516
15-6 2SLS with Heteroskedasticity 518
15-7 Applying 2SLS to Time Series Equations 519
15-8 Applying 2SLS to Pooled Cross Sections
and Panel Data 521
Summary 522
Key Terms 523
Problems 523
Computer Exercises 526
chapter 16 Simultaneous Equations
Models 534
16-1 The Nature of Simultaneous Equations
Models 535
16-2 Simultaneity Bias in OLS 538
16-3 Identifying and Estimating a Structural
Equation 539
16-3a Identification in a Two-Equation System 540
16-3b Estimation by 2SLS 543
16-4 Systems with More Than Two Equations 545
16-4a Identification in Systems with Three or More
Equations 545
16-4b Estimation 546
16-5 Simultaneous Equations Models with Time
Series 546
16-6 Simultaneous Equations Models with Panel
Data 549
Summary 551
Key Terms 552
Problems 552
Computer Exercises 555
chapter 17 Limited Dependent Variable Models
and Sample Selection Corrections 559
17-1 Logit and Probit Models for Binary
Response 560
17-1a Specifying Logit and Probit Models 560
17-1b Maximum Likelihood Estimation of Logit and
Probit Models 563
17-1c Testing Multiple Hypotheses 564
17-1d Interpreting the Logit and Probit Estimates 565
17-2 The Tobit Model for Corner Solution
Responses 571
17-2a Interpreting the Tobit Estimates 572
17-2b Specification Issues in Tobit Models 578
17-3 The Poisson Regression Model 578
17-4 Censored and Truncated Regression Models 582
17-4a Censored Regression Models 583
17-4b Truncated Regression Models 586
17-5 Sample Selection Corrections 588
17-5a When Is OLS on the Selected Sample
Consistent? 588
17-5b Incidental Truncation 589
Summary 593
Key Terms 593
Problems 594
Computer Exercises 596
chapter 18 Advanced Time Series Topics 604
18-1 Infinite Distributed Lag Models 605
18-1a The Geometric (or Koyck) Distributed Lag
Model 607
18-1b Rational Distributed Lag Models 608
18-2 Testing for Unit Roots 610
18-3 Spurious Regression 614
18-4 Cointegration and Error Correction Models 616
18-4a Cointegration 616
18-4b Error Correction Models 620
18-5 Forecasting 622
18-5a Types of Regression Models Used for
Forecasting 623
18-5b One-Step-Ahead Forecasting 624
18-5c Comparing One-Step-Ahead Forecasts 627
18-5d Multiple-Step-Ahead Forecasts 628
18-5e Forecasting Trending, Seasonal, and Integrated
Processes 631
Summary 635
Key Terms 636
Problems 636
Computer Exercises 638
chapter 19 Carrying Out an Empirical
Project 642
19-1 Posing a Question 642
19-2 Literature Review 644
19-3 Data Collection 645
19-3a Deciding on the Appropriate Data Set 645
19-3b Entering and Storing Your Data 646
19-3c Inspecting, Cleaning, and Summarizing Your
Data 647
19-4 Econometric Analysis 648
19-5 Writing an Empirical Paper 651
19-5a Introduction 651
19-5b Conceptual (or Theoretical)
Framework 652
19-5c Econometric Models and Estimation
Methods 652
19-5d The Data 654
19-5e Results 655
19.5f Conclusions 656
19-5g Style Hints 656
Summary 658
Key Terms 658
Sample Empirical Projects 658
List of Journals 664
Data Sources 665
Math Refresher A Basic Mathematical
Tools 666
A-1 The Summation Operator and Descriptive
Statistics 666
A-2 Properties of Linear Functions 668
A-3 Proportions and Percentages 671
A-4 Some Special Functions and Their
Properties 672
A-4a Quadratic Functions 672
A-4b The Natural Logarithm 674
A-4c The Exponential Function 677
A-5 Differential Calculus 678
Summary 680
Key Terms 681
Problems 681
Math Refresher B Fundamentals of
Probability 684
B-1 Random Variables and Their Probability
Distributions 684
B-1a Discrete Random Variables 685
B-1b Continuous Random Variables 687
B-2 Joint Distributions, Conditional Distributions,
and Independence 688
B-2a Joint Distributions and Independence 688
B-2b Conditional Distributions 690
B-3 Features of Probability Distributions 691
B-3a A Measure of Central Tendency: The Expected
Value 691
B-3b Properties of Expected Values 692
B-3c Another Measure of Central Tendency: The
Median 694
B-3d Measures of Variability: Variance and Standard
Deviation 695
B-3e Variance 695
B-3f Standard Deviation 696
B-3g Standardizing a Random Variable 696
B-3h Skewness and Kurtosis 697
B-4 Features of Joint and Conditional
Distributions 697
B-4a Measures of Association: Covariance and
Correlation 697
B-4b Covariance 697
B-4c Correlation Coefficient 698
B-4d Variance of Sums of Random Variables 699
B-4e Conditional Expectation 700
B-4f Properties of Conditional Expectation 702
B-4g Conditional Variance 704
B-5 The Normal and Related Distributions 704
B-5a The Normal Distribution 704
B-5b The Standard Normal Distribution 705
B-5c Additional Properties of the Normal
Distribution 707
B-5d The Chi-Square Distribution 708
B-5e The t Distribution 708
B-5f The F Distribution 709
Summary 711
Key Terms 711
Problems 711
Math Refresher C Fundamentals of
Mathematical Statistics 714
C-1 Populations, Parameters, and Random
Sampling 714
C-1a Sampling 714
C-2 Finite Sample Properties of Estimators 715
C-2a Estimators and Estimates 715
C-2b Unbiasedness 716
C-2c The Sampling Variance of Estimators 718
C-2d Efficiency 719
C-3 Asymptotic or Large Sample Properties of
Estimators 721
C-3a Consistency 721
C-3b Asymptotic Normality 723
C-4 General Approaches to Parameter Estimation 724
C-4a Method of Moments 725
C-4b Maximum Likelihood 725
C-4c Least Squares 726
C-5 Interval Estimation and Confidence Intervals 727
C-5a The Nature of Interval Estimation 727
C-5b Confidence Intervals for the Mean from a Normally
Distributed Population 729
C-5c A Simple Rule of Thumb for a 95% Confidence
Interval 731
C-5d Asymptotic Confidence Intervals for Nonnormal
Populations 732
C-6 Hypothesis Testing 733
C-6a Fundamentals of Hypothesis Testing 733
C-6b Testing Hypotheses about the Mean in a Normal
Population 735
C-6c Asymptotic Tests for Nonnormal
Populations 738
C-6d Computing and Using p-Values 738
C-6e The Relationship between Confidence Intervals
and Hypothesis Testing 741
C-6f Practical versus Statistical Significance 742
C-7 Remarks on Notation 743
Summary 743
Key Terms 744
Problems 744
Advanced Treatment D Summary of Matrix
Algebra 749
D-1 Basic Definitions 749
D-2 Matrix Operations 750
D-2a Matrix Addition 750
D-2b Scalar Multiplication 750
D-2c Matrix Multiplication 751
D-2d Transpose 752
D-2e Partitioned Matrix Multiplication 752
D-2f Trace 753
D-2g Inverse 753
D-3 Linear Independence and Rank of a
Matrix 754
D-4 Quadratic Forms and Positive Definite
Matrices 754
D-5 Idempotent Matrices 755
D-6 Differentiation of Linear and Quadratic
Forms 755
D-7 Moments and Distributions of Random
Vectors 756
D-7a Expected Value 756
D-7b Variance-Covariance Matrix 756
D-7c Multivariate Normal Distribution 756
D-7d Chi-Square Distribution 757
D-7e t Distribution 757
D-7f F Distribution 757
Summary 757
Key Terms 757
Problems 758
Advanced Treatment E The Linear Regression
Model in Matrix Form 760
E-1 The Model and Ordinary Least
Squares Estimation 760
E-1a The Frisch-Waugh Theorem 762
E-2 Finite Sample Properties of OLS 763
E-3 Statistical Inference 767
E-4 Some Asymptotic Analysis 769
E-4a Wald Statistics for Testing Multiple
Hypotheses 771
Summary 771
Key Terms 771
Problems 772
Answers to Going Further Questions 775
Statistical Tables 784
References 791
Glossary 797
Index 812
About the Author xxii
chapter 1 The Nature of Econometrics
and Economic Data 1
1-1 What Is Econometrics? 1
1-2 Steps in Empirical Economic Analysis 2
1-3 The Structure of Economic Data 5
1-3a Cross-Sectional Data 5
1-3b Time Series Data 7
1-3c Pooled Cross Sections 8
1-3d Panel or Longitudinal Data 9
1-3e A Comment on Data Structures 10
1-4 Causality, Ceteris Paribus, and Counterfactual
Reasoning 10
Summary 14
Key Terms 15
Problems 15
Computer Exercises 15
Part 1
Regression Analysis with
Cross- Sectional Data 19
chapter 2 The Simple Regression Model 20
2-1 Definition of the Simple Regression Model 20
2-2 Deriving the Ordinary Least Squares Estimates 24
2-2a A Note on Terminology 31
2-3 Properties of OLS on Any Sample of Data 32
2-3a Fitted Values and Residuals 32
2-3b Algebraic Properties of OLS Statistics 32
2-3c Goodness-of-Fit 35
2-4 Units of Measurement and Functional Form 36
2-4a The Effects of Changing Units of Measurement
on OLS Statistics 36
2-4b Incorporating Nonlinearities in Simple
Regression 37
2-4c The Meaning of “Linear” Regression 40
2-5 Expected Values and Variances of the OLS
Estimators 40
2-5a Unbiasedness of OLS 40
2-5b Variances of the OLS Estimators 45
2-5c Estimating the Error Variance 48
2-6 Regression through the Origin and Regression on a Constant 50
2-7 Regression on a Binary Explanatory Variable 51
2-7a Counterfactual Outcomes, Causality, and Policy
Analysis 53
Summary 56
Key Terms 57
Problems 58
Computer Exercises 62
chapter 3 Multiple Regression Analysis: Estimation 66
3-1 Motivation for Multiple Regression 67
3-1a The Model with Two Independent Variables 67
3-1b The Model with k Independent Variables 69
3-2 Mechanics and Interpretation of Ordinary Least
Squares 70
3-2a Obtaining the OLS Estimates 70
3-2b Interpreting the OLS Regression Equation 71
3-2c On the Meaning of “Holding Other Factors Fixed”
in Multiple Regression 73
3-2d Changing More Than One Independent Variable
Simultaneously 74
3-2e OLS Fitted Values and Residuals 74
3-2f A “Partialling Out” Interpretation of Multiple
Regression 75
3-2g Comparison of Simple and Multiple Regression
Estimates 75
3-2h Goodness-of-Fit 76
3-2i Regression through the Origin 79
3-3 The Expected Value of the OLS Estimators 79
3-3a Including Irrelevant Variables in a Regression
Model 83
3-3b Omitted Variable Bias: The Simple Case 84
3-3c Omitted Variable Bias: More General Cases 87
3-4 The Variance of the OLS Estimators 87
3-4a The Components of the OLS Variances:
Multicollinearity 89
3-4b Variances in Misspecified Models 92
3-4c Estimating s2: Standard Errors of the OLS
Estimators 93
3-5 Efficiency of OLS: The Gauss-Markov
Theorem 95
3-6 Some Comments on the Language of Multiple
Regression Analysis 96
3-7 Several Scenarios for Applying Multiple
Regression 97
3-7a Prediction 98
3-7b Efficient Markets 98
3-7c Measuring the Tradeoff between Two
Variables 99
3-7d Testing for Ceteris Paribus Group
Differences 99
3-7e Potential Outcomes, Treatment Effects, and Policy
Analysis 100
Summary 102
Key Terms 104
Problems 104
Computer Exercises 109
chapter 4 Multiple Regression Analysis:
Inference 117
4-1 Sampling Distributions of the OLS
Estimators 117
4-2 Testing Hypotheses about a Single Population
Parameter: The t Test 120
4-2a Testing against One-Sided Alternatives 122
4-2b Two-Sided Alternatives 126
4-2c Testing Other Hypotheses about bj 128
4-2d Computing p-Values for t Tests 130
4-2e A Reminder on the Language of Classical
Hypothesis Testing 132
4-2f Economic, or Practical, versus Statistical Significance 132
4-3 Confidence Intervals 134
4-4 Testing Hypotheses about a Single Linear
Combination of the Parameters 136
4-5 Testing Multiple Linear Restrictions: The F Test 139
4-5a Testing Exclusion Restrictions 139
4-5b Relationship between F and t Statistics 144
4-5c The R-Squared Form of the F Statistic 145
4-5d Computing p-values for F Tests 146
4-5e The F Statistic for Overall Significance of a
Regression 147
4-5f Testing General Linear Restrictions 148
4-6 Reporting Regression Results 149
4-7 Revisiting Causal Effects and Policy
Analysis 151
Summary 152
Key Terms 154
Problems 154
Computer Exercises 159
chapter 5 Multiple Regression Analysis: OLS
Asymptotics 163
5-1 Consistency 164
5-1a Deriving the Inconsistency in OLS 167
5-2 Asymptotic Normality and Large Sample
Inference 168
5-2a Other Large Sample Tests: The Lagrange
Multiplier Statistic 172
5-3 Asymptotic Efficiency of OLS 175
Summary 176
Key Terms 176
Problems 176
Computer Exercises 178
chapter 6 Multiple Regression Analysis:
Further Issues 181
6-1 Effects of Data Scaling on OLS Statistics 181
6-1a Beta Coefficients 184
6-2 More on Functional Form 186
6-2a More on Using Logarithmic Functional
Forms 186
6-2b Models with Quadratics 188
6-2c Models with Interaction Terms 192
6-2d Computing Average Partial Effects 194
6-3 More on Goodness-of-Fit and Selection of
Regressors 195
6-3a Adjusted R-Squared 196
6-3b Using Adjusted R-Squared to Choose between
Nonnested Models 197
6-3c Controlling for Too Many Factors in Regression
Analysis 199
6-3d Adding Regressors to Reduce the Error
Variance 200
6-4 Prediction and Residual Analysis 201
6.4a Confidence Intervals for Predictions 201
6-4b Residual Analysis 205
6-4c Predicting y When log(y) Is the Dependent
Variable 205
6-4d Predicting y When the Dependent Variable Is
log(y) 207
Summary 209
Key Terms 211
Problems 211
Computer Exercises 214
chapter 7 Multiple Regression Analysis with
Qualitative Information 220
7-1 Describing Qualitative Information 221
7-2 A Single Dummy Independent
Variable 222
7-2a Interpreting Coefficients on Dummy
Explanatory Variables When the Dependent
Variable Is log(y) 226
7-3 Using Dummy Variables for Multiple
Categories 228
7-3a Incorporating Ordinal Information by Using
Dummy Variables 230
7-4 Interactions Involving Dummy Variables 232
7-4a Interactions among Dummy Variables 232
7-4b Allowing for Different Slopes 233
7-4c Testing for Differences in Regression Functions
across Groups 237
7-5 A Binary Dependent Variable: The Linear
Probability Model 239
7-6 More on Policy Analysis and Program
Evaluation 244
7-6a Program Evaluation and Unrestricted Regression
Adjustment 245
7-7 Interpreting Regression Results with Discrete
Dependent Variables 249
Summary 250
Key Terms 251
Problems 251
Computer Exercises 256
chapter 8 Heteroskedasticity 262
8-1 Consequences of Heteroskedasticity for OLS 262
8-2 Heteroskedasticity-Robust Inference after OLS
Estimation 263
8-2a Computing Heteroskedasticity-Robust LM
Tests 267
8-3 Testing for Heteroskedasticity 269
8-3a The White Test for Heteroskedasticity 271
8-4 Weighted Least Squares Estimation 273
8-4a The Heteroskedasticity Is Known up to a
Multiplicative Constant 273
8-4b The Heteroskedasticity Function
Must Be Estimated: Feasible GLS 278
8-4c What If the Assumed Heteroskedasticity Function Is
Wrong? 281
8-4d Prediction and Prediction Intervals with
Heteroskedasticity 283
8-5 The Linear Probability Model Revisited 284
Summary 286
Key Terms 287
Problems 287
Computer Exercises 290
chapter 9 More on Specification and
Data Issues 294
9-1 Functional Form Misspecification 295
9-1a RESET as a General Test for Functional
Form Misspecification 297
9-1b Tests against Nonnested Alternatives 298
9-2 Using Proxy Variables for Unobserved Explanatory
Variables 299
9-2a Using Lagged Dependent Variables as Proxy
Variables 303
9-2b A Different Slant on Multiple Regression 304
9-2c Potential Outcomes and Proxy Variables 305
9-3 Models with Random Slopes 306
9-4 Properties of OLS under Measurement Error 308
9-4a Measurement Error in the Dependent Variable 308
9-4b Measurement Error in an Explanatory
Variable 310
9-5 Missing Data, Nonrandom Samples, and Outlying
Observations 313
9-5a Missing Data 313
9-5b Nonrandom Samples 315
9-5c Outliers and Influential Observations 317
9-6 Least Absolute Deviations Estimation 321
Summary 323
Key Terms 324
Problems 324
Computer Exercises 328
Pa r t 2
Regression Analysis with Time
Series Data 333
chapter 10 Basic Regression Analysis with
Time Series Data 334
10-1 The Nature of Time Series Data 334
10-2 Examples of Time Series Regression
Models 335
10-2a Static Models 336
10-2b Finite Distributed Lag Models 336
10-2c A Convention about the Time Index 338
10-3 Finite Sample Properties of OLS under Classical
Assumptions 339
10-3a Unbiasedness of OLS 339
10-3b The Variances of the OLS Estimators and the
Gauss-Markov Theorem 342
10-3c Inference under the Classical Linear Model
Assumptions 344
10-4 Functional Form, Dummy Variables, and Index
Numbers 345
10-5 Trends and Seasonality 351
10-5a Characterizing Trending Time Series 351
10-5b Using Trending Variables in Regression
Analysis 354
10-5c A Detrending Interpretation of Regressions
with a Time Trend 356
10-5d Computing R-Squared When the Dependent
Variable Is Trending 357
10-5e Seasonality 358
Summary 360
Key Terms 361
Problems 361
Computer Exercises 363
chapter 11 Further Issues in Using OLS with
Time Series Data 366
11-1 Stationary and Weakly Dependent Time
Series 367
11-1a Stationary and Nonstationary Time Series 367
11-1b Weakly Dependent Time Series 368
11-2 Asymptotic Properties of OLS 370
11-3 Using Highly Persistent Time Series in Regression
Analysis 376
11-3a Highly Persistent Time Series 376
11-3b Transformations on Highly Persistent Time
Series 380
11-3c Deciding Whether a Time Series Is I(1) 381
11-4 Dynamically Complete Models and the Absence of
Serial Correlation 382
11-5 The Homoskedasticity Assumption for Time
Series Models 385
Summary 386
Key Terms 387
Problems 387
Computer Exercises 390
chapter 12 Serial Correlation and
Heteroskedasticity in Time Series
Regressions 394
12-1 Properties of OLS with Serially Correlated
Errors 395
12-1a Unbiasedness and Consistency 395
12-1b Efficiency and Inference 395
12-1c Goodness-of-Fit 396
12-1d Serial Correlation in the Presence
of Lagged Dependent Variables 396
12-2 Serial Correlation–Robust Inference
after OLS 398
12-3 Testing for Serial Correlation 401
12-3a A t Test for AR(1) Serial Correlation with
Strictly Exogenous Regressors 402
12-3b The Durbin-Watson Test under Classical
Assumptions 403
12-3c Testing for AR(1) Serial Correlation without
Strictly Exogenous Regressors 404
12-3d Testing for Higher-Order Serial Correlation 406
12-4 Correcting for Serial Correlation with Strictly
Exogenous Regressors 407
12-4a Obtaining the Best Linear Unbiased
Estimator in the AR(1) Model 408
12-4b Feasible GLS Estimation with AR(1)
Errors 409
12-4c Comparing OLS and FGLS 411
12-4d Correcting for Higher-Order Serial
Correlation 413
12-4e What if the Serial Correlation Model Is
Wrong? 413
12-5 Differencing and Serial Correlation 414
12-6 Heteroskedasticity in Time Series
Regressions 415
12-6a Heteroskedasticity-Robust Statistics 416
12-6b Testing for Heteroskedasticity 416
12-6c Autoregressive Conditional
Heteroskedasticity 417
12-6d Heteroskedasticity and Serial Correlation in
Regression Models 418
Summary 419
Key Terms 420
Problems 420
Computer Exercises 421
Pa r t 3
Advanced Topics 425
chapter 13 Pooling Cross Sections across
Time: Simple Panel Data Methods 426
13-1 Pooling Independent Cross Sections across
Time 427
13-1a The Chow Test for Structural Change across
Time 431
13-2 Policy Analysis with Pooled Cross Sections 431
13-2a Adding an Additional Control Group 436
13-2b A General Framework for Policy Analysis
with Pooled Cross Sections 437
13-3 Two-Period Panel Data Analysis 439
13-3a Organizing Panel Data 444
13-4 Policy Analysis with Two-Period Panel
Data 444
13-5 Differencing with More Than Two Time
Periods 447
13-5a Potential Pitfalls in First Differencing Panel
Data 451
Summary 451
Key Terms 452
Problems 452
Computer Exercises 453
chapter 14 Advanced Panel
Data Methods 462
14-1 Fixed Effects Estimation 463
14-1a The Dummy Variable Regression 466
14-1b Fixed Effects or First Differencing? 467
14-1c Fixed Effects with Unbalanced Panels 468
14-2 Random Effects Models 469
14-2a Random Effects or Pooled OLS? 473
14-2b Random Effects or Fixed Effects? 473
14-3 The Correlated Random Effects Approach 474
14-3a Unbalanced Panels 476
14-4 General Policy Analysis with Panel Data 477
14-4a Advanced Considerations with Policy
Analysis 478
14-5 Applying Panel Data Methods to Other Data
Structures 480
Summary 483
Key Terms 484
Problems 484
Computer Exercises 486
chapter 15 Instrumental Variables Estimation
and Two-Stage Least Squares 495
15-1 Motivation: Omitted Variables in a Simple
Regression Model 496
15-1a Statistical Inference with the IV
Estimator 500
15-1b Properties of IV with a Poor Instrumental
Variable 503
15-1c Computing R-Squared after IV Estimation 505
15-2 IV Estimation of the Multiple Regression
Model 505
15-3 Two-Stage Least Squares 509
15-3a A Single Endogenous Explanatory
Variable 509
15-3b Multicollinearity and 2SLS 511
15-3c Detecting Weak Instruments 512
15-3d Multiple Endogenous Explanatory
Variables 513
15-3e Testing Multiple Hypotheses after 2SLS
Estimation 513
15-4 IV Solutions to Errors-in-Variables Problems 514
15-5 Testing for Endogeneity and Testing Overidentifying
Restrictions 515
15-5a Testing for Endogeneity 515
15-5b Testing Overidentification Restrictions 516
15-6 2SLS with Heteroskedasticity 518
15-7 Applying 2SLS to Time Series Equations 519
15-8 Applying 2SLS to Pooled Cross Sections
and Panel Data 521
Summary 522
Key Terms 523
Problems 523
Computer Exercises 526
chapter 16 Simultaneous Equations
Models 534
16-1 The Nature of Simultaneous Equations
Models 535
16-2 Simultaneity Bias in OLS 538
16-3 Identifying and Estimating a Structural
Equation 539
16-3a Identification in a Two-Equation System 540
16-3b Estimation by 2SLS 543
16-4 Systems with More Than Two Equations 545
16-4a Identification in Systems with Three or More
Equations 545
16-4b Estimation 546
16-5 Simultaneous Equations Models with Time
Series 546
16-6 Simultaneous Equations Models with Panel
Data 549
Summary 551
Key Terms 552
Problems 552
Computer Exercises 555
chapter 17 Limited Dependent Variable Models
and Sample Selection Corrections 559
17-1 Logit and Probit Models for Binary
Response 560
17-1a Specifying Logit and Probit Models 560
17-1b Maximum Likelihood Estimation of Logit and
Probit Models 563
17-1c Testing Multiple Hypotheses 564
17-1d Interpreting the Logit and Probit Estimates 565
17-2 The Tobit Model for Corner Solution
Responses 571
17-2a Interpreting the Tobit Estimates 572
17-2b Specification Issues in Tobit Models 578
17-3 The Poisson Regression Model 578
17-4 Censored and Truncated Regression Models 582
17-4a Censored Regression Models 583
17-4b Truncated Regression Models 586
17-5 Sample Selection Corrections 588
17-5a When Is OLS on the Selected Sample
Consistent? 588
17-5b Incidental Truncation 589
Summary 593
Key Terms 593
Problems 594
Computer Exercises 596
chapter 18 Advanced Time Series Topics 604
18-1 Infinite Distributed Lag Models 605
18-1a The Geometric (or Koyck) Distributed Lag
Model 607
18-1b Rational Distributed Lag Models 608
18-2 Testing for Unit Roots 610
18-3 Spurious Regression 614
18-4 Cointegration and Error Correction Models 616
18-4a Cointegration 616
18-4b Error Correction Models 620
18-5 Forecasting 622
18-5a Types of Regression Models Used for
Forecasting 623
18-5b One-Step-Ahead Forecasting 624
18-5c Comparing One-Step-Ahead Forecasts 627
18-5d Multiple-Step-Ahead Forecasts 628
18-5e Forecasting Trending, Seasonal, and Integrated
Processes 631
Summary 635
Key Terms 636
Problems 636
Computer Exercises 638
chapter 19 Carrying Out an Empirical
Project 642
19-1 Posing a Question 642
19-2 Literature Review 644
19-3 Data Collection 645
19-3a Deciding on the Appropriate Data Set 645
19-3b Entering and Storing Your Data 646
19-3c Inspecting, Cleaning, and Summarizing Your
Data 647
19-4 Econometric Analysis 648
19-5 Writing an Empirical Paper 651
19-5a Introduction 651
19-5b Conceptual (or Theoretical)
Framework 652
19-5c Econometric Models and Estimation
Methods 652
19-5d The Data 654
19-5e Results 655
19.5f Conclusions 656
19-5g Style Hints 656
Summary 658
Key Terms 658
Sample Empirical Projects 658
List of Journals 664
Data Sources 665
Math Refresher A Basic Mathematical
Tools 666
A-1 The Summation Operator and Descriptive
Statistics 666
A-2 Properties of Linear Functions 668
A-3 Proportions and Percentages 671
A-4 Some Special Functions and Their
Properties 672
A-4a Quadratic Functions 672
A-4b The Natural Logarithm 674
A-4c The Exponential Function 677
A-5 Differential Calculus 678
Summary 680
Key Terms 681
Problems 681
Math Refresher B Fundamentals of
Probability 684
B-1 Random Variables and Their Probability
Distributions 684
B-1a Discrete Random Variables 685
B-1b Continuous Random Variables 687
B-2 Joint Distributions, Conditional Distributions,
and Independence 688
B-2a Joint Distributions and Independence 688
B-2b Conditional Distributions 690
B-3 Features of Probability Distributions 691
B-3a A Measure of Central Tendency: The Expected
Value 691
B-3b Properties of Expected Values 692
B-3c Another Measure of Central Tendency: The
Median 694
B-3d Measures of Variability: Variance and Standard
Deviation 695
B-3e Variance 695
B-3f Standard Deviation 696
B-3g Standardizing a Random Variable 696
B-3h Skewness and Kurtosis 697
B-4 Features of Joint and Conditional
Distributions 697
B-4a Measures of Association: Covariance and
Correlation 697
B-4b Covariance 697
B-4c Correlation Coefficient 698
B-4d Variance of Sums of Random Variables 699
B-4e Conditional Expectation 700
B-4f Properties of Conditional Expectation 702
B-4g Conditional Variance 704
B-5 The Normal and Related Distributions 704
B-5a The Normal Distribution 704
B-5b The Standard Normal Distribution 705
B-5c Additional Properties of the Normal
Distribution 707
B-5d The Chi-Square Distribution 708
B-5e The t Distribution 708
B-5f The F Distribution 709
Summary 711
Key Terms 711
Problems 711
Math Refresher C Fundamentals of
Mathematical Statistics 714
C-1 Populations, Parameters, and Random
Sampling 714
C-1a Sampling 714
C-2 Finite Sample Properties of Estimators 715
C-2a Estimators and Estimates 715
C-2b Unbiasedness 716
C-2c The Sampling Variance of Estimators 718
C-2d Efficiency 719
C-3 Asymptotic or Large Sample Properties of
Estimators 721
C-3a Consistency 721
C-3b Asymptotic Normality 723
C-4 General Approaches to Parameter Estimation 724
C-4a Method of Moments 725
C-4b Maximum Likelihood 725
C-4c Least Squares 726
C-5 Interval Estimation and Confidence Intervals 727
C-5a The Nature of Interval Estimation 727
C-5b Confidence Intervals for the Mean from a Normally
Distributed Population 729
C-5c A Simple Rule of Thumb for a 95% Confidence
Interval 731
C-5d Asymptotic Confidence Intervals for Nonnormal
Populations 732
C-6 Hypothesis Testing 733
C-6a Fundamentals of Hypothesis Testing 733
C-6b Testing Hypotheses about the Mean in a Normal
Population 735
C-6c Asymptotic Tests for Nonnormal
Populations 738
C-6d Computing and Using p-Values 738
C-6e The Relationship between Confidence Intervals
and Hypothesis Testing 741
C-6f Practical versus Statistical Significance 742
C-7 Remarks on Notation 743
Summary 743
Key Terms 744
Problems 744
Advanced Treatment D Summary of Matrix
Algebra 749
D-1 Basic Definitions 749
D-2 Matrix Operations 750
D-2a Matrix Addition 750
D-2b Scalar Multiplication 750
D-2c Matrix Multiplication 751
D-2d Transpose 752
D-2e Partitioned Matrix Multiplication 752
D-2f Trace 753
D-2g Inverse 753
D-3 Linear Independence and Rank of a
Matrix 754
D-4 Quadratic Forms and Positive Definite
Matrices 754
D-5 Idempotent Matrices 755
D-6 Differentiation of Linear and Quadratic
Forms 755
D-7 Moments and Distributions of Random
Vectors 756
D-7a Expected Value 756
D-7b Variance-Covariance Matrix 756
D-7c Multivariate Normal Distribution 756
D-7d Chi-Square Distribution 757
D-7e t Distribution 757
D-7f F Distribution 757
Summary 757
Key Terms 757
Problems 758
Advanced Treatment E The Linear Regression
Model in Matrix Form 760
E-1 The Model and Ordinary Least
Squares Estimation 760
E-1a The Frisch-Waugh Theorem 762
E-2 Finite Sample Properties of OLS 763
E-3 Statistical Inference 767
E-4 Some Asymptotic Analysis 769
E-4a Wald Statistics for Testing Multiple
Hypotheses 771
Summary 771
Key Terms 771
Problems 772
Answers to Going Further Questions 775
Statistical Tables 784
References 791
Glossary 797
Index 812