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utzWtY
working paper
department
of economics
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
FLEXIBLE SIMULATED MOMENT ESTIMATION OF
NONLINEAR ERRORS -IN -VARIABLES MODELS
VHITNEY K. NEWEY
Massachusetts Institute of Technology
93-18
Nov.
1993
mTt. LIBRARIES
DEC -7 1993
RECEIVED
Flexible Simulated
Moment Estimation of
Nonlinear Errors-in-Variables Models
Whitney
K.
Newey
MIT
October, 1989
Revised, November 1993
This research was supported by the National Science Foundation and the Sloan Foundation.
David Card and Angus Deaton provided helpful comments.
-
1
-
1.
Introduction
Nonlinear regression models with measurement error are important but difficult to
estimate.
(e.g.
Measurement error
is
a
common problem
in
discrete choice) models are often of interest.
are not consistent for these models, as discussed
approaches must be adopted.
microeconomic data, where nonlinear
Instrumental variables estimators
in
Amemiya
The purpose of this paper
is
(1985), so that alternative
to develop
cin
approach based on
a prediction equation for the true variable, that uses simulation to simplify
computation.
The approach allows for
flexibility in the distribution being simulated,
and could be used for simulation estimation of other models.
The measurement error model considered here
Hausman, Ichimura, Newey, and Powell
(1991)
is
the prediction model analyzed in
and Hausman, Newey, and Powell (1993).
model has a prediction equation for the true regressor with a disturbance that
independent of the predictors.
This previous work shows
how
This
is
to consistently estimate
polynomial regression models, and general regression models via polynomial
approximations.
This paper avoids polynomial approximation by working directly with
certain integrals, estimating them by simulation methods.
Other work relies on the
assumption that the variance of the measurement error shrinks with sample
Wolter and Fuller (1982) and Y. Amemiya (1985).
This approach
is
size,
applicable
including
when there
are a large number of measurements on true regressors, but this situation does not occur
often in econometric practice.
Flexibility in distribution of the prediction error is desirable, because
consistency of the estimator depends on correct specification.
meinage computation costs, so that the estimator
models.
is
It
is
also important to
feasible for a variety of regression
These goals are accomplished by combining simulated moment estimation with a
linear in parameters specification for distributional shape.
Simulated moment estimation
provides a convenient approach when estimation equations are integrals, e.g. Lerman and
Manski
(1981),
Pakes (1986), and McFadden (1989).
- 2
This approach uses Monte Carlo methods
to
form an unbiased estimators of integrals
distribution
in
moment
incorporated by multiplying by a linear
is
equations.
in
Flexibility in
parameters function that
approximates the ratio of the true density to the simulated one.
This approach
is
similar to the importance sampling technique from the simulation literature.
Section 2 describes the errors in variables model and some of its implications for
conditional moments.
Section 3 lays out the estimation method and discusses parametric
Section 4 gives a semiparametric consistency
asymptotic inference for the estimator.
result.
Section 5 presents results of a small Monte Carlo study.
Section 6 describes an
empirical example of Engle curve estimation of the relationship between income and
consumption.
The Model
The model considered here
(2.1)
y = f(w .5q) + C.
EIClx.v] = 0.
w
E(t)|x,v,(^]
-
w
where
cr
is
y
w
+
T),
= n'x + (TV,
and
C,
are conformable matrices.
errors, and
w
unobserved.
The
S
,
Here
observed regressors.
0,
independent of
V
are scalars,
=
w
,
w
w,
t),
x,
x.
and
are vectors,
v
represents true regressors,
The
x
are observed and
w
,
tj,
and
last equation is a prediction equation for the true regressors,
are observed predictor variables,
v
is
an unobserved prediction error,
scaling matrix, a square root of a variance matrix.
allowed to be observed, with
w
and
ti
measurement
t)
i!;,
and
and
<r
may
v
where
is
x
a
Some of the true regressors can be
equal to an element of
x,
by specifying that
be
corresponding elements of
element of
tt'x
w
is
cind
t)
are identically zero, and the corresponding
v
a-
This model was considered by Hausman, Ichimura, Newey, and
= w.
•
Powell (1991) (HINP henceforth), for the special case where f(w
,5)
is
a polynomial in
»
w
e.g.
x
As long as
.
E[v] =
as
includes a constant, the location and scale of
Var(v) =
and
when the second moment of
I
v
v
can be normalized,
exists.
Instrumental variables (IV) estimators can be used to estimate this model when
f(w
is
,5)
w
linear in
Substituting
.
w-tj
be valid instruments, because the disturbaince
w
for
is
in the first
linear in the
equation leads to
measurement error
the nonlinear case this substition leads to residuals that are nonlinear in
x
Consequently,
An approach
Let
error.
will not be valid instruments,
be the density of
gp>(v)
E[y|x] = /f(7r^x +
(2.3b)
E[w.y|xl = Sin^x +
(2.3c)
E[w|x] =
first is a regression of
on
X,
that
tj.
and another approach has to be adopted.
o-^v,
Integrating leads to three equations:
v.
5Q)gQ(v)dv,
cr^v]f(n'^x + cr^v,
d^)g^{v)dv,
ir^x.
unobserved variable
wy
In
to consistent estimation can be based on integrating out the prediction
(2.3a)
The
t).
x
v
is
y
on
x,
analogous to the usual one, except that the
has been integrated out.
less familiar.
The second equation
The third equation
is
is
a regression of
a standard regression
equation.
The second equation
is
important for identification of nonlinear models.
components of this equation correponding to unobserved
w
(i.e.
those not corresponding
to observed covciriates) provide information additional to the first equation.
in
HINP for polynomial regression, the
identification.
Intuitively, there are
As shown
first equation does not suffice for
two functions that need
regression function and the density of
The
v,
so that
- 4 -
to be identified, the
two equations are needed for
identification.
It
was shown
HINP that the parameters of any polynomial regression
in
equation are identified from these two equations,
sind
one expects that identification of
the regression parameters will hold more generally.
is
It
beyond the scope of this paper to develop fully primitive identification
conditions for this model, but some things can be said.
from
identified
so
distribution with a finite support and
b) provide
2m
m
2m
x,
w
case where
nonsingular.
Also,
has a discrete
i.e.
that a "rank condition"
paraimeters from these equations, including
if
f(w
a scalar and
is
parameters are identified
,5)
v.
is
moment matrix
the second
some of the moments of
For example, HINP showed
a polynomial of degree
of
are identified
v
should be identified.
Assuming that the
distinct functions, these equations give
functions to be identified.
So,
left
and
6
(I.ti' x,...,(7i'
x)
in this case.
If
is
ti'
,5
the
p,
)'
f(w
a continuous distribution, then a simple counting airgument suggests that
(v)
on
points of positive probability, then (2.3 a -
parameters of a parametric family of distributions for
in the
ir'x
If
Assuming that none are redundant,
equations.
holds, one could identify
g
are
ti
w
(2.3c) as the coefficients of a least squares regression of
considered by focusing on equation (2.3a) and (2.3b).
5
the parameters
can be treated as known and identification of the other pieces of the model
n
that,
First,
has
x
)
and
hand sides of (2.3a) and (2.3b) are
two functional equations, and there are two
by an analogy with the finite dimensional case,
it
should be possible, under appropriate regularity conditions, to identify both the
w
regression function for
and the density function for
v.
Making this intuition
precise would be quite difficult, because of the nonlinear, nonparametric
functional) nature of these equations, but
it
is
cin
(i.e.
important problem deserving of future
attention.
Independence of
model of equation
moments
of
v
(2.1)
and
x
it
is
v
is
a strong assumption, but in the general nonlinear
difficult to drop this assumption.
can depend on
x,
then
it
is
much more
regression function from the distribution.
- 5 -
Intuitively,
if
some
difficult to separate the
3.
Estimation
To describe the estimator
moment
conditional
parameters,
g
vector, and
Let
setup.
z
denote a data observation,
a density function of a random vector
H(z,p,v)
a
r
x
1
v,
in a
q x
a
p
more general
a
p(z,p,g)
r
1
x
vector of
1
residual
vector of functions, related as in
p(z,p.g) = J-H(z,p,v)g(v)dv.
(3.1)
Suppose that there
parameter value
is
and density
p
x
a set of conditioning variables
g
such that for the true
,
E[p(z,^Q,gQ)|x] = 0.
(3.2)
The nonlinear errors-in-variables model
is
a special case of this one, where
H^(z,p,v) = y - fdr'x-KTV.S),
(3.3)
H
(z,3,v)
=
H2(z,P,v) =
and
embed the model
helpful to
is
it
L
is
Uw.y
w
- [Tr'x+(rvl«f(n'x+(rv,6)),
- tt'x,
(3
= {d'.a-.n'V.
a selection matrix that picks out those elements of
w
that include
measurement error.
The common approach to using equation
variables.
One difficulty with this approach
Another difficulty
compute.
(3.2) in estimation is nonlinear instrumental
is
that the residual
is ein
is
that the density
integral that
may be
g(v)
is
unknown.
difficult to
Here, these difficulties are dealt with simultaneously, by choosing a
flexible parameterization for the density that
estimator of the integral.
To describe
makes
it
this approach,
the density function.
- 6 -
easy to use a simulation
we begin with
a specification of
For now, suppose that the density
g(v,r) =
(3.4)
where
and
^(v)
is
= V
p.(v)
member
a
is
of a parametric family, of the
P(v.r) = Zj^^yjPj(v).
P(v,yMv),
some fixed density function.
For example,
if
^(v)
then this would be an Edgeworth approximation.
,
form
need not be positive, but leads to residuals that are linear
in
were standard normal
The function
g(v,y)
the shape parameters
y
and that can easily be estimated by simulation.
For a density
like that of
equation (3.4), a simulated residual can be constructed
by drawing random variables from
combination
[v. ,...,v.
H
and the
P(v,y)
y(v)
is
<p{v)
if
observation
pAe) =
(3.5)
is
denote a single observation and
[v.
v.
]
could be computer
Then an estimator of the residual
s~V
^ =^,H(z.,3,v.IS )P(v.is ,r),
p{z.,P,g{y))
for
e = O'.y')'.
1
essentially an importamce sampling estimator of the residual,
sampling density and
ip(.v).
is
1
1
This
z.
a standard normal pdf, then
generated Gaussian random numbers.
i
Let
functions.
denote a vector of random variables, each having marginal density
]
For example,
the
and then evaluating the product of the linear
P(v,y)
approximates
g(v)/^(v).
where
^(v)
The simulated residual
is
is
an
unbiased estimator of the true residual, because
E[p.(e)|z.] = p(z.,p,g(y)).
Therefore, by the results of McFadden (1989) and Pakes and Pollard (1989), an
instrumental variables (IV) estimator with
p.O)
as the residual should be consistent
1
if
the IV estimator with the true residual
familiar way.
estimated.
Let
A(x)
Suppose that
denote a
9
q x r
is.
An IV estimator can be formed
in
vector of instrumental variables, that
solves
- 7 -
a
may be
n ^Ij"jA(x.)3.(e) =
(3.6)
This
is
a simulated, nonlinear IV estimator like that of McFadden (1989).
Because equation (3.6)
density
0.
linear in
is
to integrate to one.
P(v,y)^(v)
and scale normalization on this density.
P{v,9r),
it
Also, it
is
important to normalize the
may be important
There are different ways to impose
For example,
normalizations by imposing constraints on the coefficients.
the standard normal density and
pJv), p„(v),
...
cu'e
=
for
j
*
k),
then
integrates to one, and has zero
P(v,y)v(v)
=1,
y
9'^
mean and
^(v)
if
is
the Hermite polynomials that are
orthonormal with respect to the standard normal density
J'p.(v)p, (v)^(v)dv
to impose a location
=
2
(i.e.
Jp.(v) <p{v)dv =
and
0.
=
y
unit variance.
1
and
will imply that
is
It
also possible
to impose such constraints using the simulated values, by requiring that
,v? )P(v. ,r) =
I-",L^,(l.v.
^1=1^=1
IS
is
0.
IS
In the nonlinear errors in variables model, it is convenient to
estimator, where the first step consists of estimation of
tt
work with a two step
by least squares (LS), and
an instrumental variables estimator using the first two residuals of
the second step
is
equation (3.3).
The
first order conditions for such cin estimator can be formulated as a
solution to equation (3.6), if A(x)
is
chosen
in
a particular fashion.
Let
tt
be the
LS estimator and
A(x) = diag[B(x),x],
(3.7)
where
(.8' ,<r,Tf' ).
y
_.P(v.
has two columns and number of rows equal to the number of elements of
B(x)
,3r)
Suppose that constraints are imposed on the
=
0.
Then the solution
equation (3.7), requires that
ir
y
to equation (3.6), with
coefficients such that
A(x)
specified as in
be the least squares estimator, and that the other
paremieters solve the equation
- 8 -
n A
=
(3.8)
a =
j:.;:^B(x.)p.(a),
iS' ,a;r'
)'
f(Tr'x. -HTV.
y;
p.(a) =
1
equation as
min
Specifically,
The
a
s
,5)
P(v. ,r)
lis
L(7i'x.-Hrv.
IS
)f (ti' x.+o-v. ,6)
1
IS
order condition, although the normalization
for
C(x)
a
solves
a positive definite matrix,
(3.9)
1
example the estimator minimizes a quadratic form that has this type of
its first
not imposed.
ri
s=l
Lw.y.
In the empirical
1
T
=
-i^'^- '^^
is
equal to a vector of instrumental variables and
W
n A,
[r.",C(x.)p.(a)]' W[j:.",C(x.)p.(a)]
^1=1
^1=1
1
1
1
1
first order conditions to this minimization
problem are as given
equation (3.8),
in
with
n
B(x) = [ar. ,C(x.)p.(a)/aa)'W.
^1=1
(3.10)
1
1
Standard large sample theory for IV can be used for asymptotic inference procedures.
If
the simulated values
(v.,,...,v._)
observation for the
data point, then the usual IV formulae can be used to form a
i
consistent variance estimator.
independent observations as
see
are included with the data to form an augmented
iS
il
Newey and McFadden,
For example, suppose that
i
vairies.
(z.,v. ,...,v.
are
)
Then under standard regularity conditions
1994), the asymptotic variance of
^11(8
-
6
(e.g.
can be estimated
)
by
V = G'^nc'^'
(3.11)
,
G = n"V.",A(x.)ap.(e)/ae,
^1=1
1
If
the simulation draws
given
i,
v.
1
111
n = n"V.",A(x.)p.(e)p.(e)'A(x.)'.
^1=1
are mutually statistically independent as
then one could also use the estimator
- 9 -
s
1
varies for a
,
V = G"^nG
(3.12)
A.
1
n =
^',
n~^j;."^A(x.)A.A(x.)'
= s"t^,H(z.,p,v.
*^=1
1
)H(z.,p,v. )'P(v. ,y)^.
IS
IS
1
IS
Both of these variance estimators ignore estimation of the instruments, which
Because the large sample theory for these
under standard regularity conditions.
estimators
we do
straightforward,
is
is valid
not give regulcirity conditions here.
Consistent Semiparametric Estimation
If
the functional form of the density
unspecified, then the
Models where identification
model becomes semiparametric.
moment
is left
gf^(v)
is
acheived by conditional
restrictions like those of equation (3.1) are nonlinear, nonparametric
simultaneous equations models.
Newey and Powell
such models, and their result can be applied here.
previous estimator, but with
(1989) have considered estimation of
The basic idea
is
to apply the
chosen to be a member of an increasing sequence of
P(v,y)
approximating families and the IV equation
(3.6) replaced
by a nonparametric conditional
expectation equation.
&
Let
density
be a set of functions of
gf>(v)
{P (v.y)}
function.
and satisfy other regularity conditions given below.
be a sequence of families such that
Let
Euclidean vector
(3.5)
with
&. = <P.(v,y)y(v)}n§',
p
Pj(v,9f)
and a density
replacing
9 = 0,g)
g,
P(v,9r).
of the conditional expectation of
estimator.
that will be assumed to include the true
v
p. (6)
and
Let
P.(v,3')y(v)
be the simulated residual of equation
E[p.(e)|x.]
x.,
Then a minimum distamce estimator of
- 10
can approximate any
be the peirameter consisting of the
p.(0)
given
Also, let
be a nonparametric estimator
such as a series or kernel
9
= 0-,g^)
is
„ „
e = argmin^
(4.1)
Q(e) =
Q(e);
oCioXj';
j;."
11
11
DEl^.O)
E[p.(e) x.]'
|
1—1
|x.l/n.
n
D
where
is
a positive definite matrix and
The objective function
size.
equation
in
CEin
J
(4.1)
depend on the data and on sample
a sample analog of
is
Q(6) = E[E(p(z.0)|x]'DE[p(z.e)|x]],
D
where
and
e
is
is
the limit of
identified
p(z,9) = jH(z,P.v)g(v)dv.
and
If
from the conditional moment equation
then
a unique solution),
D
Q(e)
extremum estimator reasoning
is
(i.e.
positive definite
that equation has
have a unique minimum of zero at
will
Newey and McFadden,
(e.g.
(3.2)
D
8
The general
.
1994) then suggests that
should be consistent.
sets,
The estimator can be shown
to be consistent
similarly to Gallant (1987).
The compact function set assumption
but the results of
Newey and Powell
if
and
p
g
are restricted to compact
(1989) indicate its importcince for
estimators of the form considered here.
A =
For a matrix
[a..],
let
is
a strong one,
minimum distance
=
IIAII
ij
[trace(A'A)]
1/2
,
and for a function
g(v)
let
denote a function norm, to be
llgll
further discussed below.
Assumption
4.1:
p
6 S,
which
is
compact, and
In the primitive regularity conditions given below,
The following dominance condition
Assumption
M(z,v),
4.2:
There exists
M(z,v)
g
e ^,
will be a Sobolev
llgll
will be useful in
such that for
a compact set in a
0,
norm
norm.
showing uniform convergence.
g € B,
IIH(z,^,v)ll
s
IIH(z,p.v)-H(z,^.v)ll < M(z,v)llp-pll.
Moment conditions for the dominating function
To show uniform convergence
M(z,v)
will be specified below.
of the objective function of equation (4.1),
useful to impose a strong condition on function norm, that
-
11
-
it
it
is
dominates a weighted
llgl
supremum norm.
llglL,
V,W
Below
will be
it
V
Let
denote the support of
= sup,,|g(v)|tj(v).
cj(v)
<p{v),
and
0.
>
V
assumed that
UglU,
dominated by
is
llgll,
so that Assumption 4.1
V,(*)
"gUy
implies that
on the
faster
in
behavior of
tail
moves outwcird,
v
order to guarantee that
restriction on
this assumption is a
the faster the tails of
sup^< |g(v)|tj(v)}
sampling imposes a restriction on the
baseline density.
The import of
&.
uniform bound
imposed by the presence of the weight function; the
g(v),
grows as
w(v)
bounded on
is
is finite.
must go to zero
g(v)
Also, the nature of importance
tail thickness of the
true density relative to the
For second moment dominance this restriction will translate into a
w(v)
relative to
<p(v).
These considerations lead to the following
assumption:
Assumption
4.3:
^ HgH
HgH,,
for
g e
^
and there exists
e
such that
>
V,£J
ElX^VKz.v)
u(v)
lu(v)(p(v)]
In order to
dv]
<
oo.
guarantee that the parametric approximation suffices for consistency, the
following denseness condition will be imposed.
Assumption
4.4:
g e ^
For any
limj_^IIPj(.,y)«> -
gll
=
necessciry to
g
there exists
).
&.
such that
make some assumption concerning the conditional expectations
is lifted
from Newey and Powell
changing notation assume that the data observation
(v. ,...,v.
P.{v,f)(p(v) e
can be approximated by the family.
The following condition
estimator.
J
0.
This condition specifies that
It is
and
Assume that the data are stationary.
- 12 -
z.
(1989).
Without
includes the simulation draws
>
Assumption
and
For
4.5:
e
11
if
E[|^(z.)|^*^''^] <
finite,
T.", IIEl^tz) |x.] -
^1=1
.i/»(z.).
iJ
J
U
^1
r.",E[^(z)|z.]/n =
^1=1
then
oo
^.^i^^^i'^"
|x.lll^/n
E[.//(z)
(1);
z.
iJ
or
b) E[0(z)|z.]
it
is
K/n
(1977),
b),
iii)
—
)
0,
while
ii)
iii)
with P
Lemma
in
P'.
and and
ii)
if
positive definite,
is
is
Elf/ztz)^""^]
a) E[i/»(z)|x.]
=
if
E[
|i//(z)
I
^^^''^l <
oo,
.P'.j'j]." ,P ..//(z.).
(J^.",?
^j=l
^j=l j"^
J
some cases and
holds by construction.
containing
Newey
A. 10 of
either
iii)
D
D,
J
is
J
For instsmce,
quite general.
K
1
—
K
For K-nearest-neighbor estimators with
8 of Robinson (1987) and Proposition
of Stone
For a series estimator of the form given
in
elements such that any function with finite mean square
can be approximated arbitrarily well
from Lemma
^
D —
easy to use known results to show that Assumption 4.5 holds
follows by
a)
i)
E[«//(2.)l;
n).
=
for nearest neighbor and series estimators.
00,
0;
^11
p
then
i.i.d.
is
"^
(s,t=l
1,
J=l
Assumption 4.5 can easily be checked
if
-^
4.3,
1
I.",w.. =
w.. i 0.
r.",w.
J=l
from Assumption
>
in
mean-square for large enough
(1993a) and the arguments for
Neither of these results allow for data based
Assumption 4.5 restricts the form of randomness.
K.
It
Lemma
A.
11
K,
ii)
follows
—
K
as long as
>
m
should be noted that
Implicitly the
form of the weights
w
st
in
Assumption 4.5 and the approximating functions
P
are restricted to not depend on
i//.
Thus, while they could be chosen based on some fixed
of
p(z,e)
(i.e.
for some prelimineu-y estimator
with
9
p(z,9)).
in
9),
ip
(e.g.
a
lineair
combination
they are not allowed to vary with
\p
Assumption 4.5 should also be "plug-compatible" with future
results on nonparametric conditional expectation estimators, such as those for time
series.
The
last
Assumption
assumption specifies that
4.6:
As mentioned
J
-^
eairlier,
oo
as
n
—
)
J
must go to infinity with the sample
size.
oo.
the degree of approximation
- 13
J
can be random,
in
a very general
and
way.
However,
should be noted that
it
it
not restrictions on the growth rate of
is
J
that are used to obtain consistency, but rather the restriction of the function to a
compact
Often, the compactness condition will require that higher order derivatives
set.
be uniformly bounded, a condition that will have more "bite" for large values of
J,
imposing strong constraints on the coefficients of higher order terms.
These assumptions and identification deliver the following consistency result:
E[p(z,Q)\x] =
Theorem
4.1:
4.1 - 4.6
are satisfied, then
It
If
has a unique solution on
-^
lip-3_ll
and
llg-g_ll
Bx'S
-^
at
and Assumptions
G_
0.
should be noted that the hypotheses of this theorem are not very primitive until the
norm
Once that
is specified.
llgll
is
specified,
may require some work
it
to check the
the other assumptions.
The following set of Assumptions
is
sufficient to demonstrate that the assumptions
are not vacuous, and do cover cases of some interest.
Assumption
4.7:
fixed constant
V
is
cmd
I- qT-V^;
vi)
Corollary
4.2:
and
£
is
This result
is
B
is
^(v)
for
is
all
v e
v,
:
Vh
There
ii)
g(v)
=
iii)
is
for
llgll
If Assumptions
]
<
3.1,
IIP-/3qII
V
a compact interval
\ i V, sup
= sup^|g(v)|;
|g(v)|
iv)
continuous and bounded away from zero on
Elsup^^^M(z,v)
compact then
is
one-dimensional;
such that & = {g(v)
s B|v-v|
|g(v)-g(v)|
(p{v)
v
i)
^ B,
The support of
V;
v)
4.2, 4.5 - 4.1
-^
and
are satisfied,
llg-g_ll
It is
-^
p
e S,
satisfied,
0.
easy to relax the assumption that
one dimensional, using the results of Elbadawi, Gallant, and Souza (1983).
noncompact support for
using the results of Gallant and Nychka (1987).
quite thick tails, with
P(v,y) =
00.
restrictive in several ways.
difficult to allow for
and a
w(v) = C(l+v'v)
in
v,
although this extension
is
more
possible
Unfortunately, their result allows for
Assumption
- 14
is
It
v
4.3.
This tail behavior does
not allow Assumption 4.3 to be satisfied when
fiv)
is
the standard normal density.
Of
course, there are fast computational methods for generating data from densities
(l+v'v)
proportional to
densities.
Also,
so that one could easily use such thick-tailed baseline
,
should be possible to develop intermediate conditions that allow for
it
more general simulators.
5.
A Sampling Experiment
A small Monte Carlo study
is
useful in a rough check of whether the estimator can
Consider the model
give satisfactory results in practice.
•
(5.1)
S^w
y = 6^ +
w
w
=
=
w
+ 5^e
-w
+ c,
= 5^ =
5^
1.
C
is
N(O.l),
»
+
+
TT
T),
71
N(0,1),
is
T)
X +
The regression equation for
=
n
V,
this
71
model
=
is
1,
one that
relationship between consumption and income.
discussed in Section
6,
where
it
is
used
in
and
X
is
v
are
useful in estimating the
This specification will be further
the empirical example.
were set so that the r-squared for the prediction equation for
signal to noise ratio
was
1.
N(0,.5).
The parameter values
w
The number of observations was set to
observations was chosen to be small relative to typical sample sizes
make computation
thcin typical in
easier.
was
The r-squared for the regression of
w
1/2,
100.
in
on
and so the
The number of
economics, to
x
was
set higher
order to offset the small sample size, so that the estimator might be
informative.
Table One reports the results from 100 replications.
15 -
Results for three different
estimators of
8
,
and
5„,
6„
squares (OLS) regression of
The second estimator
are reported.
on the mismeasured right-hand side variables
y
R. = (l,h.,h.,h.
111
1
and
is
W
=
is
a simulated
I®(J^._ R.R'.
)
first estimator is an ordinary least
an instrumental variables estimator
is
side but with instruments
estimator
The
momemt
,
where
)'
,
where
h.
=
(IV)
n,
+ ir„x..
2
is
The third
a two dimensional identity matrix.
C(x.) = I®R.
This estimator
a system two-stage least squares estimator where the instrumental variables are
P(v,y)
Also,
and
was a Hermite polynomial of the third-order, where
was estimated.
7
y
=1,
R..
7^ = 7- =
0,
There were two simulations per observation.
In one replication out of the 100 the estimator did not
converge to a stationary
This replication was excluded from the results, that are reported in Table One.
point.
The estimator shows promise.
The standard errors of the IV and simulated moment
estimators are much larger than the OLS estimator, but the biases
As previously noted the IV estimator
smaller.
it
).
1
estimator (SM) from equation (3.9), with
I
~w
with the same right-hand
1
1
(l,w,e
leads to bias reduction.
estimator
is
It
is
is
aire
substantially
inconsistent, although in this example
interesting to note that the standard error of the
smaller than that of the IV estimator.
Thus, in this example the valid
SM
SM
correction for measurement error leads to both smaller bias and variance than the
inconsistent IV correction.
6.
An Application
to Engel Curve Estimation
The application presented here
is
long been of interest in econometrics.
estimation of Engel curves, a subject that has
Measurement error has recently been shown
Hausman, Newey, and Powell (1993) to be important
curves.
in the
in
estimation of nonlinear Engel
This section adds to that work by estimating a nonlinear, nonpolynomial Engle
curve for the model of equation
(2.1),
which was not considered by Hausman, Newey, and
- 16 -
The functional form considered here
Powell (1993).
S.
(5.1)
=
5,
is
S.
2
1
1
where
»
•
+ 5_ln(I.) + 5_(1/I.) +
3
1
is
that preferred by Leser (1963),
c,
1
1
the share of expenditure on a commodity and
is
I.
the true total
As suggested by the Hausman, Newey, and Powell (1993) tests of the Gorman
expenditure.
rank restriction, a rank two specification such as this may be a good
(1981)
specification, once the
a specification
In addition,
is
results
from the fact that
if
results
is
1.
= Y./l.,
S.
for.
considered that accounts for the presence of
denominator of the left-hand side of this equation.
in the
Thus,
measurement error has been accounted
where
Y.
is
1.
This "denominator problem"
the expediture on the commodity.
measured with error, another nonlinear measurement error problem
from using the measured shares.
This problem can be dealt with by bringing
1.
out of the denominator, giving
Y. = 5,1.
(5.2)
1
1
If
e.
+ 6^1.1n(l.) + 5^ + I.e..
2
1
1
11
3
1
Elcll.] =
satisfies the usual restriction
11
1
0,
then equations
(5.1)
aind
(5.2)
are equivalent statistical specifications, in that running least squares on either
Covariates will also be allowed
equation should give a consistent estimator.
specification by allowing additional variables
equation, corresponding to inclusion of
equation
1.x
in this
to enter linearly in this
.
as additional regressors in the share
x
(5.1).
The measurement error
will be
assumed to be multiplicative,
i.e.
for
1.
equal to
the observed total expenditure,
(5.3)
•
ln(I.)
1
In the empirical
11111
•
•
= w. =
1
tt'x.
1
+
v.,
ln(I.)
1
1
work the predictor
= Ind.)
vEiriables
squared for household head and spouse,
aind
x.
«
+
t?.
= w. = w. +
ij..
will be a constant,
age and age
dummies for educationational attainment.
- 17
spouse employment, home ownership, industry, occupation, region,
With this specification for the
total of 19 variables, including the constant.
measurement and prediction equations,
f(w
black or white, a
eind
= 5
,5)
+
5„w
+ 6
exp(-w
as in the
),
Monte Carlo exeimple.
The measurement error
denominator can be accounted for as
in the left-hand side
*
f(w
equation (5.2), leading to a specification with
m
m
,5)
= 5 exp(w
+ 6
)
in
»
exp(w )w
+ 6^.
*
It
is
that
interesting to note that even
5
=
0,
if
the shaire equation
is
linear in
ln(I.),
this equation is nonlinear, so that IV will not be consistent.
measurement error
in the
denominator of the
shao'e suggests the
so
Thus,
need for the estimators
developed here.
The data used
in
estimation are from the 1982 Consumer Expenditure Survey (CES).
The basic data we use are total expenditure and expenditure on commodity groups from the
Results were obtained for four commodity groups, food, clothing,
first quarter of 1982.
transportation, and recreation.
The number of observations
empirical results were reported as elasticities,
econometrics.
To compaire shapes,
elasticities
i.e.
in the
dlnf(x)/dlnx,
data set
as
is
1321.
common
is
The
in
were calculated at the quartiles of
observed expenditure.
The results are given
in
Tables
Two through
Five.
income distribution.
statistics, including the quartiles of the
include estimated expenditure elasticities at these quartiles.
information on the prediction regression.
R
The
2
quite sizeable for such a cross-section data set.
Two
Table
in this
gives
some sample
The other tables
Table
Two
regression
also gives
.23,
is
The other information
will
is
which
is
useful in
calculating the magnitude of the measurement error and bounding the size of the variance
of the prediction error
standard error
.45
In particular, the
v.
of the residual is
ain
model we have assumed implies that the
upper bound on the standard deviation of both
the measurement error and the variance of the prediction error
estimator
a-
of
Var(v)
1/2
,
an estimator of the
R
2
v.
Also, given an
of the measurement equation,
that determines the magnitude of the measurement error bias in a linear model,
- 18 -
is
Vartrt'x + v)/Var(w) = [Vardr'x) + o-^]/Var(w) = [(.25)^ + ^^]/(.51)^ = .24 + (3.8)o^^.
Tables Three to Five give results for each commodity for three different
specifications of the share equation and four different estimators.
results for the share equation,
left-hand side
ignored.
is
where measurement error
This specification
is
Table Three gives
the denominator of the
in
the same as
in
the Monte Carlo study.
Table Four changes the specification to account for the left-haind side denominator by
multiplying through the original equation by total expenditure, as described above.
Table Five adds covciriates
regional price effects.
dummy
three regional
Four
in
+ 5
to the share equation to allow for
The equation estimated
variables.
•
exp(w )w
demographic and
There are six covariates; own and spouse age, family
is
size,
and
analogous to that of Table
accounting for the left-hand side denominator, with
•
5
x
f(w
,x ,5)
=
•
+ 6
+
exp(w
)x
restricts fcimily size to be absent
'
6
.
It
should be noted that this specification
from the prediction equation.
Tables Three to Five report results for four different estimators, ordinary least
squares (LS), two stage least squares
(IV)
with instruments described below, the
simulated moment estimator with Gaussian
v
(SMO), and the simulated
with one Hermite polynomial term (SMI), of the third order, included
functions.
p. (a)
The simulated moment estimators are each obtained as
as given in equation (3.8),
simulation draws, and
10
an estimated asymptotic variance of
W
(5.4)
0.
1
where
a
is
= f:"\
^._ C(x.)p.(a)/v^.
W
in
in
moment estimation
the
moment
equation (3.9), with
equal to the inverse of
Specifically,
Z = n"V.",U.O'..
= C(x.)p.(i) + [5j;.",C(x.)3.{a)/S7i](^.",x.x'.)"^x.(w.-TC'x.).
"^j
1
an
^j=l
'^i
^j=l
J
initial consistent estimator.
minimizing choice of
W,
2
This
J
is
J
11
1
an asymptotic variance
that accounts for the presence of
n
2
in
p..
The procedure used to obtain the intial consistent estimators was to begin with an
identity weighting matrix, use a few iterations to obtain "reasonable" parameter values,
calculate
W
as in equation (5.4), and then minimize to get
- 19 -
a.
The standard errors for LS and IV were calculated from heteroskedasticity consistent
The standard errors for simulated moment
fromulae, e.g. as given in White (1982).
estimators were calculated from the
H =
GMM
asymptotic variance estimator
(H'Z
H)
,
where
ay;.",c(x.)p.(a)/aa.
''1=1
1
1
A preselection process was used
moments
powers of the predicted
Starting at the second order, the
value to include in the instruments.
to have enough
to choose the order of
minimum needed
was
to allow estimation of distribution paremieters, the order
chosen by cross-validation on the food equation, Gaussicin, simulated moment estimator
(SMO), using the cross-validation criteria for choice of instruments suggested in
(1993b).
Newey
Inclusion of higher order powers did not result in any decrease in the
Consequently, in Tables Three and Four the instrumental
cross-validation criteria.
variables were
(1,
x'jr,
(x'lr)
In
).
Table Five
was added
exp(x'ir)«x
to the
instruments, because of the presence of the covariates.
The number of Hermite polynomial terms to include was chosen essentially by an
upwards testing procedure, applied
in the
model of Table Three.
order term was tried in each case, as reported in Table Three.
asymmetry
term was
in the distribution of
tried.
In none of the cases
one, third order, Hermite polynomial
This term allows for
it
was
statistically significant, a fourth order
was
this
term significant, so only results for the
If
v.
Inclusion of a third
term are reported
in the tables.
For each estimator, elasticities at the quartiles, the estimate of
and the estimator of the coefficient
standard errors
(in
y
cr
= Var(v)
1/2
of the Hermite polynomial term, as well as
parentheses below the estimates) are reported.
The (asymptotic)
t-statistic on the coefficient of inverse expenditure (t-stat) and the over identification
(minimum chi-square) test
reported.
The t-statistic
(Q) statistic for the simulated
is particulau'ly
estimator would be consistent
if
moment estimator are
relevant in Table Three because the 2SLS
the coefficient on inverse expenditure were zero.
The degrees of freedom of the overidentification test statistic are
respectively for
SMO and
also
SMI, in Tables Three and Four, and
- 20 -
8
2
and
and
7
1
respectively in
Table Five.
The difference between these
statistics for
SMO and SMI
is
a one-degree of
freedom chi-squared test of the Hermite coefficient being zero.
Even though the IV estimator
estimator
number of
in a
inconsistent,
is
it
gives results similar to the
When the share denominator
cases.
with error there are larger differences between IV and SM.
smaller thain those of
which
IV,
allowed to be measured
is
The standard errors of SM are
consistent with the Monte Carlo results of Section
is
There are large differences between the OLS and SM estimators, as
presence of measurement error.
SM
consistent with the
interesting to note that the elasticities for
is
It
is
4.
transportation go down rather than up, unlike linear regression with measurement error.
In
comparing Tables 3 and
4,
it
is
the denominator leads to some changes
food equation
Table 4 than
in
more precisely estimated
larger
in
Table
in
3.
the results.
In
more nonlinearity
The overidentification test
in
Table
3,
although Table 4
is
most cases SMO
is
in
There
is
quite similar to SMI, except for
in
the
a-
is
statistics are
a levels equation that is
thought to be more heteroskedastic than the share equation.
nonnormality.
is
The prediction error standard deviation
these equations.
in
TTiere
Suprisingly, the estimated standard errors in Table 4 are not
4.
larger than those
Table
in
apparent that accounting for measurement error
much
sometimes
little
evidence of
much larger standard
errors.
In
summary, although allowing for nonnormality does not change the empirical
results, correcting for
measurement error maikes a big difference.
simulated moments estimator
is
In several
cases the
quite different than the inconsistent IV estimator,
suggesting that the inconsistency of IV estimator
may
not be uniformly small.
Furthermore, the simulated moment estimators seem quite accurate, having small standard
errors.
These results illustrate the usefulness of using simulated moment estimation to
correct for measurement error, while allowing some flexibility
in
prediction error to asses the impact of allowing for nonnormality.
21
the distribution of the
Appendix
The proof proceeds by verifying the hypotheses of Theorem
Proof of Theorem
4.1:
Newey and Powell
(1989).
= Sx5
S
compact by
is
-1
S
&
amd
Z =
augmented data vector, with
S
For
compact.
(z,v ,...,v
Note that for
,H(z,fi,v )P(v ,y).
y
^^=1
s
s
9 = O.g)
Let the norm for
).
be
11611
lipil
simulations let
S
+
Z
llgll.
Note that
denote the
p(Z,e) =
Also, let
p_(v) = g_(v)/cp(v),
u
=
5.1 of
it
u
follows by Assumptions
4.2 and 4.3 that by the triangle inequality
£ J:^^^IIH(z,Pq,v^)II|Pq(v^)|/S s <j:^f^M(z.v^)[(j(v^)v(v^)rVs>llgQll,
(A.l)
llpCZ.e^)!!
(A.2)
{E[llp(Z,e_)ll^''^]>^''^^"'^^
£ C-r^,<E[m(z,v )[u(v )v)(v )] h'^''^]}^^^^''^^/S
*^=1
s
s
s
= C.'(E[J^M(z,v)^*^w(v)"^"%(v)"^"^dvl}>^^^^''^^
It
follows similarly to equation A.l that that for
iip(z,e)-p(z,e)ii
(A.3)
so that Assumption 5.1 of
Newey and Powell
and 5.3 then follow by Assumptions 4.4 and
4.5.
Furthermore, by the fact that
in the
sup norm.
by
The conclusion then follows by
QED.
(1989).
4.1.
by hypothesis and the Arzela theorem, which gives compactness of
Assumption 4.3 follows with
zero and Assumption 4.6,
g(v)/^p(v)
Newey and Powell
2.1.
Newey amd
The proof proceeds by verifying the hypotheses of Theorem
Proof of Corollary 4.2:
W
Assumptions 5.2
(A.2).
for an unbiased simulator, as noted in the text,
the conclusion of Theorem 5.1 of
4.1 follows
6 6 8,
(1989) follows by eq.
Powell's (1989) Assumption 3.1 holds by Assumption
Assumption
CO.
c«{j;^^^M(z,v^)[w(v^)^(v^)l Vs>iie-eii,
:£
E[p(Z,e)|x] = E[p(z,^,g)|x]
6,
<
vi).
w(v)
=
1
by
(p(v)
bounded away from
Assumption 4.4 follows by a Weirstrass approximation of
PAy). The proof then follows by the conclusion of Theorem
22 -
4.1.
QED.
6
References
Y. (1985): "Instrumental Variables Estimator for the Nonlinear Errors in
Variables Model," Journal of Econometrics 28, 273-289.
Amemiya,
Elbadawi, I., A.R. Gallant, and G. Souza (1983): "An Elasticity Can be Estimated
Consistently without a Priori Knowledge of Functional Form," Econometrica, 51,
1731-1751.
Gallant, A.R. (1987): "Identification and Consistency in Nonparametric Regression," in
T.F. Bewley, ed.. Advances in Econometrics: Fifth
University Press, 145-169.
World Congress, Cambridge: Cambridge
Gallant, A.R. and D.W. Nychka (1987): "Semi-Nonparametric
Maximum
Likelihood Estimation,"
Econometrica, 55, 363-390.
Gorman, W.M.
(1981):
"Some Engel Curves,"
Measurement of Consumer Behaviour
in
in
A.
Deaton, ed.. Essays in the Theory and
Sir Richard Stone, Cambridge: Cambridge
Honor of
University Press.
J. A., H. Ichimura, J.L. Powell, and W.K. Newey (1991):
"Estimation of
Polynomial Errors in Variables Models," Journal of Econometrics, 50, 273-295.
Hausman,
J. A., W.K. Newey, aind J.L. Powell (1993): "Nonlinear Errors in Variables
Estimation of Some Engle Curves," Journal of Econometrics, forthcoming.
Hausman,
Laser, C.E.V. (1963): "Forms of Engel Functions," Econometrica
31,
:
694-703.
S. and C. Manski (1981): "On the Use of Simulated Frequencies to Approximate
Choice Probabilities," in Structural Analysis of Discrete Data with Econometric
Applications, ed. by C. Manski and D. McFadden, Cambridge: MIT Press.
Lerman,
McFadden, D. (1989): "A Method of Simulated Moments for Estimation of Discrete Response
Models Without Numerical Integration," Econometrica, 57, 995-1026.
Newey, W.K. (1993a): "Series Estimation of Regression Functionals," Econometric Theory,
forthcoming.
Newey, W.K. (1993b): "Efficient Estimation of Models with Conditional Moment
Restrictions," forthcoming in G.S. Maddala and C.R. Rao, eds., Handbook of Statistics,
Volume 11: Econometrics. Amsterdam: North-Holland.
Newey, W.K. (1994): "Large Sample Estimation and Hypothesis Testing," forthcoming
R. Engle and D. McFadden (eds.). Handbook of Econometrics, Vol. 4, Amsterdam:
in
North-Holland.
J.L. Powell (1989): "Instrumental Variables Estimation for Nonparametric
Models," preprint. Department of Economics, Princeton University.
Newey, W.K. and
Pakes, A. (1986): "Patents as Options: Some Estimates of the Value of Holding European
Patent Stocks," Econometrica, 54, 755-785.
Pakes, A. cind D. Pollard (1989): "Simulation and the Asymptotics of Optimization
Estimators," Econometrica, 57, 1027-1057.
- 23 -
Robinson, P. (1987): "Asymptotically Efficient Estimation in the Presence of
Heteroskedasticity of Unknown Form," Econometrica, 55, 875-891.
Stone, C.J. (1977): "Consistent Nonparametric Regression" (with discussion), Annals
Statistics, 5, 595-645.
of
White, H. (1982): "Instrumental Veiriables Regression with Independent Observations,"
Econometrica 50, 483-499.
Wolter, K.M. and W.A. Fuller (1982): "Estimation of Nonlinear Errors in Variables
Models," Annals of Statistics 10, 539-548.
- 24 -
Table One: Monte
^
Bias
OLS
IV
-1. 10
- .30
SM
-
.09
SE
RMSE
.25
3.31
2.27
3.32
2.27
1
.
B
13
i
as
.
67
.
13
.
07
'2
SE
RMSE
Bias
67
26
05
.85
.47
13
.
1.25
1.05
25th
50th
75th
3373
4574
6417
Sample standard error of log of expenditure
.51
Standard error of predicted
.25
Standard error of residual
.45
R-squared
.23
Table Three:
^3
Some Sample
Table Two:
Income Quartiles
Carlo Results
.
1
.
1
.
SE
.
.31
Statistics
Elasticity Estimates for Share Equations
Food
25th
50th
75th
LS
.72
(.02)
.66
(.02)
.59
(.03)
2SLS
.82
(.05)
.78
(.04)
.74
(.06)
SMO
.82
(.04)
.78
(.04)
.74
(.06)
.84
(.20)
.78
(.09)
(.36)
SMI
.71
7
a-
t-stat
Q
4.52
.
.61
47
3.67
6.36
.04
6.08
(.08)
-.01
.31
(1.49)
- 25 -
(
.05)
12
2.01
1.62
RMSE
.
2.
1.
86
07
65
Clothing
LS
25th
50th
75th
1.21
(.05)
1.08
.97
(.05)
18.66
(.04)
2.25
2SLS 1.61
SMO
SMI
r
<r
1.42
1.30
(.12)
(.09)
(.10)
1.63
1.40
1.26
(.20)
(.10)
(.18)
.02
(.38)
1.62
1.28
1.07
-.0009
(.46)
(.30)
(.56)
(.0018)
.15
t-stat
.56
6. 11
.20
1.99
(.11)
Tr an s p ortation
LS
2SLS
SMO
25th
50th
75th
t-stat
1.28
1.44
1.50
11.19
(.07)
(.06)
(.07)
.99
(.08)
1.06
1.12
(.08)
(.12)
1.02
1.01
(.06)
1.01
(.06)
1.71
(2.01)
.98
(.07)
.63
(.18)
.10
(.07)
(.07)
SMI
1.40
(.27)
1.00
.028
(.018)
.04
11.28
3.10
7.54
Recreation
LS
25th
50th
75th
1.40
1.20
1.06
(.07)
(.06)
(.07)
1.31
(.12)
1.07
1.33
2SLS 1.70
(.15)
SMO 2.97
SMI
7
a-
t-stat
16.59
11.45
(.15)
6.48
.02
(.34)
(.63)
(.12)
.39
(.48)
6.98
2.32
1.28
.71
(4.40)
(.36)
(.12)
(.18)
(
- 26 -
.024
002)
.
5.58
.09
Table Four:
Elasticity Estimates for Level Equations
Food
a
25th
50th
75th
.68
.04
.63
.03
.57
.03
.31
.90
(.08)
.80
.70
3.34
(.05)
(.05)
SMO
.98
(.08)
.81
(.04)
.63
(.05)
.35
(.02)
SMI
1.21
(.21)
.83
(.06)
.46
(.13)
.24
(.06)
LS
2SLS
r
t-stat
12.4
.008
5.09
Q
18.16
16.42
(.005)
CI othing
LS
25th
50th
75th
1.40
1.11
(.06)
.89
(.04)
53.95
1.50
1.21
(.13)
4.92
(.11)
2SLS 2.04
SMO
SMI
(.26)
(.12)
2.07
1.36
(.21)
(.08)
2.14
1.37
(.58)
(.11)
r
a-
.96
(.07)
.34
(.02)
.93
(.18)
.33
(.09)
.001
(.009)
t-stat
39.04
17.15
3.19
17.13
Transportation
LS
2SLS
SMO
25th
50th
75th
3.14
1.95
1.48
(1.06)
(.19)
(.09)
.23
(.65)
.93
(.12)
1.53
1.16
.94
(.05)
.76
(.04)
.64
(.04)
.97
(.20)
.74
(.09)
.58
(.27)
(.08)
SMI
7
<r
1.25
(.59)
t-stat
1.68
2.04
(.53)
.004
(.021)
- 27 -
38.27
10.63
.69
10.52
Recreation.
LS
25th
50th
75th
1.74
1.26
(.09)
.95
(.05)
42.87
(.19)
1.01
(.15)
14.85
2SLS 1.84
1.32
(.28)
SMO 2.35
SMI
(.16)
1.44
.95
(.07)
.36
(.03)
1.93
.33
(.34)
(.15)
.25
(.02)
(.26)
(.09)
7.85
(4.76)
Table Five:
t-stat
y
cr
(
.024
005)
48.62
33.23
15.88
21.83
.
Elasticity Estimates for Level Equations with Covariates
Food
25th
50th
75th
LS
.72
(.05)
.66
(.03)
.58
(.03)
1.90
2SLS
.97
(.09)
.85
(.06)
.74
(.06)
4.40
SMO
1.00
.86
(.05)
.73
(.05)
.33
(.02)
1.25
.90
(.29)
(.07)
.58
(.15)
.21
(.07)
(.08)
SMI
y
cr
.009
t-stat
Q
7.14
43.51
2.47
41.59
(.007)
Clothing
LS
25th
50th
75th
1.35
1.08
(.11)
(.06)
.88
(.04)
1.50
1.22
(.14)
(.15)
(.22)
1.31
(.09)
.94
(.08)
.33
(.02)
2.01
1.33
.002
(.14)
.92
(.23)
.31
(.73)
(.11)
(.011)
2SLS 2.01
(.31)
SMO
SMI
1
.
94
y
<r
t-stat
32.76
4.22
- 28 -
29.32
43.23
1.59
42.95
Transportation
LS
2SLS
SMO
SMI
25th
50th
75th
2.54
1.83
1.50
(.63)
(.14)
(.08)
.05
(.55)
.70
(.16)
1.38
1.07
(.09)
.87
(.06)
.69
(.04)
(.04)
1.86
1.
13
(.26)
.61
(.07)
.40
(.08)
(.87)
r
<r
t-stat
Q
1.30
3.09
(.45)
.61
.016
(.009)
28.01
30.12
2.61
28.92
Recreation
LS
25th
50th
75th
1.73
1.25
(.09)
.95
(.05)
35.96
(.19)
1.01
(.16)
10.69
2SLS 1.78
SMO
SMI
1.31
(.30)
(.
2.40
1.41
(.32)
(.
11)
(.08)
8.38
5.45
1.91
(.32)
.18
.21
(.23)
(.03)
17)
.88
r
cr
.31
(.03)
.021
(.005)
29 -
t-stat
37.86
54.23
10.91
44.88
2267 U45
Date Due
SEP
1 1999
nr