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FLEXIBLE SIMULA TED MOMENT ESTIMA TION OF
NONLINEAR ERRORS-IN-VARIABLES MODELS
Whitney K. Newey
No.
99-02
February, 1999
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
FLEXIBLE SIMULATED MOMENT ESTIMATION OF
NONLINEAR ERRORS-IN-VARIABLES MODELS
Whitney K. Newey
No.
99-02
February, 1999
MASSACHUSETTS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
Flexible Simulated
Moment Estimation of
Nonlinear Errors-in-Variables Models
Whitney K. Newey
2
MIT
First Version, October 1989
Revised, February 1999
Abstract
Nonlinear regression with measurement error
microeconomic data.
is
important for estimation from
One approach to identification and estimation
where the unobserved true variable
paper
is
is
a causal model,
is
predicted by observable variables.
This
about estimation of such a model using simulated moments and a flexible
disturbance distribution.
parametric models.
estimator
is
An estimator of the asymptotic variance
is
given for
Also, a semiparametric consistency result is given.
demonstrated
in a
The value of the
Monte Carlo study and an application to estimating Engle
Curves.
JEL
Classification:
Keywords:
C15, C21
nonlinear regression, errors-in-variables, simulated moments.
This research was supported by the National Science Foundation and the Sloan Foundation.
David Card and Angus Deaton provided helpful comments.
2
Department of Economics, E52-262D, MIT, Cambridge, MA 02139. Phone: 617-253-6420, Fax:
617-253-1330, Email: wnewey@mit.edu.
1.
Introduction
Nonlinear regression models with measurement error are important but difficult to
estimate.
Measurement error
models are often of interest.
is
common problem
a
microeconomic data, where nonlinear
in
For example, flexible functional forms often lead to
Instrumental variables estimators are not
inherently nonlinear specifications.
consistent for these models, as discussed in
Amemiya
(1985),
The purpose of this paper
approaches must be adopted.
computationally feasible and also allows for flexibility
disturbances.
This purpose
is
in
is
so that alternative
to develop an approach that is
the distribution of
accomplished by using simulated moments estimation with
flexible distributions, an approach that
may be
useful for simulated
moments estimation
of other models.
The measurement error model considered here has a prediction equation for the true
regressor with a disturbance that
is
independent of the predictors.
The estimator
is
based on the conditional expectation of the dependent variable, and the conditional
expectation of the product of the dependent variable and mismeasured regressor.
model for measurement error
in
nonlinear models has previously been considered by
Hausman, Ichimura, Newey, and Powell
(1991)
and Hausman, Newey, and Powell (1995), but
only for the case of polynomial regression or approximation, and simulated
not considered.
This
moments was
This paper allows for general functional forms, significantly extending
the scope of the previous work.
Much
of the other
work on measurement error
in nonlinear
the assumption that the variance of the measurement error
size.
is
models relies heavily on
small relative to the sample
These papers include Wolter and Fuller (1982) and Amemiya
(1985).
In
econometric
practice the measurement error often seems quite large relative to the sample size, and
has big effects on the coefficients.
Thus,
it
seems important to consider approaches
that allow for relatively large measurement error, as does the one here.
Simulated moments estimation provides a computationally convenient approach when
estimating equations involve integrals, as discussed in
in
Lerman and Manski
(1981),
Pakes (1986), McFadden (1989), and Pakes and Pollard (1989).
This approach uses Monte
Carlo methods to form an unbiased estimators of integrals
moment
in
equations.
Allowing
flexibility in disturbance distributions is desirable, because consistency of the
estimator depends on correct specification of the distribution.
Also,
preserve the computational convenience of simulated moments.
These goals are
accomplished by combining simulated moment estimation with a linear
specification for distribution shape.
density to the simulated one.
it
in
is
useful to
parameters
The specification parameterizes ratio of the true
This approach
is
similar to the importance sampling
technique from the simulation literature.
The parametric simulated moments estimator we propose
method of moment estimator.
Here the moments are smooth
standard asymptotic theory applies.
is
essentially a generalized
in the
parameters, so that
For that reason we just give large sample inference
procedures with an outline of the asymptotic theory for the parametric case.
We pay more
attention to conditions for consistency for the nonparametric case, giving a consistency
result
when the number
grow with sample
of parameters in the distribution approximation is allowed to
size.
The paper also includes Monte Carlo and empirical applications, to evaluate the
potential impact of this approach for applied work.
The empirical application
estimation of Engel curves from household expenditure data.
is
The measurement error
correction makes a big difference in the application, with a Gaussian specification for
prediction error sufficing in most cases.
having small standard errors.
moment estimation
Also, the estimator
seems quite accurate,
The results illustrate the usefulness of using simulated
to correct for
measurement error, while allowing some
flexibility in
the distribution of the prediction error.
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 Engel curve estimation of the relationship between income and
consumption.
The Model
The model considered here
y = f(w ,5
(2.1)
w
=
w
where
W
E[<|x,v] = 0,
C
+
)
*
+
E[7?|X,V,^]
7},
= ti'x +
and
y
Q
is
cr
are scalars,
^
5
are conformable matrices.
cr
errors, and
w
unobserved.
The
w
,
Here
0,
independent of
v
v,
=
w,
,
w
t),
are vectors,
v
represents true regressors,
The
x
w
are observed and
and
and
Tt
measurement
7)
,
£,
tj,
and
last equation is a prediction equation for the true regressors,
v
is
an unobserved prediction error,
scaling matrix, a square root of a variance matrix.
allowed to be observed, with
w
corresponding elements of
and
tt'x
and
*
observed regressors.
are observed predictor variables,
element of
x,
x.
is
w
tj
v
cr
may be
where
is
x
a
Some of the true regressors can be
equal to an element of
cr
and
v
x,
by specifying that
are identically zero, and the corresponding
*
= w.
This model was considered by Hausman, Ichimura, Newey, and
#
Powell (1991) (HINP henceforth), for the special case where f(w
w
.
e.g.
As long as
as
E[v] =
x
,5)
includes a constant, the location and scale of
and
Var(v) =
Instrumental variables
(IV)
I
when the second moment of
v
is
v
a polynomial in
can be normalized,
exists.
estimators can be used to estimate this model when
*
*
*
f(w
w
linear in
is
,5)
Substituting
.
w-tj
being valid instruments, because the disturbance
is
in the first
linear in the
equation leads to
measurement error
case this substitution leads to residuals that are nonlinear in
In the nonlinear
x
Consequently,
w
for
will not be valid instruments,
x
17.
tj.
and another approach has to be
adopted.
An approach
Let
error.
to consistent estimation can be based on integrating out the prediction
be the density of
g n (v)
Integrating over the prediction error leads
v.
to three useful conditional expectation equations:
(2.2a)
E[y|x) = JTdr^x +
(2.2b)
E[wy|x] =
(2.2c)
E[w|x] =
The
Jln' x +
Q
first is a regression of
on
x,
that
Q
S
v,
<r
)g
(v)dv,
v]T{n'x +
<r
v,
5 )g (v)dv,
tt^x.
unobserved variable
wy
cr
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 corresponding to unobserved
w
(i.e.
those not corresponding
to observed covariates) provide information additional to the first equation.
in
HINP for polynomial regression, the
identification.
Intuitively, there are
identification.
It
was shown
in
As shown
first equation does not suffice for
two functions that need to be
regression function and the density of
The
v,
identified, the
so that two equations are needed for
HINP that the parameters of any polynomial regression
equation are identified from these two equations, and one expects that identification of
the regression parameters will hold more generally.
It is
beyond the scope of this paper to develop fully primitive identification
conditions for this model, but some things can be said.
identified
so
7r
from
n
w
are
on
x,
can be treated as known and identification of the other pieces of the model
m
distribution with a finite support and
provide
2m
2m
*
Also,
that a "rank condition"
v.
and
5
For example, HINP showed
#
a scalar and
is
parameters are identified
nonsingular.
i.e.
parameters from these equations, including
w
where
has a discrete
points of positive probability, then (2.2 a -
parameters of a parametric family of distributions for
that, in the case
ti'x
If
Assuming that none are redundant,
equations.
holds, one could identify
8
parameters
(2.2c) as the coefficients of a least squares regression of
considered by focusing on equation (2.2a) and (2.2b).
b)
First, the
if
f(w
,5)
is
moment matrix
the second
some of the moments of
v
a polynomial of degree
of
are identified
(I.tc'x
^n x
in this case.
the
p,
^'
*
s
has
rc'x
If
*
a continuous distribution, then a simple counting argument suggests that
g
(v)
should be identified.
Assuming that the
distinct functions, these equations give
functions to be identified.
So,
left
f(w
,5
)
and
hand sides of (2.2a) and (2.2b) 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
(i.e.
an important problem deserving of future
attention.
Independence of
model of equation
moments of
v
(2.1)
x
it
and
v
a strong assumption, but in the general nonlinear
is
is difficult
can depend on
x,
to drop this assumption.
then
regression function from the distribution.
it
is
much more
Intuitively,
if
some
difficult to separate the
3.
Estimation
To describe the estimator
moment
conditional
g
vector, and
H(z,£,v)
a
p(z,P,g) =
Suppose that there
parameter value
denote a data observation,
z
r
x
a set of conditioning variables
is
and density
(z.P.v)
= y -
H
(z,/3,v)
=
Uwy
(z,/3,v)
=
w
and
L
is
3
q x
a
p(z,/3,g)
r
1
x
vector of
1
residual
g
x
such that for the true
,
)|x] = 0.
,g
H
H
a
/3
more general
vector of functions, related as in
1
The nonlinear errors-in-variables model
(3.3)
v,
in a
rH(z,/3,v)g(v)dv.
£
E[p(z,(3
(3.2)
embed the model
a density function of a random vector
parameters,
(3.1)
Let
setup.
helpful to
is
it
is
a special case of this one, where
f(7r'x+crv,S),
- [7r'x+o-v]-f(7r'x+crv,5)>,
- jt'x,
/3
=
(S'.o-.ji' )',
a selection matrix that picks out those elements of
w
that include
measurement error.
The common approach to using equation
variables.
One difficulty with
Another difficulty
is
this
approach
that the residual
is
(3.2) in estimation is nonlinear instrumental
is
that the density
g(v)
is
unknown.
an integral that may be difficult to compute.
Here, these difficulties are dealt with simultaneously, by choosing a flexible
parameterization for the density that makes
integral.
function.
To describe
this approach,
it
easy to use a simulation estimator of the
we begin with a
specification of the density
For now, suppose that the density
g(v,y) =
(3.4)
where
and
is
<p(v)
p.(v)
= v
member
a
is
of a parametric family, of the
P(v,y) = £. jr.p.Cv),
PCv.rMv),
some fixed density function.
For example,
if
<p(v)
then this would be an Edgeworth approximation.
,
form
need not be positive, but leads to residuals that are linear
were standard normal
The function
g(v,?-)
shape parameters
in the
n
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.
and the
P(v,y)
H
and then evaluating the product of the linear
<p{v)
Let
functions.
z.
denote a single observation and
denote a vector of random variables, each having marginal density
]
For example,
if
(p(v)
is
a standard normal pdf, then
[v
,...,v.
]
could be computer
Then a simulated residual for the
generated Gaussian random numbers.
<p(v).
observation
i
is
-1
p\(e)
(3.5)
This
is
= s £j\H(z.,(3,v
)P(v
e = (p'.r')'.
*),
essentially an importance sampling estimator of the residual,
sampling density and
P(v,y)
approximates
g(v)/<p(v).
where
is
<p(v)
The simulated residual
is
an
unbiased estimator of the true residual, because
E[p.(G)|z.]
= p(z
,0,g(y)).
Therefore, by the results of McFadden (1989) and Pakes and Pollard (1989), an
instrumental variables (IV) estimator with
p. (6)
as the residual should be consistent
l
if
the IV estimator with the true residual
familiar way.
estimated.
Let
A(x)
Suppose that
denote a
9
solves
q x r
is.
An IV estimator can be formed
in
vector of instrumental variables, that
a
may be
n
(3.6)
This
^.^ACxJp.O) =
0.
McFadden
a simulated, nonlinear IV estimator like that of
is
Because equation (3.6)
density
linear in
is
to integrate to one.
P(v,3")<p(v)
P{v,y),
Also,
is
it
it
important to normalize the
may be important
normalizations by imposing constraints on the coefficients.
p.(v), p~(v),
for
j
*
k),
then
integrates to one, and has zero
P(v,y)<p(v)
=1,
y
For example,
if
is
<p(v)
are the Hermite polynomials that are
...
orthonormal with respect to the standard normal density
Jp.(v)p (v)<p(v)dv =
to impose a location
There are different ways to impose
and scale normalization on this density.
the standard normal density and
(1989).
j
mean and
=
2
(i.e.
J"p.(v) <p(v)dv
and
0,
y
unit variance.
=
It
to impose such constraints using the simulated values, by requiring that
=
and
1
will imply that
is
also possible
y
S
2
n
,(l,v. ,v
)P(v. ,r) = 0.
is
^i=l^s=l
is
is
T
satisfy£.
In the nonlinear errors in variables model,
it
is
convenient to work with a two step
estimator, where the first step consists of estimation of
n
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 an estimator can be formulated as a
solution to equation (3.6),
if
A(x)
is
chosen
in a particular fashion.
Let
u
be the
LS estimator and
A(x) = diag[B(x),x],
(3.7)
where
(5',o\y').
£
_iP( v
has two columns and number of rows equal to the number of elements of
B(x)
-
»y)
Suppose that constraints are imposed on the
=
0.
Then the solution to equation
equation (3.7), requires that
parameters solve the equation
n
(3.6),
y
with
coefficients such that
A(x)
specified as in
be the least squares estimator, and that the other
n a
= £"B(x.)p.(a),
(3.8)
y
p. (a)
*
equation as
s=l
min
The
P(v
V
*-
order condition, although the normalization
l
Specifically, for
a
3-)
is>
example the estimator minimizes a quadratic form that has this type of
a
a positive definite matrix,
(3.9)
,5)
L(7r'x.+crv. )f (n'x.+o-v. ,5)
1
is
1
is
'
l
l
is
l
-i£
Lw.y.
its first
not imposed.
f(7r'x.+crv.
:
=
1
In the empirical
a = (5',<r,r')\
l
,P(v. ,y)
=
is
is
equal to a vector of instrumental variables and
C(x)
W
solves
n a
[£ " C(x.)p.(a)]'
^1=1
S
s=l
W[£"
l
^i=l
C(x.)p.(a)].
l
l
first order conditions to this minimization
problem are as given
equation (3.8),
in
with
n
B(x) = [aX. ,C(x.)p.(a)/Saj'W.
^1=1
l
l
(3.10)
Standard large sample theory for IV can be used for asymptotic inference procedures.
If
the simulated values
observation for the
v.)
(v
data point, then the usual IV formulae can be used to form a
i
For example, suppose that
consistent variance estimator.
independent observations as
see
are included with the data to form an augmented
Newey and McFadden,
i
varies.
v.)
(z.,v
are
Then under standard regularity conditions
1994), the asymptotic variance of
v^nte - 9
(e.g.
can be estimated
)
by
_1
V = G nG
(3.11)
_1/
,
n
G = n"V.
,A(x.)s3.(e)/ae,
^1=1
1
If
the simulation draws
given
i,
v.
1
n
111
a = n"V. A(x.)p.(e)p.(G)'A(x.)'.
^1=1
i
are mutually statistically independent as
then one could also use the estimator
10
s
1
varies for a
,
V = G
(3.12)
_1
Q = n V.^AtxJA.Atx.)'
^1=1
£2G *',
1
=
A.
i
l
l
S
,H(z.,/3,v. )H(z.,/3,v. )'P(v. ,£)
S"V
is
is
is
^s=l
2
.
i
1
Both of these variance estimators ignore estimation of the instruments, which
Because the large sample theory for these
under standard regularity conditions.
estimators
4.
straightforward,
is
valid
is
we do
not give regularity conditions here.
Consistent Semiparametric Estimation
If
the functional form of the density
becomes semiparametric.
g n (v)
left unspecified,
is
Models where identification
is
then the model
achieved by conditional
moment
restrictions like those of equation (3.1) are nonlinear, nonparametric simultaneous
Newey and Powell
equations models.
their result can be applied here.
with
and the IV equation
density
&
function.
with
to apply the previous estimator, but
by a nonparametric conditional expectation equation.
that will be assumed to include the true
v
be a sequence of families such that
Let
&. =
/3
Pj(v,y)
9 = (£,g)
{P.(v,y)<p(v)}n§',
and a density
replacing
g,
P(v,y).
of the conditional expectation of
estimator.
is
and satisfy other regularity conditions given below.
Euclidean vector
(3.5)
(3.6) replaced
be a set of functions of
g n (v)
{Pj(v,^)>
The basic idea
chosen to be a member of an increasing sequence of approximating families
P(v,y)
Let
have considered estimation of such models, and
(1991)
p. (8)
and
Let
be the parameter consisting of the
be the simulated residual of equation
be a nonparametric estimator
E[p.(G)|x.]
x.,
Then a minimum distance estimator of
11
can approximate any
P Sv,j)<p{v)
p.(0)
given
Also, let
such as a series or kernel
9
=
(/3
,g
)
is
6 = argmin
(4.1)
0eSx
^
Q(6) = £. E[p\(e) x.]' DE[ P .(e) |x.]/n,
=1
Q(9);
|
n
where
D
is
a positive definite matrix and
The objective function
size.
equation
in
can depend on the data and on sample
J
(4.1)
a sample analog of
is
Q(9) = E[E[p(z,e)|x]'DE[p(z,e)|x]],
where
and
6
D
is
is
D
the limit of
identified
a unique solution),
p(z,9) = TH(z,/3,v)g(v)dv.
and
from the conditional moment equation
then
Q(8)
extremum estimator reasoning
(3.2)
positive definite
is
(i.e.
that equation has
have a unique minimum of zero at
will
(e.g.
D
If
Newey and McFadden,
6
The general
.
1994) then suggests that
should be consistent.
The estimator can be shown to be consistent
sets, similarly to Gallant (1987).
if
and
fi
are restricted to compact
g
The compact function set assumption
but the results of Newey and Powell (1991) indicate
estimators of the form considered here.
a strong one,
is
importance for minimum distance
its
A =
For a matrix
[a.
.],
let
II
A
=
II
ij
Itrace(A'A)]
1/2
,
and for a function
g(v)
let
denote a function norm, to be
llgll
further discussed below.
Assumption
4.1:
/3
€ B,
which
is
compact, and
In the primitive regularity conditions given below,
The following dominance condition
Assumption
M(z,v),
4.2:
There exists
IIH(z,/3,v)-H(z,/3,v)ll
<
M(z,v)
e
g
i*,
will be a Sobolev
llgll
will be useful in
such that for
a compact set in a
norm
norm,
showing uniform convergence.
£,
e S,
IIH(z,J3,v)ll
^
M(z,v)IIJ3-j3ll.
Moment conditions for the dominating function
M(z,v)
will be specified below.
To show uniform convergence of the objective function of equation
useful to impose a strong condition on function norm, that
12
it
(4.1),
it
is
dominates a weighted
llgli
supremum norm.
it
will be
implies that
on the
faster
HgH
assumed that
in order to
HgH
moves outward,
v
sampling imposes a restriction on the
restriction on
llgll,
this
so that Assumption 4.1
assumption
the faster the tails of
sup {|g(v)|w(v)>
guarantee that
baseline density.
dominated by
The import of
§\
and
0.
>
is
v
<p(v),
is
a uniform bound
imposed by the presence of the weight function; the
g(v),
grows as
u(v)
w(v)
bounded on
is
v
behavior of
tail
denote the support of
= sup.,|g(y)|w(v),
llgll, ,
Below
V
Let
is
finite.
g(v)
must go to zero
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
These considerations lead to the following
<p{v).
assumption:
Assumption
4.3:
EIJ M(z,v)
-
HgH,,
V
t
dv]
w(v)
[u(v)<p(v)]
In order to
g6
for
llgll
?
and there exists
e
such that
>
0)
<
oo.
guarantee that the parametric approximation suffices for consistency, the
following denseness condition will be imposed.
Assumption
lim
J
_
IIP
>aj
J
4.4:
For any
(-,r)*> - gll
=
g €
J
there exists
such that
g
can be approximated by the family.
necessary to make some assumption concerning the conditional expectations
The following condition
estimator.
is lifted
from Newey and Powell
changing notation assume that the data observation
(v. ,...,v.
P Av,y)<p(v) e ^
0.
This condition specifies that
It is
and
*§
).
Assume that the data are stationary.
13
z.
(1991).
Without
includes the simulation
draws
>
Assumption
€
from Assumption
>
1+e/2
and
if
finite,
y
For
4.5:
E[
|tf/(z.)
]
|
£." IIEty(z)
n
,w..0(z.),
J=l iJ
J
IXj]
w.. £
<
-
E[i/»(z) |x.]ll
E[0(z)|z ]/n = 0(1);
l"
z.
then
is i.i.d.
it
is
-^
U
or
b) Etyr(z)|z
in
]
K/n
—
>
0,
(1977), while
b),
iii)
ii)
iii)
with P
Lemma
by construction.
containing
K/n
e/(€+2)
A. 10 of
->
either
and and
is
E[^/(z)
if
if
"
E[
is
]
a) E[i//(z)|x.]
.P' .)"£
=
1+€/2
|i//(z)
,P.0(z
]
|
<
«,,
.).
For instance,
quite general.
K
Newey
1
—
K
of Stone
For a series estimator of the form given
in
elements such that any function with finite mean square
in
mean-square for large enough
(1994a) and the arguments for
Lemma
K,
ii)
follows
—
K
A. 11 as long as
>
K.
It
P
p(z,6)
p(z,G)).
in
6),
form of the weights
w st
are restricted to not depend on
of
for some preliminary estimator
should be noted that
Implicitly the
Thus, while they could be chosen based on some fixed
\p
(e.g.
a linear combination
they are not allowed to vary with
i//
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
oo
0.
ip.
B
ii)
For K-nearest-neighbor estimators with
Assumption 4.5 and the approximating functions
with
positive definite,
is
2+e
(£ " P
Assumption 4.5 restricts the form of randomness.
(i.e.
D
D,
>
E[(ft(z.)];
iii)
Neither of these results allow for data based
in
—
8 of Robinson (1987) and Proposition
can be approximated arbitrarily well
from Lemma
P'.
D
easy to use known results to show that Assumption 4.5 holds
follows by
a) holds
=
i)
some cases and
for nearest neighbor and series estimators.
oo,
0;
(s,t=l,...,n),
1,
J =1
Assumption 4.5 can easily be checked
if
/n
n
V. ,w.. =
0,
iJ
-^
[."^(z.)/n
then
oo
4.3,
assumption specifies that
4.6:
J
—^->
oo
as
n
—
>
J
must go to infinity with the sample
size.
m.
As mentioned earlier, the degree of approximation
14
J
can be random,
in a
very general
and
However,
way.
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
E[p(z,G)\x] =
Theorem
4.1:
4.1 - 4.6
are satisfied, then
It
If
II
has a unique solution on
0-/3
II
-^
and
llg-g
Bjc?
-^
II
at
result:
and Assumptions
6
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
it
require some work 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
is
I- Q r
-v
V
J
and
£
4.2:
is
This result
is
and
vi)
;
Corollary
B
is
for
y>(v)
E[sup
is
all
:
v e V};
v,
There
ii)
g(v)
=
iii)
is
v g V, sup |g(v)|
for
llgll
If Assumptions
]
<
3.1,
11/3-pJI
V
a compact interval
= sup
|g(v)|;
iv)
continuous and bounded away from zero on
M(z,v)
vgV
compact then
is
one-dimensional;
such that ^ = <g(v)
^ B|v-v|
|g(v)-g(v)|
<p(v)
v
i)
^
The support of
V;
v)
4.2, 4.5 - 4.7
-^
Hg-gJI -^>
and
It
are satisfied,
is
(3
e B,
Ptv,^) =
satisfied,
0.
easy to relax the assumption that
one dimensional, using the results of Elbadawi, Gallant, and Souza (1983).
noncompact support for
v,
using the results of Gallant and Nychka (1987).
quite thick tails, with
B,
oo.
restrictive in several ways.
difficult to allow for
and a
w(v) = C(l+v'v)
in
although this extension
is
more
possible
Unfortunately, their result allows for
Assumption
15
is
It
v
4.3.
This tail behavior does
w
when
not allow Assumption 4.3 to be satisfied
<p(v)
is
the standard normal density.
Of
course, there are fast computational methods for generating data from densities
(1+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
give satisfactory results in practice.
Consider the model
*
*
(5.1)
y = 5
+ 5
{
2
w
—
+ 5 e
+ e,
3
= 8
8
[
2
=
I,
<
is
N(0,1),
*
w
w
=
=
w
tt
+i),
+
x +
tt
N(0,1),
is
7)
v,
The regression equation for
= n
=
model
is
tt
this
1,
x
one that
relationship between consumption and income.
discussed in Section
6,
where
it
is
used
and
is
v
are
useful in estimating the
This specification will be further
in the
observations
was
was chosen
1.
The parameter values
empirical example.
were set so that the r-squared for the prediction equation for
signal to noise ratio
N(0,.5).
w
*
was
The number of observations was set to
1/2,
100.
and so the
The number of
to be small relative to typical sample sizes in economics, to
*
make computation
than typical
in
easier.
The r-squared for the regression of
w
on
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.
16
Results for three different
estimators of
5
,
5
and
,
squares (OLS) regression of
The third estimator
7r„x..
with
C(x.) = I®R.
This estimator
matrix.
W
and
is
instrumental variables are
third-order, where
y
is
=
(l,w,e
)
that are
an instrumental variables estimator (IV)
is
with the same right-hand side but with instruments
+
first estimator is an ordinary least
on the right-hand side variables
y
The second estimator
measured with error.
The
are reported.
5
R. = (l,h.,h. ,h.
)'
,
where
h.
a simulated moment estimator (SM) from equation
I®(£.
R.R'.
)
where
,
I
is
=
7i
(3.9),
a two dimensional identity
a system two-stage least squares estimator where the
Also,
R..
=1,
was a Hermite polynomial
P[v,-y)
y_ = y„ =
0,
and
y
was estimated.
of the
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 are substantially
As previously noted the IV estimator
smaller.
it
leads to bias reduction.
estimator
is
It
is
is
inconsistent, although in this
example
interesting to note that the standard error of the
smaller than that of the IV estimator.
SM
Thus, in this example the valid 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 (1995) to be important
curves.
in the
in
estimation of nonlinear Engel
This section adds to that work by estimating a nonlinear, nonpolynomial Engel
17
curve for the model of equation
Hausman, Newey, and Powell
a specification that was not estimated in
(2.1),
Also, the results here take account of
(1995).
measurement
error in the denominator of the share equation.
The functional form considered here
S.
(5.1)
=
5,
is
that preferred by Leser (1963),
+ 5„ln(I.) + 5_(1/I.) + £.,
*
where
the share of expenditure on a commodity and
is
S.
I.
is
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
measurement error has been accounted
for.
*
a specification
In addition,
in
is
considered that accounts for the presence of
the denominator of the left-hand side of this equation.
I.
This "denominator problem"
*
results
from the fact that
ill
= Y./I.,
S.
where
Y.
is
the expenditure on the commodity.
l
*
Thus,
if
results
is
I.
measured with error, another nonlinear measurement error problem
from using the measured shares.
This problem can be dealt with by bringing
out of the denominator, giving
Y.
(5.2)
=
l
If
e.
5,1.
1
l
+ 6_I.ln(I.) + 5
2
l
l
3
+ I.e..
11
E[e.|I.] = 0,
satisfies the usual restriction
then equations
(5.1)
and
(5.2)
are equivalent statistical specifications, in that running least squares on either
equation should give a consistent estimator.
Covariates will also be allowed in this
*
specification by allowing additional variables
equation, corresponding to inclusion of
equation
x
I.x
.
to enter linearly in this
as additional regressors in the share
(5.1).
The measurement error
will be
assumed to be multiplicative,
the observed total expenditure,
18
i.e.
for
I.
equal to
I.
*
ln(I.)
(5.3)
*
= w. = n'x. +
1
i
In the empirical
ln(I.)
v.,
1
1
=
11111
*
ln(I.) +
l
work the predictor variables
*
= w. = w. +
T).
tj..
will be a constant,
x.
age and age
squared for household head and spouse, and dummies for educational attainment, spouse
employment, home ownership, industry, occupation, region, and black or white, a total of
19 variables,
With this specification for the measurement and
including the constant.
f(w
prediction equations,
= 8
,<5)
The measurement error
+ 5
w
+ 5
exp(-w
in the left-hand side
as in the Monte Carlo example.
),
denominator can be accounted for as
*
*
f(w
equation (5.2), leading to a specification with
It
is
that
interesting to note that even
5
=
0,
if
,5)
*
= 5 exp(w
the share equation
is
+ 5
)
linear in
this equation is nonlinear, so that IV will not be consistent.
measurement error
in the
*
exp(w )w
ln(I.),
in
+ S
.
so
Thus,
denominator of the share suggests the 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
first quarter of 1982.
Results were obtained for four commodity groups, food, clothing,
transportation, and recreation.
The number of observations
empirical results were reported as elasticities,
econometrics.
To compare shapes,
i.e.
in the
dlnf(x)/dlnx,
data set
as
is
common
is
1321.
The
in
were calculated at the quartiles of
elasticities
observed expenditure.
The results are given
in
Tables
Two through
statistics, including the quartiles of the
Five.
Table
income distribution.
include estimated expenditure elasticities at these quartiles.
information on the prediction regression.
R
The
2
in this
Two
gives
The other tables
Table
Two
regression
The other information
quite sizable for such a cross-section data set.
some sample
will
also gives
is
is
.23,
which
is
useful in
calculating the magnitude of the measurement error and bounding the size of the variance
of the prediction error
v.
In particular, the
model we have assumed implies that the
19
standard error
.45
of the residual
an upper bound on the standard deviation of both
is
the measurement error and the variance of the prediction error
Also, given an estimator
Var(v)
of
cr
1/2
v.
R
an estimator of the
,
2
of the
measurement
equation, that determines the magnitude of the measurement error bias in a linear model,
is
Var(7r'x + v)/Var(w) = [Var(ir'x) +
0-
2
]/Var(w) = [(.25)
2
+
^
2
2
]/(.51)
tt
.24 + (3.8)<?
2
Tables Three to Five give results for each commodity for three different
specifications of the share equation and four different estimators.
where measurement error
results for the share equation,
left-hand side
ignored.
is
This specification
in the
Table Three gives
denominator of the
the same as in the Monte Carlo study.
is
Table Four changes the specification to account for the left-hand side denominator by
multiplying through the original equation by total expenditure, as described above.
Table Five adds covariates
regional price effects.
dummy
three regional
x
to the share equation to allow for demographic and
There are six covariates; own and spouse age, family
The equation estimated
variables.
is
size,
and
analogous to that of Table
*
Four
in
*
5
+ 5
f(w
accounting for the left-hand side denominator, with
=
*
*
exp(w )w
,x ,5)
+ exp(w )x
+ 8
'
5
It
.
should be noted that this specification
from the prediction equation.
restricts family size to be absent
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
(5.4)
W
U.
i
= E
Z = n T."
,
^i=l
£._ C(x.)p.(a)/v n.
l
J=l
J
equal to the inverse of
Specifically,
1
i
^J=l
J
20
x.x'.)~ x.(w.-7r'x.),
J
J
li
moment
equation (3.9), with
l
i
i
in the
U.U'.,
= C(x.)p.(a) + [S^ n C(x.)p.(a)/57r](y n
i
W
in
moment estimation
l
where
a
is
an
initial consistent estimator.
minimizing choice of
W,
3
This
is
an asymptotic variance
that accounts for the presence of
n
in
p..
The standard errors for LS and IV were calculated from heteroskedasticity consistent
The standard errors for simulated moment
formulae, e.g. as given in White (1982).
estimators were calculated from the
H =
GMM
asymptotic variance estimator
(H'Z
H)
where
,
n
aY. ,C(x.)p.(a)/aa.
i
l
^i=l
A
selection process
was used
to include in the instruments.
to choose the order of
powers of the predicted value
Starting at the second order, the
minimum needed
to
have enough moments to allow estimation of distribution parameters, the order was
chosen by cross-validation on the food equation, Gaussian, simulated moment estimator
(SMO), using the cross-validation criteria for choice of instruments suggested in
Newey
Inclusion of higher order powers did not result in any decrease in the
(1994b).
cross-validation criteria.
variables were
(1,
x'n,
Consequently, in Tables Three and Four the instrumental
-2
(x'rc)
).
In
Table Five
exp(x'n:)'X
was added
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
asymmetry
term was
in the distribution of
tried.
In
v.
If it
was
in
Table Three.
Inclusion of a third
This term allows for
statistically significant, a fourth order
none of the cases was this term significant, so only results for the
one, third order, Hermite polynomial
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
o-
= 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 overidentification
3
The procedure used to obtain the initial consistent estimators was to begin with an
identity weighting matrix, use a few iterations to obtain "reasonable" parameter values,
choose
W
as in equation (5.4), and then minimize to get
21
a.
(minimum chi-square)
test (Q) statistic for the simulated
The t-statistic
reported.
the coefficient on inverse expenditure were zero.
if
The degrees of freedom of the overidentification test
Table Five.
SMO and
also
particularly relevant in Table Three because the 2SLS
is
estimator would be consistent
respectively for
moment estimator are
statistic are
SMI, in Tables Three and Four, and
The difference between these
statistics for
and
2
and
8
respectively in
7
SMO and SMI
is
1
a one-degree of
freedom chi-squared test of the Hermite coefficient being zero.
Even though the IV estimator
estimator
in a
number of
inconsistent,
is
When
cases.
it
gives results similar to the
the share denominator
with error there are larger differences between IV and SM.
smaller than those of IV, which
is
allowed to be measured
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
apparent that accounting for measurement error
the denominator leads to some changes in the results.
food equation in Table 4 than
more precisely estimated
larger in Table
4.
in
in
Table
3.
There
these equations.
Surprisingly, the estimated standard errors in Table 4 are not
most cases SMO
is
in the
cr
is
The overidentification test statistics are
is
a levels equation that
thought to be more heteroskedastic than the share equation.
In
more nonlinearity
The prediction error standard deviation
larger than those in Table 3, although Table 4
nonnormality.
is
There
is
quite similar to SMI, except for
is
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 makes a big difference.
simulated moments estimator
is
in
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
22
errors.
These results illustrate the usefulness of using simulated moment estimation to
correct for measurement error, while allowing some flexibility in the distribution of the
prediction error to asses the impact of allowing for nonnormality.
23
Appendix
The proof proceeds by verifying the hypotheses of Theorem
Proof of Theorem
4.1:
Newey and Powell
(1991).
=
"B-xS
S
compact by
is
and
-1
S
Y,
(z,v ,...,v
Note that for
_,H(z,/3,v )P(v ,y).
II
ell
=
+
11/311
Note that
llgll.
Z
simulations let
S
Also,
).
be
(/3,g)
For
compact.
"S
Z =
augmented data vector, with
S
9 =
Let the norm for
5.1 of
denote the
p(Z,6) =
let
Pq( v ) = g (v)/<p(v),
it
follows by Assumptions
4.2 and 4.3 that by the triangle inequality
^ ^
IIH(z,P
sf1
(A.l)
llp(Z,9
(A.2)
{Etllp(Z,e n )ll
)ll
,v )ll|p
s
2+e 1/(2+e)
^
]}
= C-{E[J [M(z,v)w(v)]"
It
1_€
V (v)~
dv]»
iip(z,e)-p(z,e)n ^ c-<y
s
<
S
Newey and Powell
so that Assumption 5.1 of
and 5.3 then follow by Assumptions 4.4 and
E[p(Z,6)|x] = E[p(z,/3,g) |x]
6,
S
Theorem
J*
in the
4.1 follows
sup norm.
by
oo.
6 e 0,
))
/s}iie-en,
(1991) follows
4.5.
by
eq.
(A.2).
Assumptions 5.2
Furthermore, by the fact that
for an unbiased simulator, as noted in the text,
5.1 of
/S
Newey and Powell
2.1.
Newey and
The conclusion then follows by
QED.
(1991).
4.1.
by hypothesis and the Arzela theorem, which gives compactness of
Assumption 4.3 follows with
zero and Assumption 4.6,
g(v)/op(v)
]>
The proof proceeds by verifying the hypotheses of Theorem
Proof of Corollary 4.2:
Assumption
>
ll,
S
Powell's (1991) Assumption 3.1 holds by Assumption
the conclusion of
)]" /S}llg
s
1
,m(z,v )[u(v )<ph
^5=1
)]
?)(v
2+e 1/(2+e)
1
)[u(v )<p(v
1
)
1/(2+6)
follows similarly to equation A.l that that for
(A.3)
<I s f 1 M(z,V s )[w(v s
=s
s
C-£ ? <E[<M(z,v
2-e
v
)|/S
(v
PjW- T ne
vi).
w(v) =
1
by
<p{v)
bounded away from
Assumption 4.4 follows by a Weirstrass approximation of
proof then follows by the conclusion of Theorem
24
4.1.
QED.
References
Amemiya,
1985, Instrumental variables estimator for the nonlinear errors in variables
Y.,
model, Journal of Econometrics 28, 273-289.
A.R. Gallant, and G. Souza, 1983, An elasticity can be estimated
consistently without a priori knowledge of functional form, Econometrica 51, 1731-1751.
Elbadawi,
I.,
Gallant, A.R., 1987, Identification and consistency in nonparametric regression, in T.F.
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University Press, 145-169.
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26
Table One: Monte
5
5
1
OLS
-1.
10
.25
IV
- .30
- .09
3.31
2.27
SM
RMSE
SE
Bias
1
3
.
.
Bias
13
.
67
.
.
13
1.
.
07
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:
RMSE
Bias
67
26
05
.85
.47
.31
13
25
1.05
Some Sample
Table Two:
Income Quartiles
6
2
SE
32
2.27
Carlo Results
.
1
.
1
.
.
Elasticity Estimates for Share Equations
Food
50th
75th
LS
.72
(.02)
.66
(.02)
.59
(.03)
4.52
2SLS
.82
(.05)
.78
(.04)
.74
(.06)
47
SMO
.82
(.04)
.78
(.04)
.74
(.06)
(.08)
.84
(.20)
.78
(.09)
.71
.31
SMI
(.36)
y
o-
.61
-.01
(1.49)
(
27
.05)
t-stat
12
2.01
1.62
Statistics
25th
3
SE
Q
3.67
6.36
.04
6.08
RMSE
.
2.
1.
86
07
65
Clothing
LS
25th
50th
75th
1.21
(.05)
1.08
.97
(.05)
18.66
(.04)
2.25
SMI
IS
1.42
1.30
(.12)
(.09)
(.10)
1.63
1.40
1.26
(.20)
(.10)
(.18)
.02
(.38)
1.62
1.28
1.07
-
2SLS 1.61
SMO
o-
(.46)
(.30)
(.56)
(
.56
0009
.15
.0018)
(.11)
.
t-stat
.20
Q
6. 11
1
.
99
Transportation
LS
2SLS
SMO
25th
50th
75th
1.28
1.44
1.50
(.07)
(.06)
(.07)
.99
(.08)
1.06
1.
(.08)
12
(.12)
1.02
1.01
(.06)
1.01
(.06)
1.71
(2.01)
.98
(.07)
.63
(.18)
10
(.07)
(-07)
SMI
1.40
(.27)
t-stat
o~
11.19
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
K
cr
t-stat
16.59
11.45
(.15)
.02
(.34)
(.63)
(.12)
.39
(.48)
6.98
2.32
1.28
.71
(4.40)
(.36)
(.12)
(.18)
6.48
(
28
.024
002)
.
5.58
09
Table Four:
Elasticity Estimates for Level Equations
Food
25th
50th
75th
LS
.68
.04
.63
.03
.57
.03
2SLS
.90
.80
(.05)
.70
(.05)
(.08)
.81
(.04)
.63
(.05)
.35
(.02)
1.21
(.21)
.83
(.06)
.46
(.13)
.24
(.06)
(.08)
SMO
SMI
.
98
y
cr
t-stat
Q
.31
3.34
12.4
.008
(.005
5.09
18.16
16.42
Clothing
LS
25th
50th
75th
1.40
1. 11
(.06)
.89
(.04)
53.95
(.11)
1.21
(.13)
4.92
2SLS 2.04
SMO
SMI
1.50
H
<r
(.26)
(.12)
2.07
1.36
.96
(.21)
(.08)
(.07)
.34
(.02)
2.14
1.37
(.58)
(.11)
.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
1.25
(.59)
y
<r
t-stat
1.68
2.04
(.53)
.004
(.021)
29
38.27
10.63
.69
10.52
Recreation.
LS
50th
75th
1.74
1.26
42.87
(.19)
(.09)
.95
(.05)
1.01
(.15)
14.85
2SLS 1.84
1.32
(.28)
SMO 2.35
SMI
t-stat
25th
(.16)
1.44
(.26)
(.09)
7.85
1.93
(4.76)
(.34)
Table Five:
0"
n
.95
(.07)
.36
(.03)
.33
(.15)
.25
(.02)
.024
48.62
33.23
15.88
21.83
(.005)
Elasticity Estimates for Level Equations with Covariates
Food
25th
50th
75th
.72
(.05)
.66
(.03)
(.03)
2SLS
.97
(.09)
.85
(.06)
.74
(.06)
SMO
1.00
.86
(.05)
.73
(.05)
.33
(.02)
.90
(.07)
.58
(.15)
.21
(.07)
LS
(.08)
SMI
1.25
(.29)
1.08
.88
(.11)
(.06)
(.04)
1.50
1.22
(.14)
(.15)
(.22)
1.31
(.09)
.94
(.08)
.33
(.02)
SMI
2.01
1.33
.002
(.14)
.92
(.23)
.31
(.73)
(.11)
(.011)
1
.
94
43.51
2.47
41.59
r
1.35
(.31)
SMO
.009
7.14
(.007)
50th
2SLS 2.01
Q
4.40
25th
75th
t-stat
1.90
.58
Clothing
LS
7
<r
K
cr
t-stat
32.76
4.22
30
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
(.09)
.87
(.06)
.69
(.04)
(.04)
1.86
1.
13
(.26)
.61
(.07)
.40
(.08)
1
.
07
(.87)
<r
If
t-stat
Q
1.30
3.09
(.45)
.61
.016
28.01
30.12
2.61
28.92
.009)
(
Recreation
LS
25th
50th
75th
1.73
1.25
(.09)
.95
(.05)
35.96
(.19)
1.31
(.17)
1.01
(.16)
10.69
(.32)
1.41
(.11)
.88
(.08)
8.38
5.45
1.91
(.32)
.18
.21
(.23)
(.03)
2SLS 1.78
(.30)
SMO
SMI
2.40
7076
K
cr
.31
t-stat
37.86
54.23
10.91
44.88
(.03)
(
DIO
31
.021
005)
.
Date Dub
Lib-26-67
MIT LIBRARIES
3 9080 01972 0975
i
6
7
8
9
10
11
OREGON RULE
CO.
U.S.A.