1. No category

themes
in
m odern
econom etrics
Unit Roots,
Cointegration,
and
Structural Change
G. S. Ma d d a l a a n d I n - M o o Ki m
U n it R o o t s , C o in te g r a tio n , a n d S tr u c tu r a l C h a n g e
T im e series analysis has undergone m any changes in
recent years w ith th e advent of u n it roots and
cointegration. M addala an d K im present a com prehensive
review of these im p o rta n t developm ents an d exam ine
stru c tu ra l change. T he volum e provides an analysis of
u n it ro o t tests, problem s w ith u n it ro o t testing,
estim ation of cointegration system s, cointegration tests,
an d econom etric estim atio n w ith integ rated regressors.
T h e au th o rs also present th e Bayesian approach to these
problem s an d b o o tstra p m ethods for sm all-sam ple
inference. T h e ch ap ters on stru c tu ra l change discuss th e
problem s of u n it ro o t te sts an d cointegration under
stru c tu ra l change, outliers and robust m ethods, the
M arkov sw itching m odel, an d H arvey’s stru c tu ra l tim e
series m odel. U nit Roots, Cointegration, and Structural
Change is a m ajo r co n trib u tio n to T h e m e s in M o d e r n
E c o n o m e tr ic s , of in terest b o th to specialists and
g rad u ate a n d u p p e r-u n d e rg ra d u a te students.
G. S. M ADDALA is U niversity E m inent Scholar a t th e
O hio S ta te U niversity an d one of th e m ost distinguished
econom etricians w riting today. His m any acclaim ed
p ublications include L im ited D ependent and Qualitative
Variables in E conom etrics (C am bridge, 1983) and
E conom etrics (M cG raw -H ill, 1977) an d Introduction to
E conom etrics (M acM illan, 1988, 1992).
IN -M O O K IM is P rofessor of E conom ics a t Sung K yun
K w an University, Seoul, K orea.
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UNIT ROOTS
COINTEGRATION
AND STRUCTURAL CHANGE
G . S. M a d d a la
The Ohio State U niversity
In -M o o K im
Sung K yu n K w an U niversity
iC a m b r i d g e
UNIVERSITY PRESS
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http://www.lrc-tnu.edu.vn
PUBLISHED
BY T H E
PRESS SYNDICATE
OF THE
UNIVERSITY OF
CAMBRIDGE
T he P itt Building, TYumpington S treet, C am bridge C B2 1RP, U nited Kingdom
CAMBRIDGE
UNIVERSITY
PRESS
T he E dinburgh Building, C am bridge CB2 2RU, UK h ttp ://w w w .c u p .c a m .a c .u k
40 W est 20th S treet, New York, NY 10011-4211, USA h ttp ://w w w .c u p .o rg
10 Stam ford Road. Oakleigh. M elbourne 3166, A u stralia
© C am bridge U niversity P ress 1998
T his book is in copyright. Subject to s ta tu to ry exception
and to th e provisions of relevant collective licensing agreem ents,
no reproduction of any p art may take place w ith o u t
th e w ritten perm ission of Cam bridge U niversity Press.
F irst published 1998
P rin te d in th e U nited K ingdom at th e U niversity Press. C am bridge
T ypeset in C om puter M odern 10/13pt, in L^lfeX2e [TAG]
A catalogue record o f this book is available from the B ritish 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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To m y p a re n ts
G. S. M addala
To Jo n g H an , Ju n g Y oun, an d So Y oun
In-M oo Kim
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Contents
page xii
xiii
xvii
Figures
Tables
Preface
P art I Introduction and basic concepts
1
1
In tro d u ction
References
3
6
2
Basic concepts
S tochastic processes
Some com m only used sta tio n ary m odels
Box Jenkins m ethods
In teg rated variables and cointegration
Spurious regression
D eterm inistic tren d an d stochastic tren d
D etrending m ethods
VAR. E C M . an d ADL
U nit root te sts
C oin teg ration te sts an d EC M
8
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
Sum m ary
References
8
11
17
20
28
29
32
34
37
39
41
42
P art I I Unit roots and cointegration
45
3
3.1
3.2
3.3
3.4
47
47
49
60
65
U nit roots
In tro d u ctio n
U n it ro o ts an d W iener processes
U n it root te s ts w ith o u t a determ inistic tren d
D F test w ith a linear determ inistic tren d
vii
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viii
Contents
72
3.5
3.6
3.7
3.8
3.9
3.10
3.11
Specification of determ inistic trends
U n it root te sts for a wide class of erro rs
S argan B hargava an d B hargava tests
V ariance ratio te sts
T ests for T S P versus D S P
Forecasting from T S versus DS models
S um m ary an d conclusions
References
4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
Issues in unit root testing
In tro d u ctio n
Size d isto rtion and low power of u n it root te sts
S olutions to th e problem s of size an d power
P roblem of overdifferencing: MA roots
T ests w ith sta tio n a rity as null
C onfirm atory analysis
Frequency of observations an d power of u n it ro o t te s ts
O th er types of n o n sta tio n arity
P anel d a ta u n it ro o t te sts
U ncertain u n it ro o ts an d th e p re-testin g problem
O th er u n it root te sts
M edian-unbiased estim atio n
S um m ary an d conclusions
References
5
5.1
5.2
E stim atio n of cointegrated system s
In tro d u ctio n
A general C l system
5.3
A tw o-variable m odel: E n g le -G ran g er m eth o d s
156
5.4
5.5
5.6
5.7
5.8
A tria n g u la r system
System estim atio n m eth o d s
T he identification problem
F in ite sam ple evidence
F orecasting in cointegrated system s
160
165
5.9
5.10
M iscellaneous o th er problem s
S um m ary an d conclusions
lg 7
References
191
6
T ests for cointegration
ig g
6.1
6.2
In tro d u c tio n
Single eq u ation m ethods: residual-based te s ts
igg
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92
98
100
103
116
120
126
129
131
133
139
140
141
145
146
155
155
155
173
175
lg 4
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ix
Contents
6.3
6.4
6.5
Single equation m ethods: ECM tests
T ests w ith cointegration as null
M ultiple equation m ethods
203
205
211
6.6
C ointegration te sts based on LCCA
O th er tests for cointegration
M iscellaneous other problem s
O f w hat use are cointegration tests?
Conclusions
References
222
226
228
233
241
242
6.7
6.8
6.9
6.10
7
Econom etric m odeling w ith in teg rated regressors
249
7.1
7.2
7.3
7.4
1(1) regressors not cointegrated
1(1) regressors cointegrated
U nbalanced equations
Lagged dependent variables: th e A RD L m odel
249
250
251
252
7.5
7.6
7.7
U ncertain u n it roots
U ncertain u n it roots and cointegration
Sum m ary and conclusions
References
254
256
258
258
P art I II E xtensions o f the basic model
261
8.8
T he Bayesian analysis of stochastic tren d s
In troduction to Bayesian inference
T he posterior distrib u tio n of an autoregressive p a ra m e te r
Bayesian inference on th e N elson-P losser d a ta
T he d eb ate on th e a p p ro p riate prior
Classical te sts versus Bayesian te sts
P riors and tim e u n its of m easurem ent
O n testin g point null hypotheses
F u rth e r com m ents on prior d istrib u tio n s
263
264
266
268
271
277
277
278
284
8.9
8.10
8.11
Bayesian inference on cointegrated system s
Bayesian long-run prediction
Conclusion
287
290
291
References
292
9
9.1
F ractional unit roots and fractional cointegration
Some definitions
296
296
9.2
9.3
9.4
U n it root te s ts ag ain st fractional altern ativ es
E stim atio n of A R F IM A m odels
E stim atio n of fractionally co integrated m odels
298
300
302
8
8.1
8.2
8.3
8.4
8.5
8.6
8.7
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C ontents
9.5
9.6
E m pirical relevance of fractional unit roots
S um m ary and conclusions
References
305
306
10
10.1
10.2
10.3
10.4
10.5
10.6
Sm all sam ple inference: b o o tstra p m ethods
In troduction
A review of th e b o o tstra p app ro ach
T he AR(1) m odel
B o otstrapping u n it ro o t te s ts
T he m oving block b o o tstra p an d extensio ns
Issues in b o o tstrap p in g cointegrating regressions
309
309
309
322
325
328
332
10.7
10.8
Miscellaneous o th er applications
Conclusions
References
335
336
336
11
11.1
11.2
11.3
11.4
C ointegrated system s w ith 1(2) variables
D eterm ination of th e order of differencing
C o integration analysis w ith 1(2) an d 1(1) variables
E m pirical applications
S um m ary and conclusions
R eferences
342
342
348
355
358
359
12
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
Seasonal u n it roots and seasonal cointegration
Effect of seasonal ad ju stm en t
Seasonal integration
T ests for seasonal u n it roots
T h e unobserved com ponent m odel
Seasonal cointegration
E stim atio n of seasonally cointegrated system s
E m pirical evidence
Periodic autoregression an d periodic integration
362
364
365
366
371
375
376
37g
379
12.9 Periodic cointegration an d seasonal cointegration
12.10 T im e aggregation an d system atic sam pling
381
3g j
12.11 Conclusion
References
3 g2
ggj
P art I V Structural change
13
13.1
13.2
S tru c tu ra l change, u n it roots, an d cointegration
T ests for s tru c tu ra l change
T ests w ith know n break points
13.3
T ests w ith unknow n break points
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C ontents
xi
13.4
13.5
13.6
13.7
13.8
13.9
13.10
13.11
A sum m ary assessm ent
T ests for u n it roots under stru ctu ral change
T he Bayesian approach
A sum m ary assessm ent of th e em pirical work
Effect of stru c tu ra l change on cointegration te sts
T ests for stru ctu ral change in cointegrated relationships
M iscellaneous other issues
P ractical conclusions
References
398
399
402
407
410
411
414
416
418
14
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
O utliers and u n it roots
In tro d uction
Different types of outliers in tim e series m odels
Effects of outliers on u n it ro o t te s ts
O utlier detection
R obust u n it ro o t te sts
R obust estim atio n of cointegrating regressions
O utliers an d seasonal u n it roots
Conclusions
References
425
425
425
428
437
440
445
448
448
449
15
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
15.9
15.10
15.11
15.12
Regim e sw itching m odels an d stru c tu ra l tim e series m odels
T he sw itching regression m odel
T he M arkov sw itching regression m odel
T h e H am ilton m odel
O n th e usefulness of th e M SR m odel
E x tensions of th e M SR m odel
G rad u al regim e sw itching m odels
A m odel w ith param eters following a random walk
A general sta te -sp ace m odel
D erivation of th e K alm an filter
H arvey's s tru c tu ra l tim e series m odel (1989)
F u rth e r com m ents on stru c tu ra l tim e series m odels
S um m ary an d conclusions
References
454
454
455
457
460
463
466
469
470
472
475
477
479
479
16
F u tu re directions
References
A ppendix 1 A brief guide to asym ptotic theory
A u th o r index
Subject index
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