Panel and Pseudo-Panel Estimation of Cross-Sectional and Time Series Elasticities of Food Consumption: The Case of U.S. and Polish Data Author(s): François Gardes, Greg. J. Duncan, Patrice Gaubert, Marc Gurgand, Christophe Starzec Source: Journal of Business & Economic Statistics, Vol. 23, No. 2 (Apr., 2005), pp. 242-253 Published by: American Statistical Association Stable URL: http://www.jstor.org/stable/27638815 . Accessed: 27/10/2011 05:21 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

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and

Panel

Pseudo-Panel

Estimation

and

Cross-Sectional

Time

of Food

The

Consumption: and Polish Data

U.S.

Fran?ois Gardes

IPantheon-Sorbonne

Universit? de Paris

Greg J. Duncan

Elasticities

Case

Paris, 75005,

(Cermsem),

IL 60208

Northwestern University, Evanston,

Series

of

of

France

([email protected])

([email protected])

Patrice Gaubert sur Mer Cedex,

Universit? du Littoral (LEMMA), Boulogne Marc

France

62321,

(gaubert?

univ-paris1 .fr)

Gurgand

CNRS-Delta,

France

Paris, 75014,

Christophe Starzec

France

Paris, 75013,

CNRS-TEAM,

([email protected])

(starzec@univ-paris1

.fr)

This article addresses

the problem of the bias of income and expenditure elasticities estimated on pseudo panel data caused by measurement error and unobserved heterogeneity. We gauge these biases empirically by comparing cross-sectional, pseudo-panel, and true panel data from both Polish and U.S. expenditure surveys. Our results suggest that unobserved heterogeneity imparts a downward bias to cross-section

estimates of income elasticities of at-home food expenditures and an upward bias to estimates of income elasticities of away-from-home food expenditures. "Within" and first-difference estimators suffer less bias, but only if the effects of measurement error are accounted forwith instrumental variables.

KEY WORDS:

1.

AIDS

Individual and grouped data; Unobserved

model;

INTRODUCTION

to estimate

time

series,

panels

demand

systems. time

within-group

using

Data

types

series, and

data,

aggregated

include

as theirbirth year or the education level of the reference per son of a household. Estimation with pseudo-panel data dimin

can be

A multitude of data types and econometric models used

ishes

and

panels

using individual data. Aggregate time series data frequently produce aggregation biases because of composition effectsdue to the change of the population or the heterogeneity of price and

income

different

effect between

social

classes.

These

prob

lems have led the vast majority of empirical studies in labor economics to use individual data (Angrist and Krueger 1999). In contrast, and

periods els

individual

are

on countries

attrition can

sectors

industrial

span

generally

to nonresponse

subject or

data

panel

bias.

suffer from

short Even

time

ternative, riods

or

to estimate

data

grouping

even when

panel

to compare

data

different

on

exist, countries.

a

pseudo-panel on

to estimate

Pseudo-panel

same

reference and

In pseudo-panel to criteria

population sometimes analyses,

that do not

change

in different

but

periods,

from

one

are

the cross-equation

survey

as

of measurement,

period

imposed

by consumption

ture elasticities.

by

article

addresses

elasticities

the

issue

caused

by

to income

of bias errors

of

ex

and

mea

specification,

static

estimating

expenditure

models

using

cross-sectional,

pseudo-panel, and true panel data from both Polish and U.S.

pe

expenditure

are

parison

sometimes

timations

surveys.

between based

To

our

cross-sectional, on

the

same

?

according

to another,

time

restrictions

this

knowledge, dataset.

of Business

The

and use

com

is the first

pseudo-panel,

2005 American

Journal grouped

show

theory.Using different types of data helps reveal the nature of thebiases that they impart to estimates of income and expendi

not. individuals

but we

dimension,

surement, and omitted variables or by heteroscedasticity in grouped and individual-based models. We gauge these biases

typically constructed from a time series of independent surveys thathave been conducted under the same methodology on the consecutive

or

units

consumption as

penditure

is an al

data

across well

This

structural

longer

the cross-sectional

Static and dynamic demand models have been developed for these different types of data, with each model adopting a differ ent approach to problems caused by unobserved heterogeneity

pan

changes or composition effects thatmake itdifficult tomaintain the stationarityhypotheses for all variables. Thus,

on

dimension.

pseudo

cross-sections

efficiency

herein that it also gives rise to a heteroscedasticity in the time

aggregate

cross-sections,

heterogeneity.

panel

of one

Statistical

es our

Association

& Economic April 2005,

of

Statistics

Vol. 23, No. 2

DOI 10.1198/073500104000000587

such

242

243

of Food Consumption

Elasticities

et al.: Estimation

Gardes

Panel Study of Income Dynamics (PSID)? is motivated by the numerous expenditure studies based on it (Altug and Miller 1990; Altonji and Siow 1987; Hall and Mishkin 1982; Naik and Moore 1996; Zeldes 1989). Our sec ond dataset is fromPoland in the late 1980s, which enables us to capitalize on large income and price variations during the transitionperiod inPoland. Section 2 presents a background discussion. Section 3 two dataseis?the

the econometric

presents

and

problems

the methods

describes

results.

these

6 discusses

Section

and

evaluate

systems

static

"within"

forms.

However,

rather are

than

No

matter

how

survey

complete,

come

and

For

elasticities.

price

the value

example,

of time dif

fers across households and is positively related to a household's Because

income.

observed

activities

consumption

(e.g.,

rise

give

1996). We estimated instrumen

usual

using

to those

estimated

However,

these

to consider

prefer

and pseudo-panel

for the static models are

versions

dynamic

panel

the static

only

each

methods

specification.

offers

advantages

and disadvantages for handling the estimation problems inher ent

in expenditure models. error. Survey

concerns

set of

first

A

measurement

eating

dynamic

of

variety

factors

in terms of the specification and the econometric

so we

problems, True

similar

quite

questionable

expen

ditures and demographic characteristics lack explicit measures of all of thepossible factors thatmight bias the estimates of in

observations.

(Arrelano 1989) and found that elasticity esti

in this article.

presented

on household

data

were

and other

as

from

and

a wide

from

the static models

of

and

treated

be

household-level

(e.g., Naik andMoore

versions

tationmethods

mates

also

cross-sections

estimates

persistence

to dynamic models

needed

data

in first-difference can

data

independent

individual

to compare

able

types. Habit

dynamic

2. BACKGROUND

from

ex

panel

us with

models these

to esti

approaches

sets of household

provide

panels

expenditure

came

they

though grouped Thus we

two

using

two

The

data.

penditure to estimate

of alternative

implications

demand

mating

data

used in this study. Section 4 describes the data. Section 5 gives the results,

We

reports

on

centers

are

income

of household

meals) often involve inputs of both goods (e.g., groceries) and measured with error; differencing reports of household income across waves increases the extent of error. Instru time (e.g., time spent cooking and eating), households will face undoubtedly can to caused the biases mental variables be address used even if the prices of the different (full) prices of consumption by error instrumenta measurement Siow Like and case are the identical. If?as is likely in 1987). (Altonji goods-based inputs are asso at home?these tion, aggregation in pseudo-panel data helps reduce the biasing of meals positively prices prepared

and

income

with

ciated

a

have

themselves

effect

negative

on

consumption, then theomission of explicit measurement of full prices

a negative impart same The argument

bias

will

ticities.

to the estimated

income

to the case

can be applied

elas

effects

of measurement

those

prices arising from, say, liquidity constraints thataremost likely in low-income households (Cardoso and Gardes 1996). Taking

case

on

estimated

elastic

the income

expect

error

of variables

like

we

is not

location,

confine

true panel using to be serious likely

social

age,

our

data.

income

instrumented

measurement

Because

of virtual

so we

error,

ityparameters estimated with pseudo-panel data tobe similar to

and

category, variables

instrumental

in the family

adjustments

into account

the virtual

composition, to our income

sources

as

errors in our dependent expenditure variable are included in

such

re from nonmonetary prices appearing con to or due the choice time, space restricting

straintsapplying only to subpopulations, or changing from one period to another,may help us better identify and understand and

cross-sectional

series

time

differences.

estimation

because

biases,

on

information

they contain

in

changes

expenditures and income for the same households. Differencing successive

panel

waves

nets

out

effects

the biasing

of unmea

sured persistent characteristics. But while reducing bias due to omitted variables, differencing income data is likely tomagnify source

another

error. Altonji

measurement

of bias:

and

Siow

(1987) demonstrated the likely importance ofmeasurement er ror in the context of first-differenceconsumption models by showing

that estimates

of

elasticities

income

are

several

times

higherwhen income change is instrumented thanwhen it is not. Deaton (1986) presented the case for using "pseudo-panel" data

to estimate

searcher

has

demand independent

He

assumed

cross-sections

with

systems.

expenditure and demographic cross-sections

in successive

years

that the

the

re

required

information and shows how can

be

grouped

into compa

rable demographic categories and then differenced to produce many of the advantages gained from differencing individual panel data. Grouping into cells tends to homogenize the in dividual effects among individuals grouped in the same cell, so that the average specific effect is approximately invariant between two periods and is efficiently removed by within or first-difference

transformations.

residuals

total

and,

errors

data when

expenditure

correlated

should

not bias

of our

the levels

appear

do not contain

cells

corresponding

with

in

estimates.

the coefficient

can

in measurement

Measurement

predictors.

unless

variables,

dependent Special

Panel data on households provide opportunities to reduce these

model

and

in pseudo-panel individ the same

uals in two differentperiods. Thus if the firstobservation for cell 1 during the firstperiod is an individual A, then itwill be paired with a similar individual B observed during the second period,

error arises

so that measurement

this observa

between

tion of B and the truevalues forA ifhe or she had been observed during the second period. Deaton (1986) treated this problem as a measurement error; cohort

sample-based cohort

averages.

He

sure convergence

averages proposed

measures

are error-ridden a

Fuller-type estimates.

of pseudo-panel

of true to en

correction However,

Verbeek

and Nijman (1993) showed thatDeaton's estimator converges with thenumber of timeperiods. Moreover, Le Pellec and Roux (2002) matrix

showed used

that the measurement in Deaton's

estimator

error is often

correction not

variance

positive

defi

nite in the data. The simpler pseudo-panel estimator used in this article has been shown to converge with the cell sizes (Verbeek and Nijman 1993; Moffit 1993), because measure ment

error

becomes

negligible

when

cells

are

large.

Based

on

simulations, Verbeek and Nijman (1993) argued that cells must contain about 100 individuals, although the cell sizes may be smaller if the individuals grouped in each cell are sufficiently homogeneous.

244 error

the measurement

Resolving

problem

by

using

large

samples within cells creates another problem: loss of efficiency of the estimators. This difficultywas shown by Cramer (1964) and Haitovsky (1973) with estimations based on grouped data and by Pakes (1983) with the problem induced by an omitted variable with a group structure,which is similar to the problem error.

of measurement

to a minimum

but

also

error

measurement

keeping

ignorably

small (Baltagi 1995). Grouping methods were developed by Cramer (1964), Haitovsky (1973), Theil and Uribe (1967), and Verbeek and Nijman (1993) that involve carefully choosing co horts to obtain the largest reduction of heterogeneity within each

cohort

but

same

at the

time maximizing

the heterogene

itybetween them. Following these empirical principles, using leads to consistent pseudo-panels out the problems with associated

and

efficient

true panels.

with

estimators Our

work

groups

individuals into cells thatare both homogeneous and large. a

duces

inherent

the aggregation

Second,

systematic

heteroscedasticity.

in pseudo-panel can be This

data

corrected

exactly by decomposing the data into between and within di mensions and computing the exact heteroscedasticity on both dimensions.

But

because

the heteroscedastic

factor

on

depends

can

result

in serious

errors

estimation

to, cell

size.

Thus

the

least

squares

coefficients

on

computed

the grouped data may differ slightly from those estimated on individual data. As described in the next section, this approxi mate and easily implemented correction uses GLS on thewithin and

between

dimensions

as computed structure eroscedastic

with

a common

the between

matrix,

variance-covariance of

transformation

the het

to aggregation.

due

Third, unmeasured heterogeneity is likely to be present in both

panel

and

pseudo-panel

data.

In

the case

of panel

data,

the individual-specific effect for household h is a(h), which is assumed

to be

constant

through

time.

In

the case

of pseudo

panel data, the individual-specific effects for a household (h) belonging to the cell (H) at period t can be written as the sum of two orthogonal effects,a(h,t) = ?(H) + v(h, t). Note that the second component depends on time,because the individuals composing the cell H change through time. The specific effect ?jlcorresponding to the cell H [?(H)] rep resents the influence of unknown explanatory variables W(H), constant through time, for the reference group H, which is defined here by the cell selection criteria. Here v(h,t) are individual-specific variables

planatory

effects Z(h,t).

effects containing In the pseudo-panel

gregated specific effect ? (H) for the cell H aggregation of individual-specific effects,

of unknown data,

ex

the ag

is defined as the

a part

cancels

aggregation

of

Therefore, with panel data, thewithin and thefirst-difference operators suppress all of the endogeneity biases. With pseudo data,

panel

the same

operator

the endogeneity

suppresses

due

to ?i,but not thatdue toJ2 Y (h, t) x v(h, t). For each individual, this part of the residual may be smaller relative to ?i, because cell homogeneity is increased. Conversely, the aggregation into cells is likely to cancel this same component v across individ uals, so it is not easy to predict the effectof the aggregation on the endogeneity bias. Our search for robust results is facilitated by the fact that two

that we

dataseis

panel

use

cover

different

extremely

societies and historical periods. One dataset is from theUnited States for 1984-1987, a period of steady and substantialmacro economic growth. The other is from Poland for 1987-1990, a turbulentperiod that spans the beginning of Poland's tran sition

from a command

to a free-market

economy.

3. SPECIFICATION AND ECONOMETRICS OF THE CONSUMPTION MODEL

and

Gardes,

(Gurgand,

because

data,

aggregated

this effect.

time, correcting it by generalized least squares (GLS) makes individual-specific effects vary with time, thus canceling the spectral decomposition inbetween and within dimensions. This Bolduc 1997). The approximate correction of heteroscedastic ity thatwe use involves weighting each observation by a het eroscedasticity factor that is a function of, but not exactly equal

April 2005

Statistics,

operators, whereas the individual effect v(h,t) may be largely eliminated by the aggregation. Thus it can be supposed that the endogeneity of the specific effect is greater on individual

the

pro

& Economic

The within and first-difference operators estimated with panel data cancel the individual-specific effectsa(h). The com ponent ?i(H) is also canceled on pseudo-panel data by the same

than on

The answer to the efficiency problem is to define group ings that are optimal in the sense of keeping efficiency losses

Journal of Business

Data

constraints

force

us

a demand

to estimate

system

on

only two commodity groups over a period of 4 years: food con sumed

at home

from-home estimates

and

food are

food

consumed

not

very

reliable,

from

away

are rare

expenditures

home.

in the Polish

but we

keep

them

data,

Away so our

to compare

estimates. We use the almost-ideal demand system (AIDS) developed by Deaton and Muellbauer (1980), with a quadratic form for the natural logarithm of total income them with PSID

or

expenditures

to take nonlinearities

into account.

that the true quadratic system proposed by Banks, Blundell, and Lewbel (1997) implies much more sophisticated econometrics if the nonlinear effect of prices is taken into ac count. Itmay be difficult to precisely estimate the price effect because of the short estimation period in the PSID data. On the other hand, our Polish dataset contains relative prices for food that change both across 16 quarters and between 4 so Note

cioeconomic

classes.

Thus

we

estimated

the linearized

version

of Quadratic Almost Ideal System (QAIDS) (with the Stone index) on the Polish panel, using the convergence algorithm proposed by Banks et al. (1997) to estimate the integrability parameter e(p) in the coefficient of the quadratic log income [see (1)]. We obtained very similar income elasticities for food consumed at home and food consumed away from home to those obtained by the linearAIDS model. The AIDS estimates for both countries are presented in Tables 1 and 2. The ad ditivity constraint is automatically imposed by ordinary least

squares (OLS). The possible correlation between the residuals of food con t) xa(h, Y?,Y(h, sumed at home and food consumed away from home would where t indicates theobservation period and y is theweight for suggest the use of seemingly unrelated regression (SUR). the aggregation of h within cells. Note that the aggregate, but We tested this possible correlation on Polish data and found no significant difference between OLS and SUR estimations. not individual-specific, effects depend on time. = ?(H, t) ^y(h,

= t) ?(H)+

t) x v(h,t),

et al.: Estimation

Gardes

245

of Food Consumption

Elasticities

1. Income Elasticities

Table

at Home

forFood

Food away Not instrumented Instrumented Whole population Robust estimation

Pseudo-panel Food at home

Age,

.963 (.019)

.872 (.039)

and

away

1.387

.265 (.056)

1.265

(.068) (.060)

1.384

1984-1985-1986-1987 90 Log of age and itssquare,

Surveys n

(.043)

.062 (.063)

.375 (.166) .387 (.166)

.431 (.260) .447 (.258)

.240 (.095) .460 (.062)

.382 (.114)

(.083) 1.125

.800 (.150) (.127)

.847 (.172)

scale

and

its square

Control

variables

NOTE:

of totalexpendituresby theusualmethod. The values inparentheses are standarderrorsadjusted forthe instrumentation

relative

do

prices

not

waves

within

vary much

of

equivalence

of prices, shocks

variations

other macroeconomic

we

account

with

for price

survey

year

he or she was

Table 2.

Elasticities

forFood

at Home

Between Panel Food at home

Not instrumented Instrumented

Food

away

Not instrumented Instrumented

and Away From Home: Cross-sections

Polish

Surveys

(1987-1990)

Within

First differences

.466 (.005) .755 (.013)

.451 (.010) .788 (.028)

(.073) (.143) 14,520

1.239 (.164) 1.326 (.329) 14,520

2.618 (.556) 4.195(1.080) 14,520

1.460 (.185) 1.315 (.393) 10,890

.583 (.011) (b) .591 (.010) (c) .591 (.011) (d) .589 (.013)

.572 (.017) .584 .581 (.018) .581 (.018)

.549 (.020) (.022) .526 (.020) .965 (.023)

.864 (.033)

.820 (.203)

.890 (.258)

-.218 (.318)

.696 (.331)

level, location, Log of age, proportion of children, education and yearly dummies price forall commodities, cross-quarterly

log of relative

(not instrumented)

(a)

Food

is the expenditure budget share on good /by house wlht at hold h time t, Y ht is its income in the case of theU.S. data or logarithmic total expenditure in the case of the Polish data,

where

.536 (.009) .567 (.023)

1.119 1.216

variables

Pseudo-panel Food at home

(1)

(.004) (.010)

.579 .494

n Control

= d + blMYht/p,)+ + ZhA1+ u\v c'/eipnHYht/pt)]2

effects

surveyed.

Total Expenditure


30%).

from our various

models

are

produced

in their purchasing

pseudo

of the sample.

has

heteroscedasticity

been

corrected

panel

data.

The

between

and

cross-sectional

estimates

are similar for the differentcorrection methods, especially for food

at home,

but

the within

and

first-differences

tained under the false correction, which estimations,

pseudo-panel those rection,

computed or no correction.

Thus

estimates

estimates

than

the exact

correction,

correction

ob

is currently used in

different

very

gives

for the approximate

cor

for heteroscedasticity

seems to be an importantmethodological point in the estimation

on

data. The false correction gives very different for time series estimations. in However, especially our case the exact correction rise to estimated gives parameters to those obtained so we close correction, by the approximate pseudo-panel

estimates,

principally discuss these estimates thatcan be easily compared under the spectral decomposition into the between and within dimensions.

Looking first at the PSID results for at-home food expen ditures, it is quite apparent that elasticity estimates are very sensitive

to adjustments

error

for measurement

Cross-sectional

heterogeneity.

estimations).

the cross-sections Pseudo-panel cross-sectional

in Tables

data,

pseudo-panel

robust

of very-low-income

presented

4%

represent

estimates

and

unmea in

of at-home

come elasticities are low (between .15 and .30) but statistically significantwithout or with instrumentation (when performing

5. RESULTS Estimates

observations

Rejected

by the approximate method (GLS with a heteroscedasticity fac tor 8h, that is constant through time) [Tables 1 and 2, (a)], and the exact method [Tables 1 and 2, (b)] presented inAppendix A. Also given inAppendix A are the estimates obtained without correction [Table 2, (c)] and with a false correction (GLS with a heteroscedasticity factor 2/y/n,where n is the number of ob

re

and

Journal of Business

1

(PSID) and 2 (Polish surveys). Respective columns show in come forPSID and total expenditure for thePolish data, along

The and

data

between

also

effectively

low estimates

produce similar

produces

estimates.

estimates

elasticities some

Despite

average

of elasticities. for between

variations

among

and the

different estimations, the relative income elasticity of food at home is around .20 based on this collection ofmethods. Within and first-difference estimates of PSID-based in come around

elasticities .40 with

are

around

instrumentation.

0 without A

instrumentation

Hausman

test strongly

and re

jects (p value < .01) the equality of within and between esti with estimates for between, cross-sectional (com elasticity mates (Table 3). The test compares within and GLS estimates; as the means on of cross-sectional estimates obtained puted equivalently, it can be built from thewithin and between es each cross-sectional and first-difference mod survey), within, timates (see Baltagi 1995, p. 69). The test is computed by are also for models els. Results in which the usual quadratic form,with a chi-squared distribution,with presented separately income or total expenditure is and is not instrumentedusing the V defined as the variance of the difference between the esti models detailed inTables A.2 and A.4. We expect thebetween mators tested, = or (?b j8w),where ? ?wyCV~{)(?b (?iy) = to the average to be similar of cross-sectional estimates esti on for the estimation the Polish ? quadratic panel (?ly, ?^) mates. Compared with thewithin estimates, the first-difference and V = to all the variables. V?, + Vw corresponds

explanatory

Note that a testwith V as a matrix 2x2 computed only for the specific effectsmay be better taken into account whenever they two income variables would be biased. Because thewithin and firstdifferencemodels adjust forper change within theperiod. For the PSID, we eliminated some observations to obtain sistent heterogeneity and the instrumentationadjusts formea robust estimations using theDFBETAS surement of the estimate error, .40 is our preferred explained by Belsley, approximate

estimates

may

be biased

by greater

measurement

error, but

the

et al.: Estimation

Gardes

Elasticities

of Food Consumption

249

Table 3. Hausman

Without

Instrument

instrumental

Test for Income Parameters

variables

With

and

cross-sectional

413.7 343.4

estimates,

parameter

that

suggesting

failure to adjust forheterogeneity imparts a considerably down ward

to cross-sectional

bias

estimates.

PSID-based

across

the full set of pseudo-panel

are

on

computed

total

estimates.

so

expenditures,

they must

be

multiplied by the income elasticity of total expenditures, which is around .7, to be compared to the PSID income elasticities.) Higher elasticities are to be expected for a country in which food constitutes a share of total expenditures that is three times higher than that in the United States (Tables A.l and A.3). Their consistency probably stems from the smaller degree of measurement error in the Polish expenditure than in the PSID income

data.

Before were

10 households was

On

of food

the panel some

pressing

The

suppressed.

the prediction

mates.

were

the households

robust

data,

outliers

for suppression

in cross-sectional

estimations

similar

gave

produced

by

to those

results

a

little

smaller

than

estimates

on

based

income are

expenditures to specification timates.

elasticities

for

away-from-home even more

from

and

than the at-home

food

expenditure

quite

Between

and

cross-sectional

food

es

around

1.0

in both individual and pseudo-panel data. In contrast with the case

of

at-home

expenditures,

for heterogeneity

adjustments

through use of within and first-differenceestimates produce much lower estimates.We speculate on why thismight be the case in Section 6. Polish the survey

food-away-from-home relatively

expenditures are very low

and

rarely

are even

reported when

in

com

pared with those for other countries with comparable income levels.

teens with

about

Moreover,

served periods and

70%

cafeterias.

with

in the ob

expenditures

is spent on highly subsidized business can Thus

the estimations

the food-away-from-home

countries

these

of

should

be

compared

estimation

expenditure

in other

caution.

Price data inPoland enabled us to compute price elasticities. Quarterly

price

indices

for four

puted from themonthly GUS

social

categories

(Polish Main

from

time

were

of the

estimates

good

com

Statistical Office)

es

the cross-sectional

elasticities,

series.

As

prices

between

change

quarters

and house

levels.

a correlation

Such

is less

than

probable

the correlation

between relative income and the specific component of food consumption (which produces the endogeneity bias on income elasticities). However, if systematically high (or low) price lev are

els

correlated

with

household

type

(e.g.,

as

in segmented

markets), and if these types of households are characterized by a

systematically

positive

or

negative

specific

then

consumption,

the endogeneity bias can appear. This is not the case forPoland in 1987-1990.

6. DISCUSSION have

estimates

expenditure to adjust

Failure cases,

to assess

attempted

elasticity

food

sensitive

elasticity are

estimates

the income

error and unobserved

the microdata.

different

with

sup

obtained

elasticities.

PSID-based

provides

holds of different types, the only explanation of an endogeneity bias would be a correlation between household types and price

We

the estimations on Polish data also produced higher within and first-differenceelasticities than between and cross-sectional

categories

timates of direct price elasticities are close to those obtained

ture

the whole,

On

social

esti

for thewhole panel. Time series Polish pseudo-panel estimates

were

over

and

into cells,

grouped

criterion

consumption

time

In contrast

pseudo

Expenditure elasticities for at-home food estimated with the Polish data are much higher in value than the income elastic ity estimates based on PSID data. (Note that the Polish elas ticities

publication Biuletyn Statystyczny and imputed at the individ ual level to the dataset. The variability of the prices both over direct compensated elasticity for food at home (around ?1).

panel data also produce significantly (according to a Hausman test) higher elasticity estimates for first differences compared with between and cross-sections models, although there is little consistency

variables

46.8 .5 292.4 2.2

income elasticity of at-home food expenditures in theUnited States. This is roughly double the sizes of the corresponding be tween

_^anei_ Pseudo-panel without instrumental

instrumental variables

107.7 PSID Poland 336.6

Poland (QAIDS)

(food at home)

measurement

the bias

caused

by

direction

income

for estimates

of

expendi

adjustments operate in the

the at-home

ticityfound in thePolish data. In the case

and

to measurement

In the case of U.S. at-home heterogeneity. our estimate around .40. is elasticity, preferred some for unmeasured in and, heterogeneity error appears to impart a substantial down

ward bias to this estimate. These same

on

inattention

of U.S.

food

away-from-home

elas

expenditure

our

expenditures,

preferred elasticity estimate is less certain but similar inmag nitude to the .40 elasticity for at-home expenditures. (Note that it is considerably higher in pseudo-panel estimate.) Surprising here is themagnitude and sign (upward) of the apparent bias inherent in both individual and pseudo-panel estimates thatdo not adjust for unobserved heterogeneity. should unmeasured heterogeneity induce an upward

Why bias

in away-from-home

in the U.S.

expenditures

and

Polish

data? Earlier, we speculated that a likely downward bias for the at-home food elasticity estimates may be caused by fail ure to account for the fact that the value of time differs across households served

and is positively related to the household's

income.

The

time

input

to producing

at-home

ob

meals

leads households to face different (full) prices of consump tion even if the prices of the goods-based inputs are identi cal. If these prices are positively associated with income and themselves

have

a

negative

effect

on

consumption,

then

their

omission will impart a negative bias to the estimated income elasticities.

Note

that the bias

seems

somewhat

less pronounced

Journal of Business

250 on

The

data.

pseudo-panel

income

between

correlation

and

In the case there

rants, service

spent

food

and

restau

full-service

consuming

restaurant

fast-food households

this case higher-income to such time value positive to control for this source Failing

a more

attach

households.

low-income

in

but

meals,

well

may

of heterogeneity will probably impart a positive bias to esti mates of income elasticities. (Note that the endogeneity bias is much more pronounced for food away fromhome than for food at home.)

The use of time can be measured with the shadow price of

time. This

shadow

household

with

increases

price

so

income,

that the complete price for food, computed as the sum of the and

monetary distribution. and

time

shadow

prices,

series

increases

the difference

Therefore,

can be

elasticities

income

also

the

along

between

income

cross-sectional to the change

related

of this complete price. The argument is formalized inAppen dix B. Calibrating the price effect as half of the income effect (as suggested by Frisch) and using (B.2) inAppendix B pro duces an estimate of the income elasticity of the food shadow price (Table 4). It is remarkable that these elasticities are simi lar in the two countries, positive and around 1 for food at home straint

larger

for food

stronger

relative

and much

and negative

much

imparts

than at home,

in restaurants

because

parts

the ratio of their complete

when

a similar

changes

prices

but greater

in the expenditures,

change

cross-sectional as

im

changes

in the food away fromhome budget share. Elasticity estimates from thePolish data also proved some what sensitive to adjustments for heterogeneity. This is not surprising given the very differentprice regimes in the Polish economy during this period, with one (presumably low) set of prices faced by themany farm families who have the option of growing theirown food; official, subsidized prices set by the Polish government; and higher black market prices for the same such

grew

food

a

were

as bread, for

their own

set so low

that very

consumption,

queuing

was

not

problem.

Table

4.

Income Elasticity

of Food

PSID

Shadow

Incomeelasticity

.19 Food at home Food away from home Direct price elasticity Income elasticity of the shadow price

1.00 (FH/FA) (a) FH (b) FA

-.19 1.00 .71 -4.78 -3.13

three

from

we

model,

2-year

independent

our

treated

panels

inclusion

of the short-run

the values

changed

income

elasticity

very little. An important result of thiswork is that pseudo-panel esti are often

mates

to estimates

close

on

based

data.

panel

genuine

Large and similar apparent endogeneity biases were found in both

countries.

estimations

Cross-sectional

elasticities

produce

that are systematically higher for food at home and lower for food

from home.

away

lowers

the endogeneity

search

should

data.

expenditure

for food

bias

to verify

aim

it seems

Moreover,

Once

these

that the aggregation

re Future consumption. on a set of complete

results

for the biases,

corrected

elastic

income

ities for food at home and food away from home become very

close

to each

other

in the U.S.

a result

data,

seems

that

rea

sonable to us and highlights possible errors that can arise from estimations

using

data.

cross-sectional

ACKNOWLEDGMENTS The authors gratefully acknowledge helpful suggestions from participants at seminars at the University of Michigan, Northwestern University, Universit? de Paris I, Universit? de Gen?ve, Erudite, CREST, Journ?es de Micro?conomie Ap pliqu?e (1998), Congress of the European Economic Associ ation,

and

International

on Panel

Conference

as

as well

Data,

The research support from INRA, INRETS, and CREDOC. Polish data were made available by Professor G?recki, Uni versity

of Warsaw,

thankCREST,

of Economics.

Department

INSEE, andMATISSE

The

authors

for support.

APPENDIX A: HETEROSCEDASTICITY THE PSEUDO-PANEL MODEL Write

the pseudo-panel trix form as

model

also

presented

IN

in the text in a ma

where y is anNT column vector with N the number of cohorts

panel

and T

3,630 By social category .38 .39

ver

dynamic

y = X/? + Za + e,

1987-1990

TS

CS

Polish

a

diture and found that the coefficient on the lagged dependent variables were significant in these dynamic models, but their

Prices

(U.S.)

Period 1984-1987 n2,430 No Prices

came

they

though

few farm families

when

and

(1984-1985 through 1986-1987 for the PSID and 1987-1988 through 1989-1990 for thePolish expenditure survey).We es timated this dynamic model as a system using SURE and both with and without instrumentationfor log total income or expen

or similar products. In fact,until 1988 the official prices of sta ples,

of our dynamic

estimates

the budget

shares of food away are much smaller in both countries (espe cially in Poland, where the ratio of the price elasticities is the largest). A substitution between food at home and food away from home

model

consumption

this by estimating

investigated

in expenditures

changes

is normal

which

We

persistence.

the time con

Thus

away.

a static

April 2005

riskof bias due to theomission of dynamic factors, such as habit

data

is typically longer than time spent consuming

rant meals

than

Time

components.

on

Statistics,

sion of ourmodel that included lagged consumption. To obtain

in restau

consumed for meals of expenditures are of in the mixture variations large

are based

estimates

Our

spe

cific effectmay be decreased by aggregation, as we suggested in Section 2.

& Economic

CS

TS

.49 1.22 -.38/-.18

.76 .36

the number

of periods,

X

is a matrix

of explanatory

vari

ables, ? is a vector of parameters, Z is an (NT x N) matrix that contains

cell

dummies,

a

is a vector

of cohort

effects,

and

e

is

a heteroscedastic residual. Call D a diagonal (NT x NT) ma trix such thatD = diag( 0 Budget share for food away from home Percent with away-from-home share = 0

In head's

age

In family size

-.097 (.047) -.093 (.051) .005 (.042)

Wage growth'Promotedf Wage growth*Promotedi_i Wage growth*Promotedi_2

.043 (.074) -.149 (.075) -.035 (.069) .037 (.021)

.034 (.028) -.016 (.026)

-.054 (.026) .138 (.006) .012 (.003) .022 (.003)

City size>3 head Education Wage Wagef_-|

Deviations

Level .484

(.14)

(.15) 100

100

.006

(.02)

(.03)

10.65

(.45)

3.789

(.33) 1.140

(.59)

The values inparenthesesare standarderrors.

Used

in the Polish

(.14)

(.02)

(.16)

.020

-.019

(.24)

Level

Dit

.554

.068

(.18)

(.17)

(.15)

(.17)

100 .005

(.02)

12.25

( 02)

(-62)

(.32)

( 15)

1.095

(.61)

.005

(.03)

.0002

( 03)

20.5 -.18

(.79)

3.824

100

100

-.001

26.9 .50

(.32) (.60)

1990 Dit .003

100

(.38)

1.121

Analyses

.486

-.001

(.49)

3.809

Level

-.024

29.7 11.17

Panel

1989 Dit

100

.006

28.4

of Variables

1988 Dit

.508

for

expenditure

Wage growth*Layofft Wage growtrfLayoff^-i Wage growth*Layoffi_2

Region-|_2 Region>3 City size-|_2

1987 Level

In household

error)

The values inparentheses are standarderrors.

Table A.3.

Budget share food at home

for

.001 (.015) .073 (.004) -.0007 (.00004) -.063 (.033) -.003 (.035) -.005 (.032)

Age head head squared

Age

.037 (.029) .202 (.030)

Marriage

(standard

Birth

.050 (.024) -.005 (.022) -.005 (.016)

growtht Wage Wage growthf_-| Wage growtht_2 Divorce

variable

Independent

-.271

hrsf_i hrsf_2

Unemp Unemp Unemp

Level

Coefficient

error)

.025 (.022) .047 (.021) .017 (.021)

Promoted^

NOTE:

51.3 .033

3.7801

(.013)

(.081)

.002

(.686)

.0267

(.359)

-.003

.033

10.1238

(.320)

3.7573

(.013)

.134

(.096)

10.3

.1729

10.1714

(.280)

.008

(.082) 41.5

5.5

9.6

.0731

(.657)

3.7044

head

5.7

.137

-.003

(.033)

Dit

Level

(.100)

74.0

.001

9.9254

(.086)

0

53.2

1987 Dit

Level

-.015

(.095)

.034

(.648) Inage

.129

(.084)

.033

9.5

Analyses

1986 Dit

Level

-.003

0

0

Budget share for food away from home

In household

.147

in the PSID

Used

1985 Dit

Level

(.103)

for

of Variable

Deviations

1984

1983 Level Budget share food at home

251

of Food Consumption

.014

-.026

(.21)

14.14

-.03

(.50)

(.58)

(.32)

(.15)

3.842

1.081

(.61)

.019

-.014

(.22)

Journal of Business

252 Coefficient and Standard Errors for Instrumental Regression for Level and Change for Total Expenditure Variables Equation Panel the Polish Expenditure

Table A.4.

Coefficient variable

Independent

(standard

Log Children

Log age Location

-.270

in family

percent

-.0299

squared

1.704

age

Log

(Ref: countryside) Large

.028

city Average Small city Social category . (Ref: wage earners) earners-farmers Wage

.017

-.096

Two

.004

income squared

-.028

sectional

D

matrix

=

]8GLS analysis,

between

and within

be

as

expressed which

estimators,

sum

a weighted

is called

of

decom

spectral

Gurgand et al. (1997) showed that this can be the case only ifmatrix Bf? is symmetric,which implies that theweights inD are time-invariant.The intuition of this result is that if, in the effects

individual

the variances,

scaling

no longer

transformation

a

to the mation

is no

longer

over

time,

thus

is constant

individual

resolves

heterogene

possible.

In contrast,

this argument

does

as

long

not hold,

as cell and

size

the de

obtains.

composition

A corollary is that the efficientwithin estimator cannot be based simply on the original model weighted by heteroscedas ticity

factors,

time-varying

because

this would

create

time

varying cohort effects.Gurgand et al. (1997) showed that the within estimator that is both efficient and consistent when a

are correlated

and X

?w where when

A

=

D"1

the number

= -

is

(X/WAWX)"lX/WAWy,

is de all

contains

the residual

effect

variables

an

?

cross

o?ih +

?iht- The

a correlation by

between

some

specific

effect

Sint: utht

can be biased

Znt

this

and

1978). Such a correlation is due to latent per (e.g.,

event

during

characteris

infancy,

specific of

effect

permanent

variables

the explanatory

to the between

and

transfor