Table Of ContentADVANCED TEXT S I N ECONOMETRICS
General Editors
C. W. J. GRANGE R G . E. MIZO N
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CO-INTEGRATION, ERROR
CORRECTION, AND
THE ECONOMETRI C
ANALYSIS O F
NON-STATIONARY DAT A
Anindya Banerjee, Juan J. Dolado,
John W. "Galbraith, and David F . Hendry
OXFORD UNIVERSITY PRESS
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ISBN 0-19-828810-7
Preface
This book is intended as a guide to the literature on co-integration and
modelling of integrated processes . Time-series econometric s ha s devel-
oped rapidly during the past decade, but especially so in the analysis of
non-stationarity. I n particular , th e stud y o f integrate d processe s ha s
grown in importance from the status of an exotic topic, discussed only in
technical journals , t o bein g a n essentia l par t o f th e econometrician' s
collection o f techniques . I t ha s thereb y develope d int o a n are a o f
interest fo r econometri c theorist s an d applie d econometrician s alike .
This book is aimed at graduate students in economics, applie d econo -
metricians, econometri c theorists , an d the genera l audienc e o f econo -
mists who use empirical methods to analyse time series.
Despite the growing importance o f the literatur e o n integration and
co-integration, mos t account s o f thi s literatur e remai n confine d t o
journals, edited collections of papers, o r survey papers. While some of
the surveys are quite detailed, space restrictions usually do not allow a
full expositio n of many of the theoretical points. This book attempts to
bridge th e ga p betwee n account s suc h a s surveys , which ar e mainly
descriptive, an d account s tha t ar e mainl y theoretical. I t explain s th e
important concept s informall y an d als o present s the m formally . Th e
asymptotic theor y o f integrate d processe s i s describe d an d th e tool s
provided b y thi s theor y ar e use d t o derive , i n som e detail , th e
distributions of estimators. By taking readers step by step through some
of th e mai n derivations , ou r hop e i s t o mak e th e theor y readil y
accessible to a wide audience.
We hav e trie d t o mak e th e boo k a s self-contained a s possible. A
knowledge of econometrics, statistics , and matrix algebra at the level of
a final-year undergraduate or first-year graduate course in econometrics
is assumed, but otherwis e all of the importan t statistica l concepts an d
techniques are described.
A book suc h as this one, which discusses an area that is developing
rapidly, i s inevitably incomplete an d run s the ris k of not bein g quit e
up-to-date. To limit the time taken in writing and revising, we did not
seek to chase a frontier that was expanding in many directions. Rather ,
the topics covered reflect our views of issues, models, and methods that
are likely to remain important for some time to come, many of which
will continue to provide the platform for future research.
Acknowledgements
Our book was written in two continents, three years, and four univer-
sities, s o the lis t o f people, acros s time, space , an d departments , t o
whom we owe extensive debts of gratitude has grown formidably large.
A major part of this debt is owed to the Departments of Economics at
the Universitie s o f Californi a a t Sa n Diego , Florid a i n Gainesville ,
McGill, and Oxford, and the Bank of Spain, where the authors either
worked or visited for substantial periods. Their generous support of our
work is much appreciated.
The boo k ha s als o benefite d greatl y from th e patien t scrutin y of
several o f ou r colleagues , wh o rea d th e entir e typescript an d mad e
detailed comments . We hav e pleasure i n thanking Michael Clements ,
Rob Engle, Neil Ericsson, Tony Hall (and several of his students), Colin
Hargreaves, S0re n Johansen , Katarin a Juselius , Teu n Kloek , Jame s
MacKinnon, G . S . Maddala, Grayham Mizon, Jean-Fran9ois Richard,
Mark Rush , Nei l Shephard , Tim o Terasvirta , an d fou r anonymou s
referees fo r thei r help. The y hav e made a grea t contributio n t o this
book, and found many infelicities in earlier versions, but of course ar e
not responsible for any that remain.
Early version s o f the boo k wer e inflicted by us upon ou r graduate
students. Amon g thos e wh o suffere d fro m th e confusio n cause d
by obscur e notatio n an d prose , bu t continue d unflinchingly , Hughe s
Dauphin, Carol Dole, Jesu s Gonzalo, Catherine Liston, Claudio Lupi,
Neil Rickman, and Geeta Singh deserve special thanks.
We ar e als o indebte d t o Juli a Campos , Michae l Clements , Steve n
Cook, Neil Ericsson and Claudio Lupi for proof reading.
The financia l suppor t of the Economi c an d Social Research Counci l
(UK) unde r grant s B0125002 4 an d R23118 4 an d th e Fond s pou r l a
Formation de s Chercheurs et 1'Aide a la Recherche (Quebec ) i s grate-
fully acknowledged. Finally, we thank Andrew Schuller and the editors
of thi s series, who remained encouraging about the projec t despite its
many difficulties.
Oxford A . B.
Madrid J . J. D.
Montreal J . W. G.
Oxford D . F. H.
Contents
Notational Conventions, Symbols, an d Abbreviations x i
1. Introduction and Overview 1
1.1. Equilibrium relationships and the long run 2
1.2. Stationarity and equilibrium relationships 4
1.3. Equilibrium and the specification of dynamic models 5
1.4. Estimation of long-run relationships and testing for orders
of integration and co-integration 8
1.5. Preliminary concepts and definitions 1 0
1.6. Data representation and transformations 2 8
1.7. Examples: typical ARM A processes 3 2
1.8. Empirical time series: money, prices, output, and interest
rates 4 0
1.9. Outline of later chapters 4 2
Appendix 4 3
Linear Transformations, Erro r Correction, and the Long
Run in Dynamic Regression 4 6
2.1. Transformations o f a simple model 4 8
2.2. Th e error-correction model 5 0
2.3. A n example 5 2
2.4. Bdrdsen and Bewley transformations 5 3
2.5. Equivalence o f estimates from different transformations 5 5
2.6. Homogeneity and the ECM as a linear transformation
oftheADL 6 0
2.7. Variances of estimates of long-run multipliers 6 1
2.8. Expectational variables and the interpretation of
long-run solutions 6 4
3
Properties of Integrated Processes 6 9
3.1. Spurious regression 7 0
3.2. Trends and random walks 8 1
3.3. Some statistical features of integrated processes 8 4
3.4. Asymptotic theory for integrated processes 8 6
3.5. Using Wiener distribution theory 9 1
3.6. Near-integrated processes 9 5
viii Content s
4. Testin g for a Unit Root 9 9
4.1. Similar tests and exogenous regressors in the DGP 10 4
4.2. General dynamic models for the process of interest 10 6
4.3. Non-parametric tests for a unit root 10 8
4.4. Tests on more than one parameter 11 3
4.5. Further extensions 11 9
4.6. Asymptotic distributions of test statistics 12 3
5. Co-integratio n 13 6
5.1. A n example 13 7
5.2. Polynomial matrices 14 0
5.3. Integration and co-integration: formal definitions and
theorems 14 5
5.4. Significance o f alternative representations 15 3
5.5. Alternative representations of co-integrated variables:
two examples 15 3
5.6. Engle- Granger two-step procedure 15 7
6. Regressio n wit h Integrated Variable s 16 2
6.1. Unbalanced regressions and orthogonality tests 16 4
6.2. Dynamic regressions 16 8
6.3. Functional forms an d transformations 19 2
Appendix: Vector Brownian Motion 20 0
7. Co-integratio n i n Individual Equations 20 4
7.1. Estimating a single co-integrating vector 20 5
7.2. Tests for co-integration in a single equation 20 6
7.3. Response surfaces fo r critical values 21 1
7.4. Finite-sample biases in OLS estimates 21 4
7.5. Powers of single-equation co-integration tests 23 0
7.6. A n empirical illustration 23 6
7.7. Fully modified estimation 23 9
7.8. A fully modified least-squares estimator 24 0
7.9. Dynamic specification 24 2
7.10. Examples 24 4
Appendix: Covariance Matrices 25 2
8. Co-integration i n Systems of Equations 25 5
8.1. Co-integration and error correction 25 7
8.2. Estimating co-integrating vectors in systems 26 1
8.3. Inference about the co-integration space 26 6
8.4. A n empirical illustration 26 8
8.5. Extensions 27 1
Contents i x
8.6. A second example of the Johansen maximum likelihood
approach 29 2
8.7. Asymptotic distributions of estimators of co-integrating
vectors in 1(1) systems 29 3
9. Conclusio n 29 9
9.1. Summary 29 9
9.2. Th e invariance of co-integrating vectors 30 0
9.3. Invariance of co-integration under seasonal adjustment 30 1
9.4. Structured time-series models and co-integration 30 3
9.5. Recent research on integration and co-integration 30 4
9.6. Reinterpreting econometrics time-series problems 30 7
References 31 1
Acknowledgements for Quoted Extracts 32 1
Author Index 32 3
Subject Index 32 5
