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working paper
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ESTIMATION OF STRUCTURAL CHANGE BASED ON
WALD-TYPE STATISTICS
Jushan Bai
94-6
Dec.
1993
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
ESTIMATION OF STRUCTURAL CHANGE BASED ON
WALD-TYPE STATISTICS
Jushan Bai
94-6
Dec.
1993
APR
5
1994
Estimation of Structural Change Based on
Wald-Type
Statistics
Jushan Bai*
Massachusetts Institute of Technology*
Revised: December 1993
Abstract
This paper addresses structural change in a linear regression model with
an unknown change point. The change point is estimated by maximizing a
sequence of Wald-type
of the estimator.
statistics.
T-consistency
This paper focuses on the convergence rate
is
Asymptotic distributions
obtained.
for
the estimated change point and the estimated regression parameters are also
considered.
The
analysis applies to both pure
Key words and
and partial structural changes.
phrases: Structural change,
Wald
statistics,
weak conver-
gence, change point, T-consistency.
JEL
Classification Codes: C13, C21.
thank Jerry Hausman and seminar participants at Brown University for very useful comments.
Mailing Address: E52-274B, Department of Economics, Massachusetts Institute of Technology,
jbaitait .edn.
Cambridge,
02139. Phone: (617) 253-6217. Fax: (617) 253-1330. email:
"I
f
MA
Introduction
1
In a regression framework, structural change
with a
shift in
some of the
of the shift point
point
is
when
unknown and must be estimated. Estimation
it is
statistics.
These
sequence of Wald
unknown
statistics
is
maximum.
is
(1960) for
When
shift point.
known
thejhift point
used in constructing the
from sequential estimation carried out
estimator
statistics are typically
Chow
the existence of a structural change; see
for
considered a regression equation
regression coefficients. This paper focuses on the inference
based on Wald-type
Andrews (1993)
may be
for every possible
test.
of the shift
used
for testing
shift point
is
not known, a
This sequence results
sample
split.
The
shift-point
then defined as the point where the Wald statistic achieves
For ease of reference, we shall
call this
and
its
global
estimator the W-estimator. Under
normality with independent and identically distributed errors, the Wald statistic
just the F-statistic.
to
When
a single parameter changes, the W-estimator
maximizing a sequence of
The Wald type
statistics
is
is
identical
t-statistics in absolute value.
can be thought of as a measure of the distance between
the estimator of pre-shift parameters and that of post-shift parameters. Correctly
identifying the shift point will tend to
may
also consider the pre-shift
this distance
and vice
versa.
One
sample drawn from one distribution and post-shift
from another whereby the Wald
ance."
maximize
statistics
can be viewed as "between sample
vari-
Correctly classifying the observations into two samples will maximize the
"between samples variance" relative to "within samples variance,"
cation framework.
It is
interesting to note
and
is
as in the classifi-
straightforward to prove that,
when
the Wald-statistics are computed using a sequence of least squares estimators, the
W-estimator
is
equivalent to minimizing the
because LM-type
statistics
statistics in linear
and
LR
sum
statistics are
of squared residuals. Furthermore,
monotonic transformations of Wald
models, the W-estimator can also be obtained by maximizing a
sequence of these latter
statistics.
This
may
not be true for nonlinear models or for
estimation methods other than least squares.
1
There are
tics.
First
at least
two advantages
for basing the
and most important, Wald
consistency proof given later.
We
estimation on Wald-type statis-
statistics are particularly
explore the structure of the
convenient for the
Wald
statistic as a
distance between pre-shift and post-shift estimated parameters and are able to show
that this distance
advantage
and are
is
globally
is
maximized only near the true
Wald
the computational convenience.
many
built into
and estimating the
null hypothesis of
shift point
can be performed concurrently.
no change, one may wish
Andrews
statistics are
easy to compute
software packages. In addition, testing for a parameter shift
Upon
Wald
statistics
is
tests statistics
from
studied by Hawkins
(1993).
Structural change in linear regressions was considered early on by
The problem has
rejecting the
to estimate a shift point simply
the test statistics. Testing for a change using
(1987) and
The second
shift point.
Quandt
(1958).
received considerable attention recently in the literature. Various
have been put forward in
Well known examples include the op-
use.
timal test of Andrews and Ploberger (1992) and the fluctuation test of Ploberger,
Kramer and Kontrus
(1989), in addition to the
Wald type
of tests
mentioned
earlier.
Tests for change in nonstationary time series models have been developed by Per-
ron (1989), Banerjee, Lumsdaine and Stock (1992),
(1992),
and Vogelsang (1992), among
others.
comprehensive bibliography on the subject of
tion of structural change has also been
Chu and White
(1992),
Hansen
Hackl and Westlund (1989)
offer a
The
estima-
its
early development.
examined by many
researchers.
given by Krishnaiah and Miao (1988). Methods of estimation include
A
survey
is
maximum like-
lihood (MLE), nonparametric, least squares (LS), least absolute deviation (LAD),
and Bayesian estimation, among
others.
MLE
is
studied the most, examples
cluding Hinkley (1970), Picard (1985) and Yao (1988).
i.i.d,
on
two
i.i.d.
Nonparametric estimation
samples. LS
is
These authors deal with
Picard (1985) and Yao (1987) focus
or autoregressive models with a shift.
local shifts.
in-
is
considered by
Duembgen
(1991) for
studied by Hawkins (1986) and Bai (1993a) and
LAD
by
Bai (1993b). Bayesian estimation
and
Tsummi
summarized
monograph by Broemeling
in the
(1987) (also see Zivot and Phillips (1992)).
be found in Bai (1992).
type
is
These
statistics.
In this paper,
statistics are
we estimate the
is
Despite the large body of
been obtained for linear regressions.
Furthermore, we consider the problem
in
by using Wald
shift point
based on least squares estimation. Our aim
to establish T-consistency for the proposed estimator.
literature, T-consistency has not
Further references can
context of partial structural change
in the
which some of the regression parameters hold constant throughout the sample.
Thus these parameters should be estimated using the
The
efficiency.
entire
sample
in order to gain
difficulty of the consistency proof increases dramatically
structural change in contrast to pure structural change.
under partial
Bai (1993a) essentially
considers a pure change problem and offers a simple argument of consistency. Despite
the increase in difficulty,
this
we shall work with a partial
model includes pure structural change
pure and partial problems
parameters.
in
The Wald type
and
The use
We
deal with the
a unified way by concentrating out the unchanged
of statistics serves this purpose well
post-shift parameters
and Wald
method
is
statistics are
and allows us
also allow heterogeneous
is
necessary for
based on these estimators.
known
likelihood estimation.
We
and dependent disturbances. The design of the regressors
virtually arbitrary but satisfies
estimation.
maximum
to
used to estimate the pre-
of least squares estimation enables us to relax the assumption of a
error distribution function as
is
model because
as a special case.
focus on the shifted parameters. Least squares
shift
structural change
some standard assumptions under
least squares
and time trends are allowed.
In particular, stochastic regressors
In
addition to the T-consistency of the shift point estimator, root-T consistency for the
estimated pre-shift and post-shift parameters
Asymptotic distribution
problem
is
magnitude,
for the
is
W-estimator
also established.
is
also considered in this paper. This
studied for shifts with shrinking magnitudes. Under a shift with a fixed
it
is
mathematically intractable to obtain the asymptotic distribution
except for some
i.i.d.
samples with
shift, as
shown by Hinkiey
The asymp-
(1970).
for the shift point.
We
also derive the asymptotic distribution for the estimated regression parameters.
We
totic distribution allows
one to construct confidence intervals
show that the asymptotic distribution
for the latter
normal and
is
is
the
same
as
if
the shift point were known.
This paper
is
organized as follows. Section 2 specifies the model and the underly-
ing assumptions. Section 3 proves the T-consistency of the W-estimator.
The root-T
consistency and asymptotic normality for the regression parameters are also derived
Section 4 examines the asymptotic distribution of the shift-point
in this section.
Concluding remarks are provided
estimator.
Some
in Section 5.
technical matters
are collected in the Appendix.
Models and Assumptions
2
Consider the following linear regression with a structural change:
where {e
t
} is
t
unknown
the
=
xtf
+ ei,
yt
=
x't
+
+ eu
z't 6
vectors of regressors (x t
1
shift point,
and
/?
and
is
all
we
shall
model
assume zt
is
=
obtained
B!x t
t
Let us introduce
=
(£i,£2,--,£r)',
XQ =
1,2,
(*
=
fco
...,*<>)
+
(1)
l,...,T)
.
shift at
[see
time
some
X
x
1
(2)
and
zt
R
with
where p
zt
is
<
q);
further notation. Let y
=
(yy,
fc
The matrices X\
.
e t are
For dependent
When
x
t
=
zt
,
,
column rank
(x 1 ,x 2 ,...,i ,0,...,0)',
.
k
a sub-vector of x t a partial
More
a discussion].
for
full
zt
6, ko,
(0, ...,0,Xfco+ i,...,xr)'.
1
and
/?,
is
x
q
yt
to estimate
=
is
may include lagged
When
ko.
Andrews (1993)
The purpose
p x
e t are uncorrected with x t
some matrix
for
transformation of x
£
assume the
parameters of the model
shift
=
unknown parameters. When the
6 are
uncorrected over time, the regressors x t and z t
disturbances,
(r
a sequence of unobservable disturbances that are weakly dependent;
x and z t are p x
is
yt
and
...,
generally,
so that z t
<j
yr)',
2
is
a linear
with a focus on k
X
=
(x\, X2,
...,
X = (0,...,0,xfc + i,...,xr)\
and X depend on k, but
2
2
we
.
xt)'>
and
this
dependence
Z\
be displayed
will not
X
= X\R, Z2 =
2
R, and Zq
for notational succinctness.
= XqR.
Equations
and
(1)
(2)
Furthermore,
let
can then be rewritten
as
y
We
= X0 + Z
=
consider the problem of testing 6
+ e.
S
(3)
based on the Wald
unknown, the matrix Z cannot be constructed. The strategy
obtain a sequence of
Z2
(recall
depends on
mean
such as the
3 and
Wald
6,
the dimension of i t
is
is
=
to
of testing 6
for k
=
p,
p + 1,
and then using
.
..
is
,
T—p
as a test
and
6k
on
be the
X
least squares estimators
and
Z2 The
corresponding
.
then
1
——
m—
J
q
M — I — X{X' X)~
and S(k) can be written
X
x
and S(k)
the
is
w
km ** +1 >-"> T -*
'
sum
of squared residuals.
Note that
Si,
as
6k
S(k)
is
,
Because k
value of the sequence, or other functional of this sequence,
respectively, obtained by regressing y
statistic
where /()
is
value. For each fixed k, let 0k
t() =
where
where p
Z2
by replacing Zq with
statistics
k),
maximum
statistic the
of
Wald
test.
=
(Z'2
MZ
)- l
2
Z2 My
= J>« - xtk The
the indicator function.
WT =
z t S k I(t
>
k))
2
test statistic for testing 8
=
is
WT {k)
sup
(5)
iT<k<(\-i)T
where
e
>
dard, see
is
a small positive number. The limiting distribution of
Andrews (1993)
Wj
is
nonstan-
for details.
Now assume 6^0. Our
objective
is
to estimate the shift point k Q
the shift-point estimator as the location where the
maximum
achieved, namely,
k
= argmax ,<
J
fc
< T _ p VVT (fc)
of
.
Wald
We
define
statistic
is
where p
is
the
number
of
We
columns of X.
k the W-estimator for ease of
call
reference. This approach to estimating of a shift has been used in empirical applications. Christiano (1992) looked for potential
Note the restriction k €
[p,
T — p]
is
GNP
changes in U.S.
needed to ensure the existence of
estimators. However, no restriction of the form k € [eT, (1
—
e)T]
Although both the numerator and denominator of Wr(k) of
is
using F-statistics.
is
(4)
least squares
required.
depend on
k,
it
not difficult to show that k can be obtained by maximizing the numerator alone
and can
also be obtained
We state
by minimizing the denominator alone.
simple
this
result as a proposition.
Proposition
1
A
A
k
That
as
is,
= argmax
fc
W:r(fc)
= argmax
fc
6 k (Z2
A
MZ
2 )6k
=
argmin
fc
5(fc).
the W-estimator obtained by maximizing Wald-type statistics
minimizing the sum of squared residuals. To see
squared residuals under restricted estimation (8
=
0),
this, let
is
S denote
same
the
the
sum
then the Wald statistic
of
is
^» = (^P)(^P).
Because S does not depend on k and the Wald
formation of S(k),
k
it
follows immediately that k
= argmax {5 — 5(A;)}. The
fc
[see
Amemiya
statistic
=
is
a strictly decreasing trans-
argmin
proposition then follows from
fc
5(fc).
Or
S — S(k) =
equivalently,
8k(Z'2 MZ2)Sk
(1985, p. 31-33)].
For a linear model with least squares estimation,
tonic (increasing) transformations of the
be obtained by maximizing
LR and LM
Wald
LR and LM
statistics,
statistics.
statistics are
mono-
thus W-estimator can also
Despite these facts,
with Wald statistics for two reasons. First and foremost, Wald
we
shall
work
statistics lead nat-
urally to the consistency arguments, as will be seen from the proofs below.
Other
forms of test statistics do not have this advantage. Second, under estimation methods
other than least squares
(e.g.
instrumental variable estimation or
GMM estimation),
the results of Proposition
LR
equivalence of Wald,
may
1
and
not hold. In particulax, for nonlinear models, the
LM
no longer holds, but our analysis which
statistics
is
based on Wald statistics has the potential to be extended both to other estimators
and nonlinear models
as well as simultaneous equations systems.
For example, Lo
and Newey (1985) and Andrews and Fair (1989) consider structural change problems
based on Wald-type test statistics
and nonlinear simultaneous equations.
for linear
In the case of pure structural change, that
k
=
argmax,
{(/?,
= (X[X\)~ X[y
x
where
f3\
- k)'[{X[X
x
)- 1
is,
x
=
t
+ (Xyra )- l ]-»(A - &)}
= {X^Xt)" X'2 y
The term
1
and $2
W-estimator becomes
z t , the
.
inside the braces
represents a weighted average of the distance (squared) between the pre-shift and
post-shift
parameter estimators. The W-estimator maximizes
In addition to the shift point,
Let
is,
=
J3
we
(3(k)
and
6
=
Z2
replace Zq by
same
In
8(k) be the estimators of
with k
=
limiting distribution as
what
are also interested in the regression parameters.
follows,
we use
to zero in probability
for
1 € TV-
||A||
=
•
||
||
6 corresponding to k.
We
(3).
That
shall establish
and
if
the shift point to were known.
is
random
op (l) to denote a sequence of
p (l) to
denote a sequence which
Bj = O p (l)
if
is
For a matrix A, we use
||A||
stochastically bounded.
each of
used to denote the Euclidean norm,
variables converging
i.e.
its
elements
||i||
=
is
V {\).
'
1/
(5Zf=i I ?)
to denote the vector- reduced
norm,
2
i.e.,
sup r ^||Ax||/||x||.
We make
Al.
ifco
=
the following assumptions:
[tT],
A2. The data
where t €
(0, 1)
{(y t T,XtT,ZtT)\
1
notational simplicity, the subscript
where
and
k and then estimate model
For a sequence of matrices Bj, we write
The notation
/?
and 6 are root-T consistent, asymptotically normal, and have
subsequently that
the
we
this distance.
R
is
p x
9,
rank(iZ)
=
q, z t
and
<
T
t
[•]
is
<
T,
will
the greatest integer function.
T >
form a triangular
be suppressed.
€ IV, x € TV,
t
1}
q
<
p.
array.
In addition, zt
=
For
R'x u
A3.
The matrix
sup ; >i
j T,ttu
\\-
x x
Yfttu
t
't\\
x x
t
zero
-jtj
bounded
stochastically
is
—Q
xx
where
,
Ylt=i x t £ t =* -^( 5 )i
mean and
positive definite for large values of
is
bounded away from zero
1S
A4. j(X'X)
A5.
't\
x t x't
J2t=i
Q xx
for large
—
j\
and
Furthermore,
I.
;'.
and positive
is finite
where B(s)
every fixed
for
\i
definite.
multivariate Gaussian process with
is
covariance matrix
E{B(s )B(s 2 )'}=n( Sl As l
1
)
where
lTi][T,)
i
=
n(s)=Km-'£'£E{x
T-oo T
i=l ;=1
G.(*\
The notation "=»" stands
for the
real
number
j^XtCt
r
>
x' e i e j ).
j
weak convergence
A y" stands
topology, see Pollard (1984) and "x
A6. For some
i
r/2
C>
for all 1
z')
1]
with the Skorohod
minimum
for the
2 and constant
< C{j -
in D[0,
value of x and y.
0,
<
i
<
j
<
T.
t=l
Assumption Al assumes that the
shift point is
This assumption can be slightly more general,
and tj,t €
(0,1).
form {t/T) e
(£
g(t/T).
>
Assumption A2 allows
0) or,
more
for
for
bounded away from the end
example, &o
=
['TrT'h
where tj
—
r
trending regressors written in the
generally, written as
any function of the time trend,
Expressing trending regressors in this way avoids a scaling matrix that
would otherwise be required when deriving limiting distributions. The
A3
points.
requires that a
positive definite.
sum
The
involving an increasing
latter half
is
number
typically implied
numbers. Assumptions A4 and A5 are standard
independence of the errors
see e.g., Wooldridge
is
not assumed.
A5
is
(1980) and
Andrews and Pollard
8
by the strong law of large
satisfied for a
and White (1988). Assumption A6
(1990).
half of
must become
Note that
for linear regressions.
martingales or strongly mixing sequences with some
Yokoyama
of observations
first
is
wide range of settings,
satisfied for
moment
x e that are
t
t
conditions, see, e.g.,
Define f
p
=
We shall show
k/T.
that f
is
consistent for r and prove that
T(t-t) =
(l) in the following section.
The Consistency
3
In this section,
tain
its
we
first
convergence
of f
establish the consistency of the W-estimator
The
rate.
latter result
and then ob-
also used for obtaining the limiting
is
distribution of the W-estimator as well as the limiting distribution of the estimated
regression parameters. First, the consistency result:
Proposition 2 Under assumptions A1-A5, for any
To
>
0,
when
T > TQ
we
this proposition,
-
r\
Proposition
1,
k
behavior of Vj{k).
and n >
0,
there exists
=
>
<
n)
6'
k (Z'2
MZ
2
argmax Vj{k). To obtain
We
show
can only be maximized near k
.
that,
The
if
^
6
)6 k
we examine the
global
then with high probability, Vj(fc)
0,
basic idea
.
consistency,
fc
shall
e.
define
VT (k) =
By
>
,
P(\r
To prove
e
to
is
decompose Vr(k) — Vr(ko)
into
two parts: a "deterministic" part and a stochastic part. The deterministic part
maximized near
ko,
whereas the stochastic part
we
the deterministic part. To see this,
first
Sk
=
{Z'2 MZ2)- {Z'2
8*
=
6
x
+ (ZLMZo)-
uniformly small
(in k) relative to
notice that
MZ
l
is
is
)S
+
{Z'2
MZ
2
)- 1 Z' Me,
2
Z'Q Me,
therefore
VT (k)-VT (ko) =
=
6k
(ZWZ
6'{(Z
)6k-6ko(Z'Q
2
MZ
2
+h(k,S,e),
)(Z'2
MZ
2
MZ
)- l (Z'
2
)6 ho
MZo)-(Z
(6)
MZ
)}6
(7)
(8)
where
=
h(k,6,e)
26'{Z'Q
MZ
2
){Z2
MZ
l
+ e'MZ2 {Z'2 MZ2)-
2
)- x
Z2 Me-26'Z'Q Me
Z'2 Me-e'MZo{Z'Q
Expression (7) constitutes the deterministic part and h(k,
tic paxt.
MZ
(9)
)- x Z' Me.
Q
(10)
e) constitutes the stochas-
8,
Denote
X& =
XA
and define
X
2
-
X =
= -(X2 -Xo) =
X&
,
(0,...,0,Iito+1 ,...,x fc ,0...,0)
to be a zero vector
if
k
=
k so that
X
2
the distinction between the positive and negative signs
X = X ±X±.
2
ZA =
Set
for k
(0, ...,0,ifc+i,...,Xjkp,0,...,t))'
XA R,
and
for
<
k
>
k
k
=
X
is
immaterial, we simply write
+ Xasign(fc — k). When
let
'
m=
7W
When
we
k
=
k
,
*'{(^MZq) -
{Z'
MZ
2 ){Z'
2
is
=
Note that
S'6.
semi-positive definite.
must
shift-point estimator k
k\f(k).
))6
•
*y(k) is
We
(U)
satisfy Vr{k)
non-negative because the matrix
have the following identity
VT (k)-VT (ko).--\ko-k\i(k) +
—
MZ
both the numerator and denominator of f(k) are zero. In that case
inside the braces
l^o
2 )-\Z'
2
\k^k\
arbitrarily define 7(^0)
The
MZ
>
h{k,S,£)
for all k.
Vr{ko), or equivalently, h(k,8, e)
Thus we have
F(f-r|>7 =
7)
<p(
sup
P(|fc-*
J
|>r 7)
J
\h(k,6,e)\>
inf
|*o
-
*b(*0
t
<p(
sup
\h(k,6,e)\>T V
\p<k<T-j>
= P(lT
l
sup
r-
1
inf
\ko-k\>Tv
|Mfc,*,e)l>'7)
p<jt<r-p
10
(12)
7 (*))
J
>
where
=
7r
Lemma A. 2 shows
that 77
J"
Next we verify
at
at the rate of
of h taken over
shall
\/T
all
k,
is
it
O p {T~ l/2 log(T)),
is
a stronger
not difficult to see that h(k,6,s) grows
We
not enough.
is
(13)
(13)
T
follows that h divided by
It
.
show that
converges to zero in
must prove that the maximum value
possible k grows at a slower rate than T. This
the precise argument. Divided by T, the
is
Thus consistency
= op (l).
sup
\h(k,8,e)\
p<k<T-p
we
(13). In fact,
probability. However, this
Here
1
than needed. For each fixed
most
zero.
from
will follow
result
0.
and bounded away from
positive
is
>
inf
-f(k)
l*-*ol>ru
indeed the case.
RHS
term of the
first
is
of (9) can be
written as
26'^=^=(ZiMZ2 )(Z'2 MZ2 )- l/2 (Z'2 MZ2 )-^Z2 Me.
The law
(14)
of iterated logarithm implies that
y 1/2 Z
sup \\(Z2.MZ2
2 Me\\
= O p (log T).
(15)
k
where the supremum
is
taken over
supBT B'T =
sup
k
by
{Z'Q
T
MZ
is
Q)
T~
l
l
M
(14)
M£ = O p (T-
T is obviously O v {T~
O p {T~ l,2 \ogT). We
It
1
/2
2
is
).
l
).
)(Z'2
M
O p {TFrom
1
2
'2
)~ x (Z'
2
log T).
(15), su Pjt
to the
first
statistics.
M
T.
Next we
k, or equivalently,
Combining these
Q)
<
{Z'Q
MZ
for all k
)
The second term
and
of (9) divided
T- e'MZ2 {Z2 MZ2 )-'Z'2 Me =
term of
x
(10).
results,
The
last
term of (10)
we have T~ sup t
l
\h(k,
di-
8, e)\
thus establish the consistency of the W-estimator.
should be pointed out that the consistency
Wald-type
<
l
O p (T -1 (log T) 2 ), which corresponds
vided by
k
T- {ZoMZ2 )(Z2 MZ2 )- {Z'MZo) = O p (l).
= O p (l). Thus
Z'Q
uniformly in
<
k
But the above follows from (Z'
l
values of k such that p
A^{ZqMZ2 ){Z2 MZ2 )~ xI2 = Op (l)
show Bt —
T-
all
The
is
obtained by globally searching the
consistency argument does not require that the searching
11
be limited within a positive fraction of the data points such that k €
The
implication
Wald type
that the sequence of
is
the two ends, attaining
maximum
global
its
statistics
is
well
[eT, (1
—
e)T}.
behaved even
at
near the true shift point. In a recent
paper by Nunes, Kuan, and Newbold (1993), they prove the consistency but without
obtaining any rate of convergence. Also their arguments require restricted searching.
Next we establish the T-consistency
result:
Proposition 3 Under assumptions A1-A6, for any
P (\T{f -
t}\
>C) = P
> Vr(kQ by
Again because Vr(k)
(
it
VT (k) >
sup
-
(|ifc
definition,
)
P
Jfcol
T.
Thus
any
2, for
to prove (16),
e
it is
>
—k
where K(C)
=
Finding the
maximum
but this
is
0,
:
\k
\
<
Vr(ko)]
e.
show
(16)
«.
/
and a >
sufficient to
Pi=p(
{k
C >
there exists a
0,
> C) <
suffices to
\l*-*ol>C
Proposition
>
T
such that for large
By
e
we have P(\k — k
0,
> Ta) <
t
for large
show that
VT (k) > VT (k
sup
\
> C and Ta <
k
))
<
< (1— a)T}
e
for a small
K(C) amounts
value over the set
legitimate only after establishing consistency.
number
a
>
0.
to restricted searching,
Note Vr(k)
>
Vr{kQ )
is
for large
C
equivalent to
Thus
P\
< P[
h(k,6,e)
sup
\keK(C)
By Lemma
=
A.2, inf fc6 K-(c) l(k)
and large T. Thus
it
=P
—
>
for
is
sup
\k€K(C)
ko
12
—
k
7 (fc)V
bounded away from
any fixed
h(k, 8,e)
\
inf
k
fcr, which
show that
suffices to
P2
Jcq
A>
>A )<e
0,
(17)
when
C
is
Consider the terms
large.
data (at
itive fraction of
in k.
Thus
e'
e'MZQ {Z' MZo)-
O p {l)/\k -
k\
l
M
Z'
2
{Z'2
Next consider
S\Z'Q
2
)~ x Z'
2
Me
because
2
MZ
){Z'2
2
- O p (l)
—
\k
)- x Z'
2
>
k\
A>
Z2 = Z ± Z&
Use
MZ
involves a pos-
/T)~
X
A5, where
= O p (l)
Op {\)
uniformly on K{C).
-
\k
C
Choose
C.
k\ is
large
is
by
uni-
Similarly,
bounded by
enough
so that
0.
deduce that
to
Me
Me ± 6'(Z'A MZ2 ){Z'2 MZ2 )-
6'Z'2
2
Therefore (10) divided by
e/3 for any pre-given
(9).
=
M
Me = O p (l).
< O p {l)/C
P(O p {l)/C > A) <
MZ
(Z2
observations), thus
T~ xl2 Z'2 Me = O p (l) by assumption
assumptions A2-A4 and
form
Ta
least
Z2
For k € K(C),
in (10).
l
Z'2
Me
= 6'Z^Me ±6'Z'^Me ±6'{Z':,MZ2 ){Z2 MZ2 )-
1
Z'2 Me.
(18)
Therefore,
6'{Z'Q
MZ
2
)(Z'2
MZ
2
ko
\8'Z'A Me\
Me - 8'Z' Me
)- x Z'
2
—
(19)
k
+ \h\Z^MZ2 ){Z'2 MZ2 )-
x
Z'2 Mt\
(20)
\ko-k\
S'Z'^Me
—
ko
Now
k
+
6'{Z'A
MZ
2
)
(Z2 MZ2 \~
X
(Z'2 Me
(21)
ko-k
by assumptions A3 and A4,
MZ
6'{Z'A
2
—
ko
Z'^Zt^
)
k
ko
—
k
Z'^X*
—
ko
(X'Xy-l
k \
1
T
J
X'Z2
T
P (1),
which together with the functional central limit theorem implies that the second
term of
(21)
is
(the case of k
1
ko
—
O p (T>
ko
is
Z'Me =
k
x
l2
).
Next, Z'A Me
= Z*e -
x
Z'A X(X'X)- X'e. Suppose k
<
similar), then
1
*0
1
£
E **-TTk
Ko- K
k0~ K
k K
t
k+ i
I
\
t=k+1
ko
1
*°
Y, **t +
~ K tsfc+1
O v {T-" 2
13
).
«<] (X'X/T)'\X' £ /T)
J
k
It
remains to be shown that
P
for a given
A. 3, there exists a
P
than
less
is
Proposition
t
such that
> A) <t.
L >
0,
such that
for
1
A>
any
>A
<
,k<ko-C
which
0,
K t=k+i
"Q
sup
I
C>
1
sup
I
,k<ko-C
By Lemma
A, there exists a
when
C
and
C>
A r Cr / 2
large for a given A. This proves (17)
is
0,
and therefore
3.
T-consistency seems to be typical for shift-point estimators, just as root-T consistency
is
typical for other parameters under regularity conditions. T-consistency
obtained under nonparametric estimation in the case of two
bgen (1991). The same rate of convergence
and
a
mean, see Bai (1993a,
also established for
LAD
Duem-
estimation
b).
Given the convergence rate of
$ and
samples, see
a strongly mixing sequence or a linear process with
least squares estimation in
shift in
is
i.i.d.
is
k, it is relatively
easy to show that the estimators of
6 are not only root-T consistent, but asymptotically normal. In estimating the
regression parameters,
we use k
in place of
Icq.
Because of the
fast rate of convergence,
treating k as ko does not affect the limiting distribution of the estimated regression
=
parameters. Recall that
Corollary
1
=
0(k) and 6
6(k).
We
have
Under assumptions A1-A6, together with
(M-!!)-™>
e t being uncorrelated,
n
where
V=
,.
plim
—1
( T,t=i x t x t
\
Ht=ko x t zi
(22)
„
For serially-correlated disturbances, the variance-covariance matrix of the limiting
distribution
is
given by
v- uv~\
x
14
where
„
£/
=
,.
Zl>i
/
1
hm—
£(*^e,-e,)
£?>*„
~
jB(*««Je*«i)
\
(23)
I
Methods developed by Newey and West (1987) and Andrews (1991) can be used
estimate the matrix U.
When
a and $ have the same
that
sion here
is
estimating
t
a and
may be
after a structural
Lemma
A.l and
possibility
the
can be constructed
possible to have a regressor in z
An
Lemma
in the
determined
conventional
does in fact exist.
shift
by using the identity
variable
but not in x
The
play.
t
That
.
is,
only
current proof does
extension to cover this case requires similar results to
A. 2 without assuming z t
M,
variable in the matrix
new
t
change does a new variable enter into
not allow this scenario.
new
Notice
were known. The conclu-
This conclusion assumes that a
t-statistics).
.
required that the vector z t be a sub-vector of x t or a linear transformation
It
.
ko
if
in place of k
that, although the estimators of regression coefficients are
way (based on
of x
and U, one uses k
limiting distributions as
sequentially, confidence intervals for
We
V
to
in
Lemma
which
and including
it
in
t
One may
.
A. 4. Another solution
explore this
to include the
is
equivalent to assigning a zero coefficient to
is
x
= Rx
t
But
.
this
method produces an
inefficient
estimation of regression parameters and consequently an inefficient estimation of the
shift point as well.
i
4
Asymptotic Distribution and Confidence Sets
We now
consider the limiting distribution of the W-estimator for small shifts (shifts
of shrinking sizes as
T increases).
There are two reasons
intractability for shifts of fixed magnitude.
dependent
for fixed shifts
and thus
indicates that, even for an
distribution
is
is
One
difficult to obtain.
The
result of
normal sample with a mean
of less practical importance.
similar to that of Picard (1985) and
Yao
(1987).
15
is
limiting distribution
enormously complicated. Second, because of
the limiting distribution
is
i.i.d.
The
for this.
this
the technical
is
highly data
Hinkley (1970)
shift,
the limiting
data dependence,
The approach we take here
We find a limiting distribution
for
small changes. This limiting distribution can then be used as an approximation to
the underlying distribution for moderate shifts. Of course, the approximation
not be satisfactory for large
insight
in
on how
Nevertheless, the limiting distribution provides
shifts.
the precision of the shift point estimator and
serial correlation affects
what way the magnitude
may
of the shift
and regressors influence the precision of the
shift point.
We
assume the magnitude
of shifts,
-4
||6r||
In the previous section,
we treated
\\
=
fco
=
- oo.
+
for the
of Vr{k)
argmax
=
2
0(||6r||
)
—
varies in a
an<^ u
is
a real
KT (V) =
For any given
V>
0,
we
(25)
may be found
Bai (1993a,b). Given the
in
may be
obtained by the local
Vr{ko) in conjunction with the continuous
By
functional.
weak convergence when k
where X\
T and established:
0„(l).
convergence rate (25), the limiting distribution of k
theorem
(24)
+ Op (\\6T \\- 2 ).
ko
omit the details here. Similar results
weak convergence
such that
we can show
modifications,
k
We
Vf\\6T
0,
T
depends on
6 as a constant not varying with
fc
With some
\\6\\,
the local weak convergence
neighborhood of k such that k
number
{k: k
=
kQ
in a
compact
+ [v\t%
\v\
<
set.
mapping
we mean the
=
k
+
[uA^
2
],
Let
V}.
derive the limiting process of
VT (ko + M? 2 ]) - VT (ko)
for v
€ [— V,
V].
Suppose Vr(ko
theorem then implies Xj(k— ko)
of an
argmax
functional,
On Kt(V), we
we
+ [vX??]) - Vj(fco)
=* G(v), the continuous
— argmax
For a discussion on the continuity
refer to
v G(v).
Kim and
Pollard (1991).
have a more simplified expression
16
for
Vr(k)
—
Vr(ko).
mapping
Proposition 4 Under assumptions of A1-A6
VT (k) - VT (ko) =
w/iere o p (l)
The proof
is
is
is
1:
±
26T Z'A e
+
o p (l)
uniform on Kj{V).
provided in the Appendix. Thus the limiting process of
determined by
Case
-S'jZ'^Z^t
— 6tZ'a Z&6t ±
We
ISjZ'^e.
Vj(jfc)
—
Vj(/fc
)
consider two leading cases.
Nontrending regressors but possibly correlated
errors.
Assume
that the
regressors satisfy
[T>]
Y
plimT-00 *
for all r. Let
Aj
=
£
Ziz'x
i
= sQ
plim-^^ -
and
J
1=1
T
T
££
8'
t Q6t- Consider the limit of S^Z'^Z^Sj
OjZ^Z^Ot — A T
Since Z'^Z^ involves |[uAj
2
]|
E(z,-zJ-e,-Cj
)
=Q
t=ij=i
=
for k
Icq
+ [uAj 2
Now
].
—3
(the absolute value of [vAj
2
])
observations of z t
,
we
have
7'.
7.
MO
i-a
uniformly in v € [—V, V] (note that A^
—
oo). This implies
»
8'
t Z'a Z&6t —*
\v\
uniformly in u € [—V, V].
Next consider the
By
limit of
6' Z'^e.
T
Assume v <
the functional central limit theorem of
S'T Z'A e
=
6'
T
£
A5
0,
that
is,
k
=
k
+
[vX?
(re-scaled),
z«e t =»
^Wi(-»)
t=fc+i
where W^-)
is
a Brownian motion process on [0,oo) with Wi(0)
17
=
and
2
]
<
k
.
The weak convergence
on
Jfc
[-V,0],
>
ko
space D[-V,0], the set of cadlag functions denned
in the
is
endowed with the Skorohod topology
(i.e.,
>
v
[see Pollard (1984)]. Similarly,
when
we have
0),
fc
t=ko+l
where
W
2 (i;) is
processes
another Brownian motion process on
W\ and
W
2
and W(v)
0,
W (v)
—
2
for v
(— oo,+oo). Then combining these
VT (k +
2
The
-
ko)
-i*
definition for
—
we have
- VT (k
)
])
2
<f>
,
we can
its
{6'
T Q6T)
For uncorrelated errors, because
special case
is
follows that #(ib
Case
2:
a
shift in
-
argmax
ifco)
<r
2
2
0(||6t||
).
first
0,
-
|w|/2}.
given in (35). In view of the
is
write
2
~
l
T n6T)- {k
{6'
ft
=
<r
2
M -1 V
*
(26)
Q, (26) becomes
fc)
-£
V-
(27)
t
=
1,
so
Q=
1.
It
V\
Trending regressors. Assume
Let us
<
Denote the random variable
the intercept only. In this situation, z
-*->
for v
functional,
a
density function
z<
=
the functions ft(x) have bounded derivatives on
is
The two
2<t>W{v).
^ argmax v {W(t;)
in variable.
(Jl9hl { k -
A
+
=> -|v|
for the
from a change
by V*,
|u|/2}
Aj and
0.
a two-sided Brownian motion process on
0,
argmaxJ-M + 2<j>W{v)} =
last equality follows
argmax„{W(i;)
^(0) =
W(v) = W\{— v)
Let
results,
2
[v\t
>
From the continuous mapping theorem
\ T {k
oo) with
are independent because they are the limits of non-overlapping
disturbances that are only weakly dependent.
W(v) =
[0,
g(t/T)
[0, 1].
show that
S'tZ'^ZoSt
18
-
\v\
=
Let
(gi(t/T),...,gq (t/T))'
A?r
=
and
T g(T)g(T)''8j which
S'
uniformly in v € [— V, V]. Consider the case of v
S'T
Z'^ST =
£
6T
<
k
(i.e.
<
ko).
Then
g(t/T)g(t/T)'6T
(28)
t=k+i
=
{ko
-
+8'
T
t
in v
is
equal to {ko
We
€ [— V,0\.
Note that (30)
X
£
(30)
WIT) - 9(r)][g(t/T) - g(r)}'6 T
£
[g{tlT)-g{T)\g{r)'6 T
— k)\\ = — [uAj 2 ]A^,
(31)
which converges to
—v
uniformly
next argue that (30) and (31) are o p (l) uniformly on Kt(V).
bounded by
is
2
dg(x)
*o
£
8'
T 6T
sup
(29)
= k+l
+26'T
Expression (29)
k)6'T g{T)g{T)'6 T
dx
(<
- hf/T 2 < B
\\6r\?{ho
x
- kf/T 2 < B 7 (T 2 \\6t\\*)-
1
-.
0,
t=k+l
(32)
for
some constants B\ and
5j.
The proof
=
T g(ko/T)
that (31)
is
also o p (l)
is
similar.
Next
S T Z'A e
=
Suppose the
hand side
£
S'T
£j
g(t/T)e
t
6'
£
et
+ 4 J)
[j(i/T)- 5 (VT)k t
.
(33)
are uncorrelated, then the variance of the second term on the right
is
*% £
^)-</(wr)][j(t/r)-s(io/r)]^
which converges to zero by (30) and
determined by
6'
Tg(r) J2tLk+i
e
t
-
Because k
by the functional central limit theorem
6'
T9(t)
The argument
for v
>
is
Thus the
(32).
£
=
k^
+
[uAf
(see A5),
£<
= »Wk(-v).
similar. In particular,
^Z^e
limiting distribution of (33)
= <rW {v)
2
19
2
]
and
A?r
=
6'
is
T g(T)g(T)'6T,
where
W
2 {-) is
another Brownian motion process independent of W\{-). Define W(v)
as in the previous case,
we have
VT (ko +
[uAf
- VT {ko)
2
])
=> -\v\
+
2aW(v).
This implies, by the continuous mapping theorem and a change in variable,
\f( k - ko)
_^
where
A?r
=
convergence
f>T9{ T )9{ T )'fostill
_
argmaXu{v^ (u)
a2
u|/2})
(34)
|
For stationary and serially correlated errors, the above
holds but with
a 2 replaced by
h
*2
h
l
}™
h->oo iZZE(^)h
=
1=1 J=l
Confidence Intervals.
To construct confidence
asymptotic results (26), (27), and
intervals for k
Under case
(34).
distribution function for the
G(x)
for
>
i
=
1
+
W
and G(x)
1 '2
=
1
^-*'* — G(— x)
function of a standard normal
are easy to obtain. All
a2
.
However, 6j
as fJ2t=i z t z t-
is
The
a2
.
l
-{x
[see
random
we need
variable
Yao
(1987)],
variable.
variance
it is
So quantiles
=
is
random
t Q6t and
S'
Q
(35)
variable
for the variance
can be simply taken
^5(fc).
Similar to the
a root-T. consistent estimator
For serially correlated errors, one also needs to estimate the matrix
the methods of
Newey and West
trending regressors, Ay
=
(1987) and
S'T g{T)g{T)' Sj
by g(k/T)g(k/T)', which only uses the
.
</(f )<7(r)'
observation.
for constructing confidence intervals are easy to obtain.
interval
is
given by
[k
-
2
a-
c o/2 /A^, k
20
Cl
Andrews (1991) can be used.
The matrix
k-th.
is,
the distribution
is
for this
The matrix
b2
|u|/2}
x
a 2 can be estimated by a 2 =
not difficult to show
given by (27).
|e *(-3v£/2)
where $(x)
are estimates for Xj
1.
is
V — argmax„{W(t;) —
+ 5)*(-v^/2) +
estimable, see Corollary
result of Corollary 1,
of
random
we use the
(non-trending regressors)
1
together with uncorrected errors, the limiting distribution of k
The
,
+ a 2 ca/2 /\T]
A
and
For
can be estimated
All quantities necessary
100(1
— a)%
confidence
where
ca / 2
is
V.
the a/2 percentile of the distribution of
Because the estimated change point k
close to k Q
is
,
one can estimate Aj
in either
case by
(
M^
.
\
some
for
m
difficult to
true for
m
>
0.
k+m
\
is
and
5
.
? H*t
£=fc-m
/
In the case of trending regressor such that z t
show that
^ (EJ2U, 2 2
:
<)
=
g(t/T),
m/T
(E*^ m z
-*
't
z t)
(£,
for
m=
example,
should be close to
Finally, the confidence intervals for regression
relatively large.
it
~> 2( r )s( r )'> for each fixed m. This
growing unbounded but satisfying
contains no trending regressors, then
j^
m
\
2
Q
not
is
is
\ff
.
also
If z,
provided
parameters
6 are constructed in the usual way.
Concluding Remarks
We have considered
part or
all
shift point
the structural change problem in a linear regression model where
of the coefficients have a shift occurring at an
is
unknown
estimated by maximizing a sequence of Wald
the T-consistency for the estimated shift point.
time.
statistics.
The unknown
We established
In the case of partial structural
change, the unchanged regression parameters are estimated with the whole sample,
whereas the shifted parameters are estimated with subsamples. With the presence
of structural change,
we have demonstrated
that the sequence of
are well behaved in the sense that the sequence achieves
the true shift point. Thus the maximization
the search for a
maximum
in the
global
statistics
maximum
near
done via global searching. Confining
is
middle of the sequence (ignoring some fractions of
the sequence in the beginning or at the end)
for a linear regression
its
Wald type
is
not necessary.
We
also noted that
with least squares estimation the shift-point estimator can
be equivalently obtained by maximizing a sequence of F-statistics, LM-statistics, or
LR-statistics.
When
only a single parameter
be obtained by maximizing a sequence of
21
is
allowed to change, the estimator can
t-statistics in absolute value as well.
In
addition,
we studied the asymptotic
distributions for the shift-point estimator as
well as the regression-parameter estimators.
These
results enable
one to construct
confidence intervals. Most importantly, the asymptotic results provide insight on
how
the precision of the shift-point estimator depends on the regressors, the magnitude
of the shift,
and
serial correlations in the data.
of regressor designs, with time trend
also permit heterogeneous
It will
Our
results hold for a
and stochastic regressors
wide range
as special cases.
We
and dependent disturbances.
be interesting to extend the argument of
by Andrews (1993), who employs Wald,
LM
and
this
paper to the setup considered
LR
statistics to test for the exis-
tence of a structural change in nonlinear models. These statistics are constructed
using instrumental variable estimation or more generally the Hansen-type
timation.
estimator.
tor.
By maximizing
An
models and
hope
this
a sequence of these statistics, one also obtains a shift-point
important unresolved topic
Much work
for
paper
is
GMM es-
is
the consistency of the resulting estima-
needed to make the analysis of
this
paper applicable
for nonlinear
models with nondifferentiable objective functions such as LAD.
will stimulate further research in this area.
22
We
A
Appendix
Lemma
A.l The following two
MZQ
MZ
inequalities hold:
MZ )>R {X'£,XA ){X' X r
(Z^MZo) - {Z' MZ ){Z' MZ ){Z'
)
-
{Z'Q
2
){Z'2
,
x
2
2
2
X
2
l
2
2
MZ
{X' XQ )R,
{Z2
)
k<k
{Z2 MZo)
> RfX'^XA (X'X - X'2 X2 )-\X'X.- X^X
Proof of (36). Write
H=
{X'2
X
2
- {X'X)~
)- X
X
2
2
A = H X/2 (X2 X2 )R.
X
A'
>
0.
2
2
kQ
(37)
.
jfco,
(38)
X
)R}- R!{X'2 X2 )H{X' XQ )R.
1
2
2
-A{A'A)~
Since I
A'
is
H
x
I2
is
identical to (39).
Z^MZo - R (X'Q Xo)H(X
,
,
112
A{A' A)~ A'
x
Thus
=
R!{X'
X )R > R {X'£,XA )(X^X
,
2
X
)- X
last equality follows
and
is
from
x
2
2
X
X'Q XQ
omitted.
23
-
(X'2
{X'
+
Xr
2
{X'Q
XQ )R > 0.
show
)- X (X'
side of (40)
X
XQ )[{X'Q Xo)-
XX =
hand
- (X'X)- - H]
= R!{X'M{X2 X2 )The
'2
l
suffices to
it
In fact, the equality holds in (40) because the left
R!{X'Q Xo) [{X'Q
XQ )H xl2 from the left and
(X'q Xq)R, we obtain
Xq )H{X'q Xq)R- R!(X' Xo)H
The second term above
(39)
a projection matrix, we have
Multiplying this inequality by R'(X'Q
multiplying from the right by
R!{X'q
>
)
2
2
2
- A{A'A)-
k
)R,
x
2
/
/
<
For k
.
MZ )(Z MZ y (Z MZ =
R (X' X )H(X X )R {R(X' X )H{X' X
(Z'
Let
(36)
l
{X'
X
)R.
(40)
is
Xq )R
](X'Q Xo)R
XQ )R
X'A X A
.
The proof
of (37)
is
similar
Lemma
A. 2 Under assumptions A2-A4, for
bounded away from
That
defined in (11).
is
notice that for any
number n >
=
inf
\k-k<,\>c
<
Proof: For k
X'A X&
j(k)
inf
k
Lemma
by
,
be written as R'[(X
(41)
is
positive.
far apart (i.e.,
£~3£ X'^ X&
A.l part
=
such that ko
—
X
when
X&
Also
in probability.
inf
~t(k)
V
'
|*-*o|>C
RHS
£j3£ 5Zjl fc+1 x t x't
k
of (36)
is
)R6.
if
X'A XA >
zero,
0,
when
in fact positive definite,
is
bounded away from
is
(41)
positive definite since
contains enough nonzero elements).
the
can
it
LHS
of
k and ko are
Also when k
by A2 and A3,
<
k
,
for all k
large. In addition,
is
v
v
bounded away from zero
~
=
for all k
,Jgf>c
bounded away from
zero.
<
ko.
{**%=!
The
P
j
sup -
i
^2-^2
\
Thus we can choose
>
ko can
there exists
Y,2
t
X Xo
T-k
C
sufficiently large
ww«)
case for k
A. 3 Under assumption A6,
°
T-k \T-kJ
so that
Lemma
>
-
+ (X'^X^-^R. Thus
)- X
v \-i/(A AoJ
v \
[A 2 A.2)
is
7c >
8'B!^±{X'2 X2 )-\X'Q X
Note that X'^X^
/
is
=
(i),
positive definite then the
is
and
positive
for large T.
7 (jfc) >
If
is
f(k)
IK '
lim inf t~<x Icr
is,
\k-ko\>Tr,
Tn > C
for
0,
=
7T
because
0,
where
zero,
for
and i{k)
C >
large
et
t=l
24
> A
be proved similarly.
L >
<
such that for any
m 7M'
A>
0,
Proof:
This
is
essentially proved by Serfling (1970)
omitted. Note for z t e t
i.i.d.
(Theorem
lemma
or martingale differences, the
Details are
5.1.).
is
the Hajek and
Renyi (1955) inequality. Extension of the Hajek and Renyi inequality to a linear
process of martingale differences
Proof of Corollary
the estimators
$ and
is
given in Bai (1993a).
Let us use Zq to denote Zq
1.
when
X
by regressing y on
6 are obtained
k
replaced by
is
and
Zq.
Then
k.
Equation
can
(3)
be written as
=
y
with e'
=
+ (Z —
e
we have
when Zo =
to
show
6
*""'
X'X X'Zo
ZQ X ZqZq
*1_ /
T
\
ko (the case of k
>
-Lr(Z
Since the
sum
we have
J_(
VT\
X'e
Z'Q e
+ X'(Zo-Z
+ Z' {Zo-Z
the
is
same
)6
)6
as the limit
show
plim-^X'(Z -
<
m
that the limit of the right-hand side
is
Zq. Let us
Supposing k
+e
Zq)S. Proceeding in the usual way,
*({:»
All
+Z
X/3
only involves ko
The
(43) converges to zero.
—
k
is
Zo)6
similar),
-Z
)
=
=
0.
(42)
we have
E
^=
k terms, and ko
—
x
k
t
z't .
(43)
= O p (l)
by Proposition
zero limit for (43) certainly implies plim
3,
=
fX'Z
plim fX'Zo- All other terms involving Zq can be treated similarly and the derivations
are omitted.
Remarks:
Corollary
not only holds for fixed
1
and
-v/ril^Tll
—^ oo, as in the setup of Section
and
(42) can
be proved as
Since the
sum
x {Z° ~ *°
involves about
sumption, (v/rPrl!)
-1
—
*
but also holds when
=
In this case, k
k
+
\\$t\\
2
||<$r||"
—*
O p (l),
follows. Note:
'
"jr
4.
6,
0,
m
2
||<Jt||~
and
-
vm\ ,£
11
terms,
||
I,2; "
Ejl i+1 xt z't
\\
mi
\\6
so the desired result follows.
25
T
2
\\
'-
= O p (l). By
as-
Lemma
A. 4 The following
S'T {{Z'
MZ
identity holds
)
-
(Z'
T (Z'±MZA )6T
Lemma
follows simplify
A. 5 Under
(z)
Proof of
2
*H^*)|| =
(Hi)
||T-
(iv)
e'MZ2 {Z'2 MZ2 )-
1/
)-\Z'2 MZo)}8T
=
fact that Z'
MZ
2
.
= Z2 MZ2 ± Z'±MZ2
.
A1-A6,
op (i)
VA/ZA ||=op (l)
l
Z'2
Me - e'MZQ {Z'Q MZQ )-
x
Z'Q
M£ =
op (l)
uniform on Kj{V).
(i).
11^(^)11 =
<£Mo
Proof of
2
'%(Z'tMZ2 )\\=op (l)
\\T-
is
MZ
1
from the
(zz)
where the op (l)
)(Z'2
the assumptions of
||r-V
l
2
- ST {Z'A MZ2 )(Z'2 MZ2 )- {Z^MZA )6T
6'
The proof
MZ
11
^^^
p (i)
11
=
_L_o (i),
-
—7=6'T (Z'±X)
P
0p( i).
(ii)
-l
-j=6'T Z'±MZ2
By
part
(i),
=
-j=6'T Z±Z2
the second term
is
op (l)O p (l)
be proved in exactly the same way as
k
<
kQ and equal to zero for k
Proof of
>
=
op (l).
in part (i)
ko).
(iii).
26
(
-jr~j
That the
first
~Y~'
term
is
op (l) can
(Note that Z'^Z2 equals Z'^Z^
for
2
Because e'Z& consists of only A^ observations, the functional central
=
implies that At||£'^a||
term on the
right
and \(ko-k)\/T
is
=
KT (V).
being uniform on
the second term
Proof of
is
Use
(iv).
P
(1)
Z2 =
(X'ZA )/T =
follows that
it
Finally,
•
Zo
theorem
This together with y/TXj -* oo implies that the
p (l).
op (l). Consider the second term. Because
o(l),
limit
=
o p (l)
± Z&
from e'X/y/f
X'Z&/(kQ — k)
op (l), where
= O p (l)
O p (l)
first
= O p (l)
and o(l) both
and (X'X/T)"
1
= O p (l).
op (l).
to obtain
e'MZ2 {Z'2 MZ2 )- Z2 Me =
x
£'MZ {Z2 MZ2 )- ZoMe +
The
the
result of
left
hand
(iii)
MZ±{Z2 MZ2 y Z2 Me + £'MZQ {Z'2 MZ2 r Z^Me.
x
1
e'
'
imphes that the
last
1
two terms above are op (l)0 p (l)
=
o p (l).
Thus
side of (iv) can be written as
Z2 MZ2 \~
e'MZp
l
(Z'Q MZo\~
X
Z'
Me
Because the two matrices inside the bracket converge to the same limit on Kt(V),
(iv) is
proved.
Proof of Proposition
Lemma
By
4:
the definition of Vj(fc)
—
Vr(kc) [see (7) and
(8)]
and
A. 4,
VT (k) - VT (ko)
=
From the
(44)
_
-6' {Z^MZ^)6T
T
definition of
M,
+
the
6'
T (Z'A
first
MZ
2
){Z^MZ2 )-\Z2 MZ^)6T + h(k,6T ,e).{45)
term of
6y(Z^MZ^)St = SjZ^Zk&T —
(45)
is
S'fZ'^X
y/T
By Lemma
(A.5)(i), the second
term on the
right
is
(X'X\
\ T )
-l
X'Z&St
y/f
op (l), leaving S'jZ'^Z^St-
Now
the second term of (45) can be written as, upon appropriate scaling,
6T
{Z^MZ2 ){Z2 MZ2 )-
1
{Z2 MZ^)6T
= T- 1/2 6t (Z^MZ2 ){Z'2 MZ2 /T)-\Z2 MZ£,)6tT- 1/2
27
.
(46)
On
KT (V),
(Z^MZi/T)-
1
=
P (1)
and by
Lemma
A.5
(ii),
consider the last term of (45) h(k,6T ,e) [see (9) and (10) for
A.5
(iv) says (10) is o p (l).
Lemma
A.5,
it is
26' {Z'
T Q
Combining these
Using {Z'
MZ
2)
=
{Z'2
MZ
2)
±
(46)
o p (l).
is
its definition].
Z'A
MZ
2
Next,
Lemma
together with
easy to show
MZ
2
){Z'2
MZ
2
)- x Z'
2
Me - 26'T Z'Q Me =
±26' Z*e
T
+
op (l).
results yields Proposition 4.
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