Econometric Theory

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ISBN: 9780631215844 | 0631215840
Cover: Paperback
Copyright: 4/7/2000

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This book surveys recent developments in the rapidly expanding field of asymptotic distribution theory, placing special emphasis on the problems of time-dependence and heterogeneity. It is technically self-contained, with all but the most basic mathematical prerequisites being explained in their context.

Figures

Symbols and Abbreviations

xvi

Preface

I Basic Regression Theory

(56)

The Linear Regression Model

(14)

The Model

(1)

The Least Squares Estimator

(5)

Derivation of the Estimator

(2)

Goodness of Fit

(1)

The Projection Matrices

(1)

Linear Transformations

(1)

The Partitioned Linear Model

(1)

The Statistical Model

(3)

Model Specification

(5)

Linearity

(1)

Included Variables

(2)

Relevant Variables

(2)

Statistical Analysis of the Regression Model

(20)

Statistical Assumptions

(1)

The Properties of OLS

(5)

Mean and Variance

(2)

The Residuals

(1)

The Gauss--Markov Theorem

(3)

Generalized Least Squares

(3)

Failure of the Assumptions

(1)

Aitken's Theorem

(1)

Interpreting Residual Autocorrelation

(1)

The Gaussian Linear Model

(6)

Interval Estimation

(2)

Testing Hypotheses

(1)

The F Test of Linear Restrictions

(1)

The Constrained Estimation Approach

(1)

Significance Tests

(2)

Stability Analysis

(5)

Structural Breaks

(1)

Prediction

(4)

Asymptotic Analysis of the Regression Model

(20)

Stochastic Convergence

(3)

Modes of Convergence

(2)

More on Testing Exogeneity

188

(2)

Testing Overidentifying Restrictions

190

(1)

Nonlinear Simultaneous Equations

191

(6)

Fisher's Identification Analysis

191

(2)

Example

193

(1)

Kelejian's Approach

194

(3)

III Advanced Estimation Theory

197

(138)

Optimization Estimators I: Theory

199

(35)

Preliminaries

199

(3)

Numerical Optimization

202

(11)

Line Search

203

(2)

Gradient Methods for Multivariate Optimization

205

(3)

Quasi-Newton Methods

208

(2)

Direct Search Methods

210

(2)

Practical Implementation

212

(1)

Asymptotic Properties

213

(8)

Existence of OEs

213

(2)

Consistency

215

(2)

Identification

217

(2)

Asymptotic Normality

219

(2)

Other Issues

221

(8)

Model Selection

221

(3)

Two-step Estimation

224

(2)

Estimation of Covariance Matrices

226

(3)

Appendix: Additional Proofs

229

(5)

Optimization Estimators II: Examples

234

(28)

Linear Least Squares

234

(1)

Nonlinear Least Squares

235

(10)

Computation

236

(1)

Asymptotic Properties

236

(2)

The AR(1) Error Model: Computation

238

(2)

The AR(1) Error Model: Asymptotics

240

(3)

The MA(1) Error Model

243

(1)

The Adaptive Expectations/Stock Adjustment Model

243

(1)

The Error Correction Model

244

(1)

Generalized Method of Moments

245

(4)

Asymptotic Theory

245

(2)

A Nonlinear Example

247

(1)

Application to Rational Expectations

247

(1)

Nonlinear 2SLS

248

(1)

Modelling Second Moments

249

(7)

Residual Variance Estimation

249

(3)

Conditional Heteroscedasticity and ARCH Models

252

(4)

Misspecified Regression Models

256

(4)

Appendix: Proof of Theorem 10.5.3

260

(2)

The Method of Maximum Likelihood

262

(21)

Introduction

262

(2)

Examples

264

(7)

The Classical Gaussian Regression Model

264

(1)

The Stationary Gaussian AR Model

265

(2)

The Dynamic Regression Model

267

(1)

Conditional Heteroscedasticity

267

(1)

Discrete and Censored Data Models

268

(3)

Properties of the MLE

271

(9)

Preliminaries

271

(1)

Consistency and Identification

272

(2)

Asymptotic Normality

274

(2)

Model Selection

276

(1)

Computational Issues

277

(1)

Asymptotic Efficiency

278

(2)

General Distributions

280

(3)

Testing Hypotheses

283

(25)

Basic Ideas

283

(4)

Concepts and Definitions

283

(2)

Test Principles

285

(2)

The Wald Test

287

(2)

Tests Based on Constrained Estimation

289

(8)

Distribution of the Constrained OE

289

(1)

The Likelihood Ratio Test

290

(2)

The Lagrange Multiplier Test

292

(2)

Testing for Autocorrelation

294

(2)

Testing for Heteroscedasticity

296

(1)

Power Calculations

297

(4)

Pitman Drift

297

(2)

Asymptotic Distributions under HA

299

(2)

M Tests

301

(7)

Theory

301

(3)

M Tests for Heteroscedasticity

304

(1)

Tests of Functional Form

304

(2)

The Information Matrix Test

306

(2)

System Estimation

308

(27)

Three-stage Least Squares

308

(6)

Construction of 3SLS

308

(1)

Distribution of 3SLS

308

(3)

Special Cases

311

(3)

Full Information Maximum Likelihood

314

(3)

Set-up and Computation

314

(1)

Properties

315

(2)

Least Generalized Variance

317

(1)

Least Generalized Variance Ratio

317

(4)

Subsystem Estimation

317

(1)

LIML

318

(1)

Asymptotic Equivalence of LIML and 2SLS

319

(2)

Nonlinear FIML

321

(6)

Preliminaries

321

(1)

Computation

322

(1)

Consistency of the MLE

323

(2)

Quasi-FIML

325

(2)

System GMM

327

(8)

The One-stage Estimator

327

(1)

The Efficient Case

328

(2)

The Two-stage Estimator

330

(2)

Nonlinear SUR

332

(3)

IV Cointegration Theory

335

(92)

Unit Roots

337

(23)

The Random Walk Model

337

(1)

The Probability Background

338

(7)

Function Spaces

338

(2)

Brownian Motion

340

(2)

The Functional CLT

342

(3)

The Unit Root Autoregression

345

(7)

Basic Convergence Results

345

(1)

Tests of the I(1) Hypothesis

346

(3)

Serial Correlation

349

(2)

Including an Intercept

351

(1)

Allowing Deterministic Trends

352

(3)

Distribution of the AR Coefficient

352

(2)

Tests of I(1)

354

(1)

Testing the Null of I(0)

355

(3)

Appendix: Proof of Theorem 14.3.1

358

(2)

Cointegrating Regression

360

(28)

Cointegrated Time Series

360

(4)

Basic Concepts

360

(1)

Cointegrated Equation Systems

361

(1)

The VECM Representation

362

(2)

Limit Theory for Cointegrating Regressions

364

(14)

Limit Theory Preliminaries

364

(2)

Static Least Squares

366

(3)

When Static Least Squares is Optimal

369

(2)

Including I(0) Regressors

371

(4)

The Error Correction Model Approach

375

(1)

Fully Modified Least Squares

376

(2)

Testing for Cointegration

378

(4)

Regression Without Cointegration

378

(1)

Residual-based Tests

379

(2)

Correcting for Autocorrelation

381

(1)

Drawbacks with Residual-based Tests

382

(1)

Trended Variables

382

(4)

Simple Regression

382

(2)

Detrending the Data

384

(1)

Testing for Cointegration

385

(1)

A Postscript on OE Analysis

386

(2)

Cointegrated Systems

388

(39)

The VECM Framework

388

(3)

Modelling Issues

388

(2)

Least Squares Analysis

390

(1)

Johansen's Analysis

391

(4)

The General Dynamic Model

391

(2)

Reduced Rank Regression

393

(2)

Inference in the Cointegrating VAR

395

(8)

Tests for Cointegrating Rank

395

(3)

Likelihood Ratio Tests

398

(1)

Inference on β

399

(3)

The Other Parameters

402

(1)

Allowing Deterministic Trends

403

(7)

Modifications to the Analysis

403

(2)

The Cointegrating Vectors

405

(2)

Restricting and Testing Trends

407

(2)

Trends in the Cointegration Space

409

(1)

Partial Models and Exogeneity

410

(5)

The Conditional Model

410

(2)

Testing Cointegrating Rank

412

(1)

Estimating β

413

(2)

Structural Cointegrating Relations

415

(9)

Identification

415

(4)

Single Equation Analysis

419

(2)

System Analysis

421

(3)

Appendix: Additional Proofs

424

(3)

V Technical Appendices

427

(42)

A Matrix Algebra Basics

429

(12)

A.1 Vectors and Matrices

429

(2)

A.2 Matrix Inversion

431

(2)

A.3 Linear Dependence and Rank

433

(1)

A.4 Equation Systems

433

(1)

A.5 Eigenvalues and Eigenvectors

434

(1)

A.6 Diagonalization of Symmetric Matrices

435

(1)

A.7 Definite Matrices

435

(2)

A.8 Generalized Inverses

437

(1)

A.9 Matrix Calculus

438

(1)

A.10 Vecs and Kronecker Products

439

(2)

B Probability and Distribution Theory

441

(20)

B.1 Set Theory Basics

441

(1)

B.2 Probability

442

(1)

B.3 Random Variables

443

(3)

B.4 Expectations

446

(4)

B.5 Joint Distributions

450

(3)

B.6 Conditional Expectations

453

(2)

B.7 Independence

455

(1)

B.8 Random Vectors

456

(1)

B.9 Change of Variable

457

(1)

B.10 Conditioning on a Sigma-field

458

(3)

C The Gaussian Distribution and its Relatives

461

(8)

C.1 The Univariate and Multivariate Cases

461

(3)

C.2 The Conditional and Marginal Distributions

464

(1)

C.3 The Chi-squared, F and t Distributions

465

(1)

C.4 Quadratic Forms in Normal Variables

466

(3)

References

469

(16)

Author Index

485

(4)

Subject Index

489

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Econometric Theory

Summary

Author Biography

Table of Contents

Supplemental Materials

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Econometric Theory

Summary

Author Biography

Table of Contents

Supplemental Materials

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