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HB31
.M415
working paper
department
of economics
Testing for Parameter Constancy
in Linear Regressions:
Empirical Distribution Function Approach
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
Testing for Parameter Constancy
in Linear Regressions:
Empirical Distribution Function Approach
Jushan Bai
No.
93-9
July 1993
Testing for Parameter Constancy
in Linear Regressions:
Empirical Distribution Function Approach
Jushan Bai
Department
Economics
Massachusetts Institute of Technology
of
July 29, 1993
Abstract
This paper proposes some tests for parameter constancy
models with possible
infinite variance.
in linear regression
Both dynamic and trending regressors are
The
tests are based on the empirical distribution function of estimated
and are shown to have non-trivial local power against a wide range of
alternatives. Within a certain class of alternatives including simple shifts, the
tests have higher power for testing the simple shift alternatives. These tests are
allowed.
residuals
formulated
residuals
in
such a way that the limiting variables are distribution-free.
may be
obtained based on any root-n consistent estimator (under the
null) of regression parameters.
for
The
As part
of these results,
weighted sequential empirical processes of residuals
some weak convergence
is
established.
Key words and phrases: structural change, empirical distribution function,
weak convergence, nonparametric test, fluctuation test, CUSUM test.
1
Introduction
There are various sources
in
economics that could cause a parametric model to be
unstable over a period of time.
in policies
and regulations
all
Changes
in taste, technical progress,
are such examples.
A
change in the economic agent's
expectation can induce a change in the reduced-form relationship
variables, even
though no change
in the
and changes
among economic
parameters of the structural relationship
present, as envisioned by the Lucas critique.
1
The
shifts of
is
the Phillips curve over
time perhaps serve as the best illustration (Alogoskoufis and Smith, 1991).
model
result,
been an important concern
stability has always
Earlier studies of parameter constancy include
perhaps a consequence of diagnostic
instability
failures,
Chow
in
econometric modeling.
(1960) and
Quandt
(1960).
have constantly spawned out. The random- coefficients model of Cooley and
(1973a,b) and numerous others find widespread use in economics.
is
As
models capable of handling parameter
Prescott (1973), for instance, the switching-regression models of Goldfeld and
paper
As a
Quandt
The purpose
of this
to provide additional tools for the diagnosis of parameter instability in linear
regressions.
Recent work
in econometrics
parameter changes occurring
large
body
at
on
been directed toward detecting
this topic has
an unknown time, hardly a new problem given the
of related literature in econometrics
and
statistics.
particularly concerned with parameter instability in
Econometricians are
dynamic models with trending
and with
regressors, cointegrated variables
and perhaps a unit
or heteroskedastic disturbances.
Various test statistics that are capable of detecting
changes in those situations have been developed;
Chu and White
(1992),
Hansen
root,
see, for
(1992), Perron (1991),
serially correlated
example, Andrews (1990),
and Ploberger, Kramer and
Kontrus (1989). Empirical applications together with supporting theory can be found
Lumsdaine and Stock
in Bai,
(1991), Banerjee,
tiano (1992), Perron (1989), and Zivot and
paper,
we propose some
Lumsdaine and Stock (1992), Chris-
Andrews
(1992),
among
others.
In this
some
of these
tests able to detect structural instability for
models. In addition, these tests are applicable to infinite variance regressions.
Two classes of tests are proposed,
two-sample
test.
of residuals.
The first
class
is
resembling the prototypical Kolmogorov-Smirnov
based on non- weighted sequential empirical processes
This class was previously considered by Picard (1985) and Csorgo and
Horvath (1988), among others. However, these authors only consider the case of
observations under the null.
We
i.i.d.
extend the tests to apply to regression models with
estimated parameters.
The
first class
of tests has limited applicability in time series econometrics since
if
trending regressors are included in the regression model, the tests will no longer
In this case, the second class of tests can
be asymptotically distribution-free.
be
considered, obtained by constructing a weighted empirical process of residuals with
weights equal to the regressors, apart from some weighting matrices.
tests
is
This class of
asymptotically distribution-free whether or not a trend regressor
Our procedure may be regarded
in light of the
as nonparametric, yet
it is
present.
not fully nonparametric
By way
need to estimate the regression parameters.
is
of construction,
our tests are robust against heavy-tailed distributions and data aberrations. Recent
work of Carlstein (1988) and Duembgen (1991), who consider the estimation of the
shift point
under the single
shift alternative
(two
samples) and obtain good
i.i.d.
may
convergence rate, indicates that the tests of the Kolmogorov-Smirnov type
be
powerful.
The
classical statistical literature
shows that goodness- of- fit
processes involving estimated parameters will depend
ical
It is
somewhat
surprising that
we can eliminate
based on empir-
upon both the estimated
parameters and the underlying error-distribution function even
(1973)).
tests
this
in the limit (see
Durbin
dependence by choosing
weighting vectors, in a natural way, in the construction of the empirical processes upon
which our
tests are based,
totically distribution-free
(see
whereas
classical goodness-of-fit tests
can be
made asymp-
merely by essentially abandoning part of the observations
Durbin (1976)).
The
finite
tests
proposed
in this
paper are quite general
in
the sense that
we
require no
variance for the disturbances. Both dynamic and trending regressors are allowed
in the regressions.
Within certain
classes of alternatives,
powerful when used for testing simple
we show the
shift alternatives, as
tests to
be more
expected. Moreover, these
tests exhibit non-trivial local power.
As a
related result that
may
be of independent interest, a weak convergence for
randomly-weighted sequential empirical processes has been obtained.
result to obtain the
weak convergence
We then use this
for its counterpart for the regression residuals,
laying the theoretical foundation of our tests.
This paper
is
organized as follows.
Section 2 specifies the models and describes
the assumptions. Section 3 defines the test statistics. Section 4 provides alternative
expressions for the test statistics that are suitable for computation. Section 5 examines
the local power of the tests. Trending regressors are considered in Section
comments and
possible extensions are discussed in Section
6.
Some
Technical materials are
7.
collected in the appendix.
Models and Assumptions
2
The
regression
model under the
yt
where y
x't /3
of the independent variables, e t
the p x
1
The
+e
is
t
is
(1)
apx
1
vector of observations
ft is
vector of regression coefficients.
/3 t
may
=
we
Changing variance:
i)
where Ai and
and
A
+ el.
2
e*
=
e ( (l
may
not be iden-
are interested in the following local alternatives.
Changing regression parameters:
Both
x'tPt
not be constant over time and the disturbances e*
tically distributed. In particular,
iii)
l,2,...,n)
non-null hypothesis specifies the following model:
where the
ii)
=
an unobservable stochastic disturbance, and
Vt
i)
(<
t
is
an observation of the dependent variable, x
is
t
=
null hypothesis
j3 t
=
(3(1
+
Aig(2/n)n
+ A 2 /i(2/n)n -1 / 2
-1 ' 2
).
).
ii).
are two real numbers; the functions h and g are
assumed to be
bounded.
In
what
follows, the notation o p (l)
(O p (l))
is
used to denote a sequence of random
variables converging to zero in probability (being stochastically bounded).
•
||
||
represents the Euclidean norm,
i.e.
denotes the greatest integer function.
We
impose the following assumptions:
||x||
=
2
(Ya=i x 1Y^ f° r x €
-R p
-
The norm
Finally,
[•]
Under the
(A.l)
null hypothesis, the e t are
which admits a density function
uniformly continuous on the real
< L and
such that \xf(x)\
/ >
/,
F,
(d.f.)
Both f(x) and xf(x) are assumed to be
0.
Furthermore, there exists a
line.
< L
|/(x)|
with distribution function
i.i.d.
The mean
for all x.
of e t
is
number L
finite
zero
this
if
mean
exists.
(A. 2)
The disturbances
(A. 3)
The
e t are
independent of
t
Q
is
uniform in
contemporaneous and past regressors.
regressors satisfy
plim— >J x x
n t=i
where
all
— SQ
\
f° r s
G
[0, 1]
a p x p nonrandom positive definite matrix. The convergence
because the
s,
sum
"monotonic"
is
is
necessarily
in s.
(A.4)
max n _1/2 ||x
l<t<n
t
||
=
op (l)
random
(A. 5) For every fixed Si, there exists a sequence of positive
Op (l)
Zn =
such that
\ns]
i
n
for all s
>
J2
4
W
Xt
W
-
(
~
5
In addition, the tail probability of
Si.
Zn may
probability
(A. 6) There exist
7
>
1,
a>
1
where
and
K
A;
=
i
Zn
<
J2\=i
[nsi]
is
\\
some p >
satisfies, for
M/C 2(1+p)
-1
be taken to be max^
also satisfied,
is
a. 5.
r
\
Note that
5 i)Zn
=[nsi]l
P(\Zn > C) <
tail
variables
0:
.
x t\\ provided the condition on the
fixed.
00 such that for
<
all
s'
<
s"
<
1,
and
for
all n,
n
where
Also
=
i
if
V
[ns'],
E(x' x
because
£(x;x
7
t
)
<-A'(5"
-
5')
E{- Y\
and
t
2
t
)
=
j
<
[ns"].
M
The assumption
for all
2,
t
't
satisfied
is
then the assumption
E(jy =i x x t) 2 < {Ylt^i^i^t) 2
1^2
]
x[x
n i<t<j
~i
r~i
:<<<J
2
}
is
if
t
f<
K(s"
-
s')°,
the x are bounded regressors.
t
satisfied
with 7
=
2
and a
=
by the Cauchy-Schwartz inequality.
2,
(A.7)
{X'Xfl 2
where
X = (a^,^, ...,x„)'.
When
then least squares estimator
robust estimation such as
(A. 8) There exist a 8
>
-
LAD method
P (1)
and have
i.i.d.
assumption.
finite variance,
For infinite variance models,
has to be used to assure (A.7).
M < oo such that
and an
<
)
±
the disturbances are
satisfies this
3(1+{)
£(-£lM
~
"
0)
M and £(-£lNI
n
1+5
3
)
<
M
Vn.
(=1
(A.9) Finally
i
t
ns
J
plim— y^ x
n t=i
where x
is
a p x
1
t
=
sx
uniformly in
When
constant vector.
5
£
[0, 1]
a constant regressor
is
included, (A.9)
is
implied by (A. 3).
Assumptions (A. 3) and (A.9) exclude trending
regressors,
which
The
based on the estimated
will
be discussed
in Section 5.
3
Let
The
$ be an
residuals £ t
on the the
.
Test Statistics
estimator of
and put
For each fixed
k, define
first
=
it
yt
—
x't 0.
test
is
the empirical distribution function
k residuals as
K t=\
and the
e.d.f.
based on the
—
n
last
P:_ k (x)
=
k residuals as
-J—
n
£
i(i
t
<
x).
K t=k+l
Further define
Tn (-,x) =
n
and the
-(1
tj
n
- *)Vn (Pk (x) - P:_ k (x))
n
>
.r>.
test statistic
M
n
= max sup
k
x
\Tn (k/n,x)
'
(e.d.f.)
based
max
where the
is
taken over
taken over the entire real
<
1
<
k
n and the supremum with respect to x
For each fixed
line.
the
k,
supremum
Tn
of
is
with respect to
the second argument gives the weighted Kolmogorov-Smirnov two-sample test with
weights [(k/n)(l — Jc/n)]
Kolmogorov-Smirnov
We
1 ?2
Thus the
.
M
n looks for
test
sample
statistics for all possible
have the following
maximum value of weighted
the
splits.
identities:
Tn (-,x) = n-^J2l(e <x)--n-^±I(i <x)
t
n
= nWriting in the form
M
and hence of
As
scale
will
n
i l2
t=\
Y,{^t<x)-F{x)}--n-
(3) will
be convenient
for
test
M
e.d.f. of residuals.
Let
X
k
=
like
(3)
t
parameters
if
the
(xi,
...,
x^)'
A k = {X'X)-
CUSUM
mean
the
we introduce a new
this undesirable feature,
1
J2{I{e <x)-F{x).}
Tn
has non-trivial local power against changes in the
n
shifts in the regression
Define the p x
'2
studying the limiting distribution of
parameter of the disturbances. However,
cumvent
l
.
be shown, the
power against
(2)
t
n
t=i
test,
it
has no local
regressor
is
zero.
class of tests
To
cir-
based on weighted
and
l l2
{X'k
X
k
){X'X)-
1l2
(4)
.
vector process T*,
rn*(-,x) = (x'xy
1
n
'2
J2Mi(£t<x)-Fn (x)}
t=i
-A
k
{X'X)-
l
l2
Yx
j
t
{I{e
<x)-Fn {x)}
t
(5)
t=\
and the
test statistic
M
n
where
||y||oo
reduce to
If
=
Tn
there
is
max{|y 1
and
M
n
,
|,
...,
k
=rn a xsup||Tn*(-,x)
k
the
|y p |},
respectively,
x
maximum
n
norm. The process T* and
when the weights x =
t
1
for all
test
M*
t.
a constant regressor, then the following identity holds,
(X'X)- 1 ' 2
£x
t=i
t
- A k (X'X)-^ 2
£^ =
t=i
0,
V
k
(6)
so that the value of
T*(k/n,x)
change when
will not
Fn (x)
in (5) is replaced
by an
arbitrary function of x. In particular, T* can be written as
(X'X)-^
Equation
£
(7) is
x't {I(i t
<x)~
F(x)}
a weighted version of
the test A/*, as F(x)
is
(3).
test,
This expression cannot be used to compute
T*
for
P {\)
(5)
and
(7)
uniformly
it
will
be useful
one should omit
included. However, whether or not there
is
is
in deriving the limiting
Fn (x)
a constant regressor
if
a constant regressor, the two expressions
have the same null limiting process, because n 1 ^ 2 {Fn (x)
a well
in x,
known
(7)
t
t
unknown; however,
When computing the
process of T*.
MX'X)-^ £ x' {I(i <x)- F(x)}.
-
—
F(x)}
=
(Shorack and Wellner
result for residual e.d.f.
1986, Chapter 4), and
n-^{(X'X)-^ 2
J> - A
k
{X'X)-" 2
£x
t
}
=
o p (l)
uniformly in k by assumptions (A. 3) and (A. 9). Finally, we remark here that
of the regressors are trending, then
Ak
we may
none
substitute the scalar k/n for the matrix
.
Let B(u, v) be a Gaussian process on
E{B(s,u)B(t,v)}
which we
"
=$
D(T) x D(T) x
"
is
•
=
[0, l]
2
with zero
(min(s,£)
—
mean and
st)(m'm(u,v)
two-parameter Brownian bridge on
shall call a
the notation
or
if
•
[0, l]
used to denote the weak convergence
x D(T) where
T =
2
[0, l]
1
Under model
(1)
and assumptions (A.1)-(A.9),
Tn ^-,-)^B(-,F(-))
[
(i)
(
n
and
(ii)
T„*(— ,)=»/?•(-, F(-))
n
uv),
2
in
.
In
what
follows,
the space of
D(T)
under the (extended) Skorohod
topology.
Theorem
—
covariance function
J\
—
where B"
bridges on
(B\, B2,
[0, l]
...,
)'
p independent two-parameter Brownian
a vector of
is
2
.
Let G(-) denote the
in
Bv
of the r.v. sup 0<u<1 sup 0<v<1 \B(u,v)\,
d.f.
which
is
tabulated
Picard (1985).
Corollary
1
Under
the assumptions of
Theorem
\imP(Mn <x) =
1,
G(x)
and
lim
n—*oo
The proof
of the
theorem
is
P(M* <x) =
.
based on the limiting behavior of the process A'*,
= (X'X)-^ 2 f^x
I<:(s,x)
[G(x)} p
t
{I(e t
<
x)
-
F(x)}
t=i
which we
shall call the
weighted sequential empirical process of residuals (w.s.e.p.).
Note
T:( ^,z)
[
= K:(s,z)-A
[ns] i<:(i,z).
Introduce
Hn (s,x) =
{X'X)- l l 2 Y,x {I{e
t
t
<x)-
F(x)}
t=i
which
is
a non-residual version of w.s.e.p.
Theorem A. 2
Appendix implies that
in the
A'*(s,x) can be written as
Hn (s, x) + f(x)(X'X)-^(X{ X
ns]
plus an op (l) term which
is
is
uniformly small
identical to the corresponding
T*(s,x)
=
in
both
s
[ns]
)(P
and
-
x.
fi)
The second term above
term of A[ ns ]K*(l,x), so that
Hn (s,x) - A
[ns
]Hn (l,x)
+ op (l).
Corollary A. 2 in the Appendix gives the limiting process of T*.
9
(8)
The
limiting process of K*,
of the estimated parameters
from
The
if it
exists, will
depend on the limiting distribution
and on the error density function
However, parameter estimation does not
(8).
fact that the limiting process of
dependence on F. Further,
that .F(O)
=
if
easily seen
is
affect the limiting process of T*.
F
T* depends on
The sup-type
construct distribution-free tests.
/, as
rather than / allows us to
example, transforms out this
test, for
the error e t has a symmetric distribution about zero, so
1/2, then tests based
on T*(s,0)
will also
be asymptotically distribution-
free.
Besides the sup-type tests, the mean-type test can also be used. Let
An
=
^EEl^,li)|
k
The
result of
implies that
1
in distribution to
convergence of Theorem
Many
is
among
converges in distribution to /J /J B(s,
who
2
t)
ds dt
in-
other tests can be constructed based on the weak
of empirical processes based
on estimated residuals has
authors, see, for example, Mukantseva (1978), Boldin (1982,
and Kopecky (1982), and Kreiss (1991).
first
-
j
2
/ / Y7i=\Bi(s,t) dsdt, where B\,...,BP are
It
appears that Koul (1970)
studied weighted empirical processes and followed by Withers
Weighted empirical processes
(1972).
1991).
the
many
An
2
1.
The weak convergence
1989), Pierce
K = ^ttK&i)\\
k
dependent copies of B(-,-).
been studied by
and
J
Theorem
and A* converges
2
of residuals
have been studied by Koul (1984,
Shorack and Wellner (1986) give more references on residual empiricals. The
weak convergence for the sequential
version, which
is
essential for the structural
change
problem, has not been widely examined: Bai (1991) considered the sequential empirical
process for
4
ARMA
residuals.
Computing the
We now
derive
are suitable for
some
tests
alternative expressions for the test statistics
programmed computation.
10
We
shall focus
M
n
on M*; the
and
test
M*
that
M
a
n is
Now
special case.
at £i,£2)
be found at x
=
£; (i
the i-th ordered
is
£(,)
each fixed
when x
?£n
•
•
for
\\T*(-,
k,
maximum
varies, therefore, the
=
us evaluate T£(-,x) at x
=
First
£(_,).
I3"=i x tl{£t
5: £(j)) is
the
sum
we
define a sequence of
numbers
Z%,k (t
for
1
Zt,k
for
Then
=
Zd,,A:
1 if
and only
if
is
where
1,2, ...,n),
<
D{
rn*(-,£ i)) = (i'i)- 1/2
(
77
—
=
t =
i
For a fixed
.
.
.
,£ n
j, let
a constant regressor, so that
omitted due to
of those vectors Xj such that i t
<
i.
t
Fn (x) =
can be written as Ya=\ x d,- Similarly, J2t=i x tl{£t
if
=
(i
£(,-)
= D# =
Zfo,
assume there
equivalent to the expression (5) with
is
=
Let Ri,R.2, ...,R n denote the ranks of £i,£2,
statistic.
value
its
value with respect to x can
1,2, ...,n) or equivalently, at x
and D\^D2i----,D n denote the anti-ranks so that
T*
can only possibly change
a:)||oo
=
£(j))
Since
(6).
not larger than era,
is
X],=i
x D I{Di
x
<
k).
it
Thus
1,2, ...,n) such that
1,2,
k
...,
k
+ l,k + 2,...,n
Thus
k.
E^.^.a
- (a^X*'*)" 1 5>*
(9)
1=1
,t=l
and
m:
max max (X'X)-^
k
where /
3
^{ZD „
kI
(X'k
X
k
)(X'X)- l }x D
(10)
,
,t=i
the p x p identity matrix. Taking Xf
is
-
=
1 in
the above formula for
all 2,
we
obtain
Mn
k
1
= max max —= Yl ZD
k
y/n
i
t
,k
77.
i=i
yielding an easily computable formula, see similar formula in Hajek (1969, p.
63).
When
there
subtracting the
left
hand
is
left
side of (6)
no constant regressor, the expression
hand
and
side of (6) multiplied by j'/n,
Fn (i^)).
The
test
M*
is
has to be adjusted by
(9)
which
A SAS
program
for
computing the
statistics
is
11
is
the product of the
adjusted accordingly and
the same. 1
:
available
upon
62-
request.
M
n stays
Local Power Analysis
5
We
consider model (2) with the Kramer, Ploberger, and Alt (1988) (henceforth
KPA)
type local alternatives:
f3 t
where e t are
=P+A
i.i.d.
g(t/n)n-
1
1
/2
and
with distribution function
g and h are defined on
[0, 1]
=
e*
F
e t (l
+ A 2 h(t/n)n- 1/2 )-
and density function
1
(11)
.
The function
/.
and are integrable. Define the vector function
M
=/
s)
-s
9{v)dv
g{v)dv
(12)
[ h(v)dv-s [ h(v)dv.
(13)
J
and the function
S
X h (s)
=
Jo
If
h
is
Jo
=
a simple shift function such that h(x)
where r £
(0, 1),
Theorem
2
then \h{s)
=
(r
A
s)(l
—
r
V
for
s).
Under assumptions (A.1)-(A.9) and
M
n -i
sup
sup \B{s,t)
0<s<l 0<<<1
x
<
Similar
r and h(x)
is
=
true for A s
1
for
x
>
r,
.
the local alternatives (11),
we have
+ A lP (t)x'X g {s) + A 2 q(t)X h (s)\
(14)
and
sup
MZ^ 0<s<l
where p{t)
The
tests
and
have nontrivial
=
X^
in the
+ A lP {t)Q^ 2 X g {s) + A 2 q(t)Q- 1/2 x\ h {s)\\^
=
q(t)
local
for all s
= /(F^ityF^it).
power
if
as long as X g (s)
and only
if
^
or Xh(s)
^
for
some
s.
In
g and h are constant functions, implying
parameters.
Corollary 2 (Changing regression parameters
orem
(15)
0<<<1
= f(F- l (f))
addition, X g
no change
sup \\B*{s,t)
only).
Under
the assumptions of The-
2,
M
n -i
sup
sup \B(s,t)
+ Aip(t)x'Xg(s)\,
0<s<\ 0<t<l
sup sup \\B*{
M *-i a<s<\
o<t<\
d
n
Sl t)
12
+ A lP {t)Q 1/2 X g
(
s)
The
corollary
A =
obtained by simply taking
is
M
in the regression parameters,
and Evans (1975)
behaves
n
in
Theorem
2.
like the
CUSUM
test of
2
in the sense of lacking local
KPA. The
term
drift
is
Thus
g.
is
M*, however, does
test
have local power irrespective of the relationship between x and
PKK. The
Brown, Durbin,
power when the mean of regressors x
orthogonal to the vector function g, as shown by
the fluctuation test of
In testing for changes
it
behaves
like
form to the fluctuation
also similar in
test.
There
following
a danger of misinterpreting the result of Corollary
is
model under the alternative hypothesis:
=
yt
where g
there
the
is
Let us examine the
2.
is
x't (3
+ Ag{t/n)n- l/2 +
a constant regressor, but for
is
zero.
(16)
,
This model would be a special case of (11) provided
a scalar function.
mean regressor x
et
Then
it is
now assume
M*
not
M
n
there
is
no constant regressor and
has no local power, which seemingly
This situation arises because there
change in the parameter
contradicts Corollary
2.
of a regressor that
not considered under the null. To see this,
is
is
we
consider a
more
general situation:
yt
where
null of
s
zt
is
q x
A=
where
is
some p x
is
+ Az' g{t/n)n- ^ +
1
x't f3
t
The x
a vector function.
Suppose that n
0.
R xz
and g
1
=
_1
YJt=\ z t
q matrix.
M
n -i
~~*
t
sz an ^ n
are the only regressors under the
~1
sup
sup \B{s,t)
+
d
n
if
the
zt
=
mean
course, for z
Now
1,
so that (17) reduces to (16).
value of the regressor
t
=
taking
x
t
,
Ai
(18)
=
and
in
Ht=i x z
t
~
t
*
s Rxz
uniformly in
Ap(t)z'X g (s)\
sup ||B*(a,t).+.Ap(t)gM *^sup
0<a<\ 0<t<l
let
(17)
Then
0<s<l 0<t<l
Now
et
is
zero,
M*
1/2
i2 r ,A,(a>|| 00 ,
Moreover, 2
2 yields:
13
=
1
and
(19)
R xz —
has no local power but
(19) coincide with Corollary 2.
Theorem
(18)
M
x.
Thus
n does.
Of
Corollary 3 (Changing
M ^
d
d
sup \\B*(s,t)
sup
2,
+ A 2 q(t)X h (s)\,
0<«1
0<s<l
^
Theorem
the assumptions of
sup \B(s,t)
sup
n
M'n
Under
scale only).
+ A 2 q(t)Q- ^x\ h (s)
1
0<s<l 0<t<l
When
testing for a shift in the scale parameter, the situation
test of a shift in regression parameters;
x,
zero whereas
is
M
n
summary, the
In
of the angle
Moreover,
power
test
M
n
has non-trivial local power
power when testing
mean
M
for
changes
when
in
if
the regressor mean,
value.
testing for changes in
The
test
the regression parameter
shift, see
has local power for testing changes in
n also
(3
KPA
and
for
M*
M*
/3,
has non-
regardless
comparison.
also has local
changes in the disturbances, except for some special circumstances
discussed earlier. Whereas the conventional
in regression
parameters and
heteroskedasticity, see Ploberger and
performs for a change
test
mean
value of the regressors.
between the regressor and the structural
for testing
changes
has no local power
has local power irrespective of this
the disturbances regardless of the
trivial local
M*
reversed from the
is
CUSUM test only
CUSUM-SQ
Kramer
test only has local
(1990).
in the disturbances, as
has local power against
It is
it is
not clear
power against
how the fluctuation
not intended to be used for this
purpose.
The
test statistics
shift alternatives.
3.
Let
H
a
be the
To
M
n
M*
and
fix ideas,
are
more powerful when used
for testing the
simple
consider the scale change alternatives as in Corollary
set of functions h defined
Ha = {h;
<
h
<
by
/ h(v)dv
1,
=
1
-
a}
Jo
for
some a
from
of h
satisfying
<
a
<
1.
|A 2
the value of
|.
1
—a
represents on average the deviation
0.
Consider the test statistic
ity,
The number
M
n
Thus with high
is
M
n
.
Since B(s,t)
is
mainly determined by the
probability, in order for
14
uniformly bounded in probabildrift
Tn (s,x)
term A.2q{t)\h{s),
for large
to be maximized, the following
quantity needs to be maximized with respect to
—
h(v)dv
/
We
is
determine the h G
H
and
a
s
h(v)dv
/
Jo
s
(20)
Jo
£
that maximize the objective function (20).
[0, 1]
It
easy to show that there are two set of solutions, depending on whether the quantity
inside the absolute value sign
is
7
./
x
h*(v)
v
=
5*
a.
The other
solution
is
v
it
v
if
v
if
v
<
>
I
1
I
1
=
s*
1
—
a.
more powerful
But both
=
is
given by
a
,„„,
(21)
v
'
a
-a
—a
1
1
of these h* imply a simple shift alternative, so the test
is
against a simple shift. Furthermore, the value of the objective function
evaluated at the optimal solution
for a
solution
given by
h*(v)
and
<
~
>
if
f
=
'
and
One
positive or negative.
1/2, implying a higher
is
a(l
power
—
both cases. But a{\
a) in
— a)
for detecting a shift that occurs
is
maximized
near the middle
of the observations.
To
see (21)
is
a solution, consider the objective function (20) without the absolute
value sign. For each fixed
imized by choosing h(v)
with
fs
h(v)dv
=
(1
s,
=
term
since the second
for v
<
s.
The
is
non-positive, (20) will be
objective function becomes s Js h(v)dv
— a). To maximize the objective function,
one needs to choose
large as possible. In order to choose the largest s such that /s
to choose h as large as possible.
which implies
s*
=
a.
Thus h(v)
The second
=
solution
max-
1.
is
The
hdv
constraint
=
1
— a,
s as
one needs
becomes f^dv
= l—a
obtained by maximizing the negative
function inside the absolute value sign.
6
We
Trending Regressors
consider the following model:
yt
=
z't
a + 7o + n(t/n) +
15
...
+
l q {t/n)
q
+
et
(22)
where z t
of
St.
a r x
is
Let x t
=
vector of stochastic regressor and {z s
1
l,t/n,
(z't ,
The polynomial
through by
(A.3')
all
(t/n) q )' be a p x
<
trends {(i/n)';l
<
i
s
<
vector, with p
1
t
—
1} are independent
= r + q + 1.
q} could be written without dividing
Writing in this fashion saves notations by eliminating the weighting
n.
matrix such as diag(n
maintain
...,
;
-1 / 2
...,n~^ q+1 ^
,
2
assumptions (A.1)-(A.8) of Section
1,
[ns]
M
^
I
t
t
't
t=i
where Q(s)
except changing (A. 3) to
is
= Q( s )i
't
>
positive definite for s
implies uniform convergence in
class of
uniformly in
s
€
models than
and Q(0)
If
0.
each element of Q(s)
show that pointwise convergence
Assumption
s.
—
(A.3') actually
We
1
'2
n
Theorem
assume that there
J2 x
t
as in (4)
I(i t
<
x)
and T*
M*,
- A k (X'X)-^ 2 J2 x
HT^^/rZjx)!^. The computation of
2
Q(s)Q(l)- 1 ' 2 uniformly
a vector Gaussian process defined on
t
I(i t <x).
M*
is
s)
-
Corollary 4 Under assumptions of Theorem
3,
v)'}
M*n
(23)
given by (10).
in s.
[0, l]
2
1
=
±
{A{r A
sup
A(r)A(s)}{u
HS-^tOHoo.
0<s,u<l
16
),
with zero
covariance matrix
E{B*(r, u)B*(s,
Let
t=l
^ A(s) = Q{l)-^
is
asymp-
as follows:
3 Under assumptions (A.1-A.8) with (A.3) replaced by (A.3
where B*(s,u)
is
a constant regressor.
is
t=l
M* = max^ sup x
A [ns]
shall
and and define Ak
rn*(-, x) = {X'X)-
Note that
in s
(22).
totically distribution-free.
let
is
admits a much wider
In the presence of trending regressor only the weighted version,
(x'1 ,x'2 ....,x'k y
[0, 1]
t=i
a continuous function on [0,1], then one can
Again
shall
.
plim— 22 x x = h m — & /J x x
n
n
Xk =
We
that would otherwise be needed.
)
Av-uv}.
we have
mean and
The behavior
Extending
of the test under the local alternatives (11) can again
Lemma
4 of
KPA, we can show
n "
variation on
[0,1].
When
Q(v)
Jo
av
w
where e
=
Theorem
t
x't g(t/n)±
uniform
is
that
Sd
-jr,x
and the convergence
be analyzed.
=
in s.
j
Jo
^g{v)dv
(24)
av
The above
integral exists
if
vQ(l), (24) reduces to the result of
Jo
g has bounded
KPA.
Let
av
(1,0, ...,0)'.
4 Under the local alternative (11),
M;A
sup
\\B*( S ,u)+p(u)A 1
Q(l)-^X;(s)
+
2
q (u)A 2 Q(l)-^ Xl(s)\\ 00
0<s,u<l
where
p(-)
and
q(-) are
given in Theorem
Of course, when Q(v)
=
2.
vQ(l), the theorems and corollaries in previous sections
can be derived from the results of this section.
obtained here
is
However, the limiting distribution
generally regressor-dependent, so critical values of the tests have to
be found case by case, though leading cases can be tabulated. Also, the result of
this
section requires the existence of a constant regressor.
Tests allowing trending regressors have been proposed by MacNeill (1978), Sen
(1980),
Kim and Siegmund
(1991), Vogelsang (1992).
hood
ratio, or
in this
7
Wald-type
(1989),
Those
Hansen
tests are
statistics.
It
paper perform relative to those
(1992),
Chu and White
(1992), Perron
connected with the partial sums, the
likeli-
remains to be studied how the proposed tests
in the literature.
Some Comments
We have maintained the assumption
This could be extended to
that the disturbances {e t } are independent
ARMA models.
r.v.'s.
Although further extension to more general
17
dependence structure such
mixing
as
is
critical values of the tests are difficult to
a complicated correlation structure.
where B(L)
ut
is
2
obtain because the limiting process will have
ARMA
For the
models such as e
a ratio of two polynomials of the lag operator L, one can
This could be done with a two-step procedure. The
.
first
first
The two-step procedure
step residuals.
ARMA
obtained similar results for pure
The
proposed
tests
in this
models
paper are similar
In fact, consider regressing I(i
<
t
e^(x) = (x'k x k )- 1
Using the expression (23)
for T*,
sample, and
0^ k '(x)
6^ T \x)
/
t
i(i t
from
details
M
n
k,
PKK's
in their
way
The
using the whole sample.
PKK's
notation by the matrix
A
t
Sn = max op<K<n
where a and fiW (£
=
p^
.. #)
test of
PKK.
and write
#">(*))
can be considered to be an estimate of
to extend
Bai (1991)
one obtains,
.
Q~ xF(x)
1
test here has
test does not include trending regressors
also suggests a
remain to
.
using a partial
one more dimension
than PKK's, and thus can be viewed as a two-dimensional fluctuation
that
estimate
<x).
= A k {X'Xfl 2 (§W(x) -
T;(k/n, x)
The quantity
Yx
....,
,
yields estimates of u t
form to the fluctuation
x) on x t for t=l,
t
B(L)
for the test
in
still
expected to hold.
is
= B(L)u
coefficients of
which empirical distribution function can be constructed. Although
be worked out, the results of the previous sections
t
step involves estimating
and the second step estimating the
the regression coefficients
using the
terms of weak convergence,
also possible in
l
Simply replace t/T
let
\\A k {X'X)
n ) are defined
Sn
Notice
but T* does. This comparison
test to trending regressors.
and
test.
in
l
l\^ _ ^H)^
PKK. Then
it is
not difficult to show
-» SUp ||B(s)||oo
0<s<l
2
There
is
a large literature on empirical process based on mixing sequences; see the early study
of Billingsley (1968, p. 240) and the recent study of
18
Andrews and Pollard
(1990).
where
B
is
mean
zero vector Gaussian process with covariance matrix
E{B(u)B(v)'}
Other issues that are
and
size
left
= A(u At;)-
unaddressed
paper include cointegrated regressors,
in this
and power comparisons with other
A(u)A(v).
tests in the literature.
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Wahrsch. verw.
Gebiete, 52 203-218.
[43]
Shorack, G.R. and Wellner,
to Statistics. Wiley,
[44]
New
J. A.
(1986). Empirical Processes with Applications
York.
Vogelsang, T.J. (1992). Wald-type tests for detecting shifts in the trend function of a
dynamic time
series.
Manuscript, Department of Economics, Princeton
University.
[45]
Withers, C.S. (1975). Convergence of empirical processes of mixing rv's on
The Annals of
[46]
Zivot, E.
oil
Statistics, 3 1101-1108.
and Andrews, D.W.K.
price shock
[0, 1].
(1992). Further evidence
on the great crash, the
and the unit root hypothesis, Journal of Business
Statistics, 10 251-271.
23
&
Economic
A
Appendix
In view of the mathematical structure of parameter changes in regression models,
shall first present
and prove a
we
concerning the weak convergence of
series of results
weighted sequential empirical processes. These results are of independent interest and
be used subsequently
will
Let D*[0,
1]
be the
continuous and have
Z)'[0,
1]
is
proving the theorems stated in the body of the paper.
in
/ = (/1?
set of functions
...,/p )
defined on
that are right
[0, 1]'
Endowed with the extended Skorohod
left limits.
J\ topology,
a separable and complete metric space, so that finite dimensional conver-
gence plus tightness implies weak convergence for a sequence of random elements of
ZMO,
1];
see Bickel and
Wichura
(1971).
The space D^O,
1]
for
p
=
q
=
1 is
extensively
studied by Billingsley (1968).
Theorem A.l
,Un
Let U\, U2,
variables on [0,1]
and
=
x, (i
1,2, ...,n) be a sequence of
assumptions (A. 5) and (A. 6). Assume that U{
the process
Yn (s,u)
random
be a sequence of i.i.d. uniformly distributed
random
vectors satisfying
<
independent of Xj for j
is
i.
Then
defined as
[»]
Yn (s,u)
=
n- l ' 2 Y,Xt{I{U
<u)-u)
t
t=\
with
Yn (0,u) = Yn (s,0) =
Remarks: The
process
Yn
is
is
ment
of uniform distribution
i.i.d.
random
where
F
is
.
a multivariate and multiparameter process.
is
only for convenience.
In this case,
I(U <
t
the distribution function of e
addition, the
Un \, ,Unn
variables £ t
tight in D%[0, 1].
i.i.d.
assumption on
[/,
t
.
u)
Then
—
u
(with u
require-
holds for arbitrary
replaced by I(et
is
Yn (s,u)
=
<
F(x))
x)
is
—
F(x)
tight. In
can be relaxed to a triangular array such that
are independent variables on [0,1] with
maxi<,<„ |Fm (u2)
The theorem
The
— Fn j(ui)| < C|u 2 —
u\\,
can be easily seen from the proof.
24
where
C
Un
is
i
having a d.f.
Fm
generic constant.
-
such that
This claim
Lemma
A.l Assume
such that for
<
all Si
and
s2
<
u\
—
S\\
<
>
0,
<
where
ux )
a
-
{s 2
sx
,
s,, u,
Ul )
a
-(7-l)/2(a-l)
lemma
then the
This inequality
is
Proof. Write v
and j
,Ui)
=
=
1 (i
+ Yn (s uUl
<
<
oo,
1,2)
2^
)\\
+ n-^-^K(u 2 -
)
<
U2
_
-
Ul )(s 2
—
7, since \u 2
and T n -(7-D/2(a-l)
Ul
s a ).
<
«i|
and
1
t
- Yn ( Sl ,u 2 - Yn (s 2
)
,
analogous to (22.15) of Billingsley (1968,
p.
+ Yn (si,ui)
< U <
t
for the
Note that {x
[TIS2].
a
-
I(u x
gale differences, where
(25)
+ Yn (s uUl )\\^
Ul )
(s 2
=
<^_^
implies
< K[\ + t- 2 ^-^}(u 2 -
2
<
K
there exists
Moreover, when
1.
E\\Yn (s 2 ,u 2 )
Yn (s
Then
hold.
one can assume that a
loss of generality,
Tn
for r
,
)
< K{u 2 -
1-52
u2
- Yn {s u u 2 - Yn (s 2
E\\Yn (s 2 ,u 2 )
Without the
Theorem A. I
the conditions of
J-\ is
u2)
—
u2
+
u^
,^}
is
)
= n~
1
a
Sl )
(26)
.
198).
= Yn (s 2 ,u 2 - yn (si,u 2 -
and Y*
moment. Then Y*
t 7/ t
Ul
)
'
2
J2i<t<j
)
x tilt with
i
=
[nsi]
a sequence of (nonstationary) vector martin-
the u-field generated by
...,Xt,
x t +i;
...,
Ut-u
Ut-
inequality of Rosenthal (Hall and Hedye, 1980 p. 23), there exists a constant
By
the
M < 00
only depending on 7 and p such that
n
{
ME
<
Note that x
is
t
i<h<j
'«<J
I-
£
j
—
<
(A. 6) provide
bounds
u2
M(u 2 -
U\
and En
for the
u^E
'
r
t
<
u2
—
T -i
t
\
U
£
i<*<3
(x't x )\
t
I
25
and n
These
U\.
two terms on the
(-
+Mn-"
E{(x[x t )r)^t-i})
measurable with respect to
addition, Erj
1
\
t
J2 E{(x' x r^}.(27)
t
t
independent of Tt-\-
is
results together with
right of (27).
< MK(u 2 -
The
Ul
first
y(s 2 -
term
a
Sl
)
In
assumption
is
bounded
and the second term
bounded by
is
Renaming M/"f
Lemma
\\Yn (s, u)
-
Yn
lemma then
as if, the
A. 2 Under
( Sl
,
where the term
P {\)
<
s
- Yn (s uUl
uniform in
s (s
,
>
2
-
-
Ul )(s 2
a,).
from (u 2 — Ui) 7 < (u 2 — Ui) a
\\Yn (s 2 u 2 )
<
is
<
S\
M Kn^-^ (u
<
follows
we have for
(A. 5),
Ul )\\
^W
£
Mn-^-^ua - ui)-
and
s2
)\\
<
U\
u
<
u2
+ O p (l)n^ 2 [(u 2 -
,
for
7 >
+
(s 2
-
a.
,
Ul )
s,)]
does not depend on u and u\ and
S\),
satisfies
2
P{\O p (l)\>C)<M/C ^ +p \
Proof.
x
ti
>
linear
components of x
First notice that all
Otherwise write x
and x?(i)
—
t
=
—
Y%=ivt{i)
(0, ..0,
—x
t
Yfi-\
i,0, ...,0)' if
combination (with coefficients
1
\\.
x T{i) where xf(i)
it
<
,
take a
functions x
t
<
I(U <
mean
b to
a,
<
6,
for all
||xt"(i)||
t
W
1
/2
2
,u 2 )
M
(-j:x
n =
(
t
)(u 2
-u) + n
It is
- Yn (s u u 2
/
1
'2
\
can be written as a
<
and
||o;(||
and
||z^~(0ll
xj~(i).
It is
—
thus
Since x
>
t
0,
the vector
easy to show
)
[n«2]
1
[-
1
0, ...,0)' if
piece of notation, for vectors
components.
u) and x u are nondecreasing in u.
Yn (s,u) - Yn (s u ui) < Yn (s
x ti,
most 2p processes with each process
A new
t
6,
0.
(0, ..0,
Yn
satisfied for xf(i)
enough to assume that the x are nonnegative.
a and
=
In this way,
0.
In addition,
So assumptions (A. 5) and (A. 6) are
>
for some P
can be assumed to be nonnegative.
t
or -1) of at
having nonnegative weighting vectors.
\\x t
VC>0,
n
£
u2 )
-
u2 )
x {I(U
<
u)
x {I(U
t
<
t
t=[n5]
and
Yn {s u u )-Yn {s,u)<n
x
l2
x
){u-u,) +
{-Y,x
n
t
t=l
The lemma
follows from the
Remarks:
Bickel and
n
l
l2
(- jr
\
n
t
t
-
t=[n Si ]
Ul } )
.
J
boundedness of the indicator function and (A. 5).
Wichura (1971) provided a general framework
26
for
showing the
2
tightness of a sequence of multiparameter stochastic processes. Their conditions are
hard to verify and probably do not hold because of the dependence and unboundedness
of x t
Although there are empirical process theories
.
variables, (see
Andrews and Pollard (1990) and the
are directly applicable. Also, the presence of the
to
make
proof
in the
is
for
mixing and nonstationary
P (1)
term
Lemma
our
in
A. 2 seems
necessary for us to evaluate directly the modulus of continuity.
it
also instructive.
The arguments
them
references therein), none of
A
direct
and Wichura inspire the ideas used
of Bickel
remaining proof.
Proof of Theorem A.l.
Define
us (Yn ) = S u ? {\\Yn (s',u')-Yn (s",u")\\;
\s'-s"\
We
rj
shall
show that
for
any
>
e
and
>
<
\u'-u"\
8,
<
>
there exist a 8
0,
8,s',s",u',u"
e
[0,1]}.
and an integer
no,
show that
for
such that
P{u s {Yn >
Since
2
[0, l]
<
e)
)
n > n
n,
has only about 8~ 2 squares with side length
every point (sx,Ui) €
[0, l]
2
,
every
e
>
and
>
rj
.
to
8, it suffices
there exist a 8 € (0,1) and an
0,
integer no such that
P (sup
\\Yn (s,u)-Yn {s 1 ,u l
)\\
>
5e)
u
<
Ui
<28 2
>
n
t),
n
(28)
.
(*)
where
(8)
=
{(s, u); Si
For given 8
Lemma
>
<s <
and
si
>
r)
+ 8, u <
x
0,
C
choose
+ 8}
C\ [0, l]
2
.
enough so that
large
O p (\)
in
A.
>C)<
P(\Op (l)\
By Lemma A.2
sup||yn (s,u)
2
8 v
(29)
.
(see also (22.18) of Billingsley, 1968, p. 199),
- yn (si,"i)|| <
3
max
l<:j<m
\\Yn (s<i
+it n ,ui +
when |O p (l)| < C,
jt n )
- Yn (si,u.\)\\ +
[6]
where
for the
tn
=
c/(n^ 2 C) and
m
X{i,j)
=
=
[n^ 2 C8/e]
Yn(si
+
+
ie„,U!
27
1.
Write
+ je n - Yn (s u ui).
)
2e
Then
>
\\Yn {s,u)-Yn ( Sl , Ul )\\
P(sup
5e)
< P(\0 P (1)\ > C)+P( max
Now
for fixed
i
and k
>
(i
which follows from n -h- l )/ 2 i"- 1
-
2^
Z(l)\\
=
where, from (26) with r
C =
[1
e
Thus by Theorem
P( max
where K\
<
(mt„)
is
+
=
k) write Z(j)
(e/Cjn-^ 1 '^-
E\\Z(j)
1
)
KC
<
< tl{Cn^) =
< n'
)
(
-
[(i
1'2
—
X(i,j)
because
k)e n
a
-
[(j
)
l)t n ]
(C/e)
2{a -V}
2(C/e) 2(Q
<
K KC
-^[(i -
>e)<
a generic constant and
let
Ki
V(i)
= maxj
Thus by Theorem
is
the partial
-
V(*)|
>
e)
fc)e n
=
1
a
Q
]
Thus by
a
<
1
,
and
(25)
~ 1)
<
/
for small
j
<
(26),
m
e.
(31)
we have
KC
-
(me n )° < -^-i[(i
K\K. The
k)e n }°6°
(32)
last inequality follows
from
sum
is
- X(kJ)\\ =
max||Z(j)||,
j
< ^^[(t -
fc)e„r<T,
12.2 of Billingsley once again
Si of
random
|V(i)|
l<i<m
A' {
By
7.
1,
then (32) implies
||X(i,_7')||,
P( max
where
>
j
,
j
j
P(|V(i)
V(i)
(30)
26 for large n. Because
j
we
e).
e/C,
max\\X(iJ)\\-max\\X(kJ)\\ < max\\X(i,j)
if
jen
<a <
1
>
X(k,j). Notice that
<
tn
12.2 of Billingsley (1968, p. 94),
\\Z(j)\\
\\X(i,j)\\
l<«,J<m
[S]
>
e)
variables
<
£/,
1
[let £/,
=
en
(_
a generic constant and A3
=
2° A{ A' 2
.
lb
V{h)
<
i
— V(h —
^
\\Yn (s,u)
- Yn (s uUl )\\ >
28
1),
so that
we obtain
2°
£*~
Note that max, V(t) =max, maXj
|
|
(30)
P(sup
< m.
in Billingsley's notation],
£Mi( m )^ <
i~
<
56)
<
S
2
r,
+
^<$
2 °.
\\X(i, j)
By
(31), the
second term on the right hand side above
-^-6
By Lemma
hand
A. 2, one can choose
side (33)
<
6
t2{l+a _ x)
{CS)
^
^
n,
)
(33)
to assure (29)
a constant and a
is
,
bounded by
'.
(M/r)) 2 1+p) 6~ {T
C =
becomes K(t,rj)8 a where K(t,Tj)
a
Choose 8 such that K(e,r])8 <
is
The proof
then (28) follows.
and the
=
>
0.
theorem
is
ilgrllg
of the
left
completed.
Corollary A.l Under assumptions
(A. 2), (A. 3'), (A.5)-A.6), the process
Hn
defined
as
[«]
Hn (s,x) =
{X'X)- l ' 2 Y,xt{I{e < x) - F(x)}
t
t=i
converges weakly to a Gaussian process
=
E{H(r, x)H(s, y)'}
Proof.
Hn (s,x) =
H
mean and covariance matrix
with zero
Q(l)- 1/2 Q(r A s)Q(l)-
(X'X/n)-^ 2 Yn (s,F(x))
if
l '2
one
[F(x
lets
Ay)-
U =
A.l.
The
finite
F(e,)-
{
converges in probability to the matrix Q(l), the tightness of
Hn
<
and u
s
=
F(x) < v
=
is
(34)
Since (X'X/n)
follows from
dimensional convergence to a normal distribution
the covariance matrix, consider for r
F(x)F(y)].
obvious.
F(y) and
Theorem
To
verify
utilize the
martingale property,
[nr]
1
f
\
E{Yn (r,u)Y^s,v)} = -E X>x' (u-uv)
(35)
(
which tends to Q(r)(u
—
Corollary A. 2 Under
uv) by (A. 3').
the assumptions of the previous corollary, the process
Vn
de-
fined as
Vn (s,x) =
converges weakly
to a
Hn (s,x) - A
Gaussian process
E{V{r,u)V(s,v)'}
=
V
{A(r A
with
s)
29
[ns
]Hn (l,x)
mean
zero
and covariance matrix
- A{r)A{s)}{u Av -
uv}.
(36)
Proof. The tightness of
Vn
follows
Hn
from the tightness of
A[ na to a deterministic matrix A(s) uniformly in
The
s.
]
and the convergence of
limiting process of
Vn
is,
by
Corollary A.l,
=
V{s,x)
Now
from
(36) follows easily
Note that (A. 3)
A(s)H(l,x).
(34).
When
a special case of (A. 3').
is
covariance matrix of
-
H(s,x)
V
A
then becomes (r
s
—
= sQ
Q(s)
rs){F(x A
y)
for
some
Q>
— F(x)F(y)}I
0,
the
where /
the p x p identity matrix, yielding a multivariate Kiefer process with independent
is
components.
We
model
next study the asymptotic behavior of the residual empirical process. Under
(1),
£t
<
z
if
and only
if
<
et
z
+ x' {$ —
t
0), thus the residual s.e.p.
K*
is
given
by
[ns]
K:(s, z)
= {X'X)-" 2 52 x
t
{I(e t
<
+
z
-
x't
/3))
-
F(x)}.
t=i
Under the
et
<
z{\
local alternative of (11), it
+ A 2 h(t/n)n- 1/2 } +
x't
<
z
if
{0 - 0) +
and only
if
^x'.gitln)^
1!2
-^ 2
}.
}^ + A 2 h(t/n) n
Thus K* becomes
[ns]
K'n (s,z) = (X'X)"
1 /2
]>>{/(£, <
+
z{\
a
t
n-" 2 ) +
b t n~
-
ll2
)
F(x)}
(37)
t=\
where
at
= A 2 h(t/n),
and
bt
Choosing the weights x t
als.
We
shall introduce
=
=
x't
{y/^0 -
1 in (37),
P)
then
=
(ai,a 2
(ci,C2, ...,c n )'
,
A,x' flr(</n)}{l
K*
f
is
+ A 2 /i(i/n)n _1/2 }.
...,a n ), b
—
(38)
just the non-weighted s.e.p. of residu-
a more general process that can accommodate
and examine the asymptotic behavior of
Let a
+
all
above cases,
this general process.
(bi,b 2 ,...,b n )
be a n x q random matrix (q
>
be two
1).
1
x n random vectors, and
Define
[ns]
Kn (s, z, a, b)
= (C'C)-
l/2
2d
{/(£< <*(!.
(=i
30
+
atn-
112
)
+
b^
1
'2
)
- F(z)}
.
C =
E
For c
=
t
xu a
—
t
and
0,
Kn (s,z,a,b)
=
bt
=
x't n
1
^2
—
($
/?),
E
or a t
and
we have
Kn (s, 2,0,0) = Hn (s, z).
and moreover,
K*(s,z)
b t in (38),
(39)
Define
Zn (s, «, a, 6) = 4='E c
n
V
7
l
<
^^
2(!
+
a
^ _1/2 + ^ _1/2 - F
)
)
t=l
W
+
a n-
1
'2
t
+
)
b
t
n~ ll2 )\
Assume
(B.l)
The
variable e
t
independent of
is
=
Tt-i
(B.2)n^Y^i\\ct\\
(B.3)
(B.4) There exist a 7
E N|
—
field{a s+ i,6 s+ i,c s+ i,e s
2
\r] t
>
=
\
and
1
(M +
m=
op (l), for
<
/I
|fc|)F
a,,
t
—
1}.
< A
all
n
and -X>{||q||
replaced by
|6<|
Note that under (B.l) the summands
Under
6,-.
00 such that for
(B.5) Condition (B.3) and (B.4) with
Theorem A. 2
<
s
;
= O p (l).
n" 1/2 maxi<,<„
£{-
cr
where
J-t-\,
Zn
in
||x t
2
(H +
7
|6t|)}
<
A
||.
are conditionally centered.
the assumptions of (A.l), (B.1)-(B.5)
Kn (s,z,a,b) = Kn (s,z, 0,0) +
(cc/n)where the o p (l)
in
x't
a £ D, an
n^ 2
-
is
1
uniform
/2
{/(*)*
in s
and
f
-(3)
<)
+
HP
/(*)
(^
c
+
<M
<*U)
J
and for
in z,
arbitrary compact set of
l 2
0) as long as n '
c< G
^
bt
=
x't a, the o p (l) is a/so
In particular, the result holds for b t
.
= O p (l).
Proof: By adding and subtracting terms,
Kn (s,z,a,b)
=
A' n (s,z,0,0)
+Zn {s, z, a, b) - Zn (s, z, 0, 0)
+ (C'C)-^ 2
£
ct
{F(z(\
t=\
31
uniform
+
a
t
n-" 2 +
)
b
t
n-" 2 - F(z)}
)
.
=
Theorem A. 2 now
from Theorem A.3(i) and
follows
below and the Taylor
(ii)
series
expansion.
Theorem A. 3
Under assumptions (A.l) and (B.1)-(B.4),
(i)
sup
- Zn (s,z, 0,0)\\ =
\\Zn (s,z,a,b)
o<s<i,zeR
Let b
(ii)
=
t
x't a
a
for
Then under assumptions (A.l) and
sup
sup
D
compact set
in a
Rv
of
- Zn (s,z, 0,0)|| =
aeDO<s<l,zeR
(Hi) Let a t
=
(r[T,...,r'n T).
r't r;
G
r t ,r
Assume
Re
for some £
>
r G S,
1;
(B.3) and (B.4) hold with
=
and denote b(a)
(x[a,
...,x'n a).
and (B.5)
(B.l), (B.2)
\\Zn (s,z,a,b{a))
o p (l).
\a t
a compact set.
=
\
op (l).
\\r t
=
Denote a(r)
Then under (A.l), (B.l)
\\.
and (B.2)
sup
sup sup
reS a€DO<s<l,zeR
Note that part
(i) is
However, each of the
(ii)
6,
allows b
=
t
x\y/n{$
\\Zn (s, z, a(r), b(a))
a special case of part
latter
to depend, in a particular way,
—
/?)
Similarly,
(ii).
consequence of
also a
is
- Zn (s, z, 0,0)|| =
as long as y/n($
—
/?)
(ii) is
a special case of
(iii).
former, as will be shown. Part
its
on the entire data
= O p (l).
op (l).
Similarly, part
parameter to be estimated. In our application, part
(ii)
(iii)
that
all
is
An example
set.
allows scale
required.
is
is
To
prove the theorem, we need the following lemma.
Lemma
A. 3 Under assumption (A.l) and (B.l)-(B.J ), for every d £
i
su P
where y*
=
y{\
extends over
+
all
/2
defined in
l2
—
a n~
pair o/(y,z) such that \F(y)
—
F(z)\
)
(i).
Lemma
Let
+ b n~
(
+
1
t
t
mean
l
,
z'
1/2)
F(z *)||=op (l)
tJ
z{\
a n~
Proof: Follows from the
Proof of
4=£l|c,F(y;)-c
(0,
1
t
^2
)
+
< n~
6( n
1
'
-1 / 2
and
the
supremum
2~d
.
value theorem.
N(n) be an integer such that N(n) =
[n
1
/ 2+d
]
+
1,
where d
is
A. 3. Following the arguments of Boldin (1982), divide the real line
32
N(n)
into
As explained
Cj
>
Then
0.
— oo =
parts by points
Lemma
in the proof of
<
c t I(e t
z)
<
z
and
<
<
z\
A. 2, there
ZN( n
no
is
=
)
Thus when
zr
iiV(n)
-1
.
by assuming
loss of generality
c t F(z) are nondecreasing.
=
oo with F(z,)
<
z
< x r+1 we
,
have
Zn (s,z,a,b) - Zn (s,z, 0,0)
< Zn (s,z T +i,a,b) - Zn (s,z T+1
[™1
1
E
+ ~7=
C
^ 7 (£/ <
- ^r+l) -
*r+l)
5=5>{F(z r+1 (l + a^"
The
\y\
when
reverse inequality holds
<
0,0)
,
max(|c|,
c
\d\) for
<
y
<
1/2
)
z r+1
+
/(£(
b t n-
1 '2
)
Z)
-
+
F(Z)}
F(z(l
replaced by z T
is
.
+
a
t
n~^ 2 +
)
b^
1 '2
)}.
Therefore, by the inequality
d,
- Zn (s, z, 0,0)|| < maxsup
sup||Zn (s,z,a,6)
<
||Zn (5,z r ,a, 6)
- Zn (s, zr ,0,0)||
I
1
+
sup
s,|ii-v|</V(n)-l
1
last
of
t
y/n
5
Because
£c {F(zr+1
_
+ sup
||
\/n
+ a^-
1/2
+
)
b t n-'l
2
)
- F(z T (l + a^-
-||
<
E?=i
term on the right
A.l.
It
is
"
II
II
o p (l)
and \F{z r +i)
by
Lemma
- F(z r )\ < n~
A. 3.
Zn (s,z r ,a,b) - Zn (s,z T
where Z*(j/n,zT ) :=
max
max
0<r<N(n) l<J<n
The remaining
£<
=
(hi I
U
t
)
+
bt
n^
2
)}
task
<
-d
last
by construction, the
term
is
op (l) because
remains to show
m^
(
l l2
The second
x H ZnOV n '^)ll =
nJ^,
0<r<N(n) l<J<n
p
1 '2
t=i
e!=1
Theorem
(l
z r {\
is
||Z;(j/n,2 r )ll
to
t
)
-J=6
T
\\Z*{j/n, z T )\\
>
c).
J
probability. Let
- F
(
(40)
But
0,0).
< -W(n) max P(max
>
bound the above
+ -j=a +
,
o P (1)
J
33
(z T (l
+ -^ a< + -j=M )
/(e,
<
zr )
+ F{z
T)
then
{it,
Ft)
an array of martingale differences and
is
t=\
By
Doob
the
inequality,
PCmaxHn-
1/2
3
where M\
is
^!!
p. 23),
Because
2
(a,, &,, c,) is
is
<
2
\\ci\\
{F(zr (l
an upper bound
(42), for
M
3
^-
=
+
a in
such that
0,
2
i^-i)}
7
+M
2
f:Eii6ii^
(42)
t=\
£{£(||6'||
27
|/(x)|
1 /2
)
£, is
independent
- F(zr )} < -L|M| 2 £(H +
and |x/(x)|
for all x.
|6,|)
Using the above
|^-i)}, we have
l
|6 :
|
|)r.
M L^,
=
1/2
2
HE6||)
27
<
last inequality follows
M
4
= 2M
M n-^E{-±\\c
n
3
2
t \\
(\a t
+
\
\b t
\)r
(=1
<
for
6,-n"
)
<n- 7/2 ^^{||c,|| 2 (|a +
+
Thus
-" 2 +
both
t=\
.
(41)
the Rosenthal inequality (Hall
measurable with respect to ^i_i and
for
2 'y
on z T
>
2
2
^||6H
The
By
t=l
inequality and £||6|| 27
By
'2
(B.l),
|^-i)
where L
M
<M ^{x:^(ii6ii
t=\
J5(||6||
1
t=i
there exists
27
by
e^A^n- ^!*,
t=i
^(iiE6ii)
of Ti.x
<
«)
a constant only depending on p and 7.
and Heyde, 1980,
for all n.
>
t
M n-^
3
2
-^-^-J2E{\\c
n
2M3 An"
2
t
\\
(\a t
\
+
\b t
\)r
t=\
t/2
.
from assumption (B.4). The above bound does not depend
M A,
3
P(maxmax|Z;(;/n,2 r )| >
c)
<
t~
2 'y
M
4
34
N(n)n-'' /2
=
t~
21
M n- ^ )l2+d
{
A
= n
because N(n)
l
The above
.
A. 3. The proof of
Proof of
^ 2+d
is
o(l)
we choose d £
if
This really follows from the compactness of D.
(ii).
can be partitioned into
not greater than
Dk
ak e
Pick
.
Dk
A||x n ||)
||ajt
— q|| <
<
D
is
compact,
any 8
for
The proof
is
the set
D
>
0,
of subsets such that the diameter of each subset
Thus
8.
Dk
a £
all
if
then assuming again
of ct I(e t
Lemma
Denote these subsets by D\, D2, ...,D m ^)- Fix k and consider
8.
For
.
number
finite
{x't a k
because
l)/2) in
completed.
(i) is
standard, see Koul (1991), for example. Since
is
—
(0,(7
we
x't a
<
(x't a k
+
8\\x t
define the vector b(k, A)
>
ct
<
S\\x t \\)
for all
t,
we have for
\\)
=
all 0:
+ A||xi||, ...,x'n a k +
(x\a k
Dk
£
,
by the monotonicity
z),
Zn (s, z, a, 6(a)) < Zn (s, z, a, b(k, 8)) +
1
l
n5 l
^E
c<
{
F
(
z(l
+
a <"~
1/2
)
+
(
W
+
and a reversed inequality holds when 6
theorem and assumption (A.l),
is
bounded (with respect
all s
£
[0, 1], all
z
to the
£ R, and
all
it is
is
1/2
)
-F
(z(l
replaced by —8.
+
a.n"
1/2
)
+
xjan"
1/2
Using the mean value
easy to verify that the second term on the right
norm
by
•
||
||)
80 p (l), where
the
P
(1) is
uniform
in
a £ D. Thus
supsup||Zn (s,2,a,i(o))
a
*||xt||)n-
- Zn (s,z, 0,0)||
s,z
< max sup
k
||Z„(s, z, a, b(k, 8))
— Zn (s, z, 0, 0)||
s,z
+ maxsup||Zn (s,z,a,6(fc, -8)) - Zn (s, 2, 0,0) + 80p (l)
||
«
3,2
where the supremums are taken over a £ D,
respectively.
a small
8.
The term
Once
8
is
<50 p (l) can be
made
(iii).
is
[0, 1],
z
£ R, and k < m(8),
arbitrarily small in probability
The
(i),
first
by choosing
two terms on
(i).
Follows from the same type of arguments as
Instead of using the result of part
theorem
£
chosen, m{8) will be a bounded integer.
the right hand side are then op (l) by part
Proof of
s
one uses the result of part
now completed.
35
in
(ii).
the proof of
The proof
(ii).
of the
)}
We now
Theorem A.l
the preliminary results,
satisfied
prove Theorems
in the position to
under (A.1)-(A.9)
to
through
4.
Conditions required for
Theorem A. 3 and
their corollaries, are all
1
for various choices of a t ,b t
and
c t below. Conditions (A. 3)
and (A. 9) can be replaced by (A. 3') when weighted empirical processes are under
consideration.
Proof of Theorem
<
z
if
x't y/n(J3
—
/?),
so i t
and Theorem
1
and only
and
=
ct
<
et
if
x
t
\
in
z
+
x't (fi
view of
K'n (s,z) - A [ns] K'n (l,z) =
1
Under the
3.
—
null hypothesis, i t
=
e
Apply Theorem A. 2 with a
/?).
t
—
t
0,b
Hn (s,z) - A Hn (l,z)
is
M
n
•)
t
n
t=i
t
bt
(44)
t=i
(45)
identically zero for all s
G
when
[0, 1]
bt
=
x't y/n(/3
from Corollary A. 2. Theorem
t
=
and A[ ns
1
i
(
ns
=
]
l(ii)
follows as a special case.
n
which
is
r
n
i
n
t=i
t
=i
/
\
$
bt
t
ns l
2.
3
That
now
is,
follows
To prove Theorem
iE»tn
Under the
—
t=i
n
i
r
l(i),
\
Z*t)yft0-f>),
n =i
The
l(i)
limiting process of
when x =
t
local alternatives (11),
(46)
j
t
Theorem
A[ ns ]Hn (l, z) reduces to the one stated in
and
1
o p (l) under assumptions (A. 7) and (A. 9).
Proof of Theorem
/?).
[ns]/n in the above proof, then (44) becomes
/(^EWW— E*« = /W
l
—
Theorem
the drift terms of K*(s,z) and A[ ns ]K*(l,z) are canceled out.
take x
=
(43)
[ns]
+op (l).
Expression (44)
t
(39),
+f(z)(X'X/n)-^-J2^b - f(z)A [ns] (X'X/n)-^ 2 -J2x
n
— x' (f3 — /3)
t
K*
is
1
for all
Hn (s,z) —
t.
given by (37) with a
t
given by (38). Note that under these local alternatives, the root-n consistency of
generally prevails. For example, assuming the e t have a finite variance, least squares
estimator of
(3
is still
The
root-n consistent.
a non-explosive limit (otherwise the tests
Note that
b
t
is
dominated by
x't y/n(/3
negligible in the limit. Moreover,
drift
term of
A'*(s, z)
—
be consistent even
will
— l3) + Aix' g(t/n),
t
when
A[nj]A'*(l, z)
root-n consistency allows us to obtain
is
b
t
=
x'(V/n(^
with the remaining term being
— /?), from
negligible for either
36
for local changes).
tt
the previous proof, the
=
x or c
t
t
=
1.
We
can
thus assume b
K*(s,z)
=
t
Aix' g(t/n). Let
ct
t
—
.
Now
by Theorem A. 2,
-E^M - AM (—
)- 1/2
{X'X )-
t
n
ns
n
X'X
'
-Z^x'
t9 (t/n)-A
n
(
)- 1/2
n
t=\
'
1
l/2
= A 2 h(t/n)
for a t
[na]
n
[
x
Hn (s,z) - A Hn (l,z)
- A [ns] ICn (l,z) =
+f(z)zA 2 }(
=
[ns] (
n
t=1
-f:x
n
h(t/n))
t
t=\
"
1
(47)
J
1
)-^-j:x
nf^
t
x't9 (t/n)
(48)
J
+oP (l)
By
the results of
KPA, under
(A.3) and (A.9)
[ns]
r
1
- J2 x
71
t
h(t/n) -A x
(=1
^
/
%)du,
tni]
1
-Y,x
t x't g(t/n)
^Q
r
(49)
s
g(v)dv.
/
(50)
Furthermore, under assumption (A.3),
A
is
obtained and (14)
Proof of (18) and
t
=
assume
is
-2,
sQ- x '\
(51)
l
f(z)A\Q 1 ^ 2 X g (s) where A 5
similarly, (48) converges to
Let a
(X'X/n)-^
1 2
these results, (47) converges to f(z)zA 2 Q~ ^ x\i (s) where A^
From
and
[ns]
0, b t
bt
=
is
obtained similarly by choosing
(19).
x't y/n(/3
= Az' g(t/n).
t
Now
£
<
t
z
if
and only
— (3) + Az' g(t/n).
=
t
1
or c t
et
=
<
1.
given by (12). Thus (15)
The proof
-
x't
Again we can ignore
t
For c
if
ct
is
=
x
( ,
given by (13);
is
/?)
is
completed.
+ Az' g{t/n)n-^ 2
.
t
x't y/n(f3
the drift term of A'*(s, z)
—
(3)
in b t
and
— A[ ns ]K*(l,z)
given by
—
{n<n
(
plus an op (l) term.
i
)- i/2
n
Finally,
I
nj ]
-E*<9(t/n) - A
n
t=i
if
c
t
=
1
M —y
C'C
(
n
1
l/2
then (18) follows; and
follows.
37
n
-E^'Mt/n)
n
(=1
if
c
t
=
x
t ,
then (19)
—
Proof of Theorem
4.
The proof
is
virtually identical to that of
Theorem
2,
except
under (A. 3'), (49)-(51) are replaced by
[ns]
—
>
n
jT[
x t n(t/n)
-2^x
A
respectively,
where Q(v)e
[ns]
is
—
>
h(v)dv,
/
dv
Jo
t
x t g{t/n) -»
(x'x/n)-^ 2
the
-*>
column
first
/
—j
g{v)dv,
2
q(i)->/ q( s )q(i)-\
of Q(v).
The
first
case of the second due to the presence of a constant regressor.
38
7579
;
convergence
is
a special
Date Due
hl c
'
Lib-26-67
MIT LIBRARIES DUPL
3
TQflO
Q0fi32TE0
6
7
8
9
10
11
IToREGON RULE
CO.
1
U.S.A.
2
3
4
I
OREGON
RULE
CO.
1
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(M
2
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10