Unit Roots, cointergration, and Structural Change

th e m e s in m o d e r n e c o n o m e t r i c s

Unit Roots,

Cointegration,

and

Structural Change

G. S. Maddala and I n - M o o Kim

U n it R o o ts, C o in te g ra tio n , an d S tr u c tu ra l C h an ge

Tim e series analysis has undergone m any changes in

recent years w ith the advent of unit roots and

cointegration. M addala and Kim present a comprehensive

review of these im portant developm ents and exam ine

structural change. The volume provides an analysis of

unit root tests, problem s w ith unit root testing,

estim ation of cointegration systems, cointegration tests,

and econom etric estim ation w ith integrated regressors.

T he authors also present the Bayesian approach to these

problem s and boo tstrap m ethods for small-sample

inference. T he chapters on structural change discuss the

problem s of unit root tests and cointegration under

stru ctu ral change, outliers and robust m ethods, the

Markov sw itching model, and H arvey’s stru ctu ral tim e

series model. Unit Roots, Cointegration, and Structural

Change is a m ajor contribution to T h em es in M od ern

E c o n o m etrics, of interest bo th to specialists and

graduate and u p p er-undergraduate students.

G. S. MADDALA is University Em inent Scholar a t the

Ohio S tate U niversity and one of th e m ost distinguished

econom etricians w riting today. His m any acclaimed

publications include L im ited Dependent and Qualitative

Variables in Econom etrics (Cam bridge, 1983) and

Econom etrics (M cGraw-Hill, 1977) and Introduction to

Econom etrics (M acM illan, 1988, 1992).

IN-M OO KIM is Professor of Economics at Sung Kyun

Kwan University, Seoul, Korea.

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UNIT ROOTS

COINTEGRATION

AND STRUCTURAL CHANGE

G . S. M a d d a la

The Ohio State University

In -M o o K im

Sung K yun Kwan University

i C a m b r i d g e

UNIVERSITY PRESS

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PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The P itt Building, TYumpington Street, Cam bridge CB2 1RP, U nited Kingdom

CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge CB2 2RU, UK http://w w w .cup.cam .ac.uk

40 W est 20th Street, New York, NY 10011-4211, USA h ttp ://w w w .cu p .o rg

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© Cambridge University Press 1998

This book is in copyright. Subject to statu to ry exception

and to th e provisions of relevant collective licensing agreem ents,

no reproduction of any part may take place w ithout

the w ritten permission of Cam bridge U niversity Press.

First published 1998

Printed in th e United Kingdom at the University Press. C am bridge

Typeset in C om puter M odern 10/13pt, in L^lfeX2e [TAG]

A catalogue record o f this book is available from the British Library

Library o f Congress cataloguing in publication data applied fo r

ISBN 0 521 58257 1 hardback

ISBN 0 521 58782 4 paperback

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G. S. M addala

To m y p a r e n ts

T o J o n g H a n , J u n g Y o u n , a n d S o Y o u n

In-M oo Kim

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Contents

Figures

Tables

Preface

Part I Introduction and basic concepts

1 Introduction

References

2 Basic concepts

2.1 Stochastic processes

2.2 Some commonly used stationary models

2.3 Box Jenkins m ethods

2.4 Integrated variables and cointegration

2.5 Spurious regression

2.6 D eterm inistic trend and stochastic trend

2.7 Detrending m ethods

2.8 VAR. ECM . and ADL

2.9 U nit root tests

2.10 Cointegration tests and ECM

2.11 Sum m ary

References

Part II Unit roots and cointegration

3 U nit roots

3.1 Introduction

3.2 U nit roots and W iener processes

3.3 U nit root tests w ithout a determ inistic trend

3.4 DF test w ith a linear determ inistic trend

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page xii

xiii

xvii

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8

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3.5 Specification of determ inistic trends

74 3.6 U nit root tests for a wide class of errors '

82 3.7 Sargan Bhargava and Bhargava tests

3.8 Variance ratio tests

3.9 Tests for T SP versus DSP ^

3.10 Forecasting from TS versus DS models

3.11 Sum m ary and conclusions ^2

References 92

4 Issues in unit root testing

4.1 Introduction 98

4.2 Size distortion and low power of unit root tests 100

4.3 Solutions to the problem s of size and power 103

4.4 Problem of overdifferencing: MA roots 116

4.5 Tests with stationarity as null 120

4.6 C onfirm atory analysis 126

4.7 Frequency of observations and power of unit root tests 129

4.8 O ther types of nonstationarity 131

4.9 Panel d ata unit root tests 133

4.10 U ncertain unit roots and the pre-testing problem 139

4.11 O ther unit root tests 140

4.12 M edian-unbiased estim ation 141

4.13 Sum m ary and conclusions 145

References 146

5 E stim ation of cointegrated system s 155

5.1 Introduction 155

5.2 A general C l system 155

5.3 A two-variable model: E ngle-G ranger m ethods 156

5.4 A triangular system 160

5.5 System estim ation m ethods 165

5.6 The identification problem 173

5.7 F inite sam ple evidence 175

5.8 Forecasting in cointegrated system s lg 4

5.9 M iscellaneous other problem s lg 7

5.10 Sum m ary and conclusions 191

References 191

6 Tests for cointegration igg

6.1 Introduction igg

6.2 Single equation m ethods: residual-based tests j<^

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72

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Contents ix

6.3 Single equation m ethods: ECM tests 203

6.4 Tests w ith cointegration as null 205

6.5 M ultiple equation m ethods 211

6.6 C ointegration tests based on LCCA 222

6.7 O ther tests for cointegration 226

6.8 Miscellaneous other problems 228

6.9 Of w hat use are cointegration tests? 233

6.10 Conclusions 241

References 242

7 Econometric modeling w ith integrated regressors 249

7.1 1(1) regressors not cointegrated 249

7.2 1(1) regressors cointegrated 250

7.3 Unbalanced equations 251

7.4 Lagged dependent variables: the ARDL m odel 252

7.5 U ncertain unit roots 254

7.6 U ncertain unit roots and cointegration 256

7.7 Sum m ary and conclusions 258

References 258

Part III E xtensions o f the basic model 261

8 The Bayesian analysis of stochastic trends 263

8.1 Introduction to Bayesian inference 264

8.2 The posterior distribution of an autoregressive param eter 266

8.3 Bayesian inference on the N elson-Plosser d a ta 268

8.4 The debate on the appropriate prior 271

8.5 Classical tests versus Bayesian tests 277

8.6 Priors and tim e units of m easurem ent 277

8.7 On testing point null hypotheses 278

8.8 F urther com m ents on prior distributions 284

8.9 Bayesian inference on cointegrated system s 287

8.10 Bayesian long-run prediction 290

8.11 Conclusion 291

References 292

9 Fractional unit roots and fractional cointegration 296

9.1 Some definitions 296

9.2 U nit root tests against fractional alternatives 298

9.3 E stim ation of ARFIM A models 300

9.4 E stim ation of fractionally cointegrated m odels 302

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X Contents

9.5 Em pirical relevance of fractional unit roots

9.6 Sum m ary and conclusions 305

References 306

10 Small sam ple inference: boo tstrap m ethods 309

10.1 Introduction 309

10.2 A review of the boo tstrap approach 309

10.3 The AR(1) model 322

10.4 B ootstrapping unit root tests 325

10.5 The moving block bootstrap and extensions 328

10.6 Issues in bootstrapping cointegrating regressions 332

10.7 Miscellaneous other applications 335

10.8 Conclusions 336

References 336

11 C ointegrated systems w ith 1(2) variables 342

11.1 D eterm ination of the order of differencing 342

11.2 C ointegration analysis w ith 1(2) and 1(1) variables 348

11.3 Em pirical applications 355

11.4 Sum m ary and conclusions 358

References 359

12 Seasonal unit roots and seasonal cointegration 362

12.1 Effect of seasonal adjustm ent 364

12.2 Seasonal integration 365

12.3 Tests for seasonal unit roots 366

12.4 T he unobserved com ponent model 371

12.5 Seasonal cointegration 375

12.6 E stim ation of seasonally cointegrated system s 376

12.7 Em pirical evidence 37g

12.8 Periodic autoregression and periodic integration 379

12.9 Periodic cointegration and seasonal cointegration 381

12.10 Tim e aggregation and system atic sam pling 3g j

12.11 Conclusion 3 g2

References g g j

Part I V Structural change

13 S tructural change, unit roots, and cointegration 339

13.1 T ests for stru ctu ral change jqq

13.2 Tests w ith known break points ;jgQ

13.3 Tests w ith unknown break points

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13.4 A sum m ary assessment 398

13.5 Tests for unit roots under structural change 399

13.6 T he Bayesian approach 402

13.7 A sum m ary assessment of the em pirical work 407

13.8 Effect of structural change on cointegration tests 410

13.9 Tests for structural change in cointegrated relationships 411

13.10 Miscellaneous other issues 414

13.11 Practical conclusions 416

References 418

14 Outliers and unit roots 425

14.1 Introduction 425

14.2 Different types of outliers in tim e series models 425

14.3 Effects of outliers on unit root tests 428

14.4 Outlier detection 437

14.5 Robust unit root tests 440

14.6 Robust estim ation of cointegrating regressions 445

14.7 Outliers and seasonal unit roots 448

14.8 Conclusions 448

References 449

15 Regime sw itching models and stru ctu ral tim e series models 454

15.1 The sw itching regression model 454

15.2 The M arkov sw itching regression model 455

15.3 T he H am ilton m odel 457

15.4 On the usefulness of the MSR model 460

15.5 Extensions of th e MSR model 463

15.6 G radual regime sw itching models 466

15.7 A model w ith param eters following a random walk 469

15.8 A general state-space model 470

15.9 D erivation of the K alm an filter 472

15.10 Harvey's stru ctu ral tim e series m odel (1989) 475

15.11 F urther com m ents on stru ctu ral tim e series models 477

15.12 Sum m ary and conclusions 479

References 479

16 Future directions 486

References 488

Contents xi

Appendix 1 A brief guide to asym ptotic theory

A uthor index

Subject index

490

492

500

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Figures

2.1 Correlogram of an AR(2) model 16

2.2 Exam ples of two AR(1) processes w ith a drift 23

2.3 The variances of x t and yt 24

2.4 The autocorrelations of x t and yt 25

2.5 C ointegrated and independent 1(1) variables 27

2.6 ARIM A(0,1,1) and its com ponents 31

8.1 M arginal posterior distributions of p w hen p = 1 272

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Tables

2.1 Regression of integrated variables 32

3.1 Critical values for Dickey-Fuller tests 64

3.2 A sym ptotic distributions of the ¿-ratios for different D G Ps 71

3.3 Critical values for the Schm idt-Phillips LM test 85

3.4 Nelson and Plosser’s results 89

4.1 Critical values of D F max statistics 112

4.2 C ritical values for the E lliott-R othenberg-S tock D F-G LS

test 114

4.3 C ritical values for the H w ang-Schm idt D F-G L S test (¿-test) 116

4.4 C ritical values for th e KPSS test 122

4.5 Q uantiles of th e LS estim ator in an AR(1) model w ith drift

and trend 143

6.1 C ritical values for the A D F ¿-statistic and Z t 200

6.2 C ritical values for th e Z a 200

6.3 Response surface estim ates of critical values 201

6.4 C ritical values for the H arris and Inder test 210

6.5 Q uantiles of the asym ptotic distribution of the Johansen’s

LR test statistics 213

6.6 C ritical values of the LCCA-based tests 224

7.1 Features of regressions am ong series w ith various orders of

integration 252

8.1 Posterior probabilities for the Nelson Plosser d ata 270

12.2 C ritical values for seasonal unit roots in m onthly d ata 371

13.1 A sym ptotic critical values for the diagnostic test 412

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N othing is so powerful as an idea whose tim e has come.

Victor Hugo

The Gods love the obscure and hate the obvious.

B rihadaranyaka U panishad

Undue emphasis on niceties is a disease to which persons w ith m athe￾m atical training are especially prone.

G. A. B arnard, “A Com m ent on E. S. Pearson’s P aper,”

Biom etrika, 1947, 34, 123-128.

Simplicity, simplicity, simplicity! I say, let your affairs be as two or three,

and not a hundred or a thousand. Simplify, simplify.

H. D. T horeau: Walden

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Preface

The area of unit roots, cointegration, and stru ctu ral change has been

area of intense and active research during the past decade. Developme

have been proceeding a t a fast pace. However, alm ost all th e books

technically oriented and do not bring together the different strands

research in this area. Even if m any new developm ents are going to t;

place, we thought it tim e to provide an overview of this area for 1

benefit of em pirical as well as theoretical researchers. Those who

doing empirical research will benefit from th e com prehensive cover;

of the book. For those who are doing theoretical research, particula

graduate students starting on their dissertation work, the present b(

will provide an overview and perspective of this area. It is very ei

for graduate students to get lost in the intricate algebraic detail o

particular procedure and lose sight of the general framework their w

fits in to.

Given the broad coverage we have aim ed at, it is possible th a t we h,

missed several papers. T his is not because they are not im portant 1

because of oversight a n d /o r our inability to cover too m any topics.

To keep the book w ithin reasonable length and also to provide acc<

ibility to a broad readership, we have om itted the proofs and derivati'

throughout. T hese can be found by interested readers in the pap

cited. P a rts of th e book were used a t different tim es in graduate coui

a t the U niversity of Florida, the Ohio S tate University, Caltech, St

University of New York a t Buffalo, and Sung K yun Kwan University

Korea.

We would like to thank Adrian Pagan, P eter Phillips, and an ano

m ous referee for m any helpful com m ents on an earlier d ra ft. T hanks

also due to professor Chul-Hwan Kim of Ajou University, Young Se P

of Sung K yun Kwan University, and M arine C arrasso of the Ohio St

xvii

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XVU1 Preface

University for their helpful comments. Responsibility for any remaining

errors is ours. We would also like to thank Patrick M cC artan at the

am ri ge University Press for his patience in the production of this

G. S. M addala

The Ohio S tate I Diversity, U.S.A.

In-M oo Kim

Sung Kyun Kwan University, Seoul, Korea

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