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working paper
department
of economics
ESTIMATING AND TESTING LINEAR MODELS WITH
MULTIPLE STRUCTURAL CHANGES
Jushan Bai
Pierre Perron
95-17
Sept. 1995
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
ESTIMATING AND TESTING LINEAR MODELS WITH
MULTIPLE STRUCTURAL CHANGES
Jushan Bai
Pierre Perron
95-17
Sept. 1995
MASSACHUSETTS INSTITUTE
1 4 1995
BRARiES
95-17
Estimating and Testing Linear Models with
Multiple Structural Changes 1
Jushan Bai
Pierre Perron
Massachusetts Institute of Technology
Universite de Montreal
First Version:
October 1994
This Version: September 1995
1
This paper has benefited from comments by Maxwell King and Serena
Ng
as well as
from seminar presentations at Harvard/MIT, Northwestern University, McGill University,
the University of Copenhagen, the University of Illinois at Urbana-Champaign, the Universite de Montreal, the University of
University, the 1995
Sao Paulo,
PUC
at Rio de Janeiro, Hitotsubashi
Winter Meeting of the Econometric Society and the 1995 joint Meeting
of the Institute of Mathematical Statistics and the Canadian Statistical Association.
Jushan Bai: Department of Economics, E52-247B, Massachusetts Institute of Technology,
Cambridge, MA, 02139
E-mail: jbai@mit.edu. Partial financial support is acknowledged
.
from the National Science Foundation under Grant SBR-9414083.
Pierre Perron:
Succ.
Departement de Sciences Economiques, Universite de Montreal, C.P. 6128,
Centre- Ville, Montreal, Canada, H3C-3J7.
E-mail:
perronp@plgcn.umontreal.ca.
acknowledged from the Social Sciences and Humanities Research Council of
Canada, the Natural Sciences and Engineering Research Council of Canada and the Fonds
pour la Formation de Chercheurs et l'Aide a la Recherche du Quebec.
Support
is
Abstract
This paper considers issues related to multiple structural changes, occurring at un-
known
dates, in the linear regression
model estimated by
least squares.
The main
aspects are the properties of the estimators, including the estimates of the break
and the construction of tests that allow inference to be made about the presence of structural change and the number of breaks. We consider the general case of
a partial structural change model where not all parameters are subject to shifts. We
show convergence at rate T of the estimates of the break fractions. We also discuss
a procedure that allows one to test the null hypothesis of, say, £ changes, versus the
alternative hypothesis of t + 1 changes. This is particularly useful in that it allows a
specific to general modeling strategy to consistently determine the appropriate number of changes present. An estimation strategy for which the location of the breaks
dates,
need not be simultaneously determined
is
discussed. Instead, our
method
successively
estimates each break point. Empirical applications are presented to illustrate the usefulness of the various procedures.
Keywords: Asymptotic Distribution, Change
lection, Dynamic programming.
JEL
Classification: C20, C22, C52.
point,
Rate of convergence, Model
se-
Introduction.
1
This paper considers issues related to multiple structural changes in the linear regression
model estimated by minimizing the sum
treat the dates of the breaks as
unknown
of squared residuals.
variables to be estimated.
Throughout, we
The main
aspects
considered are the properties of the estimators, including the estimates of the break
dates,
and the construction of tests that allow inference to be made about the presence
of structural change
null hypothesis of
and the number of breaks. To that
statistics
discuss tests of the
and econometrics
2
.
The econometric
literature contains a vast
it
amount
of
work
specifically designed for the case
literature has witnessed recently
an upsurge of
models with an unknown change point,
interest in extending procedures to various
thereby offering serious alternatives to the
as well
£+1 changes.
on issues related to structural change, most of
of a single change
we
no structural change versus an arbitrary number of changes
as tests of the null hypothesis of, say, I versus
Both the
effect
CUSUM
test of
Brown, Durbin and Evans
(1975).
With
respect to the problem of testing for structural change, recent contributions
include the comprehensive treatment of
Andrews (1993) who considers sup Wald,
Like-
lihood Ratio and Lagrange Multiplier tests. Weighted versions of these tests satisfying
some asymptotic optimality
criterion are discussed in
Andrews and Ploberger
(1992).
Recent studies also consider econometric models with trending regressors, unit root,
cointegrated variables and serial correlation
3
Methods allowing the investigator to be
.
agnostic about the presence or absence of integrated variables are presented in Perron
(1991) and Vogelsang (1993).
The
of attention in the recent debate
issue of structural change has also received a lot
on unit root versus structural change
function of a univariate time series
4
.
Yet,
all
in the trend
these recent developments consider only
the case of a single structural change.
Issues about the distributional properties of the parameter estimates, in partic-
ular those of the break dates, have received
importance.
The work
somewhat
less
attention despite their
of Bai (1994,1995a) contains general results concerning the
2
For a survey, see Knshnaiah and Miao (1988) and Zacks (1983) as well as the comprehensive
treatment of Deshayes and Picard (1986).
3
See, among others, Christiano (1992), Chu and White (1992), Kim and Sigmund (1989) and
Perron (1991) (trending regressors), Kramer, Ploberger and Alt (1988) (serial correlation), Bai,
Lumsdaine and Stock (1994) and Hansen (1990, 1992) (models with integrated variables).
4
See Perron (1989, 1993, 1994), Banerjee, Lumsdaine and Stock (1992), Zivot and Andrews (1992),
Perron and Vogelsang (1992) and Gregory and Hansen (1993).
asymptotic distribution of the estimated break date when a single break occurs, in
particular the fact that the estimated break fraction converges to
its
true value at rate
T.
In comparison, the literature addressing the issue of multiple structural changes
is
Recent developments include Andrews, Lee and Ploberger (1992)
relatively sparse.
who
consider optimal tests in the linear model with
known
Perron (1994) study the sup Wald test for two changes
our knowledge, the most comprehensive treatment
who
(in
consider, as
we
in
a dynamic time series 5
Wu and
that of Liu,
is
Garcia and
variance.
model estimated by
do, multiple shifts in a linear
the context of a more general multiple thresholds model).
.
To
Zidek (1994)
least squares
They study the
rate
of convergence of the estimated break dates, as well as the consistency of a modified
Schwarz model selection criterion to determine the number of breaks. Their analysis
considers only the so-called pure-structural change case where
all
the parameters are
Wu
and Zidek (1994).
subject to shifts.
Our assumptions
are less restrictive than those of Liu,
Furthermore, we consider the more general case of a partial structural change where
not
all
parameters are subject to
We
Concerning the asymptotic behavior of the
we improve on the rate they
estimates of the break dates,
convergence at rate T.
shifts.
report,
T/ln 2 (T), by showing
also consider the asymptotic distribution
and confidence
intervals of the estimates of the break dates.
Our study
considers, in addition, the important
structural changes.
esis of
To that
effect
we present sup Wald type
We also
tests for the null
hypoth-
discuss procedures that allow one to test the null hypothesis of, say,
£ changes, versus the alternative hypothesis of
it
criterion
number
changes. This
is
particularly useful
of changes present (thereby avoiding the use of a
which requires the estimation of the model
up to some a
Some
^+1
allows a specific to general modeling strategy to consistently determine the
appropriate
5
for multiple
no change versus an alternative hypothesis containing an arbitrary number of
changes.
in that
problem of testing
priori specified
maximum).
contributions include Fu and
Curnow
for all possible
model
number
selection
of breaks
Finally our paper contains a discussion of an
(1990)
who
discuss
maximum
likelihood estimation
binomial model. Yao (1988) considers estimating the
number of breaks in the mean of a sequence of normal random variables based on the BIC criterion.
Yao and Au (1988) treat the estimation of multiple mean breaks in a sequence of random variables
of multiple shifts in a
somewhat
restrictive
and consider estimating the number of breaks using the BIC
window nonparametric technique
criterion.
Yin (1988) uses the moving-
to estimate the breaks in a sequence of
random
variables.
Also,
Feder (1975) considers estimating the joint points of polynomial type segmented regressions (nondiscrete shifts). Other relevant contributions include Kim (1993) and Lumsdaine and Papell (1995).
estimation strategy for which the location of the breaks need not be simultaneously
determined. Rather our method successively estimates each break point.
There are many practical advantages arising from the estimation and inference of
models with structural changes. To mention a few, we
identification of events that
may have
first
note that
it
allows the
fostered the structural changes. For example,
an approach often used to examine the effectiveness of policy changes involves
and inference on the corresponding regression
variable regressions
alternative
is
compare the estimated break date with the
to
may
recent regime
The
many
if
An
coefficient.
effective date of
change (or policy implementation). Another potentially useful aspect
of forecasting. Indeed,
dummy
is
a policy
in the field
regimes are present in a given sample, using the most
lead to better forecasts.
rest of this
paper
is
structured as follows. Section 2 discusses the model and
the assumptions imposed on the variables and the innovations.
Section 3 contains
and the asymptotic
results pertaining to the consistency, the rate of convergence
distribution of the estimates of the break dates (as well as the others parameters of the
model). Section 4 proposes test statistics, derives their asymptotic distributions and
methods used to estimate the
presents critical values. Section 5 discusses sequential
model without treating
all
Appendix
A
applications.
break points simultaneously. Section 6 presents empirical
contains
some mathematical
derivations
and Appendix B
a description of a procedure to obtain estimates with multiple changes based on the
principle of
2
dynamic programing
.
The Model and Assumptions.
m breaks
Consider the following multiple linear regression with
Vt
= x' P + z' 6 + u
= x'tP + z'th + ««,
yt
=
yt
where y
t
is
t
x't j3
t
+
1
+u
t
,
<
= Ti + l,...,T2
t
= Tm +
the observed dependent variable at time
vectors of covariates
of coefficients;
u
t
is
and
/?
and
6j (j
=
1,
t.
The
1, ...,
regimes):
t
,
T,
x (p x
1)
and z
t
(q
x
1) are
are the corresponding vectors
indices (7\, ...,Tm ), or the break
unknown. The purpose
regression coefficients together with the break points
are available.
t;
...,m+l)
the disturbance at time
points, are explicitly treated as
1
i=l,2,...,Ti,
t,
zt'<5TO+1
(m +
is
to estimate the
when T observations on
unknown
(y t ,
z t zt )
,
Note that
vector
When
/? is
p
=
this
a partial structural change model in the sense that the parameter
is
not subject to shifts and
0,
we obtain a pure
A
subject to change.
is
effectively estimated using the entire sample.
model where
structural change
partial structural
change model
all
the coefficients are
therefore
is
more general and
includes the latter as a special case.
To proceed,
convenient to introduce some terminologies. First,
it is
partition (or simply a partition) of the integers (1,
(7i,
Tm
...,
)
such that
<
1
T =
the convention that
7\
...,
we
an m-
call
T), an m-tuple vector of integers
Tm < T. Note also that, throughout, we shall use
Tm+ = T. Second, define the block-diagonal matrix
<
•
and
•
•
i
( Zi
\
Z=
zm+1
\
=
with Zi
The matrix Z
(zTi^+u—fZTi)'-
m ). Using
(zi, ...,zt)' at (7i,...,T
tem
(1)
may be
Z
is
Y=
expressed in matrix form as
(y x ,...,y T )',X
=
X/3
+ Z8 +
U,
=
U = (u 1 ,...,u T
partitions Z at (Ti, ...,Tm
(x u ...,x T )',
the matrix which diagonally
)',
<5
{8[,8'2 •••,<^l+1 )
,
In particular, £°
=
(8°
,
...,(^ +1 )'
and (T®,
Z
is
the one which diagonally partitions
generating process
is
assumed
suming
goal
is
to estimate the
8f 7^ 8f+1 (1
<%<
Z
Xj3°
unknown
+Z
8°
+
methods
The
Hence, the data-
at (T°, ...,T^).
U.
m). In general, the number of breaks
of estimating
used to denote,
...,J^) are
coefficients (0°,8°, ...,8^ +1 ,T°, ...,T^), as-
an unknown variable with true value m°. However,
discuss
and
to be
Y=
The
,
superscript or
and the true break points.
respectively, the true values of the parameters 8
matrix
/
).
Throughout, we denote the true value of a parameter with a
subscript.
Z =
said to diagonally partition
is
these definitions, the multiple linear regression sys-
Y=
where
J
it
in later sections.
for
We
m
can be treated as
now, we treat
also
it
as
known and
postpone the problem of
testing for the presence of structural change to Section 4.
The method
of estimation considered
is
that based on the least-squares principle.
For each m-partition (Ti,...,Tm ), the associated least-squares estimates of
f3
and
8j
,
sum
are obtained by minimizing the
of squared residuals
m+l
T,
£
(Y-XP- Z6)'(Y -XP-Z6)=Y:
~
bit
~ z'M 2
xtf
-
t=l t=T,_i+l
Let (3({Tj}) and S({Tj}) denote the resulting estimates based on the given m-partition
(7i,...,rm ) denoted {Tj}. Substituting these estimates in the objective function
denoting the resulting
sum of squared
residuals as St{Ti,
...,
Tm ),
and
the estimated break
points (Ti,...,rm ) are such that
(fi,
(2)
where the minimization
...,
is
fm =
)
Tm ST {Ti,
argmin Tl
taken over
...,
partitions (2i,...,Tm ) such that T,
all
Thus the break-point estimators are global minimizers
q.
Finally, the regression
P=
(3)
T,_i
>
of the objective function.
least-
i.e.
= 6{{fj}).
S
fc{fj}),
Since, the break points are discrete parameters
model, an
—
parameter estimates are obtained using the associated
squares estimates at the estimated m-partition {Tj},
values, they can
Tm ).
and can only take a
finite
number
of
be estimated by grid search. In the case of a pure structural change
efficient
procedure to obtain global minimizers can be obtained using a
dynamic programming approach. This allows the estimates to be calculated using a
number of sums of squared
that
is
of order
0(T 2
)
for
residuals (corresponding to the different possible partitions)
any
m>
2.
The
calculations needed can be significantly
reduced further using standard updating formulae for recursive residuals.
the procedure amounts to computing
pairwise comparisons of the associated
easily
be extended,
in
T
sets of recursive residuals
sum
an iterative fashion, to cover the case of a partial structural
Al. Let
6
is
i
w =
t
(x't ,z't )',
W=
=
1,
...,
m + l, that
...,
(lOl,...,u^^)
and
,
and
7^) such that
W
W° =
W°Wf/(Tf — Tf^)
A GAUSS program
available
tests
.
next section
set of assumptions.
at the true break points (7?,
each
B6
statistical properties of the resulting estimators are studied in the
under the following
for
and performing
The method can
of squared residuals.
change model. These issues are discussed in detail in Appendix
The
In effect,
be the diagonal partition of
diag(W?'
...,
W
W%+1 We assume
).
converges to a non-random positive
that calculates global minimizers using this dynamic programing approach
from the authors upon request. This program also has procedures to compute the other
statistics discussed in this paper.
same
T° =
matrix (with
definite
for all
T).
The
limiting matrices need not be the
i.
A2. For large
the
£,
minimum eigenvalues
bounded away from zero
(i
Au =
The matrix
A3.
=
and T£ +1
1
=
1,
+
...,m
Yli z t z
of \ T.^o
+x
A4. The sequence of errors {u t }
ancl °f
i
£t°-*
u^ are
1).
invertible for £
1S
't
"W
—
satisfies either of
k
>
the dimension of z t
q,
.
the following two sets of condi-
tions:
i)
Let {Fi
i
:
=
•
•
•
,
—2, —1,0, 1,2,
and
and Andrews
That
(1988)].
(a)
:
such that
r
=
4
+6
for
a-fields.
some
8
>
m—
J.
co and for alH
>
and
Assume
[McLeish
there exist nonnegative constants {c,
is,
as
^m
\\E{ui\^ m )\\ T < *il>m
\\m - E(Ui\Fi+m )\\ r < C.V'm+l,
{i/>m
m > 0,
:
z
>
1}
we have
,
(b)
where
m > 0}
be a sequence of increasing
•}
L r -mixingale sequence with
that {u{, Ti] forms a
(1975)
• •
=
||X|| r
(c) max,- C{
(E\X\ t ) 1/t
<
K
E~= -oo i>m
<
(d)
.
We
assume
in addition that
oo,
<
OO,
t
and
The disturbance u
(e)
ii)
Tl =
Let
a)
fying
b)
We
is
independent of the regressors {z s ,x s } for
cr-iie\d{...,w t -i,Wt, ...,u t _ 2 ,
assume that {u
£(u 4 |^"t*_i)
We
t
=
t
} is
t
and
s.
u -\}.
t
a martingale difference sequence relative to {F*} satis-
and sup t E\u 4+S
0,
all
t
\
<
oo.
have
_ PV]
plimT'"
1
^^ =
Q(v),
t=i
uniformly in v €
[0, 1],
where Q(v)
—
Q(u)
is
Q{v)
in u (i.e.
c) If
s,
positive definite for v
is
positive definite for v
and
strictly increasing
u).
the disturbances u t are not independent of the regressors {z s } for
the minimization problem defined by (2)
that T{
>
>
—
T,_i
under part
A5. T?
> eT
(i
=
l,...,m-f 1) for
is
taken over
some
e
>
all
all t
and
possible partitions such
(note that this
is
not required
(i)).
= [TX%
Assumption Al
where
is
<
A?
<
•
•
•
< A^ <
1.
standard for multiple linear regressions. Assumption
A2 requires
that there be enough observations near the true break points so that they can be
identified.
Now
consider A3.
Because the break points are estimated by a global
we
least-squares search,
otherwise obtained.
in
some
A3
that
fixed
If
we impose the number of
number h
(h
>
is
all
an exact
as
In
q.
is
fit
observations in each segment to be at
combinations
k) for
(£,
which £
—
k
>
Note
h.
actually for technical convenience and could be dispensed with. This would
require the use of generalized inverses.
inverse of
9,
>
k
not depending on T), the invertibility requirement
q,
can be weakened to hold for
A3
—
Y?k z t z \ to be invertible for £
no segment should contain fewer observations than
particulax,
least
sum
require the
We
for simplicity, the existence of the
assume,
generalized inverses at the expense of
Am, but the proof goes through with
a greater technical burden.
The assumptions
stated in
or absence of a lagged
A4
pertain to two specific cases related to the presence
dependent variable
in z t
The
.
conditions described in part
pertain to the case where no lagged dependent variables are allowed in z t
.
(i)
In this case,
the conditions on the residuals are quite general and allow substantial correlation.
mixingale sequence includes
ferences, strong
see
processes as special cases, such as martingale dif-
mixing processes, linear processes, and functions of mixing processes,
Andrews (1988)
for details.
istence of a uniformly
The sequence {u
bounded moment
noting that condition (c)
(e)
many
A
is,
in
most
t
need not be stationary but the ex-
}
+8
of order 4
is
required (this can be seen by
cases, equivalent to sup,-
||u,|| r
<
Condition
00).
precludes the presence of lagged dependent variables in the regressors.
Part
(ii)
of
Assumption A4 considers the case where lagged dependent variables
are allowed as regressors. In this case, no serial correlation
The requirement
{u t }.
permitted
of {z t u t } forming a martingale difference
convergence of the partial sums
at the
is
T
-1 / 2
Ht = rrui+i z u
t
t
is
obtained
a lagged dependent
,
is
not constraining from a practical point of view since
be arbitrarily small. Note that no such restriction
is
If
is
weak
present in the z t each segment considered must contain a positive fraction
of the total sample. This
variable
to permit
This extra generality
.
expense of some restrictions on the admissible partitions.
varaible
is
in the errors
present in the
it's.
is
necessary
if
t
can
a lagged dependent
In both cases, the assumptions are general
enough
to
allow different distributions for both the regressors and the errors in each segment.
The
possibility of lagged
dependent variables
is
potentially quite useful
if
the pa-
rameters associated with the dynamics of the dependent variables are not subject
to structural change.
into account either in
In this case, the investigator can take these
a direct parametric fashion
(e.g.
dynamic
effects
introducing lagged dependent
variables so as to have uncorrelated residuals) or using an indirect nonparametric ap-
proach
(e.g.
leaving the dynamics in the disturbances and applying a nonparametric
correction for proper asymptotic inference). This trade-off can be useful to distinguish
gradual from sudden changes the same way a distinction
and additive
innovational
outliers.
Assumption A5
totic theory
made between
is
is
a standard requirement to permit the development of an asymp-
and allows the break points
to be asymptotically distinct.
It
essentially
requires the asymptotic experiments to be carried under the assumption that each seg-
ments increase proportionately
(A°,
...,
as the
sample
A£J as the break fractions and we
let
We refer to
A^ +1 = 1.
size increases.
A°
=
and
the quantities
Consistency and Limiting Distributions.
3
we
In this section,
are interested in the consistency property of the estimated break
and especially the rate of convergence. The
fractions
result
about the rate of con-
vergence will allow us to derive results about the asymptotic distribution of the estimates of the break dates as well as the estimated regression coefficients.
A
=
(A l7
that A
A m ) with corresponding true values A
...,
is
and
consistent for A
of notation,
we
and
tribution,
"=*>"
(A°,
...,
A^J.
later that the rate of convergence
"—" denote convergence
let
=
weak convergence
in probability,
in the space D[0,
1]
We
is
We
shall first
T.
let
show
As a matter
"—" convergence
in dis-
under the uniform metric,
see Pollard (1984, Chapter 5).
Consistency.
3.1
The main
result of this section
the consistency of A for A
Proposition
tent.
That
is,
1
Under assumptions A1-A5,
for each n
outline the
appendix.
summarized
main
Note that
T,
in the following proposition
which states
.
>
P(|A
We
is
e
>
0,
i?)
<
e,
and each
fc
-
A2|
>
the estimated break fractions are consis-
we have, when
(*
=
T
is
large:
l,...,m).
steps of the proof using a few
lemmas that
are proved in the
need not equal Tf so that estimated segments
(or regimes)
need not correspond to true regimes and the two partitions {T,} and {T°} are allowed
to overlap.
By showing
that Xj
—
»
X° we, in effect,
bound the degree
of overlap.
Denote by
between the
the estimated residual for the r-th observation and by d t the difference
lit
and the true regression
fitted regression "line"
u
=
t
yt
-
x
t
-
J3
zt 6k
t
,
e
[7jt-i
+
line.
fk
1,
That
is,
]
and
for
A;,
j
dt
=
=
l,...,m
-
x't
$°)
+
+
z't
(h -
«9),
Note that,
1.
t
[7U +
€
d
in general,
t
is
1,
n pj., +
f)t]
defined over
1,
(m +
r°]
l)
2
segments corresponding to each of the possible m-partitions {T,} and {T
t
different
}.
Using
elementary properties of projections,
£x;fi?<ii>?,
j
(4)
-*
and using u
t
=
u
t
—
dt
t=i
,
^E^=^E«*+ii:^-4^ u^-
(5)
j
The proof
of
t=i
T~
l
of Proposition
J2t=i <%
Lemma
an d
±
t=i
1
j «=i
t=i
j
simply uses relations
T~ l J2t=i u d
t
t
We
.
and
(4)
t=\
(5)
and the associated limits
start with the latter.
Under assumptions A1-A5, we have
1
T
1
u tdt =
-J2
1
op {l).
t=i
=
In the absence of structural changes (6j
{3)'J2
x u t, which
t
The proof
is
of this
occurring at
readily seen to be
lemma, presented
unknown dates
exist.
result applicable in the case of
Corollary
T—
>
1 If
E{u\)
= a2
Op {T~
ll2
for all j),
In this case,
).
in the appendix,
Lemma
1
is
T~
l
Yl
u tdt
Lemma 1
= T~
l
(J3
—
holds trivially.
quite involved
when breaks
allows us to state directly the following
homogeneous disturbances.
for
then under Assumptions A1-A5, we have as
all t,
oo:
1
T
2
p
-
t
-
i
a2
.
Proof: Inequality (4) implies that a2
a =
< a2 + op (l)
D
+ £<f?/T + op (l)>cr + 0p (l).
Lemma 1 together with (4) and (5) implies
2
cr
2
2
while (5) implies, by
Lemma
1,
under A1-A5
that,
^xx^o.
1
(6)
t=i
The proof
A
.
More
some
specifically,
This
i.
Lemma
of the consistency of A for A
is
we prove
in the
follows
by showing that
appendix that
(6) implies
cannot hold
(6)
if
A, -/+ p
A -^
A° for
stated in the following lemma.
2 Suppose that assumptions
A1-A5
are satisfied and that
some break
date,
say A°, cannot be consistently estimated, then
limsu P
T
for some constant
We
are
now
p(i|:^>CK-6?+1
)
C >
and some
in the position to
to
>
0.
prove Proposition
and under the supposition that some break date
2,
2
||
Using
1.
is
(5)
and Lemmas
and
1
not consistently estimated,
we
have the inequality
rp
rp
1
i
i
which holds with probability no
i
less
than some
the inequality (4), which holds with probability
eo
1
>
0.
This
for all T.
is
in contradiction
Hence,
all
with
break dates are
consistently estimated.
The consistency
estimates
in the
and
result for A allows us to state consistency results for the
6k (k
=
l,...,m).
The proof
parameter
of the following proposition, presented
appendix, uses arguments similar to those involved
in
the proof of
Lemma
Proposition 2 Under assumptions A1-A5, we have
he
-S°
m
y
}
That
Sk
is,
k
^°
-^0
(Jfc=l,...,m).
the least squares estimators of the regression coefficients are consistent.
10
2.
Rates of convergence.
3.2
We now
consider the rate of convergence of the estimates.
Xk converge to
its
true value at rate T.
More
Proposition 3 Under assumptions A1-A5, for every
such that for
all large
-\°k )\>C)<r,,
presented in the appendix.
is
77
>
by showing that
start
have:
0,
C <
there exists a
00,
T,
P(\T(X k
The proof
we
precisely,
We
=
(*
It is
l,...,m).
important to remark that the rate
convergence pertains to the estimated break fractions
A,
and not to
T,,
T
the estimates
of the break dates. For the latter, our result states that with high probability the bias
is
bounded by some constant
high probability,
we have
C
—
\T{
that
If\
is
independent of the sample
size
T,
with
i.e.
< C.
This result about the rate of convergence of the break fractions allows us to derive
the rate of convergence and obtain the asymptotic distribution of the estimated coefficients
proof
is
$ and
8.
The
relevant results are stated in the following proposition
similar to Corollary
1
and
of Bai (1994b)
Proposition 4 Under assumptions A1-A5,
consistent
is
whose
therefore omitted.
the estimates
(3
and 6
in (3) are
vT
and asymptotically normal such that
where
$ = pKm^(X,Zo)'n(X,Z
(9)
),
and
n=
Note that when the errors are
$ =
cr
2
V
e{uu').
serially uncorrelated
and the asymptotic covariance matrix reduces to a 2 V~ l which can be con,
sistently estimated using a consistent estimate of
heteroskedasticity
lines of
and homoskedastic we have
is
a
2
.
present, a consistent estimate of
Newey and West
sible serial correlation
(1987) and
can be
Andrews
(1991).
made assuming
When serial correlation and/or
$ can be constructed along the
Note that the correction
identical distributions across
or allowing the distributions of both the regressors and the errors to differ.
11
for pos-
segments
Limiting Distributions of Break Dates
3.3
While the consistency of the estimates of the break fractions depends on the global
behavior of the objective function, their limiting distribution depends on the local
behavior around the true break dates. For this reason, the technical apparatus required
to prove consistency
rather different in the multiple breaks case
is
However, the analysis of the limiting distribution
single break case.
is
similar in both
In the single break case, results concerning the limiting distribution of the
cases.
We
break dates are discussed in Bai (1994b).
discuss the relevant results
We discuss
and
provide, in this section, the appropriate
Since the methods are similar,
extensions to the case of multiple changes.
The
compared to the
Bai (1994b) for more details.
refer the reader to
two types of limiting distributions
we only
for the estimates of the
break points.
corresponds to shifts of fixed magnitude and the second to shifts of shrinking
first
magnitudes as the sample
size increases.
Limiting Distributions with Fixed Shifts.
3.3.1
To study the
we need the
limiting distribution in the case of fixed shifts,
following
stronger assumptions.
For regime
A6.
+
l,...,m
We
{z t ;T°_ 1
i,
say that z follows process
t
posed on the
<
0,
To
if's.
W^(k)
i
when the time index
<
t
w}>\k)
=
is
may
=
(i
wjp{k)
=
(£f+1
For example,
=
>
WJ'^Jfe) for k
£
-a;.
— Sf) and
+
2a;.
random walk with
,u\
we
t
=
1,
define a stochastic
first
0,
W^(k) = W^(0 (A:)
for
...,m:
£ zM\
*
=
-1, -2,
...
t=k+l
"
,
+1,
A.-
-2a;.£z!' +1) u!' +1)
,
*
=
1,2,...
t=i
where
1
(z\
belongs to the ith regime.
W^(0) =
where, for
z<'Vi'>A,-
= -A;£*f +1 Vj
when
t
follow different stochastic processes and,
set of integers as follows:
t=i
with A,
a strictly stationary process
is
allowed. Also, no stationarity assumption need be im-
t=k+l
(ii)
T°}
characterize the limiting distribution,
on the
and W«(Jb)
(io)
<
for different regimes, {z t ,u t }
hence, piecewise stationarity
process
1
l).
Note that
k
+
') is
{z\
,
u\*'} follows the ith stochastic process for all
independent over time, the process
(stochastic) drift.
12
W^
is
t.
a two-sided
Proposition 5 Under assumptions A1-A6, and assuming
continuous distribution (for
2
± A'^tUt
has a
then
all t),
fi-Tf
that [A^*]
-^argmaxH^W(jfc)
(i
= l,..., m
).
Furthermore, the estimated break points are asymptotically independent of each other
and of
the estimated regression parameters.
that {[AJz^] 2
The assumption
uniqueness of the
maximum
of
± A|z u
t
W^'
t
has a continuous distribution ensures the
}
(with probability
1).
Note that the limiting
distributions of the break points estimates in the multiple break
as in a single break model.
This
is
basically
due to the
model are the same
fact that these limiting
distributions are determined by the local behavior of the objective function because
of the fast rate of convergence of the estimated break points.
view the limiting distribution of T,
P?-,
+ hT?+1
—
T°
Accordingly, one can
as being solely determined
by the segment
).
Because the estimates of the break fractions are consistent at rate T, they are
sentially
determined by a bounded
no matter how large
is
finite
number
es-
of observations with large probability
the sample. This explains the asymptotic independence of the
estimated break points since each contributing segments are increasingly apart as the
sample
size increases.
The
regression coefficients, on the other hand, are estimated
using the entire data set or at least a positive fraction.
deleting the finite
points
number
think of possibly
of observations that determine the distribution of the break
when estimating the
tribution.
One can then
regression coefficients without affecting their limiting dis-
Hence, the information determining the distribution of the break points
and the remaining
coefficients
can be viewed as non-overlapping observations from
which the asymptotic independence of the break-point estimates and the
coefficients
estimates follows.
The above
result,
though of definite theoretical
practical use because of the
interest,
is
perhaps of limited
dependence of the limiting distribution of
distribution of {z\f\ uj' '} for each segment
i
(though
it is
T,
— Tf on
independent of x and
t
the
/?)
.
Hinkley (1971) provides an analytical expression for the probability density function
in the case
where z t
=
\
and u t
i.i.d.
of {z\[*', u\ *'}, the stochastic process
normal. In general,
W^
if
one knows the distribution
and the location of
easily simulated.
13
its
maximum
can be
Limiting Distributions with Shrinking Shifts.
3.3.2
An
alternative strategy
is
to consider an asymptotic
framework where the magnitudes
of the shifts converge to zero as the sample size increases.
Even though the setup
is
particularly well suited to provide an adequate approximation to the exact distribution
when the
shifts are small,
it
remains adequate even
for
moderate
This
shifts.
alter-
native asymptotic framework also allows a substantial relaxation of the assumption
concerning the distribution of {z ,u
t
t
Moreover, the resulting limiting distribution
}.
independent of the specific distribution of this
We
pair.
provide, in this section, a description of the results
trending.
The
A7. a) Let
AT? = T?-T?_ v The
pWmiAT?)-
b)
The
when the data
are not
required conditions are stated in the next assumption.
process {(zt ,u t );
+1 <t <
Tf^
T?_ 1+ [sAT°]
uniformly in
is
1
€
s
for
[0, 1]
=
i
such that
is
Tf_ 1+ [s-AT9]
Ez
Yl
T°}
t
1,
z't
= sQi&nd
...,m
+
V Yim{ATf)-
£
1
Eu 2 =s*?,
t
1.
following limit exists:
T°_! +[sAT°] 2?_j +{s AT°]
plimiAT?)-
uniformly in
c)
A
€
s
£
Yl
T=Tf_ 1 +l
t=T°_ 1 +l
1
E(zT z't u T u t )
= sty
(z
=
1,
...,m
+
1).
l
[0, 1].
functional central limit theorem holds for {z t u t }. That
is,
as
AT, —*
oo:
Tf^+lsAT?]
(AT, )"
where B(s)
is
EBi(s)Bi(u)
1
£
/2
z t u t =>Bi{s)
a multivariate Gaussian process on
=
min{s,u}ft
the fact that
if
[0, 1]
=
l,...,m+l),
with
mean
zero and covariance
.
t
The next assumption concerns the behavior
lies in
(t
the shifts decrease as
around each break point to discern
it.
T
of the shifts as
increases
T
increases.
more observations
Its role
are needed
Hence, this make possible the application of a
central limit theorem.
A8.
Let Ax,i
=
^j-t+i
—
independent of T, where vj
&Ti
is
(*
=
l,--,
77*)-
a scalar satisfying
14
Assume
A^,-
=
u^A,
for
some A,
1
v T -*
For
i
=
0,
T 1/2
and
~a
[0,
i.
let
W^
'(s)
and
W
2
Define, for
i
=
a;.q i+1 a,/a:q,a„
*?,
= AlaAi/A^.A,-,
#
= A;nt+1 A
2
t
/A;.g l+1 A,-,
be independent standard Wiener processes denned on
(s)
when
oo), starting at the origin
=
s
0.
These processes are also independent across
1, ...,m:
Z«(s)
(12)
=
rfuwtfk-')
- M/2,
jf«<o,
<
V£foW w (a)-&M/2,
ifa>0.
2
We
are
(0, 1/2).
l, ...,m, let:
6 =
and
a€
v T -» oo, for some
now
in
a position to state the following
result.
Proposition 6 Under assumptions Al-A5, A7-A8,
The
The
£2,
4 arg max.Z«(a)
(A;<9,-A,-)4(T;
-
2?)
limiting distribution
is
the same as that occurring in a single break model.
(13)
density function of argmax s ZW(.s)
and a\ are the same
for adjacent
is
t's,
(i
=
1,
=
1,
and ^,4
=
=
<£ ti2
m)
When
derived in Bai (1994b).
£,-
...,
^, in
the limits Qi,
which case the
limiting distribution (13) reduces to:
(AJQAOt^Cfi-I?)
argmax.{^W( a )_| 5 |/ 2 }
-i
=
-
^ 2 argmaxs {^«( S )
|s|/2}.
or
W^
(14)
Finally,
when the
reduces to
(15)
4{fi " 7f)
" ars m ax
,
errors are uncorrelated,
(A& A,-)
^_
^
we have
£2
(,)
(
5)
"
= a2 Q
^ argm^^')^) _
In this case, the limiting density function
is
'
and the limiting
|
result
5 |/ 2 }.
symmetric about the
has been analyzed by Bhattacharya (1987), Picard (1985) and
15
5 l/ 2 >-
Yao
origin.
This case
(1987) for a single
The cumulative
break.
distribution function of arg
max s {W^(s) —
|s|/2}
is
Yao
(see
(1987)):
H(x) =
>
x
for
+
1
(2 7r)-
1 /2
=
and #(x)
v^e-
1/8
- |(x + 5)*(-v^/2) +
— H(—x),
1
standard normal variable.
where $(x)
For instance, the
|
C **(-3>/i/2),
the distribution function of a
is
95% and 97.5%
quantiles are 7.7 and
11.0, respectively.
The
the break dates. All that
parameters;
T
_1
correlation
is
Andrews
needed
is
J2t=i z tz[ f° r
sistently estimated
in
above allows easy construction of confidence intervals
results discussed
^
Qi
is
present
to construct consistent estimates of the various
_1
"t f° r
St=i
by the differences
for
°2 The parameters ujA,- can be con-
in the coefficient estimates Si
— 6,_i When
serial
.
can be estimated using a kernel-based method as discussed
ft
Note that when the segments are not homogeneous, obtaining
(1991).
consistent estimates of these quantities
is still
possible using data over the relevant
subsamples only.
The
limiting distribution in the case of trending regressors
is
discussed in Bai
(1994b) for a single structural change model. His results remains valid for multiple
breaks.
We
omit the details and
paper
refer the reader to that
for
more
details.
Test Statistics for Multiple Breaks.
4
A
4.1
We
F type
consider the sup
hypothesis that there are
[TXi\
(i
=
Let
R be
test of
m=
Again
1, ...,&).
(Ti,...,T/t).
let
no structural break (m
k breaks. Let (7\,
Z
...,Tfc)
FT (X U
Here
SSRk
is
...,
the
A,; q )
sum
=
(T
^
denote the matrix which diagonally partitions
to change.
The
statistic
analysis,
we need
6,
to
=
^
-
(k
+
!)<?
(S[
—
—8'
2,
...,S'k
—
T,
=
Z
at
8'
k+1 ).
- p \ 8'R\R{Z'MX Z)-*R')-'R8
j
.
of squared residuals under the alternative hypothesis, which
(Ti,...,Tjt); q is
8k+i against
0) versus the alternative
the conventional matrix such that (RS)'
depends on
=
=
of breaks.
be a partition such that
Define
(16)
number
Test of no break versus some fixed
Ft
8i +i
number
the
is
for
of regressors
whose
coefficients are subject
simply the conventional F-statistic for testing
some
impose some
i
with given T\,
restrictions
16
...,
I*.
8i
=
To carry the asymptotic
on the possible values of the break dates.
we need
In particular,
to restrict each break date to be asymptotically distinct
bounded from the boundaries
for
some
of the sample.
arbitrary small positive
Ac =
The sup F type
{(Ai,
...,
test statistic
A
is
fc
number
);
-
|A )+1
>
<
define the following set
A,|
>
Ax
e,
e,
A
fc
1
- e}.
FT (\i,...,\ k ;q).
A*)eA<
a generalization of the supF test considered by Andrews (1993) and others
for the case
The
we
e:
sup
(Aj
is
this effect,
then defined as
supF T (k]q)=
This test
To
and
=
k
1.
limiting distribution of the test depends on the nature of the regressors
and
the presence or absence of serial correlation and heterogeneity in the residuals.
We
consider the case where the following additional assumptions are imposed.
w —
A9. Let
(x't z't )'
t
,
and
Q
be some positive definite matrix, we assume that
[Ts]
.
P umT->oo r_1 J2 W W =
t
't
5<5'
t=l
uniformly in
s
€
Note that A9 precludes the presence of trending
[0,1].
Extensions to the general case where
-1
plimj^^T
Y/tJi
w
t
=
w't
in
Q(s), which allows
They
trending regressors, are beyond the scope of the present paper.
regressors.
will
be discussed
a separate paper by the authors.
A10. The disturbances {u t } form an array of martingale differences relative to {Ft}
where
T
t
—
<r-field {•
•
•
,wt -i,w u
•
•
,u t _ 2 ,?it_i}.
functional central limit theorem holds for
{w u
t
t
}
Also, E[v%\
= a2
for all
t
and a
such that
[TV]
t=i
where W*(r)
The
is
is
a (p
+ q)
vector of independent Wiener processes.
case where {u t } satisfies the general conditions stated in Assumption
discussed in Section 4.4 below.
appropriate modifications are
made
We
show how the
results
A4
remain valid provided
to account for the effect of serial correlation
on
the asymptotic distributions.
The
following proposition, proved in the appendix, relates to the asymptotic dis-
tribution of the sup Fr(k; q) test.
17
Proposition 7 Under assumptions A9-A10:
supF T (fc;
q) -i
supF^ =f
F(X U
sup
...,
A*; g),
A )eA,
(*i
fc
where
(Aj,
fFn
...,
Ai.„x
fc) gj
^ [W,(A.— 2^
+1
1
-
)
-
A t+1 ^(A,)r[A,^(A, + i)
r-r
tt
A t A, +1 (A 1+1
«9,=i
W
is
q (-)
A
e
As
.
e
for further details.
values for
supF
who provide a
to that in
In
what
>
tests for k
follows,
critical values are
=
=
2 and q
A*; q)
of replications
critical values
(see the
-1 / 2
£»=i
10,000.
is
with respect to (Ai,
programming algorithm
...,
^
(q
=
A
4.2
The
=
1)
double
value
(17)
=
0.05.
m, under the
No
critical
10)
(1994).
e,-
The Wiener
with e
A*) over the set
appendix
1, ...,
1.
-
t
i.i.d.
A
t
is
N(0^Iq ) and n
=
supremum
of
obtained via a dynamic
for further details).
whose
process W,(A)
(i.e.,
We
present, in Table
up to 10 regimes, k
—
1,
1,...,9)
coefficients are the object of the test.
The
The column corresponding
can also be found in Andrews (1993).
maximum
test.
alternative hypothesis.
structural break against an
test,
e
Andrews (1993)
test discussed in the previous section requires the specification of the
breaks,
the
Thus a small
infinity.
significantly, see
reported values are scaled up by q for comparison purposes.
to one break (k
1.
depends on the value of
For each replication, the
covering cases with up to 9 breaks
and up to 10 regressors
=
obtained via simulations, using an approach similar
Andrews (1993) and Garcia and Perron
The number
...,
,
2 are available except those of Garcia and Perron (1994)
approximated by the partial sums n
F(Xi,
power
+1 W,(A,-)]
A,)
test statistic
we have adopted a
partial tabulation for k
Asymptotic
1,000.
-
converges to zero, the test statistic diverges to
positive value instead of zero can improve the
is
A.-
a q-vector of independent Wiener processes on [0,1] and A^+i
Note that the asymptotic distribution of the
e in
-
r-r
maximum number
It is
number
of interest to consider a test of no
unknown number of breaks given some upper bound
of breaks permitted.
A new
test, called
the double
can now be defined as
DmaxFT(M,q) = max
sup
l<m<M (Ai,...,A m )€A«
18
of
Fr(Ai,...,
\m ;q).
M on
maximum
m dependent chi-square random variables
with q degrees of freedom, each one divided by m. This scaling by m can be viewed,
in some sense, as a prior imposed to account for the fact that as m increases a fixed
For a fixed m,
F(A l5
...,
A m q)
sample of data becomes
The
e
=
last
sum
of
informative about the hypotheses being confronted.
less
column of Table
reports the critical values of this test for
1
M = 5 and
This should be sufficient for most empirical applications. In any event, the
0.05.
critical values
vary
Test of
4.3
the
is
;
£
versus
£+1
M larger than
bound
choices of the upper
little for
5.
breaks.
This section considers a test of the null hypothesis that £ unknown breaks exist against
the alternative that an additional break exists.
sum
the difference between the
that obtained allowing £
however,
is,
+
1
difficult to use.
sum
one would base the
test
of squared residuals obtained allowing £ breaks
breaks.
The
Here we pursue a
different strategy. For the
by
obtained by a global
Ti,...,Ti, are
Our
of squared residuals.
The
for the presence of
test
amounts
test strategy
proceeds conditional
tural change versus the alternative hypothesis of a single change.
T
applied to each segment containing the observations 7i_i to
=
a model with (^+1) breaks
the overall minimal value of the
the
all
sum
1
and Tf+i
T.
segments where an additional break
of squared residuals
FT (£ + l\£) =
1)
no struc-
is
(i
The
—
test
is
1,...,£
therefore
+
1)
using
We conclude for a rejection in favor of
included)
is
sum of squared
residuals
sufficiently smaller
than
from the £ break model. The break date thus selected
the one associated with this overall
(18)
=
-
t
again the convention that To
(over
+
(£
an additional break.
to the application of (^+1) tests of the null hypothesis of
if
and
model with
on the £ estimated break points under the null hypothesis by testing each
segments
on
limiting distribution of this test statistic
£ breaks, the estimated break points denoted
minimization of the
Ideally,
{ST (fu ...,fe)-
minimum. More
imn
inf
precisely, the test
is
is
defined by:
ST (f1 ,...,f - 1 ,T,f„...,fe )}/a\
t
where
A,,,
(19)
and a 2
1,
is
= {rjiU +
(f
{
- f^T)
<r<fi- (fi - fi-M,
a consistent estimate of a2 under the null hypothesis.
5t(Ti,...,T',_i,t, Ti, ..,!/)
.
is
understood as St(t, Ti,
ST (fi,...,ft ,T).
19
...,T*)
Note that
and
for
i
—
£
for
+
i
1
=
as
.
In the case of a pure structural change, an alternative interpretation of the test
can be given as follows.
sum
Let D(i,j) be the minimized
the segment containing observations from
(i
i
+
of squared residuals for
then the test statistic can be
1) to j,
written as
FT (£+1\£) =
(20)
sap{P(T . ll Ti)-D{T . U T)-D(T T )}/a-2
sup
i
i
i
i
l<i<t+l t6A;,,
5T (Ti, ..., ft ) = D(0, 2\) + D(fu f2 ) +
+ D(ft T) and
St(Ti, ..., T,, k, T,+i,
Tt) so that many common terms are
This follows from
expression for
out.
,
-
•
,
Under assumptions A9-A10, standard arguments show
(2l)a^
sn P
{D(T^Tf)-D(T^r)-D(r^)}^
reA^
is
as defined in (19)
asserts that T,
ISfell gSSll!!
sup
Ml 1
=
a q— vector of independent Wiener processes on
is
is
the
,
Mj
[0, 1]
and A°
iT?
with J replaced by T®. Under the null hypothesis, Proposition 3
,-
T° + O p (l). Using
this result,
In addition, because over different regimes D(-,
of (20)
-
1
it is
not
difficult to
show that the weak
convergence in (21) also holds with Z£.j and If replaced by r,_i and
observations, the
canceled
that,
»?<^<i-t!
where, as before, W,(-)
a similar
weak
are
•)
limits in (21) for different
maximum
of £
and we have the following
+
i's
computed using non-overlapping
are independent.
independent random variables
1
T,, respectively.
Thus the
limit
form of
(21),
in the
result:
Proposition 8 Under assumptions A9-A10:
lim
(22)
T—>oo
where
G
q
^(x)
is
P(FT (£+1\£) <
the distribution function of the
1).
in
DeLong
Gq>v (x). A
(1981) and
:
Gq ^{x) t+l
Table 2 calculated with
rj
partial tabulation of
Andrews (1993)
However, the grid presented
percentage points of
in
- pWq (l)\\>
M 1- /^
=
.
is
•
can be obtained from the
some percentage points can be
(see also the first
column
of our Table
not fine enough to allow obtaining the relevant
Accordingly, we provide a
.05.
,
variable
critical values of this test for different values of £
distribution function
found
random
7-
v<»<i-v
+1
G,,„(x)'
=
\\Wq fr)
sup
The
x)
full set
of critical values
These were obtained using a simulation method
similar to that used for the construction of Table
20
1
Note that a 2
is
only required to be consistent under the null hypothesis for the
The
validity of the stated asymptotic distribution.
power
if
a
2
is
also consistent
under the alternative hypothesis.
constructed under the null hypothesis will overestimate
biased downward, thereby decreasing
=
°
2\,
...,
A
power.
its
the null and the alternative hypothesis
where
may, however, have better
test
If
a 2 The
.
the latter
is
test statistic
true,
is
a2
then
consistent estimator under both
given by
is
j;St(Ti,...,Ti + i),
£+1 break
Tt+\ are the estimates of the
points. Note, finally, that the re-
sults discussed in this section, including (22), hold true for partial structural changes.
Also,
it is
important to note that the results carry through allowing different distribu-
tions across segments for the regressors
valid
4.4
The
under A7(a and
c)
and the
errors.
instead of A9-A10, provided
That
a2
is
is,
Proposition 7 remains
a2
replaced by
in (20).
Extensions to serially correlated errors.
tests discussed
above can be applied without the imposition of
rected errors as specified
in
A 10.
Assumption
In this case,
serially uncor-
some modifications
are
necessary to take account of the change in the limiting distribution of the statistics
A
under the null hypothesis.
simple modification
is
to use the following version of the
F-test instead of that specified in (16):
Fft\ u
(23)
...,\ k ;q)
= I -"
{k
+
(
where V{8)
is
A
" P 6'K(EV(6)B!))
and heteroskedasticity;
V{8)
i.e.
e.g.,
R6,
is
robust to
a consistent estimate of
= p\\wT(Z'MxZ)-' Z'MxnMx Z(Z'MxZ) -l
l
consistent estimate of V(8) can be obtained using
gested by,
l
an estimate of the variance covariance matrix of 8 that
serial correlation
(24)
1)g
Andrews
(1991).
methods such
as those sug-
Again, note that this estimate can be constructed
allowing identical or different distributions for the regressors and the errors across
segments. In some instances, the form of the statistic reduces in an interesting way.
Consider the case of a pure structural change model
(/?
=
variables are such that
(25)
phm^Z'nZ =
h u (0)plim^Z'Z
21
0)
where the explanatory
with h u (0) the spectral density function of the errors u t evaluated at the zero frequency.
In that case,
V(6)
and the robust version
FZ(\ u
= MO)plim(^)-
1
,
of the F-test can be constructed as:
r"
=
...,X k] q)
(fc
"P
1)g
^
(
S'B!(R(z'Z)-
1
Ii!)-
1
R6/h u (0).
)
with h u (0) a consistent estimate of h u (0). In that case, we have the following asymptotically equivalent test
a2
F}(\i,
...,
h;
= j——FT {X
q)
1
,
...,
Xk
;
h u (0)
with a 2
= T -1 J2t=i
"t
q),
a consistent estimate of the variance of the residuals. Hence,
the robust version of the test
is
simply a scaled version of the original
condition (25) holds, for example, where testing for a change in
statistic.
mean
The
as in Garcia
and Perron (1994).
The computation
ally involved
when
of the robust version of the F-test (23) can
considering
data dependent method
of S.
An
is
all
be quite computation-
combinations of possible break points, especially
if
a
used to construct the robust asymptotic covariance matrix
asymptotically equivalent version
F-test to obtain the break points,
is
to first take the
imposing
i.e.
Q = a2 I
.
supremum of the
original
This can be done since the
break fractions are T-consistent even in the presence of correlated errors.
One can
then obtain a robust version of the test by evaluating (23) and (24) at these estimated
break dates.
The
extensions for the
PmaxFj
test
and the sequential Fr(£
+
1\£) are similar
since they are simply functions of the sup Fr(k,q) tests.
4.5
Consistency of the
A
is
test
consistent
if,
tests.
under the alternative hypothesis, the associated
diverges to infinity as the sample size increases. Because supF T (l; q)
A;supF T (A;;
results of
is
9),
the consistency of the supFr(/:; q) (k
Andrews (1993) who proved that the
test
>
2) follows
test statistic
< 2supF T (2; q) <
immediately from the
based on the statistic supFr(l;
q)
consistent for various alternatives including multiple breaks. Consequently, the test
based on the statistic Dmax-F^A:; q)
is
also consistent.
22
We
next argue that the test based on Ft(£
more than
and a model with only
£ breaks
one break that
is
+
£ breaks
1\£) is also consistent.
is
Hence, at least one segment contains a nontrivial
not estimated.
the true break point by a positive fraction of the total
number
is
separated from
For
of observations.
segment, the supF T (l;g) test statistic converges to infinity as the sample size
this
increases since
+
there are
estimated, there must be at least
break point in the sense that both boundaries of each segment
£
If
Accordingly, the statistic Ft(£
consistent.
it is
segments) also converges to
1
infinity.
+
l\£)
(computed
for
This shows consistency.
Sequential Methods.
5
In this section
We
points.
mates
we
discuss issues related to the sequential estimation of the breaks
with some results about the limit of break point
start, in section 5.1
in underspecified models,
number
when the
i.e.
is
The Limit
An
interesting by-
the possibility of a sequential algorithm to estimate models
with an unknown number of breaks. This
5.1
regression structure allows for a smaller
of breaks than contained in the data-generating process.
product of this analyses
esti-
is
discussed in section 5.2.
of Break Point Estimates in Underspecified
Models.
In this section,
we show
that the estimate of the break fraction in a single structural
change regression applied to data that contain two breaks converge to one of the two
true break fractions. In independent work,
also Bai (1994c) for
an
Chong
(1994) obtains a similar result (see
earlier exposition).
To present our arguments, we consider a simple three-regime model:
t
=
(ii
+ eu
if*
Vt
=
(*2
+
£t,
if
[T\i]
+
yt
=
fi 3
+
et ,
if
[TX 2]
+ l<t<T.
y
(26 )
with
et
~
i.i.d.(0, of).
Assume
points in the model. Let
Ta
fii
^
(i 2
,
fi 2
<
^
[TAi]
f*3,
1
<
and
*
X\
<
denote the estimated single
show that despite the misspecification of the number
for either Ai or A2
< [TX 2
]
A 2 so there are two break
,
shift point.
of regimes,
depending on the relative magnitudes of the
23
Our aim
Ta /T
shifts
is
is
to
consistent
and the
spell of
each regime. To verify this claim, we examine the global behavior of S(t), defined as
the limit of
T^StUTt]). We
for the full
sample without a break
In this way, St([Tt])
the convergence of
is
T~
l
(i.e.
St{[Tt}) to S(t)
-
Y^(yt
well defined for all r
^([TAx])
(27)
and St(T)
define St(0)
6
2
j/)
A 5(A0 = a\ +
(1
~ A2
It is
1
€
in r
(A 2
LA
sum
where y
,
[0, 1].
uniform
is
as the
"
of squared residuals
is
the sample mean).
not difficult to show that
In particular
[0, 1].
Al)
(^ ~ *)'
1
and
1
c nm\ i\
^5t([TA
2 ])
(28)
Without
Lemma
P
2
\
)
=
- ^) 2
+ ^(A 2 - A\w
)( z
2_,*l/i
2
<r
/ 1
< £(A 2 ),
case where 5(Ai)
our claim
< 5(A 2 ),
3 Suppose that the data are generated by (26) and that S{\\)
Ta /T
the
consistent for Ai.
is
Proof: This can be proved by showing that S(t)
The
is
lemma
estimated single break point
at Ai.
x
1
we consider the
loss of generality
stated in the following
CM
A S(A
€
for r
[0, 1]
function S(t) has different expressions over
has a unique
Some
[0,1].
minimum
algebra reveals
that
- S(X
S(t)
1 )
\~ T
=
(1
which
is
-
-M)
-Tj(l
- Ax)^ -
[(1
is
nonzero, so S(t)
— S(Xi)
is
(regarded as reversing the data order), S(t)
g
+
[A 2
,
1],
S(t)
-
=
SiX,)
S(t)
-
(1
nonnegative. Under the assumption that S(\\)
bracket above
for r
to)
X 2 )(to
< S(X 2 ),
S(X 2 )
is
2
to))
r
,
<
Ax
.
A l5
By symmetry
nonnegative for r
>
- S(X 2 + S(X 2 ) - S(X X > S(X 2 - 5(A
)
)
<
the expression in the
strictly positive for r
—
-
)
A2
X )
.
>
Thus
0.
It
remains to consider the case where r € (Ai,A 2 ). Again, simple algebra shows
S(r)-S(A0 =
(r
-
2
X^ito - in) -
2
^~)
{
= {r-X,)-\~{to-to) Aa(1
i
_
-
{tl3
Xl
T)(1
~ toW
)
_
*,)(/*
-^)J
> (r-A^^AO-^Ai)]
where the
from -*
<
first
1.
inequality follows from
Thus 5(r)
—
S(Xi)
is
^~_'|
<
1
and the second inequality follows
strictly positive for r
24
€ (Ai,A 2 ). Thus we have
shown that S(t) has a unique global minimum
<
because St(To)
St([T\\]),
The assumption
that S(Xi)
pronounced or dominating
The above lemma
spells.
can the
for Ai,
sum
sum
follows that
it
Ta /T
< £^2) imphes
when S(Xi) <
at A!
is
consistent for Ai.O
that the
break point
first
terms of the relative magnitude of
in
shifts
is
Ta /T
of squared residuals be reduced the most. Given that
is
more
and the regime
when the dominating break
implies that only
Now
5"(A 2 ).
is
identified
consistent
one can use the subsample [Ta T] to estimate another break point such that the
,
of squared residuals
then consistent for A 2
is
minimized for
this
subsample. The resulting estimate
This follows from the same type of argument as in the preceding
.
paragraph because only A 2 can be the dominating break in the sample [Ta T], even
,
Ta <
[TX\]. Hence,
for A 2
if
one knows that
Ta /T
is
break model
principle
consistent for Ai, a consistent estimator
is
straightforward to extend the argument to the case where a one-
fitted to
is
the same and the estimate of the break fraction converges to one of the
squared residuals.
break model
also conjectured that a similar result holds
It is
fitted to
is
general result
a relationship that has
not, however,
is
needed
for the
m
2
breaks (with
when,
7712
>
say,
sum
of
an mi
mi). Such a
arguments that follow.
Sequential estimation of the break points.
The arguments
in section 5.1
Ta /T is
showed that
consistent for one of the true break
point, the one that allows the greatest reduction in the
Suppose, as above, that this break point
we do not know
if
the other break
break point either in the interval
squared residuals for
1
The
a relationship that exhibits more than two breaks.
true break fraction, namely the one which allows a greatest reduction in the
(i.e.,
if
can also be obtained.
It is relatively
5.2
is
as the
sample
all
[1,
Ta
]
is
may
before or after). In that case,
not be
,
sum
break in that segment will not significantly reduce the
will
sum
be
[1,
in the interval [Ta T].
Ta
,
],
allowing one
more
of squared residuals.
inf
5T (r,fa ),
inf
On
in the interval [Ta ,r] will permit a large
reduction given the presence of a break. Notationally, with probability tending to
min{
of
minimized. With probability tending to
is
estimated break point
more break
known
we choose one
or in the interval [Ta T], such that the
This follows since, in the absence of a break in the interval
the other hand, allowing one
of squared residuals.
A l5 which, in general,
is
observations [1,T]
size increases, the
sum
ST (fa ,T)
25
}
= inf ST (fa ,r) = ST (fa ,t)
1,
where f
is
subsample [Ta ,T], and thus t/T
that
we can obtain
Similarly,
Ta
if
second estimated
is
is
be in
shift point will
N
t
<
N
2
-
[1,jT
].
in a sequential way.
first
the case of a
identified, the
is
> 5 (A 2 )),
the
the ordered
consistent for (\ u A 2 ).
is
known number
S(\i)
(Ni,N2 ) be
Generally, let
Then (Ni/T,N2 /T)
,
if
of break points.
number
known number
of break points
is
sample
split into
is
is
m. The idea
Once the
sum of squared
residuals.
and again a third break point
is
squared residuals. The process
is
minor modifications.
to
break
this first
estimated and the
is
The sample
is
then partitioned
selected as one of the three estimates
from three estimated one-break model that allow the greatest reduction
The procedure
first
is
chosen as that break point (of the two obtained) which allows
the greatest reduction in the
in three regimes
unknown.
or
two subsamples separated by
estimated break point. For each subsample, a one break model
second break point
known
of break points, say
estimate the breaks sequentially rather than simultaneously.
point
implies
analysis suggests a straightforward algorithm for estimating models with
multiple break points whether the
Consider
and A2
actually consistent for A 2 (this will be true
Sequential estimation with a
The above
of squared residuals for the
The preceding argument
consistent for A2.
consistent estimates of Ai
version of (f„,f) such that
5.2.1
sum
equivalently obtained by minimizing the
is
continued until the
m
in the
sum
of
break points are selected.
simple to implement using existing least squares routines with
It
yields consistent estimates of the break points;
though the
estimates are not guaranteed to be identical to those obtained by global minimization.
Interestingly,
allows the estimation of models with any
it
using a
number
5.2.2
Sequential estimation with an
Consider
now
of least-squares estimation that
the case of an
A
only of order 0(T).
unknown number
unknown number
particular relevance in practice.
is
is
number of structural changes
of breaks
of breaks.
which
is
likely to
be of
standard problem with any estimation procedure
that an improvement in the objective function
is
always possible by allowing more
breaks. This naturally leads to consider a penalty factor for the increased dimension
of a model.
Yao (1988) suggests the use
of the Bayesian Information Criterion
defined as
BIC(m) =
In
a 2 (m)
26
+ p' ln{T)/T,
(BIC)
where
= (m +
p'
number
the
R
adjusted
AIC
+m+
l)q
p,
and
<7
2
r.
-1
,
Many
the prediction error criterion and Mallows
is
not recommended because, as
to overestimate the dimension of a model.
Zidek (1994)
is
An
Cp
that
other criteria such as the
can be used as well.
widely known,
is
He showed
S|r(Ti, *..,£»).
of breaks can be consistently estimated.
2
criterion
it
The
has a tendency
alternative proposed by Liu,
Wu
and
a modified Schwarz' criterion that takes the form:
MIC(m) =
They suggest using
We
=
(m)
S
]n(ST (fu ...,fm )/(T-p*))
=
0.1
and
=
Co
+
(
P 7r)co(ln(r))
2+6 °.
0.299.
propose an alternative method to determine the number of breaks. The ap-
proach
directly related to the sequential procedure outlined above.
is
Start by esti-
mating a model with a small number of breaks that are thought to be necessary (or
start
with no break). Then perform parameter-constancy tests for each subsamples
(those obtained by cutting off at the estimated breaks), adding a break to a subsample associated with a rejection with the test
increasing £ sequentially until the test Ft(£
of
no additional structural changes. The
number
of rejections obtained with the
breaks used in the
It is
context
8
is
initial
+
final
+
is
l\£).
This process
number
of breaks
parameter constancy
is
repeated
the null hypothesis
l\£) fails to reject
thus equal to the
is
tests plus the
number
of
round.
important to note that the application of the
test Fj{£-\- \\£) in this sequential
rather different from that discussed earlier. Indeed, the result of Proposition
based on having the
of the
Ft(£
sum
first
£ breaks obtained simultaneously,
as global minimizers
i.e.
assuming £ breaks. The reason
of squared residuals
for this is that the
stated limiting distribution of the test requires convergence of the estimates of the
break factions at rate T.
Fortunately, this rate
T
convergence extends to the case
where the break points are obtained sequentially one
at a time.
This
last result is
proved in a very recent study by Bai (1995b). Hence, the limiting distribution of the
Ft(£+1\£) test in the current sequential setup
is
the same as that stated in Proposition
8.
With
probability approaching
determined this way
will
be no
1
as the
less
rejection
method
is
size increases, the
number of breaks
than the true number. The procedure does not
provide a consistent estimate of the true
sequential
sample
number of breaks,
say
mQ
.
This
is
because the
based on a test procedure which implies a non-zero probability of
under the null hypothesis given by the
level of the test, say a.
However, the
asymptotic probability of selecting a model with a larger number of breaks, say
27
m + j,
is
a3 which
given by
decreases rapidly. Hence, there
to estimate models with
no need (with large probability)
is
more than the true number
of breaks, as
is
necessary
when
using a model selection approach based on an information criterion.
The
sequential procedure could be
Ft(£
for the test
A
increases.
+
1 \£)
made consistent by adopting a
significance level
that decreases to zero, at a suitable rate, as the sample size
result to that effect
is
presented in the next proposition whose proof
is
provided in the appendix.
Proposition 9 Let
m
be the
based on the statistic Ft(£
number of breaks.
to
+
number of breaks obtained using
l\£)
applied with
If err converges to
size aj,
and
let
m
slowly enough (for the test based on
method
be the true
Ft(£+
l\£)
remain consistent), then, under assumptions A1-A5,
P(m =
as
some
the sequential
T—
oo.
That
is,
the estimated
m —
)
1,
number of breaks
is
consistent for the true number.
Empirical Applications.
6
we
In this section,
this paper.
The
discuss two empirical applications of the procedures presented in
first
analyzes the U.S. ex-post real interest rate series considered by
Garcia and Perron (1994). The second reevaluates some findings of Alogoskoufis and
Smith (1991) who analyze the
issue of changes in the persistence of inflation
and the
corresponding shifts in an expectations-augmented Phillips curve resulting from such
changes in persistence.
The U.S Ex- Post Real
6.1
Interest Rate.
Garcia and Perron (1994) considered the time series properties of the U.S. Ex-Post
real interest rate (constructed
CPI
inflation rate taken
sample
is
is
from the three-month treasury
bill
rate deflated by the
from the Citibase data base). The data are quarterly and the
1961:1-1986:3. Figure
1
presents a graph of the series.
the presence of structural changes in the
mean of the
our procedure with only a constant as regressor
(i.e.
zt
series.
=
The
To that
issue of interest
effect
we apply
{1}) and take into account
potential serial correlation via non-parametric adjustments. In the implementation of
the procedure,
we allowed up
at least 7 observations.
The
to 5 breaks
and each segment was constrained
results are presented in Table 3.
28
to
have
The
the sup
first
issue to be considered
Ft (k)
the determination of the
k between
tests are all significant for
The sup .F:r(2|l)
present.
is
test takes value 34.32
sequential procedure (using a
criterion of Liu,
Wu
1%
and
is
all select
Of
favor of the presence of two breaks.
So at
5.
of breaks.
least
Here
one break
therefore highly significant.
BIC
significance level),
and Zidek (1994)
and
1
number
is
The
and the modified Schwarz
two breaks. Hence, we conclude
in
direct interest are the estimates obtained
under global minimization. The break dates are estimated
at 1972:3
and
1980:3.
The
first
date has a rather large confidence interval (between 1971:2 and 1973:4 at the
95%
level).
The second break date
however, precisely estimated since the
is,
The
confidence interval covers only one quarter before and after.
95%
differences in the
estimated means over each segment are significant and point to a decrease of 3.16%
and a large increase of 7.44%
in late 1972
in late 1980.
These
results confirm those of
Garcia and Perron (1994).
Changes
6.2
in the Persistence of Inflation
and the Phillips
Curve.
Alogoskoufis and Smith (1991) consider the following version of an expectations-
augmented
Phillips curve:
Aw =
t
where
ut
is
w
t
is
oi
+ a 2 E(Ap
t
\It -\)
the log of nominal wages, p t
the unemployment rate.
They
is
t
_i
+£
t
,
the log of the Consumer Price Index, and
is
61 +-82Apt - l
Hence, upon substitution, the Phillips curve
an AR(l) so that
.
is:
Aw = 7i + j2 Ap ^ + 73 Au + 74 u -i + 6,
(30)
where
t
posit that inflation
E(APt \It-i) =
(29)
+ ct 3 Au + a 4 u
t
72
=
0:262
t
f
Here, a parameter of importance
measuring the persistence of
Kingdom and the United
inflation.
t
is
82 which
is
interpreted as
Using post-war annual data from the United
States, Alogoskoufis
and Smith (1991) argue that the process
describing inflation exhibits a one-time structural change from 1967 to 1968, whereby
the autoregressive parameter 82
is
interpreted as evidence that the
abandonment of the Bretton Woods system relaxed
significantly higher in the second period.
This
is
the discipline imposed by the gold standard and created higher persistence in inflation.
29
They
also argue that the parameter 72 in the Phillips curve equation (30) exhibit a
similar increase at the
same time, thereby lending support
to the empirical significance
of the Lucas critique.
Using the methods presented
in this paper,
we
reevaluate Alogoskoufis and Smith
(1991) claims using post-war annual data for the United
structural stability of the
Figure
2.
AR(l) representation
Kingdom 7 Consider
.
whose
of inflation
first
the
series is depicted in
Details of the estimation results are contained in Table 4.
When
applying
a one break model, we indeed find the same results, namely a structural change in
1967 with
82 increasing
the break
is,
remains constant. The estimate of
61
95%
however, imprecisely estimated with a
—
the period 1961
any conventional
The sup Ft(2)
is,
level.
two breaks and
selects
too strong a penalty
test
5%
level
and the sup Ft(2\1)
The sup Fj(£
+
MIC
one suggesting that the
selects
when the sample
not significant at
is
data does not support a one break model.
however, significant at the
10%
confidence interval covering
More importantly, the supi*r(l)
1973.
level indicating that the
test
significant at the
BIC
from .274 to .739 while
size
1\£) test is
is
test
is
>
2.
not significant for any £
latter
may impose
small. Overall, the tests support a
two
break model.
The estimates
data.
The
first
rather with the
of a
two breaks model reveal a rather
break date
first oil
not linked to the end of the Bretton
is
price shock in 1973.
coefficient estimates point to the
than changes
period 1973
different interpretation of the
The second break
importance of
1980,
and back
located in 1980.
to .011 after 1980.
If
from
.021 to .130 in the
anything, the data suggests a
significant decrease in the persistence of inflation in the period 1973
—
1980, while the
estimates of the autoregressive parameters are not significantly different in the
and
last
first
segments.
Since, there indeed appears to be structural changes in the inflation process,
interest to see
if
The
results are presented in Table 5.
to a single structural change.
The sup Fj (k)
procedure and the criterion of Liu,
The data
Alogoskoufis.
of
>
1.
Furthermore, both the sequential
Wu and Zidek (1994) select a one break model (only
two breaks). Again, the estimate of the break
are the
same
We refer
Here the evidence point
tests are significant for all k while the
sup Ft{£+1\£) tests are not significant for any £
BIC chooses
it is
the Phillips curve equation underwent similar changes in accordance
with the Lucas critique.
7
The
shifts in the level of inflation rather
in persistence. Indeed, the coefficient 6\ varies
—
is
Woods system but
is
associated with the
first
as in Alogoskoufis and Smith (1991) and were kindly provided by George
the reader to their paper for details on the definition and source of each series.
30
oil
price shock (1973) and not with the end of the Bretton
confidence interval
is
The estimate
small).
Woods system
of 72 indeed shows a
(the
95%
marked decrease
similar to the decrease in the persistence of inflation (the parameter estimate even
becomes negative but not
significantly so).
Given the changes
in the estimates of
the parameter 73 and 74, the data suggest that the Phillips curve itself underwent a
structural change in 1973.
The
data, however, do not support any adjustment of the
Phillips curve following the change in the inflation process in 1980.
Since
BIC
selects
two breaks, we
also present results for this specification.
These
lend even less empirical support to the Lucas critique since the breaks dates are 1966
and 1975 and do not correspond to those of the
Conclusions.
7
Our
inflation process.
analysis has presented a rather comprehensive treatment of issues related to the
estimation of linear models with multiple structural changes, to tests for the presence of
number of changes
multiple structural changes and to the determination of the
Our
results being asymptotic in nature, there
of the approximations
and the power of the
is
certainly a need to evaluate the quality
tests in finite
samples via simulations.
intend to present such a simulation study in a subsequent paper.
to be investigated,
methods
issues
present.
Among
We
the topics
an important one appears to be the relative merits of different
to select the
number
of structural changes.
There
on the agenda. For instance, extensions of the
that are optimal with respect to
some
test
are, of course,
other
procedures to include tests
criteria, extensions to
nonlinear models and the
derivation of tests that are valid in the presence of trending regressors.
31
many
A
Mathematical Appendix.
As a matter
of
random
we
of notation,
let
op (l) and
and one that
variables converging to zero in probability
bounded. Unless indicated otherwise,
all
and
op (l)
is
projection matrix, /
likewise for
— A(A'A)~
1
p (l).
We
A'.
is
stochastically
convergence are taken as T, the sample
increases to infinity. For a sequence of matrices Bt,
elements
denote, respectively, a sequence
p (l)
we
write
Ma
For a matrix A,
use
•
||
||
Bt =
op (l)
if
size,
each of
its
denotes the orthogonal
to denote the Euclidean
norm,
i.e.
= (Hi 1 ?) 2 f° r x £ R? For a matrix A, we use the vector-induced norm, i.e.
\\A\\ = sup x?t0 ||Ax||/||x||. We note that the norm of A is equal to the square root of
the maximum eigenvalue of A'A, and thus ||A|| < [fr^'A)] / 2 Also, for a projection
matrix P, ||PA|| < \\A\\. Finally, [a] represents the integer part of a.
||
x
1
''
!l
1
.
Proof of
Lemma
We
1:
start
with a series of lemmas that will be used subse-
quently.
Assumption A5
is
assumed throughout.
Lemma
A.l Let S and
V
be
matrix
S'MyS
is
two matrices having the same number of rows. Then the
non decreasing as more observations (rows) are added
to the
matrix
(S,V).
Proof:
Write
arbitrary vector
S =
(S[,S'2 )' and
V =
We
(V{,Vf)'.
a (having the same dimension
as the
need to show that for an
number
of rows of
S and V)
a'S'Mv Sa>a'S[MVl S1 a.
Note that a'S'MySa
is
sum
the
of squares of the residuals from a projection of Set
on the space spanned by V. Similarly, q'^JMv^iq:
from a projection of
o:5i
of squared residuals
is
the
number
Lemma
on
V\.
The
inequality
is
is
the
sum
of squared residuals
verified using the fact that the
sum
non-decreasing as the number of observations increases (here
of rows of S\
and S). See,
e.g.,
Brown, Durbin and Evans (1975).D
A. 2 Under assumption Al,
su
ri
.
fx'M-zxy
LHH
1
where the swpremum with respect to (Ti,...,Tm )
such that |T,_i
—
Ti|
>
q
(i
=
l,...,m
+
1); the
(7\,...,:rm ).
32
„
..
=0p(1)
is
-
taken over
matrix
Z
all possible
partitions
diagonally partitions
Z
at
We
Proof:
m+
m-break model has
regimes. Each partition (7\, ...,Tm ) leaves at least one true
1
By Lemma
(Xf'MzoXf/T)- 1 This
partitions.
It
over
The lemma now
i
M X
Zt
1
l
l
\\
\\
from assumption Al.
follows
MxZ/T)~ l
should be pointed out that (Z
all
such that (X,-,Z,) contains (Xf,Zf)
i
> Xf'M^Xf. Hence {X'MjX/T)- <
\\(X'MzX/T)- l < max,- \\{X? MzoX? IT)- for all
X'
A.l,
implies
.
.
x
regime uncut. In other words, there exists an
as a sub matrix.
+ X'm+1 MZm+l Xm+1 An
X'MjX = X[MZl X +
have the identity
possible partitions. In fact,
it is
is
D.
not uniformly stochastically bounded
easy to see that this matrix
some
is
Op(T)
<
\\X\\\\M-^Zo\\
for
partitions.
Lemma A. 3
Under assumption Al,
X'MzZ =
sup
P (T).
m
Ti,...,T
Proof: Because M-%
||X||||Zo||
is
uniformly over
a projection matrix, we have \\X'M-zZ
The lemma
all partitions.
Op {VT)
and similarly
Lemma
A. 4 The following
\\Z
\\
= Op (y/T).
||X||
<
< Jtr(X'X) =
D
identity holds:
(z'Mxzy = (z'zy +
1
1
1
(z'z)- (z'x)(x'MzX)- x'z(z'z)-\
1
.
Proof: Follows from direct verification.
Lemma
from
follows
\\
A. 5 Under assumption A4,
there exists
a <
\\Z(Z'Z)- Z'U\\
=
where the supremum with respect to (Ti,...,Tm )
is
sup
l
1/2 such that
P (T°),
T\ ,...,Tm
—
\Ti-i —
such that \Ti-i
such that
> q (i = l,...,m + l)
Ti\ > tT for some e >
T,-|
Proof: Consider
first
taken over
all possible
under assumption A^(i) and over partitions
under assumption A^(ii).
the case where part
(i)
of assumption
Because of the independence assumption between z s and u
t
,
A4
is
we can
We
shall prove that
the
summation
=
\U'PzU\
of the
m+
1
5Z
Ti+\
uniformly in
)
J2
T.+l
^ z t)
-1
(
X)
T.+l
33
«
u «)'
treat the z t 's as
.
.
Ti+i
Ti+i
2 < u *)'(
to hold.
l
terms
Ti+i
(
p (T
assumed
Pj = %(% Z)~ Z
Tu ...,Tm However, U'PjU is
nonstochastic, otherwise conditional arguments can be used. Let
2a
partitions
(»'
=
°i
-i
m
)
Thus
suffices to
it
prove that
^p
(31)
with £
—
with Akt
k
>
=
q
and where
t=k
vector defined by £t
1
=
£t (k,£)
=
{Akt)~
l ^2
zt u t
For notational simplicity, the dependence of £ on k and ^ will
-Zj-zj.
]£j=jfc
a q x
is
£t
=o,(n
£6ii
ii
l<k<l<T
t
Now
not be exhibited.
p[
£ £
||E6II>^) <
su P
p(\\J:^\>tA
t=k+q
fc=l
T
T
£
< r- 2as j2
(32)
£
t=Jfc
By
the mixingale property,
we can
write u t as
oo
ut
£
=
J
and
for
each
position,
u i<>
with Uj t
= £(u
r
,t_j_ 1
)
a sequence of martingale differences. Using this decom-
j, {ti Jt -^t— j} is
,
we have
oo
£6
= (A^) -ly,2 2(Ujf By
Minkowski's inequality,
(
£6
(33)
<
r=Jfc
key point
is
£&.
j=-oo t=i
~
Is
*
e
£
=
t=Jfc
A
|^_,-)-.E(ut|.7
= — oo
t
where £J(
t
.
£
E £6.
j=-oo
that for fixed j, k, and
£,
2s
v
2s" l/2s
e
t-k
-
{Cjt,^t-j}
(t
=
k,...,£)
form a sequence of
martingale differences. Thus by Burkholder's inequality (Hall and Heyde 1981, p. 23)
there exists a
C>
0,
only depending on q and
5,
such that
2s
<
(34)
< c (£(£iiu 2s
^(£n^ir)
i/s
)
)
t=k
where the second step follows by Minkowski's
Thus
{E\\£it
f) 11 =
'
z't
{A kl )- l z (E\ujt
t
2s yl a
.
inequality.
By
Now
A4(a), for r
2
||£jt||
=
2s,
=
2s
\
Y
/2s
<
2cti}>j
< 2(max c,)Vj <
34
Ki/tj
1
zt u
we can show
\
Hansen 1991)
(E\u jt
z't (Ake)
for all j.
2
.
t
(see
It
follows that (£||^||
2s 1/s
<
)
z't (A kt )-
l
zt
K
2
2s
£6
E
(35)
Thus from
rl,).
< C (£z' (A kt )-
l
t
K^
zt
(34),
= C(K^) 2
Y
t=k
where we have used the fact that J2t=k 2t(^«) lzt
q.
Combining
(35)
and
(33),
=
l
trace((Akt)~ Y?k z t z 't)
Note that the
right
2s
£6
of (32), that with I
hand
-
< Cq'K
A4),
0.
Let s
£
< ~.
+*)
depend on k and
L
This implies, in view
q,
sup
= 2 + 8/2
we can choose a G
(
side above does not
\\<k<t<r
some C\ >
'
U=-oo
p{
for
2
t=k
>
Jb
trace(I)
we have
2s
E
=
(0,
HE^r*)J
kct-
2 *** 2
t=k
(the
moment of order 4+6
1/2) such that
r
_2ari+2
->
0.
of u t exists by assumption
This proves (31) and hence
the lemma.
Consider
now
the case where part
this case
trp
A4
is
assumed
In
to hold.
1
£
T- 1
of assumption
(ii)
zt z't -> Q(v)
-Q(u),
t=[Tu]+l
and hence (T
v
—
u
>
t
>
_1
0.
Z^ = pyi +1
z t z 't)~
X
—*
— Q( u ))~
{Q( v )
l
uniformly in v and u such that
Also,
[Tv]
T- 1 ' 2
*tu t
J2
= Op (l)
t=[Tu)+l
uniformly using a functional central limit theorem for martingales differences.
cordingly, \U'P^U\
holds with
Lemma
a=
= O v {\)
uniformly in 7\,...,rm and the statement of the
Ac-
lemma
0.
A. 6 Under assumptions A1-A4, we have for some a <
(36)
Proof: This follows
\\X\\\\Z(Z'Z)-^U\\.
sup X'Z(Z'Z)- l Z'U
Tm
Ti
from
Lemma
A.5,
\\X\\
a
35
= Op (T Q+1/2
= Op {T^ 2
)
1/2,
).
and \\X'Z(Z'Z)- 1 Z'U\\
<
M
Note that the same argument leads to
(37)
Tj
Proof of
Lemma
sup Z' Z(Z'Z)- 1 ZU'
rm
By
1:
the definition of dt
= O p (T a+1/2
).
,
T
£
= U'X0 -
utdt
0°)
+ U'TS - U'Z
S°
i
where
T
U'Z =
P {T
suffices to
Z
diagonally partitions
1 /2
)
(these terms do not
show T~^U'X$
=
Because U'X/3°
at (fi,...,fm ).
depend on
op (l) and
T
-1
£/'Zo
(T\, ...,Tm )), to
=
op (l).
Z
Let (Ti, ...,Tm ) be an arbitrary partition and
result.
and S({Tj})
partition of Z. Also let $({Tj})
same
S corresponding to this
partition.
We
P (T
1/2
)
and
prove the lemma,
shall prove
it
a stronger
be the associated diagonal
be, respectively, the estimates of
f3
and
shall prove
sup hrxfaTi))
Tm 1
(38)
We
=
= op (l),
Tx
hrZt({Ts }) =
sup
(39)
where the supremum with respect to (Ti,...,Tm )
those in
Lemma
5.
First consider (38).
The
taken over the same partitions as
estimator 0({Tj}) can be written as
= (X'MjXy'X'MjY
fr{{Tj})
= (X'MjX^X'MjZoS + [X'MjX^X'MjU.
(40)
Using the argument of
Lemma
A. 3, we deduce that
with lemmas A. 2 and A. 3 implies that $({Tj})
Hence
is
op (l),
T^U'X^Tj}) = O p (T"
Next, consider (39).
1 /2
)
=
uniformly over
X'M-gU =
P (1)
O p (T).
uniformly over
From 6({Tj}) = (Z'MxZ)- !?
= U'ZiTMxZy'Z'MxZoS
1
+U'Z(Z'MxZ)- Z'MxU
=
B}
?
Lemma
+ (//)
A.4,
(/)
(42)
(/)
=
U'Z(Z'Z)- 1 Z'MxZoS
+U'Z(Z'Z)- 1 Z'X(X'MzX)- 1 X'PIMxZ Q S
36
partitions.
MX X
X Y and
obtain
(41)
all
partitions, obtaining (38).
all
1
U'ZS({Tj})
This together
.
=
0,
we
Because P^ and
\\X'PjMx Z Q
over
all
\\
Mx
<
\\X\\\\Z
=
\\
that ZoS°
of magnitude,
repetition).
more
< a+
and, hence, the
Proof of
a-i'2
p {T
Lemma 2:
If
Note that
)
p (T
2a
op (l) uniformly over
1 is
>
T
t
falls in
>
+
and
7^.
1
<
:
*
< T(\® + 7;).
Let
1,
-
T(AJ
—
7/),T(A°
<5°)
1T
[0-p o
namely, Tk-i
for
t
€ [T(X° -
+ »/)]. We
2
rj)]
the
<
for
a
same
as
—
rj)
T(\°
and
77),TA°]
have
t
2^2 X t X t
2J2
x t zt
Y.2 z t2 t
E
z t z t ) \ 8k
—
77)
<
t
2
< TA°
and
£
—
+¥
k
-^\\
2
(43).
<$j+i
extends over the set
2
be the smallest eigenvalue of the
\\
is
+
EiVt Ziz z'
t
77-
number
first
matrix in (43)
Then
2
}+fT\0-n +1&-&1I1
2
> min{ 7r,7T}
The
l
l
z't (6 k
be the smallest eigenvalue of the second matrix in
j2d 2 +j:d2 >
1
"
<
extends over the set T(A°
t
This proves (39)
there exists a positive
the interval [T(X°
t
'
J2i
all partitions.
classified into the fc-th regime,
]
TA°
for
is
.
*
Equivalently,
).
,
)
where
a+1/2
complete.
+ 77) < fk Then d = x' - p°) +
= x\0 - 0°) + z[(8k - 8j+1 for t e [TA? +
and T(A°
[
p (T
(without loss of generality, assume this subsequence
T). Suppose this interval
dt
of a lower order
there exists a break, say, A° which cannot be consistently
such that no estimated break
subsequence of
is
only in
differ
(the details are omitted to avoid
)
estimated, then with some positive probability to
7]
and (II)
(I)
we have U'ZSdT,}) =
1/2,
=
proof of Lemma
T- U'Z8{{Tj}) =
l
=
specifically (II)
Because 2a
)
Similar argument shows that (77)
replaced by U.
is
)
||
A. 2, A.5, and A.6.
remains to derive a bound for (//).
It
l
||
Now, we have
P {T).
Lemmas
partitions using
= O v {T l 2 and
< ||Z
a+1/2 uniformly
(/) =
p (T
||M^Z
are projection matrices,
(114
last inequality follows
-
2
<?||
from
-
\\6 k
an arbitrary positive definite matrix
(43) can be written as (Tti)y-J2t(x<>_
smallest eigenvalue of
of (Ttj)At, 7t,
is
At
is
2
> \ min{ 7r,7f }||«J - <?+1 2
(x— a)'A(x— a) + (x — b)'A(x — b) > (l/2)(a — b)'A(a — b)
+
\
6?+1
A
||
and
wtw' =
t
||
for all x.
(Tt])At, say.
bounded away from
of the order Tt).
.
)
zero.
The same can be
37
Now
the
first
matrix in
By assumption A2,
the
Thus the smallest eigenvalue
said for
jT Therefore,
.
Yli
d2
>
TriCiWq ~
than
eo
>
^°+ ill
=
2
TC\\#j
-
for
||
Because A* (k
2:
with large probability, 6k
=
x
- 0°) +
t
interval. If either
||/?
—
(3\\
>
a or
{3
\\6k
or 6k
Lemma
analysis for Ai
By
a
will
l,...,m)
less
consistent by Proposition
is
+ e), T(\°h — e)].
€ [r(A°_!
t
be true
for
with
some a >
E« ^? > ^(A" —
leads to
Without
3:
and provide an
3)
and A 3
Over
this interval,
over the same
£. extending
A°_j
0.
—
Similar argument as in the
2e)a
2
C
some
for
is
virtually the
— T°\ <
C>
0,
with
of squared residuals,
\T{
—
Jf\
same (and
1,
5x(Ti,T2,T3 ),
< tT
actually simpler) and
for
for all
i.
t
{(Tx.Ta.Ta); \T
t
Because St(Ti,T2, Z3)
enough to show that
< 5r(r
1
for
each
77
>
T2 <
where the minimum
is
C>
there exist
taken over the set
3,T2 1,
T (C).
T2 —
Now
denote
SSRx = St(Tu T2 ,T3 ),
ssr.2
=
St(Ti,t2
,
r3 ),
and introduce
SSR3 =
we have
St(Ti,
38
T2 T2 T3
,
C>
0,
define
T2° < -C).
>
o)
<
it
such that for large
v
,
Such a relation would imply that
t
probability,
C.
large,
examine the behavior
e
and
C, global optimization cannot be achieved on
<
T
to prove the proposition
for a large
T°|
to
T°. For
p(min{5T (r1 ,ra ,r3 ) - SrCri,r2 ,r3 )} <
(44)
and
thus omitted.
Also using an argument of symmetry, we
-T°\<cT,l<i<
0,
>
is
The
for those T{ that are close to the true
,T23 ,T3) with probability
each
e
we only need
can, without loss of generality, consider the case
T (C) =
there are only
explicit proof of T-consistency for A 2 only.
eT, with high probability. Therefore
sum
we assume
loss of generality,
the consistency result of proposition
breaks such that
is
with probability no
1.
(m =
three breaks
of the
>
not consistent, then, with some positive probability, either
is
— 6°\\ >
1
=
£f d? > £. <%
$)• Hence
Proof of Proposition
\Tk
d
V
positive probability. This again gives rise to a contradiction with (4) in view of
and
(5)
~
z't (6k
proof of proposition
some
C=
estimated using at least a positive fraction of the
is
observations from [T°_j, T°], say using
dt
some
0.
Proof of Proposition
1,
2
<5?+1
,
).
T (C). Thus
C
with large
T
By
definition,
we have
ST (TU T2
(45)
,
This latter relation
3)
- Sr (rl5 7* T3 = (SSR - SSR3 - (SSR 2 - SSR3 ).
is
useful because
r
)
it
)
allows us to carry the analysis in terms of two
problems involving a single structural change. Indeed, note that
difference in the
sums
sums
T2
and
T3
SSR2 — SSR3
Similarly,
.
the
is
an additional fourth break at time
of squared residuals allowing
T° between the breaks
SSR\ — SSR 3
is
the difference in the
of squared residuals allowing an additional fourth break at time
T2
between the
as the
sum
of
squared residuals from a constrained version of a more general model whose
sum
of
breaks 7\ and T°. Hence, in each case
squared residuals
It is
is
SSR3
SSR\ and SSR2 can be viewed
.
then easy to derive exact expressions for the two components on the right
hand
side of (45) in terms of estimated coefficients. Consider first
have
(e.g.,
Amemiya
,
1985, p. 31):
sT (Tu r2 t3 - st (tu t2
)
,
where
SSR\ — SSR3 we
,
t°,
r3 =
(s;
)
- 6A yzA MwzA (s; -
8A ),
W = (X,Z), with Z the diagonal partition of Z at (Ti,T2,T
the vector
3 ), 63 is
of estimated coefficients associated with the regressors (0, ...,0,z r o +1 ...,zy3 ,0,
...,0)',
and 6 A
ZA =
,
(0,
...,
0,
is
the vector of estimated coefficients associated with the regressors
zt7 +\
,
...,
2r°, 0,
...,
0)' (see
Figure
3). Similarly,
st (tu t*,t3 ) - s r (r1 ,ra ,3*,r3 ) =
,
where
(s 2
-
we have
s A yz'A
Mw zA
3 ),
(again, see Figure 3).
ssRi - ssr 2 =
Z'A ZA
,
Mjy being a
side of (46),
...,
- sA),
and 82
0, -ZTi+i,
••-,
is
2r2
the vec,
0,
...,
(s;
- sA yz'A MwzA (s; - sA )
is
bounded by
{s 2
(S2 — 6 A )'ZA
-
sA yz'A
ZA (62 — SA
projection matrix. Expanding the
first
)
Mw zA
{6 2
>
because Z'A M^rZA
term on the
right
hand
we have
(s;
- sA yz'A zA (6-3 - sA ) -
-(82
0)'
- sA )
SSRi — SSR2
(47)
:
Thus
The second term on the right
<
2
(s 2
W = (X, Z) with Z the diagonal partition of Z at (Ti, T°, T
tor of estimated coefficients associated with the regressors (0,
(46)
SSR — SSR3
for
(s;
- 8A yZA ZA (62 - SA =
)
39
- sA )zA W(W'W)- l W'zA (6'3 -
(I)
-
(II)
-
(HI).
sA )
Consider the limiting behavior of term
be close to 8% given that, on the
controlled
set
Note
(I).
T {C),
t
the distance between T, and Tf can be
and made small by choosing a small
Noting that 8 A
e.
observations from the second true regime only, 8A
is
T {C). Hence, for large C, large T and small
(l/2)(6° - 8°)'Z'A ZA {8% - 6°) with large probability.
C, on
T (C), 8* and
Z'A W/C = Op (l)
that on
and
(II) is
C
expression
the strong law of large numbers,
on
8 A are 0„(1) uniformly. Also
(because Z'A
W involves no more
(I) is
it is
no
enough
less
than
easy to argue
T {C), (WW/T)- =
than C observations).
1
P {\)
t
and small
Thus
e).
Because both 82 and 8A are close to
(III).
large probability for
>
any given small number p
(III) is
Thus
8%,
\\8 2
— 8A <
\\
T, large
(this is true for large
no larger than pi'Z'A Z&i, with
1
a vector of
p with
C
l's.
summary, we have that the inequality
SSRi - SSR 2 >
(48)
By
estimated using
no larger than (l/T)C 2 O p (l).
Consider finally
In
(II).
is
close to 8% for a large
e,
t
Next consider term
that the estimates £3 will
first
The
holds with large probability.
same order
of
-
(l/2){8°
3
magnitude
C
as
A ZA {8l -
8° )'Z'
2
first
-
8°
2)
term on the
^C
2
P {1)
hand
right
since the smallest eigenvalue of Z'A ZA /C
SSRi — SSR2 >
Proof of Proposition
7:
FT (\ u .,.,X k
0.
( T-(k
q )=\^
first
is
is
of the
bounded
term and thus
This proves (44) and thus the proposition.
Note that we can
;
pt'Z'A ZA L,
side of (48)
away from zero by A2. The other two terms are dominated by the
with large probability,
-
write:
—
+ l)q-p \ SSRo-SSR k
J
where
SSRq and SSRk
are the
sum
of squared residuals under the null hypothesis
and under the alternative allowing k breaks,
p)~ 1 SSRk
let
D
—p
u (i,j)
l)q
—
a2 Hence, we concentrate on the limit of
Now,
R
(D (i,j), resp.) be the sum of squared residuals from the unrestricted
>
.
(restricted, resp.)
observation
We have (T — (k +
F^ = SSRq — SSRk.
respectively.
T _i +
t
model using data from segments
1
i
to j (inclusively),
to Tj (these notations have different
defined in Section 4.3, where
the numbering of segments).
i
and
We
j refer to the
numbering of the observations not
Jt+i
£Du
1
«=i
40
from
meanings from the D(i,j)
can then write:
F} = DR (l,k + l)-
i.e.
(i,i),
or
(^)F^=J^[D R (l,i + l)-D R (l,i)-D u (i + l,i + l)]+DR (l,l)-n u (l,l).
Consider
the estimate of the coefficients on the
first
estimate of
{X'M-zX)-
/?
l
in the unrestricted
X'MzY
R
and
and
Let
x's.
f3
restricted models, respectively.
= {X'Mz X)- X'Mz Y
l
to introduce further notations. Let Yij, Uij,
Z=
where
R be the
have U =
u and
J3
We
J3
We
(z[,...,zT )'.
need
Xij and Zij denote the corresponding
vectors or matrices containing elements belonging to the partition from segment
to
segment j
from segments
z's
of 6j using data
on the
to j in the restricted model. Also, let
6f be the estimate
1
z's
Xj and Zj be the vectors or matrices
Now, let 8 Rj be the estimate of 8 using data
Also, let Yj, Uj,
(inclusively).
containing elements from segment j only.
on the
from segment j only
%
in the unrestricted
8i
Yj
=
82
=
=
Xj/3
...
=
+
ZjS
8k+i
=
+
Uj (with 8
8),
We
model.
have
= (zfrr'zM-x^).
Y=
Using the fact that, under the null hypothesis,
and
1
=
(£',
...
a (k
,8')'
+ Z6 + U =
Xfi
+
1) vector
X/3
+ ~Z8 + U
with 8 defined by
straightforward algebra yields
D R (l,j) =
\\
(I
- Pz^K^j - XhJ A T ) \\\
D u (j,j) = \\(I-PZj )(Uj-X, A t)\\\
where
A T = {X'Mz X)- X'Mz U,
l
A T = (X'MzX)-
x
X'M-zU.
FT
Consider the ith element in the summation defining
FT
,
t
= D R (l,i + l)-D R (l,i)-D u (i +
=
To
Kj
P* Ml )(Ui, i+ - Xu+1 At) 2 (I - Pzi+1 )(Ui+1 - Xi+1 AT ) f
\\(I-
-
\\
i
simplify the exposition,
=
+
l,i
ZJj-Xij, Lj
||
and Mj
= X'ltj Uh:
U' +i uu+i
=
^i,t+i-^i,t+i
= U
u
.
||
(/
- PZl „)(Uu -
Sj
=
lti
41
Xij
,
+ Ui+ iXi+i,
XU A T
Z[jUij, Hj
Noting that
u uu + u:+1 ui+l
u'
we have
l)
we introduce the notation
= XljXu,
in (49),
=
2
)
||
Z[jZ\
i
j,
we deduce
that
FTfi = -%+1 H£ S* + S' Hr
1
+ (Si+i-S y[Hi+1
+2S't+1 H-+\Kt+1 A T - 2S' Hr 1 K AT
l
1
Si
i
i
-
1
(Si+l
Mi)'(A T
- AT +
)
(A T
- AT )'(L i+1 -
Li)(A T
Using the stated assumptions, we have the following basic convergence
=
T-V'iXu^jyUu
1)
-S
i)
- Hi)-\Ki+l - Ki)A T
Si)'[Hi+l
+2(Mi+1 -
i
i
i
-2(S,-+i
-H ]-
a(B1 (Xj ),B2 (Xj ))' =
- A T ).
results:
aB{Xj)' where B(r)
is
a (q
+ p)
dimensional vector Brownian motion with covariance matrix
Q=
T-\X U ,Z^)'{X „ZU
2)
X
From
T-^Sj
= aB
Q21
Q22
easily the following results
2 (Xj);
2
x
T- Hj -> p a XjQ 22
c)r- /^-."<r 2 A i g 21
b)
Q12
-t'otXjQ.
we deduce
these two limits,
a)
)
Qn
;
1
T-'Lj ->» o*\jQ u
d)
T-
e)
x
!
2
Mj
;
;
=> vB^Xj);
T 1,2 A T =
{T-
l
X'X-T-
x
X'Z(T- l Z'Z)- T- ZX)l
l
1
x(T~ 1/2 X'U - T- x X'Z{T- l Z'Z)- T- ll2 Z U)
x
_1
=»
<r
(Qn - Qi2Q 22 Q2i)-\Bi(\) - Q 12 Q^B2 (l)) =
T 2 At- Let A =
matrix. We deduce that
It
remains to consider the limit of
(k
+ 1) by (k + 1) diagonal
2
i) r-^'z -»* a (A ® g 22
T~ X'Z
X
ii)
iii)
We
T-
1 /2
-> p
^
)
x'
diag{Xi,X 2
2
(e'A
<g>
...,
)
x
vector;
...,
then obtain
T
l'2
AT =
[T-
X
X'X - T- X'Z(Tl
x[T~ 1/2 X'U
(T^iQii
-
l
~Z'Z)-
- T- X'Z(T-
(e'A
1
® Q 12 )(A
42
<g>
1
x
T- l ZX)- x
Z'Z)- 1 T- 1/2 Z'U]
g 2 2)
_1
(Ae
® Q^)]"
A'.
Ai,...,l
;
Q 12 where e' = (1,1, 1), a (Jb + 1)
a(B2 (Ax), B 2 (A 2 - A ),
B2 {1 - A*))' = 5*.
a
=>
l
1
-
X k }, a
-
x[Bj(l)
(e'A
<g>
Q 12 )(A ® g 2 2)
_1
5']
-1
_1
= ^ [<3n - (e'A* ® Qi2Q 2 2 Q 21 )]- 1 [5i(l) - (e' ® QuQvW]
= ff-MQn " QuQ^Qn]- l [B!(l) - Q l2 Q 22 B2 (l)} = A'.
l
The second
equality follows since e'Ae
=
and
1
® Q\ 2 Q 22
(e'
)B"
= QwQm-^MI)-
Using the results stated above we easily deduce that
(Mt+1 - Mi)\AT - A T
(AT
SJ+j^rji^+xilr
-
=»
)
0,
- ^)(AT - Ar
Ar)'(X,+i
- S^H-'KiAr -
-
(Si+1
)
=»
0,
- ff.r^+i -
Si)'[Hi+i
K )At
{
)Q^Q 21 A' - aB2 (\i)Q^Q 21 A'
-a(B2 (X i+1 - B2 (\ ))Q£Q 21 A* = 0.
=> <tB2 (\, +1
)
Hence, we are
Ft,
left
i
with
= -S'l+1 H-+\Si+1 +
and we deduce, using the
S'.H-'Si +,(5, +1
-
Si)'[Hi+1
B2 (\j) =
fact that
-
ff.-j^S+i
- £) +
op (l),
W(\j) with W(\j) a vector
crQ 22
of q
independent standard Wiener processes,
FT ,
=*>
-B2 (X i+i yQ^B2 (X i+1 y\ i+1 + B2 (X
1
i
)'Q 22
B2 (X
i
)/X t
-1
+(£2 (A, +1 - B 2 (A,))'g 2 2 (52 (A, +1 - B2 (A,))/(A t+1 J
)
=
-a 2
+(7
=
2
2
<r
||
||
W(A, +1 )
)
2
||
/A, +1
+ a2
||W(A l+1 )-W(A i )|| 2
W(A
i+1 )
-
||
W(A;)
A,)
2
/A,-
||
/(A, + i-A,)
A, +1 ^(A,)
2
-
/A l+1 A,(A t+1
||
A,).
Finally, note that
D R (l,l)-D u (l,l) =
P^Ur - X,A T
= 2U[(I-PZl )X (AT -AT
(I
\\
-
)
1
2
\\
-
\\
(I
- Pz^U, - X.At)
)
+A T X[(I - PZl )X A T - A T X[(I - PZl )X A T
x
=
1
0.
Hence
f:
-
a2
E
,=i
w(A +i) - *sagw
'"
"
A,-
+ iA;(A, +1
43
—
2
n
,
A,)
2
\\
1
and the
result of Proposition 7 follows.
Proof of Proposition
+
Ft(£
1\£)
P(Ft (£ +
£.
Now
P(Ft(£
m
{m <
c(ar,£,T) be the
1\£)
>
c T<l
+
l\£)
>
\
< aT
is
m
By
.
,
the consistency of the
m
breaks]
—
=
for £
1,
—
0, 1, ...,m
1,
enough given the assumption on the rate of decrease of ax-
be the number of breaks estimated by the sequential procedure.
The event
77io} satisfies
where Aj,k
=
{Fr{k
+
<
l\k)
m
or,k)-
}
C Ug^Ar,*,
That
is,
the case that the hypothesis of k against k
<
definition,
conditional on £ breaks)
conditional on
ct,i\
{m <
k
By
value of the test
critical
we have
since ct,( increases slowly
Let
=
ct,i
suppose the true number of breaks
sequential test,
(51)
Let
corresponding to a given size aj.
(50)
for all
9:
tuq.
By
(51), P(AT,k\Tno) —*
as
T—
»•
in order to obtain
+
m
<
m
,
it
must be
breaks cannot be rejected for
1
oo for each k
m
<
It
.
some
follows that
mo—
P(rh
(52)
<m
Q)
< j^ P(A Ttk \m
)
-»
0, as
T
-* oo.
k=o
Next, consider the event
case that the hypothesis of
in large
{m > mo}.
mo
In order to obtain
breaks against
m
+1
breaks
is
samples, with probability no more than or, given
(50) with £
=
m
Combining
(52)
>
mo,
it
must be the
rejected. This happens,
m
breaks.
Formally, by
,
P(rh
(53)
m
and
>
m
(53),
)
< P(FT {m + l|m
we
see that
P(m ^
)
>
c T mo
\m
,
m —
)
as
j
< aT
T
—* oo.
estimate of the number of breaks obtained using the sequential test
44
.
is
That
is,
consistent.
the
B
Computational Appendix.
In this section,
we discuss an algorithm based on the principle of dynamic programming
that allows the computation of estimates of the break points as global minimizers of
the
sum
The method
of squared residuals.
directly applicable to the case of a pure
is
structural change model. Useful references include Guthery (1974),
Bellman and Roth
(1969) and Fisher (1958).
A standard
grid search procedure to obtain global minimizers with
require least squares operations of order
provides an efficient
most
method that
requires least squares operations of order
when estimating a model with more than two breaks
some savings
in the case of
computation
is
two breaks). The basic reason
once
fairly intuitive
number of possible segments is
at
it is
most'T(T-f l)/2 and
is
minimum sum
T(T +
distance between each break
may be
of segments to
be considered of (h
when the segment
is
starts at
T — hm when
m
all
to
h
>
q.
compare possible
of squared residuals.
zero; for simplicity
is
we
This implies a reduction in the number
and h
breaks are allowed
number
some minimum
l)/2.
Now
the largest possible
other segments before or after.
1
0(T 2 ). The
distance be denoted by h. Note that
— 1)T — (h — 2)(h —
m
also allows
done in the construction of the
is
of squared residuals
a date between
further reduction in the total
Hence
minimum
loss of generality that
segment must be such as to allow
segment
imposed, as
sum
possible in which case the
suppose without
way
l)/2 segments are permissible. First,
tests discussed in Section 4. Let this
is
method
therefore of order
combinations of these partitions to achieve a
In practice, less than
(the
realized that, with a sample of size T, the total
as an efficient
q
at
for this possible reduction in
dynamic programming algorithm can be seen
<
0(T 2 )
any number of breaks; hence substantial savings in computations can be
for
achieved
h
m breaks would
0(T m ). The dynamic programming approach
—
(i.e.,
1,
the
m+1
For example,
maximal length
of this
regimes). This allows a
of segments considered of h
2
m(rn
+
l)/2
— mh
.
the relevant information can be obtained from the examination of the sums
of squared residuals associated with
T(T +
segments.
l)/2
-
(h
-
l)T
+ (h- 2){h -
l)/2
-
h
2
m(m +
We therefore need to evaluate the sum of squared
45
l)/2
+ mh
residuals associated with
h
segments having the following starting and ending dates:
starting date
i
=
1, ...,
ending date
hm —
1
i=£h,...,(£+A)h-l
i
= hm + 2,...,T- hm +
1
+ —
j
=
j
= h + i-l,...,T-{m-£)h
j
=
h
h
1,
i
+ i-
...,
T — hm
=
(£
l,...,m-
1)
1,...,T
This can be achieved using standard updating formulae to calculate recursive residuals. Indeed, all the relevant
information can be calculated from
recursive residuals and the fact that the
vations
is
the
sum
of squared residuals using
recursive residual at time
of an order 0(T).
sum
To be
t.
T — hm +
of squared residuals using, say,
t
—
1
using a sample that starts at date
i,
and
We
All the relevant information
is
=
is
simply
SSR(i,j) be the sum of squared residuals
let
note the following recursive relation
SSR(i,j)
obser-
be the recursive residual at time j obtained
obtained by applying least-squares to a segment that starts at date
j.
t
observations plus the square of the
Hence, the number of matrix inversions needed
precise, let v(i,j)
sets of
1
-
and ends
at date
Brown, Durbin and Evans (1975)):
(e.g.,
SSR(i,j
i
1)
+ v(i,j) 2
.
contained in the values SSR(i,j) for the combinations
(i,j) indicated above.
Once the sums
of squared residuals of the relevant
segments have been computed
and stored, a dynamic programming approach can be used to evaluate which partition
achieves a global minimization of the overall
essentially proceeds via a sequential
ments) partitions. Let
sum
of squared residuals. This
examination of optimal one-break
SS R({TTtn }) denote
the
sum
(or
first
n observations.
optimal partition can be obtained solving the following recursive problem:
It is
SSR({Tm T })=
,
min
mh<]<T—
[SSR({Tm . ltj })
+ SSR(j +
1,T)].
instructive to write (54) in the following way:
SSR({Tm T }) =
,
+ l,T)+
min{m-i)h<h<h-h[SSR(J2 + 1, ji)+
m^(m-2)h<j <h-h[SSR(J3 + 1, j 2 )+
minm A<i <T-/
1
,
l
[5'5 i?(ji
3
min*< im ^-m _ 1 _fc[55i2(l,j m )
46
two
seg-
of squared residuals associated
with the optimal partition containing r breaks using the
(54)
method
+ SSR(jm +
1,
jm -i )]-]]]
The
Looking at the
minimization problem, we see that the procedure
last displayed
by evaluating the optimal one-break partition
starts
possible break ranging from observations h to
a set of
sum
T — (m +
+
l)h
T — mh.
Hence, the
first
step
is
to store
optimal one break partitions along with their associated
1
of squared residuals.
sub-samples that allow a
for all
Each of the optimal partitions correspond to subsamples
T — (m —
ending at dates ranging from 2h to
l)h. Consider
Such partitions have
proceeds in a search for optimal partitions with two breaks.
T — (m —
ending dates ranging from 3h to
2)h.
step which
now the next
For each of these possible ending
dates, the procedure looks at which one-break partition can be inserted to achieve a
minimal sum of squared
residual.
The outcome
The method
breaks (or three segments) partitions.
of
T — (m +
l)h
ranging from
-\-
(m —
optimal
1
(m — l)h
to
1)
T — (m + l)h +
sum
The method can
continues sequentially until a set
of squared residuals
when combined
therefore be viewed as a sequential
segments into optimal one, two and up to
1
partitions (or into two, three
m
optimal two
T — 2h. The final step is to see which of these optimal (m — 1)
with an additional segment.
a single optimal
T — (m + 1)^ + 1
breaks partitions are obtained with ending dates
breaks partitions yields an overall minimal
updating of
a set of
is
and up to
breaks (or
m+
1
m—
1
breaks
m sub-segments); the last step simply creating
segments) partition.
This dynamic programming method to obtain global minimizers of the
sum
of
squared residuals cannot be applied directly to the case of a partial structural change
model. This
is
basically
due to the
fact that
we cannot concentrate out the parameters
without knowing the appropriate partition,
global minimization
sum
is
available. Let 6
=
(6, 7\,
and
of squared residuals as a function of the vectors
discussed in Sargan (1964),
follows.
the estimate of
First
respect to
we can minimize SSR(0,6)
minimize with respect to 6 keeping
keeping 6 fixed, and iterate.
objective function.
associated with a
depend on the optimal partition which we are trying
However, a simple iterative procedure
the
i.e.
The convergence
Each
in
fixed
...,
to obtain.
Tm we can
),
6, i.e.
write
SSR(0,8). As
an iterative fashion as
and then minimize with
iteration assures a decrease in the
properties of this scheme are discussed in Sargan
(1964).
Note that the
first step,
minimizing with respect to 6 keeping
fixed,
applying the dynamic programming algorithm discussed above with y t
dependent variable. Since
change model. Let 0*
{T*}
=
(rx*,
...,7^)).
=
(6*,
is
fixed this
is,
step
is
x't f3 as
to
the
indeed, a step involving a pure structural
{T*}) be the associated estimate from this
The second
—
amounts
first
stage (with
a simple linear regression with dependent
47
— z' 8*
variable y t
t
for
t
in
regime j
(j
=
partition {T*}) and independent variable x t
An
issue that remains
iteration.
We
is
that then minimizes the
=
taken as the
1, ...,
.
the choice of the initial value of the vector
all coefficients
pure structural change and
(j
(the regimes being defined by the
suggest the following procedure. First apply the
algorithm treating
/3 is
m + 1)
1, ...,
OLS
let
sum
9
a
as subject to change,
=
(6°,
{T
a
})
i.e.
of squared residuals.
The
—
treat the
initial
z't S^
even when some of the
model
}.
as
one of
(7\, ...,Tm )
value of the vector
on x
m + 1), the regimes being defined by the partition {T a
is
to start the
dynamic programming
be the estimates of S and
estimate in a regression of y t
a choice of the startup value
/?
t
for
t
in
regime j
The reason
for such
that the estimates of the break fractions are consistent
coefficient of the
parameter vector 5
=
(<$i, ...,
8q ) do not change
across regimes provided at least one does change at each break date. Hence, this choice
of the starting value for
/5 is
asymptotically equivalent to the estimate associated with
the global minimization and should, accordingly, allow obtaining global minimizers
with respect to
all
the parameters in a few iterations.
48
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52
Table
The
1:
Asymptotic Critical Values of the Multiple-Break Test.
x such that P(supF
< x/q) = a.
entries are the quantiles
fc
123456789
Number
of Breaks, k
DmaxF
9
a
1
.90
8.02
7.87
7.07
6.61
6.14
5.74
5.40
5.09
4.81
8.78
.95
9.63
8.78
7.85
7.21
6.69
6.23
5.86
5.51
5.20
10.17
.975
11.17
9.81
8.52
7.79
7.22
6.70
6.27
5.92
5.56
11.52
.99
13.58
10.95
9.37
8.50
7.85
7.21
6.75
6.33
5.98
13.74
.90
11.02
10.48
9.61
8.99
8.50
8.06
7.66
7.32
7.01
11.69
2
3
4
5
6
7
8
9
10
.95
12.89
11.60
10.46
9.71
9.12
8.65
8.19
7.79
7.46
13.27
.975
14.53
12.64
11.20
10.29
9.69
9.10
8.64
8.18
7.80
14.69
.99
16.64
13.78
12.06
11.00
10.28
9.65
9.11
8.66
8.22
16.79
.90
13.43
12.73
11.76
11.04
10.49
10.02
9.59
9.21
8.86
14.05
.95
15.37
13.84
12.64
11.83
11.15
10.61
10.14
9.71
9.32
15.80
.975
17.17
14.91
13.44
12.49
11.75
11.13
10.62
10.14
9.72
17.36
.99
19.25
16.27
14.48
13.40
12.56
11.80
11.22
10.67
10.19
19.38
.90
15.53
14.65
13.63
12.91
12.33
11.79
11.34
10.93
10.55
16.17
.95
17.60
15.84
14.63
13.71
12.99
12.42
11.91
11.49
11.04
17.88
.975
19.35
16.85
15.44
14.43
13.64
13.01
12.46
11.94
11.49
19.51
.99
21.20
18.21
16.43
15.21
14.45
13.70
13.04
12.48
12.02
21.25
.90
17.42
16.45
15.44
14.69
14.05
13.51
13.02
12.59
12.18
17.94
.95
19.50
17.60
16.40
15.52
14.79
14.19
13.63
13.16
12.70
19.74
.975
21.47
18.75
17.26
16.13
15.40
14.75
14.19
13.66
13.17
21.57
.99
23.99
20.18
18.19
17.09
16.14
15.34
14.81
14.26
13.72
24.00
.90
19.38
18.15
17.17
16.39
15.74
15.18
14.63
14.18
13.74
19.92
.95
21.59
19.61
18.23
17.27
16.50
15.86
15.29
14.77
14.30
21.90
.975
23.73
20.80
19.15
18.07
17.21
16.49
15.84
15.29
14.78
23.83
.99
25.95
22.18
20.29
18.93
17.97
17.20
16.54
15.94
15.35
26.07
.90
21.23
19.93
18.75
17.98
17.28
16.69
16.16
15.69
15.24
21.79
.95
23.50
21.30
19.83
18.91
18.10
17.43
16.83
16.28
15.79
23.77
.975
25.23
22.54
20.85
19.68
18.79
18.03
17.38
16.79
16.31
25.46
.99
28.01
24.07
21.89
20.68
19.68
18.81
18.10
17.49
16.96
28.02
.90
22.92
21.56
20.43
19.58
18.84
18.21
17.69
17.19
16.70
23.53
.95
25.22
23.03
21.48
20.46
19.66
18.97
18.37
17.80
17.30
25.51
.975
27.21
24.20
22.41
21.29
20.39
19.63
18.98
18.34
17.78
27.32
.99
29.60
25.66
23.44
22.22
21.22
20.40
19.66
19.03
18.46
29.60
.90
24.75
23.15
21.98
21.12
20.37
19.72
19.13
18.58
18.09
25.19
27.28
.95
27.08
24.55
23.16
22.08
21.22
20.49
19.90
19.29
18.79
.975
29.13
25.92
24.14
22.97
21.98
21.28
20.59
19.98
19.39
29.20
.99
31.66
27.42
25.13
24.01
23.06
22.18
21.35
20.63
19.94
31.72
.90
26.13
24.70
23.48
22.57
21.83
21.16
20.57
20.03
19.55
26.66
.95
28.49
26.17
24.59
23.59
22.71
21.93
21.34
20.74
20.17
28.75
.975
30.67
27.52
25.69
24.47
23.45
22.71
21.95
21.34
20.79
30.84
.99
33.62
29.14
26.90
25.58
24.44
23.49
22.75
22.09
21.47
33.86
Table
2:
Asymptotic
The
Ft(£+
Critical Values of the Sequential Test
+1
x such that
=
G q>v (xY
entries are the quantiles
lK)-
a.
I
9
a
1
.90
8.02
9.56
10.45
11.07
11.65
12.07
.95
9.63
11.14
12.16
12.83
13.45
14.05
.975
11.17
12.88
14.05
14.50
15.03
15.37
.99
13.58
15.03
15.62
16.39
16.60
16.90
.90
11.02
12.79
13.72
14.45
14.90
.95
12.89
14.50
15.42
16.16
16.61
.975
14.53
16.19
17.02
17.55
17.98
.99
16.64
17.98
18.66
19.22
20.03
.90
13.43
15.26
16.38
17.07
17.52
.95
15.37
17.15
17.97
18.72
19.23
.975
17.17
18.75
19.61
20.31
21.33
.99
19.25
21.33
22.01
22.73
23.13
.90
15.53
17.54
18.55
19.30
.95
17.60
19.33
20.22
.975
19.35
20.76
.99
21.20
22.84
.90
17.42
19.38
2
3
4
5
6
7
8
9
10
1
2
3
4
7
8
9
12.47
12.70
13.07
13.34
14.29
14.50
14.69
14.88
15.73
16.02
16.39
17.27
17.32
17.61
15.81
16.12
16.44
16.58
17.27
17.55
17.76
17.97
18.15
18.46
18.74
18.98
19.22
20.87
20.97
21.19
21.43
21.74
17.91
18.35
18.61
18.92
19.19
19.59
19.94
20.31
21.05
21.20
21.59
21.78
22.07
22.41
22.73
23.48
23.70
23.79
23.84
24.59
19.80
20.15
20.48
20.73
20.94
21.10
20.75
21.15
21.55
21.90
22.27
22.63
22.83
21.60
22.27
22.84
23.44
23.74
24.14
24.36
24.54
24.04
24.54
24.96
25.36
25.51
25.58
25.63
25.88
20.46
21.37
21.96
22.47
22.77
23.23
23.56
23.81
23.90
24.34
24.62
25.14
25.34
25.51
26.84
5
15.35
17.02
6
15.56
17.04
.95
19.50
21.43
22.57
23.33
.975
21.47
23.34
24.37
25.14
25.58
25.79
25.96
26.39
26.60
.99
23.99
25.58
26.32
26.84
27.39
27.86
27.90
28.32
28.38
28.39
.90
19.38
21.51
22.81
23.64
24.19
24.59
24.86
25.27
25.53
25.87
.95
21.59
23.72
24.66
25.29
25.89
26.36
26.84
27.10
27.26
27.40
.975
23.73
25.41
26.37
27.10
27.42
28.02
28.39
28.75
29.13
29.44
.99
25.95
27.42
28.60
29.44
30.18
30.52
30.64
30.99
31.25
31.33
27.06
27.46
27.70
.90
21.23
23.41
24.51
25.07
25.75
26.30
26.74
.95
23.50
25.17
26.34
27.19
27.96
28.25
28.64
28.84
28.97
29.14
.975
25.23
27.24
28.25
28.84
29.14
29.72
30.41
30.76
31.09
31.43
33.25
33.85
.99
28.01
29.14
30.61
31.43
32.56
32.75
32.90
33.25
.90
22.92
25.15
26.38
27.09
27.77
28.15
28.61
28.90
29.19
29.49
.95
25.22
27.18
28.21
28.99
29.54
30.05
30.45
30.79
31.29
31.75
.975
27.21
29.01
30.09
30.79
31.80
32.50
32.81
32.86
33.20
33.60
.99
29.60
31.80
32.84
33.60
34.23
34.57
34.75
35.01
35.50
35.65
.90
24.75
26.99
28.11
29.03
29.69
30.18
30.61
30.93
31.14
31.46
.95
27.08
29.10
30.24
30.99
31.48
32.46
32.71
32.89
33.15
33.43
35.07
.975
29.13
31.04
32.48
32.89
33.47
33.98
34.25
34.74
34.88
.99
31.66
33.47
34.60
35.07
35.49
37.08
37.12
37.23
37.47
37.68
.90
26.13
28.40
29.68
30.62
31.25
31.81
32.37
32.78
33.09
33.53
35.65
.95
28.49
30.65
31.90
32.83
33.57
34.27
34.53
35.01
35.33
.975
30.67
32.87
34.27
35.01
35.86
36.32
36.65
36.90
37.15
37.41
.99
33.62
35.86
36.68
37.41
38.20
38.70
38.91
39.09
39.11
39.12
Table
3:
Empirical Results: U.S. Ex- Post Real Interest Rate
Specifications
zt
=
q=
{1}
p
1
=
h=2
M=5
SupFT {4)
SupFT (5)
51.88
44.76
Tests 1
SupFT (l)
SupFr (2)
SupFT (3)
58.53
44.16
53.76
Number
Sequential Procedure
2
LWZ
2
BIC
2
of Breaks Selected
Parameter Estimates with
=
=
=
*i
<5 2
S3
=
=
T,
T2
1
The supir7'(fc)
tests
Two
2
Breaks 3
1.36 (.16)
-1.80 (.50)
5.64 (.59)
72:3
(71:2-73:4)
80:3
(80:2-80:4)
and the reported standard
and confidence
errors
vals allow for the possibility of serial correlation in the disturbances.
inter-
The
het-
and autocorrelation consistent covariance matrix is constructed
following Andrews (1991) and Andrews and Monahan (1992) using a quadratic
kernel with automatic bandwidth selection based on an AR(1) approximation.
The residuals are pre-whitened using a VAR(l).
2
We use a 1% size for the sequential test supFr(^ + 1|^).
eroskedasticity
3
(i
=
In parentheses are the standard errors (robust to serial correlation) for
1,2,3) and the
95%
confidence intervals for T^ and
T2
-
£,-
Table
4:
Empirical Results: U.K. CPI Inflation Rate 1948-1987
Specifications
zt
=
q=2
{l,y t -i}
=
p
h
=
5
M=5
Tests
Su P FT (l)
SupFr (2)
SupFT (3)
SupFr(4)
SupFT (5)
5.34
12.69
12.74
10.76
8.58
SupFr (2|l)
SupFT (3|2)
SupFr (4|3)
SupFT (5|4)
SupFT (6|5)
10.70
7.47
4.58
1.71
1.66
Number
of Breaks Selected
Sequential Procedure
LWZ
1
BIC
3
Parameter Estimates with one Break
=
=
4,i
<$2,1
=
.025 (.013)
4,2
.274 (.316)
4,2
4,1
=
=
-021 (.010)
4,2
-488 (.220)
4,2
f2
=
=
.024 (.013)
.739 (.125)
1967 (1961-1973)
Parameter Estimates with
4,i
=
=
=
=
-130 (.029)
4,3
-115 (.206)
4,3
1973 (1972-1974)
1980 (1979-1981)
=
=
Two
-011 (-019)
-633 (.223)
Breaks
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cm
to
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Figure
3:
A
particular configuration of (7\,
T2 T3
,
)
in the set
T (C)
£
defined in the proof
of Proposition 3
T,
6a
Si
*?
6°
°2
7^0
To
L
/
2554
)
1
%
°3
2
TO
J
3
Date Due
gift
2 8 1390
Lib-26-67