~ (mbhaiies) n
working paper
department
of economics
A UNIFIED APPROACH TO ROBUST, REGRESSION-BASED
SPECIFICATION TESTS
Jeffrey M. Wooldridge
No. 480
Revised
November 1988
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
A UNIFIED APPROACH TO ROBUST, REGRESSION-BASED
SPECIFICATION TESTS
Jeffrey M. Wooldridge
No. 480
Revised
November 1988
Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium Member Libraries
http://www.archive.org/details/unifiedapproacht00wool2
A UNIFIED APPROACH TO ROBUST, REGRESSION- BASED
SPECIFICATION TESTS
Jeffrey M. Wooldridge
Department of Economics
Massachusetts Institute of Technology, E52-262C
Cambridge, MA 02139
(617) 253-3488
January 1988
Revised:
November 1988
I am grateful tc Adrian Pagan, Peter Phillips, Jim Fcterba, Danny Quah,
Tom Stoker, Hal white, and an anonvmous referee for helpful comments and
suggestions on a previous version.
JAN 9
19&8
ABSTRACT:
This paper develops a general approach to robust, regression-based
specification tests for (possibly) dynamic econometric models.
The key
feature of the proposed tests is that, in addition to estimation under the
null hypothesis, computation requires only a matrix linear least squares
regression and then an ordinary least squares regression similar to those
employed in popular nonrobust tests.
For the leading cases of conditional
mean and/or conditional variance tests, the proposed statistics are robust to
departures from distributional assumptions that are not being tested.
Moreover, the statistics can be computed using any vT-consistent estimator,
resulting in significant simplifications in some otherwise difficult
--contexts.
Among the examples covered are conditional mean tests for models
estimated by weighted nonlinear least squares under misspecif ication of the
conditional variance, tests of jointly parameterized conditional means and
variances estimated by quasi-maximum likelihood under nonnormality
new,
,
computationally simple specification tests for the tobit model.
and some
1
.
Introduction
Specification testing has become an integral part of the econometric
model building process.
The literature is extensive, and model diagnostics
are available for most procedures used by applied econometricians
The most
.
popular specification tests are those that can be computed via ordinary least
squares regressions.
Examples are the Lagrange Multiplier (LM) test for
nested hypotheses, versions of Hausman's
[24]
specification tests, White's
information matrix (IM) test, and regression-based versions of various
nonnested hypotheses tests.
[26]
[13]
In fact, Newey [17], Tauchen [21],
and White
have shown that all of these tests are asymptotically equivalent to
particular conditional moment (CM) test.
a
In a maximum likelihood setting
with independent observations, Newey [17] and Tauchen [21] have devised outer
product- type auxiliary regressions for computing CM tests.
extended these methods to
a
White [26] has
general dynamic setting.
The simplicity of most popular regression-based procedures currently
employed,
cost.
including the Newey-Tauchen-White (NTW) procedure,
is not
without
In many cases the validity of these tests relies on certain auxiliary
assumptions holding in addition to the relevant null hypothesis.
example,
For
in a nonlinear regression framework where the dynamic regression
function is correctly specified under the null hypothesis, the usual LM
regression-based statistic is invalid in the presence of conditional cr
unconditional hetercskedasriciry
.
product statistic is also invalid.
for heteroskedasticity:
Except in special cases the NTW outer
Other examples include the various tests
currently used regression forms require constancy of
the conditional fourth moment of the regression errors under the null
hypothesis.
In addition, the Lagrange Multiplier and other CM tests for
jointly parameterized conditional means and variances are inappropriate under
various departures from normality.
The above situations are all characterized by the same feature:
validity of the tests requires imposition of more than just the hypotheses of
interest under H
.
In addition,
traditional econometric testing procedures
require that the estimators used to compute the statistics are efficient (in
some sense) under the null hypothesis.
is
It is important to stress that this
not merely nitpicking about regularity conditions.
Due primarily to the work of white
[7],
[22,23,24,26], Domowitz and White
Hansen [11], and Newey [17], there now exist general methods of
In the context of linear regression models,
computing robust statistics.
Pagan and Hall [19] discuss how to compute conditional mean tests that are
robust to heteroskedasticity
.
Their discussion centers around the use of the
White [22] heteroskedasticity-consistent standard errors.
For one degree of
freedom tests the Pagan and Hall suggestion leads to easily computable tests,
and it is certainly an alternative to the current approach for regression
models.
But computation of the statistics for tests with more than one
degree of freedom requires explicit inversion of the White covariance matrix
estimator;
this matrix must then be used in a quadratic form to obtain the
neteroskedasticity-robust Wald statistic.
much in the spirit cf the LM approach:
the model only under the null,
The tests proposed here are very
computation requires estimation of
so that any particular model can be subjected
to a battery of robust specification tests without ever reestimating the
.
model.
.
More importantly, the tests can be computed using any standard
regression package.
Although there are some fairly general formulas available for robust LM
statistics (e.g. Engle [9], White [24,25]), formulas for general nonlinear
restrictions involve an analytical expression for the derivative of the
implicit constraint function and a generalized inverse.
In specific
instances computationally simple robust LM statistics are available.
notable example
is
A
the paper by Davidson and MacKinnon [6], which develops a
regression-based heteroskedasticity-robust LM test in a nonlinear regression
model with independent errors and unconditional heteroskedasticity
It is a safe bet that the substantial analytical and computational work
required to obtain robust statistics is a primary reason that they are used
infrequently in applied work.
Evidence of this statement is the growing use
of the White [22] heteroskedasticity-robust t-statistics
computed by many econometrics packages.
a Hausman test,
test,
,
which are now
Only occasionally does one see an LM
or a nonnested hypothesis test carried out in a manner
that is robust to second moment misspecif ication.
This is unfortunate since
these tests are inconsistent for the alternative that the conditional mean is
correctly specified but the conditional variance is misspecif ied.
words
,
In other
the standard forms of well known tests can result in inference with
the wrong asymptotic size while having no systematic power for testing the
auxiliary assumptions tnat are imposed in addition to h_
This paper develops a unified approach to calculating robust statistics
via least squares regressions which
econometricians
.
I
believe is easily accessible to applied
The general method suggested here can be viewed as an
extension of the Davidson and MacKinnon
[6]
approach.
In fact,
in the
context of nonlinear regression models, their procedure is shown to be valid
for quite general dynamic models with conditional as well as unconditional
heteroskedasticity
.
In the the same context the approach here can be viewed
as the Lagrange Multiplier version of the robust Wald strategy suggested by
Pagan and Hall [19].
This paper also extends Wooldridge's [30] robust,
regression based conditional mean and conditional variance tests in the
context of quasi-maximum likelihood estimation in multivariante linear
exponential families.
The current framework is more general because it
applies anytime a generalized residual function (defined in section 2) is
the basis for the "test.
For the leading cases of conditional first and second moments,
the
regression-based tests proposed maintain only the hypotheses of interest
under the null, and they are applicable to specification testing of dynamic
multivariate models of first and second moments without imposing further
assumptions on the conditional distribution (except regularity conditions).
Moreover, in classical situations, these tests are asymptotically equivalent
under the null and local alternatives to their traditional counterparts.
Robustness is obtained without sacrificing asymptotic efficiency.
For some specification tests the current approach does impose auxiliary
assumptions under the null hypothesis.
regression-based nature cf the tests.
far,
This is the price one pays for the
Still,
in most cases encountered so
the current framework imposes fewer auxiliary assumptions under the nui]
than popular nonrobust tests.
This does not mean that robust tests are not
available in such circumstances, but only that regression-based forms of
these tests are not known.
The goal here is to provide a unified approach to
robust, repress ion -based specification tests, and not to robust tests in
Nevertheless, the coverage is fairly broad.
general.
A general treatment of
robust tests is contained in White [26].
A second aspect of the proposed statistics is that they may be computed
using any ,/T-consistent estimator.
The asymptotic distribution of the test
statistic under the null and local alternatives is invariant with respect to
the asymptotic distribution of the estimators used in computation;
be viewed as another kind of robustness.
this can
Consequently, the methodology leads
to some interesting new tests in cases where the computational burden based
on previous approaches can be prohibitive.
This is true whether or not
robustness to violation of auxiliary assumptions is an issue; in fact, the
procedure can be profitably applied to situations which assume correct
specification of the entire conditional distribution provided that the test
statistic can be put into the form considered in section
the proposed tests have properties similar to Neyman's
2.
[18]
In such cases
C(q) tests, but
they are applicable even whether or not the score of the log- likelihood is
the basis for test statistic.
When restricted to LM tests, the new
statistics offer generalized residual alternatives to outer product-type C(a)
statistics, provided of course that the score of the log- likelihood can be
put into the appropriate generalized residual form.
Section
section
3
2
cf the paper discusses the setup and the general results,
illustrates the scope cf the methodology with several examples, and
section 4 contains concluding remarks.
contained in an annendix.
B.egularity conditions and proofs are
2
General Results
.
Let
lxJ
y
,
{
(y
z
,
):
lxK.
z
t— 1,2,...) be a sequence of observable random vectors with
y
is
context) past values of y
(z
v
v ,,z
,
t'^t-l' t-1'
V
z
e (y
For time series applications,
.
can be excluded from x
z
one may take x
z
and
variables.
denote the predetermined
v
y.,,z,)
,J
1'
,
current
and (in a time series
in terms of the explanatory variables z
explaining y
Interest lies in
the vector of endogenous variables.
y
.
.
.
,y
let x
=
Note that
or,
if there are no "exogenous" variables,
)
For cross section applications set x
.
and assume that the observations are independently distributed.
The conditional distribution of y given x always exists and is denoted
t
t
Assume that the researcher is interested in testing hypotheses
D (-|x ).
about a certain aspect of D
the conditional variance.
contains
(
(y^.z
,
for example the conditional expectation and/or
Note that, because at time
.),..., (y.,z)
}
or [y
.
,y t _
2
.
•
•
t
.Y^
the conditioning set
the assumption is
that interest lies in getting the dynamics of the relevant aspects of D^
correctly specified.
For cross section applications this point is
irrelevant.
For motivational purposes and to illustrate the notation,
to introduce a couple of examples.
testing cf a conditional mean.
.
y^_
(taken to be a scalar for simplicity)
The parametric model is
(m (x ,a):
c
t
where A c R
The first example concerns specification
Suppose interest lies in testing hypotheses
about the conditional expectation cf
given x_
it is useful
,
a e A.
t=l,2,...},
and the null hypothesis is
(2.1)
2
H
E(y |x
:
Q
t
t
- m (x ,a
)
t
t
o
some q
),
q
g A, t-1
2
(2.2)
,
A
If a_ is a VT-consistent estimator of a
I
A
defined as u (y
,x
under H„ then the residuals are
U
o
A
,q
^ y
)
m
-
(x
,a
)
.
can be based on the
A test of H
sample covariance
^ vvvVWW
T
t"
1
A
where
X
(x
A
1
-
- T
I
t-1
A
(2
3)
A
A'u
(2.4)
is a lxQ vector function of "misspecif ication indicators"
.q.tt)
A
A
that can depend on a
and a nuisance parameter estimator n
approachleads
-
The standard LM
(uncentered) r-squared from the
to a test based on the
regression
A
A
on
u
t
If a
A
V m
Q t
t-1
A
,
(2.5)
T.
t
asymptotically equivalent to the NLS estimator then under H. and
is
conditional homoskedasticity
J
TR
,
2
u
2
is asymptotically
J
y^..
Thus,
the LM
Q
approach effectively takes the null hypothesis to be
HI: H. holds and V(v
U
(J
t
|x
t
- a
)
1
o
for some
1
a
o
> 0,
t=l
,
(2.6)
but it is inconsistent for the alternative
H'
K n holds but
:
does not.
K'
It also essentially reauires that
a„.
be the NLS estimator.
i
The Nevey-Taucher.-White regression fcr the same problem is
1
u V m
or.
t Q
,
t
u
t=l ,....T.
A
t
(2.7)
t
A
In general,
R'
is also
required for
from this regression to be
TP.
i
2
asmDtoticaily x^> although there are some cases, such as testine for serial
Q
.
correlation in a static regression model with static conditional
heteroskedasticity (i.e. V(y
|x
)
V(y
«=
|z
where the NTW regression is
)),
A
being
The validity of the NTW procedure also generally relies on a
robust.
the NLS estimator.
As pointed out by Pagan and Hall [19], a robust test is available from
The White [22] heteroskedasticity-robust covariance
the regression (2.5).
matrix estimator can be used to compute a robust Wald statistic for the
A
A
hypothesis that
can be excluded from the regression (2.5).
A
When
is a
A
scalar this is simple because a robust test statistic is simply the robust
A
A
t-statistic on
A
When
.
vector computation of the robust Wald
is a
A
statistic is somewhat more complicated since it involves inversion of the
White covariance matrix estimator as well as explicit construction of the
appropriate quadratic form.
the Wald procedure is valid
In addition,
A
essentially only when a
is the NLS estimator.
The regression-based heteroskedasticity-robust form of the test, which
in addition is valid for any 7T-consistent estimator,
Example 3.1 discussed in section
is a special case of
3.
As a second example, consider testing for heteroskedasticity.
The null
hypothesis is taken to be
H
E(yjx
•
b
i_
)
t
ttO
= m (x
,a
)
and V(v |x
t
)
t
- a
2
00
,
a
e A, c
2
> 0,
t=l
Again let u^(y ,x^,a) be the residual function, and let A(x_,£,7r) be
_
vector or Heteroskedasticity indicators, where
of tests is based
A^(x^,f„,r_)' iu_(y_,x^,Q_)
I
^
= (a' ,a
or.
T
T
6
u
l.
i
1
Z.
Z.
Z.
T
-
O
T
]
2
)'
.
,
2
,
.
.
.
a IxQ
A general class
.
V
T
a
- t"
A
where a
2
= T
A
A
J-
1
y
I
t«l
-12
X u
r
A
O
r.
(u
-
f.
A standard LM-type statistic is obtained from the
•
t=l
centered r-squared from the regression
A
Art
u
TR
2
t
on
asymptotically v
is
X
1,
C
2
t
,
t=l
(2.8)
T.
under
L^
H': H n holds and,
where u
y
-
m^Cx
in addition,
,q
).
of the Breusch-Pagan [2]
E[(u
l
o A
t
)
|x
'
t
]
= k
2
o
> 0,
t=l
,
2
,
.
.
Regression (2.8) yields the "studentized" version
test as derived by Koenker [16].
The studentized
form of the test is robust to certain departures from normality, and it is
now widely used in the literature (see, e.g.. Engle [8], Pagan and Hall [19],
and Pagan, Trivedi, and Hall [20]).
Unfortunately, this form of the test is
not completely robust in the sense defined in this paper.
o
E[( u _)
4
(2.8)
is an auxiliarv
assumption imposed under H A that is required for
J
u
|x
t
The constancy of
t
to lead to a valid test.
conditional on x
Normality of u
heterokurtosis under H_
,
but it is easy to construct examples to illustrate
that the auxiliary assumption of homokurtosis is binding.
o
.
,.
rules out
t
t
If the regression
.
errors u_ nave a conditional t- distribution witr. constant variance but
decrees of freedom that otherwise deoend on x
then K_ holds but
K'
does not.
t
Hsieh [14] and Pagan and Hall [19] have noted that just as with conditional
mean tests, the white [22] covariance matrix
car.
be used tc compute
heteroskedasticity tests that are robust tc heterokurtosis.
when A^
is a scalar,
However, except
computation of the statistic requires several matrix
'
operations.
Pagan, Trivedi, and Hall [20] report the White
[22]
t-statistic
A
in a model for the variance of inflation when X
An alternative
is a scalar.
robust form of the test is provided in Example 3.2 in section
It is
3.
A.
almost as easy to compute as the nonrobust form even when
is a vector,
A
but
it allows for heterokurtosis under the null.
There are other examples where the goal is to test hypotheses about
certain aspects of a conditional distribution but auxiliary assumptions are
maintained under the null hypothesis in order to obtain
regression-based test.
simple
a
Because the limiting distributions of test statistics
are usually sensitive to violations of the auxiliary assumptions,
it is
important to use robust forms of tests for which H n includes only the
hypotheses of interest.
To be attractive these tests must be easy to compute
under reasonably broad circumstances.
The remainder of this section develops
a general approach to constructing robust,
regression-based tests.
Many specification tests, including those for conditional means and
variances, have asymptotically equivalent versions that can be derived as
follows.
Let ^>,_(y^,x
t
set 9 C R
H
•
U
p
.
t
t
be an Lxl random function defined on a parameter
,8)
The null hypothesis of interest is expressed as
El© (v
,x
"
'
t
t
By definition,
t
,8
O
)|x
t
«
]
0,
is the "true"
S
o
for some
S
o
€ 9,
t-1,2,
parameter vector under H„.
-
hypothesis specifies that the conditional expectation of
_ne prece tcnrinSw
VansD^6s
x_
(2.9)
is zero,
c
Because the null
(y
.x
it is natural tc Cci. c
,
6
)
given
a
For the conditional near, tests in a nonlinear
regression model,
1=1,
6
= a, and a (y^,x
10
,
0)
u„(y ,x
,o)
= y
-
.
The tests for heteroskedasticity take L
m (x ,a).
<^
t
(y t .x
E u 2 (a)
.0)
-
t
a
1,
8
=
Q ' >a
(
2
)'
,
and
2
.
The validity of (2.9) can be tested by choosing functions of the
predetermined variables x
between these functions and
and checking whether the sample covariances
,x
(y
<f>
,6
are significantly different frc
:om
)
In order to cover a broad range of circumstances that are of interest
zero.
to economists
depend on
it is useful to allow the misspecif ication indicators to
,
and some nuisance parameters.
8
nuisance parameters.
Let n G
IT
denote a Nxl vector of
Let A (x ,8,n) be an LxQ matrix of misspecif ication
symmetric and positive semi-definite
indicators and let
C
weighting matrix.
Assume the availability of an estimator
T
1/2
'
(x
be an LxL,
,8 ,ir)
such that
8
"
(#„
-
8
o
1
estimator
)
-=
p
(1)
U
is such that T
jr_
T
Also assume that the nuisance parameter
under H_.
(7r_
-
T
tt_)
=
T
(1)
under H
p
rt
,
u
where
(?r _:
T
A
T=l
,
2
,
.
.
.
}
is a
nonstochastic sequence in
II.
need not have
It is because r
an interpretable probability limit under H_ that n is called a nuisance
parameter
A computable test statistic is the Qxl vector
-,
T
1
l
t-1
AAA
A'CA
u
u
(2-10)
"
A
A
where "*" denotes that each function is evaluated at
or
8
1
(
dependence of the summaries in (2.10)
convenience)
.
or.
A
(6',-k')'
1
x
the sample size T is suppressed for
For the conditional mean tests and the heteroskedasticity
tests, A_(x_,£,7r) is the IxQ vector denoted ^(x^,
8 ,?r)
.
From the point of view of simply obtaining tests with known asymptotic
size under H
the p.s.d. matrix C
"
t
could be absorbed into A
.
t
But the
structure in (2.10) is exploited below to generate regression-based tests
with the additional property that they are asymptotically equivalent to
In the examples discussed
better known tests in classical circumstances.
thus far C (x
to allow C
=
,ff,n)
Section
1.
covers some cases where it is profitable
3
to be random.
To use (2.10) as the basis for a test of (2.9),
the limiting
distribution of
m
£_
S
t"
T
1/2
T
AAA
V
£
A'C 6
(2.11)
t t*t
A
under H n is needed.
under H
finding the asymptotic distribution of
In general,
£
entails finding the limiting distribution of
C
-
T"
1/2
I A
t«l
C
'CV
(2.12)
(values with "o" superscripts are evaluated at
limiting distribution of T
1/2
or (f
6
,
-k
'
)'
and the
)
A
(8
-
6
(the limiting distribution of T
)
A
1/2
(.n
A
-
n
)
does not affect the limiting distribution of
under H^)
£
.
Because
£
is the standardized sum of a vector martingale difference sequence under H
n
,
its limiting distribution is generally derivable from a central limit theorem
(provided that {A^'C^<p^} is also weakly dependent in an appropriate sense).
L.
In standard cases T
l_
1/2
l_
A
(.6-
-
6
)
will also be asymptotically normal.
the asymptotic covariance matrices of
differentiability assumptions on
asymptotic covariance matrix of
A^_,
£,,,
o
£T
C_
test statistic is easy to compute.
1/2
and e^.
(0
-
8
)
and
it is possible to derive the
by a standard mean value expansion.
principle, deriving a quadratic form in
distribution is straightforward.
,
and T
Given
A
£„,
In
which has an asymptotic chi-square
But nothing guarantees that the resulting
In certain instances test statistics based on £
simple OLS regressions.
can be computed from
The Newey-Tauchen-White approach can be applied when
a
#
the maximum likelihood estimator and the conditional density of y
is
A
given x^ is correctly specified under H_.
In addition to
<j>
A
,
A
A
and
,
the
C
A
score
of the conditional log-likelihood is needed for computation.
s
The
NTW regression is simply
1
and one uses TR
2
on
A
AAA
S{_,
4>'
t
C
A
t=l,...,T
,
as asymptotically v
2
(2.13)
If interest lies in the case where
•
A
the entire conditional density is correctly specified under H
maximum likelihood estimator of
8
,
and
is
8
the
then the Newey-Tauchen-White approach is
computationally easier than the present approach.
It should be noted,
A
however, that the NTW regression is valid only when
8
is the MLE
,
whereas
A
the procedure described below is valid when
of
6
8
is any Jl- consistent estimator
Wooldridge [31] discusses a C(q) version of the NTW statistic that
.
A
allows
8
to be any
,/T-
consistent estimator.
Another possible drawback to
the NTW regression is that there is growing evidence that it can yield tests
with poor finite sample properties even in the best possible circumstances
(Davidson and MacKinnon [6], Bollerslev and Wooldridge
Neumann [15]).
[1]
,
Kennan and
This is at least in part because the NTW regression ignores
the generalized residual structure in (2.2) in always using the outer product
of the gradient in computing an estimate of the information matrix.
A relatively simple statistic that typically imposes fewer assumptions
A
than the NTW approach is available if |„ is appropriately modified.
that
8
G int(9) and that
<?_
is
differentiable on int(S)
1
1.
.
Assume
Define $ (x
,
6
)
=
1
E V fl^
[
v
t
(y
t
>
x
>#
t
O
)|
x
t
instead of basing
Then,
]•
test statistic on the
a
A
covariance of the weighted misspecif ication indicator C
1/2
A
A
A
"1/2"
generalized residuals C
linear projection
onto
f
j
*
*rt
A
A
$
C
the idea is to first purge from C
,
<f>
A
and the weighted
t
A
A
its
A
= $ (x
where $
,
t
t
t
1/2
,
t'
0„).
T'
That is, consider the
modified statistic
~
L
t
t
T
t
J
t
r
(2.14)
t
where
A
1
A
^,
A
1
t
t-i
T A A A
y *'c a
1
A
y *'c $
B„
c
t
(2.15)
z
is the PxQ matrix of regression coefficients from the matrix regression
"
1/2
C/
A
t
:i/2
C ' *
t
t
on
t-1,.
(2.16)
.,T.
can be written more concisely as
|
"
T
*i :i/2:
[A
= C
where A
-
1/2
$ B
by
C
4>
t
T are the LxQ matrix residuals from the
t=l
],
A
regression (2.16) and
(2.17)
JA't
t=l
m C 1/2
t
A
Note that by construction A
.
<}>
t
1/2
.
It is important to realize that
£
and
£
are not always asymptotically
A
A
eouivalent in the sense that £_
-
r
A
A
[A
-
t
weighted
is
t
£_
i
-*
under H„
o
.
The indicators A
t
and
A
$ B_]
L.
generally vield tests with different power functions.
I
Nevertheless, the robust form of the test almost always has a straightforward
interpretation.
I
return to this issue below.
A
Ever,
when
~^,
and
£„.
are not asympotically equivalent
the basis for a useful specification test.
14
£„.
can be used as
The computational simplicity of a
;
,
limiting x
quadratic form in
Theorem 2.1
(i)
consequence of the following theorem.
is a
f
Assume that the following conditions hold under H
:
Regularity conditions A.l in the appendix;
For some
(ii)
e int(6)
6
o
(a)
E[* (y ,x ,0 )|x
t
t t o
t
(b)
*t
(c)
T
(vV
1/2
(0
" E[
V
]
(y
-
,x
t
lop
-
6
=
)
t-1 2
0,
0(1),
,
f
T
V
1/2
,
|x
t
.
.
]
'
.
t=1
>
2
----;
Tip
(tt_
-
tt°)
=
(1).
Then
,0,0 .,^,0,0
t
T
J
t
r
(2.18)
(l)
o
t
p
t-=l
where
-1 T
c
tii
c
I E[$°'C°A°-
z
In addition,
TP
TR
where R
2
is the
is given
d
2
"*
*Q'
uncentered
on
and B
2
u
r-
squared from the regression
[cy ij rcy (l
2
2
-
;j T
)
t-l,...,T
by (2.15)
Equation (2.18) has a very useful interpretation.
action of
?,
(2.19)
?r,
and E evaluated at the estimators
6^,,
Viewing |_ as
r.^,
:nH B
e Gua t icr.
(2. IS)
demonstrates that the asymptotic distribution of this vector is
un.cn £
id
;
a
when the estimators are replaced by their probability limits.
original statistic {^ does not generally nave this property.
15
Note
.
Theorem (2.1) can be applied as follows:
AAA
A
A
LL-tJ.
- C^, and ^ = C^;
Given A
(1)
C
,
,
<f>
6- and n _,
,
compute A
?/%
=
t
*
t
A
,
tf>
and $
w
l»
t-
i-
A
C
,
.
Define
I—
Run the matrix regression
(2)
on
A
t=l,...,T
*
and save the residuals, say A
(2.20)
;
Run the regression
(3)
1
2
and use TR
- T
on
£'
t-1
A
t t
T
2
-
RSS as asymptotically x n under H
assuming that A
does
not contain redundant indicators
It must be emphasized that condition (ii.b),
EfV
<j>
c
U
(y t ,x t ,6 O )|x
c
"
be computable under the null hvpothesis, can impose
additional restrictions on
t
,8
i_
o
)
that must be satisfied in order for (l)-(3) to
$
be a valid procedure under H n
$ (x
which requires that
.
If additional assumptions are used in forming
then the "implicit null hypothesis" includes more than just (2.9).
But as shown in Wooldridge [30], $^ is always computable under the relevant
null hypothesis for conditional mean (hence conditional probability) or
conditional variance testing in a linear exponential family.
leading
-
but certainly not the only
-
cases where one would like to be
robust agair.st other distributional misspecifications
section
3
These are
.
Example 3.3 in
snows that no auxiliary assumptions are needed to compute
regression-based specification tests of jointly parameterized mean and
variance functions that are robust to nonnormalitv.
16
,
In many other situations $ (x
,
8
computed if some additional
is easily
)
and in many cases standard assumptions are imposed under H
the nonlinear regression example suppose that
where
4>
,x
(y
For example,
E [y
,6)
contains a and any conditional variance parameters.
6
seen to be $
to
(6
-=
)
-3V m (a )v(y |x
ato
tt
m (x ,a)]
-
$
(6
in
3
is easily
)
Most tests for skewness in the
).
literature impose homoskedasticity or some other conditional variance
assumption under the null, and
V(y
|x
)
has been specified.
<J>
(8
readily computed once
is
)
a
model for
Tests for skewness are typically carried out
after the first two moments are thought to be correctly specified.
is
If this
the case, Theorem 2.1 imposes no auxiliary assumptions under the null.
However,
it should be noted that the choice of
by
J the form of $
indicator A
If A
.
t
is
A^_
L
limited in this example
A
A.
is
t
linearly
J related to $
t
then the modified
In a linear model with conditional
is simply zero.
A
A
homoskedasticity and repressors w
,
$
is
cannot contain linear combinations of w
proportional to w
.
t
unity then the choice A t =
1
for unconditional skewness based on
if w
In particular,
F
is unavailable;
Thus
.
t
,
A
contains
this rules out a standard test
^ Newey-Tauchen-White test would
X u Z-t=l
allow more flexibility in the choice of A^ for this example.
consider testing for nonconstancy of the conditional
As another example,
first absolute moment of the regression errors.
E[jy^
!y_
-
n^(>-_.Q
m_(x_.a)|
)
-
differentiabie in
|
jx
k
a,
]
— n
where
> C.
9
The generalized residual is
= (a' ,*)'
.
at (6)
<j>
(y^.x^.tf) =
Although e_(f) is not strictly
it is differentiabie almost surely under the usual
assumptions imposed in these contexts.
is V 6
Under H„
tto
- (liy -m (a
)
>
0]
-
The quasi -gradient with respect to a
tto
lfy -m (a
)
'ato
< 0])V m (a
)
.
Under
.
conditional symmetry of the distribution of y
E(l[y -m (a
t
t
o
Also, E[V
0.
> 0]|x
)
t
= E(l[y
)
)|x
/clol
(8
<j>
=
]
m (o
-
t
t
and * (x
1,
t
o
t
< 0] |x
)
,
6
o
)
is
given x
t)
,
so that
E[V^
simply (0,1).
t
(«
o
If m
)
=
|x
t
]
is the
l.
A
is an M- estimator other than the NLS
conditional mean function and a
estimator (e.g. the least absolute deviations estimator) then conditional
A
symmetry is needed anyway for a
Assumption (ii.c)
to be consistent for a
.
perhaps more properly listed as a regularity
is
condition, but it is placed in the text to emphasize the generality of
Theorem 2.1.
Having
./T-
cons is tent estimators of
8
and
ix
requirement, and allows relatively simple specification tests when
well as n
has-been estimated by
)
assumptions).
an-
weak
is a fairly
(as
8
inefficient procedure (under classical
An application to the tobit model is given in section
3.
The
tobit example has the feature of imposing correct specification of the
conditional distribution under H n but, unlike the usual LM or NTW
regressions, Theorem 2.1 can be used when an estimator other than the MLE is
available
A yet unresolved issue
a
the relationship between £
is
and
£
There is
.
simple characterization of their asymptotic equivalence under H n
The
.
proof of the following lemma follows immediately from the construction of
Lemma 2.2
Let the conditions of Theorem 2.1 hold.
:
I
(iii)
l'
T—
/ 1-\
l/l
a
a
in addition,
a
VC^
I t
t-1
If,
u
-
u
0(1),
p
tnen
C
-
T
£
T
=
o
(2.21)
(l),
p
18
|
.
.
When (iii) holds,
and
£
are asymptotically equivalent under H_
f
Condition (iii) is usefully interpreted as the sample covariance between
"1/2 A
'
$
(C
A
:
t=l,..,T) and
(C
1/2
'
A
:
tf>
t=l,...,T) being asymptotically zero.
It is
trivially satisfied if
T
X *.(*)'C
t-1
U
(0,»r
(0)
-
(2.22)
x
is the defining first-order condition for
6
.
This is frequently the case,
A
in particular when
6
is a quasi -maximum likelihood estimator
(QMLE) of the
parameters of a conditional mean (see Wooldridge [30]) or of the parameters
of a jointly parameterized conditional mean and conditional variance (see
Example 3.3 below).
In these examples (2.21) also holds (trivially) for
local alternatives, so that the difference between the test based on
£
and,
A
say,
the NTW test based on £
is
,
simply that different estimators have been
used for the moment matrices appearing in the quadratic form.
Consequently,
under the conditions required for the classical test to be valid, the two
procedures are asymptotically equivalent under local alternatives
is
achieved without losing asymptotic efficiency.
known asymptotic size under H_
,
;
robustness
In addition to having
the robust test has a limiting noncentral
chi-square distribution even when the auxiliary assumptions are violated
under local alternatives (e.g. heteroskedasticity is present in a dynamic
regression model).
Lemma 2.2 does not directly provide a description of the local behavior
of £„ when (iii) fails tc held under local alternatives, but viewed from a
slightly different angle it provides useful insight.
implies that the quadratic form in
19
£
Note that Theorem 2.1
has an asymptotic chi-square
:
distribution under H n regardless of whether or not (iii) holds; the issue is
has power
how to characterize the directions of misspecif ication that £
Fortunately, it is frequently the case
against when (iii) does not hold.
that £
asymptotically equivalent to some well-known statistic under local
is
This facilitates interpreting
alternatives, when classical assumptions hold.
a rejection when (iii)
fails to hold.
To characterize the local behavior of
f
it is useful to be somewhat
,
more explicit about the nature of the local alternatives.
/
nonstochastic sequences such that JT(6
{#„,:
*
Let
/
-
8
- 0(1) and jT(n
)
and n
6
*
-
n
O
be
- 0(1).
)
T— 1,2,...} indexes the sequence of local alternatives, but, as with
under H_
,
6
need not uniquely index the nonnull probability measure.
the plim of the estimator n
T— 1,2,...}.
6
k
under the sequence of local alternatives (H
is
,
Assume that the conditions of Theorem 2.1 are supplemented with
conditions of the form
%
as T
-<
co
f-liT
v
*
r
^
1
*
.
,
f
J
t=l
for various functions G
o
.
T
i,
,
-l
v
ol
t=l
This corresponds to standard assumptions
in the analysis of the local behavior of test statistics.
The arguments of
Theorem 2.1 can be used to show that under the sequence of local alternatives
Tl-1/2
£
7
i~ -
T
*
B
T
!i
* *
*
I [A_
$_B T ]'
-
* *
C^
+
o^(l)
(2.23)
where
=
^ El£
Lt-i
*
* *
"
I
^-*-J
j
x
***
i
I EL** C i-A
t-i
J
and values with a "*" superscript are evaluated at
20
or (0
,?r
).
Equation
'
is the extension of (2.18)
(2.23)
to local alternatives and implies that the
local limiting distribution of £
their plims
6^.
when
is the same
and n^, provided that JT(8
-
6
T
A
0(1) under (H^)
This implies that if
.
,/T-consistent estimators of
*T1
under (H
A
A
(0
„,7r
)
,
where
|
.
"
,x
(6
(8
Tl
(iii),
=
)
>*
.
are replaced by
A
Tl
and
)
(
VT^
and
(1)
A
A
,n
that
t^) =
-
A
7T
^T2
T2 ^
are b ° th
then
}
(2
is evaluated at (6
in addition,
Suppose,
and n
(1)
^T2 " °p
A
„).
A
8
A
under {H
)
A
6
)
and
24)
evaluated at
is
£
-
A
„
are chosen to satisfy
and w
i.e.
T*
T
1/2
t
I^ t (^ 2 )'C t (J T2 ,; T2 )^(? T2
)
= o
p(
(2.25)
l).
A
Then, by the analog of Lemma 2.2 for local alternatives,
A
where
£
evaluated at
(6
9
).
,7r
=
*T1
-
9
£
o
«=
„
(1)
A
A
is
£
Along with (2.24) this implies that
(2
°p (1)
^T2
'
26)
Conclusion (2.26) is simple yet very
under H n and local alternatives.
A
powerful.
It means that for any Vi-consistent estimator
asymptotically equivalent to
£
6 _..
,
£_..
is
A
A
„
because
£„,„
has been evaluated at an
estimator that satisfies the asymptotic first order condition (2.22).
Whenever such an estimator is available the interpretation of
straightforward:
£
is
£T
is
asymptotically equivalent to the vector that
originally motivated the test statistic, £_, when |„ is evaluated at the
estimator that solves the first order condition.
estimator is used in computing
the interpretation of
|
£
,
It does not matter which
provided that it
is
JI-co-sLs ter.t.
Thus,
does not depend on the estimator used in computing
21
it.
In many situations there is available an estimator that satisfies
(2.22), and typically it solves a well-known problem.
Interpreting £
even
whether or not (iii) holds typically reduces to interpreting an LM-type
statistic in a particular weighted nonlinear regression model or in a model
estimated by MLE under normality.
An example of a case where an estimator
satisfying (2.22) does not have a simple interpretation involves testing for
skewness as discussed above.
In a linear model with homoskedasticity
,
the
estimator that solves the first order condition (2.22) sets the correlation
between the regressors and the third moment of the errors equal to zero.
This method of moments estimator is not particularly easy to interpret.
The reasoning of the previous paragraph is applied to the tobit model in
the next section.
There it is seen that a test for the conditional mean
using, e.g., Heckman's
[12]
two-step estimator, is asymptotically equivalent
to a standard Davidson-MacKinnon [5]
test for comparing two readily
interpretable weighted nonlinear regression models.
The results of Theorem 2.1 and Lemma 2.2 are asymptotic.
Very little is
known about the finite sample performance of the statistics of Theorem 2.1,
especially for nonlinear dynamic models.
It should be emphasized, however,
that even though the regression in step (3) uses unity as the dependent
variable,
these statistics do not necessarily have the same finite sample
biases sometimes exhibited by outer product-type regressions.
Unlike
standard outer product regressions, the robust form does exploit the
generalized residual form cf the test statistic.
Davidson and MacKinnon
[6]
In fact,
the simulations of
for a static regression model and of Bollerslev
and wooidridge [1] for an AR-GARCH model suggest that the orthogonalization
22
2
A
of C
1/2
A
A
A
with respect to C
1/2
:
A
in step (2) improves the finite sample
4>
performance relative to the NTW outer product regression, even under
classical assumptions.
That this might be the case was previously suggested
to me by Peter Phillips.
3
.
Examples of Robust
Example 3.1
be a scalar and let (m (x ,q)
Let y
:
Regress ion- Based Tests
.
a G A)
:
parametric family for the conditional expectation of y
A c R
,
given x
p
be a
,
The null
.
hypothesis is
H
Let {h (x
> 0,
E(y
:
Q
7)
:
|x
t
t
- m (x ,q
)
t
{7
)
,
some a
t'
t
)
t-1
(3.1)
,
estimator such that T
is an
t
i
of V(y
|x
J
e A,
Q
It is not assumed that (h (x ,7):
C T.
}
q
7 e T) be a sequence of weighting functions such that h
and suppose that 7
where
t
or that h (x ,7
to
t
)
t
(7
-
|x
t'
)
=
,7)
(1),
contains a version
7 £ T)
to V(v
is proportional
F
F
J
7
(x
for some 7
)
t
o
e T
.
The researcher chooses a set of weights (h (x ,7 )] and performs weighted NLS
(WNLS)
,
or uses some other 7T-consistent estimator for a
matter which estimator for a
o
is used,
.
However, no
o
the tests are motivated bv the WNLS
first order condition
T
I V a m . (Q)' [y t
fc
-
t_ 1
A general class of diagnostics
iMq)]/M7
t
t
u
T
is
- 0.
)
(3.2)
obtained by replacing 7 m_(a) with a 1x0
vector of cisspecif icatior. indicators evaluated at the estimators
T±
l
A
t
A
A
(a
,7r
T
A
T
)'
[y t
-
23
A
m (a )]/h (7 )
t T
t T
(3.3)
and other nuisance parameters
can contain 7
where n
Theorem 2.1,
6
1/h (7)-
is
It
= a,
<t>A e
m v
)
m
'
t
t
(
A ({,1) = A (a.jr), and C (0,7r) =
t
Q ).
easy to see that computation of $ (x
auxiliary
assumptions under H_
J
,
u
tto
and in fact $ (x
The usual LM-type statistic, which is TR
A
A
A
on
u A/h.
t
t
A
A
2
.0
,
6
requires no
)
-V
)
QttO
(x
ni
.0.
).
from the regression
A
A//h,
t
t
V m /7h
a t
t
In the notation of
.
t-l,...,T,
A
be asymptotically equivalent to the WNLS estimator and that
requires that a
A
h (7
)
be a consistent estimator of V(y
|x
)
up to scale.
The following
A
procedure is valid under H n for any
assumptions about V(y
|x
,/T-
without any
cons is tent estimator a
):
A
(i)
Let q
be a ,/T-consistent estimator of a
A
A
u
,
t
V m (cO
the gradient
b
a t T
and the indicator
,
_
~
:
'
e ;-i/2:
m
V^m^ = h^ /2„
Vjn^,
and A^
x
h„ ' X
t
(ii) Regress A
t
Compute the residuals
.
A
A^*/-\A
A
(a_,7r_).
A
t
T
T
Define u
(iii) Regress 1 on u A
t
u
t
,
;
and save the lxQ residuals, say
on V m
t
= h
••
and use TR
2
= T
-
A
;
RSS from this regression as
2
asymptotically x n under H_.
A
This procedure with h
[6]
=
1
was first proposed by Davidson and MacKinnon
in the context of a nonlinear regression model with independent errors
and unconditional heteroskedasticity
.
It was independently suggested by
Vooldridge [29] for nonlinear, possiblv dynamic regression models with
conditional or unconditional heteroskedasticiry under a martingale difference
assumption on the regression errors.
A
a
Theorem 2.1 further demonstrates that
A
need not be the NLS estimator.
24
The indicator A^ can be chosen to yield LM
tests, Hausman tests based on two WNLS regressions, and tests of nonnested
hypotheses
such as the Davidson-MacKinnon
,
test, which do not require
[5]
correct specification of the conditional variance of y
given x
.
Conditional mean tests in the more general context of multivariate linear
exponential families are considered in more detail in Wooldridge [30].
The estimator that satisfies condition (iii) of Lemma 2.2 is the WNLS
A
estimator based on weights 1/h (7„)
From the remarks following Lemma 2.2,
.
the robust test statistic employing any ,/T-consistent estimator is
asymptotically equivalent to the LM statistic based on (3.3) when (3.3) is
A
evaluated at the WNLS estimator, h
is a ,/T-consistent-estimator of 7
(7
)
G T, h (x ,7
2
= a h (x ,7) where a
:
E(y
|x
t
and 7
),
.
Suppose now, in the context of Example 3.1, the goal is to test
:
whether for some 7
H
|x
For efficiency reasons it is prudent to
.
o
put some thought into the choice of h
Example 3.2
V(y
is proportional to
t
)
=
2
proportional to V(y |x
is
)
is absorbed into 7.
m^x^),
V(y
t
|x
t
)
7
o
%
e k
(3
'
"
4)
e T, t-1,2,....
A
A
Let a
7)
The null hypothesis is
VVV'
-=
Let v (x
).
be the WNLS or some other ,/T-consistent estimator of a
any Jl- consistent estimator of 7
indicators where
6
a statistic or tne
.1
= (a' ,7')'
.
Let
.
A
(x
,6 ,t)
,
and let 7 T be
be a lxQ vector of
Most tests for variances can be derived from
rom
T
A
I
Kl<
"2
-
vj/v_
25
(3.5)
Choosing A(x
,8,
it)
vechfV m (a)'V m
q t
q t
nonredundant elements of
to be the nonconstant,
(a)
information matrix test in the
leads to the White [24]
]
context of quasi-maximum likelihood estimation in a linear exponential
is a lxQ subvector of x
where w
2
(7) = a
When v
familiy (see Wooldridge [30]).
choosing
A
,B,n)
(x
(u
2
(q),...,u
= w
,8,-k)
,
leads to the Lagrange Multiplier test for
,
a general form of heteroskedasticity (see Breusch and Pagan
A
(x
test for ARCH(Q) under
gives Engle's [8]
(q))
Setting
[2]).
a
null of conditional homoskedasticity.
The correspondences for Theorem 2.1 are L
u*(a)
v (7),
-
V v (7).
7
Under H
*-
-V v (7 )
o
7 t
t
{$,*) - l/v?( 7 ).
C
t
ULOL
E[u (a )|x
Note that V
=
]
«=
so that $
<j>
1,
(a' ,7')'
,
(8)
<f>
- -2V m (a)u (a)
($)
uuo
(x
=
8
6
)
-
ptou
- E[V *
(0
)|x
-
]
=
no additional assumptions are needed under H„ to compute
;
o
The choice C
(8 ,n)
e 1/v 2 (7)
in (3.5)
is
motivated by the structure of
the score of the normal log- likelihood with mean function m (a) and variance
"2
function v
(7)
appears in the the variance
In particular, the scaling 1/v
.
tests of Godfrey [10] and Breusch and Pagan [3].
in this context is TR
o
/\ f\
(u
t
2
u
from the regression
°
A
V AV
-
The standard LM statistic
A
A
A
A
VV
Vt/V
t=1 ----- T
C3.6)
-
In addition to (3.4) this test imposes
£[(*0
t-
|x,_]
- k [v 4_(x_,7
C
«-
w
^
o
)]
,
under the null, so that it is ncnrobust.
and Pagan [3],
normality.
(3.6)
is
some k
>
Moreover, as pointed out by Breusch
generally valid only
if 7
J
Breusch and Pagan
[3]
26
(3.7)
o
T
is the QMLE of 7
'
o
under
offer a computationally simple C(a) test
.
that allows 7
to be any 7T- cons is tent estimator of 7
that (3.7) hold under the null.
,
but it still requires
The NTW procedure applied to this case is
valid essentially in the same cases as the usual LM statistic.
The robust procedure obtained from Theorem 2.1 is easy to compute,
A
imposes only (3.4) under the null, and allows 7
estimator of 7
to be any Jl- consistent
.
o
01
A«AAA«
A
A
Let q_ be a
(i)
,/T-
1
cons is tent estimator of a
,
and let 7_ be a
A
,/T-consistent estimator of 7
Compute the residuals u
.
AA
A
V v (7
t
T
and the indicator
),
A
A
7
7 t
V
and
,
t'
(ii) Regress
A
6
A /v
A
v
t
on V v
7 t
t
tt
(fl
the gradient
A«
Define
).
4>
= (u
-
A
v )/ v - u / v
t
t
t
t
1
A
A
V v /v
V v
A
,
t
;
and save the 1x0
x residuals, say
J
- •
2
(iii) Regress
1 on 6
and use TR - T
Y A
6
t t
u
-
A
;
t'
RSS from this regression
as
6
2
asymptotically x n under H_
Interestingly, when v (x
homoskedasticity
Given u
t
,
2
,
so that the null is conditional
the regression in (ii) simply demeans the indicators.
,
and a choice for
a_,
T
a
,7)
A
,
t
the Ay^ statistic is obtained as TR from
u
Q
the regression
A A
on
1
A -
A
T = T
£
A
U
1
L.
where
A
0(V
(uf-
.
-
t-I,...,T
A_)
(3.8)
I
This procedure is asymptotically equivalent to the
traditional regression form (2.8) under the additional assumption that
E[u^_(a
)
|x
]
is constant.
Note that (3.6) and the regression (2.8) usually
yield different test statistics that are not asymptotically equivalent under
27
'
The demeaning of the indicators may not seem like much of a
H
modification, but it yields an asymptotically chi-square distributed
statistic without the additional assumption of constant fourth moment for u
In the case of the White test in a linear time series model,
.
the demeaning of
the indicators yields a statistic which is asymptotically equivalent to
Hsieh's [14] suggestion for
robust form of the White test, but the above
a
statistic is significantly easier to compute.
In the case of the ARCH test, TR
asymptotically equivalent to TR
1
(VV Vl"V
on
(
2
(
2
from the regression in (3.8) is
from the regression
V
The regression based form in (3.9)
a
T
)(u
t= Q +1 ---- T
t-Q"V
is robust to
<
-
3
-
9
)
departures from the
conditional normality assumption, and from any other auxiliary assumptions,
such as constant conditional fourth moment for u
.
Nevertheless, it is
t
asymptotically equivalent to the usual ARCH test under normality.
Example 3.3
Theorem 2.1 can also be applied to models that jointly
:
parameterize the conditional mean and conditional variance.
Again,
and consider LM tests that are robust to nonnormality.
a scalar,
let y
The
unconstrained conditional mean and variance functions are
(M
M
where A C R
E(yJx
tt )
t
(x
t
,5),
» (x ,«):
6 A)
6
t
t
(3.10)
It is assumed that
.
tto
- M^(x_,<5
),
V(v
|x
t
t
)
-
a-
tto
(x
,6
some
),
S
o
€ A.
(3.11)
Take the null hypothesis to be
H
•
5
o
« r(0
o
)
for some
28
6
o
e 6 c R
P
(3.12)
be
where P < M and r is continuously dif ferentiable on int(9)
H (r(0)) and
v
(0) - w
Let m (6)
.
(r(0)) be the constrained mean and variance functions.
A
QMLE is carried out under the null hypothesis.
under H.
and let
,
t
8
T
T
are the lxP gradients
of m
&
A
ttO
and v
under H_
of the quasi-log likelihood evaluated at
s
t
(5)'
V^ t (S)'u t (£)A> t (0
-
Vt (5)
lV t
Evaluating
s
i/«
(*),
t
u>
t
s
2
-
T"
V„m
.
Note that
u>
t
8
= v
t
and
t
and
u
«=
*t
The transpose of the score is
.
(6)'[u (S)
-
w (fi)]/2u£(«)
t
t
u (5)
t
(ff)
t
(fi)^]J
\(S}_- w
t
(3.13)
(3.14)
(«)
J
at r(S) gives
-
A
[V r /i
t
t
v (#)].
1/2
I
t-1
(3.15)
(^)'C (5)^ (^)
(r(0))'
t
|
t
V.w (r(0))'],
C
t
the middle of (3.14) evaluated at v(8)
u^(r(£))
S
2
+ V
l/[2w
t
where A (8)' =
t
[
(r(0))'
s
AAA
'o
.
8
The LM test of (3.12) is based on the unrestricted score
by definition.
m
be the estimator of
8
= r(0_) be the constrained estimator of 7
£„,
A
V„v
Let
and
,
t
4>
(0)
{&)'
is the diagonal
=
[u
(r(0)),
The standardized score evaluated at r(0
s
(r(L))
matrix in
,-1/2
)
is
(3.16)
t«l
C
2
Under H„ and the assumption of conditional normalitv,
TP." from the regression
.ity, TR
e
u
•
on
is asvmotioticaliv v
>
t=l
vnere Q
M
,
-
?
(3.17)
T
is
tne number cr restrictions under
fa_.
Unfortunately, this procedure is invalid under nonnormality, nor does it have
systematic power for detecting nonnormality.
form of the test. In this case,
29
Theorem (2.1) suggests a robust
u
U
2x1
t
t
V
-
'
t
t
Vt
Vt
=
A
2xM
l/v.
t
where u
v
J
t
m
-
t
t
T'
u //v
/v
t
[u£
V
f
*
2xP
tJ
6
-
t
v ]/(72v
t
t
)
,
k
v
t
/(y2
v
rt /7v t
Vt/<V2v t
I
J
are
The transformed quantities
H
(#„,).
v
>
t
l/[2v£]
.
2x2
)
J
The robust test statistic is obtained by first running the regression
A
on
t
$
t=l,...,T
t
and saving the matrix residuals [A
on
1
t'
2
= T
-
t=l,...,T).
:
Then run the regression
t-l,...,T
A
t t
and use TR
(3.18)
(3.19)
RSS as asymptotically x
2
under H 0"
n
.
Note that the
regression in (3.19) contains perfect multicollinearity since A V r(8 „) =
t
where V r(6) is the MxP gradient of
compute a RSS;
b
0,
i
Many regression packages nevertheless
r.
for those that do not, ? regressors can be omitted from
(3.19).
a
Note that the first order condition for 8— is simply
1a
a
a
(0.)
I * c.(*.AP )'C.(Oi
c
u
x
i
-
0,
1
t-l
so that the robust indicator is asymptotically equivalent to the usual LM
indicator.
The matrix regression in (3.18) is the cost to the researcher in
.
guarding against nonnormality
Example 3.4
x
y
Suppose that y
:
random scalar censored below zero, and let
is a
be a lxM vector of predetermined variables from x
Among other things, the tobit model implies that
is the tobit model.
E(y
ly
VJ
IJ >0,x
t
t
'
A popular model for
.
)
t'
— x -a
+ a FV
p(x ,q /a )
o
tl o' o'
tl o
- m (0 )
t
o
(3.20)
and
V(y
>0,x )
t ly t
t
-a*(l
- v
-
(6
t
o
Cx
a /a )p(x
tl o
o
a /a )
tl o o
[p^^/a^h
-
(3.21)
)
where p(-) is the Mill's ratio and
8
ooo
- (a
,a
)
.
Here a
2
is the conditional
o
JO
variance usuallv
associated with the underlying "latent" variable, and w a
J
is
conditional mean of the latent variable.
t
From a statistical standpoint,
the tobit model is no more sensible than
log
yjy t >0,x t
N(x
-
(3.22)
£ ,r^)
tl Q
((3.22) also seems reasonable for many economic applications; see Cragg [4]),
If (3.22) is valid, B
log y
oo
2
and
on
rj
x
L.
can be estimated bv OLS of
.
Li
using only the positive values of y^
Recall that (3.22) implies
.
i.
E(y^_ jy^>0 ,x_)
2
- exp(rj /2 + x
_/5
).
2
c
c
and
V(v
J
t"t >G,X t
ly
)
= [eXD(r?
'
-
2
-o
)
uAB.A
too
-
2
r
' eXD(r; o /2
I1
A
+ X -0
tl o
) ]
(3-24)
A
Let a
A"2
,
T
,
a
T
be any VT-consistent estimators of a
and a
o
2
o
under H A
.
These
include the MLE's, Heckman's [12] two-step estimators, and various WNLS
Define the Davidson-MacKinnon
estimators.
indicator to be the following
[5]
weighted difference of the predicted values from (3.23 and (3.20):
AA
A
A
t
A
where v
A«
= v
A
A
(a
,a
A
-
A
= w
and u
)
AA
A
X^^]
- (v / Wt ){exp[r^/2 +
t
[x^
AA
+ t^p
(x^/a^J
]
}
(3.25)
A «
r/
(j3
)
if the tobit model is true,
Then,
.
A
A
should be asymptotically uncorrelated with
A
u
A
E y
J
t
t
A
A
x ,a_
-
-
tl T
A
,a^/a^,).
o^pix
p
T
tl
T
(3.26)
T
A test which can be shown to be consistent against the alternative (3.23)
A
based on the weighted correlation between u
is
A
and
A
.
Unfortunately, the
A
usual LM statistic is invalid even when the weighting
reason is that the estimators
A
A
(a
a
)
is employed.
1/,/v
The
need not have been obtained from the
weighted nonlinear least squares problem
1
-
min
I (y t
a a t=l
-
x
tl
Q
-
^P(x
A
Q / £7 )) / v
t
tl
(3.27)
-
,
Nevertheless, a statistic is available from Theorem 2.1.
q>
t
(a,o) = y
-
x
tl
Q
-
erp(x -a/cr)
Let
= u (^)
t
A
and let V m^ denote the
1
x (M+l)
L.
A
to a and a,
evaluated at
asymptotically valid:
gradient of x .a + ap(x .a/a) with respect
tl
ti
A
(a^,o„,)
.
Then the following procedure is
T
.
Define
(i)
as in (3.25), u
A
quantities are
The weighted
M
6
A
= A /v
/7v
t
t
A
on
t
and save the residuals
1
V^m
•=
T
t=l
t
A
,
t
A
V„m /7v
/v
t
.
t
T
.
t
on
u
t=l
A
t
2
t't
as above.
A
^ u /v
/7v
V„m
t
t'
u
,
and V m
A
Run the regression
(iii)
u
A
Run the OLS regression
(ii)
and use TR
as in (3.26),
A
A
,
t
,
.
RSS as asymptotically
r
J
J
-
.
.
,
'
2
y,
1
under H_
.
This test takes the null hypothesis to be correct specification of the
conditional density
given x
J of y
J
t b
t
,
i.e.
the tobit model holds under H„.--Tn
particular, it relies on linearity of the conditional expectation in x
conditional homoskedasticity
,
.
and conditional normality in the underlying
latent variable model; it is not intended to be robust to departures from any
of these assumptions.
Instead the test is devised to have power against
departures from the tobit model that invalidate the conditional expectation
(3.20).
Equation (3.20) is of course only one of many consequences of the
tobit model that could be tested.
The test is most useful if interest lies
in determining the effect of explanatory variables on positive values of the
dependent variable
The discussion following Lemma 2.2 implies that (i)-(iii)
is
asymptotically equivalent to the Davids on -MacKinnon test fcr weighted NLS
estimation of (3.20) and (3.23), provided a T and c
estimators.
are Jl- cons is tent
If g„ and a_ are the MLE's then these estimators are more
i
I
efficient than the WNLS estimators that solve (3.27).
33
But Heckman's
[12]
two
.
step estimators yield a test that is asymptotically equivalent to the test
One is essentially doing
employing the MLE's or the WNLS estimators.
specification testing of the nonlinear regression model (3.20) with variance
This makes interpretation of procedure (i)-(iii) straightforward.
(3.21).
It must be emphasized that if the MLE's are available then a
Newey-Tauchen-White regression (2.13), which requires the estimated scores of
the conditional log-likelihood,
The procedure obtained from
is available.
Theorem 2.1 is more flexible albeit less efficient.
A similar test could be based on competing specifications for E(y
that is,
the zero as well as positive observations for y
would require specifying P(y >0|x
Cragg
)
in the competing model
|x
);
This
can be used.
(3.22) such as in
[4]
Before leaving this example, it is useful to note that simple tests for
exclusion restrictions can be developed along similar lines.
unrestricted mean function for E(y
w
X
!y >0,x )
t
t i; t
is
+ X Q
+ ap([x Q
+ X a
Q
tl ol
t2 o2
tl ol
t2 o2 ]
The null hypothesis is that a
„
The
(3
'V-
= 0, which reduces to
(3.20).
"
28)
The indicator
A
is now the gradient of (3.28) with respect to a
A
evaluated at the
restricted estimates:
A
A
t
= x
A
+ c„V p(x ,a„,,/a„,)x .
t2
T z
t^ Tl
T
t2
.
A
where V p(-) is the derivative of the Mill's ratio.
z
When this
X
t
is
used in
(i)-(iii) a test asymptotically equivalent to the LM statistic in the context
A
of WNLS is obtained.
a
,
ol
and a
Again, a„,
A
and c
.
o
34
are any Jl consistent estimators of
4
Conclusions
.
This paper has developed a general class of regression-based
specification tests for (possibly) dynamic multivariate models which, in the
leading cases,
imposes under H_ only the hypotheses being tested (correctness
of the conditional mean and/or correctness of the conditional variance)
The
.
framework can be applied to testing other aspects of a conditional
distribution under a modest number of additional assumptions.
It is hoped
that the computational simplicity of the methods proposed here removes some
of the barriers to using robust test statistics in practice.
The possibility of generating simple test statistics when
T
1/2
'
A
(6
-
6
has a complicated limiting distribution should be useful in
)
several situations.
The tobit example in Section
is only one case where
3
the conditional mean parameters are estimated using a method other than the
efficient WNLS procedure or the even more efficient MLE.
is to choosing
Another application
between log-linear and linear-linear specifications.
case, both models can be estimated by OLS
,
In this
and then transformed in the manner
of the tobit example to obtain estimates of E(y
]x
t
t
)
for the separate models.
Theorem 2.1 applies directly to linear simultaneous equations models
(SEM's).
Computation of $
(x
t
,6
t
o
)
is
straightforward provided that the
reduced form for the endogenous right hand side variables is available.
parameter vector
8
The
contains the reduced form parameters of the relevant
endogenous variables as well as the structural parameters in the equation(s)
If all equations of a simultaneous system are being tested
of interest.
jointly then
6
is simply all structural parameters.
35
An immediate application
is to testing for serial correlation in linear dynamic
simultaneous equations
models in the presence of heteroskedasticity (conditional or unconditional)
of unknown form.
Also,
tests for multivariate ARCH in SEM's that are robust
to heterokurtosis are also easily constructed.
The scope of applications to
nonlinear SEM's is limited by one's ability to compute $ (x
E[V
4>{y
u
,x
t
t
,6
O
)|x
t
].
,
6
)
=
This is exactly the problem of computing the optimal
instrumental variables for nonlinear SEM's.
Theorem 2.1 can be extended to certain unit root time series models.
The initial purging of C
effectively stationary.
1/2
A
4
from
m
./-.A
1/2
C
A
can produce indicators A
that are
This happens for the LM test in linear time series
models when the regressors excluded under the null hypothesis are
individually cointegrated (in a generalized sense) with the regressors
included under the null.
In this context the statistics derived from Theorem
2.1 have the advantage over the usual Wald or LM tests of being robust to
conditional heteroskedasticity under H
Extending Theorem 2.1 to general
nonstationary time series models is left for future research.
36
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Tauchen, G. Diagnostic testing and evaluation of maximum likelihood
models. Journal of Econometrics 30 (1985): 415-443.
22.
White, H. A heteroskedasticity-consistent covariance matrix estimator
and a direct test for heteroskedasticity Econometrica 48 (1980)
817-838.
,
.
23.
White, H. Nonlinear regression on cross section data. Econometrica 48
(1980): 721-746.
24.
White, H. Maximum likelihood estimation of misspecif ied models.
Econometrica 50 (1982): 1-26.
25.
White, H. Asymptotic Theory for Econometric ians
Fress, 1984.
26.
White, H. Specification testing in dynamic models. In T.
Advances in Econometrics - Fifth World Congress Vol. I,
Cambridge University Press (1987): 1-58.
New York: Academic
.
,
Bewley (ed.),
New York:
27. White,
H. and I. Domowitz. Nonlinear regression with dependent
observations. Econometrica 52 (1984): 143-162.
28.
Wooldridge J.M. Asymptotic properties of econometric estimators
Ph.D. dissertation, 1986.
,
38
.
UCSD
29. Wooldridge, J.M. A regression-based Lagrange Multiplier statistic that
is robust in the presence of heteroskedasticity
MIT Deparment of
Economics Working Paper No. 478, 1987.
.
30.
Wooldridge, J.M. Specification testing and quasi -maximum likelihood
estimation. MIT Department of Economics Working Paper No. 479, 1987.
31.
Wooldridge, J.M. On the application of robust, regression-based
diagnostics to models of conditional means and conditional variances.
Manuscript, MIT Department of Economics, 1988.
39
1
,
Mathematical Appendix
For convenience,
I
include a lemma that is used repeatedly in the proof
of Theorem 2.1.
Lemma A.
:
Assume that the sequence of random functions (Q (w
T— 1,2, ...}, where Q T (w
R
p
,
e 8,
0):
is continuous on 8 and 8 is a compact subset of
•)
G 8, T=l 2
and the sequence of nonrandom functions {Q (0):
,
,
.
.
.
)
satisfy the following conditions:
(i)
sup |Q (w
0e8
(ii)
Let
{Q
(0):
0)
-
Q (0)]| B 0;
e 8, T=l
,
2
,
.
.
.
}
is continuous on 8
be a sequence of random vectors such that
uniformly in
-0*0
_
T.
where (0„!
Then
C 8.
Q T (w T ,0 T )
Proof:
-
Q T (0°) 2 0.
see Wooldridge [28, Lemma A.l, p. 229].
A definition simplifies the statement of the conditions.
Definition A.l
A sequence of random functions
:
(q
(y
,x
,0):
6 8,
t— 1,2,...}, where q^(y ,x ,-) is continuous on 8 and 8 is a compact subset of
t
t
t
R
P
,
is
said to satisfy the Uniform Weak Law of Large Numbers (UWLLN) and
Uniform Continuity (UC) conditions provided that
(i)
sup
0£8
1
IT"
T
Y q^(y x 0)
^
^
C-l "
E[q (y .x .0)]| 2
-
and
T
1
(ii)
{T " Y E iq^(>V.^ .*)]:
Z
Z
Z
t-1
continuous on 8 uniformly in T.
U0
8
e 6, T-1,2,...} is 0(1) and
1
;
In the statement of the conditions,
variables y
a(9)
the dependence of functions on the
is frequently suppressed for notational convenience.
and x
is a lxL function of the Pxl vector
then, by convention, V a(0)
6
If
is the
8
LxP matrix V {a(8)'].
If k(6)
QxL matrix then the matrix V A(0) is the
is a
P
8
LQxP matrix defined as
h(ey
v
-
[Vfaier
g
where A. (8) is the
j
IVq
•••
I
°'
th row of A(0) and V.A.(B)
as defined as above.
]
gradient of A. (9)
is the LxP
0J
J
derivative of
(
J
Also, for any Lxl vector function
<p,
define the second
to be the LPxP matrix
<p
^<P<*) - V [V^(*)'].
fl
Finally, define the parameter vector
Conditions
(i)
A.
(iii)
that
P
and
c R
II
e int(6),
6
b
(#' ,»r')'
(a)
are compact and have nonempty interiors;
U°: T-l
(y ,x
[<f>
N
,8):
,
2
,
.
.
}
c int(II) uniformly in T;
is a sequence of Lxl
e 6)
9
.
t
functions such
ft (y t ,x t ,•)
6 6 and
is Borel measurable for each
4>^{- ,6)
.
:
6 c R
(ii)
S
continuously differentiable on the interior of 6 for all y
(b)
int(8)
.
Assume that
<*>
t
)
(x
is continuously
interior of 6 for all x^
(c)
ppttwOUJ
ttO
Define 9Ax.,B
,
(C j_(x j_,c):
,.•)
= E,
,
t=l
,
2
,
.
.
.
)
|x
for all
t=l
6
o
,
2
,
.
.
.
e
differentiable on the
t
t-1,2,...;
S
semi-definite for all x_ and
t
.
,
o
£ A} is a sequence cf LxL matrices satisfying
the measurabiiity requirements, C (x ,6)
for ail x
x
[V,i (y
,x
is
5.
and C (x
;
41
symmetric and positive
is
,
•
)
is
differentiable on int(A)
;
(d)
(x
(A
,
;
,8):
e A) is a sequence of LxQ matrices satisfying
8
the measurability requirements, and A (x
for all x
(iv)
,
t-1,2,
T
(a)
T
(b)
(v)
.
1/2
.
is
)
differentiable on int(A)
.
To6)
=
tt°)
=
(0„
-
1/2
(tt
-
t
(1);
p
0(1);
(6)* (0)) and {$ (0)'C (5)A (5) satistfy the UWLLN
($ (0)'C
(a)
•
and UC conditions;
(T
(b)
T
-1
'"
Y,
E[$
(vi)
l$ (0)'C
(a)
t
{* (9)' [I.
t
(0'
[I
t
Z
(5)V^
t
T"
(a)
1/2
t
is uniformly positive definite;
]}
(0)},
{[I
-
r
I 9>°'C%°
^i t t t
p
(6)V^
{A (5)'C
t
t
(A
C $
®
p
t
(0)'C («)]V *
t
fl
(fl)}
t
and
I
t
(0)},
(1);
"
v
{
® ^.(0)' C
[I
® * («)' ]V C (6)}, and (* (0)'
t
5 t
t
L
^
® ^ (0)' ]V C (5)} satisfy the UWLLN and UC conditions;
(b)
(vii)
'
*~
t-1
t
(S)
® 4>J6)'
[^
JV^*)'
]^C t (fi)
)
}
satisfy the
UWLLN and UC requirements
T
i
(viii)
{E° - T"
(a)
1
I E[(a!-*°b!)'cV^'c!(aJ-*°b2)]} is uniformly
P-d.;
H^ 1 / 2 !" 1 / 2
(b)
(c)
(A
t
I-
L.
=
t
^
N(0,I
Q
);
J.
(5)'C (5)^ (0)^ (0)'C (5)A (5)},
t
t
t
[A_(S)' C^(8)<f>^(8)4>(6)'
t-
-*° B?)'C°<
I (A°
<,
t
C^(8)§(6))
U
t
,
t
and
t
{<J-^(f)'
L.
satisfy the UWLLN and UC conditions.
42
C_(5)^_(^)^_(£ )'C_(5)* (0)}
;
t
t
t
t
t
,
Proof of Theorem 2.1
First, note that assumptions (i)-(vi) ensure existence
:
lip
of B„ and imply that B_
T
B_
-
o
•=
Lemma A.l.
(1) by
Therefore,
T
rT
-T-^z
[a
*
(L
-
(-D
v°]'v t
-
t=1
b°)-t"
-
1/2
T A A A
y J'c J
.
A
Consider the term post-multiplying
value expansion about
6
,
B
-
)'
.
A standard mean
assumption (vi.a), and Lemma A.l yield
r l/2 Al $°'CV
r
t
t t
1/2
J'C i I
^, t t*t
t=l
T'
(B
(a.
v
2)7
t-1n
T
T-2 {^'[l L »*°']VjC°}T 1 / 2 (« T -« o
+
)
+
(l).
o
p
i
The first term on the right hand side of (a. 2) is
(1)
and (iv.a,b), the terms in lines two and three of
(a. 2)
by (vi.b).
By (vi.a)
are also
(1).
P
Therefore
1/9
A
A
A
'
1 *'C
6
T
-
(a. 3)
(1).
P
t-1
A
Along with B_
B_ - o (1),
-
r
L
p
r
T
- T" 1/2 X [A
t-1
-
this establishes that under H n
u
*
B°]'C^
„-l/2r'
-
_1
+ T
T
- 1
r
0(1).
+
(a. 4)
P
A mean value expansion, assumption (vii)
C
,
o
I [A t
-
,
and Lemma A.l yield
_c_o,,_c.o
* B ] 'C 4
t T
t t
ttTtet
X . ([a!-^b!]'C°V,c^
-
IP
B°' ri_ ®
43
,
CN
(a. 5)
*!'C°]V,$°} T
t
t
6
t
i/2
(fl
Ti
-
B
o
)
,
^
t=l
•T
+
1/2
(5
-
6°)
T
(l)
o
p
Consider the second line of
appearing there is
o
under H.
(1)
It must be shown that the average
(a. 5).
First, note that by definition of $
.
and
the law of iterated expectations,
E([A°-*°b£]'C°V^°) - E{E([A°-f°B°]'C°V^°|x
L
t T
t
J
t
t
]))
(a. 6)
-
(a. 7)
t
Therefore
I-^E([A°.t°B°,-C°7^°,
by definition of B
T-^EUA^XrcV,
-
The regularity conditions imposed imply that each of
.
the averages appearing in (a. 7) satisfy the WLLN.
_1
Y
I [A t -$°B°]'C°V^
~
t t
t e t
T
L
Because Ef^^Jx
t
t
]
j
=
sample averages in
under K
=
(a. 8)
(1).
p
it is even easier to show that the remaining
,
u
are o (1).
(a. 5)
o
Therefore
Combined with T
(5_
T
p
-
5_)
T
=
p
(1)
this
establishes the first conclusion of the theorem:
l
T
- T"
1/2
J
[A°
-
+ o (l).
p
*l*°]'Cyt
Given (viii.a), the asymptotic covariance matrix of
definite.
Moreover, E_
(viii.c) ensures that
x
tT
-*
N(0,I
)
(a. 9)
£
is
uniformly positive
under H- by (viii.b).
Condition
-•JT
T"
AAAAA
AA
A
I [(A t
t-1
-
*t
C
V' tVt
is a consistent estimator of S_.
CT rT
where R
2
is the
(f>
and A
regression
-
"
t
]
(a 10)
-
It is easy to see that
Tr2,
(a. 11)
uncentered r-squared from the regression
1
and
\
t
W
AA
(A
C
on
J'
A
t-1
t t
are as defined in the text.
(a. 12)
is unity,
squares from the regression
TR
2
- T
(a. 12).
45
-
T,
(a. 12)
Because the dependent variable in
RSS, where RSS is the residual sum of
26 10
UI6
I _7 7
-
*i
6-5"-© 1
Date Due
1
2
1989
^26 199V
ll
Lib-26-67
MM
3
L
[BRARIES
TOAD DOS MOT
fl^fl