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http://www.archive.org/details/nonlinearerrorsiOOhaus
HB 31
.M415
M
working paper
department
of economics
NONLINEAR ERRORS IN VARIABLES:
ESTIMATION OF SOME ENGEL CURVES
J. A. Hausman
W. K. Newey
J. L. Powel
No.
504
November 1988
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
9
1
989
NONLINEAR ERRORS IN VARIABLES:
ESTIMATION OF SOME ENGEL CURVES
J
.
W.
J.
No.
504
Hausman
Newey
L. Powel
A.
K.
November 1988
Revised Draft
November 1, 1988
Comments Welcome
Nonlinear Errors in Variables:
Estimation of Some Engel Curves
by J. A. Hausman, W.K. Newey, and J.L. Powel
1988 Jacob Marschak Lecture of the Econometric Society
Australian Economics Congress
Canberra, Australia
August 31, 1988
Estimation of Some Engel Curves
Nonlinear Errors in Variables:
by J. A. Hausman, W.K. Newey
The
Adcock
(1878)
simple
the
errors
is
in
perhaps
bivariate
variables
the
,
and J.L. Powell 1
problem has
been
long
known
statistics;
in
first reference which points out the problem.
regression model
the
result
of
errors
in
variable
downward bias (in magnitude) of the estimated regression coefficient:
law"
of econometrics as known to MIT students.
econometrics
in
variable problem.
research
the
the
1930' s,
considerable
In
is
a
the "iron
During the formative period of
attention was
given to
the
errors
in
However, with the subsequent emphasis on aggregate time series
errors
econometric research.
in
variables
problem
decreased
in
importance
to
most
In the past decade as econometric research on micro data
has increased dramatically,
the errors in variables problem has once again moved
to the forefront of econometric research.
Solutions to the errors in variables problem for the linear regression
model have been well explored and are often used by econometricians
.
The most
common solution is the use of instrumental variable estimation (IV) which depends
on
the
existence
of
an
appropriate
instrument or
repeated observation of the
MIT, Princeton University, and University of Wisconsin.
We thank Greg Leonard
for excellent research assistance and the National Science Foundation for
financial support.
A. Deaton, A. Lewbel, R. Pollak, D. Jorgenson and J. Poterba
made helpful suggestions.
Presented as the Jacob Marschak Lecture of the
Econometric Society at the 1988 Australian Economics Congress.
2
The notion of the "iron law" is that the estimated effect is (almost) never as
large as economic theory or the applied researcher expects it to be.
Of course,
in the multiple regression situation with many right hand side variables the
result need no longer hold true.
Nevertheless, the folklore in econometrics plus
years of reading students econometrics papers results in the belief that downward
bias in coefficients estimates is a pervasive problem in micro data parameter estimates.
3
Griliches (1986) discusses micro data problems which
variables problems in many typical econometric data sets.
lead
to
errors
in
•2-
infrequently
been
used
by
exist,
Two other solutions
variable measured with error.
econometricians
.
The
first
but
they have only
alternative
solution
involves knowledge or an estimate of the variance of the measurement error(s) of
the right hand side variable(s) or its relative size compared to the variance of
the
stochastic
disturbance,
measurement error
in
the
which
can
left hand
partly
be
side
variable.
usually not available to econometricians.
entirely
or
This
type
composed
of
of knowledge
the
is
The second alternative solution is to
use distributional properties of the right hand side variables or to use higher
order moments which depend on distributional assumptions,
moments,
to
estimate the parameters.
beyond the first two
This approach again has only rarely been
used by econometricians.
Thus,
the
IV
appropach
is
by far the most widely used technique
for
dealing with errors in variables problems in linear multiple regression problems.
The
linear model with measurement error is
isomorphic
equation model, so that two stage least squares or
a
to
a
linear simultaneous
closely related estimator is
Zellner (1970) and Goldberger (1972) extend the single equation errors in
variables problem to the multiple equation context.
Geraci (1977), Hausman
and Hsiao (1976) consider the errors in variables problem in the
(1977)
simultaneous equations situation. An excellent survey is given by Aigner, Hsiao,
Kapteyn, and Wansbeek (1984).
Reiersol (1950) demonstrated identification of the errors in variables problem
using distributional assumptions.
Kapteyn and Wansbeek (1983) generalize the
result to the multiple regression context.
Bickel and Ritov (1987) apply the
method in an adaptive estimation framework.
Geary (1942) originally proposed
using higher order moments in estimation in the errors in variables problem.
Higher order moments may offer a useful methodology given the increasingly large
data sets of many thousands of observations used by econometricians.
They can be
applied in a straghtforward method of moments procedure.
However, the technique
has been little used to date and J. Hausman has been unsuccessful in a few
previous attempts.
.
;
3-
However,
used.
this
relationship no
framework as recently noted by
Y.
longer holds
Amemiya (1985).
in the nonlinear
regression
The reason that 2SLS no longer
leads to a consistent estimator in the nonlinear errors in variables problem is
because the error of measurement is no longer additively separable from the true
variable
2SLS
in
the
Application of 2SLS or nonlinear
nonlinear regression model.
(N2SLS) leads to inconsistent estimates.
A staightforward way in which to see why 2SLS or N2SLS does not yield
consistent estimators
the
in
nonlinear errors
the linear in parameters and nonlinear in
(1.1)
where g(z) is
the
linear
a
yi -
/So
tj^
is
variables specification:
+ Pi g(2i) + *i
in
variables
instead the observed varible
where
model is to consider
i
~
1
n
sufficiently smooth function to do Taylor approximations.
errors
(1.2)
in variable
X£ - Zi +
r?i
Xj_
we
framework,
assume
that
z
is
As in
unobservable
takes the form:
i
-
1
n
assumed to be uncorrelated with z^.
with x in equation (1.1) and taking
a
Replacing the unobservable
z
Taylor expansion leads to:
Fuller (1987) discusses many of these other estimators which may improve the
finite sample performance of IV-type estimators in the errors in variables context.
The failure of additive separability also arises in the reduced from of the
nonlinear
simultaneous
stochastic
equations
problem
where
additive
the
disturbance of the structural form enters nonlinearly into the reduced form.
This situation leads to N2SLS and N3SLS being inefficient relative to ML in the
nonlinear simultaneous equations model as demonstrated by T. Amemiya (1977)
^
4-
+ Pi g(xi) + «i
(1.3) y L - P
where
[
j
Taylor
]
denotes the
expansion
j
-
N2SLS or other IV techniques.
be uncorrelated with
contains both
framework
first
the
derviative
fundamental
the
equation
in
Taylor expansion were
variable would exist.
(1.3)
absent,
However,
problem with
of
g(x^)
so
that,
it
is
but
,
In the linear errors in variables
.
is
unity
even
if
observation error
the
so
higher order terms of the
the
unlikely that an appropriate
the additional
is
This linear separation is
linearly separable from the right hand side variable.
not present
fu.J/J'1
the first term of the Taylor expansion
However,
and the derivative of g(x^)
r/^
Xi )
The instrument must be correlated with g(x^)
and e^.
rj^
(
Inspection of the first term of the
demonstrates
(1.3)
gUl
2
0!
-
th derivative of g.
equation
in
1
ft g (x i )r?i
instrumental
factor of the added terms in the
Taylor expansion beyond the first make the problem even more unwieldy to solve
with the usual instrumental variable techniques.
To
date
variable model
the
methods
depend on very
proposed
strong
to
estimate
restrictions
on
nonlinear
the
the
measurement errors of the unknown regression coefficients.
errors
in
distribution of the
However, knowledge of
the parametric form of the distribution function of the measurement errors is not
sufficient for consistent estimation.
the
true
values
of
the
regressors,
An additional assumption is needed that
e.g.
the
Zj_
in
assumed to be random drawings from a distribution with
Instead,
equation
a
(1.1),
are
also
known parametric form.
if the true regressors are treated as fixed but unknown constants,
then
maximum likelihood estimation is inconsistent due to the "incidental parameters"
problem of Neyman and Scott (1948).
8
Aigner et. al
variables models.
.
(1984)
have a brief discussion on ML for nonlinear errors in
.
,
5-
An alternative appi"oach assumes that a large number of measurements on
each
regressor
true
exist,
that
so
errors
for
then
the
follows because
regressors
average
these
of
measurements
closely
Consistent estimation of nonlinear errors in
approximates the true regressors.
varibles models
the
the
approaches
covariance matrix
zero
as
the
the
measurement
size
increases.
of
sample
Estimators under this type of assumption have been proposed by Villegas (1969)
Dolby and Lipton (1972), Wolter and Fuller (1982b), Powell and Stoker (1984), and
Y.
Amemiya
situation
This
(1985).
seems
unlikely
to
occur
very
often
in
econometrics
Lastly,
Griliches
and
Ringstad
(1970)
analyzed
quadratic
a
specification and demonstrated that the bias of least squares can be exacerbated
by the nonlinearity
Wolter and Fuller
.
(1982a)
propose a consistent estimator
for the quadratic specification so long as the errors are normally distributed.
Neither
instrumental
variables
nor
additional
measurements
are
required
for
estimation.
In this paper we discuss consistent estimators for nonlinear regression
specifications when errors in variables are present.
Our estimators depend on
the existence of instrumental variables or a single repeated observation.
Thus
we do not require the large number of measurements or shrinking covariance matrix
assumption of much previous research.
In Section 2 we discuss an estimator for polynomial specifications in
the
true
Powell
regressors.
(HINP)(1986)
This estimator proposed by Hausman,
leads
to
Ichimura,
Newey,
and
consistent and asympotically normal estimators so
long as either instrumental variables or an additional measurement of each true
regressor
are
present.
An
interesting
result
emerges
in
the
instrumental
•6-
variable case because the model turns out to be overidentif ied.
tests of
Thus,
the model specification are possible.
In
Section
3
discuss
we
an
estimator
for
Newey
and Powell
,
(HNP)(1988)
yields
a
nonlinear
This estimator proposed by
specification when errors in variables are present.
Hausman,
general
the
consistent
additional measurement of each true regressor is present.
estimator when
To date,
an
we have not
been able to establish asymptotic normality of the estimator or to extend it to
the instrumental variable situation.
However, Monte Carlo evidence provides some
indication that the distribution of the estimator is not badly behaved so that we
use bootstrap estimates of the precision of our estimates.
In Section 4 we apply our methodology to estimation of Engel curves on
household data,
research on.
a problem which econometricians
have done considerable previous
Here we find a number of interesting results.
First, we find that
the "Leser-Working" specification of budget shares regressed on the log of income
or expenditure should be generalized to higher order terms in log income.
Also,
we find that errors in variables in either reported income or expenditure should
be accounted for.
However, we do not find evidence that more general functional
forms beyond polynomial specifications in income improve estimation of the Engel
curve significantly.
Lastly, and perhaps most interesting, we find rather strong
support for the Gorman (1981) rank restriction on the matrix of coefficients for
the
polynomial
analysis,
estimation.
findings
a
terms
in
income.
restriction
from
This
is
result
Thus,
economic
remarkable
after
theory
if
over
100
may
future
years
affect
research
of
the
leads
Engel
curve
econometric
to
similar
-7-
Identification and Estimation of the Polynomial Functional Model
II.
estimation
consider
first
We
parameters
the
of
of
polynomial
the
specification
K
yt =
(2.1)
S
0*-
(
J
j=0
Zi )J + £i
- l,...n
i
which is a kth order polynomial in the unobservable variable z^.
Zj_
as
{zj_}
a
interpreted
be
modifications
in
the
as
sequence
a
(2.2)
first
xj_
regularity conditions.
- Zi +
estimator
fixed
of
same relationship to the unobserved variable
Our
alternatively,
random variable with unknown distribution function;
can
uses
rji
i
-
1
The
constants
the
appropriate
with
observed variable x^ has the
as in Section I:
Zj_
n.
additional
the
We will treat
information
of
repeated
single
a
measurement w^ of z^ with an additional measurement error v^ defined by
(2.3)
w^-z^
+
v-^
i-l,...,n.
Two points of interest arise from the specification and assumptions of
equation (2.3).
is
independent
nonlinearity
.
First, we will assume that v^ is uncorrelated with
of
These
Zj_.
The
assumptions
independence
are
assumption
analogous
to
the
is
e
and
£
required
linear
case
so
rjj_
by
and
the
that w^
could be used as an instrumental variable if the specification of equation (2.1)
were linear.
Second, we will not impose the usual restriction E(v^) -
a constant term can be present in the measurement equation (2.3).
so that
Alternatively,
vj can be assumed to have zero mean, but the slope coefficient of zj in equation
-8-
The details of the derivation of the estimator in
can then be non-unity.
(2.3)
this latter case is left to the interested reader.
We now turn to sufficient regularity conditions to allow identification
estimation of the
and
maxi
>
j
|
ajj
|
.
i
.
d.
1
random variables
The
:
Vi) - E
E
(ii)
vj is independent of z^
(iii)
E
£i
(
1
matrix norm
the
|A|
|
=
|
e^,
r?^,
vj_
,
and
are
z^
jointly
1
z i(
|
(e it
Vi
,
Vi
2K
(r/i
Zi
,
|
2K
Zi,
V£) -
2
cc
)||
<
All necessary moment matrices are nonsingular.
(iv)
assumption
i.i.d.
heterogeneity
assumptions
or
linear
derivation
of
allow
to
A.l(i),
both
for
(iii),
is
necessary
dependence
either
and
identification
Only assumption A.2(ii)
Independence
case.
specification.
relaxed
be
Assumptions
both.
allow
to
can
distribution of the estimator.
usual
Define
with
(i)
The
(2.1).
We make
.
Assumption
i
equation
in
/3j
is
because
are
(iv)
and
the
or
standard
asymptotic
stronger than in the
of
the
nonlinear
Note that assumption A.l(i) could be strengthened to independence
for purposes of symmetry; we require only the weaker no correlation assumption.
For
equations.
identification we
Define the moments
for m - 0.....2K.
£p
consider the population analogue of the normal
- E[y^ (z^)P] for p - 0.....K and
The normal equations z'y - (z'z)
take the form
/J
K
(2.4)
Cp -
Z
Q
Pi
* j+p
P -
K
4>
m
m - E[(zi)
]
.
,
-9-
Both sides of equation
depend on the unobservable variables z^.
(2. A)
However,
the unobservable moments can be derived from the observable moments E[x^ (w^)P]
E[(wi)P],
and E[yj (w^)P]
= E[( Zi )°]
-=
E[(vi)°] =
E -
(2.5)
[
We now use assumption A.l and the fact that E[(wj_)°]
.
1
to find:
Xi (wi)J-
1
J
-
]
J
2
p-0q
<I>
]
p
[
j
Equations
allows
(2.6)
E [(Wi)J] -
(2.7)
E
(2.5)
[y i (w i )J]
and
L
=
which
have
a
S
[
Q
p
p ^j.p
Cp i/j-p
]
That
the
parameter
for
j
for
relationship
-
1
j
-
[
(v t ) J
]
2K.
K.
and equation (2.7)
(2.5)-(2.7)
between
the
then
defined (5K +
moments
of
1)
the
elements of the unobservable moment vectors
1)
$ and v each of which have 2K elements and
of
,2K where v$ - E
1
equations
is,
one-to-one
observable variables and the (5K +
derive recursive
-
for
^_ pl
allow identification of z'z,
(2.6)
identification of z'y.
equations
*
[J]
p+1
£
HINP (1986)
which has K elements.
relationships which permit convenient solution of the elements
vector
6
-
($'
,
u'
£').
,
Once
6
is
computed,
/3
then
is
identifiable as a solution to the normal equations (2.4).
To derive the asymptotic distribution of the estimator define the (5K +
1)
dimensional data vector
(2.8) mi - [x i ,...,x i (w i ) 2K
Define
the
population
moments
to
1
be
Wi
,
fi
—
(wi)
E[m^]
.
2K
,
y i ,...,y i (w i )
The
moment
K ].
vector
/i
is
-10-
consistently estimated by the sample average moment vector m and application of
the Lindeberg-Levy CLT yields the asymptotic distribution of m
d
Jn (m
(2.9)
The
elements
of
-
are
9
diffentiable relationship
the delta method,
N (o,
-»
/*)
estimated
then
6
for
fi)
= h(/i)
fi
= E [m^mi']
by
w'
-
.
continuous
the
continuously
and
also known as
First order approximations,
.
lead to the asymptotic distribution of G
d
(2.10)
The
elements
of
Jn
(5
-
6)
+ N
(0,
Jacobian
the
HfiH')
matrix
H
for H
-=
can
be
the
normal
dh(fi)/8^'
calculated
recursively
with
computational details given in HINP (1986).
Lastly,
estimated
0.
we
solve
for
fi
using
Let D be the second moment matrix of (1,
D(9) based on the estimated 6.
(2.11)
fi
We estimate
fi
...,
(z^)^)
and
the
and D -
- fr 1 i.
matrix which gives S$ 6 - vec(D)
7n
V - D" 1
{
and S$ as the selection
HINP (1986) derive the asymptotic distribution
.
d
i_
(2.12)
z^,
(2. A)
by
Define S| as the selection matrix which gives Sc 6 -
We have
equations
(fi
[
-
S^
fi)
-
- N
(fi-
(0,
®
I
V) where
K+1
)
S$
]
H
fi
demonstrated identification and derived
H'
a
[S£
-
(/3'®I
R+1
)
S^'D"
1
consistent and asymptotically
11-
normal
estimator
case
the
in
of
measurement
replicated
single
a
for
the
unobservable variable z^.
We
now
turn
estimation
and
identification
to
when
instrumental
variables which allow prediction of the unobserved regressor z^ are available.
Thus, we assume that z^ is determined by the p dimensional vector of instruments
zi - qja +
(2.13)
To
sufficient
state
assumption
A.l.(ii)
for
v^
that
sufficient
the
n.
1
estimation
and
identification
assumption
an
to
-
j_
conditions
Otherwise
independent.
vj_
and
conditions
instruments
the
quite
are
we
similar
change
q^
are
to
the
previous case that we considered:
Assumption
i
.
i
.
d.
(i)
2
random variables
The
:
e^,
rj^,
and q^
,
are
jointly
with
E
(«i
I
qi, vj_) - E
(f/i
qi,
|
v^
-
,
E
(ci^i qi
[
(ii)
vj is independent of q^ with E [vjj -
(iii)
E
(iv)
All necessary moment matrices are nonsingular.
Again,
vj_
[HCci, r,0\\
the i.i.d.
2
]
<-,
E
tMvi, qi
2(K + X)
]
||
Vi) - a €r?
<«
assumption can be relaxed to more general situations.
For purposes of identification we assume
that a is known since it is
identified from
(2.14)
,
xj[
- q^a +
»7i
-
V£
i
- l,...,n.
12-
Let wj - qia and again denote
We must again determine the
- E[(v^)J].
«/j
i/j
for
identification and estimation.
Substitution of the instrumental variables into equation (2.1) yields:
(2.15)
K
S 7i
j-0 J
(i)
yi -
(ii)
7j
(iii)
ei - ei +
K
Equation (2.15)
(iii)
-
p
S
[
]
/?
j-0
implies that E(e^
from
j
wj)
|
/3
]
[
p
F
[(vi)
K
p -J-^
]
P
(wi)J
J
so that 7 is identified by the
-
(i).
f)
and v in 7 which must be sorted out
Before proceeding to do so, note that since vq —
for identification.
identified
P
J
P-J
2
We now have the convolution of
we have 7^ —
j-0
p yp.j
least equares projection of equation (2.15)
0,
where
(wi)J + ei
/S^
and 7^-1 — ^k-1-
equation
(2.15)
(i)
1
and v\ —
two highest elements of
Thus,
the
alone
which
will
subsequently
/9
lead
are
to
over identification.
To complete the identification, we now multiply through equation (2.1)
by the observable variable x^ and we substitute z^ - w^ +
(2.16)
(i)
K+l
x i v i " -Sn
<
H( w i)-'
+ ui
where
K
6
i
-^
[P]
ViVj
"
J
1
K+1
v^_
-13-
K+l
K+l
jS
pSj
+ z^e*
Again,
disturbance
the
expectation so that E[u^
|
w^
- 1
j_
term
,
/9
from
Sj^+l
vjj - 0.
we
Thus,
c$0
,
-
-
•
,
<$k+1
We
•
parameters in
we
can
u.
have
(2.16)
(2K
(Wi)J +
)
(i).
5
iyi
a
-
in
\
conditional
follows from the least
We can again identify the two highest
Over identification of these two
coefficients.
6
+
reduced
3)
form
coefficients
have
K+l
Thus,
we have overidentif ication of order
discard two equations
[r,
zero
has
(i)
The estimate of
~ ^K anc^ ^K " ^K-l-
parameters follows from the 7 and
Vj
•
n.
equation
in
squares projection of equation (2.16)
order terms of
J
sP'J
1-
t(vi)
£ _l
p
t'l
coefficients
unknown
and still
identify
the
and
70>---i7K>
unknown
K
nuisance
or equivalently,
2,
unknown parameters.
The
solution to the equations once again follows a convenient recursion relationship.
HINP (1986) give the recursion formulae.
Estimation
parameters 7 and
procedure
research.
need
The
6.
not
proceeds
from
Given 7 and
be
initial
we
6,
asymptotically
derivation
of
interested reader to HINP
then estimate
efficient;
asymptotic
the
estimator is straightforward, but tedious.
the
estimation
(1986)
the
f)
of
the
and
v.
topic
distribution
is
of
reduced
form
This two step
left
the
to
future
resulting
We give only a sketch here and direct
who give the complete derivation.
Taking
account of the fact that a must be estimated, the asymptotic distribution of the
reduced form parameter is normal, say
(2.17)
/n"
7
7
6
8
d
.
.
1
|
14-
Given
equation
minimum
we
(2.17)
chi-square
can
Denote
estimation.
estimates
efficient
obtain
then
the
(2K
+
vector
2)
of
of
the
/9's
reduced
by
form
coefficients, after elimination of Sq by
(2.18)
ft
- (7,
-=
(tli,
S)
fl
where
2)
ft
-
x
(
7K
,
7^,
*K+1
'
^ k)
'
Similarly, denote the 2K vector of b and u parameters as
(2.19)
6 - (£, v)
- (e lt 6 2 ) where & 1 = (0 K
,
0^)
.
The unknown 6 parameters follow from the reduced form
(2.20)
II
parameters
n - h(6)
so rearrange the covariance matrix M from equation (2.17)
to conform to equation
(2.20) and denote a consistent estimate of the rearranged matrix, V.
The minimum chi square estimator is then
(2.21)
Q - min
[ft
-
h
(e)
'
]
v" 1
[ft
-
h(e)].
e
The
value
of Q
from
equation
(2.21)
can
be
used to
test
the
overidentifying
restrictions since it is distributed as a central chi square random variable with
2
degrees of freedom if the specification is correct.
The test of identification
•15-
is
equivalent to
coefficient
of
a
zj
test of equation (2.2):
cause
system
the
be
to
non-zero mean and non-unity slope
a
just
The
identified.
asymptotic
covariance matrix of the minimum chi square estimator takes the usual form
(2.22)
P
Tn"
-
ft
N
(HV'l-H')"
(0,
specification of equation
The
besides the polynomial terms.
However,
1
)
where H - 3h(e)/d6'.
contain other variables
does not
(2.1)
in many applications we might expect the
appropriate specification to be
yi -
(2.23)
K
2
0j
Q
(
Zi )J +
R^
+ ei
where we assume that the Rj are measured without error.
technique does not work for equation
variables
specification.
replicated
variables
equation
R.
measurement
Instead,
and
accounted for
are
(2. A),
(2.23)
we
apply
in
the
the normal equations.
two
The
matrix R'z
depends
different
setups.
replicated measurement
approaches
The
case
for
additional
by
the
R^
considering
We need to augment z'y and z'z to include
Thus we need to form the matrices R'z and R'y.
observable.
The usual partialing out
because of the nonlinear errors in
variables
instrumental
n.
1
on
the
The latter matrix is directly
unobservable variable
z;
however,
Because 62 is just identified, equation (2.21) may be further simplified using
partitioned inverses. 62 follows from II2 while the overidentif ied parameters 6^
are estimated from 11^
See HINP(1986) for computational details.
,
.
16-
equation
(2.5)
with
R^
included
permits
computation
estimate
an
of
of
the
required moment matrix.
Inclusion of additional regressors in the instrumental variables case
The addition of
is also quite straightforward.
R^
to equation (2.15)
effect on the disturbance e^ so that 7j in equation (2.15)
Similarly,
to the
5-;
is estimated from equation (2.16)
-y*
and
has no
remains the same.
(ii)
after the term
w^R^
is added
In both cases least squares projections
right hand size of the equation.
yield consistent estimates of
(i)
(i)
so
6a
that
estimates
of
/3
and v
can be
calculated from the estimated reduced form parameters.
In this section we have proven identification and developed consistent
estimators
for
the
measurement
or
measurements
can be
estimated
polynomial
instrumental
included
parameters
offers
a
specification
when
present.
variables
are
easily;
minimum chi
a
convenient
single
replicated
Additional
replicated
either
square
a
combination of the
Similarly,
approach.
replicated
a
measurement and instrumental variable situation can be combined using
chi square approach.
a
minimum
In both cases we increase the efficiency of our estimator,
or alternatively, we can test the specification of our model.
However, we do not
claim to have found the most efficient estimator since there exists an infinite
of unconditional moment
class
estimation.
We
leave
the
restrictions which can,
construction of feasible
in principle,
efficient
be used in
estimators which
attain an efficiency bound as a topic for future research.
w -
This approach is equivalent to using the transformed replicated measurement
R(R'R)" 1 R)w together with equations (2.4)-(2.7).
(I
:
-17-
III.
Errors in Variables in a General Nonlinear Specification
identification and consistent estimation of
We now discuss
nonlinear errors
Hausman,
Newey
measurement
variables specification with errors
in
and Powell
,
have
we
case;
been
instrumental variables case.
unable
Also,
we
normal distribution for the estimator.
extend
to
do
in variables
following
limited to the replicated
Our estimator is
(1988).
estimation
as
yet
investigations
Carlo
to
the
not currently have an asymptotically
We lack the centering correction for the
estimator which would permit derivation of the asymptotic distribution.
limited Monte
general
a
indicate
that
bias
the
of
However,
estimator is
the
small and that bootstrap estimates of the sampling distribution of the estimator
provide a good indication of the precision of the estimates.
consider
We
general
the
regression
nonlinear
model
with
additive
disturbance
yi - f(z i? 0) + ej
(3.1)
where
we
take
without error
variable
is
zj_
z
is
be
to
scalar.
-
n.
1
Inclusion of additional variables measured
straightforward using the approach of the last section.
unobservable
replicated measurement
make
a
i
assumption A.l
w^_
;
is
instead we observe x^ as
in equation
determined similarly by equation (2.3).
(i)-(ii)
of
Section
II
where
independent of z^ is crucial for our estimator.
the
assumption
3
:
2
and E
[
|f(zj_,
T>0 such that E
[
exp (T
E
[
|
«
i
1
]
P)\
|
2
(z^
The
(2.2).
Lastly, we
that
vj_
is
The moment assumptions that we
make are
Assumption
The
]
are finite and there exists
,
r)^,
Vj_)|}]
is finite.
18-
assumptions
these
Using
coefficients
unobservable
of
,
and
results
of
projection of y^
linear
the
the
Section
II,
we
polynomials
on
can
of
estimate
the
true,
the
but
regressors.
We denote this polynomial approximation using the estimated moments for
the normal equations (2.4) as
(3.2)
where
£p-
projection coefficients
p-0....K.
ft
j+p
Once we have the estimated n(K)
we
can
estimate
the
true
regression
function
fo( z )
"
by
(3.3)
which
jK
stands for a Kth order polynomial.
II(K)
f(z,/?)
ft
j£o
is
a
£ K (z)
- .S
nonparametric
n jK (zJ)
estimate
of
the
regression function.
Equation
(3.3)
provides an estimate of the true least squares projection
(3. A)
of y^ on z-^(K)
f K (z) -
iQ
II
jK (zJ)
and also provides an estimate of the least squares projection of
fg(z^) on z^(K) because e^ has zero mean conditional on z^ by Assumption A.l (i).
Furthermore,
A. 3,
is
existence of the moment generating function of
sufficient
polynomials
for
to
form
a
integrable functions, which implies that
(3.5)
lim^
E
[(fo(zi)
-
f K Ui)) 2
]
-
basis
for
Zj_,
given assumption
the
space
of
square
19-
provides an arbirarily good mean square approximation
For K large enough fj^(z^)
of f (zi>.
Given
fR( z )
estimate
nonparametric
the
of
regression
true
the
function,
we estimate P by imposing the restrictions implied by the parametric model
-
on f^( z )
^ e use
•
a
minimum distance approach and obtain
P - argmin
(3.6)
^
J
fi
[fj( X j.)
«( Xl )
1
f( Xi
-
from
ft
P)]
,
2
where lo(x^) is a nonnegative weight function and B is a feasible set of parameter
values.
the
Note that the observed variable x^ is used in equation (3.6) in place of
unobserved variable z^.
Other observed variables could also be used which
We restrict our attention here to the
could lead to more efficient estimators.
xj_
with an analysis of other variables left to future research.
the weight
The purpose of
function w(x^) is to take account of the substitution of x^ for the
unobservable z^ so that the mean square approximation holds
lim^
(3.7)
Also,
we
require
E
[w( Xi )
regularity
{
(x t )
f
-
conditions
fK
(Xi))
and
an
2
]
-
identification
assumption
to
proceed
Assumption k
;
(i) B is compact.
(ii)
with probability one.
SUP
j9eB
I
f(Xi
'
/9)
I
f(x^, P)
(iii) There exists d(.) such that
is finite
~ d(Xi) and E [d(x i )2j
all p in B such that p / p
where Pq is the true
is continuous at each P in B
p.
,
E
[w(x i )(f(x i
,
Pq)
-
<
-
f( Xi
,
iv ) for
P))
2
]
>
•20-
Our last assumption imposes a condition on the weight function
Assumption
5
distributions
The
:
of
z^
and x^
are
absolutely continuous with
densities g z (.) and g x (-) respectively such that there is a positive constant
with w(.)g z (.) <
C
C
gx (.).
While estimation of the weight function may sometimes require careful attention,
in the
common case where the density of z^ is continuous and nonzero everywhere
and the the density of x^ is bounded,
then any w(x^) which is zero outside some
bounded set will suffice.
Hausman, Newey and Powell (1988) then prove that given the assumptions
and the additional condition that the density of z^ is bounded away from zero on
an interval,
determined from equation (3.6) is consistent, plim
then
/3
-=
fi
,
if
K which is chosen as a function of the sample size K(n) has the properties that
K(n)
-*
» and K(n)^
ln[K(n) /ln(n)
]
-»
0.
Note
that
the
growth rate
for
K
is
somewhat slower than the square root of the natural log of the sample size.
In
this
section
we
have
discussed
a
consistent
general nonlinear errors in variables specification.
together with
data.
the
estimators
of
Section II
to
estimator
for
the
We now apply this estimator
estimate
Engel
curves
on micro
Derivation of an estimator with two or more mismeasured variables and
derivation of the asymptotic distribution of the estimator of this Section are
both quite complicated problems which we defer to future research.
21-
IV.
Estimation of Some Engel Curves
Estimation
econometricians
Many of
.
Surveys
led
has
the
early
information
section
detailed cross
curves
Engel
of
collected in
interest
used British data,
investigations
further
considerable
to
been an area of
long
has
annual
the
Family
Expenditure
investigate the best specification for the form of the Engel curves;
Houtthaker
Working"
income
Leser
and
(1955,1971)
studies
Prais and
"Leser-
The
form of Engel curve in which budget shares are regressed on the log of
expenditure has
or
been widely
translog specifications of Engel curves,
and
examples.
notable
are
(1963)
and the
Many of these
investigation.
among
AIDS
the
specification
specification.
adopted
Deaton
of
Lau and Stoker
Jorgenson,
e.g.
and
An alternative specification,
Both
research.
recent
in
Muellbauer
(1982),
use
(1980)
the
this
the quadratic expenditure system,
which specifies budget shares as a function of both the inverse of expenditure
and it square has been estimated by Pollak and Wales (1980).
Economic
Engel curves.
theory gives
almost no general guidance
(1981)
shares
are
expenditure.
considered
specified as
curves
Engel
polynomials
Gorman makes
demand curve literature,
the
in
in
which
However,
functions
key assumption,
expenditure,
of
does
as
aggregable
Gorman
for
,
the
Lewbel (1986,1987)
curve specifications.
or
e.g.
budget
log
of
most of the previous
that the polynomial functions which contain expenditure
in the demand curve specifications.
function"
and
in a notable paper
expenditure
either
do not depend on price
coefficients
specification of
"Adding-up" of budget shares to one is the only restriction,
this restriction is typically enforced in the data.
Gorman
in
demonstrates
polynomial
further
terms
considers
in
that
the
income
Gorman's
is
Given this "exactly
rank
of
at
most
results
the
matrix
three.
for additional
of
We
Engel
-22-
investigate Engel curve specifications of the Gorman form and provide tests of
his rank three restriction.
Few
squares
studies
nonlinear
or
least
curves
in variables model
income
current
studies.
into "permanent"
most
estimators
other
exception
notable
than
is
least
Liviatan
as
the
as
endogeneity.
(1959)
predetermined
variable
instrumental
variables
used
He
inappropriate
predetermined variable
a
instrument for current expenditure.
income made the use
income and "transitory"
Liviatan also noted Summer's
expenditure
is
The
squares.
used
have
Liviatan noted that Friedman's (1957) relabelling of the classic errors
(1961).
of
Engel
of
^
family
in
budget
objection to the use of current
because
with
reasons
of
current
income
of
joint
used as
an
Livitan's assumption that current income
uncorrelated with the stochastic disturbances in an Engel curve specification
seems
highly
questionable.
assertion that
"permanent"
with each other.
justified
He
income
the
assumption based
on
Friedman's
and "transitory" consumption are uncorrelated
However, subsequent econometric research, e.g. Attfield (1978),
has demonstrated that the Friedman assumption is unlikely to hold true in family
budget
data.
Thus,
alternative
instrumental variables
investigate two alternative sets of instruments:
determinants
of
income
these alternative
necessary.
instrumental variables
in the
Engel curve
Here we
expenditure in other periods or
and expenditure such as education and age.
sets of
stochastic disturbance
are
Neither of
should be correlated with the
specifications
,
although we test the
assumptions subsequently.
]
2
Liviatan used IV on a linear Engel curve specification.
Leser (1963) applied
a variant of Liviatan' s procedure.
However, straightforward IV is inapplicable
to all of Leser 's Engel curve specifications,
except his first two linear
specifications, because of the nonlinearity of his specifications.
Inconsistent
estimates will result for reasons discussed in Section I.
In particular, Leser's
best fitting specification (1963, p. 699) contains terms in both log expenditure
and the inverse of expenditure which makes application of regular IV inappropriate.
23-
We first consider
a
quadratic generalization of the Leser-Working Engel
curve specification where the budget share of commodity
is a
i
function of both
log expenditure and the square of log expenditure:
Wi - £
(4.1)
where
stochastic
the
is
(^
+ 0! log(z) + p 2 log 2 (z) + ei
disturbance.
we
However,
expenditure, but we instead have data on log x — log
error
the
observation
of
satisfies
the
z
properties
do
+
observe
not
where we assume that
rj
Assumption
of
actual
A.l
(ii).
Alternatively, a permanent income explanation can be attached to equation (4.1);
however,
we
unwilling
are
to
make
any
assumption
permanent income and transitory consumption.
the
Gorman rank
3
about
the
relationship
of
satisfies
Note that equation (4.1)
condition, while the usual translog-AIDS specifications based
on the Leser-Working specification have rank
2
coefficient matrices.
Our first results are from the 1982 Consumer Expenditure Survey (CES).
The
CES
collects
data
from families
over 4
quarters so
that we
repeated measurement techniques discussed in Section II.
can apply the
The basic data we use
are budget share and total expenditure for each family from 1982:1.
We initially
use
The
as
the
repeated measurement
total
observation estimator of equations
on
5
commodity
transportation.
3
groups:
food,
expenditure
(2. 5) -(2. 7)
clothing,
from
is used.
1982:2.
repeated
We estimate Engel curves
recreation,
health
care,
and
We report elasticity estimates and asymptotic standard errors at
quartiles so that the shape of the Engel curves can be compared:
.
-24-
Expenditure Elasticity Estimates Using 1982 CES Data
Repeated Measurement Estimator using 1982:2
(Asymptotic Standard Errors)
Table 4.1:
IV Estimates
OLS Estimates
Percentile
Percentile
Commodity
25th
Food
.83
Clothing
25th
50th
.74
.63
.73
.68
75th
.60
(.03)
(.06)
(.04)
(.05)
(.03)
1.44
(.16)
1.43
1.41
1.14
(.04)
(.04)
1.08
(.06)
(-18)
Transpor
75th
(.02)
Recreation 1.47
Health
50th
(.08)
(.13)
1.29
(.06)
1.28
(.09)
1.12
1.51
1.28
(.09)
(.06)
(.14)
.99
.44
.50
.56
.68
(.14)
(.21)
(.09)
(.07)
(.09)
1.19
1.11
1.18
(.06)
1.48
(.05)
1.02
(.12)
1.35
(.11)
(.04)
(.04)
.009
(.21)
.10
Number of observations- 1324
Overall,
with the exception of health the IV and OLS estimates are reasonably
close and both accurately estimate the elasticities.
Working specification, that all the 02'
IV
estimates.
s
are zero,
A joint test of the Leser-
is computed to be 9.75 for the
The marginal significance level for a x
degrees of freedom is about .08.
random variable with
5
Similarly, a test based on the OLS estimates is
computed to be 73.1 which has a marginal significance level of less than .001.
Thus, both the estimates, especially for food and recreation, and the statistical
tests give some indication that a quadratic term gives better estimates of Engel
curves
individual
on
data.
The
usual
assumption
of
constant
budget
share
elasticities which the Leser -working specification imposes appears inconsistent
with the 1982 CES data.
While
differences
do
the
IV
occur.
estimates
and
OLS
For
instance
the
are
reasonable
close,
some
sizeable
estimated food elasticity at the 25th
25-
percentile
and
elasticities
clothing
the
the
at
and
50th
75th
percentile
are
quite different with both sets of elasticities estimated with a high degree of
Also,
accuracy.
25%
to
50%
possible
the estimated transportation elasticities differ by a
50th and 75th percentile between OLS and IV.
the
at
difference
estimates versus
we
the
do
Hausman
a
estimates.
OLS
type
(1978)
To test for a
specification test of the
The estimated statistic
is
IV
87.39 which is
random variable with 15 degrees of freedom.
distributed as a \
range of
Thus,
we
find
strong evidence that use of current expenditure in estimation of Engel curves on
micro
data
consider
leads
the
errors
to
problem
estimated var(z) is
to
is
variables
in
note
Thus,
.150.
that
the
problems.
An
alternative
estimated var(f|)
is
.108
method
while
to
the
the measurement error in current expenditure is
about 42% of the total variance of .258 of the logarithm of measured expenditure.
The substantial proportion of measurement error in measured expenditure can lead
to significant problems in OLS-type estimators.
We
now
the
Gorman
results.
higher order polynomial terms
in log
income will have a linear relationship to
explore
Gorman's
theorem
implies
that
the lower order terms since the matrix of coef f icieiits is at most of rank three.
For the polynomial generalization of equation (4.1),
the rank restriction takes
the form that the ratio of coefficients of the cubic terms to the coefficients of
the quadratic
terms will be constant across equations.
First,
we estimate
a
generalization of equation (4.1) with a third degree term in log income included:
(4.2)
Wi - £
+ p l log(z) + p 2 l°g 2
U)
+ £3 log 3 (z) + ££.
IT
This rank restriction result follows from Gorman (1981), p. 16, equation (1)
26-
Th e estimated Engel curve and elasticities
specification of equation (4.1)
quadratic
the
order
terms
are
all
are
zero
for
the
IV
quite similar to those based on
statistic that the third
The \
estimator
2.59
is
with
the corresponding statistic for the OLS estimates is
freedom;
IV estimates
level of about
.07,
degrees
10.57.
Thus,
of
the
for more than a quadratic term in
do not demonstrate much evidence
the budget share specification.
five
The OLS estimates, with a marginal significance
are more ambiguous about the cubic terms.
However, we will
use the quadratic specification in our subsequent estimation because we believe
that the IV estimates are likely to be better than the OLS estimates.
We then estimate the "Gorman statistic" to see whether the coefficients
in
the
cubic
remarkable
specific
result
of
equation
(which we hope
is
(A. 2)
have
not due
to
rank
find
We
three.
computational error)
considerable variation in the estimates of the ^2
s
an<^
t ^ie
i$3' s
>
.
a
rather
Despite
we find their
ratios to be remarkably close in actual values and estimated precisely although
we cannot estimate the
individual coefficients very precisely.
Thus,
we find a
more special result than Gorman's result- -not only is the coefficient matrix of
rank three, but the linear dependency takes on a remarkably simple form.
Table 4.2:
Estimated Ratios of ^3/^2 for Equation (4.2)
Commodity
IV Ratio
OLS Ratio
Food
-24.98
(0.47)
-25.24
(0.56)
-25.12
(0.32)
-23.31
(5.18)
-25.57
(0.31)
1.60
-129.35
(10739)
-22.6
(2.46)
-22.97
(2.17)
-28.57
(2.38)
-34.09
(9.05)
7.85
Clothing
Recreation
Health Care
Transportation
X (4) Statistic
.
27-
Thus
both the IV results and the OLS results demonstrate that the Gorman results
,
on rank
holds
3
food where
to
the
in the
The one anomalous result for OLS is for
CES data.
1982
estimated quadratic term is very near zero.
This estimate leads
the high estimated Gorman ratio as well as the very high estimated standard
Note that the OLS results are not as good as the IV results
error of the ratio.
since
However,
statistic
test
the
as before,
has
a
significance
marginal
we tend to prefer the IV estimates.
level
of
about
.10.
Perhaps the results are
"too good" given the variation in prices faced by families in the sample which we
have no data on.
We now reestimate
from
1982:3
in
place
equations
of expenditure
(4.1)
from
and
1982:2
(4.2)
to
by
form
IV
using expenditure
the
instrument.
The
results are very similar to the results in Table 4.1:
Table 4.3:
Expenditure Elasticity Estimates Using 1982 CES Data
Repeated Measurement Estimator using 1982:3
IV Estimates
Commodity
Food
Clothing
E.ecreation
Percentile
25th
50th
.72
.69
.65
(.06)
(.04)
1.44
(.07)
(.06)
1.50
(.12)
1.41
(.17)
Health
Transpor
rank
1.47
(.11)
.10
.23
(.18)
1.23
(-09)
(.13)
1.09
(.05)
2
X (4) Statistic
The x
Gorman Statistic
0.44
75th
-25.18
(.38)
-25.18
(2.23)
-24.65
(1.01)
-28.34
(7.37)
-26.60
(7.10)
1.38
(.13)
1.50
(.20)
.60
(-23)
.94
(.10)
Number of Obs-1324
statistic for the Gorman ratios again shows no evidence of rejecting the
3
restriction.
A Hausman
(1978)
specification test type statistic for IV
•28-
with 15 degrees
versus OLS is estimated to be 135.71, which is distributes as \
of freedom.
error
Again, a strong indication is found of the importance of measurement
micro
the
in
estimators.
extremely
data
potential
and
problems
with
use
the
estimated variance of the measurement error is
The
close
to
previous
estimate
and
represents
which
41%
of
OLS-type
.106
which is
of
the
total
Thus, both sets of repeated
variance of the logarithm of measured expenditure.
measurement estimates yield numerically consistent sets of coefficient estimates.
Up to this point, we have used the replicated measurement estimator for
our
Engel
Section II,
curve
specifications.
equations
used include age,
(2.13)
education,
Here
we
and equations
race,
use
(2
.
15)
-
(2
union membership,
and region and industry dummy variables.
predicted
the
.
16)
,
estimator
of
where the instruments
spouse age and employment,
we use a
Thus,
IV
"predicted value" for
expenditure to form the instruments to use in the nonlinear specifications where
the R/ of the prediction equation is about 0.3.
Table 4.4:
Expenditure Elasticity Estimates Using 1982 CES Data
Using Predicted Expenditure Estimator
IV Estimates
Commodity
Food
Clothing
Recreation
Health
Transpor.
Percentile
Gorman Statistic
Overid Statistic
-25.39
3.12
25th
50th
.69
.62
.51
(.15)
(.06)
1.47
(.01)
1.51
(.11)
(.15)
1.28
(.25)
-25.22
.93
-25.94
1.71
(.35)
2.35
(.45)
.15
(.31)
1.59
(.23)
2
X (4) Statistic
2.43
75th
(.15)
(.31)
.24
.50
(.15)
1.02
(.08)
(.34)
.39
(.25)
1.35
(.21)
0.59
(.66)
-25.26
4.68
(.15)
-26.61
(1.84)
Number of obs-1324
1.46
•29-
Except for recreation and for transportation at the 75th percentile the estimated
elasticities are quite close to the repeated measurement estimates.
quadratic terms are zero is estimated to be 11.78 which has a marginal
the
all
significance
level
should
terms
A test that
about
included
be
again we
Thus,
.04.
in
that higher
specification.
curve
Engel
the
find evidence
A
order
Hausman
specification test statistic is calculated to be 73.34 which again indicates that
n
The \ (2) test for correct
the IV estimates are better than the OLS estimates.
specification from equation (2.21) is well below its critical value of 6.0 at the
5
percent
level
The
test for overidentif ication does not
the
two
each commodity.
for
reject our specification.
We
now
do
a
test
\
estimates are the same.
that
sets
repeated measurement
of
IV
This test is equivalent to a test of the overidentifying
restrictions on the instruments.
statistic is estimated to be 12.4, and
The \
since it has 15 degrees of freedom, no evidence is found to reject the hypothesis
of
orthogonality
predicted
of
IV
which
instruments.
the
expenditure
measurement
both
of
form
of
the
estimators yield \
indicate
that
either
IV
However,
the
estimator
in
statistics
of
relation
35.6
mutually
curve specifications.
consistent
with
each
Since
other,
and
the
we
tests
to
91.6,
repeated measurement
the
value instruments are not mutually orthogonal to the
the Engel
equivalent
or
the
for
the
repeated
respectively,
the
predicted
stochastic disturbance in
repeated measurement estimators are
tend
superior estimators in the current situation.
to
believe
that
they
are
the
We cannot be more specific about
the relative superiority of the estimators without further research.
We repeat the IV estimation of equations (4.1) and (4.2) using 1972 CES
data
where
we
predict
expenditure
using
similar
instruments.
These data were kindly provided to us by Professor Dale Jorgenson.
Repeated
.
-30-
The 1972 CES data
observations on family expenditure are not available for 1972.
set
sufficiently large
is
families
minimize
to
that we
potential
estimated the
family
effects
size
curve
Engel
on
the
only on 4 person
estimates.
The
estimated elasticities are reported in Table 4.5:
Table 4.5:
Expenditure Elasticity Estimates Using 1972 CES Data
Predicted Expenditure Estimator
IV Estimates
Commodity
Food
Clothing
Recreation
Health
25th
50th
75th
.76
.67
.54
(.12)
1.43
(.89)
(.07)
1.36
(.83)
1.41
(.10)
(.13)
1.22
(.80)
1.49
(.17)
1.32
(.13)
1.07
(.20)
.54
(.18)
Transpor
2
X (4) Statistic
0.54
Overid Statistic
Gorman Statistic
Percentile
.78
.44
(.11)
(.21)
.59
.65
(.10)
(.19)
IV
-42.1
(.48)
-41.8
(1.14)
-41.3
(2.23)
-41.1
(1.67)
-42.2
(.22)
OLS
-44.5
(4.73)
-42.8
(.66)
-43.5
(1.65)
-42.0
(.50)
-40.6
(2.60)
1.23
3.82
2.39
0.41
2.31
Number of obs - 992
The estimated elasticity for transportation is below the 1982 estimates which may
well
arise
1982.
to
come
from
the
extremely large
rise
in
gasoline prices between 1972
The estimated IV Gorman ratios are again very close,
close to a rejection of equality.
*
prices.
15
The
CPI
test fails
Note that the estimated values of
the Gorman ratios differ from their estimated values in 1982.
be expected since the slope coefficients are,
and a x
and
in general,
This result is to
nonlinear functions of
increased by over 130% between 1972 and 1982 with significant
Note that the estimated OLS Gorman ratios are also very close.
The x 2 (4)
statistic for the OLS estimates is 1.87 which indicates no grounds for rejection.
For completeness, the X (5) statistic that all the quadratic terms are zero is
17.2 which is strong evidence against the Leser-Vorking Engel curve specification
on micro data.
-
.
.
-31-
increases
expenditure
across
categories.
Thus,
ratios
of
nonlinear functions of prices would be expected to change as prices change.
A
differences
in
the
Hausman (1978) type specification test of IV versus OLS is estimated to be 117.8.
Thus, we again find strong evidence of the importance of measurement error in the
1972
data
CES
as
we
did
in
the
1982
Lastly,
data.
the
2
x (2)
of
overidentif ication of equation (2.21) once more does not reject our specification
of the Engel curve for any commodity.
One potential problem that we have not yet accounted for is errors in
variables in the left hand side variable,
and (4.2).
given
the budget shares,
in equations
(4.1)
To the extent that the errors in variables occurs in expenditure on a
commodity,
problem arises.
which
the
numerator
of
the
budget
share,
no
special
However, to the extent that the denominator of the budget share,
total expenditure,
measurement
forms
is
measured with error, estimation problems arise.
error enters
solution exists.
But
the
the
problem in a non-polynomial variable,
Since the
no
obvious
problem can be eliminated by respecifying equations
(4.1) and (4.2) with commodity expenditure as the left hand side variable instead
of
the budget
share.
The estimation procedure
remains
the
same
except for an
adjustment to the estimated standard errors of the coefficients to account for
heteroscedasticity
The results are presented in Table 4.6.
variables in the left hand side variable presents
this
Engel
curve
specifications
specification
does
not
fit
the
We do not find that errors in
a
significant problem although
data as well
as
the
earlier
.
32-
Expenditure Elasticity Estimates Using 1982 CES Data
Commodity Expenditure is Left Hand Side Variable
IV Estimates
Table 4.6:
Gorman Statistic
Percentile
Commodity
25th
50th
.89
.73
.62
(.08)
1.72
(.23)
(.04)
(.07)
1.61
(.11)
1.26
(.11)
1.36
(.16)
1.05
(.20)
.15
.68
(.24)
1.22
(-13)
(.32)
1.16
Food
Clothing
Recreation
1.55
(.17)
-.41
(.39)
1.06
(.33)
Health
Transpor
2
* (4) Statistic
2.29
75th
-33.38
(33.18)
-15.26
(1.49)
-14.59
(27.04)
-16.61
(1.13)
-17.69
(0.82)
(.30)
Number of Obs-1324
The elasticity estimates are quite close to the elasticity estimates derived from
the
budget
estimated
share
health
imprecisely.
results
of Tables
elasticity
The
at
Gorman ratios
and
4.1
25th
the
are
all
4.3.
percentile
quite
only
The
close
which
to
one
exception
not
come
close
to
a
estimated
is
another with
exception of food, which again is measured quite imprecisely.
does
is
The \
rejection of the Gorman restriction.
Thus,
the
very
the
statistic
when we
estimate the Engel curves in commodity expenditure form, rather than budget share
form,
the
results remain essentially unchanged.
We again find support for the
Gorman restriction on the specification of the Engel curve.
Up
to
this point we have
budget share data.
following
Engel
considered only polynomial Engel curves for
However, Leser (1963) found evidence which indicated that the
curve
specification
superior
was
specification:
(4.3)
Wi - p
+
/9i
log(z) + £ 2 / z +
f
i-
to
the
Leser-Vorking
.
-33-
Thus
,
he generalized the Working specification to include the
as well
We consider this extended Leser specification as well
its logarithm.
as
inverse of income
as another generalization of the Leser-Working specification:
(4. A)
Note
both
that
-
wj[
+ Pi log(z) + p 2 zlog(z) +
/3
equations
(A. 3)
and
(4.4)
are
ej..
rank
two
specifications.
The
coefficients of these generalized Engel curve specifications are estimated using
the
general
(3.6).
nonlinear
errors
in
variables
estimator
of
Section
III,
equation
Recall that the estimation strategy of Section III involves fitting the
Engel curves with polynomials followed by estimation of the coefficients of the
nonlinear specifications using the predicted values of the budget shares from the
polynomial coefficient estimates.
The
estimated coefficients of equations (4.3)
and (4.4) follow from the best fitting polynomial.
fitting polynomial
health care
for
is
the
a
In
9
out of 10 cases the best
second degree polynomial with the sole exception being
specification of equation (4.4)
which uses a third degree
polynomial
The estimates of the nonlinear Engel specifications are given in Table
4.7
34-
Estimates Using 1982 CES Data- -General Nonlinear Specifications 16
Table 4.7:
Eqi jation
Clothing
Recreation
50th
75th
25th
50th
.82
.71
.59
.82
.76
.67
(.060)
1.45
(.13)
1.43
(.15)
(.031)
1.45
(.11)
1.23
(.11)
(.068)
1.42
(.15)
1.09
(.17)
(.057)
1.45
(.13)
1.45
(.17)
(.043)
1.41
(.076)
1.33
(.10)
(.045)
1.39
(.084)
1.20
(.10)
.28
(.13)
.60
(.27)
1.08
(.07)
1.01
(.14)
Health
.06
(.19)
1.16
Transpor
(.08)
that
the
estimates
.04
(.17)
.12
(.15)
1.17
(.09)
the
closeness
of
Furthermore,
fit
.15
(.22)
1.07
(.08)
1.12
(.06)
the estimated elasticities
estimated elasticities for the polynomial specifications
compare
75th
elasticities are quite similar between the two
of the
nonlinear specifications.
the
•
25th
Food
Note
Equation (4 4)
Percentile
(4 .3)
Percentile
Commodity
of
in Table 4.1.
nonlinear specifications
the
are close to
to
We
predicted
the
values of the underlying polynomials since that is the criterion used to estimate
the
coefficients
Leser
in equation
specification
of
equation
specification of equation (4.4).
Engel
curve
specifications
For 4 of
(3.6).
do
the
fits
(4.3)
5
commodities,
better
than
the
the
extended
generalized
The exception is health care where none of the
very
well.
However,
to
the
extent
that
estimated elasticities are so similar, the choice of the "best" specification
the
is
probably a rather unimportant exercise.
We
for
possible
shares.
now consider an additional nonlinear specification which accounts
measurement
errors
in
the
left
hand side
variables,
the
budget
We take the quadratic generalization of the Leser-Working Engel curve of
equation (4.1) and multiply both sides of the equation by total expenditure:
16
Standard errors are calculated by the bootstrap method here.
.
-35-
ei -
(4.5)
In equation
(4.5)
j8
z
+ 01 zlog(z) + p 2 zlog 2 (z) + ei
commodity expenditure is now the left hand side variable which
from measurement
possible problems
eliminates
.
budget shares in equations (4.1) and (4.2).
error
the
in
denominator
of
the
Our estimates of equation (4.5) are
For instance, the
very similar to the estimates in Table 4.7 and earlier tables.
estimated elasticities at the 50th percentile are (0.69, 1.46, 1.17, 0.37, 1.13).
Thus,
Engel
the nonlinear specification of the quadratic version of the Leser-Working
curve
yields
estimates
measurement error in the
very
close
previous
our
to
left hand side variable
estimates
that
so
again does not seem to be an
important problem.
Our
final
exploration of
Engel
the
addition of demographic variables in equation
curve
specification involves
(4.1).
Differences
the
in household
size have often been a focus of attention in the specification of Engel curves;
here we also include region of the U.S. to account for regional price differences
of the commodities as well as age of the household head to account for life cycle
effects.
variables
The
with
demographic
4
family
variables
size
groups,
are
4
all
entered
region groups,
as
and 4
indicator
age
(dummy)
groups.
We
believe that this specification is preferable to the non- identified approach of
family equivalence scale specifications.
to
include
the
additional
right
hand
The approach of equation (2.23) is used
side
variables
in
the
Engel
curve
specifications
We now reestimate Tables 4.1 and 4.2 where we
variables and used the repeated measurement estimator.
include the demographic
36-
Expenditure Elasticity Estimates Using 1982 CES Data
Repeated Measurement Estimator using 1982:2
IV Estimates
Table 4.8:
Clothing
Recreation
50th
.72
.66
.59
(-05)
1.47
(.14)
1.17
(.17)
(.06)
(.07)
1.39
(.12)
1.15
(.17)
-.33
(.39)
Health
1.43
(.13)
1.16
(.17)
-.09
(.23)
1.07
(.10)
.21
(.17)
1.07
(.10)
Transpor
2
X (4) Statistic
Family Size,
The
75th
25th
Food
3
Gorman Statistic
Percentile
Commodity
estimated
Age, and
quartile
1.06
(.11)
Number of Obs-1321
1.90
3
-9.93
(6.17)
-10.44
(8.23)
-14.80
(.43)
-14.01
(.52)
-15.34
(.93)
Region variables are included
3
elasticities
change very
little
from
the
specification
which omits demographics, with the sole exception of the health equation.
health
equation
coefficient
estimates.
elasticities
estimates
estimated very
are
accompanied
by
quite
imprecisely with
large
the
negative
standard
asymptotic
The
error
The Gorman ratios are not as close as in Table 4.2 although the test
statistic takes on almost the same value which indicates no reason to reject the
Gorman restrictions.
The food and clothing ratios are smaller in magnitude than
the other three commodities.
However, the Gorman ratio for food and clothing are
estimated very imprecisely.
Working
specification
of
We
equation
specification of equation (4.2).
The
Hausman
indicates
(1978)
strong
test
evidence
again find
of
of
IV
and
(4.1)
The x (5)
versus
strong evidence against the Leser-
OLS
measurement
evidence
in
favor
of
the
cubic
statistic is estimated to be 48.2.
with
error
X (10) random variable under the null hypothesis.
demographics
since
it
is
is
473.0
distributed
which
as
a
-37-
Despite the closeness of the quartile elasticity estimates without and
without demographic variables included, we do find
of
demographic
variables
on
expenditures
a
shares.
quite significant influence
We
present
the
estimates
results in Table 4.9:
Table 4.9:
Coefficient Estimates for Demographic Variables
Commodity
Food
Agel (19-29)
-.04
(.01)
Age2 (30-39)
Age3 (40-49)
Regl (NE)
Reg2 (W)
Reg3 (MW)
Sizel (2)
Size2 (3)
Clothi
Mean Share
.01
(.01)
-.00
(.01)
-.02
(.01)
Transportation
.01
(-02)
.04
.01
.01
.07
.03
(.02)
(.01)
(.01)
(.02)
(.03)
-.00
(.01)
.00
.00
.01
.01
(.00)
(.01)
(.01)
(.01)
.01
(.01)
.00
.01
(.00)
(.00)
-.00
(.01)
-.00
(.01)
-.01
(.02)
.08
-.01
(.04)
.01
.01
(.03)
(.01)
-.01
(.03)
.01
.02
.01
.01
(.01)
(.02)
(.03)
(.04)
-.00
(.01)
-.00
(.00)
-.01
(.00)
.01
(.01)
.01
(.01)
.01
-.01
(.00)
-.01
(.01)
.02
.01
(.01)
(.01)
(.00)
-.00
(.00)
(.00)
.00
(.00)
(.01)
.22
.06
.06
.21
(.01)
Size3 (4)
-
HC
.01
.00
05
-
(.03)
.00
We find fairly sizeable age effects in the estimated share equations.
The region
effects are not particularly large which helps support the necessary assumption
of constant prices across
the US
in the Engel curve specification.
The notable
exception
The health care equation is
for health care.
the Western region
is
the estimated effect here may arise from the much
difficult to estimate overall;
larger share of health maintenance organizations in the West in 1982.
The family
size effects are statistically significant although they have only a small effect
compared to the shares in expenditure of the five commodities.
We then reestimated the Engel curve specifications for 1982 using the
The estimated elasticities, not reported here,
second repeated measurement.
are
quite similar to Table 4.7 and the demographic effects are quite similar to Table
5
Gorman ratios
Thus,
the ratios
The
A. 8.
11.86).
statistic
test
of
estimated
are
be
to
-8.10,
(-8.26,
-9.51,
-7.68,-
are once again quite close to each other with a x C-0
3.97.
The
test
for
quadratic
the
against
the
cubic
specification is 86.4 which once again gives strong evidence against the Leser-
Working specification.
to be 675.8.
The Hausman test statistic of IV versus OLS is estimated
However, when we include the demographic variables the two sets of
replicated measurement results
before.
The x (10)
are
longer nearly
no
so
mutually consistent as
statistic for the test of overidentif ication is estimated to
be 50.4 which easily rejects the overidentifying restrictions.
In
section we have estimated various Engel curve specifications
this
where we have taken account of errors in measurement in income or expenditure.
We find very strong evidence that substantial measurement error exists in the CES
data.
We
also
find strong evidence
that
equation
Leser-Working specification of equation (4.1).
for
the
hypothesis
polynomial terms
that more
general nonlinear
than cubic are needed.
(4.2)
is
preferable to the
However, we do not find support
specifications
or
higher order
We find strong support for the Gorman
rank condition which limits polynomial Engel curve specifications to rank
results also show that demographic variables have
3.
Our
significant effects in Engel
39-
curve specifications of the share.
Lastly, we have demonstrated the feasibility
and importance of using consistent instrumental variable estimators in nonlinear
econometric model specifications.
-40-
References
Adcock, R.
(1887),
"A Problem in Least Squares", The Analyst
Aigner, D.J., Hsiao, C, Kapteyn A., Wansbeek, T.
and M.
Griliches
Econometrics",
in
Z.
in
1323-1393
Econometrics vol. II,
,
5,
53-54
"Latent Variable Models
Handbook of
Intriligator
(1984),
D.
-
,
.
Amemiya, T. [1974], "The Nonlinear Two Stage Least Squares Estimator," Journal of
105-110
Econometrics 2:
.
Amemiya, T. (1977), "The Maximum Likelihood Estimator and the Nonlinear ThreeStage Least Squares Estimator in the General Nonlinear Simultaneous Equation
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