An Introduction to Classical Econometric Theory

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ISBN: 9780195111644 | 0195111648
Cover: Hardcover
Copyright: 3/23/2000

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In An Introduction to Classical Econometric Theory, Ruud does not bypasskey geometric intuitions underlying econometrics, revealing 'why' things aretrue; however, he also does not provide a list of how to solve the problems. Thetext shows the practical value of a thorough intuitive approach to the field:the how and why things work.Through a strong concentration on linear algebra, tools of this field becomelogical parts of the whole presentation. The author makes use of ideas such aslinear vector space, linear dependence, orthogonality, and distance. The text iscomprehensive and pedagogically methodical: the main results of a chapter arepresented at the outset, followed by examples of special cases to provide a adetailed significance for the results.

List of Figures

xvii

List of Tables

xxi

Preface

xxiv

PART I ORDINARY LEAST SQUARES

(100)

The Least-Squares Linear Fit

(16)

Earnings and Attributes of Workers

(4)

Generalizing the Sample Average

(7)

OLS Regression

(1)

Overview

(1)

Exercises

(1)

Data Collection

(2)

The Geometry of Least Squares

(28)

Introductory Example

(2)

Ordinary Least Squares

(4)

Examples of OLS

(3)

Orthogonal Projection

(6)

The Projection Theorem

(1)

Orthogonal Projectors

(3)

Exact Multicollinearity

(2)

Mathematical Notes

(4)

Gram--Schmidt Orthonomalization

(1)

Properties of Orthogonal Projectors

(1)

Proofs

(2)

Overview

(1)

Exercises

(6)

Review

(3)

Extensions

(3)

Partitioned Fit

(27)

Introductory Example

(7)

Partitioned Fit

(7)

Projection

(5)

Projectors

(2)

Overview

(1)

Exercises

(5)

Review

(3)

Extensions

(2)

Restricted Least Squares

(23)

Introduction

(3)

Linear Restrictions

(2)

Restricted Least Squares

(5)

Generalized Distance

(4)

Translation

(1)

Generalized Projection

(3)

Overview

(1)

Exercises

(5)

Review

(2)

Extensions

(3)

Overview of Ordinary Least Squares

(4)

Geometric Theory

(2)

Econometric Specifications

(1)

Econometric Method

100

(1)

PART II LINEAR REGRESSION

101

(142)

Linear Unbiased Estimation

105

(20)

Experimental Example

105

(5)

First Moments

110

(3)

Conditional Means

113

(1)

Projection of Random Variables

114

(4)

Mathematical Notes

118

(2)

Methodological Notes

120

(1)

Overview

121

(1)

Exercises

122

(3)

Review

122

(1)

Extensions

123

(2)

Variances and Covariances

125

(29)

Introduction

125

(4)

Second Moments

129

(2)

Spherical Distributions

131

(2)

The Variance Ellipse

133

(2)

Minimum MSE Linear Prediction

135

(5)

Mathematical Notes

140

(8)

A Square Root of the Variance Matrix

140

(3)

The Cauchy-Schwarz Inequality

143

(1)

Linear Transformation of Variance Ellipses

144

(2)

A Quadratic Decomposition

146

(2)

Overview

148

(1)

Exercises

149

(5)

Review

149

(3)

Extensions

152

(2)

Variances and Covariances of Ordinary Least Squares

154

(19)

Experimental Example

154

(3)

Second-Moment Properties

157

(2)

Variance and Covariance Matrices

159

(4)

Estimation of the Variance Parameter

163

(4)

Methodological Note

167

(1)

Overview

168

(1)

Exercises

168

(5)

Review

168

(2)

Extensions

170

(3)

Efficient Estimation

173

(22)

Introduction

173

(2)

Design and Precision

175

(7)

Dispersion

177

(1)

Sample Size

177

(1)

Near Multicollinearity

178

(2)

Forecast Variance

180

(2)

Restricted Estimation

182

(4)

The Gauss--Markov Theorem

186

(3)

Geometry of the Gauss--Markov Theorem

187

(1)

Proof of the Gauss--Markov Theorem

188

(1)

Mathematical Notes

189

(1)

Overview

190

(1)

Exercises

191

(4)

Review

191

(2)

Extensions

193

(2)

Normal Distribution Theory

195

(27)

Introduction

195

(3)

Distribution Theory for OLS Estimators

198

(2)

Interval Estimators

200

(5)

Variance

200

(1)

Coefficient Vector with Known Variance

201

(2)

Coefficient Vector with Unknown Variance

203

(1)

Linear Functions of Coefficients

204

(1)

Efficiency of OLS

205

(1)

Basic Distribution Theory

205

(10)

The Multivariate Normal Distribution

206

(4)

The Chi-Square and F Distributions

210

(1)

Singular Variances and Generalized Inverses

211

(3)

Singular Multivariate Normal Distributions

214

(1)

Methodological Notes

215

(1)

Overview

216

(1)

Exercises

217

(5)

Review

217

(2)

Extensions

219

(3)

Hypothesis Testing

222

(18)

Introduction

222

(2)

Hypothesis Testing

224

(4)

Statistical Power

228

(4)

Power Comparisons for Tests

229

(1)

t Statistics

229

(2)

Optimal Power and the F Test

231

(1)

Basic Distribution Theory

232

(4)

Methodological Notes

236

(1)

Overview

237

(1)

Exercises

238

(2)

Review

238

(1)

Extensions

239

(1)

Overview of Linear Regression

240

(3)

Statistical Theory

240

(1)

Probability Distribution Theory

241

(2)

PART III GENERALIZATIONS OF THE LINEAR MODEL

243

(370)

Nonnormal Distribution Theory

245

(39)

Introduction

245

(1)

Nonnormal Parametric Distributions

246

(5)

The Student t Distribution

247

(2)

Laplace (Double Exponential) Distribution

249

(1)

Logistic Distribution

250

(1)

Power Exponential Distribution

250

(1)

LAD Estimation

251

(5)

The Sample Median

252

(1)

LAD Linear Regression

253

(3)

Asymptotic Distribution Theory

256

(14)

Convergence in Distribution

259

(3)

Law of Large Numbers

262

(3)

Central Limit Theorem

265

(2)

Sample Size

267

(3)

Mathematical Notes

270

(8)

The Density of an Order Statistic

270

(1)

Properties of LAD Fit

271

(2)

Convergence Proofs

273

(5)

Overview

278

(1)

Exercises

279

(5)

Review

279

(2)

Extensions

281

(3)

Maximum Likelihood Estimation

284

(34)

Introduction

284

(1)

Probability Model Specification

285

(3)

The Likelihood Function

288

(5)

The Maximum Likelihood Estimator

293

(2)

Identification

295

(5)

The Score Function

300

(2)

The Information Matrix

302

(3)

The Cramer--Rao Lower Bound

305

(6)

Mathematical Notes

311

(1)

Overview

312

(2)

Exercises

314

(4)

Review

314

(1)

Extensions

315

(3)

Maximum Likelihood Asymptotic Distribution Theory

318

(29)

Introduction

318

(2)

Consistency

320

(4)

Asymptotic Normality

324

(5)

Score

325

(1)

Information

326

(1)

Asymptotic Distribution

327

(2)

Variance Estimation

329

(2)

Efficiency

331

(2)

Linearized MLE

333

(1)

Restricted Estimation

334

(2)

Mathematical Notes

336

(4)

Overview

340

(2)

Exercises

342

(5)

Review

342

(1)

Extensions

343

(4)

Maximum Likelihood Computation

347

(33)

Introduction

347

(2)

Grid Search

349

(1)

Polynomial Approximation

349

(2)

Line Searches

351

(11)

The Method of Steepest Ascent

353

(2)

Quadratic Methods

355

(2)

Quadratic Methods and the MLE

357

(4)

LMLE

361

(1)

Convergence Criteria

362

(2)

Transformations of Parameters

364

(4)

Concentrating the Likelihood Function

368

(3)

The Gauss--Seidel Algorithm

371

(1)

Mathematical Notes

372

(4)

Proofs

372

(2)

Stochastic Order

374

(1)

Uniqueness of the MLE

375

(1)

Overview

376

(1)

Exercises

377

(3)

Review

377

(2)

Extensions

379

(1)

Maximum Likelihood Statistical Inference

380

(36)

Introduction

380

(4)

The Classical Hypothesis Test Statistics

384

(9)

The Wald Test

384

(1)

The Score Test

385

(3)

The Likelihood Ratio Test

388

(1)

A Graphic Description of the Test Statistics

389

(4)

Asymptotic Distribution Theory

393

(4)

The Likelihood Ratio Test

394

(1)

The Score Test

395

(1)

The Wald Test

395

(1)

The C(α) Test

396

(1)

Limiting Distribution

396

(1)

Parameter Transformations and Invariance

397

(5)

Power

402

(6)

Local Power

403

(3)

Neyman--Pearson Lemma

406

(2)

Interval Estimation

408

(1)

Overview

409

(1)

Exercises

410

(6)

Review

410

(2)

Extensions

412

(4)

Heteroskedasticity

416

(39)

Heteroskedasticity in Wages

417

(4)

Heteroskedasticity and OLS

421

(2)

Testing for Heteroskedasticity

423

(4)

The Goldfeld--Quandt F Test

424

(1)

The Breusch--Pagan Score Test

424

(3)

Adjustments to OLS

427

(2)

Heteroskedasticity and WLS/GLS

429

(13)

Maximum Likelihood

433

(2)

FGLS

435

(5)

Adaptive Estimation

440

(2)

Methodological Notes

442

(1)

Mathematical Notes

443

(8)

Score and Information

444

(1)

The Maximum Likelihood Estimator

444

(2)

The Breusch---Pagan Score Test

446

(1)

Regularity

447

(1)

Asymptotic Theory for Heteroskedasticity

448

(3)

Overview

451

(1)

Exercises

452

(3)

Review

452

(1)

Extensions

453

(2)

Serial Correlation

455

(31)

The Phillips Curve

455

(3)

The Basic Autoregressive Model

458

(4)

The Autocorrelation Function

458

(2)

The Log-Likelihood Function

460

(2)

Autocorrelation and OLS

462

(2)

Testing for Autocorrelation

464

(2)

Breusch---Godfrey Score Test

464

(2)

The Durbin---Watson Test

466

(1)

Variance Estimation for OLS

466

(2)

Serial Correlation and GLS

468

(3)

Maximum Likelihood Estimation

469

(2)

FGLS

471

(1)

Prediction

471

(1)

Methodological Notes

472

(3)

Mathematical Notes

475

(6)

Score and Information

475

(1)

Breusch---Godfrey Score Test

476

(1)

Asymptotic Distribution Theory

477

(3)

OLS versus GLS

480

(1)

Overview

481

(1)

Exercises

482

(4)

Review

482

(1)

Extensions

483

(3)

Instrumental Variables Estimation

486

(45)

The Phillips Curve Revisited

487

(4)

Latent Variable Models

491

(2)

Omitted Explanatory Variables

493

(6)

Consistent Estimation

499

(3)

Two-Stage Least Squares

502

(3)

Two-Step Variance Estimation

505

(4)

Efficiency

509

(5)

Simultaneous Equations

509

(2)

Dynamic Regression

511

(1)

IV and GLS

512

(2)

Issues in Small Samples

514

(1)

Methodological Notes

515

(1)

Mathematical Notes

516

(5)

Covariance Stationarity

517

(1)

Dynamic Regression Log-Likelihood

518

(1)

Hatanaka's Estimator

518

(1)

Two-Step Estimation

519

(1)

Optimal Instruments

520

(1)

Overview

521

(1)

Exercises

522

(9)

Review

522

(4)

Extensions

526

(5)

The Generalized Method of Moments

531

(33)

A Random Walk

532

(4)

Definition of GMM

536

(6)

Turning Moments into Estimators

537

(2)

Nonlinear Least Squares

539

(2)

Two-Stage Least Squares

541

(1)

Identification

542

(3)

Distribution Theory

545

(10)

Proof of Consistency

546

(1)

Proof of Asymptotic Normality

547

(1)

Variance Matrix Estimation

548

(2)

Efficiency

550

(5)

Methodologial Notes

555

(1)

Mathematical Notes

555

(2)

Uniform Convergence of the GMM Criterion Function

555

(1)

Nonsingularity of the GMM Hessian

556

(1)

GMM Efficiency

557

(1)

Overview

557

(1)

Exercises

558

(4)

Review

558

(3)

Extensions

561

(1)

Appendix: Data Collection

562

(2)

Generalized Method of Moments Hypothesis Tests

564

(44)

Tests of Parameter Restrictions

565

(7)

Wald Test

566

(1)

Gradient Test

567

(1)

Distance Difference Test

567

(1)

Minimum Chi-Square Test

568

(1)

Special Identities

569

(1)

Generalizing Likelihood-Based Diagnostics

570

(2)

Tests of Moment Restrictions

572

(6)

Overidentifying Restrictions Tests

576

(2)

Hausman Specification Tests

578

(7)

Equivalence among Test Statistics

585

(5)

A Trinity of GMM Test Statistics

586

(1)

Minimum Chi-Square

587

(3)

Statistical Power

590

(2)

Sequential Testing

592

(2)

Minimum Distance Estimation

594

(3)

Mathematical Notes

597

(4)

Methodological Notes

601

(1)

Overview

601

(1)

Exercises

602

(6)

Review

602

(3)

Extensions

605

(3)

Overview

608

(5)

PART IV LATENT VARIABLE MODELS

613

(220)

Panel Data Models

615

(30)

Introduction

615

(1)

Fixed Individual Effects

616

(2)

Random Individual Effects

618

(4)

Fixed versus Random Effects

622

(1)

Generalizations

623

(5)

Individual-Specific Explanatory Variables

624

(1)

Time-Specific Effects

625

(1)

Dynamic Models

626

(2)

Specification Tests

628

(2)

Linear Projection

630

(5)

Identification and OLS

631

(1)

Efficient Estimation

632

(2)

Diagnostic Tests

634

(1)

Additional Moment Restrictions

635

(3)

Identification

635

(1)

Estimation

636

(2)

Mathematical Notes

638

(2)

Overview

640

(1)

Exercises

641

(4)

Review

641

(2)

Extensions

643

(2)

Autoregressive Moving-Average Time Series Models

645

(52)

Introduction

645

(4)

Autoregressive Processes

649

(9)

Stationarity

651

(5)

Restricted Estimation

656

(1)

Sequential Testing for Order

657

(1)

Moving-Average Processes

658

(15)

Identification

660

(3)

Kalman Filter

663

(4)

Estimation

667

(3)

Testing Serial Correlation

670

(3)

ARMA Processes

673

(7)

Identification and Invertibility

675

(4)

Kalman Filter and Estimation

679

(1)

Hypothesis Tests

679

(1)

Wold Decomposition

680

(4)

Linearly Deterministic Processes

681

(1)

Wold Decomposition Theorem

682

(2)

Methodological Notes

684

(1)

Mathematical Notes

685

(6)

Yule---Walker Equations

685

(1)

Kalman Filter

686

(3)

MA(q) Identification

689

(1)

Score Test Equivalence

690

(1)

Overview

691

(2)

Exercises

693

(4)

Review

693

(3)

Extensions

696

(1)

Simultaneous Equations

697

(50)

Introduction

697

(1)

Seemingly Unrelated Regressions

698

(8)

Estimation of a Cost Function

699

(1)

Assumptions

700

(1)

OLS versus GLS

701

(3)

Feasible GLS Estimation

704

(1)

Maximum Likelihood Estimation

705

(1)

Simultaneous Equations

706

(5)

Definitions

709

(1)

Assumptions

710

(1)

Identification

711

(7)

Equation Identification

714

(3)

System Identification

717

(1)

Estimation

718

(9)

Limited Information

719

(2)

Full Information

721

(2)

Maximum Likelihood

723

(4)

Hypothesis Tests

727

(3)

Mathematical Notes

730

(4)

Score Functions

730

(2)

Information Matrix

732

(2)

Overview

734

(2)

Exercises

736

(11)

Review

736

(6)

Extensions

742

(5)

Discrete Dependent Variables

747

(44)

Bernoulli Dependent Variables

748

(9)

Bernoulli Regression

748

(2)

Estimation

750

(5)

A Latent Variable Interpretation

755

(2)

Additional Univariate Models

757

(7)

Ordered Data

758

(3)

Count Data

761

(3)

Multivariate Models

764

(7)

Multiple Choice

764

(6)

Rank-Ordered Multiple Choice

770

(1)

Latent Variables and Computation

771

(6)

Score Functions

772

(1)

Hessian and Information Functions

773

(1)

EM Algorithm

774

(1)

Simulation

775

(2)

Methodological Notes

777

(1)

Mathematical Notes

778

(6)

Katz Family of Distributions

779

(1)

Logit Probabilities

780

(2)

EM Algorithm

782

(2)

Simulation

784

(1)

Overview

784

(2)

Exercises

786

(5)

Review

786

(2)

Extensions

788

(3)

Censored and Truncated Variables

791

(40)

Labor Supply

791

(3)

Mixed Probability Functions

794

(4)

Censored Moments

798

(2)

Estimation

800

(2)

Prediction and Truncated Means

802

(1)

Truncated Regression

803

(3)

Nonrandom Sample Selection

806

(4)

Log-Likelihood

807

(1)

Moments

808

(1)

Estimation

809

(1)

Specification of Distribution

810

(7)

Censored Regression

810

(4)

Truncated Regression

814

(3)

Mathematical Notes

817

(7)

Integrals

817

(2)

Censored Moments

819

(3)

Nonrandom Sample Selection

822

(2)

Overview

824

(1)

Exercises

824

(7)

Review

824

(3)

Extensions

827

(4)

Overview

831

(2)

PART V APPENDICES

833

(102)

Appendix A: Abbreviations and Acronyms

835

(2)

Appendix B: Notation

837

(30)

B.1 Limits

837

(1)

B.2 Sets

838

(1)

B.3 Functions

838

(1)

B.4 Linear Vector Spaces

838

(1)

B.5 Matrices

839

(1)

B.6 Random Variables

840

(1)

B.7 Optima and Roots

840

(1)

Appendix C: Linear Algebra and Matrix Theory

841

(1)

C.1 Linear Vector Spaces

841

(6)

C.2 Linear Transformations

847

(4)

C.3 Inner Products and Orthogonality

851

(4)

C.4 Normed Linear Vector Spaces

855

(1)

C.5 Determinants

856

(1)

C.5.1 Volume of a Parallelogram

857

(3)

C.5.2 Determinant of a Matrix

860

(2)

C.5.3 The Cofactor Expansion

862

(3)

C.6 Eigenvalues and Eigenvectors

865

(2)

Appendix D: Probability

867

(35)

D.1 Fundamental Concepts

867

(1)

D.2 Random Variables

868

(7)

D.2.1 Mathematical Notes

875

(4)

D.3 Joint and Conditional Probability

879

(4)

D.3.1 Mathematical Notes

883

(1)

D.4 Special Distributions

884

(7)

D.5 Limiting Approximations

891

(2)

D.5.1 A Sequence of Densities

893

(2)

D.5.2 Sequences of Moments

895

(1)

D.5.3 Sequences of c.f.s

896

(2)

D.5.4 Mathematical Notes

898

(4)

Appendix E: Classical Statistics

902

(14)

E.1 Sampling

902

(2)

E.2 Classical Statistical Inference

904

(1)

E.2.1 Estimation

904

(2)

E.2.2 Hypothesis Tests

906

(1)

E.2.3 Estimation Methods

907

(6)

E.2.4 Asymptotic Distribution Theory

913

(3)

Appendix F: Noncentral Distributions

916

(6)

Appendix G: Multivariate Differentiation

922

(9)

G.1 Basic Notation

922

(2)

G.2 Vectorization and Kronecker Products

924

(1)

G.2.1 Kronecker Products

925

(1)

G.3 Derivative Vectors

926

(1)

G.4 Derivative Matrices

927

(1)

G.5 The Normal Log-Likelihood Function

928

(3)

Appendix H: Characteristic Functions

931

(4)

Bibliography

935

(10)

Index

945

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An Introduction to Classical Econometric Theory

Summary

Table of Contents

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