Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium IVIember Libraries
http://www.archive.org/details/nonparametricestOOhaus
HB31
.M415
kfif*.::zrt.-
2_
working paper
department
of economics
NONPARAMETRIC ESTIMATION OF EXACT
CONSUMERS SURPLUS AND DEADWEIGHT LOSS
Jerry A. Hausman
Whitney K. Newey
No.
93-2
Dec.
1992
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
'
NONPAEAMETRIC ESTIMATION OF EXACT
CONSUMERS SURPLUS AND DEADWEIGHT LOSS
Jerry A. Hausman
Whitney K. Newey
No.
93-2
Dec.
1992
1S93
NONPARAMETRIC ESTIMATION OF EXACT CONSUMERS SURPLUS AND DEADWEIGHT LOSS
by
Jerry A.
Hausman and Whitney
K.
Newey
Department of Economics, MIT
October 1990
Revised, December 1992
This research was supported by the NSF.
Helpful comments were provided by T. Gorman,
Jewitt, A. Pakes and participants at several seminars.
Sarah Fisher-Ellison and
Christine Meyer provided able research assistance.
I.
Abstract
We
apply nonparametric regression models to estimation of demand curves of the type
most often used
in applied research.
From the demand curve estimators we derive
estimates of exact consumers surplus and deadweight loss, that are the most widely used
welfare and economic efficiency measures in areas of economics such as public finance.
We
also develop tests of the
demand.
We work
symmetry and downward sloping properties of compensated
out asymptotic normal sampling theory for kernel and series
nonparametric estimators, as well as for the parametric case.
The paper includes an application to gasoline demand.
interest here are the shape of the
from a te« on
gasoline.
Empirical questions of
demand curve and the average magnitude of welfare
In this application
loss
we compare parametric and nonparametric
estimates of the demamd curve, calculate exact and approximate measures of consumers
surplus and deadweight loss, and give standard error estimates.
sensitivity of the welfare
We
also zmalyze the
measures to components of nonpeirametric regression estimators
such as the number of terms in a series approximation.
Keywords:
Consumer surplus, deadweight
loss,
consumer
tests, nonpsirametric estimation.
Authors:
Professor Jerry A. Hausman
Professor Whitney K. Newey
Department of Economics
Department of Economics
MIT
MIT
Cambridge,
MA
02139
Cambridge,
MA
02139
Introduction
1.
Nonparametric estimation of regression models has gained wide attention
few years
the past
Nonparametric models are characterized by a very large
in econometrics.
numbers of parameters.
in
Often they may be difficult to interpret, and their usefulness
We
applied research has been demonstrated in a limited number of cases.
in
apply
nonparametric regression models to estimation of demand curves of the type most often
used
applied research. After estimation of the demand curves,
in
we
will then derive
estimates of exact consumers surplus and deadweight loss which are the most widely used
welfare and economic efficiency measures
We
also
in
areas of economics such as public finance.
work out asymptotic normal sampling theory for the nonparametric
case, as well as
the pau-ametric case (where except in certain analytic cases the results are not known).
The paper includes an application to gasoline demand.
interest here are the shape of the
from a
taix
on gasoline.
Empirical questions of
demand curve and the average magnitude
In this application
of welfare loss
we compare parametric and nonparametric
estimates of the demand curve, calculate exact and approximate measures of consumers
surplus and deadweight loss, and given estimates of standard errors.
sensitivity of the welfare
such as window width
where p
level.
aind
number of terms
.u"^)
= e(p
,u'^)
are initial prices,
The case
EV(p
where
,p
y
r
=
,p ,y)
1
also analyze the
measures to components of nonparametric regression estimators
in a series
The definition of exact consumers surplus
CS(p
We
- e(p
is
based on the expenditure function:
,u'^),
are new prices, and
p
approximation.
is
u*"
the reference utility
corresponds to the case we focus on here, equivalent variation:
= e(p
,u
)
- e(p ,u
)
= y - e(p
,u
denotes income, fixed over the price change.
),
It
is
also easy to carry out a
similar analysis for compensating variation.
The measure of exact deadweight loss (DWL) used corresponds to the idea of the
in
consumers surplus from imposition of a tax
compensated
less the
teix
under the implicit assumption that they are returned to the consumer
(1974) definition of deadweight loss
where
equivalent variation the definition of exact
DWL(p
h(p,u)
is
,p ,y)
= EV(p
,p ,y) - (p
is
p
- p
DWL
is
the vector of taxes.
Then, for
is:
-p )'h(p
,u
)
= y - e(p
,u
)
the compensated, or Hicksian, demand function and
Marshallian (market) demand function.
it
a lump sum
Here we use the Diamond-McFadden
loss (also called the excess burden of taxation).
DWL
revenues raised,
See Auerbach (1985) for a discussion of the various definitions of deadweight
mamner.
where
in
loss
- (p -p )'q(p
q(p,y)
is
,y),
the
Thus, to estimate exact consumers surplus or exact
necessary to estimate the expenditure function and the compensated function
demand which are related by the equation,
5e(p,u)/ap. = h.(p,u).
(Shephard's Lemma).
The expenditure function and compensated demand curve
on the market, or Marshallian, demand curve,
aire
estimated from observable data
q.(p,y).
Various approximations h"'ve been proposed for estimation of the exact welfare
measures.
is
Willig (1976) demonstrated that the Marshallian
measure of consumers surplus
often close to the exact measures, and he derived bounds as a function of the income
share.
Various authors have also recommended higher order Taylor-type approximations to
the exact welfare measures.
Limitations of the approximation approach are that they do
not yield measures of precision given that they
Eire
most cases and the approximations may do poorly
specify a priori.
(1981)
in
based on estimated coefficients
some situations which are
Deaton (1986) discusses these problems
in
more
detail.
Also,
in
difficult to
Hausman
demonstrated that while the approximations may often do well for the consumers
surplus measure, they often do very poorly in measurement of DWL; Small and Rosen (1981)
also demonstrated a similar proposition in the discrete choice situation.
Two approaches have
been proposed to estimation of the exact welfare measures from
estimation of ordinary demand functions.
Hausman
(1981)
demonstrated that for many
widely used single equation demand specifications, the necessarily differential equation
could be integrated to derive the expenditure function and also the compensated demand
function.
He was also able to derive the sampling distribution for the measures of
consumers surplus and DWL, given the distribution of the estimated demand coefficients.
Vartia (1983), instead of using an analytic solution to the differential equations,
proposed a variety of numerical algorithms which estimate consumers surplus and
any desired degree of accuracy.
DWL
to
While the Vartia approach can be applied to a wider
range of situations, no correct sampling distribution has been derived for the estimated
welfare measures, although some approximate results are given
in
Hayes and Porter-Hudak
(1986).
Both the Hausman approach and the Vartia approach can be applied to multiple price
changes and are path independent for the true demand function.
The various approximation
approaches do not share the path independence property which cam lead to perplexing
computational results and has led to much theoretical misunderstanding
literature.
Here,
we
in the
appropriate
consider nonparametric estimation via an unrestricted estimator
that uses some, but not
all
the implications of path independence.
us to test path independence,
i.e.
symmetry of compensated demands, and impose further
more
implications of path independence to construct
efficient estimators.
The Hausman and Vartia approaches begin with Roy's
demand function with the
Our approach allows
identity which links the ordinary
indirect utility function:
q.(p,y) = -[av(p,y)/ap.]/Ov(p,y)/ayl,
where
v(p,y) is the indirect utility function.
The partial differential equation from
this equation can be solved along an indifference curve for a unique solution so long as
the initial values are differentiable.
and
p(l)
= p
and
,
let
y(t)
Let
denote a price path with
p(t)
be compensated income, satisfying
Differentiating with respect to
y
p(0) = p
V(p(t),y(t))
= V(p
,y).
gives
[av(p(t),y(t))/ap]'sp(t)/at + [av(p(t),y(t))/ay]sy(t)/at = o.
Hausman notes that
this equation can be converted into an ordinary differential equation
by use of the implicit function theorem and Roy's identity.
denote the equivalent variation for a price change from
compensated income
is
Alternatively, this equation follows immediately
of
S(t,y).
Hausman
p(t)
S(t,y)
to
p
.
= y -
e(p(t),u
)
Then, since
y - S(t,y),
5S(t.y)/5t = -q(p(t), y-S(t,y))' Sp(t)/at.
(1.1)
Let
solves this equation for
S(l,y)
=
0.
from Shephard's Lemma and the
definition
some widely used demand curves.
The
solution gives both the expenditure function and indirect utility function, while the
right-hand side of this equation gives the compensated demands.
Vartia's numerical solutions to the differential equation also arise from equation
(1.1).
(s
=
To
solve
N),
it
Vartia uses a numerical method of Collatz.
and defines
S(t^^^)
=
S
He orders
=
t
1
- s/N,
s
iteratively by
S(t^) - .5[q(p(t^^j),y-S(t^^j)) + q(p(t^).y-S(t^))]' [p(t^^^)-p(t^)]
This algorithm consists of averaging the demand at the last price
and multiply this average by the change
in price.
eind
the current price
By the envelope theorem, the product
of the price change times the quantity equals the additional income required to remain on
the same indifference curve.
Intuitively,
dy = q*dp, where y
is
updated at each step of
the process, rather than holding y constant which the Marshallian approximation to
consumer surplus does.
One can also use alternative numerical algorithms to solve
s
equation
(1.1),
that
may
lead to faster methods.
We use an Buerlisch-Stoer algorithm
from Numerical Recipes that does not require solution to the
implicit equation in
Vartia's algorithm, has a faster (quintic) convergence rate that Vartia's (cubic), and in
our empirical example leads to small estimated errors with few demand evaluations.
The possible shortcoming of
and the Vartia approach
is
all
applications to date of both the
Hausman approach
that the parametric form of the demand function
is
required.
most applied situations, the exact parametric specification of the demand curve (up to
In
unlinown parameters) will not be known.
may
Thus, commonly used demand curve specifications
well lead to inconsistent estimates of the welfcire measures
misspecified.
This problem
is
if
the
demand curve
is
potentially quite important, especially in the case of
measuring deadweight loss which depends on "second order" properties of the demand curve.
See
Hausman
(1981) for a discussion of the
second order properties and their effect on
estimates of the deadweight loss.
Varian (1982 a,b,c) has proposed an alternative approach based on the revealed
preference ideas of Samuelson (1948) and Afriat (1967, 1973) which
is
is
nonparametric, but
only able to estimate upper and lower bounds on the welfare measures.
nonparaimetric approximation approach
bounds, because
many
is
very interesting, but
it
Varian'
often yields rather wide
price observations per individual are required for tight bounds.
Furthermore, use of sampling distributions to measure the precision of the estimates
problematical.
Here,
we
is
use a nonparametric cross-section demand analysis analysis,
imposing enough homogeneity across individuals and smoothness of the demand function that
we can estimated
it
by nonparametric regression.
consumers surplus and
DWL
sensitivity of the welfare
We
construct point estimates of exact
as well as precision estimates of the welfare measures.
measure to the amount of smoothing used
regression can be analyzed straightforwardly.
in
The
the nonparametric
Estimation
2.
Our estimator of consumer surplus
with
q(p,y)
is
obtained by solving equation
(1.1)
numerically
We
replaced by an estimator obtained from nonparametric regression.
also allow covariates
w
to enter the
demand function
these covariates are region and time dummies.
will
work
In our empirical
q(p,y,w).
In other contexts they
might be
demographic variables.
We
will try to minimize the dimension of this
nonparametric
function by restricting
w
to enter in a parametric way.
denote a one-to-one
function with range equal to the nonnegative orthant of
one-to-one function of
and
(p,y),
further discussed below.
We
t(x)
let
Let
k
IR
T(g)
such as
,
g
T(g) = e^,
x
denote a "trimming" function, to be
assume that the true value of the demand function
will
a
is
given by
g^Cx.w) = r^Cx) + w'p^,
qQ(p.y.w) = T(gQ(x,w)),
(2.1)
T"^(q) = ggCx.w) + e,
where
q
E[c|x] =
E[T(x)we'l =
0.
denotes observed quantity and the expectations are taken over the distribution
of a single observation
excluded from
q,
from the
so that
k
We assume here
data.
is
specification for the regression of
assumed to be a function of a partially linear
is
T
(q)
estimator of the demajid function will be
an estimator of
g^^,
that a numeraire good has been
the number of goods, minus one.
Thus, the true demand function
is
0,
on
p, y,
and
w.
A corresponding
q(p,y,w) = T(g(x,w)) = T(r{x)+w'3),
such as the kernel or series estimator discussed below.
estimate exact consumer surplus nonparametrically by substituting q(p,y,w)
for
where
g
We
q(p,y)
This specification does not restrict the joint distribution of the data, unlike previous
is imposed.
work on partially linear models (e.g. Robinson, 1988), where E[e|p,y,w] =
A precise description of g (x,w), for nonnegative t, is as the mean-square projection
of
Pr
E[q|x,w] on functions of the form
= E[x{x)li4)]/Pr{d), where !(•)
{id)
w'g for the probability distribution
denotes the indicator function.
r(x) +
in
equation
w
version of the paper set
A
Let
kernel estimator
mean and
equal to its sample
v
has
= h
h
q,
of
-k-1
E[B(z)|x]
J'K(v)dv =
a bandwidth parameter,
>
In
z
z,
For a matrix function
and possibly other variables.
w,
y,
h
Also, let the data be denoted by
K(v/h).
1.
elements, satisfying
k+1
and other regularity conditions discussed below,
K, (v)
t(x) =
in this
a locally weighted average that can be described as follows.
is
be a kernel function, where
J<(v)
The empirical results reported
and solving numerically.
(1.1)
,
where
B(z),
1
and
z includes
a kernel estimator
is
E[B(z)|x] = J:.",B(z.)K. (x-x.)/5:.",K. (x-x.).
^j=l
h
^j=l h
J
J
J
To estimate the
pau'tially linear specification in equation (2.1),
coefficients of
w
in
we
"partial out" the
a way einalogous to that in Robinson (1988).
The estimator of
g
IS
g(x.w) = f(x) + w'^.
(2.2)
P =
where
[Ii"ii^i(
x.
=
V^^"^
'
^i^^^
V^^"^
'
''i^^' ^'^li^i'^i^
V^fw
A convenient kernel that we consider
t(x.).
fed - v'v),
v'v <
x.l)(T'^q.)-E[T"%)
x.
)).
in the
empirical work
is
I
otherwise
,
where the
constcint
C
is
chosen so
JK(v)dv =
1.
This
is
a multivariate Epanechnikov
The asymptotic theory requires kernels with integrals over
even powers of
v
J<(v)
of certain
equal to zero, that are often called "higher order," and are useful
for reducing asymptotic bias
in
kernel estimation.
Therefore,
we
empirical work a kernel of the form
(2.3a)
I
'
K(v) =
(2.3)
kernel.
f(x) = E:lT"^(q)|x] - E[w|x]'p,
Kiv) = ^(v^)^{v2)(12 - 6v^ - 6v^ + Sv^ + lOv^v^ + 5v^),
also consider in the
for our one dimensional price (k =
It
is
application.
1)
known that the choice of bandwidth
well
In the empirical
nonparametric regression.
can have important effects on
h
work we consider a data based choice of
mean square
equal to a "plug-in" value that minimizes estimated asymptotic
also consider the sensitivity of the results to the choice of bandwidth.
we
empirical work
also allow the kernel to be data based in that
h,
error, and
In the
we normalize by the
estimated variance of price and income, although the theory does not allow for data-based
bandwidth or kernel.
A
series estimator is the predicted value
from a regression of the log of gasoline
consumption on some approximating functions for
one-to-one, smooth transformation as above, let
matrix of functions of
K
combinations of
IT.
,<j)
^1=1
(x.,w.)0 (x.,w.)']
regression of
11
T
g(x,w) =
(2.4)
where
y =
•
(i}',^')'
<f>
y.
on
K
is
.<b
=
(<p
(x.,w.)T
1
1
A
(x.,w.).
Ax,w)'y
K
(x,w) =
''1=1
-1
(q.)
K
Let
(x).
11
the idea being that
x,
^'^(x)'-;;
)'
w,
is
I,
®(^
x
For
w.
denote a
^(x))
j.(x)
a
to be approximated by linear
and
y =
be the the coefficients from a
with
g
t(x) =
1
is
+ w'3.
4>
This estimator can also be
(x).
satisfying equation (2.2) with the kernel
E[B(z)|x] =
coefficients of the least squares regression of
Power
r(x)
=
and on
y
series estimator of
conditional expectation estimator replaced by
splines.
(x)
partitioned conformably with
interpreted as "partialling out"
Two important
<f>
(x)' ,w'
(q.)
^1
and
p
B(z.)
on
4>
(x)'t)_,
<p
(x.).
where
tj^,
are the
types of approximating functions are power series and regression
series are
formed by choosing the elements of
powers of the individual components of
x.
Power series
aire
^
(x)
to be products of
easy to compute and have
good approximation rates for smooth functions, although they are sensitive to outliers
and local behavior of the approximation, and can be highly collinear.
The collinearity
problem can be overcome to some extent by replacing the powers of individual components
with orthogonal polynomials, which does not effect the estimator but may lead to easier
Regression splines are piecewise polynomials of order
computation.
x
For univariate
points.
in
[0,1]
with evenly spaced knots
regression spline approximating functions are
(x-(j-<i-l)/(K-<x])
j
,
i
<x
+ 2,
where
regression spline can be formed from
components of
(v)
all
a.^,(x)
= l(viO)v
= x
Also, the
a+1),
1
in
order to place the knots.
power
a
x,
cross-products of univariate splines
in the
is
not as fast
Unlike power series, the range
In practice,
be constructed by dropping from the data any observation where
range.
=
The spline approximation rate for very smooth functions
x.
must be known
x
(j
join points),
For multivariate
.
as power series, but they are less sensitive to outliers.
of
,
(i.e.
with fixed join
<x
such a known range could
x
is
not in some
spline sequence above can be highly collinear, but this
can be alleviated by replacing them with their corresponding B-splines:
e.g.
known
problem
see Powell
(1981).
Series estimators are sensitive to the numbers of terms in the approximation.
the empirical
work we choose the number of terms by cross-validation, and
different numbers of terms to see
how the
substituting the estimated
demand function
As
be a price path with
1,
let
p(t)
denote the solution to equation
S(y,w,g)
also try
results are affected.
Returning now to estimation of consumer surplus, our estimator
in Section
In
in
(1.1)
equation
(1.1)
p(0) = p
is
constructed by
and integrating numerically.
emd
p(l)
with demand function
Then consumer surplus at particular income and covariate values
y
= p
.
Also,
let
q(p,y) = T(g(x,w)).
and
w
respectively, with a corresponding estimator, is
Sq = S(yQ,WQ,gQ),
(2.5)
where
g
is
S = S{yQ,WQ,i).
a kernel, series, or other nonparametric estimator.
A corresponding
deadweight loss value and associated estimator can be formed by subtracting the "tax
receipts," as in
Lq = Sq
(2.6)
L =
- (p'-p°)'T(g(p'.yQ.WQ)).
§ -
(pV)'T(i(p\yQ.WQ)).
A summary measure for consumer surplus can be obtained by averaging over income
In addition,
values.
it
may sometimes be
of interest to average over different prices,
To set up such an average
to reflect the fact that individuals face different prices.
u
let
u,
index price paths, so that
with
and final prices
initial
p(t,u)
p(0,u)
is
the price at
and
For example,
for covariates.
distribution, or
y_^
solution to equation
= T(g(x,w)).
q(p,y)
means across
z,
w
and
(1.1)
u,
y
and
w
for the price path
z
Also, let
w
for income and
y
might be drawn (simulated) from some
might be the actual observations.
p(u,t)
y
at
The average surplus and deadweight
aind
loss
we
Let
S(z,g)
w
with demsind
,
be the
consider are weighted
form
of the
Mq = E[w(z)S(z,gQ)l/E[w(z)],
(2.7)
,
and values
u
for price path
e [0,1]
respectively.
p(l,u)
denote a single data observation that includes
t
A = JT^Ij^i^^^i'^^/"'
" = L^i^^^^-^^-
Xq = Mq - E[u(z){p(l,u)-p(0,u)>'T(gQ(p(l.u),yQ,WQ))]/E[u(z)l,
A = A - ir^Ej"^u(z.){p(l,u.)-p(0,u.)}'T(i(p(l,u.).yQ..WQ.))/n.
It
is
interesting to note that the consumer surplus estimator
than the deadweight loss estimator.
dimension (the variable
t
in
which depends on the value of
surplus and deadweight loss
In particular,
p(t)),
g
is
like
2
different convergence rates.
2
thaui
Similarly, average
will be the same, both being the
faster
an integral over one
and so has a faster convergence rate
at a particular point.
may have
where their convergence rates
S
may converge
L,
consumer
One important case
parametric
l/ViT
rate, is
It is known from the work on semiparametric estimation that integrals or averages
converge faster than pointwise values, e.g. Powell, Stock, and Stoker (1989). The fact
that one-dimensional integrals converge faster than pointwise values has recently been
shown
in
Newey
(1992a) for kernel estimators.
10
when the
initial
be that where the initial price for each individual
the tax rate
all
is
An example would
and final prices and income have sufficient variation.
the same across individuals.
the price they actually faced, and
is
In this
case averaging will take place over
the arguments of the nonparametric estimates, which
is
known from the semiparametric
estimation literature to result in Vn-consistency (under appropriate regulairity
conditions).
So far, our nonpaircLmetric consumer surplus estimators have ignored the residual
T
(q) - gp,{p,y,w).
econometrics, and
This approach
is difficult
One can ignore the residual
it
is
Hausman
individual heterogeneity.
consistent with current practice in applied
is
Even when
is
e
measurement error and not
all
shows that
(1985)
measurement error and heterogeneity
heterogeneity
in
is
is all
heterogeneity,
it
Brown smd Walker
for some value of
V
(e.g.
specification of
contains
possible to separate out
may
be possible to interpret the demand function
Suppose that
c = e(p,y,w,v)
of prices, income, covariates and a taste variable
independent of price and income.
value of
is
it
not identified in our nonparametric, linear in residual specification.
e(p,y,w,v)
as shown by
it
if
residual.
parametric, nonlinecir models, but the aunount of
as corresponding to a particular consumer type.
function
more information about the
to improve on without
if
c =
v
for
then
e
=
demand as
In
(1990).
g(p,y,w)
(r(p,y,w)v).
general
3
Nevertheless,
if
will
depend on
c(p,y,w,v)
where
p
v
and
is identically
y,
zero
can be interpreted as the demand function for that
In the rest of the
T(g (p,y,w)),
measurement error or to evaluation
c(p,y,w,v)
v,
for some
paper we stay with the
corresponding to an interpretation of
at a particular
consumer type.
c
as
4
3
They showed that for T(q) = q the residual c must be functionally dependent on p
and y. Also, Brown has indicated to us in private communication that the same result is
tr
true
for T(q) = e
.
4,
An alternative approach recently suggested by Brown
consumer surplus over different consumer types, when
one-to-one function of
v.
11
sind
Newey
c(p,y,w,v)
is to average
an estimable,
(1992)
is
3.
Asymptotic Variance Estimation
Under regularity conditions given
To be
asymptotically normal.
specific,
-^
Vn<r"-(B - 0^)
will be
9
•nV"^^^(e - e.)
O
n
-^
l/iVna-
V
a series estimator there will be
(3.2)
V
and
a i
such that
a =
N(0,
is
while in other cases the
slower than
1/v^
by
o-
—
>
0.
For
1).
and kernel estimators will have the same asymptotic
V
converging to
which
),
0,
such that
n
In the v^-consistent case the series
variance, with
V^
N(0. V^).
The "full-average," v^-consistent case corresponds to
convergence rate for
estimators will be
denote any one of the estimators
G
let
For a kernel estimator there will be
previously presented.
(3.1)
in Section 6, ail of the
from equation
In other cases the
(3.1).
two
estimators generally will not have the same asymptotic variance, and the series estimator
weaker property of equation
will satisfy the
convergence.
(3.2), that
does not specify a rate of
Exact convergence rates for series estimators
for v'iT-consistency, although
it
is
still
For large sample inference, suppose that there
/V
(V
If
)
-£-»
1
then
it
VnW'^^^le - G^)
n
Consequently, a
(3.4)
G ±
1-a
>x
/J
-^
N(0,
is
an estimator
large sample confidence interval will be
,Vv n /n
.„.
.
,
12
V
of
V
in
follows by equation (3.2) (and the Slutzky
1).
'^a/2
Despite this
be used for asymptotic inference.
theorem) that
(3.3)
not yet known, except
possible to bound the convergence rate.
lack of a convergence rate, equation (3.2) can
equation (3.2).
au~e
where
^
is
.„
the
1
- <x/2
form large sample hypothesis
also use equation (3.4) to
To form large sample confidence
of
V
a formula for
V.
This procedure
is
tests.
intervals, as in equation (3.4), an estimator
V
For kernel estimators, one method of forming
needed.
is
n
One could
quantile of the standard normal distribution.
n
V
would be to derive
and then substitute estimators for unknown quantities to form
cr
V
.
not very feasible, because the asymptotic variances are quite
complicated, as described in Section
6.
we use an
Instead
knowing the form of
(1992a), that only requires
method that just uses the form of
Q.
alternative method,
from Newey
For series estimators we also use a
,..,,.
6.
The asymptotic variance estimators for kernel and series estimators have some common
features.
In
each case,
(3.5)
V
= Z^.^jj^./n,
''1=1
n
ni
where the estimates
\{i
are constructed in the following way.
.
variance will be based on the form of the surplus estimator, as
(3.6)
where
G = jr^5:."^a(z..i)/n,
a(z,g)
is
ecu-lier,
and
w(y)
a function of a single observation
is
(3.7)
g
a weight function.
corresponding to each case
in
u = l-^^uiyJ/n,
g = g(x,w) = r(x) + w'p,
specification
Also, in each case, the
is
z
and the partially linear
the kernel or series estimator described
The specification of
a(z,g)
and
is
e =
S:
a(z.,g)
= S{yQ,WQ,g);
e =
L:
a(z.,g)
= S(yQ.WQ.g) - (pV)'T(g(p\yQ,WQ));
e =
fi:
a(z.,g)
= w(y.)S(z.,g);
e =
A:
a(z.,g) = u(y.){S(z..g) - (p(l,u.)-p(0,u.)]'T(g(p(l,u.),y.,w.))>,
(j(y)
13
=
1;
(j(y)
where
For both kernel and series estimation,
(p,y).
which accounts for the variability of
the variability of
we can
that
^
(3.8)
where
in
y.
0)
The
g.
component
first
.
is
that
is
the image of
have two components, one of
will
a(z.,g)
in
z.
x
evaluated at the
auid
and the other for
tj(y.),
the same for both kernel and series, so
specify
= 0^ +
.
ni
0^ = w ^a(z.,g) - e -
!/>?,
subscript on
n
aui
term
iji.
w'p
g{x,w) = r(x) +
denotes
g(p,y,w)
"Z
"-R
and
\jj.
0?
is
(J
^e[w(y.) - w],
an asymptotic approximation to the influence on
is
The term
,a(z.,g).
r"
'^1=1
of the
will account for the estimation of
!L.
observation
i
can be
It
g.
1
constructed from an asymptotic approximation to the influence on
observation in
The
suppressed for notational convenience.
of the
6
th
i
taking a different form for kernel and series estimators.
g,
For kernel estimators the idea for forming
differentiate with respect to the
developed in Newey (1992a),
.,
observation in the kernel estimator.
i
is
to
This
calculation amounts to a "delta-method" for kernels, and leads to an estimator that
is
robust to heteroskedasticity and has the same form, no matter what the convergence rate
of
G
is.
(3.9)
To describe the estimator,
h''(x)
=
J
h
J
^j=l
u
1
J
G^ = 3[Ej=ia(Zj.ip)/n]/ap|p^^,
=
j:.",K. (x-x.)/n,
^j=l h
J
1
h
1
ip(x,w) = E[T"^(q)-w'p|x] +
1
5=0_,
.
w'p
= 5:."^T.(w.-E[w|x.])(w.-E[w|x.])/n,
0| =
The term
r(x)
f(x)
J
a[n"^j;.",a(z.,{f + 5K.«-x.)r^<h^ + S{T~hq.)-vf'.p}KA'-x.) + w'.p)/n]/a5|
1
M
denote a scalar and
= j:",<T"^(q.)-w.'p}K. {x-x.)/n,
J=l
A.
5
let
(J
A.
'{A. -
in
L^jA ./n
+ g'^M
T.{w.-E[w|x.])(T ^q.)-i(x.,w.))>.
an asymptotic approximation to the influence of the
on the average of
a(z.,g),
while the second term in
14
ifi.
is
th
i
observation in
a fairly standard
i
delta-method term for estimation of
For series estimators the Idea
approximation were exact.
-'.•-'
p.
to apply the "delta-method" as If the series
Is
This results
in
correct asymptotic inference because
accounts properly for the variance, while the bias
To describe the estimator,
conditions.
G^ = a[E "
(3.10)
a(z
it
small under appropriate regularity
is
let
g^(x.w) = /(x,w)'r.
g )/n]/arL _;,
^^ = ir^G^Z"V(x.,w.)[T"^(q.)
Z = j:."^^'^(x.,w.)/(x.,w.)'/n.
- g(x.,w.)],
ACT
Here
is
\Jj.
a standard "delta-method" term for ordinary least squares estimation of
r.
/nCT
For either kernels or series, the main difficulty
the derivatives
G
and
A.
or
G
in
computing
ifi.
is
calculating
For each of the estimators described
.
in Section
1
2,
it
is
possible to derive analytical expressions for these derivatives, but the
expressions are so complicated as to make them almost useless for calculation.
these derivatives can be calculated by numerical differentiation.
requires evaluation of
J]._
many different values
for
a(z.,g)/n
Instead,
This calculation only
of
which
g,
is
quite
feasible, peirticularly for series estimators.
A procedure analogous
to that for the series estimator can be used to construct a
consistent asymptotic variance estimator for exact consumer surplus for any parametric
specification of the
dememd
depend on the parameters
9(5]._.a(z.,y)/nl/3y
1—
that
I
1
For a parametric specification,
function.
-y
demand
of the
In this case
function.
G
can be calculated by numerical differentiation.
_*
will
a(z,9r)
=
Then, supposing
"*tf
=
Vn(.y - r.)
Y.
,* (z.)/v'n + o
^1=1
1
and that
(1)
p
*T
is
an estimator of
1
*
(z.),
we
1
can form
(3.11)
ijj.
1
=
IJK
1
+ ir^G^*T.
1
Asymptotic inference for parametrically estimated exact consumer surplus could then be
carried out as described above for
V
n
as in equation (3.4).
15
Demand Conditions
Testing Consumer
4.
Tests of the downward slope and symmetry of Hicksian (compensated) demands provide
useful specification checks for consumer surplus estimates, aind are of interest in their
own
Here we consider tests that are natural
right as tests of consumer theory.
by-products of consumer surplus estimation.
surplus
An implication of symmetry
is
that consumer
independent of of the price path, which can be tested by compairing estimates
is
The downward sloping property can be tested by comparing
based on different price paths.
new
the demand at the
demand
price with compensated
at the initial price, which is easily
computed from the consumer surplus estimate.
Path independence can be tested by comparing consumer surplus for the same income
To describe
and covariates but different price paths.
path
p
with
(t),
p'^(O)
= p
and
= p.
p-^(l)
Let
this test, let
w
covariates
1,
...,
J)
should converge to the same limit.
comparing the different estimators.
Let
chi-square.
TT =
n
(4.1)
n
vector of
is
I's.
income
f$
(s,
c V
s,)'.
J
i
*.
=
A simple way
f.7,^
(i/»t
1
1
Then the test
n(II
is
and
y^,
that
all
S.,
to construct this test is by
,1,^\'
ip-.y.
=
minimum
n =j:.
=r " ,*.*'./n.
* A'
Q
1
^1=1
1
1
ft.
Let
e
denote a
- S«e)'n"^(S -
fl'e),
jl
= (e'n"^e)"^e'n"^fl.
will be
X^U-l).
S
J
statistic is given by
Under the symmetry hypothesis the asymptotic distribution of this test statistic
The estimator
(j
This implication can be tested by
an estimator of the joint asymptotic variance of
T =
(4.2)
(t),
denote the corresponding estimator from equation (3.7), and
0.
1
Here
p
An implication of symmetry of compensated demand
.
index a price
denote the equivalent
S.
variation estimator described above for the price path
j
may be
of interest in its
16
own
right.
By the usual minimum
x
1
chi-square estimation theory, under symmetry of compensated demands
as asymptotically efficient as any of the estimators
-1
1
variance matrix
(e'n
e)
S
will be at least
it
with an estimated asymptotic
.,
This efficiency improvement leads to the question of an
.
This question could be answered, in part, by deriving the optimal way
efficiency bound.
to average over all different price paths, a question outside the scope of this paper.
The downward slope of compensated demands can be tested by testing for nonnegativity
of
(p^-p )'[T{g^(.p .y^.w^)) - T(g|^(p ,yQ-S{yQ,WQ,gQ),WQ)]
To describe
incomes, cind covariates.
denote different values, let
§
this test, let
p.,
over several different prices,
p.,
w
y.^.,
(j
.,
=
J)
1
denote the corresponding equivalent variation
.
estimates, and
(4.3)
e^.
=
- p°)'[T(i(p^.,yjQ,w^.Q)) - T(i(p°.yjQ-Sj..w^.Q))],
(p^.
An implication of convexity of the expenditure function
approach of Section
Let
4.
9 = (9
^i^'*
^^^lI
be as described in equation (3.7), for
ijr:
Alternatively,
if
w(z) =
the income values
distinct then the asymptotic covariances between the
n =
5^._ diag((i/».)
distribution of
,...,(01.)
9
that the limit of
is
9
)/n
9
9
.
let
« = Ej^i
will be zero, so that
Thus, the hypothesis
Q/n.
a nonnegative vector can be tested by applying multivariate
A particularly
tests of inequality restrictions developed in the statistics literature.
simple test would be to reject
if
min .{v^9
J
./n..) ^ k
J
for some
k.
The size of
minimum
of vector of
zero normals with variance matrix equal to the correlation matrix implied by
is
possible to combine these two types of tests, of
17
this
JJ
test could be calculated by simulating the distribution of the
It
=
are mutually
y.
normal with variance
is
and
1
a(z,g)
The asymptotic approximation to the
will suffice.
then that
is
that each of these estimators
can be constructed via the
(p^.-p°)'[T(g(pJ,y^.Q,w.Q))-T(g(p°,y.Q-S.(y.Q.w.Q,g),w.Q)].
(i^.,...,i/i.)' (i^.,...,i/».)/n.
J).
1
This hypothesis can be tested using an estimator of the
should have a nonnegative limit.
asymptotic variance matrix of
is
=
(j
mean
Q.
symmetry and of downward
sloping compensated demand, into a single test of consumer
S
and
into a single vector
6
The
(§',§').
values could be stacked in the
(§',§')'
corresponding way, and an estimator of the varieince of
outer product of the stacked vector of
^
demand theory, by "stacking"
values.
Also,
formed as the average
would be possible to
it
consider versions of these tests based on the consumer surplus averages of equation
Furthermore,
(2.7).
it
should be possible to give these tests some asymptotic power
against any alternative to demand theory by letting
a
way
size in such
that different price paths and income and covariate "cover" all values in their
These extensions are beyond the scope of this paper.
support.
An Application
5.
grow with the sample
J
to Gasoline
Demand
To estimate the nonparametric and parametric demand functions for gasoline, we use
data from the U.S. Department of Energy.
The
first three
waves of the data were
collected in the Residential Energy Consumption Survey conducted for the Energy
Information Agency of the Department of Energy.
and 1981 at the household
level.
Surveys were conducted in 1979, 1980,
Gasoline consumption
we use average household
is
gallons consumed per month.
in
our analysis
is
the weighted average of purchase price over a month.
quite high during
shock.
$.
1983
most of
kept by diary for each month;
The gasoline price
Note that gasoline prices were
this period in the U.S. because of the second (Iranian) oil
Gasoline prices averaged between $1.34-1.46 for these 3 years where
Income
$).
$50,000.
is
we use
1983
divided into 12 categories with the highest category being over $50,000 (in
Here we used the conditional median for national household income above
Lastly geographical information
is
given by 8 census regions.
patterns differ significantly across regions of the U.S..
were collected by the Energy Information Agency
18
The
last three
in the Residential
Average driving
waves of data
Transportation Energy
Price and income were collected
Consumption Survey for the years 1983, 1985, and 1988.
in the
same manner as the
(The upper limit on income changed in the
earlier surveys.
surveys; however the technique used to estimate income in the top category remained the
same)
Note that the (real) gasoline price
that by 1988
had decreased to
it
before the first
divisions
oil
shock
were used.
levels,
fell
throughout this period so
about $0.83, approximately equal to prices
In the latter surveys,
in 1974.
Since
in the U.S.
we are unable
to
map
9,
rather than
8,
census
the earlier 8 region breakdown into 9
we use
regions, or vice-versa, in the empirical specifications
different sets of
indicator variables depending on the survey year.
we have
Overall,
18,109 observations which should provide sufficient observations to
Our empirical approach
do nonparametric estimation and achieve fairly precise results.
is
to do both the nonparametric and parametric estimation with indicator variables both
for survey year and for regions.
In the
specifications.
Thus,
we have 20
indicator variables in our
parametric specifications, we allow for interactions, most of
which are found to be statistically significant which
to be expected given our very
is
However, we decided to use the same set of indicator variables
large sample.
in
both the
nonparametric and parametric specifications to make for easier comparisons.
We
give four types of nonparametric estimates, using the Epanechnikov kernel
from
equation (2.3), the normal higher order kernel from equation (2.3a), a cubic regression
spline with evenly spaced knots, and
specification,
where
T(g) = e
.
power
series.
We
Also, the covariates
use a log-linear demand
w
that allow for different region and survey year effects.
are
indicator variables
In the results
evaluate demand and consumer surplus at a fixed value for
We
20
w
we present we
equal to its sample mean.
tried several different bandwidths for the kernel estimators, based on the
smoothness of the graphs discussed below.
The Epanechnikov kernel estimates used a
bandwidth of
.82
of
used to form
r(x),
order kernel
in
for the coefficients
equal to
equation (2.3a),
h = .82,
3
w,
and
h = .55, and
we used bandwidths
19
of
t(x) =
h = .45.
h = .55
1.
The bandwidths were
For the normal, higher
and
h = .45
for both
the nonpeirametric part and the coefficients of
w.
For the series estimates, we used cross-validation to help choose the number of
terms.
Cross-validation
is
the
sum of squares
observation using coefficients estimated from
a cross-validation criteria
estimates.
is
known
of prediction errors
all
other observations.
from predicting one
Minimizing
minimum asymptotic mean-square error
to lead to
Table 5.1 reports the criteria for splines and power series that are additive
in (log) price
and income.
No interactions were included because they were never found
to be significant, either in a statistical sense or in
terms of lowering the
cross-validation criteria.
Table
5.1:
Cross-validation for Series Estimators
Power series
Cubic spline, evenly spaced knots
CV
Knots
832
826
842
857
1
2
3
4
5
6
7
8
9
922
903
834
824
825
1
2
3
4
5
6
7
8
9
801
797
784
801
782
10
CV
Order
900
823
791
804
801
The criteria are minimized at
7
or
order power series.
knots and 8
9
suggests that one should choose a number of knots that
is
greater than the mean-square
error minimizing one, so the bias goes to zero faster than the variance.
we prefer
8
or
9
knots and an
knots were similar to those for
large, so
we do not report them.
8,
8
or
9
The theory
order polynomial.
For this reason
The results for
9
except that the estimated standard errors were very
Instead
we report
results for
6,
7,
and
8
knots.
For purposes of comparison we also report results for standard parametric forms for
the demand function.
The specification of the demand function
20
is:
Inq.
(5.1)
where
y
=
tIq
+ "njlnp + Tl2^ny +
household income, p
is
dummies discussed above.
.01
is
w'^
w
gasoline price, and
The estimated income
while the estimated price elasticity
To check on the
+ c,
is
sensitivity of our estimates,
are the 20 region and time
-0.80
we
with standard error
.37
elasticity is
with standard error
.09.
also estimated a "trcinslog" type of
parametric specification where we allow for quadratic terms
in log
income and log price
The income-price
as well as an interaction term between log income and log price.
interaction term has no estimated effect, but the quadratic terms do have an effect on
The sum of squared errors decreases from
the estimates.
decrease of 0.26%, but a traditional F statistic
is
degrees of freedom due to the large sample size.
8900.1
to
8877.0
which
is
a
calculated to be 16.1 with three
The estimated
elasticities for the
log-linear and log-quadratic model are quite similar at the median gasoline prices,
$1.23:
the log-linear price elasticity
log-quadratic model price elasticity
is
is
-0.81 at all income levels while the
approximately -0.87 with very
The two specifications do have different
across income levels.
higher gasoline prices.
The log linear model price
a single parameter, remains at -0.81 across
all
little
variation
elasticities at lower and
elasticity, since it is estimated as
gasoline prices while the log-quadratic
model, which has a variable price elasticity, has an estimated elasticity of
approximately -0.64 for gasoline price of $1.08 (the first quartile price) while the
log-quadratic model has an estimated elasticity of -1.14 at a gasoline price of $1.43
Since the results for log quadratic translog specification are
(the third quartile).
only approximately similar to the simpler liner specification,
both of the parametric specifications
Figures
1-3
in
we present
results for
what follows.
show the estimated nonparametric
log
demand with respect
to log price,
evaluated at mean income, for the parametric, Epanechnikov kernel, and spline
specifications.
We do
not give graphs for other income values because the graphs have
21
similar shape and the spline results discussed below strongly suggest that the log
additive in log price aind log income, in which case the shape of log
function
is
will not
depend on income.
demand
demand
There are interesting differences between the parametric and
nonparametric estimates, with the nonparametric estimates having a much more complicated
shape than the parametric ones.
generally
for
downward
h = .82
find kernel
sloping over the range of the data.
looks "too smooth," and the one for
subsequent analysis
spline
we
In Figure 2
we
demand function
will use
is
h = .55
demand curves that are
In our opinion the
h = .45
demand curve
looks "too rough" so in the
as our preferred bandwidth.
The shape of the
qualitatively similar to that the kernel estimate, although the
demand function does slope up very
slightly at
some points on the graph.
We
tested
whether this upward slope indicates a failure of demand theory using the test for of
downward sloping compensated demand described
a price chauige from
up, our
$1.39 - 1.46,
N(0,1) statistic
.90.
which
This value
above.
For
8 knots (our preferred number)
is
the range over which the demand curve slopes
is
not statistically positive at any
conventional (one-sided) critical value.
We next estimate
the exact consumers surplus, here the equivalent variation, across
our different estimated demand curves.
gasoline:
We
an increase from $1.00 to $1.30
$1.00 to $1.50 per gallon.
gasoline prices and a
50
consider two sets of price changes for
(in
1983 $) per gallon and an increase from
The starting price of
cent increase
is
$1.00
corresponds roughly to
well within the range of the data.
1992
We
estimate the equivalent variation for these prices changes at the median of income.
present the estimates in Table 5.2.
Estimated standard errors are given
in
We
parentheses,
and were calculated using the formulas given earlier.
The conventional significance levels may not be appropriate here, because we have chosen
the interval based on the estimated demand function.
However, the conventional
critical values should provide a bound when the test statistic is maximized over
choice of interval, with our test being an approximate maximum.
22
Yearly Equivalent Variation Estimates
Table 5.2:
$1.00-1.30
Parametric Estimates
1.
442.00
282.34
Log linear model
(
2.
$1.00-1.50
2.07)
285.44
Translog quadratic model
(
2.31)
(
444.16
3.90)
(
3.03)
Epanechnikov Kernel Estimates
3.
4.
h = .45
h = .55
$281.31
4.36)
(
$445.09
$284.00
$447.41
(
5.
h = .82
4.11)
4.11)
(
445.42
279.87
(
4.46)
(
3.58)
(
3.59)
Normal, Higher Order Kernel Estimates
6.
h = .45
286.44
450.14
7.07)
(
(6.56)
7.
h = .55
284.36
(
445.35
5.35)
6.34)
(
Cubic Spline Estimates
8.
(
9.
4.93)
(
282.30
7 knots
(
10.
444.63
284.58
6 knots
8 knots
441.72
5.82)
(
4.68)
447.80
287.12
(
6.31)
4.77)
(
6.04)
Power Series Estimates
11.
12.
8th order
9th order
287.31
4.76)
(
448.64
287.27
448.55
(
Note that
all
(
4.75)
(
the nonparametric estimates are quite close.
5.91)
5.88)
A choice of different
nonparametric estimator assumptions leads to virtually the same welfare estimates.
Surprisingly, although the graphs
show quite different shapes for parametric and
nonparametric estimates, the welfare estimates are quite similar.
A Hausman
test
statistic based on the difference of the kernel estimate and the difference of their
estimated variances, which
is
is
valid if the disturbance
is
homoskedastic,
is
not significant for a two-tailed tests based on the normal distribution.
23
1.3,
which
We
from a
also estimate the deadweight loss
rise in the gasoline tax of either $0.30
or $0.50 which would induce the corresponding rise in gasoline prices.
We base our
estimate of deadweight loss on the equivalent variation measure of the compensating
variation.
The results are given
Table 5.3:
in
Table 5.3:
Yearly Average Deadweight Loss Estimates
Parametric Estimates
1.
$1.00-1.30
Log linear model
26.92
(
2.
$1.00-1.50
Translog quadratic model
62.50
3.13)
(
75.30
27.07
(
6.77)
3.15)
(
7.24)
Epanechnikov kernel estimates
3.
4.
5.
h = .45
$27.70
h = .55
h = .82
$33.41
(5.65)
(8.68)
27.94
36.90
(5.01)
(7.97)
22.89
37.10
(3.88)
(6.80)
$36.38
$33.09
(6.47)
(14.06)
Normal, higher order kernel estimates
6.
7.
h = .45
h = .55
$31.62
47.61
(5.72)
(8.92)
28.60
51.05
(5.03)
(8.65)
38.68
46.84
(5.37)
(8.77)
35.72
46.95
(5.27)
(8.94)
34.92
48.66
(5.21)
(8.80)
34.86
48.74
(5.38)
(8.86)
Cubic Spline Estimates
8.
9.
10.
6 knots.
7 knots
8 knots
Power Series Estimates
11.
12.
8th order
9th order
The estimates of
DWL
are very similar across bandwidths for the kernel estimator, but are
sensitive to the use of a higher order kernel.
We
give
more credence
to the higher order
kernel estimates, because of their theoretically better property of having smaller bias
24
relative to
mean-square error, and their similarity to the spline estimates.
results are sensitive to the
number of
knots.
We prefer
The spline
the results for the larger
number of knots, because they are theoretically preferred and the results seem
sensitive to choice between
We do
7
and
8
than to
6
and
7.
find rather large differences between the nonparametric and parametric
estimates of deadweight
loss.
The estimated differences between the nonparametric
estimates and the log linear parametric estimates are
particular, the nonparametric estimates
in
the range of 40-507..
These are economically significant
differences, with the ratio of
DWL
nonparametric specifications.
The differences are also
to tax revenue varying widely
between parametric and
statistically significant:
test of based on the log-linear and normal kernel specification is
the larger price change and bandwidth
and bandwidth
.45.
In
seem to be somewhat smaller for the larger price
change and larger for the smaller price change.
Hausman
to be less
.55
and
is
5.18
A
-10.14
for
for the smaller price change
Thus, efficiency decisions on which commodities to tax and the size
of the tax might well depend on the rather sizable differences in the estimated
efficiency cost changes
from the increased taxes.
'!'),'
25
J
6.
Asymptotic Distribution Theory
In this section
we
give results for the kernel estimator of consumer surplus at
particular income and covariate values, and for power series.
The conclusions for other
These results show
cases will be described, without full sets of regularity conditions.
that the standard error formulas given earlier are correct, and describe the asymptotic
properties of the estimators, without overburdening the reader.
Let
Precise conditions are useful for stating the results.
and
g(p,y,w)
the image of
be a set of
r(x(p,y)) + w'p.
denote
p(t,u)
on
Let
W
and
[O.llxli,
p
(t,u)
be the support of
11
be the support of
w.
solving equation (LI).
The conditions
function over the set
Z = TxyxW.
will involve
Let
llgll
.
partial derivatives of all elements of
V =
Also, let
is
be
[y,y]
evaluated in
_
i
.119
»
g(p,y,w)/5(p,y,w)
i
where
II,
Z€£.,c—
denotes any vector of
d B(z)/5z
B(z),
p([0,llx1i)
supremum norms for the demand
= sup
J
for a matrix function
T =
u,
values that includes those where the demand function
y
= 3p(t,u)/St
all distinct
order
£
Let "a.e." denote almost everywhere with
B(z).
respect to Lebesgue measure.
Assumption
00,
T(g)
1:
and
W,
11,
V
and
x(p,y)
Z
are compact,
is
[O.llxlJ,
continuous a.e. and for
outside
We
"y
z.,
Hgn"?
are one-to-one and three times continuously differentiable with
nonsingular Jacobians on their respective domains,
differentiable on
contained in the support of
and
T
C = sup.^
= [Y+C+e,y-C-c],
p(t,u)
does not include zero.
^,
e
Also,
u{y)
t(x)
identically equal to one for series estimators.
26
j
,
is
bounded and
w(y} =
for
> 0.
will derive the results under an i.i.d. assumption
Let
twice continuously
^^^^^|T(gQ(p(t,u),y,w))'p^(t,u)
for some
specified in the following result.
is
and certain moment conditions
be a trimming function that will be
y
^
-1
Assumption
2:
bounded density
^q^x),
Assumptions
earlier format,
E[IIT
is i.i.d.,
z.
1
we
and
4
(q)ll
<
]
x
oo,
is
continuously distributed with
E(t(x)(w-E[w|x])(w-E[w|x])'
is
]
nonsingular.
and 2 are useful for both the kernel and series results.
Following the
The next three
will first give results for kernel estimators.
assumptions are more or less standard conditions for kernel estimators.
Assumption
on
(q)ll
|x]f (x)
is
bounded,
There
4:
is
a positive integer
1,
and for
Assumption
E[w|x]
is
J"K(u)[®.%]du =
all j < s,
5:
There
R
to all of
K(u)
is
k+1
differentiable to order
f
f^v^x),
d s s + 2
on
is positive.
X(u)
is
twice continuously
zero outside a bounded set,
is
R
d
and extensions of
(x)r_(x),
and
f
E(T
(q)|x]
(x)E[w|x]
is
k+1
.
income and covariate value,
it
is
necessary to introduce a
denote the solution to equation
-T(gQ(p(t),yQ-S(t),WQ))'p^{t),
x(p(t),y -S(t)),
and partition
S(l)
x(t)
at the truth,
(1.1)
= 0,
where
= (x (t),x
p^(t)
(t)' )'
,
i.e.
the solution to
= 5p(t)/at.
where
x
.t
= C(t)«exp{-r C(v)'[ag„(p(v),y -S(v),w ))/ay]dvK
•'o
27
at a particular
more notation.
little
Let
<(t)
An example
the Gaussian one used in the application.
To describe the result for the kernel estimator of consumer surplus
S(t)
and
are continuously
Assumption 4 requires that the kernel be a higher order, bias-reducing type.
of such a kernel
JKtujdu
0.
a nonnegative integer
such that
bounded, bounded away from zero
fp,(x)
such that
s
differentiable, with Lipschitz derivatives,
=
t(x)
and zero outside a compact set on which
z e Z,
Assumption
4
-1
E[IIT
3:
(t)
Let
is
x(t)
Let
dS/dt =
=
a scalar.
= T (gQ(p(t).yQ-S(t).WQ))'p^(t).
e(t)
Let
x(t) = x(t(T)) = (t.x^U))'.
Assumption
D
domain
zero;
6:
=
E[ee' |x]
ii)
let
x
is
(t)
p{t)
can be extended to a function with
one-to-one with derivative bounded away from
is
for
> 0,
t)
{0,y'
aind
partitioned
)
),
|x=x(T)+(0,3r))f (x(T)+(0,r))>dT
x(p(D),[y,y]),
x„(t)
such that
>
x(t) = (t,x (t)'
4
Note that
7)
CWt)).
continuous a.e. and for some
is
(1+E[llell
from zero on
is
(which will exist by Assumption 6 below),
(t)
C(t) =
and
such that
[-tj.I+t)]
-„><sup
and
There
i)
conformably with
S
x
be the inverse function of
t(T)
^^^
x
<
m.
from Assumption
y
f
iii)
bounded away
is
1.
differentiable by the inverse function theorem and the chain rule,
2
K{r) = /[JKCv.u+lSx (T)/5T]v)dv] du.
The asymptotic variance of consumer
surplus at a point will be
Vq =
Let
Xl(0£t(T):£l)K(T)fQ(x(T)r^ St(T)/St ^E[{<:(t)' e}^ X=x(T)]dT.
I
denote the indicator function for the set
1(A)
I
I
The following result gives the
A.
asymptotic distribution of the kernel estimator of consumer surplus evaluated at a
particular income and covariate value.
Theorem
with
na-^^^^/llnCn)]^ -^
addition
/ln(n)
n(r
—
/ln(n)
—
>
m
and
no-
and
oo,
>
For our application, where
ncr
1-6
If Assumptions
1:
k =
—
n<r^^
then for
m,
>
kernel be at least sixth order.
1,
0.
w(y) = l(y = y^),
are satisfied, for
-^
V
0,
in
then
Vnc^^^CS-S^)
equation
(3.5),
a-
V
the conditions on the bandwidth
These conditions require that
The normal kernel used
28
-S
^
—
<r
s > 5,
c =
(r(n)
N(0. V^).
If
K_.
are that
i.e.
in the application is
that the
such a
in
sixth order kernel.
The asymptotic distribution of a kernel estimator of deadweight
income and covariate value
straightforward to derive.
is
Because the "tax receipts" term
depends on the demand function evaluated at a particular point,
(p
-p )'T(g(p .y^.w-))
it
will have a slower convergence rate tham the
will
loss at a particular
dominate the asymptotic distribution.
consumer surplus "integral," and hence
Then standard results on pointwise
convergence of kernel regression estimators can be applied to obtain, for
(k+l)/2,,~
Vq =
Also,
means
,
—
d
.,,_,
>
,.
= x(p
.y^,),
.
N{0, Vq),
[J-J<(u)^du]fQ(x^)"^-
E[{(p'-p°)'T {g^{x\w^))c)^\x=x\
straightforward to show consistency of the asymptotic variance estimator, by
is
it
,
(L - Lq)
n
x
those used to prove Theorem
like
1.
As previously noted, average consumer surplus and deadweight loss will be
^-consistent
initial
if
and final prices are allowed to vary.
In this case the
asymptotic distribution will be the same for both kernel and series estimators.
This
asymptotic distribution will be described below.
Some of the conditions need
Assumption
from
zero.
Let
llgll
(x,w),
E[e.c'. |x.,w.]
7:
bounded and has smallest eigenvalue that
be as defined above except that the supremum
.
amd
Assumption
X
let
8:
<p,
nondecreasing
in
the order
is
to be modified for series estimators.
is
from zero on
denote the support of
{x)
X
K
is
bounded away
taken over the support of
x.
consists of products of powers of the elements of
order as
increased,
is
is
x
that are
increases, with all terms of a given order included before
a compact rectangle and the density of
X.
29
x
is
bounded away
The condition that the density
The next condition
variance of a series estimator.
Assumption
E[w|x]
and
r (x)
9:
C
a constant
bounded away from zero
is
such that for
is
order
are bounded in absolute value by
This smoothness condition
is
useful for controlling the bias.
is
all
d
integers
C
on
2),
all
orders on
X
the partial derivatives of
X.
undoubtedly stronger than necessary.
apply second order Sobolev norm approximation rates
and derivatives up to order
useful for controlling the
are continuously differentiable of
and there
d
is
(i.e.
It
is
used in order to
approximation of the function
where a literature search has not yet revealed such
approximation rates for power series except under this hypothesis.
Also, results for
regression splines are not given here because multivariate Sobolev approximation rates do
not seem to be readily available for them.
Average equivalent variation and deadweight loss will be ^-consistent under
certain conditions that
we now
Let
describe.
t.
denote a random variable that
is
1
uniformly distributed on
y.-S(t.,z.,g
),
x.
=
x{p.,y.).
Let
= S(t,z,g
)
=
p(t.,u.),
10:
T
E[a)(y.)]
C'^(x,w)
=
(gQ(p(t,u),y-S(t,z),w))'p^(t,u).
Conditional on
is
y.
and
=
a(x,w)
w =
S(t,z)
p.
?(t,z)
E[tj(y.)^(t.,z.)|x.=x,w.=w]
f(x|w)
Let
z.,
= C(t.z)«exp{-r C(v,z)'gQ (p(v,u),y-S(v,z),w))dv>,
and for the density
(x|w)
and independent of
C(t,z)
Assumption
f
and
[0,1]
is
w,
x
f(x|w)
is
of
continuously distributed with bounded density
x.
given
w,
zero outside the support of
a(x,w) =
f(x|w),
bounded.
and
= f(x|w) ^f'^(x|w)E[tj(y.)C(t.,z.)|x.=x,w.=w].
30
and
C^(x,w) = E[C^(x,w)|x] + (w-E[w|x])'M ^E[(w-E[w|x])' C^Cx.w)].
= u
\l/j
u(y.)S(0,z.,g
-^1+0)
)
Then the asymptotic variance of
^i
^l
l(j}(.y.)
will be
- w] + w" c'^(x.,w.)'c..
E[\Jf:*lr.'
].
Under an additional condition we can also derive the asymptotic variance of
deadweight
We
loss.
consider here only the case where there
requirement, by the condition that the final price
Conditional on
11:
bounded density
Let
T. =
cj
f
w,
and
(x|w),
= x(p
x.
i//*^
continuously distributed with
zero outside the support of
f(x|w).
= f(x|wr^f'^(x|w)E[cj(y.)(p^-p°)'T (g-(x..w.)) x.=x,w.=w]:
1
1
1
1
1
g
|
C^x.w) = E[C^x,w)|x]
=
/11
is
is
and
^u(y.)(p^-p°)'T(g(p^(u.),y.)))
cV.w)
continuously distributed.
is
{u.),y.)
(x|w)
cj(y)f
sufficient variation in
The next assumptions embodies this
the final price to achieve Vn-consistency.
Assumption
is
+ (w-E[w|x])'m"^E[(w-E[w|x])'c'^(x.w)].
111
- {T. - E[T.] - E[T.]w"^[(j(y.) - w] + l^^^\x.,^N.y c).
1
1
1
Then the asymptotic variance of
L
1
L 2
will be
E[(i/».)
].
The next result shows asymptotic normality of the power series estimators.
Theorem
= K(n)
L,
2:
Suppose
satisfies
K /n
Assumptions
—
>
and
Kn
- 2,
1
—X
Assumption
V
-^
V^ = E[(\l^J^]
is
11
V
> 0.
then
also satisfied and
V
—
>
9 =
equation (3.3) will be satisfied and
satisfied and
and
11
that
7 - 9 are satisfied, with
m
for some
9+0
(K/-/n).
N(0,V^)
= E[(tp.)
> 0,
then
31
0.
and
V
VrKX - X
-^
)
10 is
V^.
—
>
K
-
-
6 = 5
Then for
If Assumption
-^
.
and
1,
^
y >
VR(ii - ti^)
]
t(x) =
or
also
If
N(0,V^)
6
Proofs of Theorems
Appendix:
C
Throughout the Appendix
will denote a generic positive constant, that
may be
different in different uses.
One intermediate result that
needed for both kernel and series estimators
is
linearization of the solution to equation
(1.1)
around the true demand function.
additional notation is needed to set up this linearization.
and
u
that includes
the solution to
5S/5t = -T(g(p(t,u),y-S,w)).
C(t,z,g)
(6.1)
Let
Some
be a data observation
z
denote the corresponding solution to equation
S(t,z,g)
with respect to
g{p.yiw)
Let
a
is
(1.1),
i.e.
denote the derivative of
g (p,y,w)
and
y,
= C(t.z,g)-exp{-fr C(v,z,g)'g
(p(v,u).y-S(v,z,g),w))dv}.
'0
^(t.z.g) =
T
(g(p(t,u),y-S(t,z,g),w))'p^(t,u).
Under conditions specified below, when
g
is
near the truth
g
,
equivalent variation
can be approximated by the linear functional
_.l
A(z ,g;g)
_ I = r g(p(t,u),y-S(t,z,g).w)'C(t,z,g)dt,
J*0
n
(6.2)
g = g^.
evaluated at
Lemma
that
Al:
for
If Assumption
all
z e Z
,
Wg-g
is
1
W
satisfied then there is an
< e,
and
\\g-gA\
\u)(y)A(z,'g;g)-o>(yWz.g;gQ)\
Proof of
y e
y
.
Lemma
We
Al:
it
is the
^ C\\g-g\\^\lg-g\\^,
\oi(y)S(0,z,g)-u(y)S(0.z,g)-u)(y)h(z.g-g;g)\
\(ji(y)S(0,z,g)-u)(y)S(0.z,g^)\
< c,
c >
and for any
^ CWg-g^W^.
g
|
such
case that
w('yM('z,g;g^;
with
C
and a constant
llgll^
<
|
s CWgW^,
oo,
^ C(\\g\\^\\g-g^\\^ + WlW^Wg-g^W^).
By Assumption
1,
it
suffices to prove the result
when
w(y) =
1
for
use standard results on existence and continuity of solutions of differential
32
all
equations, e.g. in Finney and Ostberg (1976).
By
thrice continuously differentiable, its derivative up to order
T(g)
Then by
Lipschitz on any bounded set.
llg-gQiL
and
llg-g^ll^ < 5,
< 5,
In particular,
y
for
(p(t,u),y-s) e
,
TxV
for
sup.
,
^
.,
for
Furthermore, by integration of equation
y
y-S(t,z,g) e
g
replacing
Next,
for
llgll
y
y-S(t,z,g) €
q
in
g
<
00
for
t
g (p(t,u),y-S(t,z,g),w)
by
T
(g)
Z
z e
.
and
by
z e
on
(1.1)
€
t
Z
so that for
0,
1,
2).
Also, by construction
5S(t,z,g)/3t =
and
now
y
|S(t,z,g)|
[0,1],
=
Then by Theorem 12-6 of
to
,
(j
s C+e.
I
S(t,z,g)
e [0,1],
t
Clli-gll^..
[dlxl/xy x[-C,C].
(t,u,y,s) e
S(l,z,g) = 0,
s
|q(t,y,z,g)
Finney and Ostberg (1976), there exists a solution
-q(t,y-S(t,z,g),z,g),
e
small enough,
5
small enough,
5
are bounded and
2
bounded, we can choose
p (t,u)
supjQ^j^y^2l^"^'^^^'y'^>g^/^y"^ - 3Jq(t,y,z,g)/5yJ|
(A.l)
of
q(t,y,z,g) = T{g{p(t,u),y,w))'p (t,u).
Let
<
included in
C+e,
z.
so that
same existence and boundedness properties hold for
Also, the
S.
llgQll2 <
€ [0,1],
"
and
and
Z
z e
"g-gglU
,
it
<
^-
Also, by
follows that
p(t,u) e
bounded on any compact set and boundedness of
p
and
and
g(p(t,u),y-S(t,z,g),w)
Then boundedness of
are bounded on this set.
T
(t,u),
follows by
^(t,z,g)
giving the second
conclusion.
Next,
follows by Theorem 12-9 (equation 12-22) of Finney and Ostberg (1976),
it
Lipschitz on any bounded set,
(A.2)
sup^g[o,i],zeZ
bounded, and eq. (A.l) that
p (t,u)
lS(t.z,i)-S(t,z,g)
|
^ Clli-gll^,
which implies the third conclusion.
Next, for
follows that
(A.2) that
llg
all
t
e [0,1], z 6 Z
llg(p(t,u),y-S(t,z,g),w)ll
,
by
|S(t,z,g)|
^ "gUn'
^"'^
s C+c
33
:s
|S(t,z,g
)|
<
C,
it
^^ ^ mean-value expansion and eq.
lli(p(t,u),y-S(t,z,g),w) - i(p(t,u),y-S(t,z,gQ),w)ll
{p(t,u),y-S(t,z,g,gQ),w)ll|S(t,z,g)-S(t,z,gQ)|
and
<
llgll^llg-gQllQ
for an intermediate value
T(g)
g.
Then by boundedness of
CCt.z.g^),
£ "i"o^"Pt€[0
|A(z,i;g)-A(z.i;gQ)|
(A.3)
bounded for uniformly
are
all
Eire
the same expressions with
S(t,z,g
).
Also,
it
Kit,z.g)-(:(t,z,g^)\\ + CIlill^llg-gQll^.
e
and
in
t
€
[0,1],
z e
Z
llg-g
,
g
(p(t,u),y-S(t,z,g).w))
<
II
emd
e
replaced by a value in between
S(t,z,g)
llg-g_IL
as
< c,
and
S(t,z,g)
then follows by mean-value expansion arguments like those above,
including expansions of in
(^•^^
'
g (p(t,u),y-S(t,z,g),w)),
g(p(t,u),y-S(t,z,g),w)),
Also,
zeZ
1]
'
around
S(t,z,g)
S(t,z,g
IIC(t.z.g)-C(t.z.gQ)ll
^"Pt€[0.1],zeZ
that uniformly in
),
llg-g
ll„
<
e,
^ Cllg-g^l^.
e
For example, for a value
S(t,z,g,g
llg
(p(t,u),y-S(t,z,g),w)) - g
g
(p(t,u),y-S(t,z,g),w))ll +
in
)
between
S(t,z,g)
(p(t,u),y-S(t,z,gQ),w))ll
llg
£
llg
(p(t,u),y-S(t,z,g),w)) - g
llg-go"i + llgQyy(p(t,u),y-S(t,z,g.gQ),w))l|.
and
S(t,z,g
),
(p(t,u),y-S(t,z,g),w)) (p(t,u),y-S(t,z,gQ),w))ll s
IIS{t,z,g)-S(t,z,gQ)ll
£ Cllg-gQll^.
The fourth
conclusion then follows by eq. (A.3).
Finally, to
show the
first conclusion, let
notational convenience, suppress the
S(t,z,g).
(A.5)
t
and z
D(t,z,g,g) = S(t,z,g)-S(t,z,g).
arguments, and
let
S
For
= S(t,z,g)
and
S =
Differencing the differential equation gives
SD/at = -q(y-S,i) + q(y-S,g) = -[q(y-S,i) - q(y-S,g)] - [q(y-S,g) - q(y-S,g)]
- •(q(y-S,g)-q(y-S,g) - [q(y-S,g) - q(y-S,g)]}
= -C(g)Mg(y-S)-g(y-S)y - q^Cy-S.gjD - R(g,i),
R(g,g) = [q(y-S,i)-q(y-S,g)-C(g)Mg(y-S)-g(y-S)>] + [q(y-S,g)-q(y-S,g)-qy(y-S,g)D]
+ [q(y-S,g)-q(y-S,g)-q(y-S,g)+q(y-S,g)] = R^(g,g) + R2(g.g) + R3(g.g)-
The first equation here
is
an inhomogeneous linear differential equation, with final
34
=
D| _
condition
nonconstant coefficient
0,
Let
-?(g)'<g(y-S)-g(y-S)> + R(g,g).
this linear equation at
Then the solution to
DI^^Q = jJ[?(g)'{g(y-S)-g{y-S)> +
(A.6)
2—2
and
By
g(y-S)
of
a T(g(y-S))/5g
g
and
g
|S|
C+e
<
containing
€
t
and
around
q(y-S,g)
z e
[0,1],
all
t
jSj
<
with
[0,1],
|R„(g,g)|
(A.8)
i
^
S = S(t,z,g,g)
between
S
z e Z
c
z e
(A.9)
where
that
Z
e
S,
^(t,z,g)
2
5 T(g)/3g
on a line joining
g
2
).
Then by a mean-value
and
mean value
Clli-gll^.
differentiable in an open interval
mean value for an expansion of
be the
S
is
:£
so that
S,
y-S € V
of an expansion of
S
and
jS-Sj
q (y-S,g)
^ |D|.
around
A
S.
,
Iq„(y-S,g)-q (y-S,g)l|D|
y
y
<
(y-S*.g)||D|^ s Clli-gll^.
Iq
yy
'-'
q(y-S,g) - q(y-S,g)
around
for
S,
all
t
e
,
jR^tg.i)!
g,
for any
,
,
Similarly, for a mecin-value expansion of
[0,1],
to element of
q(y-s,g) - q(y-s,g)
C+e,
Let
S.
S,
€
Z
Z
£ Cllp^lllia^T(i)/Sg^llllg(y-S)-g(y-S)ll^
similar statement holds for the
Then for
is
e [0,1], z e
t
from element
differ
and
S
=
??.
—
|R^(g,g)|
(A.7)
t
q (r,y-S(r,z.g).z.g)'Cdr]
twice continuously differentiable, the elements
bounded on
will be
may
all
= exp[-
R(g,i)]i^(t.z,g)dt = A(z,i-g) + jQR(g.i)?(t,z.g)dt.
T
bounded and
g(y-S)
(that
expansion, for
By
^(t.z.g)
and nonconstant shift
-q (y-S,g),
and
is
s |q^(y-S,i)-qy(y-S,g)| IS-SI ^ CIli-gll^lli-gllQ.
S
denote mean values.
bounded uniformly
J'QR(g.g)€(t,z,g)dt ^ Cllg-gll
Proof of Theorem
1:
We
llg-gll
in
t
Then combining
e [0,1],
z €
Z
,
eqs.
and
(A.7) - (A.9)
llg-g„ll
so the first conclusion follows by eq.
first consider
S,
35
<
c,
and noting
we have
(A.6).
QED.
and proceed by using the Lemmas of Section 5 of
Newey
functions
f(x),
g(z;p,h) = f(x)
f(x)E[T
f,
h^,
P'w
(x)] +
h
cind
f(x)E[w|x]
and
(q)|x]
[h^(x)-3'h
Q
Let
(1992a) (N henceforth).
Let
here.
- e
and
and for any
m(z,p,h) = (m (z,P,h),m (z.P.h)'
m„(z,3,h) = T(x)[q-g(z;P,h)]®[w-f
D^(z,h;h,p) = A(z,L(',h;P,h);g(«;P,h))
let
denote possible values for the
-1
(x)h
h
L(x,h;/3,h)
let
p
Let
and
m
there be
)'
h = (f,h ,h
respectively, and
f(xr^[h^(x)-p'h^(x)] - f(xr^[h^(x)-p'h^(x)]f(x).
h
W
w (x)].
For
h
=
N equaUe.p'
in
= S(0,y ,w
(z.^.h)
1
with
u(y) = l(y = y
preceding Assumption
that the density of
1,
x
and
llhll
.
)
= sup
y^ e V
f^j
r
n^.119
~i\
bounded away from zero on
is
.
Let
n(x)/5x
llgll
lip-p_ll
Also,
it
are small enough then
x(?'x[y,y]),
llh-h-^ll„
(A.IO)
.
^ C llh-h
II
for
.
.
I
and
s
llh-hll
llh-h
II
1
be as defined
llh-h
if
g = g(«:h,p)
Then by the conclusion of Lemma
.
By
follows by a
it
follows by the usual mean value expansion for ratios that for such
llg-g-L(',h-h;|3.h)ll
for
llg-gll
and by
II
Ilh-h-ll
.,
h
and
llh-hll
I
£ |A(z,g-i;i)-A(z,L(-,h-h;p,h),i)| + Cllg-illQllg-ill^
s Cllg-g-L(«,h-h;P,h)ll
It
+ Cllh-hll
llh-hll
also follows by similar reasoning that for
(A.ll)
|D^(z,h;/3,h)|
= |A(z,L(',h;/3,h);i)
|D^(z,h;P,h)-D^{z,h;pQ,hQ)|
I
lip-p
£ Cllh-hll
II
and
llh-hll
lih-h
.
ll„
small enough,
s CIIL(-,h;p,h)llQ £ CIIHiIq,
= CllhllQllh-hQll^ + Cllhll^(llh-hQllQ+llp-pQll),
|m^(z,^,h)-m^(z,^Q,hQ)| s Cllh-hQil + 011^-^^11.
36
.,
g = g(«;h,p).
and
small enough,
m^(z,p,h)-m^(z,p,h)-D^(z,h-h;p,h)
(A. 2)
Then by the hypothesis
II.
straightforward application of the quotient rule for derivatives that
and
g(«;p,h))
D2(z,h;h,p) =
and
follows that
it
and
)'
from equation
A(z,g;g)
-T(x)f(x)"^q-g(z;p,h)]®[h^(x)-r^(x)h^(x)f(x)l - T(x)L(x.h;p,h)®lw-r\x)h'^(x)].
Assumption
Let
).
h,
£
llh-hll
also follows by straightforward algebra that for
It
enough, there
=
b(z)
is
lip-p
llh-h^ll
II,
llh-h
small
II
such that
C(l+llqll)
llm2(z,p,h)-m2(z,p,h)-D2(z,h-h;p,h)ll £ b(z)(llh-hQllQ)^,
(A.12)
and
,
s
b(z)llhllQ.
a = m/2,
we have
|D2(z,h;p,h)
|
IID2(z.h;^.h)-D2(z.h;/3Q,hQ)ll s b(z)llhllQllh-hQllQ,
Jlm^Cz./S.hl-m^Cz.pQ.hQJII £ bCzKllh-h^llQ + IIP-PqII).
Furthermore,
—
0,
>
v'n(T7
E[b(z)
—
)
n
4
]
0,
)
<
Vncr
t)
>
Lemma
= [ln(n)/(n(r
t)
n
0,
= a
a
and
=
a
0,
Next, by hypothesis,
m
(h)
= 0.
Then by the definition of
= JwCtlWxttDdt,
By Assumptions 5 and
A
^ ln(n)/(-/ncr
)
—
0.
^
satisfied, giving
(1).
U
=
1,
2),
f
when
t
(t)
and
is
is
t
and
)
f (t) =
uniformly distributed on
bounded and continuous
a.e.
with compact
L,
6,
w(t) = fQ(x(T)r^C(T)'[-rQ(x(T)).I,-p^]f^(T).
cj(t)
is
bounded, continuous almost everywhere, and zero outside
Also, by the inverse function
t([0,1]).
theorem and the chain
differentiable with bounded derivatives on
in its interior.
Lemma
>
)
tj
m^(h) = J"A(z,L(-,h;pQ,hQ);gQ)dF(z)
(A.14)
—
l/(yno-
^(t) = <(t(T),y ,w ,g
Let
By the inverse function theorem,
support.
0-
and
m.(h) = J'D„(z,h;h ,P )dF(z).
and
denote the density of
KO^tCx)^!) |St(T)/ST|
by
N are
s
+o-
)]
implying that
5.4 of
1/2
k+2i
v/n(r°'£^"^[m^(z.,h.pQ)-m^(z.,hQ,pQ)]/n = /n(r°'£[m^(h)-m^(hQ)] + o
(A.13)
[0,1].
—
t)
n n
Therefore, the hypotheses of
for
i
and for
co
0,
5.4 of
t(Z)),
By the above shown conditions,
N
that
Vn'cr°'[m
(h)-m
(h
and the triangle inequality that
37
)]
-^
x
rule,
(t)
is
continuously
a compact convex set containing
2k
ncr
—
N(O.V
Tncr 2^.m (z.,p
m
>
).
and
It
,h)/n
2s
no-
—
>
0.
Then
t([0,1])
it
follows
then follows by equation (A.13),
—
>
N(0,V
).
Furthermore,
2
n
by equation
m^th) =
(A.13),
Lemma
Next, note that by
for
Al,
derivative with respect to
9
equal to
straightforward to show that
n
nonsingular by Assumption
The
2.
A =
with
m
For
m„.
0, A,
1
that
i)
—
>
V„
1
ill)
—
.
N
Ill
(q.)-g(x.,w.))>,
is
cr
G
2s
and
^1
n<r
+ A. -
—
>
0.
>
.
,A./n.
^
Assumption 8 here and
N
Lemma
Lemma
a(z,g
8.4 of
bounded
)
Also, by
ip.
=
arguments
M —^
and
in probability
similsir to
so the
M,
QED.
N.
each case.
in
Newey
Assumption
Al for each case.
follows that for any
so that for
0.
(1991),
with
follows by Assumption 9 here and
equal to any positive number.
follows by
straightforward
to
^
The proof proceeds by verifying the hypotheses of Theorem
2:
Assumption 7 and boundedness of
A. 3 of
is
it
F^
12
F,,
>,
A. 4 of N,
a
,
d
Kn
such that
ll^^ll2lK^''^/Vn + K""2] £ C'K^'K^^'^/Vn + o(l)
=
Assumption A.2 of
11^
II
.
£
8.2 of
A =
2,
—
>
*
CK
A
oo
n iK zK
'^,
j
Newey
=
0,
A
for some
d
—
>
0.
N
follows by
=
0,
1,
(1991),
= A
r
>
2.
with
=
0,
1,
it
Therefore,
o(l).
V^il^^llQJI^'^iyK^^^/v^ + K'"o)(K^^^/v7r + K""i) £ CK-K^-K/i/n + o(l) =
38
A.l of
Assumption A.l of N follows by
Lemma
with
Furthermore, by
there are
5.5 of
1
(1992b), that will henceforth be referred to as
Assumption
Also,
by
1/2,
J
m„,
/n —
Jli/».ll
^1=1
is
for both
Lemma
then follows by
It
^j=l
J
satisfied for
).
N
5„ =
pn
Also, part iv) is satisfied with
second conclusion follows by the triangle inequality.
Newey
is
it
of Assumption 5.2 follow by eqs. (A. 10) - (A.ll),
U. = 0.
1
those for asymptotic normality,
Proof of Theorem
and
p,
useful to verify Assumption 5.2 of
0.
oo
>
for
check that Assumption 5.2 of
11
-
1
T.(w.-E[w|x.])(T
is
and
h,
linear in
(z.p.h)
converges in probability to
(z.,p,h)/ajS
is
it
A„ =
o
/ln(n)
no-
(T^). ,U./n
^1=1
parts
,
= A„ = 1, and
z
3k+4
the conditions that
N
is
and
p
in
m„(z,3,h)
Also,
-1.
m
small enough,
ll„
0.
first conclusion then follows by a Taylor expansion.
To show the second conclusion
and
llh-h
-^
h)/n
V^cr"j;.m (z.,p
with a first column of zeros, and the remaining columns being
E[5m„(z,p^,h^)/S3],
m
and
bounded uniformly
is
9m
J]]._
follows that
it
IIP-^qII
with derivative that
p
differentiable in
m^Ch^) =
and
0,
o(l).
a,
s CK^^'^K^/i/n =
K^'''^\\4>^\\^/Vn
Theorem
so that the rate hypotheses of
9 = S
show that for
9 = A,
or
fi
For
values with
N
Assumption
3 =
satisfied.
is
0.
It
For
L,
N
N
are satisfied.
T
x_ = x(p .y^)
p
* p
continuously distributed and
T
T
and
with
r.(x
)
=
converges to zero for
so there exists
r,
that
(x) = $
r
K
>
- (p -p )'T
For
y.,
A(z,g;g
j
—
>
in
N
is
E[ll$
Lemma
(p
r * 0.
-p )'T
)
=
2
II
]
(j(y)A(z,g;g
)
N
is
Also by
0.
A. 4 of N, there exists
in
j,
/^(x)' r(x(T))f (x)dx
K
(x)'-n
g
)),
—
>
and
—
>
0,
E[A(z,r (x);g
K.
u
K
satisfied.
here, so that
wE[A(z,g;gQ)] = E[u(y.)g(x.,w.)'C(t.,z.)] = E[E[tj(y.)C(t.,z.)' |x.,w.]g(x..w.)]
= E[J'E[w(y.)C(t.,z.)' |x.=x,w.=w]g(x,w)f(x|w)dx]
= E[/C(x,w)'g(x,w)f(x|w)dxl = E[C(x.,w.)'g(x.,w.)] = E[C(x.,w.)'g(x.,w.)],
where the
last equality follows by straightforward calculation.
similarly that Assumption A. 5 of
Theorem
A.l of N.
N
is
6 =
that Assumption A. 6 of
Prob(x(T)=x
and hence
oo,
so that Assumption A. 6 of
)
o(l).
qFIx^).
(0,1),
K.
r * 0,
(x^.w
such that
r
the proof of
such that
(x)'t)
K.
—
as
,
=
suffices to
Consider
everywhere continuous, bounded uniformly
x * x
all
in
(g
N
A. 4 of
(pS°)'T
one-to-one on
p(t)
is
T
=
there exists
Therefore, by reasoning similar to that
r.(x)
Lemma
follows as in the proof of
nonsingular and
-^
it
satisfied, while for
is
J^tx)' [r{x(p(T),yQ-S(T))+w^p]dT.
is
E[A(z.r(x);gQ)] = JCCt)' r(x(T))f^(T)dT -
By
Therefore,
N
v^""o
o(l).
satisfied.
is
from N
)
A.l of
Assumption A. 6 or A. 7 of
A. 5 of
A(z,g;g
the
S,
9 = L,
or
CK^^V/'/n =
K^'^'^H^W^/Vn £
o(l),
satisfied for
QED.
39
X.
It
also follows
Then the conclusion follows by
U
)]
References
S. (1967): "The Construction of a Utility Function from Expenditure Data",
International Economic Review, 8, 67-77.
Afriat,
Afriat, S. (1973): "On a System of Inequalities on Demand Analysis:
Classical Method", International Economic Review, 14, 460-472.
An Extension of the
Auerbach, A. (1985): "The Theory of Excess Burden and Optimal Taxation", in A. Auerbach
and M. Feldstein, eds.. Handbook of Public Economics, vol. I, Amsterdam: North Holland.
(1992):
"Consistent Bandwidth Selection
Journal of the American Statistical Association.
Bickel, P.J.
in
Nonparametric Regression,"
Bierens, H.J. (1987): "Kernel Estimators of Regression Functions", in T. Bewley, Advances
in Econometrics, Fifth World Congress, vol I, Cambridge: Cambridge University Press.
Brown, B.W. and Walker (1990): "Individual Heterogeneity and Heteroskedasticity
Equations," Econometrica.
in
Demand
Brown, B.W. and W.K. Newey (1992): "Efficient Semiparametric Estimation of Expectations,"
working paper. Department of Economics, Rice University.
Deaton, A. (1986): "Demand Analysis", in Z. Griliches and M. Intriligator, Handbook of
Econometrics,
Diamond,
P.
vol.
Ill,
Amsterdam: North Holland.
and D. McFadden (1974): "Some Uses of the Expenditure Function
of Public Economics, 3, 3-21.
in Public
Finance," Journal
Finney, R.L. and D.R. Ostberg (1976): Elementary Differential Equations with Linear
Algebra, Reading: Addison-Wesley.
Hardle, W. (1990): Applied Nonparametric Regression, Cambridge: Cambridge University
Press.
Hausman,
Review,
Hausman,
J.
J.
"Exact Consumer's Surplus and Deadweight Loss", American Economic
662-676.
(1981):
71,
(1985):
"The Econometrics of Nonlinear Budget Sets," Econometrica, 53,
1255-1282.
(1991):
"Consistency and Asymptotic Normality of Nonparametric Projection
Estimators," MIT Department of Economics Working Paper No. 584, June 1991.
Newey, W.K.
Newey, W.K. (1992a): "Kernel Estimation of Partial Means and a General Asymptotic Variance
Estimator," MIT Department of Economics Working Paper, January.
Newey, W.K. (1992b): "Asymptotic Normality of Nonlinear Functionals of Series Estimators,"
Department of Economics Working Paper, January.
Porter-Hudak, S. and K. Hayes (1986): "The Statistical Precision of a Numerical Methods
Estimator as Applied to Welfare Loss," Economics Letters, 20, 255-257.
Powell, M.J.D. (1981): Approximation Theory and Methods, Cambridge, England: Cambridge
40
University Press.
Powell, J.L., J.H. Stock, and T.M. Stoker (1989): "Semiparametric Estimation
Coefficients," Econometrica, 57, 1403-1430.
of Index
Robinson, P. (1988): "Root-N-Consistent Semiparametric Regression," Econometrica, 56,
931-954.
Samuelson,
P. A.
(1948):
"Consumption Theory
in
Terms of Revealed Preference", Economlca,
5, 61-71
Small, K. and H. Rosen (1981): "Applied Welfare Economics with Discrete Choice Models",
Econometrica, 49, 105-130.
Varian, H. (1982a): "The Nonparametric Approach to
945-974.
Demand
Analysis", Econometrica, 50,
Varian, H. (1982b): "Nonparametric Tests of Models of Consumer Behavior," Review of
Economic Studies, 50, 99-110.
Varian, H. (1982c): "Trois Evaluations de I'impact 'Social' d'un Changement de Prix",
Cahiers du Seminar d' Econometric 24, 13-30.
(1983): "Efficient Methods of Measuring Welfare Change and Compensated Income
terms of Ordinary Demand Functions", Econometrica, 51, 79-98.
Vartia, Y.
in
Willig, R.
(1976):
"Consumer's Surplus without Apology," American Economic Review, 66,
589-597.
41
N
/
/
-
o
o
CM
(£
Nv
c
x^
^\
cn
o
~
;
:
n
I
I
6.5
6.5
6.7
,-.j
7.0
6.9
6.8
:_.
'
7.1
7.2
Log Quantity
Translog
6.7
Demand
6.8
Log Quantity
FIGURE 1
6.9
7.1
6.6
I
6.7
6.8
7.0
G.9
7.3
7.1
Log Ouontily
Demand
Kernel
o
Estimate:
= .SS
h
\
e
O
.
V
d
(M
y
D.
^^^
o
x,^
\
^
o
I
o
^--....,.„^^^
^^^^^\
o
1
)
/
lO
6.6
I
6.7
6.8
7.0
6.9
7.1
Log Ouontily
Kernel
I
5.6
Demand Estimate
6.7
6.6
,
7.0
6.9
Log Ouonlily
FIGURE
2
7.2
7..1
spline
Demand
Estlnnale: 7
Knots
7.0
6.8
Log Quantity
Spline
I
6.5
6.5
Demand
6.8
Estimate: 8 Knots
6.9
7.0
Log Quantity
FIGURE 3
7.2
5939 077
Date Due
MIT LIBRARIES DUPL
3
TOflD
006Mh37fl
S