Introduction to econometrics : 2nd edition / James H. Stock, Mark W. Watson.

Contents

Preface xxvii

Part One Introduction and Review 1

Chapter 1 Economic Questions and Data 3
1.1 Economic Questions We Examine 4
Question #1: Does Reducing Class Size Improve Elementary School
Education? 4
Question #2: Is There Racial Discrimination in the Market for Home Loans? 5
Question #3: How Much Do Cigarette Taxes Reduce Smoking? 5
Question #4: What Will the Rate of Inflation Be Next Year? 6
Quantitative Questions, Quantitative Answers 7
1.2 Causal Effects and Idealized Experiments 8
Estimation of Causal Effects 8
Forecasting and Causality 9
1.3 Data: Sources and Types 10
Experimental versus Observational Data 10
Cross-Sectional Data 11
Time Series Data 11
Panel Data 13

Chapter 2 Review of Probability 17

2.1 Random Variables and Probability Distributions 18
Probabilities, the Sample Space, and Random Variables 18
Probability Distribution of a Discrete Random Variable 19
Probability Distribution of a Continuous Random Variable 21
2.2 Expected Values, Mean, and Variance 23
The Expected Value of a Random Variable 23
The Standard Deviation and Variance 24
Mean and Variance of a Linear Function of a Random Variable 25
Other Measures of the Shape of a Distribution 26
2.3 Two Random Variables 29
Joint and Marginal Distributions 29
Conditional Distributions 30
Independence 34
Covariance and Correlation 34
The Mean and Variance of Sums of Random Variables 32
2.4 The Normal, Chi-Squared,Student t and F Distributions 39
The Normal Distribution 39
The Chi-Squared Distributions 43
The Student t Distribution 44
F Distributions 44
2.5 Random Sampling and the Distribution of the Sample Average 45
Random Sampling 45
The Sampling Distribution of the Sample Average 46
2.6 Large-Sample Approximations to Sampling Distributions 48
The Law of Large Numbers and Consistency 49
The Central Limit Theorem 52
Appendix 2.1 Derivation of Results in Key Concept 2.3 63

Chapter 3 Review of Statistics 65

3.1 Estimation of the Population Mean 66
Estimators and Their Properties 67
Properties of C 68
The Importance of Random Sampling 70
3.2 Hypothesis Tests Concerning the Population Mean 71
Null and Alternative Hypotheses 71
The p-Value 71
Calculating the p-Value When sY Is Known 74
The Sample Variance, Sample Standard Deviation, and Standard Error 75
Calculating the p-Value When sY Is Unknown 76
The t-Statistic 77
Hypothesis Testing with a Prespecified Significance Level 78
One-Sided Alternatives 80
3.3 Confidence Intervals for the Population Mean 81
3.4 Comparing Means from Different Populations 83
Hypothesis Tests for the Difference Between Two Means 83
Confidence Intervals for the Difference Between Two Population Means 84
3.5 Differences-ofMeans Estimation of Causal Effects
Using Experimental Data 85
The Causal Effect as a Difference of Conditional Expectations 85
Estimation of the Causal Efect Using Differences of Means 87
3.6 Unsing the t-Statistic When the Sample Size Is Small 88
Scatterplots, the Sample Covariance, and the Sample Correlation 92
Scatterplots 93
Sample Covariance and Correlation 94
Appendix 3.1 The U.S. Current Population Survey 105
Appendix 3.2 Two Proofs That Is the Least Squares Estimator of mY 106
Appendix 3.3 A Proof That the Sample Variance Is Consistent 107

Part Two Fundamentals of Regression Analysis 109

Chapter 4 Linear Regression with One Regressor 111

4.1 The Linear Regression Model 112
4.2 Estimating the Coefficients of the Linear Regression Model 116
The Ordinary Least Squares Estimator 118
OLS Estimates of the Relationship Between Test Scores and the Student-Teacher
Ratio 120
Why Use the OLS Estimator? 121
4.3 Measure of Fit 123
The R2 123
The Standard Error of the Regression 124
Application to the Test Score Data 125
4.4 The Least Squares Assumptions 126
Assumption #1: The Conditional Distribution
of ui Given Xi Has a Mean of Zero 126
Assumption #2: (Xi,Yi), i 5 1, . . . , n Are Independently and
Identically Distributed 128
Assumption #3: Large Outliers Are Im;ole;y 129
Use of the Least Squares Assumptions 130
The Sampling Distribution of the OLS Estimators 132
4.5 Testing Hypotheses About One of the Regression Coefficients 110
Two-Sided Hypotheses Concerning b1 111
One-Sided Hypotheses Concerning b1 114
Testing Hypotheses About the Intercept b0 116
4.6 Conclusion 135

Appendix 4.1 The California Test Score Data Set 143
Appendix 4.2 Derivation of the OLS Estimators 143
Appendix 4.3 Sampling Distribution of the OLS Estimator 144

Chapter 5 Regression with a Single Regressors: Hypothesis Tests and
Confidence Intervals 148
5.1 Testing Hypotheses About One of the Regression Coefficients
149
Two-Sides Conerning B1 149
One-Sided Hypotheses Concerning B1 153
Testing Hypotheses About the Intercept B0 155
5.2 Confidence Intervals for a Regression Coefficient 155
5.3 Regression When X Is a Binary Variable 158
Interpretation of the RegressionCoefficients 158
5.4 Heteroskedasticity and Homoskedasticity 160
What are Heteroskedasticity and Homoskedasticity? 160
Mathematical Implications of Homoskedasticity? 163
What Does This Mean in Practice? 164
5.5 The Theorectical Foundations of Ordinary Least Squares 166
Linear Conditionlly Unbiased Estimators and the
Gauss-Markov Theorem 167
Regression Estimators Other Than OLS 168
5.6 Using the t-Statistic in Regression
When the Sample Size Is Small 169
The t-Statistic and the Student t Distribution 170
Use of the Student t Distribution in Pratice 171
5.7 Conclusion 171
Appendix 5.1 Formulas for OLS Standard Errorsr 180
Heteroskedasticity-Robust Standard Errors 180
Homoskedassticity-Only Variances 181
Homoskedasticity-Only Standard Errors 181

Appendix 5.2 The Gauss-Markiv Conditions and a Proof of the Gauss-Markov
Theorem 182
IThe Gauss-Markov Conditions 182
The OLS Estimator B1 Is a Linear Conditionally Unbiased Estimator. 183
Proof of the Gquss-Markov Theorem 183
The Gauss-Markov Theorem When X is Nonrandom 184
The Sample Average is the Efficient Linear Estimator of E(Y)

Chapter 6 Linear Regression with Multiple Regressors 186

6.1 Omitted Variable Bias 186
Definition of Omitted Variable Bias 187
A Formula for Omitted Variable Bias 189
Addressing Omitted Variable Bias by Dividing the Data into Groups 191
6.2 The Multiple Regression Model 193
The Population Regression Line 193
The Population Multiple Regression Model 194
6.3 The OLS Estimator in Multiple Regression 196
The OLS Estimator 197
Application to Test Scores and the Student-Teacher Ratio 198
6.4 Measures of Fit in Multiple Regression 200
The Standard Error of the Regression (SER) 200
The R2 200
The "Adusted R2" 201
Application to Test Scores 202
6.5 The Least Squares Assumptions in Multiple Regression 202
Assumpion #1: The Conditional Distribution of ui Given X1i, X2i, . . . , Xki
Has a Mean of Zero 203
Assumption #2: (X1i, X2i, . . . , Xki, Yi), i 5 1, . . . , n Are i.i.d. 203
Assumption #3: Large Outliers Are Unlikely 203
Assumption #4: No Perfect Multicollinearity 203

6.6 The Distribution of the OLS Estimators in Multiple Regression 205
6.7 Multicollinearity 206
Examples of Perfect Multicollinearity 206
Imperfect Multicollinearity 209
6.8 Conclusion 210
Appendix 5.1 Derivation of Equation (6.1) 218
Appendix 5.2 Distribution of the OLS Estimators When There Are Two Regressors and
Homoskedastic Errors 218

Chapter 7 Hypothesis Tests and Confidence Intervals in Multiple Regression 220

7.1 Hypothesis Tests and Confidence Intervals for a Single Coefficient 221
Standard Errors for the OLS Estimators 221
Hypothesis Tests for a Single Coefficient 221
Confidence Intervals for a Single Coefficient 223
Application to Test Scores and the Student-Teacher Ratio 229
The Homoskedasticity-Only F-Statistic 230
7.2 Tests of Joint Hypotheses 225
Testing Hypotheses on Two or More Coefficients 225
The F-Statistic 227
Polynomial and Logarithmic Models of Test Scores and District Income 216
7.3 Testing Single Restrictions
Involving Multiple Coefficients 232
7.4 Confidence Sets for Multiple Coefficients 230
7.5 Model Specification for Multiple Regression 235
Omitted Variable Bias in Multiple Regression 236
Model Specification in Theory and in Practice 236
Interpreting the R2 and the Adjusted R2 in Practice 237
7.6 Analysis of the Test Score Data Set 239
7.7 Conclusion 244
Appendix 7.1 The Bonerroni Test of a Joint Hypotheses 251
Bonferroni's Inequality 251
Bonifferroni Tests 252
Application to Test Scores 253

Chapter 8 Nonlinear Regression Functions 254

8.1 A General Strategy for Modeling Nonlinear Regression Functions 256
Test Scores and District Income 256
The Effect on Y of a Change in X in Nonlinear Specifications 260
A General Approach to Modeling Nonlinearities Using Multiple Regression 264
8.2 Nonlinear Functions of a Single Independent Variable 264
Polynomials 265
Logarithms 267
Polynomial and Logarithmic Models of Test Scores and District Income 275
8.3 Interactions Between Independent Variables 277
Interactions Between Two Binary Variables 277
Interactions Between a Continuous and a Binary Variable 280
Interactions Between Two Continuous Variables 286
8.4 Nonlinear Effects on Test Scores of the Student-Teacher Ratio 290
Discussion of Regression Results 291
Summary of Findings 295
8.5 Conclusion 296
Appendix 8.1 Regression Functions That Are Nonlinear in the Parameters 307
Functions That Are Nonlinear in the Parameters 307
Nonlinear Least Squares Estimation 309
Application to the Tet Score-Income Relation 310

Chapter 9 Assessing Studies Based on Multiple Regression 312

9.1 Internal and External Validity 313
Threats to Internal Validity 313
Threats to External Validity 314
9.2 Threats to Internal Validity of Multiple Regression Analysis 316
Omitted Variable Bias 316
Misspecification of the Functional Form of the Regression Function 319
Errors-in-Variables 319
Sample Selection 322
Simultaneous Causality 324
Sources of Inconsistency of OLS Standard Errors 325
9.3 Internal and External ValidityWhen the Regression Is Used for Forecasting 327
Using Regression Models for Forecasting 327
Assessing the Validity of Regression Models for Forecasting 328
9.4 Example: Test Scores and Class Size 329
External Validity 329
Internal Validity 336
Discussion and Implications 337
9.5 Conclusion 338
Appendix 9.1 The Massachusetts Elementary School Testing Data 344

Part Three Further Topics in Regression Analysis 347

Chapter 10 Regression with Panel Data 349

10.1 Panel Data 350
Example: Traffic Deaths and Alcohol Taxes 351
10.2 Panel Data with Two Time Periods: "Before and After"
Comparisons 353
10.3 Fixed Effects Regression 356
The Fixed Effects Regression Model 356
Estimation and Inference 359
Application to Traffic Deaths 360
10.4 Regression with Time Fixed Effects 361
Time Effects Only 361
Both Entity and Time and Fixed Effects 362
10.5 The Fixed Effects Regression Assumptions and Standard Errors
for Fixed Effects Regresstion 364
The Fixed Effects Regression Assumptions 364
Standard Errors for Fixed Effects Regression 366
10.6 Drunk Driving Laws and Traffic Deaths 367
10.7 Conclusion 371
Appendix 10.1 The State Traffic Fatality Data Set 378
Appendix 10.2 Standard Errors for Fixed Effects Regression with Serially Correlated
Errors 379

Chapter 11 Regression with a Binary Dependent Variable 383

11.1 Binary Dependent Variables and the Linear Probability Model 384
Binary Dependent Variables 385
The Linear Probability Model 387
11.2 Probit and Logit Regression 389
Probit Regression 389
Logit Regression 394
Comparing the Linear Probability, Probit, and Logit Models 396
11.3 Estimation and Inference in the Logit and Probit Models 396
Nonlinear Least Squares Estimation 397
Maximum Likelihood Estimation 398
Measures of Fit 399
11.4 Application to the Boston HMDA Data 400
11.5 Summary 408
Appendix 11.1 The Boston HMDA Data Set 415
Appendix 11.2 Maximum Likelihood Estimation 415
Appendix 11.3 Other Limited Dependent Variable Models 418

Chapter 12 Instrumental Variables Regression 421

12.1 The IV Estimator with a Single Regressor and a Single Instrument 422
The IV Model and Assumptions 422
The Two-Stage Least Squares Estimator 423
Why Does IV Regression Work? 424
The Sampling Distribution of the TSLS Estimator 428
Application to the Demand for Cigarettes 430
12.2 The General IV Regression Model 432
TSLS in the General IV Model 433
Instrument Relevance and Exogeneity in the General IV Model 434
The IV Regression Assumptions and Sampling Distribution of the TSLS
Estimator 434
Inference Using the TSLS Estimator 437
Application to the Demand for Cigarettes 437
12.3 Checking Instrument Validity 439
Assumption #1: Instrument Relevance 439
Assumption #2: Instrument Exogeneity 443
12.4 Application to the Demand for Cigarettes 445
12.5 Where Do Valid Instruments Come From? 450
Three Examples 451
12.6 Conclusion 455
Appendix 12.1 The Cigarette Consumption Panel Data Set 462
Appendix 12.2 Derivation of the Formula for the TSLS Estimator in Equation
(12.4) 462
Appendix 12.3 Large-Sample Distribution of the TSLS Estimator 463
Appendix 12.4 Large-Sample Distribution of the TSLS Estimator When the Instrument
Is Not Valid 464
Appendix 12.5 Instrumental Variables Analysis with Weak Instruments 466



Chapter 13 Experiments and Quasi-Experiments 468
13.1 Idealized Experiments and Causal Effects 470
Ideal Randomized Controlled Experiments 470
The Differences Estimator 471
13.2 Potential Problems with Experiments in Practice 472
Threats to Internal Validity 472
Threats to External Validity 475
13.3 Regression Estimators of Causal Effects Using Experimental Data 477
The Differences Estimator with Additional Regressors 477
The Differences-in-Differences Estimator 480
Estimation of Causal Effects for Different Groups 484
Estimation When There Is Partial Compliance 484
Testing for Randomization 485
13.4 Experimental Estimates of the Effect of Class Size Reductions 486
Experimental Design 486
Analysis of the STAR Data 487
Comparison of the Observational and Experimental Estimates of Class Size Effects 492
13.5 Quasi-Experiments 494
Examples 495
Econometric Methods for Analyzing Quasi-Experiments 497
13.6 Potential Problems with Quasi-Experiments 500
Threats to Internal Validity 500
Threats to External Validity 502
13.7 Experimental and Quasi-Experimental Estimates in Heterogeneous
Populations 502
Population Heterogeneity: Whose Causal Effect? 502
OLS with Heterogeneous Causal Effects 503
IV Regression with Heterogeneous Causal Effects 504
13.8 Conclusion 507
Appendix 13.1 The Project STAR Data Set 516
Appendix 13.2 Extension of the Differences-in-Differences Estimator to Multiple Time
Periods 517
Appendix 13.3 Conditional Mean Independence 518
Appendix 13.4 IV Estimation When the Causal Effect Varies Across Individuals 520
Part Four Regression Analysis of Economic Time Series Data 523


Chapter 14 Introduction to Time Series Regression and Forecasting 525

14.1 Using Regression Models for Forecasting 527
14.2 Introduction to Time Series Data and Serial Correlation 528
The Rates of Inflation and Unemployment in the United States 528
Lags, First Differences, Logarithms, and Growth Rates 528
Autocorrelation 532
Other Examples of Economic Time Series 533
14.3 Autoregressions 535
The First Order Autoregressive Model 535
The pth Order Autoregressive Model 538
14.4 Time Series Regression with Additional Predictors and the Autoregressive
Distributed Lag Model 540
Forecasting Changes in the Inflation Rate Using Past Unemployment Rates 540
Stationarity 544
Time Series Regression with Multiple Predictors 544
Forecast Uncertainty and Forecast Intervals 548
14.5 Lag Length Selection Using Information Criteria 549
Determining the Order of an Autoregression 551
Lag Length Selection in Time Series Regression with Multiple Predictors 553
14.6 Nonstationarity I: Trends 554
What Is a Trend? 555
Problems Caused by Stochastic Trends 557
Detecting Stochastic Trends: Testing for a Unit AR Root 560
Avoiding the Problems Caused by Stochastic Trends 565
14.7 Nonstationarity II: Breaks 565
What Is a Break? 565
Testing for Breaks 566
Pseudo Out-of-Sample Forecasting 571
Avoiding the Problems Caused by Breaks 576
14.8 Conclusion 577
Appendix 14.1 Time Series Data Used in Chapter 14 586
Appendix 14.2 Stationarity in the AR(1) Model 586
Appendix 14.3 Lag Operator Notation 588
Appendix 14.4 ARMA Models 589
Appendix 14.5 Consistency of the BIC Lag Length Estimator 589


Chapter 15 Estimation of Dynamic Causal Effects 591

15.1 An Initial Taste of the Orange Juice Data 592
15.2 Dynamic Causal Effects 595
Causal Effects and Time Series Data 596
Two Types of Exogeneity 598
15.3 Estimation of Dynamic Causal Effects with Exogenous Regressors 600
The Distributed Lag Model Assumptions 601
Autocorrelated ut, Standard Errors, and Inference 601
Dynamic Multipliers and Cumulative Dynamic Multipliers 602
15.4 Heteroskedasticity- and Autocorrelation-Consistent Standard
Errors 604
Distribution of the OLS Estimator with Autocorrelated Errors 604
HAC Standard Errors 606
15.5 Estimation of Dynamic Causal Effects with Strictly Exogenous Regressors 608
The Distributed Lag Model with AR(1) Errors 609
OLS Estimation of the ADL Model 612
GLS Estimation 613
The Distributed Lag Model with Additional Lags and AR(p) Errors 615
15.6 Orange Juice Prices and Cold Weather 618
15.7 Is Exogeneity Plausible? Some Examples 624
U.S. Income and Australian Exports 625
Oil Prices and Inflation 626
Monetary Policy and Inflation 626
The Phillips Curve 627
15.8 Conclusion 627
Appendix 15.1 The Orange Juice Data Set 634
Appendix 15.2 The ADL Model and Generalized Least Squares in Lag Operator
Notation 634

Chapter 16 Additional Topics in Time Series Regression 533

16.1 Vector Autoregressions 534
The VAR Model 534
A VAR Model of the Rates of Inflation and Unemployment 537
16.2 Multiperiod Forecasts 538
Multiperiod Forecasting: Univariate Autoregressions 539
Multiperiod Forecasting: Multivariate Forecasts 542
Which Method Should You Use? 544
16.3 Orders of Integration and Another Unit Root Test 545
Other Models of Trends and Orders of Integration 546
The DF-GLS Test for a Unit Root 547
Why Do Unit Root Tests Have Nonnormal Distributions? 551
16.4 Cointegration 552
Cointegration and Error Correction 552
How Can You Tell Whether Two Variables Are Cointegrated? 554
Estimation of Cointegrating Coefficients 556
Extension to Multiple Cointegrated Variables 558
Application to Interest Rates 559
16.5 Conditional Heteroskedasticity 561
Volatility Clustering 562
Autoregressive Conditional Heteroskedasticity 563
Application to Inflation Forecasts 564
16.6 Conclusion 566
Appendix 16.1 U.S. Financial Data Used in Chapter 14 569

Part Five The Econometric Theory of Regression Analysis 571

Chapter 17 The Theory of Linear Regression with One Regressor 573
17.1 The Extended Least Squares Assumptions and the OLS Estimator 575

The Extended Least Squares Assumptions 575
The OLS Estimator 576
17.2 Fundamentals of Asymptotic Distribution Theory 577
Convergence in Probability and the Law of Large Numbers 577
The Central Limit Theorem and Convergence in Distribution 580
Slutsky's Theorem and the Continuous Mapping Theorem 581
Application to the t-Statistic Based on the Sample Mean 582
17.3 Asymptotic Distribution of the OLS Estimator and t-Statistic 582
Consistency and Asymptotic Normality of the OLS Estimators 583
Consistency of Heteroskedasticity-Robust Standard Errors 583
Asymptotic Normality of the Heteroskedasticity-Robust t-Statistic 584
17.4 Exact Sampling Distributions When the Errors Are Normally Distributed 585
Distribution of with Normal Errors 585
Distribution of the Homoskedasticity-only t-Statistic 586
17.5 Efficiency of the OLS Estimator with Homoskedastic Errors 588
The Gauss-Markov Conditions 588
Linear Conditionally Unbiased Estimators 589
The Gauss-Markov Theorem 590
17.6 Weighted Least Squares 591
WLS with Known Heteroskedasticity 592
WLS with Heteroskedasticity of Known Functional Form 593
Heteroskedasticity-Robust Standard Errors or WLS? 596

Appendix 17.1 The Normal and Related Distributions and Moments of Continuous
Random Variables 600
Appendix 17.2 Two Inequalities 603
Appendix 17.3 Proof of the Gauss-Markov Theorem 604

Chapter 18 The Theory of Multiple Regression 606

18.1 The Linear Multiple Regression Model and OLS Estimator in Matrix Form 607
The Multiple Regression Model in Matrix Notation 607
The Extended Least Squares Assumptions 609
The OLS Estimator 610
18.2 Asymptotic Distribution of the OLS Estimator and t-Statistic 611
The Multivariate Central Limit Theorem 612
Asymptotic Normality of 612
Heteroskedasticity-Robust Standard Errors 613
Confidence Intervals for Predicted Effects 614
Asymptotic Distribution of the t-Statistic 615
18.3 Tests of Joint Hypotheses 615
Joint Hypotheses in Matrix Notation 615
Asymptotic Distribution of the F-Statistic 616
Confidence Sets for Multiple Coefficients 616
18.4 Distribution of Regression Statistics with Normal Errors 617
Matrix Representations of OLS Regression Statistics 617
Distribution of with Normal Errors 619
Distribution of 619
Homoskedasticity-Only Standard Errors 619
Distribution of the t-Statistic 621
Distribution of the F-Statistic 621
18.5 Efficiency of the OLS Estimator with Homoskedastic Errors 621
The Gauss-Markov Conditions for Multiple Regression 621
Linear Conditionally Unbiased Estimators 622
The Gauss-Markov Theorem for Multiple Regression 622
18.6 Generalized Least Squares 623
The GLS Assumptions 624
GLS When W Is Known 626
GLS When W Contains Unknown Parameters 627
The Zero Conditional Mean Assumption and GLS 628
Appendix 18.1 Summary of Matrix Algebra 634
Appendix 18.2 Multivariate Distributions 637
Appendix 18.3 Derivation of the Asymptotic Distribution of 638
Appendix 18.4 Derivations of Exact Distributions of OLS Test Statistics with Normal
Errors 639
Appendix 18.5 Proof of the Gauss-Markov Theorem for Multiple Regression 640
Appendix 642
References 651
Answers to "Review the Concepts" Questions 657
Glossary 672
Index 686


Includes bibliographical references (p. 359) and index.