Econometrics / Edition 1

Econometrics / Edition 1

by Fumio Hayashi

ISBN-10: 0691010188

ISBN-13: 9780691010182

Pub. Date: 10/30/2000

Publisher: Princeton University Press

Hayashi's Econometrics promises to be the next great synthesis of modern econometrics. It introduces first year Ph.D. students to standard graduate econometrics material from a modern perspective. It covers all the standard material necessary for understanding the principal techniques of econometrics from ordinary least squares through cointegration. The

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Hayashi's Econometrics promises to be the next great synthesis of modern econometrics. It introduces first year Ph.D. students to standard graduate econometrics material from a modern perspective. It covers all the standard material necessary for understanding the principal techniques of econometrics from ordinary least squares through cointegration. The book is also distinctive in developing both time-series and cross-section analysis fully, giving the reader a unified framework for understanding and integrating results.

Econometrics has many useful features and covers all the important topics in econometrics in a succinct manner. All the estimation techniques that could possibly be taught in a first-year graduate course, except maximum likelihood, are treated as special cases of GMM (generalized methods of moments). Maximum likelihood estimators for a variety of models (such as probit and tobit) are collected in a separate chapter. This arrangement enables students to learn various estimation techniques in an efficient manner. Eight of the ten chapters include a serious empirical application drawn from labor economics, industrial organization, domestic and international finance, and macroeconomics. These empirical exercises at the end of each chapter provide students a hands-on experience applying the techniques covered in the chapter. The exposition is rigorous yet accessible to students who have a working knowledge of very basic linear algebra and probability theory. All the results are stated as propositions, so that students can see the points of the discussion and also the conditions under which those results hold. Most propositions are proved in the text.

For those who intend to write a thesis on applied topics, the empirical applications of the book are a good way to learn how to conduct empirical research. For the theoretically inclined, the no-compromise treatment of the basic techniques is a good preparation for more advanced theory courses.

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Princeton University Press
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Economics Series
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New Edition
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7.38(w) x 10.32(h) x 1.67(d)

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Table of Contents

List of Figures xvii

Preface xix

1 Finite-Sample Properties of OLS 3

1.1 The Classical Linear Regression Model 3

The Linearity Assumption 4

Matrix Notation 6

The Strict Exogeneity Assumption 7

Implications of Strict Exogeneity 8

Strict Exogeneity in Time-Series Models 9

Other Assumptions of the Model 10

The Classical Regression Model for Random Samples 12

"Fixed" Regressors 13

1.2 The Algebra of Least Squares 15

OLS Minimizes the Sum of Squared Residuals 15

Normal Equations 16

Two Expressions for the OLS Estimator 18

More Concepts and Algebra 18

Influential Analysis (optional) 21

A Note on the Computation of OLS Estimates 23

1.3 Finite-Sample Properties of OLS 27

Finite-Sample Distribution of b 27

Finite-Sample Properties of s2 30

Estimate of Var(b | X) 31

1.4 Hypothesis Testing under Normality 33

Normally Distributed Error Terms 33

Testing Hypotheses about Individual Regression Coefficients 35

Decision Rule for the t-Test 37

Confidence Interval 38

p-Value 38

Linear Hypotheses 39

The F-Test 40

A More Convenient Expression for F 42

t versus F 43

An Example of a Test Statistic Whose Distribution Depends on X 45

1.5 Relation to Maximum Likelihood 47

The Maximum Likelihood Principle 47

Conditional versus Unconditional Likelihood 47

The Log Likelihood for the Regression Model 48

ML via Concentrated Likelihood 48

Cramer-Rao Bound for the Classical Regression Model 49

The F-Test as a Likelihood Ratio Test 52

Quasi-Maximum Likelihood 53

1.6 Generalized Least Squares (GLS) 54

Consequence of Relaxing Assumption 1.4 55

Efficient Estimation with Known V 55

A Special Case: Weighted Least Squares (WLS) 58

Limiting Nature of GLS 58

1.7 Application: Returns to Scale in Electricity Supply 60

The Electricity Supply Industry 60

The Data 60

Why Do We Need Econometrics? 61

The Cobb-Douglas Technology 62

How Do We Know Things Are Cobb-Douglas? 63

Are the OLS Assumptions Satisfied? 64

Restricted Least Squares 65

Testing the Homogeneity of the Cost Function 65

Detour: A Cautionary Note on R2 67

Testing Constant Returns to Scale 67

Importance of Plotting Residuals 68

Subsequent Developments 68

Problem Set 71

Answers to Selected Questions 84

2 Large-Sample Theory 88

2.1 Review of Limit Theorems for Sequences of Random Variables 88

Various Modes of Convergence 89

Three Useful Results 92

Viewing Estimators as Sequences of Random Variables 94

Laws of Large Numbers and Central Limit Theorems 95

2.2 Fundamental Concepts in Time-Series Analysis 97

Need for Ergodic Stationarity 97

Various Classes of Stochastic Processes 98

Different Formulation of Lack of Serial Dependence 106

The CLT for Ergodic Stationary Martingale Differences Sequences 106

2.3 Large-Sample Distribution of the OLS Estimator 109

The Model 109

Asymptotic Distribution of the OLS Estimator 113

s2 Is Consistent 115

2.4 Hypothesis Testing 117

Testing Linear Hypotheses 117

The Test Is Consistent 119

Asymptotic Power 120

Testing Nonlinear Hypotheses 121

2.5 Estimating E([not displayable]) Consistently 123

Using Residuals for the Errors 123

Data Matrix Representation of S 125

Finite-Sample Considerations 125

2.6 Implications of Conditional Homoskedasticity 126

Conditional versus Unconditional Homoskedasticity 126

Reduction to Finite-Sample Formulas 127

Large-Sample Distribution of t and F Statistics 128

Variations of Asymptotic Tests under Conditional Homoskedasticity 129

2.7 Testing Conditional Homoskedasticity 131

2.8 Estimation with Parameterized Conditional Heteroskedasticity (optional) 133

The Functional Form 133

WLS with Known [alpha] 134

Regression of e2i on zi Provides a Consistent Estimate of [alpha] 135

WLS with Estimated [alpha] 136

OLS versus WLS 137

2.9 Least Squares Projection 137

Optimally Predicting the Value of the Dependent Variable 138

Best Linear Predictor 139

OLS Consistently Estimates the Projection Coefficients 140

2.10 Testing for Serial Correlation 141

Box-Pierce and Ljung-Box 142

Sample Autocorrelations Calculated from Residuals 144

Testing with Predetermined, but Not Strictly Exogenous, Regressors 146

An Auxiliary Regression-Based Test 147

2.11 Application: Rational Expectations Econometrics 150

The Efficient Market Hypotheses 150

Testable Implications 152

Testing for Serial Correlation 153

Is the Nominal Interest Rate the Optimal Predictor? 156

Rt Is Not Strictly Exogenous 158

Subsequent Developments 159

2.12 Time Regressions 160

The Asymptotic Distribution of the OLS Estimates 161

Hypothesis Testing for Time Regressions 163

2.A Asymptotics with Fixed Regressors 164

2.B Proof of Proposition 2.10 165

Problem Set 168

Answers to Selected Questions 183

3 Single-Equation GMM 186

3.1 Endogeneity Bias: Working's Example 187

A Simultaneous Equations Model of Market Equilibrium 187

Endogeneity Bias 188

Observable Supply Shifters 189

3.2 More Examples 193

A Simple Macroeconometric Model 193

Errors-in-Variables 194

Production Function 196

3.3 The General Formulation 198

Regressors and Instruments 198

Identification 200

Order Condition for Identification 202

The Assumption for Asymptotic Normality 202

3.4 Generalized Method of Moments Defined 204

Method of Moments 205

Generalized Method of Moments 206

Sampling Error 207

3.5 Large-Sample Properties of GMM 208

Asymptotic Distribution of the GMM Estimator 209

Estimation of Error Variance 210

Hypothesis Testing 211

Estimation of S 212

Efficient GMM Estimator 212

Asymptotic Power 214

Small-Sample Properties 215

3.6 Testing Overidentifying Restrictions 217

Testing Subsets of Orthogonality Conditions 218

3.7 Hypothesis Testing by the Likelihood-Ratio Principle 222

The LR Statistic for the Regression Model 223

Variable Addition Test (optional) 224

3.8 Implications of Conditional Homoskedasticity 225

Efficient GMM Becomes 2SLS 226

J Becomes Sargan's Statistic 227

Small-Sample Properties of 2SLS 229

Alternative Derivations of 2SLS 229

When Regressors Are Predetermined 231

Testing a Subset of Orthogonality Conditions 232

Testing Conditional Homoskedasticity 234

Testing for Serial Correlation 234

3.9 Application: Returns from Schooling 236

The NLS-Y Data 236

The Semi-Log Wage Equation 237

Omitted Variable Bias 238

IQ as the Measure of Ability 239

Errors-in-Variables 239

2SLS to Correct for the Bias 242

Subsequent Developments 243

Problem Set 244

Answers to Selected Questions 254

4 Multiple-Equation GMM 258

4.1 The Multiple-Equation Model 259

Linearity 259

Stationarity and Ergodicity 260

Orthogonality Conditions 261

Identification 262

The Assumption for Asymptotic Normality 264

Connection to the "Complete" System of Simultaneous Equations 265

4.2 Multiple-Equation GMM Defined 265

4.3 Large-Sample Theory 268

4.4 Single-Equation versus Multiple-Equation Estimation 271

When Are They "Equivalent"? 272

Joint Estimation Can Be Hazardous 273

4.5 Special Cases of Multiple-Equation GMM: FIVE, 3SLS, and SUR 274

Conditional Homoskedasticity 274

Full-Information Instrumental Variables Efficient (FIVE) 275

Three-Stage Least Squares (3SLS) 276

Seemingly Unrelated Regressions (SUR) 279

SUR versus OLS 281

4.6 Common Coefficients 286

The Model with Common Coefficients 286

The GMM Estimator 287

Imposing Conditional Homoskedasticity 288

Pooled OLS 290

Beautifying the Formulas 292

The Restriction That Isn't 293

4.7 Application: Interrelated Factor Demands 296

The Translog Cost Function 296

Factor Shares 297

Substitution Elasticities 298

Properties of Cost Functions 299

Stochastic Specifications 300

The Nature of Restrictions 301

Multivariate Regression Subject to Cross-Equation Restrictions 302

Which Equation to Delete? 304

Results 305

Problem Set 308

Answers to Selected Questions 320

5 Panel Data 323

5.1 The Error-Components Model 324

Error Components 324

Group Means 327

A Reparameterization 327

5.2 The Fixed-Effects Estimator 330

The Formula 330

Large-Sample Properties 331

Digression: When [eta]i Is Spherical 333

Random Effects versus Fixed Effects 334

Relaxing Conditional Homoskedasticity 335

5.3 Unbalanced Panels (optional) 337

"Zeroing Out" Missing Observations 338

Zeroing Out versus Compression 339

No Selectivity Bias 340

5.4 Application: International Differences in Growth Rates 342

Derivation of the Estimation Equation 342

Appending the Error Term 343

Treatment of [alpha]i 344

Consistent Estimation of Speed of Convergence 345

Appendix 5.A: Distribution of Hausman Statistic 346

Problem Set 349

Answers to Selected Questions 363

6 Serial Correlation 365

6.1 Modeling Serial Correlation: Linear Processes 365

MA(q) 366

MA([infinity]) as a Mean Square Limit 366

Filters 369

Inverting Lag Polynomials 372

6.2 ARMA Processes 375

AR(1) and Its MA([infinity]) Representation 376

Autocovariances of AR(1) 378

AR(p) and Its MA([infinity]) Representation 378

ARMA(p,q) 380

ARMA(p) with Common Roots 382

Invertibility 383

Autocovariance-Generating Function and the Spectrum 383

6.3 Vector Processes 387

6.4 Estimating Autoregressions 392

Estimation of AR(1) 392

Estimation of AR(p) 393

Choice of Lag Length 394

Estimation of VARs 397

Estimation of ARMA(p,q) 398

6.5 Asymptotics for Sample Means of Serially Correlated Processes 400

LLN for Covariance-Stationary Processes 401

Two Central Limit Theorems 402

Multivariate Extension 404

6.6 Incorporating Serial Correlation in GMM 406

The Model and Asymptotic Results 406

Estimating S When Autocovariances Vanish after Finite Lags 407

Using Kernels to Estimate S 408


6.7 Estimation under Conditional Homoskedasticity (Optional) 413

Kernel-Based Estimation of S under Conditional Homoskedasticity 413

Data Matrix Representation of Estimated Long-Run Variance 414

Relation to GLS 415

6.8 Application: Forward Exchange Rates as Optimal Predictors 418

The Market Efficiency Hypothesis 419

Testing Whether the Unconditional Mean Is Zero 420

Regression Tests 423

Problem Set 428

Answers to Selected Questions 441

7 Extremum Estimators 445

7.1 Extremum Estimators 446

"Measurability" of [theta] 446

Two Classes of Extremum Estimators 447

Maximum Likelihood (ML) 448

Conditional Maximum Likelihood 450

Invariance of ML 452

Nonlinear Least Squares (NLS) 453

Linear and Nonlinear GMM 454

7.2 Consistency 456

Two Consistency Theorems for Extremum Estimators 456

Consistency of M-Estimators 458

Concavity after Reparameterization 461

Identification in NLS and ML 462

Consistency of GMM 467

7.3 Asymptotic Normality 469

Asymptotic Normality of M-Estimators 470

Consistent Asymptotic Variance Estimation 473

Asymptotic Normality of Conditional ML 474

Two Examples 476

Asymptotic Normality of GMM 478

GMM versus ML 481

Expressing the Sampling Error in a Common Format 483

7.4 Hypothesis Testing 487

The Null Hypothesis 487

The Working Assumptions 489

The Wald Statistic 489

The Lagrange Multiplier (LM) Statistic 491

The Likelihood Ratio (LR) Statistic 493

Summary of the Trinity 494

7.5 Numerical Optimization 497

Newton-Raphson 497

Gauss-Newton 498

Writing Newton-Raphson and Gauss-Newton in a Common Format 498

Equations Nonlinear in Parameters Only 499

Problem Set 501

Answers to Selected Questions 505

8 Examples of Maximum Likelihood 507

8.1 Qualitative Response (QR) Models 507

Score and Hessian for Observation t 508

Consistency 509

Asymptotic Normality 510

8.2 Truncated Regression Models 511

The Model 511

Truncated Distributions 512

The Likelihood Function 513

Reparameterizing the Likelihood Function 514

Verifying Consistency and Asymptotic Normality 515

Recovering Original Parameters 517

8.3 Censored Regression (Tobit) Models 518

Tobit Likelihood Function 518

Reparameterization 519

8.4 Multivariate Regressions 521

The Multivariate Regression Model Restated 522

The Likelihood Function 523

Maximizing the Likelihood Function 524

Consistency and Asymptotic Normality 525

8.5 FIML 526

The Multiple-Equation Model with Common Instruments Restated 526

The Complete System of Simultaneous Equations 529

Relationship between ([Gamma]0, [Beta]0) and [delta]0 530

The FIML Likelihood Function 531

The FIML Concentrated Likelihood Function 532

Testing Overidentifying Restrictions 533

Properties of the FIML Estimator 533

ML Estimation of the SUR Model 535

8.6 LIML 538

LIML Defined 538

Computation of LIML 540

LIML versus 2SLS 542

8.7 Serially Correlated Observations 543

Two Questions 543

Unconditional ML for Dependent Observations 545

ML Estimation of AR.1/ Processes 546

Conditional ML Estimation of AR(1) Processes 547

Conditional ML Estimation of AR(p) and VAR(p) Processes 549

Problem Set 551

9 Unit-Root Econometrics 557

9.1 Modeling Trends 557

Integrated Processes 558

Why Is It Important to Know if the Process Is I(1)? 560

Which Should Be Taken as the Null, I(0) or I(1)? 562

Other Approaches to Modeling Trends 563

9.2 Tools for Unit-Root Econometrics 563

Linear I(0) Processes 563

Approximating I(1) by a Random Walk 564

Relation to ARMA Models 566

The Wiener Process 567

A Useful Lemma 570

9.3 Dickey-Fuller Tests 573

The AR(1) Model 573

Deriving the Limiting Distribution under the I(1) Null 574

Incorporating the Intercept 577

Incorporating Time Trend 581

9.4 Augmented Dickey-Fuller Tests 585

The Augmented Autoregression 585

Limiting Distribution of the OLS Estimator 586

Deriving Test Statistics 590

Testing Hypotheses about [zeta] 591

What to Do When p Is Unknown? 592

A Suggestion for the Choice of pmax(T) 594

Including the Intercept in the Regression 595

Incorporating Time Trend 597

Summary of the DF and ADF Tests and Other Unit-Root Tests 599

9.5 Which Unit-Root Test to Use? 601

Local-to-Unity Asymptotics 602

Small-Sample Properties 602

9.6 Application: Purchasing Power Parity 603

The Embarrassing Resiliency of the Random Walk Model? 604

Problem Set 605

Answers to Selected Questions 619

10 Cointegration 623

10.1 Cointegrated Systems 624

Linear Vector I(0) and I(1) Processes 624

The Beveridge-Nelson Decomposition 627

Cointegration Defined 629

10.2 Alternative Representations of Cointegrated Systems 633

Phillips's Triangular Representation 633

VAR and Cointegration 636

The Vector Error-Correction Model (VECM) 638

Johansen's ML Procedure 640

10.3 Testing the Null of No Cointegration 643

Spurious Regressions 643

The Residual-Based Test for Cointegration 644

Testing the Null of Cointegration 649

10.4 Inference on Cointegrating Vectors 650

The SOLS Estimator 650

The Bivariate Example 652

Continuing with the Bivariate Example 653

Allowing for Serial Correlation 654

General Case 657

Other Estimators and Finite-Sample Properties 658

10.5 Application: the Demand for Money in the United States 659

The Data 660

(m - p, y, R) as a Cointegrated System 660

DOLS 662

Unstable Money Demand? 663

Problem Set 665

Appendix. Partitioned Matrices and Kronecker Products 670

Addition and Multiplication of Partitioned Matrices 671

Inverting Partitioned Matrices 672

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