Econometrics: a modern introduction

ECONOMETRICS

A MODERN INTROOICTION

MICHAEL P. MURRAY

ECONOMETRICS

Asian Network

for Higher Education

NoO 0 3 ^

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The A ddison-W esley Series in Econom ics

A bel/ B em anke

Macroeconomics

Bade/Parkin

Foundations o f Economics

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Game Theory with Economic Applications

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Macroeconomic Theory and Policy

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Economy

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World Trade and Payments: An

Introduction

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Application, and Policy

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An Economic Theory o f Democracy

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The Age o f the Economist

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International Economics

G h iara

Learning Economics

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Essentials of Economics

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Russian and Soviet Economic

Performance and Structure

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The Economics o f Natural Resource Use

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Money, the Financial System, and the

Economy

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American Economic History

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International Economics

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Advanced Microeconomic Theory

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A Health Economics Primer

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Mathematical Methods for Economics

K ru gm an /O bstfeld

International Economics

L aid ler

The Demand for Money

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Economics

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The Economics o f Sports

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Economics

M elvin

International Money and Finance

M ille r

Economics Today

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Understanding Modern Economics

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Urban Economics

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The Economics o f Money, Banking, and

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Econometrics: A Modem Introduction

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Microeconomics

P h elp s

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Riddell/Shackelford/ S tam os/ S c h n eid

Economics: A Tool for Critically

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Introduction to Economic Reasoning

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Principles o f Economics

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Rational Expectations and Inflation

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Industry Structure, Strategy, and Publi

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Introduction to Econometrics

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Using Econometrics

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Environmental and Natural Resource

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Environmental Economics and Policy

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Economic Development

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Macroeconom ics

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ECONOMETRICS

A MODERN INTRODUCTION

MICHAEL P. MURRAY

Bates College

T T

PEARSON

Boston San Francisco New York

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Brief Contents

Contents vii _ _________________

Contents on the Web xxiv

Preface for Students xxv

Preface for Teachers xxvii

PART I The Linear Regression Model 1

1 What Is Econometrics? 1

2 Choosing Estimators: Intuition and Monte Carlo Methods 24

3 Linear Estimators and the Gauss-Markov Theorem 70

4 BLUE Estimators for the Slope and Intercept o f a Straight Line 119

5 Residuals 164

6 Multiple Regression 212

PART II Specification and Hypothesis Testing 267

7 Testing Single Hypotheses in Regression Models 267

8 Superfluous and Omitted Variables, Multicollinearity, and Binary Variables 308

9 Testing Multiple Hypotheses 344

PART III Further Topics in Regression 385

10 Heteroskedastic Disturbances 385

11 Autoregressive Disturbances 436

12 Large-Sample Properties o f Estimators: Consistency and Asymptotic Efficiency 492

13 Instrumental Variables Estimation 535

14 Systems of Equations 593

15 Randomized Experiments and Natural Experiments 640

16 Analyzing Panel Data 678

17 Forecasting 716

18 Stochastically Trending Variables 753

19 Logit and Probit Models: Truncated and Censored Samples 811

Statistical Appendix: A Review of Probability and Statistics 849

Glossary 898

Index 918

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Contents

Contents on the Web xxiv

Preface for Students xxv

Preface for Teachers xxvii

Part I The Linear Regression Model 1

1 W hat Is Econometrics? 1

1.1 A First Example of Econometric Modeling: Financial Aid and Income 3

Example 1: Income and Financial Aid 3

What Example 1 Illustrates About Econometrics 9

R E G R E S S IO N ’ S G R E A T E S T H IT S An Econometric Top 40— Golden Oldies,

Classical Favorites, Pop Tunes 9

An Econometric Top 40—A Pop Tune: Paying for College 10

1.2 A Second Example of Econometric Modeling: Consumption and Income 11

Example 2: Income and Food Expenditure 12

R E G R E S S IO N ’ S G R E A T E S T H IT S An Econometric Top 40— A Golden Oldie:

H ow Income Influences Demand—Engel’s Law 15

What Example 2 Illustrates About Econometrics 16

1.3 Organizing Econometrics 17

What Do We Assume About Where the Data Come From? 17

What Makes a Good Estimator? 18

How Do We Create an Estimator? 18

What Are an Estimator’s Properties? 18

How Do We Test Hypotheses? 18

How Do We Forecast? 19

Summary 19

Concepts for Review 20

Questions for Discussion 21

Problems for Analysis 21

Endnotes 22

2 Choosing Estimators: Intuition and Monte Carlo Methods 24

2.1 How to Sell Econometrics 25

2 .2 Estimating a Population’s Mean 28

The Need for a Precise Statement of Assumptions 29

Sampling and Randomness 29

Precise Assumptions— the Data-Generating Process 30

Interpreting the DGP 31

Estimating Means as Estimating Intercepts 32

2.3 Estimating the Slope of a Line with No Intercept: Families of Means 33

Economic Theory and Lines Through the Origin 33

Families of Means 34

2 .4 Natural Estimators for the Slope of a Line Through the Origin 35

A First Natural Estimator 36

A Second Natural Estimator 37

A Third Natural Estimator 38

A Surfeit of Riches 40

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2 .5 The Data-Generating Process 40

The New Assumptions 41

Are Fixed X ’s Realistic? 41

REGRESSION’S GREATEST HITS An Econometric Top 4 0—A G olden O ldie:

Engel on Price Elasticity 42

2.6 Monte Carlo Comparisons 43

Building a Roulette Wheel 44

What Must the Roulette Wheel Do? 45

REGRESSION’S GREATEST HITS An Econometric Top 40— A Classical Favorite:

Friedman’s Permanent Income Hypothesis 46

Building a Roulette Wheel of Your Own: MC Builder I 49

Spinning the Roulette Wheel 50

2 .7 Picking e,’s and the Real World 51

2.8 Comparing p gl, f} g2, and p g3 52

A Monte Carlo Exercise 52

What a Monte Carlo Analysis Can Say About Unbiasedness 55

2.9 Alternative Comparisons of /Jgi, p g2, and p g3 56

Additional Monte Carlo Exercises 57

2.1 0 Graphical Lessons from the Monte Carlo Exercises 60

What DGPs Do We Study? 61

What Do We Find? 61

Summary 65

Concepts for Review 66

Questions for Discussion 67

Problems for Analysis 67

Endnotes 69

3 Linear Estimators and the Gauss-Markov Theorem 70

3.1 Linear Estimators 71

Linear Estimators and Their Weights 71

REGRESSION’S GREATEST HITS An Econometric Top 40—A Golden Oldie:

The Capital Asset Pricing Model 74

3.2 Unbiased Linear Estimators 76

The DGP 76

Unbiasedness and the Algebra of Expectations 76

Are Our Intuitive Estimators Unbiased? 78

3.3 The Variance of a Linear Estimator 79

Linear Estimators and the Algebra of Variances 80

Relative Variances of Linear Estimators 82

Revisiting Monte Carlo Exercises 82

3.4 A More Efficient Linear Estimator 84

Why Is (3g2 More Efficient Than 85

What Estimator Might Be More Efficient Than /3^2? 86

Is (Sf,4 Unbiased? 8“

The Variance o f /3^,4 88

Is There a Linear Estimator More Efficient Than /3g4? 90

3.5 A First Gauss-Markov Theorem 90

The Gauss-Markov Assumptions 91

Finding the Best Weights 91

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3 .6 Replacing Fixed X*s with Stochastic X*s 93

Extending the Reach of the Gauss-Markov Assumptions 93

The Conditional and Population Properties of /3^4 94

3 .7 Application: A U.S. Production Function 96

Estimating the Cobb-Douglas Production Function with /3gi, (3g2, Pg3 , and (3g4 96

Is Ours the Correct DGP for These U.S. Data? 98

3.8 Econometric Software Output 99

Summary 101

Concepts for Review 102

Questions for Discussion 102

Problems for Analysis 103

Endnotes 108

Appendix 3.A Finding the BLUE Estimator of a Straight Line Through

the Origin 109

3.A .1 The Gauss-M arkov Theorem 109

The Mathematical Problem 109

3.A .2 BLU E Estimation of f$ When n = 2 110

Finding the BLUE Estimator 110

3.A .3 BLU E Estimation of /3 When rt > 2 111

Solving the Constrained Minimization Problem 111

Appendix 3.B A M atrix Algebra Representation of Regressions,

Linear Estimators, and Linear Unbiased Estimators 112

3.B .1 An Alternative to Summation Notation 112

Column Vectors and Row Vectors 113

M atrix Multiplication 114

Appendix 3.B Concepts for Review 118

4 BLUE Estimators for the Slope and Intercept of a Straight Line 119

4.1 The DGP for a Straight Line with Unknown Intercept 120

REGRESSION’S GREATEST HITS An Econometric Top 40— A Classical Favorite:

The Phillips Curve 121

4 .2 The Expected Value and Variance of Linear Estimators 123

Conditions for Unbiasedness 123

The Variance of a Linear Estimator 125

4.3 BLU E Estimation of the Slope and Intercept of a Straight Line 126

A BLUE Estimator for the Slope, /3] 126

An Estimator for /3o 128

An Often Used Property of Certain Sums 129

A Relationship Between /3i and /3^4 129

An Example: The Phillips Curve 130

4.4 /3o and /3i Are Intuitively Appealing Estimators 132

The First Intuition 132

The Second Intuition 133

The Third Intuition 133

A Further Insight About BLUE Estimators 134

4.5 Logarithms in Econometrics 136

The Attraction of Logarithms 136

Double Logarithmic Specifications and Elasticities 136

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An Example: The Phillips Curve Revisited 137

Semilog Specifications and Percentage Changes 139

An Example: The Phillips Curve Yet Again 139

REGRESSION’S GREATEST HITS An Econometric Top 40— A Golden Oldie:

How Price Influences Demand 141

Summary 142

Concepts for Review 143

Questions for Discussion 143

Problems for Analysis 144

Endnotes 152

Appendix 4.A Finding the BLUE Estimator for the Slope of a Straight Line

with an Unknown Intercept 153

4.A.1 The Lagrangian Approach for the Case of n Observations 154

Appendix 4.B M atrix Algebra and Estimating the Slope of a Straight Line

with an Unknown Intercept 155

4.B.1 Constructing Appealing Linear Estimators of p i and po 156

A Matrix Representation of a Regression Model Including an Intercept 156

Multiplying Matrices Revisited 157

A Computational Path to Building fii and /3o 158

Multiplying Matrices When Neither Is a Vector 159

Extending the Notion of an Inverse Matrix 159

The Estimators fio and 160

Concepts for Review 162

4.C A Commonly Used Property of Certain Sums 162

5 Residuals 164

5.1 Estimating cr2 165

An Intuitive Estimator of cr2 165

An Unbiased Estimator of cr1 166

5.2 The Variances and Covariance of p \ and po 167

The Variances of )8i and 167

The Covariance Between /3o and 169

Estimators of Var(/3oh Var(j3i), and Cov(/8o, j&i) 170

5.3 The Gauss-M arkov Theorem and the Expected Value of Y, Given X 172

5.4 Confidence Intervals and Prediction Intervals 173

Confidence Intervals for the Slope, Intercept, and E(Y|X) 173

Prediction Intervals for Future Values of Y 175

R E G R E S S IO N ’ S G R E A T E S T H IT S An Econometric Top 40—A Golden Oldie:

The Cobb-Douglas Production Function 176

5.5 Application: A U.S. Production Function 177

R E G R E S S IO N ’S G R E A T E S T H IT S An Econometric Top 40—A Golden Oldie:

The CES Production Function 180

5.6 The Goodness of Fit of an Estimated Line 182

Decomposing the Variance of the Y, Within a Sample 182

The Coefficient of Determination, R~ 185

R E G R E S S IO N 'S G R E A T E S T H IT S An Econometric Top 40—A Classical Favorite:

The World's Surprisingly Immobile Capital Markets 189

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5 .7 Two Properties of the BLUE Estimators’ Residuals 190

The Sum of the Residuals 191

The Sum of the X ’s Times the Residuals 191

5.8 Ordinary Least Squares 192

Summary 194

Concepts for Review 195

Questions for Discussion 196

Problems for Analysis 196

Endnotes 203

Appendix 5.A The Unbiasedness of s2 204

5.A.1 Estimating the Variance of the Disturbances 205

An Infeasible Alternative Estimator 205

Some Preliminaries 205

The Unbiasedness of s2 206

Appendix 5.B The Variance of po and the Covariance Between

Linear Estimators 208

5.B .1 The Variance of the OLS Intercept Estimator 208

A Special Expression for the Sum of the X j 208

The Variance of /3o 209

5.B .2 The Covariance of Linear Estimators 210

Appendix 5.C M atrix Algebra and the Properties of OLS Residuals 211

6 Multiple Regression 212

6.1 The DGP for the Regression Model 212

Polynomials 213

REGRESSION’S GREATEST HITS An Econometric Top 40— A Pop Tune:

College Students' Misbehavior and the Price o f Beer 215

6.2 BLU E Estimation in a Multiple Regression Model 216

What Is Required for Unbiased Linear Estimation? 216

What Is the Variance of a Linear Estimator in This DGP? 218

What Are the BLUE Estimators of (3o, j3i, . . . , /3* in This DGP? 218

REGRESSION’S GREATEST HITS An Econometric Top 4 0—A Pop Tune:

The Demand for Drugs 219

A More General Gauss-Markov Theorem 221

6.3 An Application: Earnings Equations 221

BLUE Estimates for Black Women 221

Using Dummy Variables to Capture Categories 222

6.4 Estimating fr1 227

Using the Residuals to Mimic the Disturbances 228

6.5 Ordinary Least Squares 229

A Visual Approach to OLS 229

An Implication of OLS 229

6.6 R 2 in the Multiple Regression Model 231

R1— the Coefficient of Determination Revisited 231

Adjusted/?2 231

R E G R E S S IO N ’S G R E A T E S T H IT S An Econometric Top 40— A Classical Favorite:

The Solow Model, Old and New 232

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6 .7 Four Uses of “Linear” in Econometrics 235

Summary 238

Concepts for Review 239

Questions for Discussion 239

Problems for Analysis 239

Endnotes 250

Appendix 6. A Blue Estimation in a DGP with Stochastic Regressors 252

6.A.1 Unbiasedness Conditions of Linear Estimators 252

Adapted Gauss-Markov Assumptions 255

OLS BLUE in This DGP 255

Appendix 6.B M atrix Algebra and Multiple Regressions 256

6.B .1 A M atrix Representation of the Multiple Regression Model 256

The Multiple Regression Model in Matrix Form 257

The Gauss-Markov Assumptions in Matrix Form 257

6.B .2 Estimating the Multiple Regression Model’s Coefficients 258

An Intuitive Approach to Estimating /3 258

Ordinary Least Squares 259

Linear Estimators in the Multiple Regression Model 261

Unbiasedness Conditions for Linear Estimators 261

The Variance of Linear Estimators 262

The Variances and Covariances of the OLS Estimator 262

OLS Is BLUE Under the Gauss-Markov Assumptions 263

A Relationship Among Explanators and Residuals for the OLS Estimator 264

6.B .3 Estimating cr2

Appendix 6.B Concepts for Review 266

Part II Specification and Hypothesis Testing 267

7 Testing Single Hypotheses in Regression Models 267

7.1 Rejecting and Failing to Reject Hypotheses 269

7.2 Six Steps to Hypothesis Testing 270

Framing the Null and Alternative Hypotheses 271

Choosing a Test Statistic 272

The f-Statistic 273

Choosing a Significance Level 275

Choosing a Critical Region 276

The Power of a Test 278

Critical Regions and Complex Null Hypotheses 279

Computing the Test Statistic and Drawing a Conclusion 280

A Caveat About Maintained Hypotheses 282

Tests Undermined 283

7. 3 Degrees of Freedom Revisited 284

R E G R E S S IO N ’S G R E A T E S T H IT S An Econometric Top 40— A Classical Favorite:

The Capital Asset Model Revisited 286

7.4 An Application: The Capital Asset Pricing Model Revisited 287

7.5 Tests About a Linear Combination of Coefficients 289

Another ¿-Statistic 290

Two Examples of Testing for a Relationship Among Regression Coefficients 291

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A Special Case: the Expected Value of Y, Given X 292

REGRESSION’S GREATEST HITS An Econom etric Top 40— A Classical Favorite:

A Deeper Look at Discrimination 293

Summary 297

Concepts for Review 297

Questions for Discussion 297

Problems for Analysis 298

Endnotes 307

8 Superfluous and Omitted Variables. Multicollinearity. and Binary Variables 308

8.1 Including Superfluous Variables 308

Superfluous Variables and Lost Efficiency 309

A M O N T E C A R L O E X P E R IM E N T Prelude to Omitted Variables 310

8.2 Omitting Relevant Variables 311

Omitted Variable Bias 311

A Formula for Omitted Variable Bias 312

What to Include, What to Exclude 315

When Omitted Relevant Variables Are Acceptable 315

REGRESSION’S GREATEST HITS An Econometric Top 4 0—A Classical Favorite:

The Expectations-Augmented Phillips Curve 316

8.3 Multicollinearity 320

Correlated Explanators 321

The Consequences of Multicollinearity 322

Perfect Multicollinearity 323

Coping with Multicollinearity 324

R E G R E S S IO N ’S G R E A T E S T H IT S An Econometric Top 40—A Pop Tune:

Who Needs SAT Scores? 325

8.4 The Earnings Example Extended— M ore About Dummy Variables 328

Using Multiple Dummy Variables to Estimate Means 328

Using Multiple Dummies to Estimate Intercepts 330

Using Multiple Dummies to Categorize Along Several Dimensions 331

Using Dummy Variables to Allow Different Slopes for Different Groups 332

R E G R E S S IO N ’S G R E A T E S T H IT S An Econometric Top 40— A Classical Favorite:

The Rational Expectations Revolution 333

Summary 335

Concepts for Review 336

Questions for Discussion 336

Problems for Analysis 336

Endnotes 342

Appendix 8.A M atrix Algebra, Omitted Variables, and Perfect Collinearity

8.A.1 A M atrix Representation of Omitted Variables Bias 343

8.A .2 M atrix Representation of Perfect Collinearity 343

Concepts for Review 343

9 Testing Multiple Hypotheses

9.1 The Error o f Combining i-Statistics 344

REGRESSION’S GREATEST HITS An Econometric Top 4 0— A Pop Tune:

Life Is Too Short to Drink Bad Wine! 345

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9 .2 F-Tests: The Intuition 347

9.3 F-Tests for Multiple Linear Restrictions on Regression Coefficients 350

The Distribution of the F-Statistic 351

A Special Case Commonly Encountered 352

9.4 Relaxing Assumptions 353

Letting Go of Normality 353

Letting Go of Fixed Regressors 354

A Caveat About Multiple Tests 354

9.5 An Application: Linear Coefficient Constraints and the Earnings o f Blacks 355

9.6 An Application: Linear Coefficient Constraints in a M odel of Deficits 361

9 .7 Two Additional Tests for Regime Shifts 363

Chow Tests 363

Using Dummy Variables to Test for Regime Shifts 364

REGRESSION’S GREATEST HITS An Econometric Top 4 0—A Classical Favorite:

Unanticipated Money 365

Summary 368

Concepts for Review 368

Questions for Discussion 369

Problems for Analysis 369

Endnotes 377

Appendix 9.A M atrix Algebra and Hypothesis Testing 379

9.A.1 The Case o f a Straight Line Through the Origin Revisited 379

9.A .2 Testing Linear Constraints on Both the Slope and Intercept 380

9.A.3 Any Linear Constraint Can Be Expressed with R ß = c 382

9.A .4 Goodness of Fit and Deviations from the Null Hypothesis 383

9.A.5 A General Gauss-M arkov Theorem 383

Concepts for Review 384

Part III Further Topics In Regression 385

10 Heteroskedastic Disturbances 385

10.1 Visualizing Heteroskedasticity 386

A M O N T E C A R L O E X P E R IM E N T : Prelude to Heteroskedasticity 389

10.2 The Consequences of Heteroskedasticity for the OLS Estimators 390

Unchanged Unbiasedness Conditions 391

The Changed Variance of a Linear Estimator 391

10.3 Tests for Heteroskedasticity 394

The White Test 394

The Breusch-Pagan Test 397

An Example Comparing the White Test and the Breusch-Pagan Test 398

The Goldfeld-Quandt Test 401

An Example of the Goldfeld-Quandt Test 402

Omitted Variables and Heteroskedasticity Tests 402

The RESET Specification Test 405

10.4 BLU E Estimation When Disturbances Are Heteroskedastic 406

Transforming the Data to Account for Heteroskedasticity 406

Weighting Observations with Larger crf 408

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Two Special Cases of Heteroskedastic Disturbances 409

R E G R E S S IO N ’S G R E A T E S T H IT S An Econometric Top 40— A G olden O ldie:

Com plete Systems o f D em and Equations 410

10.5 An Application: GLS Estimation of the Rent—Income Relationship 412

Disturbances with Variances Proportional to Income Squared 412

Disturbances with Variances Proportional to Income 414

A Need for Caution 416

10.6 Feasible Generalized Least Squares 416

FGLS and the Rent-Income Relationship 417

10.7 White’s Heteroskedasticity-Consistent Standard Errors 418

10.8 Logarithms and Heteroskedasticity 420

Summary 422

Concepts for Review 424

Questions for Discussion 424

Problems for Analysis 425

Endnotes 431

Appendix 10.A M atrix Algebra and Generalized Least Squares I 432

10.A. 1 The Heteroskedastic Variance-Covariance M atrix 432

The Disturbances’ Variance—Covariance Matrix 432

The Heteroskedastic Variance—Covariance Matrix 433

10.A .2 OLS, GLS, and Heteroskedasticity 434

Heteroskedasticity and OLS 434

Heteroskedasticity and GLS 434

Concepts for Review 435

11 Autoregressive Disturbances 436

11.1 The Serially Correlated DGP 437

Visualizing Serial Correlation 437

The DGP 439

Stationarity 439

11.2 The Consequences of Serial Correlation for the OLS Estimators 441

The Unchanged Unbiasedness Conditions 441

The Changed Variance of a Linear Estimator 442

The Biased Standard Estimator of Var(/3j) 443

The Newey-West Serial Correlation Consistent Standard Error Estimator 444

11.3 Tests for Serial Correlation 446

The Durbin-Watson Test 446

Critical Values for the Durbin-Watson Statistic 447

The Breusch-Godfrey Test 452

An Example of Checking for Serial Correlation 453

Omitted Variables and Serial Correlation Tests 455

11.4 A DGP with First-Order Autoregressive Disturbances 457

R E G R E S S IO N ’ S G R E A T E S T H IT S An Econometric Top 40— A Classical Favorite:

Is Public Expenditure Productive? 458

Visualizing First-Order Autoregressive Disturbances 459

The DGP 461

How the Disturbances Are Correlated 461

The Mean and Variance of the Disturbances 462

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