Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/regressionbasedlOOwool
working paper
department
of economics
REGRESSION BASED LAGRAHGE MULTIPLIER STATISTIC THAT IS
ROBUST IN THE PRESENCE OP HETEROSKEDASTICITY
by
I
Jeffrey K. Wooldridge
"K^
December
/*7P
1
987
massachusetts
institute of
technology
1'',
50 memorial drive
ca mbridge,
mass 021 39
.
A REGRESSION BASED LAGRAKGE MULTIPLIER STATISTIC THAT IS
ROBUST IN THE PRESENCE OF HBTEROSKEDASTICITY
by
Jeffrey K. Vooldridge
No. 478
December 1987
LIT LIBRARIES
JUL 1 8 1988
RiOIIVED
REGRESSION-BASED LAGRANGE MULTIPLIER STATISTIC THAT IS
ROBUST IN THE PRESENCE OF HETEROSKEDASTI C I TY
Jeffrey M. Wooldridqe
DeDartment of Economics
rias5achu5ett5 Institute of Technology, E52-262C
Cambridge. MA 02139
(617) 253-3488
May 19S7
Revised:
Dece/riber 19S7
ABSTRACT:
This paper derives a form of the Lagrange hultiplier
statistic for nonlinear regression models that does not assume
conditional homoskedasticity under the null hypothesis.
to the initial nonlinear least squares estimation,
In
addition
computation of
the statistic requires only two linear least squares regressions.
The problems of computing tests of exclusion restrictions and tests
for serial correlation in dynamic linear models with unt:nown
heterosliedastici ty ^re presented as simple applications.
1
.
Introduction
This pap^f" modifies the Lagrange Multiplier
(
Ltl)
(or efficient
testing procedure to allow correct inference in the presence
5coi-e)
The test statistic is
of heteroskedastici ty of unknown form.
computable from linear least squares regressions, and is applicable
to both cross section and time series models that have been
estimated by nonlinear least squares
(
NLLS
)
The cost of the
.
heterosl: edastici ty-robust procedure is one linear
least squares
Simple regression-based procedures for computing tests
regression.
of e;;clasion restrictions and tests for serial correlation in
dynamic linear models with conditional heteroskedastici ty of unk;nown
foriT,
2.
Are presented as applications.
The Robust Lagrange Nul tipl ier Test
.
4-
l_t^^
.<
,-
-
I
V>
0-
'
h—
-- 4-
variables with Y^
a
1
scalar.
2
Y
)
I-
f
7
^^t-
V
be a sequence of observable random
}
7
t-1- ^t-1
2
prt=-determined variables.
a
t
1
;;
denote the
-
throughout.
in
tLsstiing
w
t
1
;;
For each
>
t
1,
let
L+(t-l)(L+l) vector of
The purpose of the analysis is to test
predetermined variables X^, t(Yj_|X^).
so that E(Y_^|X_^) = E(Y
L vector.
|2_^);
t
in
For cross section
this case, replace
X,
t
by
2,
I-
For time series data it is assumed that interest lies
hvpcchsses aPout the expectation concitional on curren"L
exogenous variables and all past information.
Consequently,
a
correctlv specified conditional expectation necessarilv excludes the
existence cf serial correlation.
The starting point is a correctly specified parameterized
'ersion of the conditional mean:
E(Y.
|X.
tt:
where A
on
<c
Ltioo
= m. (X,,cv. ,p
)
KP
Q
B c K
,
Q^L+Ct-l) (L+1)
€
D
A,
p
t-1,2
€ B,
o
(2.1)
and m^ is a known real-valued function defined
,
^
^^
a
)
^^^
g^
^^
and A = A
^ (c<',p')'
^
Among
B.
:•;
other regularity conditions, it is assumed that A is compact and
that m
(:•;,•
is twice continuously dif f erentiable on the interior
)
K
of^ A, ^for each
„L+(t-l)(L+l)
e K
;;
For simplicity, assume that the hypothesis of interest can be
expressed as
H
.
U
=
3
:
(2.2)
.
"^o
The LM statistic is based upon the sample covariance of the NLLB
residuals obtained under the null hypothesis and the gradient of m
with respect to
precisely,
let
p
oc_
be the NLS estimator of a
1
a
More
evaluated under the null hypothesis.
o
under H., so that
is
cx^.
O
I
solution to
T
min
2
{\\ - m
Define the residual function as U
m^(X^.. -^.0),
V m^ = V
m_^
:
t=l ,2,
.
.
the cr-fieids
,
}
(
X_^
,
= Y^ - m (X^.c-i.O).
(c;)
ct_.
and
)
Otto
The true residuals under H.
[U
(2.3)
(X\,o'.,0))-.
"
t=l
o-.^A
a^r^
U'^
= U
,
(
a
V,,m
).
=
V^,m_^
(
X^
Let U
,
c-c^
,
=
)
Note that under H.,
o-
is a martingale difference sequence with respect to
[cjC Y_^
,
X_^
)
:
t=l
,
2
,
.
.
.
>
.
The LM statistic is a quadratic form in the Qxl vector
T
.
E 7
t=l
.
n.;U
"^
^
(2.4)
in addition
If.
H^.^
it is assumed that V(Y
,
[X
)
a^ for some o^
o
o
then asymptotical 1 V equivalent versions of the LM test (under-
0,
>
to
appropriate regularity conditions) are computed as T times the
uncentered R^ from one of the followmq two regressions:
V m
on
U,
ex
,
t
V^m
,
p
t
t=l
T
(2.5)
t=l
T,
(2.6)
or
V m^U.
on
0-.
Under H
t
V_,m^U.
,
t
p
t
t
and conditional homoskedastici ty
have limiting
y*!
distributions.
,
the resulting statistics
The statistics differ in the
estimators used for the asymptotic covariance matri:; appearing in
the LM statistic.
As emphasi-ed by White
(
19B0a
.
b
,
19E2 1984
,
and White and
)
Domowitz [1984], the statistic obtained by (2.5) generally does not
have a limiting >^ distributions under H. in the presence of
heteroskedastici ty
The same is typically true of the TR" statistic
.
obtained from (2.6).
on
is not constant,
Xj_
conditional
Therefore, if the variance of Y
these regression procedures can lead to
inference with the wrong asymptotic size concerning the null
hvDothesis of interest, H..
Moreover, the standard procedures can
have Door cower properties if heteroskecasziciTiv is oreser." unoer
local alternatives.
In
closely related contexts. White
(
1930a
,
b
,
19S2 1934
,
)
Dcmowitz and White (19S2), and White and Domowitz (1934) have
derived statistics for testinq H, that are robust in the presence of
h =f t =; r w' = k (= d a s t i c i t
?cuir
t=s
cf unk;nown form.
special programming.
Com.putstion of these statistics
Moreover, the estimated covar
.
c\t
i
we
matrices used in computing the statistics are not always positive
This paper proposes
semi-definite.
regression— based method for
a
testing- H^ without imposing" homoskedasticity
To motivate the procedure, consider the regressions in
and
(2.fc).
(2.5)
Note that in each regression the first term appearing on
the right hand side is,
to the dependent
by construction, orthogonal
variable in sample, i.e.
T
J^V^^t
-
(2.7)
•='
Equation (2.7) is simply the first order condition for
ex
.
For
derivation of the robust form of the test, interest centers on
equation (2.5).
in
(2.5)
Given (2.7), the uncentered R~ from the regression
is equal
to the uncentered R^ t-esulting if V m
p
projected onto V
is first
and the resulting residuals are used as the only
m,
oc
t
t
regressors in (2.5).
More formally,
regression parameters from
be the PxQ matri:-; of
let F
multivariate regression of V m t on
a
p
V m^:
ex
t
r
T
_..._
,..,
m^
r V ex m'V
t oc tj
r^ =
T
l^^^
-1 T
T V o: m;t V^m^t
(2.B)
p
^^^
from the req:
U,
on
(
V m
p
t
- V
mi
a t T
)
t=l,
.
.
.
,T
(2.9)
is equivalently based on the D:cl vector
?
t=l
(^p-t -
(2.10)
Vt"T''^'t
which is identical, again by (2.7), to (2.4).
The advantage of
working with (2.10) instead of (2.4) is that, under H. and
regularity conditions, it can be shown that (see the appendiK)
5^ . T-l'^^E^(7^;; - V^„\?^)-G^
V^fn° =
where V ex m° = V m,(X^.o- ,0),
t
ex
t
t
D
P
t
t=l
^
(2.11)
V^m^(X.,a ,0), and
t'
O
3 t
t=l
^
The term appearing on the right hand side of (2.11)
the data and unt.nown parameters.
is a function of
interpretation of (2.11)
A useful
is that the limiting distribution of this random vector is
unaffected when the unknown parameters are replaced by VT— consistent
This feature yields a simple derivation of the
estimators.
regression-based robust test.
Under
H..
the right hand side of (2.11) has a limiting normal
,
distribution with mean zero and covariance
_-l
*-
Note that
T
matri;-;
o^o
o^o
(2.12)
—
is the correct expression whether or not the
/\_^
conditional second moment of
White (1930a,
i'S-_
b,
=
i
Thus, under H
19S2, 19S4)
_, T
T
...^
,
As pointed out by
is constant.
Y_^
A° is consistently estimated by
...
TUCvm
.^t'^t
-
\7
T'
'^
ocrT
^
'
<
"7
tT
gt -VrnT).
rr.
0-.
,
(2.14)
in
the presence of fairly arbitrary forms of heteroskedastici ty
From a computational viewpoint, it is useful to note that the
statistic in (2.14) is
on
1
times the uncentered R^ from the regression
T
-
(^off^4-
"^
t
P
ni\-r^-r)Ll'
oc
T
t
t = l,...,T.
t
(2.15)
The procedure for testing H. can be summarized:
(i)
Estimate
by NLLS imposing
c^_p
=0.
p
(ii)
U^
t
V m^ = V m^(aL^,0) and
and the qradients
"
ex t
a t T
,
Perform a multivariate regression of V_m,t on
"7
p
keep the residuals, V m
p
(2.8)
^-
-^
^.
constrained residuals
Compute the
t
= V m
P
t
- V m F
O'
t
and
is given by
where P
,
'
m.
at
1
I
;
Run the regression
(iii)
1
on
V
fii
P
U
t t
t=l,...,T
and use TR" as asymptotically XZ under H
.
R^ is of course the uncentered r— squared
.
Equivalen tly
,
use T
- SSR where SSR is the sum of squared residuals from the regression
in
(iii).
For time series applications, T might be replaced by the
actual number of observations used in the regression in (iii).
The above procedure yields computationally sim.pl e, robust
inference for nonlinear regression.
Although previously recommended
Bts.tistics are net restrictive! y difficult to program,
they
sometimes involve estimared covariance matrices that are not
positive semi— definite.
Moreover, the regression approach is
available to researchers using standard econometrics packages, and
is likely to lead to fewer progre.mming errors.
In
the linear case,
the above procedure produces a statistic
that is numerically identical to what would be obtained by applying
Theoram 4.32 in White (1934) to the cbsb of heteroskedastici ty
Therefore, in the linear case, the above procedure can be viewed as
computional 1 y simple method for calculating the White robust
a
test.
'
.
L!i
.
Before giving a specific eKample, it is interesting to compare
the LM procedure based on
(2.5)
to the robust procedure just
Suppose that, instead of estimating A
outlined.
by A
,
the
conditional homoskedasticity assumption is maintained and A
o
is
estimated by
A-^
=
ar T
--1
where a
(V m
^_iPt
J]
T
I
= T
- V m r
cr.
tl
- V m r
)'(V^m.
pt
tl
ex
(2.16)
).
T
E
'-'4- '
Using A
in
yields the statistic
place of A
t=l
\LA^T
I
P
T
.
This statistic is numerical
regression in (2.5).
equivalent
to TR^ from the
M
Iv'
7
Therefore, under H
and homoskedasticity,
the
robust statistic is asymptotically equivalent to the usual LM
statistic.
Because the statistics only differ in their estimates
o"
the moment matrix appearing in the quadratic for of the LM
statistic, they are asymptotically equivalent under local
alternatives and homoskedasticity.
asymptotically noncentral
"i=t=;;
o = ktuastici t>'
.
'X^
The robust form remains
for local alternatives under
When heteroskedastici tv' is present,
th
icz"
:
LJ
bus-
form will frequently have better power properties rhan the usual LM
statistic.
linear model
^(Y^IX^) = X^^cx^ ^
where
l;iP
vector and
X
,
X_3^
is a
*-
l;cG
-1.
^
—
«
vector, both of which may
and current and lagged values of
contain laaged values of Y
ele^Tients of
Z
Applying the methodology developed above leads to a
.
very simple heteroskedastici tv-robust test of H.;
U
'
•
(i)
on
= 0:
t=l
X^^
and save the residuals,
L).
Run the multivariate regression
°"
^t2
ind
o
Run the regression
\
(ii)
B
t=l,...,T
^tl
save the residuals,
X
.
(iii) Run the regression
1
on
t=l,
>^t2^t
and use TR~ as asymptotically
'Xi
.
.
.
,T
under H-.
The regression in step (ii) is the cost to the researcher for
being robust to heteroskedasticity
.
One application of Example 2.1
is testing for Granger causality in time series models without
imposing constancy of the conditional second moment of the dependent
variable.
Because noncausality in mean implies nothing about
conditional second moments, standard approaches can lead to
inference with the wrong asymptotic size under the null hypothesis
The above .usthodol oc v alleviates this problem.
when u =
1,
so that
X
is a scalar,
-1-.—,
cr-*-z:"*-"i^=*-''-—
^T-j-v/TM-vit-hciH
K\/
steps (i)-(iii) is the square of the LM form of the White
hstsrcsksiasti-ity-robust t-statistic.
Note that the regression in
step (ii) is now a standard OLS regression.
asymptotically equivalent version of
be computed by three OLS regressions.
a
Consequently, an
White (19B0a) t-statistic can
Example 2.2
:
linear model
(
LM test for AR
i-^ith
AR
(
1
Ip
~a
I
<
and X,,
1
tl
1
Consider
serial correlation):
)
X
is a
cx
ti o
1
-X t-1.1 cx:
+C(Y
o
t-1
^o
K subvector of
:-;
X.
t
(2.17
X,, may
tl
.
as well as current and lagged values of
lagged values of Y
a
serial correlation:
)
E(Y|X)=
t' t
where
(
contain
Z
The
.
null hypothesis is
H..
Pra
O
In
'
the above notation,
p
= p,
V
cx
'
m. =
t
X^,
tl
and V
,
are the DLS residuals from the regression of
following procedure is valid for testing H
m,
= U,
t—
on
X
P t
Y
.
,
where
CU.
t
The
in the presence of
heteroskedastici ty
(i)
Run the regression
't
""
^tl
^
,T
^'
and save the residuals U
(ii)
Run the regression of U
residuals, sav U
(ii)
(
on W^
1
,
and keep the
k_
t-1
Perform the regression
on
and use
t
;— 1)R^
^t-l^t
from this regression as asymptotically
R~ is the uncentered
r
— sauared
'k7
under H
.
.
Several features of this example Brs worth emphasizing.
First,
the robust procedure follows from a straightforward modification of
the LM principle, and the test is asymptotically equivalent to the
usual LM statistic for AR 1) serial correlation under
(
homoskedastici ty
.
Also,
there are essentially no restrictions on
what X^^ can include, so that the procedure allows specification
testing for dynamic regression models.
The usual LM approach also
but the above procedure is valid in the presence
has this feature,
of conditional heteroskedastici ty and computationally is almost as
Because certain economic time series
simple as the usual LM test.
exhibit conditional heteroskedasticity (see, for example Engle
(1982) and Bollerslev (1986)),
approach should be
a
the heteroskedasticity-robust
useful innovation.
If
the conditional mean is
the primary interest of the researcher, then it may be undesirable
to explicitly model
the second moment.
The above procedure allows
for asymptotically correct inference without worrying about the
mechanism determining the second moments.
As pointed out by White (1985),
if the procedure used
in
Example 2.2 rejects H., it is not necessarily true that (2.17)
represents the true conditional expectation of
test for AR
(
1
)
Y
given
X
.
The
serial correlation may be detecting some other form
of dynamic misspecif ication
In
.
general, the statistic wxll reject
for certain deviations from
H.
o
E(Y, |X^)
:
t
t
= X^^a
tl o
,
t=l,2
although the test is not consistent for every alternative to H
Eierens
(
19E2 19S4
,
)
(see
)
This section concludes with a formal result.
The regularity
conditions imposed are not the weakest possible; in particular, it
is assumed that moment matrices are stablized after normalization by
T
—1
.
These assuiTiStiops rule out manv nonstationa.rv time series.
They could be generalized along the lines of Wooldridge (1986) to
allow far deterministic trends
testing procedure)
,
(this would not at all change the
but the theorem cannot be expected to extend to
10
certain time series models with unit roots.
A
definition simplifies the statement of the theorem.
Definition 2.1
sequence of random functions Lq.(Y
A
:
t=l,2,...j, where q.(Y
subset of K
Numbers
(
,)
,X
,©):
,X
6 s ©,
is continuous on © and © is a compact
is said to satisfy the Uniform Uleak Law of Large
,
UWLLN
and Uniform Continuity
)
Ae©
UC
conditions provided that
)
- ECq
|T~^ Z q.(Y ,X
e)
^
^
t=l ^
sup
(i)
(
t
(Y
e)]|
X
c
t:
5
and
"T
-1
(11)
[
T
(Y^,X
EEq
Y.
^
-
continuous on © uniformly in
T.
t=l
In
6)3
c
the statement of the theorem,
e;;cluded when it is equal
T = l,2,...}
e s ©,
:
the argument
is
is often
3
The dependence of functions on the
to 0.
predetermined variables X^ is suppressed for notational convenience.
Theorem 2.1
(i)
:
Suppose the following conditions hold under H
:
A is compact.
(ii)
Cm
(;•;,.-5)
:
^ K
;c
6
,
•=
A} is a sequence of
real —v'alued functions such that
(a)
\-
m
(,£) is Barel measurable for each
;r
u
(b)
m^(;;
,
•)
interior of A for all
(iii)
(a)
-[
(m,
t
(
a.i.
•;
^
D:)-m.
r
J^
t=l
— ,...;
t = l,2....;
,
('
ex
o
)
)~} and
[ (
(
T
T~^
A and
is twice continuously dif f eren tiable on the
satify the WULLN and UC conditions:
(b)
~ — ^,
;.'^j
<=
<S
LU^~ - E(u^~):
^
11
5 0;
m
.
t
(
(x)
-m
.
t.
(
a
o
))U~']t.
(c)
is the identifiably unique minimizer
ex
and White (1985)
of
)
-1" ^
T
(v)
and
C
V
ex
o
(a)
to-
EC(m. («) - m, («
J:
^
t=l
(iv)
(see Bates
)
2
)^];
is in the interior of A;
V
C
^ni^(cx)
m. (cx)'V m. («)
cxt
},
V m («) V m
cxt
pt
'
C
.
(
oc)
},
V
C
cxcx
m.(cx)
t
},
satisfy the WULLN and UC conditions;
}
«P t
Oct
J
(b)
)'7 m^(a )] is 0(1) and uniformly
y E[V « m^(o;
t
o
a
cc
t
j^
T"-""
positive definite;
-1 T
(c)
(a )] is 0(1);
C E[V ex m.t (ex o )'V-m.
o
p t
T
_^
.
(vi)
(a)
m^(cx)'U^(cx)},
CV
t
c<cx
t
{V ^m^ a) U^ a)
'
t
cxp
(
t
(
and
}
[V m
ex
.
(
t
ex)
'
U.
t
(
a)
}
satisfy the WULLN and UC conditions;
(b)
(vii)
(a)
-
"^
-1
r,
T ^'- T 7 m,(c< )'U° =
i
^
o
t
ex
(1);
t
p
uj(a) (V^m^(cx) - V m^
ext
t
pt
(
ex)
D
'
(
V.m
pt
.
(
ex)
- V m.(cv)r)
oc
t
}
satisfies the WULLN and UC conditions;
(b)
A° =
i
^° - Vm°r°)'(V^m° - V.m°r°)]
T~^r ECU°-(V B m°
ex
t T
t
ex
t T'
t
r
,
*_
.
t=i
'
P
ii
3
0(1) and uniformly positive definite;
(c)
1 /o
o — -"'^
A^
—1
T
z"?
E
" o
o
d
(^..m° - '7.mrr°)'U° °
T'
ex t
t
P t
N(0,I^)
len
iOz'i.
where
5
3 >a
is given by (2.11) and A_ is defined in
(2.13).
Though the list of regularity conditions is long, the theorem
is widely applicable.
The WULLN and UC requirements hold under very
general conditions on the process
12
C
(
Yj_
,
Z_^
)
T
.
See Andrews (1987),
Gallant and White (19S6) and Newey (19S7) for general treatments.
The asymptotic normality assumed in (vii:c) can be established by
application of the martingale central limit theorem discussed in
Wooldridge (1986).
the conditions of Theorem 2.1 Br&
Overall,
fairly weak, and can be expected to hold in many cases where the
data are weakly dependent.
3
Cone 1 us ion 5 and Further Appl ications
'
.
The approach to specification testing taken in this paper leads
to simple robust inference for regression models that contain
unknown forms of heteroskedastici ty
The procedure should be useful
.
for both cross section data and for time series data for which
interest lies in correct dynamic specification of the regression
function.
Although it was assumed that the restrictions being
tested BTB zero restrictions,
procedure is valid when the
- r
(
ex
o
)
under H^
where
,
ex
o
a
F'-i-Q
similar analysis shows that the
;;
is a P
1
vector G
;c
1
dif f erentiable function from K P to
a
can be expressed as O
vector and
r
is a
P+D
[R
The approach developed here has several other applications.
Wooldridge (1937a), it is used to derive
In
class of regression-based
a
conditional moment tests (Newey (1985), Tauchen (1984)) for
conditional expectations estimated by GMLE
.
The resulting
regression-based tests include the LM test discussed here, Hausman
tests which do not assume relative efficiency cf either estitr:ator,
and nonns = ted hypotheses tests that
^ -=
nr. und
.-,H =
Tiisspecification
'
-^
-;
:
.
s.rB
robust to second moment
the null hypothesis.
»ppr^c(_h yields a sim.ple
Slightly modifying the
regression White test for
13
a
heteroskedastici ty which does not assume constancy of the
conditional fourth moment of the errors.
This test is derived in
Wooldridge (1987a) for the more general class of QMLE
'
s
assuming
only that the first two conditional moments are correctly specified
under the null hypothesis.
14
Appendix
For convenience, a
leiTima
is included which is used repeatedly
the proof of Theorem 2.1.
in
LeiTifTia
(W
[Q
A. 1
Assume that the sequence of random functions
:
e e ©, T=l,2,...], where
,e):
and © is a compact subset of K
functions
,
•)
is continuous on
and the sequence of nonrandom
© e ©, T=l,2,...}, satisfy the following
(©):
[Q
,
(W
'
conditions:
sup
©e©
(i)
(ii)
uniformly in
(W
- Q
©)
5 0;
(©)]|
e € ©, T=l,2,...} is continuous on ©
•CQy(©):
T.
0^(W^,©^) - 0^(e*) 2
Proof:
— ©_
->
where
Then
c ©.
}
111
be a sequence of random vectors such that ©
Let ©
i©
|Q
0.
see UJooldridge (19S6, Lemma A.l,
Proof of Theorem 2.1
:
p.
229)
The major task of the proof is establishing
the validity of equation (2.11).
ciy
i_i
rs
i_i
it
weak consistency analoQ of Bates and
(19S5, Theorem 2.2),
'--'-^
"
assumptions (i),
i-o-sistency of
•'^_-
and
c--_
LJ
i
(iv)
^
0(1)
.
(iii)
imply that
imply that
.
-»
L
J'v'^'
and
i
T
Now
(ii)
Wmte
~
\
t=l
'-"'^•-^n-^^^
°=
^
1
'-t-.s-
T
Ne;-;t,
as T
-»
00.
-
consider equation (2.11). Now
.ciently large T
(a.l)
t=l
t=l
But by (a.l), the Becond term on the right hand side equals zero
with probability approaching one, so that the second term on the
right hand side of (a. 2) has no effect on the limiting distribution
of
Given (v:a,b) and (vi:b), T 1
?T-.
Top
'^
/ r?
^(oc^ -
I
a
=
)
Using this
(1).
fact and the mean value expansion for random functions (Jennrich
(1969, Lemma 3)), expand the first term on the right hand side of
about
(a. 2)
^
and apply Lemma A.l to get
ex
1/2
=
"^
-
J^CVpm,
0„D
"
,
Vt^^'^t
"^
t=i
*-
E[(V^ m° - V <r°)'U°] - T-^ E
pa t
acx t T
t
^
To
T^'^'^Ca^ -
•
Under H., ECV^ m°'U°] = E[V
O
pDC
t
t
m°'U°] =
cxcx
m°
EHV pt
t
t
0,
ex
)
and,
- V m°r°)'7m°] =
Z E[(V-m°
t T
t
ex
P ^
- V m°r°
)
+
.
atT
o
(1)
'
V m°]l
octj
{.3..Z.)
p
by definition of r°
T'
0.
t=l
Because T 1/2
side of
Top-
'
(
(a. 3)
cv
-
ex
IS o
)
=0
(1).
(1),
the second term on the riqht hand
This shows that (2.11) holds.
P
By (viiib.c),
that A^
I'V.p
^T^
1.
a
o-i
.V^
''2'-
?
d
-»
Condition (vii:a) ensures
N(0,lp^).
consistent estimator of A^
,
an d
5^ - XT, and this completes the proof,
b t)0
<c
16
(vii:b) ensures that it
References
'
Andrews, D.W.K., 1987, "Consistency in Nonlinear Econometric
Models: A Generic Uniform Law of Large Numbers,"
EcoTi o:!}et r i ca
forthcoming.
,
Bates, C. and H. White, 1985, "A Unified Theory of Consistent
Estimation for F'arametric Models," Ecorion.etr i c Theory 1,
151-175.
Bierens, H.J., 1982, "Consistent Model Specification Tests," Journal
of EcoTtometr ics 20, 105-134.
Bierens, H.J., 1984, "Model Specification Testing of Time Series
Regressions," Jourrial of Ecoriometr ics 26, 323-353.
Bollerslev, T., 1986, "Generalized Autoregressive Conditional
Heteroskedasticity " Journal of Ecoriov)etr i cs 31, 307-327.
,
Domowitz I. and H. White, 1982, "Misspecif ied Models with
Dependent Observations," Journal of Econovtetr i cs 20, 35-58.
Engle, R., 19S2, "Autoregressive Conditional Heteroskedasticity
with Estimates of United Kingdom Inflation," Econometr i ca 50,
967-1003.
Gallant, A.R. and H. White, 1986, "A Unified Theory of Estimation
and Inference for Nonlinear Dynamic Models," UCSD Department
of Economics Discussion Paper.
Jennrich, R., 1969, "Asymptotic Properties of Nonlinear Least
Squares Estimators," Annals of f1athe::>at zeal Stat istzcs 40,
633-643.
Newsy, W., 1935, "Ma;cimum Likelihood Specification Testing and
Conditional Moment Tests," Econontetr ica 53, 1047-1070.
Newey, W., 19S7, "Expository Notes on Uniform Laws of Large
Numbers," mimeo, Princeton Universitv.
aui_hen,
3.,
1934,
i ."v
E^ooriCfiTix
lesting and Evaluation of Maximum
"Diagnostic
^
lJ
n 1 v z^ir ^, 1.^'y'
*>';_;
r
r-j
mg
.
.
o=
White, H., 199Ca,
"A Heteroskedastici ty-Consistent Covariance
Matri;; Estimator and a Direct Test for Heteroskedasticity,"
Econo:;/e t r i €3 43, 317— £3£.
White, H., 1930b, "Nonlinear Regression on Cross Section Data,"
Econ oiitst r i c B 43 721 — 746.
Whits, H., 1932, "Maximum Likelihood Estimation of Missoecified
Models." Ec on on;e tr z c a 5f) 1 — 26.
17
Whits, H., 19B4, AsyTi:ptot 1
Acamedic Press.
c
Theory for Econome
tr
i
ci
ar:
s
,
Fvew
Vorki
White, H., 19G5, "Specification Testing in Dynamic Models,"
Invited Paper at the Fifth World Congress of the Econometric
Society, Cambridge, Massachusetts, August 19E5.
I. Domowitz, 1984, "Nonlinear Regression with
Dependent Observations," Econojsetr ica 52, 143-162.
White, H. and
Wooldridge, J.M., 19B6, "Asymptotic Properties of Econometric
Estimators," UCSD Ph.D. dissertation.
Wooldridge, J.M., 1987a, "Specification Testing and Quasi-Maximum
Likelihood Estimation," mimeo, MIT.
Wooldridge, J.M., 1987b, "Regression-based Specification Tests for
Dynamic Simultaneous Equations Models," mimeo, MIT.
IS
^^J.-
Date Due
^m 219
iOV 21
FEB.
2A
199^1
^^
i
AUG
1?
9 20Q1
MIT LIBRARIES
3
TDfia
DOS 3Sb MED