Integrability and the Semiparametric Estimation of Demand

WORKING PAPER–DO NOT CITE

Krishna Pendakur, Economics, Simon Fraser University

Burnaby, BC, CANADA, V5A 1S6, pendakur@sfu.ca

Received: ; Accepted:

Abstract

I develop a semi-parametric approach to consumer demand system estimation which

provides and estimated demand system satisfying homogeneity whose estimated slutsky

matrix satis…es symmetry. Nonparametric local polynomial estimation of expenditure

share equations over normalised prices (prices divided by expenditure) satis…es homo-

geneity without the imposition of extra parametric restrictions. It also yields simple

estimates of the Slutsky Matrix at each point. I show how to impose local concavity

and local symmetry through a second stage local minimum distance procedure. Exact

consumer surplus measures are developed with local second order taylor approxima-

tions of the log-cost function. Procedures are developed for integrable demographic

e¤ects, simulating con…dence bands and testing of symmetry restrictions. Consumer

demand systems over normalised prices are estimated with American price and ex-

penditure data for the period 1980 to 1995. Imposition of symmetry does not lead to

qualitative changes in the estimated demand system, and tests of symmetry do not

reject the hypothesis that the slutsky matrix is locally symmetric.

_________________________

I gratefully acknowledge Orazio Attanasio, James Banks, Richard Blundell, Andrew

Chesher, Ian Crawford, Tom Crossley, Hide Ichimura, Brian Krauth and Joris Pinkse,

Ian Preston, Abijit Sengupta and numerous other patient colleagues for their very

important help on this work.

1

1. Introduction

Consumer demand equations over prices and expenditure must satisfy homogeneity and

have a Slutsky Matrix which satis…es symmetry and concavity in order to be useful in the

estimation of consumer surplus. However, previous non- and semiparametric approaches

to the estimation of consumer demand do not give estimates restricted to satisfy these

conditions.

This paper has …ve parts. First, without loss of generality, consumer demand equations

can be estimated as a nonparametric local polynomial equation system over normalised prices

de…ned as prices divided by expenditure. In this case, homogeneity is satis…ed trivially, and

the form of engel curves is left completely free. In comparison to local mean nonparametric

estimation, local polynomial nonparametric estimation may give better behaviour near the

boundary of the data. Further, local polynomials are characterised by easy-to-compute

derivatives yielding a simple estimated local Slutsky Matrix.

Second, minimum distance methods may be used to restrict the estimated Slutsky Matrix

so that it satis…es symmetry and concavity. Third, given local quadratic estimates that

satisfy symmetry and concavity, a second-order Taylor approximation of the log-cost function

may be used to obtain exact consumer surplus measures for large price changes. This

extends the work of Vartia (1982) and Hausman and Newey (1995) which uses …rst order

Taylor approximations of the cost function to estimate exact consumer surplus. Further,

this approach ensures that the surplus measures are locally path-independent due to the

imposition of symmetry on the estimated local Slutsky Matrix.

Fourth, I show that the model allows for the incorporation of demographic e¤ects such

2

that the estimated Slutsky Matrix satis…es symmetry for all possible vectors of demographic

characteristics. Fifth, I develop a simple procedure to estimate small-sample distributions for

estimated nonparametric demand systems and to test symmetry restrictions via an extension

of Gozalo’s (1997) Smooth Conditional Moment Bootstrap. This extension allows for cross

equation correlations in the structure of simulated errors, which is important in this context

because Slutsky Symmetry involves cross-equation restrictions.

The primary aim of this paper is to develop a methodology for the semiparametric esti-

mation of path-independent consumer surplus measures. Symmetry of the estimated Slutsky

Matrix allows the researcher to compute a second order approximation of the log-cost func-

tion.

Integrable demand systems are useful because they can be used for welfare analysis,

whereas demand systems that are not integrable cannot be so used. In particular, although

demand systems that are not integrable may be used for prediction purposes, they cannot

be used to measure consumer surplus or to adjust for price di¤erences in inequality mea-

surement. Integrable demand systems allow for the estimation of exact consumer surplus

over a price change for each household or individual in a population. Micro-level consumer

surplus measured with an integrable demand system will satisfy path-independence, that

is, will not depend on the sequence of small price changes used to approximate the large

price change (see Vartia). These micro-level consumer surpluses can then be “added up”

with a social welfare function or other aggregator to generate measures of “social surplus”

which can be used for policy evaluation. Similarly, integrable demand systems allow for the

estimation of exact economic price indices for each household or individual in a population.

Such micro-level price indices can be used to adjust for price di¤erences across households

or individuals when measuring income or consumption inequality.

3

2. Parametric Estimation of Consumer Demand

The implications of integrability are easily seen with an example. The cost function C(P; u)

gives the minimum cost necessary for a household to achieve utility level u facing an M ¡

vector of prices P = [P 1; :::;PM ]0. Consider the Gorman polar form:

C(P; u) = a(P ) + b(P )f (u): (1)

Denoting X = C(P; u) as total expenditure, applying Sheppard’s Lemma and substituting

for indirect utility f (u) = (X ¡ a(P )) =b(P ), marshallian demand equations, hj(P; x), are as

follows:

hj =@a(P )

@P j+@b(P )

@P j

ÃX ¡ a(P )b(P )

!; (2)

j = 1; :::;M . These demands are linear in total expenditure X—that is, they are quasiho-

mothetic.

The marshallian commodity demands hj are integrable if and only if the following con-

ditions hold:

1. Homogeneity is satis…ed. This requires that hj are independent of scaling P and X , or

equivalently, that hj can be written in terms of normalised prices eP j = P j=X .

2. Symmetry is satis…ed. This requires that the matrix of Slutsky terms,

Sjk =d2C(p; u)

@P j@P k=@hj(P;X)

@P k+@hj (P;X)

@Xhk(P;X); (3)

is symmetric. In terms of the Gorman polar form above, symmetry requires that a and

b be twice di¤erentiable functions.

3. Concavity is satis…ed. This requires that the matrix of Slutsky terms, Sjk, is negative

semide…nite. In terms of the Gorman polar form above, concavity requires that a and

b are concave in P .

4

A common parametric approach to the estimation of an integrable demand system is

to specify a parametric form for the cost or indirect utility function, and to estimate its

parameters. In the example above, this might involve specifying a and b as simple functions

of prices: a(P;X) =PajP j and b(P;X) =

Pbj ln P j. In this case,

hj = aj + bj³X ¡

XajP j

´: (4)

Here, symmetry is satis…ed because a and b are both twice di¤erentiable—the Hessian of a

is 0, and the Hessian of b is a diagonal matrix with elements ¡bj= (P j)2. Further, concavity

is satis…ed because a is weakly concave in P and b is concave in P . Alternatively, one might

specify a and b as more complicated functions of P , and then restrict them so as to ensure

concavity.

A major advantage of the parametric approach is the ease with which integrability is

imposed. The major disadvantage of the parametric approach is that the estimated demand

system is laden with parametric restrictions which are of no use or interest to the researcher.

In the above example, these restrictions take the form of forcing commodity demand equa-

tions to satisfy global linearity in prices and expenditure. Integrability is clearly desirable,

but possibly false restrictions on consumer demand systems are not. In this paper, I provide

a methodology for the estimation of a consumer demand system which satis…es homogeneity

and whose estimated Slutsky Matrix locally satis…es symmetry and concavity without the

imposition of global parametric restrictions on consumer demand equations.

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3. Nonparametric Estimation of Consumer Demand

3.1 Estimation Under Homogeneity

De…ne the log-price of the j’th good as pj = ln(P j ); and denote p = [p1; :::; pM ]0. De…ne x =

ln(X) as log-expenditure. De…ne the normalised price of the j’th good, eP j, as eP j = P j=X,

and denote the normalised price vector eP = [ eP 1; :::; ePM ]0. De…ne the log-normalised price of

the j’th good, epj, as

epj = ln( eP j) = ln

ÃP j

X

!= pj ¡ x; (5)

and denote the log-normalised price vector ep = [ep1; :::; epM ]0. Without loss of generality, de…ne

expenditure shares for good j, wj, as functions of log-normalised prices,

wj = wj(ep); (6)

and denote w = [w1; :::; wM ]0. These functions are not assumed to be homogeneous in ep.

(The use of ep rather than p and x gives the homogeneity of the demand system required for

integrability.) Estimation of wj (ep) may proceed using standard multivariate nonparametric

estimation techniques (see, for example, Wand and Jones 1995, or Bowman and Azzalini

1999) for each expenditure share equation.

This procedure addresses two problems in the nonparametric estimation of consumer

demand. Firstly, standard nonparametric nonparametric estimation of engel curves (see,

for example, Blundell, Duncan and Pendakur 1998) does not impose homogeneity on the

demand system. The above method imposes homogeneity from the outset. Secondly, and

perhaps more importantly, standard nonparametric estimation of engel curves allows the

researcher to use households facing only one set of relative prices. This approach allows

the researcher to pool observations facing many price regimes and to explicitly model price

e¤ects in the semi- and non-parametric estimation of consumer demand.

6

3.2 Local Quadratic Estimation

Let epi = [ep1i ; :::; epMi ]0 denote the log-normalised price vector of, and wi = [w1i ; :::;wMi ]

0 the

expenditure share vector of, the i’th household where i = 1; :::; N . Consider a local quadratic

model where in the neighbourhood of a normalised price vector, ep0,

wj(ep0; epi) = ®j +MX

k=1

¯ jk(epk0 ¡ epki ) +

1

2

MX

k=1

MX

l=1

°jkl³

epk0 ¡ epki´ ³

epl0 ¡ epli´+ ²ji : (7)

The parameters of this model may be estimated by minimising

min®j,¯jk,°

jkl

NX

i=1

­(ep0; epi)³²ji

´2(8)

for each independent expenditure share equation j = 1; :::;M¡1. I note that for any M good

demand system, there are only M ¡ 1 independent equations due to adding-up restrictions.

Here, ­ is a multivariate kernel weighting function which penalises the distance of epi from

ep0. I specify this weighting function as

­(ep0; epi) =MY

j=1

K(uji ) (9)

where K is the gaussian kernel

K(u) =exp(¡u2=2)p

2¼(10)

and the M¡ vector ui = [u1i ; :::; uMi ]

0 is given by

ui =H¡1(ep0 ¡ epi) (11)

where H is an M by M bandwidth matrix.

Given the bandwidth matrix H , estimation of the parameters can proceed by locally

weighted ordinary least squares. That is, at each point of interest in ep, the researcher

regresses wj on a constant, epj, and the unique elements of epj epk using ­(ep0; epi) as weights.

7

Local quadratic estimation gives nice boundary characteristics in comparison with local

mean estimation (see Bowman and Azzalani 1999, or Wand and Jones 1995). In general

…nite-order local polynomial estimation is biased near the boundaries of the data, but this

bias declines with the degree of the polynomial.

I note that this local quadratic nests a local Almost Ideal model given by

wj =@ ln a(ep; ep0)

@epj¡ @ ln b(ep; ep0)

@epjln a(ep; ep0) (12)

where a is translog and homogeneous of degree one in normalised prices and b is cobb-douglas

and homogeneous of degree zero in normalised prices. The functions a and bmay be di¤erent

at every p0.

The local quadratic also nests a local Extended Almost Ideal model (see Dickens, Fry

and Pashardes 1993) given by

wj =@ ln a(ep; ep0)

@epj ¡ @ ln b(ep; ep0)@ epj ln a(ep; ep0) +

@q(ep;ep0)@epj

b(ep; ep0)(ln a(ep; ep0))2 (13)

where a is cobb-douglas and homogeneous of degree one in normalised prices, b is cobb-

douglas and homogeneous of degree zero in normalised prices and q is proportional to b. The

functions a, b and q may be di¤erent at every p0.

An important feature of local polynomial models is that the estimated values and deriv-

atives are easy to compute. In particular, the estimated values of wj are given by the

estimated intercept terms and the estimated …rst derivatives of wj with respect to ep0 are the

estimated local linear parameters (see Wand and Jones 1995; Simono¤ 1995). Thus, given

an estimated model (7),

dwj (ep0) = ®j (14)

andd@wj(ep0)@ epk0

= ¯ jk: (15)

8

Due to the imposition of homogeneity through the use of normalised prices, the derivatives

with respect to log-prices and log-expenditure are also simple. They are

d@wj (ep0)@pk0

=d@wj(ep0)

@(pk0 ¡ x) =d@wj(ep0)@ epk0

= ¯jk (16)

andd@wj(ep0)@x

= ¡MX

k=1

¯ jk: (17)

The simplicity of these estimated expenditure shares and expenditure share derivatives with

respect to log-prices and log-expenditure allows a straightforward approach to restricting

the estimated Slutsky matrix to satisfy symmetry.

3.3 Local Symmetry

De…ne the cost function, C(p;u), to give the minimum cost for a household to attain utility

level u facing log-prices p. The Slutsky Matrix S with elements Sjk is de…ned as the Hessian

of the cost function with respect to (unlogged) prices, and may be expressed in terms of

log-price derivatives as follows:

Sjk =@wj

@pk0+@wj

@xwk +wjwk ¡ djwj (18)

where dj is an indicator of a diagonal element of the Slutsky matrix.

Slutsky Symmetry is satis…ed if and only if Sjk = Skj . If this condition holds over a

region of the normalised price space, then Young’s Theorem guarantees that there exists a

cost function whose derivatives could produce the observed demand system over this region.

Substituting the estimates into (18) and setting Sjk = Skj, Slutsky Symmetry requires

that, in the local quadratic model, for any given point of estimation ep0,ïjk ¡

MX

s=1

¯js®k

!¡

ïkj ¡

MX

s=1

¯ks®j

!= 0 (19)

9

for all j; k.

This framework yields a Wald test of symmetry. Under the null hypothesis that the

estimated Slutsky Matrix is symmetric, equation (19) is true; under the alternative the

di¤erence between Sjk and Skj is not zero. Thus, we may test symmetry of the estimated

Slutsky Matrix by constructing the following test statistic

¿ =M¡1X

k=j+1

M¡1X

j=1

ïjk ¡

MX

s=1

¯ js®k ¡ ¯kj +

MX

s=1

¯ks®j

!2

: (20)

Under the null hypothesis, ¿ = 0. So, to test symmetry, we ask whether or not the observed

value of ¿ is large compared to the distribution of the test statistic simulated under the null

hypothesis that symmetry is true.

A local quadratic model whose estimated Slutsky Matrix satis…es symmetry may be found

by …rst estimating the unrestricted local quadratic model, and second estimating parameters

satisfying (19) via minimum distance estimation. This symmetry-satisfying local quadratic

may be used for consumer surplus calculations and to test the symmetry restrictions.

3.3.1 Minimum Distance Estimation

The restrictions noted above may be imposed on the unrestricted local quadratic model using

Minimum Distance (MD) estimation. The local quadratic step above provides unrestricted

estimates of each of the parameters. The spirit of MD estimation is to minimise a quadratic

form di¤erence between restricted and unrestricted parameters by choice of the restricted

parameters.

In the unrestricted local quadratic model there are M¡1 independent expenditure share

equations which yieldM ¡ 1 estimated ®j’s, (M ¡ 1)M estimated ¯jk ’s, and (M ¡ 1)M(M +

1)=2 estimated ° jkl. Denote the set of (M ¡ 1)(M +1)(M +2)=2 parameters estimated from

the unrestricted local quadratic step as ©U = [®j; ¯jk; °jkl].

10

There are (M ¡ 1)(M ¡ 2)=2 restrictions for local symmetry implied by (19). In each

equation, the M ’th symmetry term is satis…ed automatically by adding up, which leaves

only the o¤-diagonal elements of the remaining M ¡ 1 byM ¡ 1 matrix of symmetry terms

to restrict.

The restrictions embodied in (19) leave (M ¡1)(M(M +2)+4)=2 free parameters in the

model. The MD step minimises a quadratic form by choice of these free parameters. Denote

the free parameters as ©0, denote the restrictions (19) as g(©U ) and denote parameter values

that satisfy the restrictions as ©R. I minimise

min©0(©U ¡ ©R)0£(©U ¡ ©R) (21)

subject to

g(©R) = 0

where £ is a weighting matrix. If £ were equal to V ¡1 where V is the covariance of ©U , this

would be the minimum chi square estimation procedure proposed by Rothenberg (1973).

However, since the restrictions g constrain both levels and derivatives of the expenditure

share equations and since nonparametric derivative estimates converge slower than nonpara-

metric level estimates, estimation which used this ‘optimal’ weighting matrix would not yield

e¢cient parameter estimates or tests of the hypothesis g(©R) = 0. For this paper, I use two

di¤erent diagonal weighting matrices—one which contains the inverse of the variance of each

element of ©U , and one which contains the inverse of the square of each element of ©U . The

former weighting matrix may be interpretted as minimising the sum of squared normalised

di¤erences between restricted and unrestricted estimates. The latter weighting matrix may

be interpretted as minimising the sum of squared proportional distances between restricted

and unrestricted parameters.

11

For example, with 4 normalised prices and 3 independent expenditure share equations

there are 45 parameter estimates in the unrestricted model, and the following 3 restrictions

could be imposed to satisfy (19):

¯21 ¡ ¡¯12 + ®2¯11 + ®2¯12 + ®2¯13 + ®2¯14 ¡ ®1¯22 ¡®1¯23 ¡ ®1¯24¡1 + ®1

= 0; (22)

¯31 ¡

¯13 + ®3®2¯11 ¡ ®3¯11 ¡ ®3¯12 ¡ ®3¯13 ¡ ®3¯14 + ®1¯33 + ®1¯34 ¡ ®2¯13 + ®1¯23

¡®3®1¯22 ¡ ®1¯13 + ®3®2¯12 + ®3®2¯13 + ®3®2¯14 ¡®3®1¯23 ¡ ®3®1¯24 + ®1®2¯13¡®21¯23 ¡ ®21¯33 ¡ ®21¯34 + ®1®3¯11 + ®1®3¯13 + ®1®3¯14

1¡ ®2 ¡ 2®1 + ®1®2 + ®21= 0;

(23)

and

¯32 ¡ ®1¯23 ¡ ¯23 + ®3¯22 + ®3¯23 + ®3¯24 ¡ ®2¯33 ¡ ®2¯34 ¡®2¯13 + ®3¯12

¡1 + ®2 + ®1= 0: (24)

Here, the 42 free parameters in ©0 are 3 ®j ’s, 9 ¯ jk’s with j · k, and all of the °jkl parameters.

Since the °jkl parameters are not involved in the restrictions, they can be dropped from the

minimisation. In this case, ©U and ©R are vectors with 15 elements each, and ©0 has 12

elements.

One might argue that since the ° jkl parameters are not constrained or even involved in the

MD step, local linear estimation would su¢ce. However, several researchers (see Fan and

Gijbels 1996) have noted that for derivative estimation, local polynomials perform better if

the polynomial is at least one degree higher than that of the derivative of interest. Thus,

for this application, local quadratics are appropriate because the derivatives of interest are

given by the linear terms.

12

One might also argue that since Young’s Theorem only gives integrability when the Slut-

sky Matrix is continuous on the domain of normalised prices, restricted estimation through

MCS might be problematic. However, since the unrestricted estimated coe¢cients ©U are

just locally weighted least squares estimates, they are continuous in normalised prices. This

implies that the unconstrained (possibly asymmetric) Slutsky Matrix is continuous in nor-

malised prices. Under the null that Slutsky Symmetry is true, the restricted estimates should

not stray ‘too far’ from the unrestricted estimates, and so the symmetry-restricted estimated

Slutsky Matrix should also be continuous in normalised prices.

3.4 Local Concavity

If estimated expenditure share equations satisfy homogeneity and the estimated Slutsky

Matrix satis…es symmetry, then estimated consumer surplus measures will satisfy path-

independence and the weak axiom of revealed preference. However, to satisfy the strong

axiom of revealed preference, the estimated Slutsky Matrix must also satisfy concavity.

Local concavity is satis…ed if the Slutsky Matrix is a negative semide…nite matrix. I use

the Ryan and Wales (1999, 2000) strategy of locally estimating the Slutsky Matrix as the

negative of the inner product of a triangular matrix and its transpose. This requires that

¯jk ¡ÃMX

s=1

¯js

!®k + ®j®k ¡ ®jdj = ¡(¤¤0)jk (25)

for some lower triangular matrix ¤.

Note that by construction ¤¤0 is symmetric. This means that imposition of concavity via

restrictions (25) implies the symmetry restrictions (19). Beyond the symmetry restrictions

(19), the concavity restrictions (25) impose only inequality constraints—for example, the

diagonal elements of S must be weakly negative—and these are embodied in (25).

13

To guarantee that the Slutsky matrix is negative semide…nite, the researcher may replace

the (M¡1)(M¡2)=2 restrictions for local second order symmetry (19) with (M¡1)(M¡1)

restrictions for the non-border elements of Sjk . The border elements are excluded because

Hicksian demands are homogeneous of degree zero in prices, which means that the row- and

column-sums of S are zero. To balance the larger number of restrictions, the researcher must

now also estimate the M(M ¡ 1)=2 nonborder nonzero elements of ¤.

Estimation of the restricted parameters given the restrictions ((25) may proceed via MD

estimation, and results in an estimated model that is locally integrable up to a concave cost

function.

4. Exact Consumer Surplus Measurement

4.1 Small Price Changes

Consumer surplus measurement from estimated demand systems is appropriate only if the

estimated Slutsky Matrix is symmetric. Exact consumer surplus for a small price change

from p0 to p1 can be estimated by evaluation of a Taylor expansion of lnC around p0 to

approximate lnC(p1; u). It is convenient to consider the economic price index, which is a

consumer surplus measure giving the ratio of expenditure needs between price situations p0

and p1. Consider a second order Taylor expansion of the log-cost function, lnC(p; u), around

a price point p0 = (p10; :::; pM0 ). The log of the economic price index is given by:

lnC(p1; u)

C(p0; u)=

MX

j=1

@ lnC(p0; u)

@pj(pj1 ¡ pj0) + (26)

1

2

MX

j=1

MX

k=1

@2 lnC(p0; u)

@pj@pk(pj1 ¡ pj0)(pk1 ¡ pk0) + R;

where R is the residual due to higher order terms.

14

The derivatives of the log-cost function with respect to log-prices are locally given by

@ lnC(p0; u)

@pj= wj (27)

and@2 lnC(p0; u)

@pj@pk=@wj

@pk+@wj

@xwk : (28)

Substituting in the local quadratic estimates gives

@ lnC(p0; u)

@pj= ®j (29)

and@2 lnC(p0; u)

@pj@pk= ¯ jk ¡

ÃMX

s=1

¯ js

!®k: (30)

Substituting back into the Taylor expansion, the second order Taylor approximation of

the economic price index for a small price change is given by

C(p1; u)

C(p0; u)¼

MX

j=1

®j(pj1 ¡ pj0) +1

2

MX

j=1

MX

k=1

ïjk ¡

ÃMX

s=1

¯js

!®k

!(pj1 ¡ pj0)(pk1 ¡ pk0): (31)

Here, the only information needed to compute the approximation of the economic price index

are estimates of expenditure shares (®j) and compensated expenditure share elasticities

(¯jk ¡³PM

s=1 ¯js

´®k). If these estimates satisfy symmetry, then the economic price index

evaluated over any sequence of price changes leading from one price vector to another will

yield the same overall economic price index.

It is notable that previous work on the estimation of consumer surplus (see, eg, Vartia

1983 and Hausman and Newey 1995) has not o¤ered estimates that satisfy local symmetry or

local path-independence. The use of symmetry-restricted parameter estimates in the above

Taylor approximation …lls this gap in the literature.

15

4.2 Large Price Changes

Following Vartia (1982), any large price change can be broken into a sequence of small price

changes if the estimated Slutsky Matrix is everywhere symmetric. Symmetry guarantees

that the price path taken will not a¤ect the exact surplus estimate. If symmetry restricted

local quadratic estimates are continuous and a second order Taylor approximation of the

logarithm of the economic price index is used, then any sequence of (su¢ciently) small price

changes which sum to the large price change will ”add up” to the same economic price index

for the large price change.

5. Demographic E¤ects

De…ne Zi = Z1i ; :::; ZTi as a T ¡ vector of demographic variables for the i’th household, with

Zi = 0T for the reference household type. Consider expenditure share equations which are

locally given by

wj(ep0; epi) = ®j +TX

t=1

®tjZ ti +MX

k=1

¯jk(epk0 ¡ epki ) +

TX

t=1

MX

k=1

¯jtk Zti (ep

k0 ¡ epki ) (32)

+1

2

MX

k=1

MX

l=1

° jkl³

epk0 ¡ epki´ ³

epl0 ¡ epli´+ ²ji : (33)

Again, the solutions are given by minimising

min®j;®tj ;¯

jk ;°

jkl

NX

i=1

³²ji

´2­(ep; epi); (34)

which is just locally weighted least squares. Clearly, this is an econometrically easy way to

include demographic e¤ects. The question is how to constrain the estimated Slutsky Matrix

to satisfy symmetry.

Allow the symmetry terms, Sjk, to depend on demographic characteristics, Zi. In this

16

case, the local estimates and estimated log-price derivatives are given by

dwj(ep0;Zi) = ®j +TX

t=1

®tjZ ti ; (35)

andd@wj(ep0;Zi)@pk0

= ¯jk +TX

t=1

¯jtk Zti : (36)

One might suspect that restricting ®tj and ¯jtk in a way analagous to the restrictions (19) on

®j and ¯jk would give symmetry, but this is not true due to the fact that the Slutsky Matrix

is not homogeneous in the parameters. For example, doubling all of the parameters in (19)

will not preserve equality of Slutsky terms.

The estimated Slutsky Matrix with demographic e¤ects is locally given by

Sjk(ep0;Zi) = ¯jk +TX

t=1

¯tjk Zti ¡

ÃMX

s=1

¯js +MX

s=1

TX

t=1

¯tjs Zti

!îk +

TX

t=1

®tkZ ti

!+

îj +

TX

t=1

®tjZ ti

!îk +

TX

t=1

®tkZ ti

!¡ dj

îj +

TX

t=1

®tjZ ti

!:

Slutsky Symmetry requires that Sjk = Skj , which implies that

¯jk +TX

t=1

¯tjk Zti ¡

ÃMX

s=1

¯js +MX

s=1

TX

t=1

¯ tjs Zti

!îk +

TX

t=1

®tkZ ti

!(37)

is symmetric.

Because Z ti appears in several places, it is di¢cult to restrict the parameters so that

Slutsky Symmetry is satis…ed for all Z ti . I o¤er two strategies: (1) restrict the demographic

normalised price interactions so that the total expenditure e¤ects of demographics are nil;

(2) restrict the demographic e¤ects to be level e¤ects only. I will consider these in turn.

Suppose that the demographic e¤ects are restricted so that

MX

s=1

¯tjs = 0

17

for all j; k; t. Here, the interaction of demographics and total expenditure is forced to zero.

Further, suppose that the restrictions (19) are imposed. Then, Slutsky Symmetry implies

thatTX

t=1

Z ti

ïtjk ¡

ÃMX

s=1

¯js

!®tk

!

is symmetric. Since it must be symmetric for all Z ti , we get

ïtjk ¡

ÃMX

s=1

¯ js

!®tk

!=

ïtkj ¡

ÃMX

s=1

¯ks

!®tj

!

or

¯tjk ¡ ¯tkj =ÃMX

s=1

¯js

!®tk ¡

ÃMX

s=1

¯ks

!®tj (38)

for all j; k; t. Here, symmetry ties the interaction of demographics and normalised prices to

the total expenditure e¤ects of the reference household type (PMs=1 ¯

js) and the level e¤ects

of demographics (®tj).

Alternatively, suppose that the demographic e¤ects are restricted so that

¯tjk = 0

for all j; k; t. Here, demographics only have a level e¤ect given by ®tj. Substituting into (37)

implies that

¯jk ¡ÃMX

s=1

¯js

! îk +

TX

t=1

®tkZ ti

!

is symmetric.

Now, suppose that the restrictions (19) are imposed. Then, Sjk = Skj implies that

ÃMX

s=1

¯js

!®tk =

ÃMX

s=1

¯ks

!®tj

for all j; k; t. This may be expressed as

®tj

®tk=

PMs=1 ¯

jsPM

s=1 ¯ks

; (39)

18

for all j; k; t.

The restrictions embodied in (39) imply that the demographic e¤ects are not completely

free. They are related across share equations by the ratio of log-expenditure e¤ects across

share equations. Further, if any expenditure share equation, j, is independent of x, so that³PM

s=1 ¯js

´= 0, then either the demographic e¤ects for that share are zero, so that ®tj = 0T ,

or all expenditure share equations are independent of x.

6. Estimation

6.1 The Data

The expenditure, demographics and region of residence data for this study are drawn from

the Consumer Expenditure Surveys of the USA for the period 1980 to 1995. These data were

put together by Sablehaus (1996) and the National Bureau of Economic Research (NBER),

and are available on the NBER website. I use expenditures on four categories of expenditure:

(1) food consumed at home; (2) food consumed outside the home (including restaurants); (3)

rent (including utilities); and (4) clothing. For the estimation without demographic e¤ects,

I use the 5584 observations of rental-tenure households containing a single adult member

for a full year residing in an urban area and reporting expenditures for four quarters in a

row. For estimation with demographic e¤ects, I use the 10,214 observations of rental-tenure

households with one, two or three full-year members residing in an urban area and reporting

expenditures for four quarters in a row. Thus, these data are for annual expenditure on

these four goods.

The price data used in this analysis are from Attanasio (1999). Monthly prices are

available for the four goods used in the analysis for the 4 BLS regions of the USA—Northeast,

19

South, West and Midwest. Prices are aggregated up to the annual level for the twelve month

period corresponding to the expenditures of each household used in the analysis.

Table 1 gives summary statistics for the data.

Table 1: The Data Single Member Households Households with 1, 2 or 3 Members

Expenditure Shares Min Max Mean Std Dev Min Max Mean Std Dev

Food at Home 0.04 0.94 0.27 0.15 0.04 0.94 0.30 0.15

Food Out 0.00 0.69 0.10 0.10 0.00 0.69 0.09 0.09

Rent 0.04 0.97 0.55 0.15 0.03 0.97 0.52 0.15

Clothing 0.00 0.50 0.07 0.07 0.00 0.50 0.08 0.07

(All Households)

Prices Min Max Mean Std Dev

Food at Home 836 1536 1165 182

Food Out 786 1514 1182 199

Rent 744 1847 1232 262

Clothing 887 1473 1119 139

6.2 Choice of Bandwidth Matrix

There is no concensus as to how to choose the bandwidth matrix H in local polynomial

regression estimation with multiple predictors. However, in the context of local polynomial

density estimation with multiple predictors, some researchers have used a strategy called

‘sphering’. Here, the bandwidth matrix used is given by

H = hb§1=2 (40)

where b§ is the estimated covariance matrix of epi and h is a scaling parameter. For multi-

variate nonparametric density estimation for a multivariate normal reference density (NRD),

20

the optimal size of h (see, for example, Simono¤ 1995) is given by

hNRD =

Ã2

(M ¡ 1) + 2

!1=(M¡1+4)

N¡1=(M¡1+4): (41)

In this paper, the bandwidth scaling parameter h is chosen by cross-validation. Cross-

validation yields a bandwidth scaling parameter that is 2.005 times as large as hNRD for the

data used in the analysis. Thus, in comparison with the normal reference density optimal

bandwidth, this bandwidth matrix oversmooths the data.

To give the ‡avour of the consequences of this choice of bandwidth scaling, I note that

the locally weighted least squares estimates would put a weight of

­(ep0 = epi) =M¡1Y

j=1

1p2¼= 0:004 (42)

on data exactly at the point of estimation. In the empirical work, with a bandwidth scaling

parameter 2.005 times as large as hNRD , approximately 20% of observations in the sample

get weight at least 1% as large as an observation exactly at the point of estimation.

6.3 Unrestricted Estimation on Normalised Prices

Figure 1 shows estimated expenditure share equations for the sample of 5576 single adult

households in the CES 1980 to 1995 microdata …le using unrestricted local quadratics. This

model is estimated locally at 91 points in the log-expenditure distribution for the sample of

single adults in the 1984 CES database. The 91 points chosen are the 5th to 95th percentiles

of the the log-expenditure distribution for these single adult households. The prices used are

those facing consumers in the West region in August 1984. These estimated regression curves

are just the ®j parameters for the local quadratic regressions at each of the 91 points in the

normalised price space. Because this consumer demand system is estimated over normalised

prices, the estimates satisfy homogeneity by construction.

21

For each local quadratic regression curve, I include pointwise 90% con…dence bands

represented by the thin lines in the …gures. These bands are computed by simulating the

distribution of the regression curves under the null that they are actually equal to the

estimated curves shown with thick lines. The simulation methodology uses the Smooth

Conditional Moment System (SCMS) bootstrap technique described in the appendix, with

1000 bootstrap iterations and employs the percentile-t method to compute con…dence bands.

That is, the con…dence bands shown are 5th and 95th percentiles of the distribution of the

simulated regression curves at each point in the domain. I note that the simulated marginal

distributions of the ®j parameters are approximately normal. Thus, using +/- 1.605 times

the standard error of ®j yields essentially identical con…dence bands.

6.4 Restricted Estimation on Normalised Prices

Figures 2 and 3 show unrestricted and restricted expenditure share equations for single

adult households. The restricted models impose the restrictions (19) which ensure that the

estimated Slutsky Matrix is symmetric at each point. Figure 2 shows results for symmetry-

restricted estimates from MD estimation which uses the reciprocal of the simulated variance

of the unrestricted parameters as the diagonal weighting matrix. Simulated variances for

unrestricted parameters at each point are computed via SCMS bootstrap (see the appendix).

This form of MD estimation is equivalent to choosing the restricted parameters by minimising

the sum of squared standardised distances between restricted and unrestricted parameters

(subject to the symmetry restrictions). Here, the restricted and unrestricted ®j are nearly

identical, due to the fact that the simulated variances for ®j parameters are small relative

to the simulated variances for other parameters.

Figure 3 shows for symmetry-restricted estimates from MD estimation which uses the

22

reciprocal of the squared unrestricted parameters as the diagonal weighting matrix. This

form of MD estimation is equivalent to choosing the restricted parameters by minimising

the sum of squared proportional distances between restricted and unrestricted parameters.

Here the restricted and unrestricted ®j are similar, but de…nitely not identical. In particu-

lar, the estimated rent share equation is quite di¤erent under symmetry restrictions than in

the unrestricted model. Not surprisingly, the di¤erences between restricted and unrestricted

estimates are large when the parameter estimates are large. However, the symmetry restric-

tions are not imposed for the purpose of estimating expenditure share equations—rather,

they are imposed to estimate consumer surplus measures.

6.5 Compensated Price Elasticities

To estimate the second-order Taylor approximation of the economic price index for a step in

a price change sequence, we require estimates of expenditure shares (®j ) and compensated

expenditure share elasticities (¯jk ¡³PM

s=1 ¯js

´®k). These compensated expenditure share

elasticities are not identical to Slutsky terms because the latter are compensated demand

derivatives rather than share elasticities.

To get some feel for the severity of the restrictions and plausibility of the estimated

demand system, it is illuminating to construct the own-price Slutsky terms given by (18)

where j = k. If the demand system is dual to a cost function which is concave in prices,

then these own-price Slutsky terms are all negative. Figure 4 shows the own-price Slutsky

terms from the unrestricted estimates and both types of restricted estimates. Lines denoted

vw are from variance weighted MD estimation and lines denoted pw are from parameter

weighted MD estimation.

The …rst thing to note from Figure 4 is that although compensated own-price demand

23

derivatives should be negative, only the food expenditure share equation has own-price

Slutsky terms which are mostly negative. For poorer households, the own-price Slutsky

terms for the rent expenditure share equation are close to zero (but still positive), but for

richer households they are positive. The own-price Slutsky terms for food out are positive

and large for almost households in almost the entire total expenditure distribution. Thus,

neither the estimated restricted nor unrestricted demand systems satisfy concavity (without

the imposition of the concavity restrictions discussed above).

The second thing to note from Figure 4 is that the weighting matrix used for the re-

strictions does not change the estimated own-price Slutsky terms very much. For example,

the variance-weighted and parameter-weighted restricted own-price Slutsky terms for food

at home are mostly within about 1% of each other. Similarly, the own-price Slutsky terms

for rent are very close to each other. However, the own-price Slutsky terms for food out

seem to depend somewhat on the weighting matrix chosen for the restricted estimation.

As noted above, to estimated the economic price index, compensated expenditure share

elasticities are needed. Figure 5 shows compensated cross-price elasticities for restricted and

unrestricted estimates, using variance-weighted MD estimation. Here, there are two cross-

price elasticities corresponding to the unrestricted system because symmetry is not imposed,

but there is only one cross-price elasticity corresponding to the restricted system.

Thin lines in Figure 5 represent unrestricted estimates and thick lines represent restricted

estimates. The dotted and black lines show compensated cross-price elasticities for the food

at home share equation with respect to the price of food out and the price of rent, respectively.

The gray lines show compensated cross-price elasticities for the food out equation with

respect to the price of rent. Eyeballing the estimates suggests that symmetry may not

hold for any of the three cross-price elasticities. The thin lines which represent unrestricted

24

estimates should coincide if symmetry holds, but for error in estimation.

Figures 4 and 5 also give some sense of how restricted estimates change over the log-

expenditure domain. Recall that Young’s Theorem uses continuity to go from satisfaction of

symmetry to integrability of the demand system and therefore path-independence of surplus

measures. Although continuity cannot be formally tested, more eyeballing suggests that

the restricted estimates may in fact be continuous, and therefore useful in consumer surplus

estimation.

6.6 Testing the Symmetry Restrictions

A key feature of Figures 2 to 5 is that the restricted and unrestricted estimates are somewhat

similar to each other, especially above the 10th percentile of the total expenditure distribu-

tion. An interesting question is whether or not these di¤erences are statistically signi…cant.

I use the test statistic ¿ given by equation (20), where ¿ is the sum of squared di¤erences

of compensated cross-price expenditure share elasticities. Under the null hypothesis that

symmetry is true, this sum is zero. Figure 6 shows the value of this test statistic and its

p-value for each type of restricted estimation. The p-values are obtained using the SCMS

bootstrap simulation (described in the appendix). For each point of estimation, the sim-

ulation takes the restricted estimates constructs a bootstrap sample of expenditure shares

equal to the restricted estimates multiplied by the normalised price distances plus a boot-

strap error vector. The bootstrap error vector reproduces the conditional heteroskedasticity

and cross-equation correlation in the data. But for the bootstrap errors, the bootstrap ex-

penditure shares correspond to a globally quadratic model whose parameters are given by

the restricted estimates. Each simulation uses 1000 bootstrap iterations, and the p-values

are computed from this bootstrap distribution. The simulation for each of the 91 points of

25

estimation is done separately.

Figure 6 shows that no single test of symmetry rejects at any conventional level of sig-

ni…cance. This does not imply that symmetry is not rejected for the joint test across all 91

observations—this simulation strategy does not allow the testing of that joint hypothesis.

That symmetry is not rejected suggests one of two things: either the test has low power, or

symmetry is in fact true.

It is interesting to note that the p-values hardly depend at all on the parameters of the

model used to generate the null distribution. The absolute value of the di¤erence in p-values

across the two symmetry-restricted parameter vectors has an average of 0.1% and a maximum

of 0.6%. If this bootstrap simulation strategy were exactly pivotal, the p-values would be

identical for the simulations from either of the two symmetry-restricted parameter vectors. If

it were asymptotically pivotal, the p-values would converge in distribution. Exactly pivotal

bootstrap tests perform better than asymptotic tests and asymptotically pivotal bootstrap

tests perform just as well as asymptotic tests.

6.7 Demographic E¤ects

Figure 7 shows separately estimated unrestricted local quadratic expenditure share equations

for food consumed at home for three household sizes: households with one, two and three

members. Each regression curve is estimated separately for the 5576 one-person households,

2940 two-person households, and 1698 three-person households in the 1980 to 1995 CES

data. The points of estimation an evenly spaced grid from the lowest used in Figure 1 to

seventy percent higher than the highest used in Figure 1. The points of estimation are

changed to re‡ect the fact that larger households have greater total expenditure.

These regression curves have somewhat di¤erent shapes across household size. In par-

26

ticular, although the share equations for households with one and two members look quite

similar, that for households with three members has a di¤erent shape. As noted above, there

are two ways to include demographic information in the symmetry-restricted model. The

researcher can interact the demographic variables with the intercept and normalised prices,

or just with the intercept. These approaches are much more general than the partially linear

inclusion of demographic e¤ects used in the non- and semi-parametric consumer demand

estimation literature (see, eg, Blundell, Duncan and Pendakur 1998), because partially lin-

ear approaches require that the demographic e¤ects be the same across di¤erent points of

estimation. Here, the demographic e¤ects di¤er at di¤erent points of estimation.

Figure 7 also shows estimated unrestricted expenditure share equations for food at home

for the three household sizes using the simple pooled estimation strategy de…ned above

applied to all 10,214 observations of one-, two- and three-person households in the 1980 to

1995 CES. Here, there is only one piece of demographic information included in the model—

the number of household members. Thus, in (32), T = 1, Zti = Zi is household size minus

1 (so that the reference household of a single adult has Z = 0). Further, the demographic

information is interacted only with the level term in the local quadratic.

Thin lines show the share equations for these three household sizes estimated separately,

and thick lines show the share equations for these three household sizes estimated with the

pooled approach. It seems that the pooled approach does do some violence to the data,

especially at the upper end of the total expenditure distribution.

Figure 8 shows the estimated expenditure share equations for food consumed at home

for the three household types in a pooled regression context, with thick lines showing un-

restricted estimates, and thin lines showing symmetry-restricted estimates satisfying the

restrictions (??). Variance-weighted and parameter-weighted symmetry-restricted estimates

27

are denoted “vw” and “pw”.

The variance-weighted symmetry-restricted estimates lie right on top of the unrestricted

pooled estimates due to the fact that the variances for the intercept and demographic

terms are much smaller than the variance for other parameters. In contrast, the parameter-

weighted estimates jump around quite a lot, and are only close to the unrestricted estimates

in the middle of the distribution of total expenditure.

7. Conclusions

This paper o¤ers several innovations to the semiparametric estimation of consumer demand

systems and exact consumer surplus measures. First, a consumer demand system may be

estimated as a multivariate nonparametric equation system over normalised prices de…ned

as prices divided by total expenditure. This strategy gives an estimated consumer demand

system which satis…es homogeneity with respect to all prices and total expenditure by con-

struction.

Second, using multivariate local quadratic nonparametric regression over log-normalised

prices allows the researcher to impose symmetry on the estimated Slutsky Matrix via a second

stage minimum distance step. A similar minimum distance strategy allows the imposition

of Slutsky Concavity as well. A consumer demand systems whose estimated Slutsky Matrix

is symmetric may be used to compute exact path-independent consumer surplus measures,

such as price indices. Estimation of a consumer demand system for single adults using

American price and expenditure data suggests that symmetry-restricted estimates are not

very di¤erent from unrestricted estimates, and the restrictions for symmetry of the estimated

Slutsky Matrix are not rejected for any pointwise test.

Third, demographic e¤ects may be included in the model as interactions in the local

28

polynomial at each point of estimation. This requires a stricter set of restrictions for the

Slutsky Matrix to satisfy symmetry. Estimation of a consumer demand system for Amer-

ican households with one, two or three members suggests that this strategy for including

demographic e¤ects does parsimoniously capture some of the variation in the data.

Fourth, a simple extension of Gozalo’s Smooth Conditional Moment Bootstrap is used

which allows for cross-equation correlation in the structure of simulated errors. This in-

novation is important in this context because Slutsky Symmetry imposes cross-equation

restrictions on the consumer demand system. In the empirical work, however, this more

general bootstrap technique does not change estimated con…dence intervals or test statistics

in comparison to bootstrapping from the empirical error distribution.

REFERENCES

Attanasio, Orazio, 2000, “Prices for the CES”, unpublished regional price data for theConsumer Expenditure Surveys of the United States, 1980 to 1998, courtesy of OrazioAnnatasio.

Banks, Blundell and Lewbel, 1997, “Quadratic Engel Curves and Consumer Demand”,Review of Economics and Statistics, 79(4), pp 527-539.

Blundell, Richard, Alan Duncan and Krishna Pendakur, 1998, “Semiparametric Esti-mation of Consumer Demand” Journal of Applied Econometrics, pp 435-461.

Blundell, Richard and Ian Preston, 1998, “Consumption Inequality and Income Uncer-tainty” Quarterly Journal of Economics, May 1998, pp 603-640.

Bowman, Adrian and Adelchi Azzalini, 1999, “Applied Smoothing Techniques for DataAnalysis”, Oxford University Press: London.

Dickens, Richard, Vanessa Fry and Panos Pashardes, 1993, “Non-linearities and Equiv-alence Scales”, Economic Journal, 103(417), March 1993, pages 359-68.

Fan, J. and I. Gijbels, 1996, “Local polynomial modelling and its applications”, London; New York : Chapman & Hall.

Gozalo, Pedro, 1997, “Nonparametric Bootstrap Analysis with Applications to Demo-graphic E¤ects in Demand Functions”, Journal of Econometrics 81(2), December 1997,pages 357-93.

29

Hausman, Jerry and Whitney K. Newey, 1995, “Nonparametric Estimation of ExactConsumers Surplus and Deadweight Loss”, Econometrica 63(6), November 1995, pages1445-76.

King, M., 1983, “Welfare Analysis of Tax Reforms Using Household Data” Journal ofPublic Economics 21, pp 183-214.

Lewbel, Arthur, 1995, “Consistent Nonparametric Hypothesis Tests with an Applica-tion to Slutsky Symmetry”, Journal of Econometrics 67(2), June 1995, pages 379-401.

Rothenberg, Thomas, 1973, “E¢cient estimation with a priori information”, NewHaven: Yale University Press, 1973.

Ryan, David and Terry Wales, 2000, “Imposing Local Concavity in the Translog andGeneralized Leontief Cost Functions”, Economics Letters 67(3), June 2000, pages 253-60.

Ryan, David and Terry Wales, 1999, “A Simple Method For Imposing Local Curva-ture In Some Flexible Consumer Demand Systems”, University of British ColumbiaDepartment of Economics Discussion Paper: 96/25, June 1996, pages 21.

Sabelhaus, John. 1996. “Consumer Expenditure Surveys: Family Level Extracts”NBER, Washington, DC (revised 1999).

Simono¤, Je¤ery, 1995, “Smoothing methods in statistics”, Series in Statistics. NewYork and Heidelberg: Springer, 1996, pages xii, 338.

Vartia, Yrjo, 1983, “E¢cient Methods of Measuring Welfare Change and CompensatedIncome in Terms of Ordinary Demand Functions”, Econometrica 51(1), January 1983,pages 79-98.

Wand, M.P. and M.C. Jones, 1995, “Kernel Smoothing”, Chapman and Hall: Mel-bourne.

8. Simulating Con…dence Bands and Test Statistics

8.1 Smooth Conditional Moment System Methods

Gozalo’s (1997) Smooth Conditional Moment (SCM) bootstrap allows the researcher to

reproduce the conditional heteroskedasticity of the data in the simulation of con…dence

bands and test statistic sampling distributions for a single equation model. However, the

SCM bootstrap is suitable for systems of equations only when there is no cross-equation

30

correlation. Here, I develop a simple modi…cation of the SCM bootstrap that allows the

researcher to reproduce the conditional cross-equation correlation in the data.

The following algorithm generates a multivariate distribution of errors conditional on an

independent variable vector ep. The algorithm generates error vectors of length M ¡ 1 for

households indexed i = 1:::N. (There are onlyM ¡1 error terms for each household because

the last error term is given by the restriction that the errors sum to zero.) These vectors,

²i(ep) = [²1i (ep); :::; ²M¡1i (ep)]0, are constructed so that E[²i(ep)]= 0M¡1, E[²i(ep)²i(ep)0]= §(ep),

and E[²ji (ep)3]= ¹j (ep). Gozalo’s (1997) SCM Bootstrap does not allow for cross-equation

correlations, and so requires that §(ep) be a diagonal matrix; here I relax that restriction. I

call this method the Smooth Conditional Moment System (SCMS) bootstrap.

Since the data are not assumed to be identically distributed, these second and third

moments depend on the independent variable ep. Following Gozalo (1997), one may estimate

¹j(ep) and the elements of §(ep) from kernel regressions. In particular, given M ¡ 1 indepen-

dent equations, one might construct the empirical error distribution, b²ji , and estimate the

§(ep) matrix by nonparametric regression of³b²jib²ki

´over ep, and estimate ¹j(ep) by nonpara-

metric regression of³b²ji

´3over ep. The strategy is to construct a distribution of uncorrelated

errors, ºi = [ º1i ; :::; ºM¡1i ]0, and to generate "i satisfying the above moment conditions as a

linear combination of ºi.

Suppress the i subscript on all functions, vectors and matrices, and consider the genera-

tion of a set of bootstrap errors for a single household. Since §(ep) is a covariance matrix eval-

uated at each point on ep, it is positive semide…nite. De…ne ¾(ep) such that ¾(ep)0¾(ep) = §(ep)

(for example, using the Choleski Decomposition of §(ep) at each point in ep). Write the el-

ements of ¾(ep) as ¾jk(ep). De…ne ¡(ep) as a matrix containing the cubed elements of ¾(ep),

so that ¡(ep) = f(¾jk(ep))3g. De…ne a vector ¹(ep) = [ ¹1(ep); :::; ¹M¡1(ep)]0. De…ne a vector

31

¸(ep) = ¡¡1(ep)¹(ep), and denote its elements as ¸j (ep). All of these matrices and vectors are

conditioned on ep.

De…ne an independent M ¡ 1 vector random variables º = [º1; :::; ºM¡1]0. The following

set of two-point distributions for ºj satis…es E[º]= 0M¡1, E[ºº 0]= 1M¡1, E[(ºj)3]=¸j (see

Gozalo 1997 for details):

cF j (ep) = ¼j(ep)µaj(ep) + (1¡ ¼j(ep))µbj (ep) (43)

where µK denotes a probability measure that puts mass one at K, ¼j(ep) in [0;1],

¼j (ep) = [1 + ¸j (ep)=T j(ep)]=2; (44)

aj(ep) = (¸j (ep)¡ T j (ep))=2; (45)

bj(ep) = (¸j (ep) + T j(ep))=2 (46)

and

T j(ep) =q¸j(ep)2 + 4: (47)

Generate " as follows:

" = ¾0º (48)

" constructed in this manner satis…es E["(ep)]=0, E["(ep)"(ep)0]=§(ep), E["i(ep)3]=¹i(ep).

Proof: Suppress the dependence of all moments and functions on ep. Since E[º]=0 and " is

linear in º, E["]=E[¾0º]= 0K. Noting that E[ºº 0]= IK , E[""0]=E[¾0ºº 0¾]= ¾0¾ = §. Noting

that " = Sº and E[ºº 0]= IK , and that all of the cross-products in the third moments of ºi

have expectation zero, E["3i ]=PM¡1j=1 ¾

3ij¸j. Denoting "3 = ["31; :::; "

3K ]0, this condition may

be expressed in matrix form as E["3]= ¡¸. Substituting in for ¸ gives E["3]= ¡¡¡1¹ = ¹.

QED.

32

8.2 Simulation of Con…dence Bands

Con…dence bands for nonparametric local polynomial estimated expenditure share equations

may be simulated as follows. De…ne a set of normalised price points of estimation e¼±,

± = 1; :::;¢. These may or may not be the same as the observations of data. Estimate the

model at every point in e¼, denoting the estimates as bwj± .

Step 1: Estimate the model at every point in the data for observations wi and epi, i =

1; :::;N . Denote estimated expenditure shares (given by the local quadratic parameters ®j)

as bwji . Construct empirical errors as b²ji = wji ¡ bwji . Estimate§(ep) and ¹j(ep) by nonparametric

local quadratic estimation.

Step 2: Construct an M ¡ 1 bootstrap error vector "i via SCMS bootstrap for every

observation. Construct the bootstrap expenditure shares as wji = bwji + "ji . Estimate the

model at each point of estimation in e¼. Denote the bootstrap expenditure shares (given by

the local quadratic parameters ®j ) as wj¤± .

Step 3: Repeat Step 2 B times, and use the distribution of wj¤± to estimate the con…dence

band for the estimated expenditure shares bwj± . For example, the 90% con…dence band for

bwj± may be given by the 5’th and 95’th percentiles of the distribution of wj¤± . In this paper,

B = 1000.

8.3 Simulation for Testing Symmetry

The test statistic

¿ =M¡1X

k=j+1

M¡1X

j=1

ïjk ¡

MX

s=1

¯js®k ¡ ¯kj +

MX

s=1

¯ks®j

!2

is equal to zero under the null hypothesis that the Slutsky Matrix is symmetric. For every

point of estimation e¼± , we can construct the test statistic ¿± from the estimated local

quadratic parameters. To test the null hypothesis, we simulate the distribution of ¿ ± with

33

symmetry holding by construction.

Denote the restricted estimates at e¼± as ©R± = [®Rj± ; ¯

Rj±k ; °

Rj±kl]. Here, R indicates that the

estimates are restricted to satisfy symmetry and ± gives the point of estimation.

Step 1: same as Step 1 above.

Step 2: Construct an M ¡ 1 bootstrap error vector "i via SCMS bootstrap for every

observation. Using the restricted estimates for e¼± , ©R± , construct the bootstrap expenditure

shares for simulating the test statistic at e¼± as

wj±i = ®Rj± +

MX

k=1

¯Rj±k epki +1

2

MX

k=1

MX

l=1

°Rj±kl epki epli + "ji :

Estimate the model, and construct the bootstrap test statistic ¿ ¤±.

Step 3: Repeat Step 2 B times, and use the distribution of ¿ ¤± to estimate the critical

values or p-values for test statistic. For example, a one-sided 5% critical value for ¿ ± may

be given by the 95’th percentile of the distribution of ¿ ¤± .

Note that the simulation strategy for testing symmetry is di¤erent than that used for

generating con…dence bands for estimated nonparametric regression curves. To simulate the

distribution of the test statistic ¿± , it is necessary to construct the bootstrap distribution so

that is satis…es symmetry by construction, deviating only by the error terms.

34

Figure 1: Expenditure Shares for %iles of the 1984 total expenditure distribution

0

0.1

0.2

0.3

0.4

0.5

0.6

7.5 7.75 8 8.25 8.5 8.75 9 9.25

Log of Total Expenditure

Sh

are

Food In Food Out Rent

Figure 2: Unrestricted versus Restricted Shares with variance weighted (standardized) MD

0

0.1

0.2

0.3

0.4

0.5

0.6

7.5 7.75 8 8.25 8.5 8.75 9 9.25

Log of Total Expenditure

Sh

are

Food In Food Out Rent

Restricted Food In (vw) Restricted Food Out (vw) Restricted Rent (vw)

Figure 3: Unrestricted versus Restricted Shares squared estimate weighted (proportional) MD

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

7.5 7.75 8 8.25 8.5 8.75 9 9.25

Log of Total Expenditure

Sh

are

Food In Food Out

Rent Restricted Food In (pw)

Restricted Food Out (pw) Restricted Rent (pw)

Figure 4: Own-Price Slutsky TermsRestricted and Unrestricted

-1

0

1

2

7.5 7.7 7.9 8.1 8.3 8.5 8.7 8.9

Log of Total Expenditure

Ow

n-P

rice

Slu

tsky

Ter

m

S11_u S22_u S33_uS11_r (vw) S22_r (vw) S33_r (vw)S11_r (pw) S22_r (pw) S33_r (pw)

Figure 5: Compensated Cross-Price Elasticities(Variance weighted) Restricted and Unrestricted

-2

-1

0

1

7.5 7.7 7.9 8.1 8.3 8.5 8.7 8.9

Log of Total Expenditure

Co

mp

ensa

te E

last

icit

y

C12_u C13_u C21_uC23_u C31_u C32_uC12_r (vw) C13_r (vw) C23_r (vw)

Singles Summary & figs Chart 12

Page 1

Figure 6: Tests of Symmetry Restrictions

0

0.25

0.5

0.75

1

7.44 7.78 7.91 8.04 8.13 8.22 8.28 8.38 8.45 8.52 8.59 8.64 8.69 8.74 8.81 8.88 8.93 9.07 9.23

Log of Total Expenditure

P-v

alu

e

0

0.5

1

1.5

2

2.5

Tes

t S

tati

stic

P-Value (vw) P-value (pw) Test Statistic

Figure 7: Food Shares for Different Family Sizes,Pooled and Separately Estimated

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

7.25

7.5 7.75

8 8.25

8.5 8.75

9 9.25

9.5 9.75

10

Log of Total Expenditure

Sh

are

1 Person Families--Separate 2 Person Families--Separate

3 Person Families--Separate 1 Person Families-Pooled

2 Person Families--Pooled 3 Person Families--Pooled

Figure 8: Food Shares for Different Family Sizes,Unrestricted and Restricted Pooled Estimates

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

7.25

7.5 7.75

8 8.25

8.5 8.75

9 9.25

9.5 9.75

10

Log of Total Expenditure

Sh

are

1 Person Families--Unrestricted 2 Person Families--Unrestricted 3 Person Families--Unrestricted

1 Person Families--Restricted (vw) 2 Person Families--Restricted (vw) 3 Person Families--Restricted (vw)

1 Person Families--Restricted (pw) 2 Person Families--Restricted (pw) 3 Person Families--Restricted (pw)