Dynamic Econometrics

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ISBN: 9780198283164 | 0198283164
Cover: Paperback
Copyright: 4/13/1995

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The main problem in econometric modelling of time series is discovering sustainable and interpretable relationships between observed economic variables. The primary aim of this book is to develop an operational econometric approach which allows constructive modelling. Professor Hendry deals with methodological issues (model discovery, data mining, and progressive research strategies); with major tools for modelling (recursive methods, encompassing, super exogeneity, invariance tests); and with practical problems (collinearity, heteroscedasticity, and measurement errors). He also includes an extensive study of US money demand. The book is self-contained, with the technical background covered in appendices. It is thus suitable for first year graduate students, and includes solved examples and exercises to facilitate its use in teaching.

List of Figures

xxiii

List of Tables

xxvii

Preface

xxix

I Concepts, Models, and Processes in Econometrics

(368)

Introduction

(28)

Empirical econometric modelling

(2)

The problems of econometrics

(1)

The aims of this book

(1)

Constructive and destructive approaches

(2)

A brief discourse on scientific method

(2)

Theories, instruments, and evidence

(2)

Economic analysis and statistical theory

(3)

Four levels of knowledge

(3)

Probability theory

(1)

Estimation theory

(1)

Modelling theory

(1)

Forecasting theory

(1)

The origins of the methodological crisis

(1)

Some economic time series

(2)

A first data-generation process

(6)

Empirical models as derived entities

(2)

Exercises

(2)

Econometric Concepts

(44)

Parameter

(1)

Constancy

(1)

Structure

(1)

Distributional shape

(2)

Identification and observational equivalence

(1)

Interdependence

(1)

Stochastic process

(1)

Conditioning

(1)

White noise

(1)

Autocorrelation

(2)

Stationarity

(1)

Integratedness

(1)

Cointegration

(1)

Trend

(1)

Heteroscedasticity

(1)

Dimensionality

(1)

Aggregation

(2)

Marginalization

(1)

A general formulation

(2)

A static solved example

(4)

Models, mechanisms, and DGPs

(1)

Factorizations

(2)

Innovation processes

(2)

Empirical models

(3)

White noise and innovations

(1)

Autocorrelated shocks

(1)

Sequential factorization

(2)

Model design

(1)

A dynamic solved example

(4)

Exercises

(3)

Econometric Tools and Techniques

(47)

Review

(1)

Estimating unknown parameters

(9)

An empirical example

(2)

Recursive estimation

(4)

A solved recursive example

(3)

Methods for evaluating models

(1)

Statistical theory

(1)

Asymptotic distribution theory

(1)

Monte Carlo

(11)

Distribution sampling

(2)

Antithetic variates

(3)

Control variables

(1)

Response surfaces

(1)

Invariance

(2)

Recursive Monte Carlo

(1)

Ergodicity

(2)

Non-stationarity*

100

(4)

A solved example*

104

(7)

Vector Brownian motion*

111

(4)

A Monte Carlo study

115

(3)

Exercises

118

(4)

Dynamics and Interdependence

122

(34)

Nonsense regressions

122

(8)

Analysing nonsense regressions*

130

(3)

Spurious detrending*

133

(1)

First-order autoregressive dynamics

134

(4)

Reduction and dynamics

138

(2)

Interdependence

140

(3)

Cointegration

143

(1)

Bivariate dynamics

143

(2)

A solved example

145

(5)

Exercises

150

(6)

Exogeneity and Causality

156

(39)

What are `exogenous variables'?

156

(1)

Two counter-examples

157

(5)

An expectations counter-example

158

(3)

A fixed-regressor counter-example

161

(1)

Weak exogeneity

162

(2)

A cobweb model

164

(2)

The counter-examples reconsidered

166

(1)

An ambiguity in strict exogeneity

167

(1)

Can the model mis-specification be detected?

168

(2)

ζt is not an innovation relative to Xt-1

169

(1)

ζt is white noise

169

(1)

Strong exogeneity

170

(2)

Super exogeneity

172

(1)

An illustration of super exogeneity

173

(2)

Causality

175

(2)

Granger non-causality

175

(1)

Invariance under interventions

176

(1)

Two solved examples

177

(4)

Weak exogeneity and unit roots*

181

(10)

A bivariate cointegrated system

182

(1)

Six cases of interest

183

(1)

Limiting distributions

184

(3)

Inference

187

(1)

A Monte Carlo study

188

(3)

Exercises

191

(4)

Interpreting Linear Models

195

(36)

Four interpretations of yt = β1 zt + &epsis;t

195

(7)

Regression

196

(1)

Linear least-squares approximation

197

(1)

Contingent plan

198

(2)

Behavioural model

200

(2)

Expectations formation

202

(5)

Rational expectations

202

(2)

Unbiased expectations

204

(1)

Data-based expectations

205

(2)

Autocorrelation corrections

207

(4)

A simple dynamic model: AD(1,1)

211

(1)

Lags and their measurement

212

(7)

Static solutions

212

(2)

Lag distributions

214

(3)

Interpreting empirical lag distributions

217

(2)

A Monte Carlo study of the AD(1,1) model

219

(4)

Coefficient biases

220

(1)

Coefficient standard errors

221

(2)

Parameter constancy tests

223

(1)

An empirical illustration

223

(4)

A solved example

227

(2)

Exercises

229

(2)

A Typology of Linear Dynamic Equations

231

(78)

Introduction

231

(2)

Static regression

233

(8)

Autoregression

241

(6)

Differenced-data model

247

(5)

Leading indicator

252

(4)

Partial adjustment

256

(10)

Common factor

266

(7)

Finite distributed lags

273

(10)

Dead-start models

283

(3)

Equilibrium correction

286

(8)

Cointegration

288

(2)

Servo-mechanistic control

290

(1)

Empirical success of ECMs

291

(3)

Solved examples

294

(10)

Partial adjustment and equilibrium correction

294

(3)

Testing cointegration*

297

(6)

Cointegration representations

303

(1)

Summary and conclusion

304

(2)

Goodness of fit

304

(1)

Distributed-lag shapes

304

(2)

Exercises

306

(3)

Dynamic Systems

309

(35)

Introduction

309

(1)

Statistical formulation

310

(1)

Theoretical formulation

311

(2)

Closed linear systems

313

(3)

Formulation

313

(1)

Invariance under linear transformations

313

(2)

Cointegration

315

(1)

Open linear systems

316

(2)

Conditional systems

316

(1)

Simultaneous equations

317

(1)

Cointegration

317

(1)

Modelling dynamic systems

318

(3)

The economy is a system

319

(1)

To test marginalization

319

(1)

Simultaneity

319

(1)

To test weak exogeneity

320

(1)

To check identification

320

(1)

Cointegration is a system property

320

(1)

To test cross-equation dependencies

320

(1)

To investigate system dynamics

321

(1)

A typology of open linear dynamic systems

321

(8)

Vector equilibrium-correction system

322

(1)

Static system

323

(1)

Vector autoregression

323

(1)

Differenced system

323

(1)

Leading indicator

323

(1)

Partial adjustment

324

(1)

Common factor

324

(1)

Finite distributed lag

324

(1)

Dead start

324

(1)

An empirical illustration

325

(4)

Models of linear systems

329

(8)

Vector autoregressive representation

329

(1)

Vector equilibrium correction

330

(1)

Simultaneous-equations model

331

(1)

Conditional model

332

(1)

Conditional simultaneous model

333

(1)

Causal chain

333

(1)

Block-recursive representation

334

(1)

Triangular representation

335

(1)

Empirical illustrations

335

(2)

Analysing dynamic systems

337

(6)

Dynamic multipliers

339

(1)

Final forms

340

(1)

Non-linearity

340

(1)

Dynamic simulation

341

(2)

Exercises

343

(1)

The Theory of Reduction

344

(25)

Introduction

344

(2)

Data transformations and aggregation

346

(1)

Parameters of interest

347

(3)

Data partition

350

(1)

Marginalization

350

(2)

Sequential factorization

352

(1)

Sequential factorization of WT1

352

(1)

Marginalizing with respect to VT1

352

(1)

Mapping to 1(0)

353

(1)

Conditional factorization

354

(1)

Constancy

355

(1)

Lag truncation

356

(1)

Functional form

356

(1)

The derived model

357

(2)

Econometric concepts as measures of no information loss

359

(2)

Implicit model design

361

(1)

Explicit model design

361

(1)

A Taxonomy of evaluation information

362

(5)

Past data

363

(1)

Present data

363

(1)

Future data

363

(1)

Theory information

363

(1)

Measurement information

364

(1)

Rival models

364

(3)

Exercises

367

(2)

II Statistical Tools for Econometric Analysis

369

(130)

Likelihood

371

(34)

Review of Part I

371

(2)

The statistical model

373

(1)

Estimation criteria and estimation methods

374

(3)

The likelihood function

377

(1)

Maximum likelihood estimation

378

(1)

Properties of the score

379

(3)

Properties of maximum likelihood estimators

382

(2)

Large-sample properties of MLEs

384

(3)

Two solved examples

387

(4)

Misleading inference when V ≠ I-1

391

(2)

Derived distributions

393

(1)

Asymptotic equivalence

394

(1)

Concentrated likelihood functions

395

(1)

Marginal and conditional distributions

396

(1)

Estimator generating equations

397

(1)

An EGE for common-factor dynamics

398

(1)

Exercises

399

(6)

Simultaneous Equations Systems

405

(37)

Introduction

405

(1)

The statistical system

406

(1)

System dynamic specification

407

(2)

System estimation

409

(3)

System cointegration estimation

412

(6)

System evaluation

418

(1)

Empirical cointegration illustration

419

(2)

The econometric model

421

(1)

Identification

422

(1)

An EGE for simultaneous equations estimation

423

(7)

Non-linear parameters

428

(1)

Vector autoregressive errors

429

(1)

A solved example

430

(3)

Simultaneous equations modelling

433

(1)

Derived statistics

434

(2)

Empirical model estimates

436

(4)

Exercises

440

(2)

Measurement Problems in Econometrics

442

(26)

Introduction

442

(1)

Errors in variables

443

(8)

Analysing the errors-in-variables model

445

(4)

Instrumental variables

449

(2)

Dynamic latent-variables models

451

(8)

Method of simulated moments

453

(2)

Simulated likelihood functions

455

(1)

A dynamic min-condition model

456

(2)

Exogeneity issues

458

(1)

Revisions to I(1) data

459

(4)

Revisions to price indices

459

(2)

An empirical illustration

461

(2)

The impact of measurement errors on ECMs

463

(3)

Exercises

466

(2)

Testing and Evaluation

468

(31)

The statistical framework

468

(2)

Non-central X2 distributions

470

(2)

Large-sample properties of tests

472

(2)

Understanding the non-central X2 distribution

474

(2)

Test power

476

(1)

Likelihood-ratio, Wald, and Lagrange-multiplier tests

477

(3)

Likelihood-ratio tests

478

(1)

Wald tests

479

(1)

Lagrange-multiplier or efficient-score tests

480

(1)

Comparing the tests

480

(2)

A solved example

482

(3)

Non-linear restrictions

485

(2)

Some methodological considerations

487

(5)

The null hypothesis

487

(1)

The alternative hypothesis

488

(1)

The test statistics

488

(2)

The significance level

490

(1)

Multiple testing

490

(2)

Exercises

492

(7)

III Empirical Modelling

499

(122)

Encompassing

501

(43)

Introduction

501

(2)

Augmenting the conventional testing strategy

503

(2)

Rejection is not final

503

(1)

Corroboration is not definitive

504

(1)

Encompassing and mis-specification analysis

505

(1)

Formalizing encompassing

506

(3)

Levels of analysis

509

(2)

Specification encompassing

510

(1)

Mis-specification encompassing

510

(1)

Selection encompassing

510

(1)

Parsimonious encompassing

511

(1)

A simple example

512

(2)

Can M1 &epsis;M2?

513

(1)

Can M2 &epsis;M1?

514

(1)

Nesting and encompassing

514

(2)

Encompassing in linear regression

516

(4)

Encompassing in stationary stochastic processes

520

(3)

A solved example

523

(2)

An empirical illustration

525

(2)

Encompassing the VAR

527

(2)

Testing the Lucas critique

529

(2)

The applicability of the critique

531

(1)

Tests for super exogeneity

532

(2)

Encompassing implications of feedback versus feedforward models

534

(2)

Does Hb &epsis;He ?

535

(1)

Does He &epsis;Hb ?

535

(1)

Incomplete information

536

(1)

Implications

536

(1)

Empirical testing of invariance

536

(3)

Testing super exogeneity

537

(1)

Testing encompassing

538

(1)

Exercises

539

(5)

Modelling Issues

544

(33)

Data mining

544

(2)

Theory dependence versus sample dependence

546

(4)

Theory-driven approaches

546

(1)

Data-driven approaches

547

(1)

Bayesian approaches

547

(1)

Data modelling using economic theory guidelines

547

(3)

Progressive research strategies

550

(4)

Identified cointegration vectors

550

(2)

Orthogonal parameters

552

(1)

Inappropriate estimation

553

(1)

Common trends

553

(1)

Residual analysis

553

(1)

Expectations and structure

553

(1)

Pyrrho's lemma

554

(3)

Dummy variables

557

(2)

Seasonal adjustment

559

(6)

Seasonal filters

561

(1)

Properties of seasonal filters

562

(1)

Cointegration

563

(2)

Approximating moving-average processes

565

(3)

A solved example: modelling second moments

568

(6)

Populations and samples

574

(2)

Exercises

576

(1)

Econometrics in Action

577

(44)

The transactions demand for money

577

(2)

Economic theories of money demand

579

(2)

Econometric formulation

581

(2)

Financial innovation

583

(2)

Data description

585

(6)

A small monetary system

591

(6)

Cointegration analysis

597

(3)

Modelling the I(0) PVAR

600

(4)

Evaluating the model

604

(2)

A single-equation money-demand model

606

(5)

Transformation and reduction

611

(3)

Post-modelling evaluation

614

(2)

Learning adjustment

615

(1)

Constancy

615

(1)

Encompassing

616

(1)

Testing the Lucas critique

616

(2)

Post-sample evaluation

618

(1)

Policy implications

618

(1)

Data definitions

619

(2)

IV Appendices

621

(198)

A1 Matrix Algebra

623

(16)

A1.1 Summary of the Appendix chapters

623

(2)

A1.2 Matrices

625

(2)

A1.3 Matrix operations

627

(6)

A1.4 Relations between operations

633

(1)

A1.5 Partitioned inverse

634

(1)

A1.6 Polynomial matrices

635

(4)

A2 Probability and Distributions

639

(38)

A2.1 Introduction

639

(1)

A2.1.1 Chance

639

(1)

A2.1.2 Empirical distributions and histograms

639

(2)

A.2.2 Events

641

(1)

A2.2.1 Random experiments, sets, and sample spaces

641

(1)

A2.2.2 Complements, unions, and intersections

642

(2)

A2.2.3 Event space

644

(2)

A2.2.4 Measurability

646

(1)

A2.3 Probability

646

(1)

A2.3.1 Probability spaces

647

(1)

A2.3.2 Conditional probability

648

(3)

A2.3.3 Stochastic independence

651

(2)

A2.4 Random variables

653

(1)

A2.4.1 Mapping events to numbers

653

(1)

A2.4.2 Image sets

654

(1)

A2.4.3 Functions of random variables

655

(1)

A2.5 Distribution and density functions

656

(1)

A2.5.1 Cumulative distribution function

656

(1)

A2.5.2 Density function

656

(1)

A2.5.3 Change of variable

657

(1)

A.2.5.4 Normal distribution

657

(1)

A2.5.5 Parameters, probability models, and distributions

658

(1)

A.2.6 Joint distributions

659

(1)

A2.6.1 Joint distribution functions

659

(1)

A2.6.2 Marginal distributions

659

(2)

A2.6.3 Conditional distributions

661

(1)

A2.7 Expectations

662

(1)

A2.7.1 Expectations, moments, and correlations

662

(1)

A2.7.2 Conditional expectations and minimum variance

663

(2)

A2.7.3 Indicator functions

665

(1)

A2.7.4 Chebychev's inequality

666

(1)

A2.8 Bivariate normal distribution

666

(1)

A2.8.1 Change of variable

666

(1)

A2.8.2 The bivariate normal density

667

(1)

A2.8.3 Conditional normal

668

(1)

A2.8.4 Regression

669

(1)

A2.9 Multivariate normal

670

(1)

A2.9.1 Multivariate normal density

670

(1)

A2.9.2 Multiple regression

670

(1)

A2.9.3 Multivariate regression

671

(1)

A2.9.4 Functions of normal variables: X2, t and F distributions

672

(2)

A2.10Exercises

674

(3)

A3 Statistical Theory

677

(13)

A3.1 Sampling distributions

677

(1)

A3.1.1Statistics

677

(1)

A3.1.2 Derived distributions

678

(2)

A3.1.3 X2, t, and F distributions

680

(1)

A3.1.4 Sufficiency

681

(2)

A3.1.5 Estimation criteria

683

(1)

A3.1.6 Consistency and asymptotic efficiency

684

(1)

A3.2 Likelihood

684

(1)

A3.2.1 Likelihood function

684

(1)

A3.2.2 Log-likelihood

685

(1)

A3.2.3 Estimation

685

(1)

A3.2.4 The score and the Hessian

686

(1)

A3.3 Maximum-likelihood estimation

687

(1)

A3.3.1 Efficiency and the information matrix

687

(1)

A3.3.2 Cramer-Rao bound

688

(1)

A3.3.3 Properties of the information matrix

689

(1)

A3.3.4 Estimating the information matrix

689

(1)

A3.4 Statistical inference and testing

690

(17)

A3.4.1 Null and alternative hypotheses

690

(1)

A3.4.2 Critical regions, error types, and power

690

(1)

A3.4.3 Significance level

691

(1)

A3.5 Powerful tests

692

(1)

A3.5.1 Neyman-Pearson lemma

692

(1)

A3.5.2 Likelihood-ratio, Wald, and efficient-score tests

693

(1)

A3.6 Non-parametric density estimation

694

(1)

A3.7 Multiple regression

695

(1)

A3.7.1 The multiple-regression model

695

(1)

A3.7.2 Ordinary least squares

696

(2)

A3.7.3 The Gauss-Markov theorem

698

(1)

A3.7.4 Distributional results

698

(2)

A3.7.5 Subsets of parameters

700

(2)

A3.7.6 Partitioned inversion

702

(1)

A3.7.7 Regression and inversion

703

(1)

A3.7.8 Multiple correlation

703

(1)

A3.7.9 Partial correlation

704

(1)

A3.7.10 Maximum-likelihood estimation

705

(1)

A3.8 Exercises

706

(1)

A4 Asymptotic Distribution Theory

707

(44)

A4.1 Introduction

707

(3)

A4.2 Orders of magnitude

710

(1)

A4.2.1 Deterministic sequences

710

(1)

A4.2.2 Stochastic sequences

711

(1)

A4.3 Stochastic convergence

712

(2)

A4.4 Laws of large numbers

714

(1)

A4.4.1 Weak law of large numbers

714

(1)

A4.4.2 Strong law of large numbers

714

(1)

A4.5 Central-limit theorems

715

(3)

A4.6 Vector random variables

718

(1)

A4.7 Solved examples

719

(1)

A4.7.1 Example 1: an IID process

719

(1)

A.4.7.2 Example 2: A trend model

720

(2)

A4.8 Stationary dynamic processes

722

(1)

A4.8.1 Vector autoregressive representations

722

(1)

A4.8.2 Mann and Wald's theorem

723

(1)

A4.8.3 Hannan's theorem

724

(1)

A4.8.4 Limiting distribution of OLS for a linear equation

725

(3)

A4.9 Instrumental variables

728

(2)

A4.10 Mixing processes

730

(1)

A4.10.1 Mixing and ergodicity

730

(1)

A4.10.2 Uniform mixing and α-mixing processes

731

(1)

A4.10.3 Laws of large numbers and central-limit theorems

732

(1)

A4.11 Martingale difference sequences

733

(1)

A4.11.1 Constructing martingales

733

(1)

A4.11.2 Properties of martingale-difference sequences

734

(2)

A4.11.3 Applications to maximum-likelihood estimation

736

(2)

A4.12 A solved autoregressive example

738

(3)

A4.13 Higher-order approximations

741

(1)

A4.13.1 Delta method

741

(1)

A4.13.2 Asymptotic expansions

742

(2)

A4.13.3 Power-series expansions

744

(2)

A4.13.4 Addendum

746

(2)

A4.14 Exercises

748

(3)

A5 Numerical Optimization Methods

751

(30)

A5.1 Introduction

751

(2)

A5.2 An overview of numerical optimization

753

(4)

A5.3 Maximizing likelihood functions

757

(2)

A5.4 Scalar optimization

759

(2)

A5.4.1 Search methods

761

(4)

A5.4.2 Gradient methods

765

(2)

A5.5 Multivariate optimization

767

(1)

A5.5.1 Gradient methods

767

(4)

A5.5.2 Step-wise optimization

771

(2)

A5.5.3 Conjugate directions

773

(3)

A5.5.4 Variable metric or quasi-Newton methods

776

(3)

A.5.6 Conclusion

779

(1)

A5.7 Exercises

780

(1)

A6 Macro-Econometric Models

781

(38)

A6.1 Introduction

781

(1)

A6.2 The skeletal structure of macro models

782

(1)

A6.2.1 Modelled variables

782

(1)

A6.2.1 Time lags

783

(1)

A6.2.3 Error terms

784

(1)

A6.2.4 Time aggregation

784

(1)

A6.2.5 Interdependence

785

(1)

A6.2.6 Size

785

(1)

A6.2.7 The economy as a system

785

(1)

A6.3 The national income accounts

786

(1)

A6.3.1 Commodity flows

786

(1)

A6.3.2 Aggregating economic transactions

786

(1)

A6.3.3 Reconciling nominal and real magnitudes

786

(1)

A6.4 The components of macro models

787

(1)

A6.4.1 Kinds of variables

787

(1)

A6.4.2 Kinds of equations

788

(1)

A6.4.3 Behavioural equations

789

(1)

A6.4.4 Identity equations

789

(1)

A6.4.5 Technical equations

790

(1)

A6.4.6 Equilibrium conditions

790

(1)

A6.4.7 Stock-flow equations

791

(1)

A6.4.8 Adjustment equations

791

(1)

A6.4.9 Expectations formation

791

(1)

A6.4.10 Observation equations

792

(1)

A6.5 A simultaneous system of equations

792

(1)

A6.5.1 Price dynamics

792

(1)

A6.5.2 Simultaneity

793

(1)

A6.6 Sectors and markets

794

(1)

A6.6.1 Households

795

(1)

A6.6.2 Firms

796

(2)

A6.6.3 Static-equilibrium solutions

798

(2)

A6.6.4 Dynamic adjustment

800

(1)

A.6.7 Additional aspects of the first model

801

(1)

A6.7 1 Financial markets

801

(1)

A6.7.2 Government sector

801

(1)

A6.7.3 Foreign sector

802

(1)

A6.7.4 Completing equations

803

(2)

A6.7.5 Revised National Income identities

805

(1)

A6.8 Industrial disaggregation

805

(2)

A6.9 A general framework

807

(2)

A6.9.1 Stable process

809

(1)

A6.9.2 Trending process

809

(1)

A6.9.3 Difference stable process

809

(1)

A6.9.4 Quadratic trend process

809

(1)

A6.9.5 Integrated process

810

(1)

A6.9.6 Cointegrated process

810

(1)

A6.10 Forecasting

810

(1)

A6.10.1 Forecast standard errors

811

(1)

A6.10.2 Model evaluation

811

(1)

A6.11 An example

812

(3)

A6.12 Addendum: static-model solution

815

(2)

A6.13 Macro-model notation

817

(2)

References

819

(28)

Common Acronyms

847

(1)

Glossary

848

(5)

Author Index

853

(6)

Subject Index

859

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Dynamic Econometrics

Summary

Table of Contents

Supplemental Materials