ERRORS IN SURVEY REPORTS OF

CONSUMPTION EXPENDITURES

Erich Battistin

THE INSTITUTE FOR FISCAL STUDIES

WP03/07
Errors in Survey Reports of Consumption

Expenditures

Erich Battistin∗

Institute for Fiscal Studies, London

6th May 2003

Abstract

This paper considers data quality issues for the analysis of consumption

inequality exploiting two complementary datasets from the Consumer Expenditure Survey for the United States. The Interview sample follows survey

households over four calendar quarters and consists of retrospectively asked

information about monthly expenditures on durable and non-durable goods.

The Diary sample interviews household for two consecutive weeks and includes detailed information about frequently purchased items (food, personal

cares and household supplies). Each survey has its own questionnaire and

sample. Information from one sample is exploited as an instrument for the

other sample to derive a correction for the measurement error affecting observed measures of consumption inequality. Implications of our findings are

used as a test for the permanent income hypothesis.

Keywords: Consumption Inequality; Measurement Error; Permanent Income Hypothesis

JEL Classification: C13, C42, D12, D91

∗First draft 11th February 2002. This paper benefited from useful discussions with Gordon

Anderson, Orazio Attanasio, James Banks, Richard Blundell, Martin Browning, Hide Ichimura,

Arthur Kennickell, Costas Meghir, Bruce Meyer, Enrico Rettore, Guglielmo Weber and from

comments by audiences at Padova University, Cemmap (London), ESPE 2002, 10th International

Conference on Panel Data and NBER Summer Institute 2002. Address for correspondence: Institute for Fiscal Studies, 7 Ridgmount Street, London WC1E 7AE - UK. E-mail: erich b@ifs.org.uk.

1
EXECUTIVE SUMMARY

This paper aims to quantify the effect of reporting errors affecting diary-based

and recall-based data on non-durable expenditure. It is likely that not all the

commodities entering non-durable expenditure are well reported exploiting only

one of these two survey methodologies. Expenditures on frequently purchased,

smaller items are presumably more accurate using diaries while recall data are more

appropriate for large expenditures or expenditures occurring on a regular basis.

It turns out that neither diary nor recall-based data alone provide a reliable

aggregate measure of total expenditure on non-durables. Ideally, the estimation of

totals at micro-level would require information on different consumption categories

obtained with the most appropriate methodology. The Family Expenditure Survey

for the United Kingdom represents a notable implementation of this strategy.

One might argue that for any practical purpose these alternative data collection strategies lead to consistent results in the estimation of economic models of

consumption behavior. Unfortunately, evidence from the literature suggests that

conclusions are strongly related to the information being used. This problem is addressed by looking at micro data from two independent samples of households from

the Consumer Expenditure Survey for the Unites States. This survey represent a

unique source of data because it consists of diary an recall information collected on

the same set of items, although it refers to separate samples of households.

The comparison of the two surveys based on the overlap in coverage of expenditures offers insights for the effects of collection modes on data quality. Pictures

emerging from the two surveys are very different, both with respect to mean expenditure and, more importantly, with respect to indices of inequality. More precisely,

while the differences in mean expenditure are roughly constant over time, consumption inequality presents different levels and trends in the two surveys. This evidence

is reconciled using integrated diary and recall information to characterize the most

likely pattern of consumption for cohorts of individuals defined by their age.

This procedure allows us (i) to define an improved measure for mean and variance of non-durable expenditure over the 1990s and (ii) to characterize the measurement error affecting the commodities whose quality is doubtful according to

other studies in the literature. The implications of our findings for the estimation

of inequality indices are discussed, with an application to permanent income models. In particular, we show that using diary and recall data to improve the quality

of household consumption the permanent income hypothesis cannot be rejected.

2
1 INTRODUCTION

Data quality is an issue of longstanding concern among researchers interested in

testing the implications of theoretical models of consumption over the life cycle.

The empirical analysis of these models requires reliable micro-data on expenditures at household or individual level. In many countries expenditure data are

regularly collected either by diaries covering purchases made within a short period

of time (typically one or two weeks) or by means of retrospectively asked questions

on the usual spending over a longer period.

There is a consensus that the time-consuming task required by diaries produces

good quality expenditure data for small items, while recall questions should be

asked for bulky items (major consumer durables: real property, automobiles and

major appliances) or for those components either having regular periodic billing or

involving major outlays (such as transportation or rent).

For this reason diary surveys are designed to obtain detailed recordings of expenditures on small, frequently purchased items which are normally difficult to

recall. Such an idea is not only intuitively clear, but it is also supported with evidence from cognitive studies and from the comparison of aggregated consumption

measures with national account data.

On the other hand, the drawback of such an evidence is that neither diary nor

recall based data alone provide a reliable aggregate measure of total consumption.

One might argue that for any practical purpose these two alternative designs lead

to consistent results, but unfortunately this is not the case. For example, there

is some evidence that recall consumption data lead to potentially misleading results in analyzing household saving behavior (Battistin et al. 2003). Other studies

demonstrate how available consumption data can be unsuitable for the analysis of

the permanent income (life cycle) hypotheses and how adjustments provide greater

consistency concerning the time series properties of consumption (Wilcox 1992 and

Slesnick 1998). Browning et al. (2002) provide a detailed discussion of these problems and review alternative survey methodologies that have been suggested to obtain reliable measures of consumption.

Ideally, the estimation of totals at micro-level would require information on

different consumption categories obtained with the most appropriate methodology.

The Family Expenditure Survey for the United Kingdom represents a notable implementation of this strategy. It consists of a comprehensive household questionnaire

which asks about regular household bills and expenditure on major but infrequent

purchases and a diary of all personal expenditure kept by each household member

(including children) for two weeks. There is some evidence - at least for the United

Kingdom - that consumption measures obtained from such a design are comparable

to aggregated values from national accounts (Banks and Johnson 1998).

Because of time constraints and survey practice, questionnaires can not cover

all the aspects of consumer behavior with the same level of accuracy. Since the

collection of diary records is highly time-consuming in itself, the issue arises of

whether consumption information based on recall questions is of comparable quality

to information based on diary records.

A related problem is the characterization of the response error across different

consumption categories in diary and recall data. Retrospectively collected information of household surveys are typically characterized by recall errors. For example,

respondents might round off the true measure causing abnormal concentrations of

values in the empirical distribution. Neter and Waksberg (1964) discuss the relative

3
importance of forgetfulness and telescoping in expenditure data from household interviews. See also Browning et al. (2002) for a review of data quality issues related

to the collection of consumption information.

It is often assumed that response errors in the variable of interest are of classical

form, so that they have limited impact on parameter estimates (if consumption is

used as a dependent variable) or can be accounted for by using suitable econometric

techniques depending on the model specification (see for example Lewbel 1996 and

Hausman et al. 1995). However, this assumption is made for ease of estimation

rather than for any theoretical conviction. In fact, Bound et al. (2001) review the

state of the art about measurement error in surveys across a wide range of areas in

economics and provide evidence that standard assumptions are likely to be violated

to various extents.

In the presence of validation data, one might be able to correct biases and derive

consistent estimates from primary data without further assumptions on the error

structure (see Lee and Sepanski 1995 and the references therein). But again this

would require the availability of complementary datasets, ideally with the same

individuals or, at least, with a set of common information rich enough to motivate

matching procedures (see Ziliak 1998 and Battistin et al. 2003 for recent applications).

The aim of this paper is to shed light on the comparison between recall based

and diary based data on household consumption using micro-level data from the

Consumer Expenditure Survey for the Unites States (CEX in the following). This

survey consists of two different components: a quarterly Interview Survey (IS) and

a weekly Diary Survey (DS), each with its own questionnaire and sample. The

most interesting feature that makes the CEX an extremely appealing source of

data is that the IS and the DS overlap for many categories of consumption for

which information is collected using different methodologies.

According to the line of thinking discussed above, neither of these two surveys

provides accurate estimates of total consumption at household level. In fact, the two

survey components are explicitly designed to collect information on different types

of expenditures (see Bureau of Labor Statistics 2002). The IS aims to obtain data

on the types of expenditures respondents can recall for a period of three months or

longer; the DS is instead designed to obtain data on frequently purchased smaller

items. Accordingly, the Bureau of Labor Statistics (BLS in the following) publishes

data integrated from the two components to provide a complete accounting of

consumer expenditures, which neither survey component alone is designed to do.

Inconsistencies between the CEX and national accounts have been already

pointed out in the literature by several papers (notably by Slesnick 1998, 2001), suggesting that the quality of these data might have deteriorated over the last decade.

Figure 1 presents aggregate expenditure on non-durable goods using published tables from the CEX and from the Personal Consumption Expenditures (PCE).1 The

two series follow a similar trend until 1992, with CEX expenditure peaking in 1989.

However, CEX data do not appear to pick up the trend in total consumer spending found in the PCE over the 1990s. Slesnick (2001) finds that only part of this

discrepancy can be explained by definitional differences, concluding that “the remaining gap is a mystery that can be resolved only by further investigation” (page

52).

1We are grateful to David Johnson at the BLS for making this graph available to us. The

contents of this figure are comparable to those of Figure 3.2 in Slesnick (2001; page 51), although

the latter figure looks at total expenditure on durable and non-durable goods.

4
 CEX PCE

85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

14000

16000

18000

20000

22000

24000

26000

28000

Figure 1: Non-durable expenditures in 2000 dollars - Consumer Expenditure Survey

(CEX) and Personal Consumption Expenditures (PCE)

Despite of the puzzle implied by these findings, the IS is currently the most

widely used source of consumption data for the United States: very many authors

have used it to validate different theoretical constructs of economic behavior over

the years (see, for example, Attanasio and Weber 1995 and Krueger and Perri

2001). The most appealing features of the IS are primarily its panel component

(individuals are interviewed every three months over five calendar quarters) and

the richer amount of household information collected with respect to the DS. On

the other hand, although the DS exists in its current format since 1986, it hasn’t

received so much attention by researchers so far (the only example we are aware of

is Sabelhaus 1996). Actually, the DS is still a relatively unknown source of data.

Given the overlap between the IS and the DS for many categories of consumption, and given that the IS is explicitly designed to collect good quality information

only on a subset of these categories, the question then arises of whether we can

jointly exploit DS and IS data to derive a superior measure of total consumption.

The answer to this question has very many empirical implications. As already

pointed out by Wilcox (1992), the imperfections of micro-data on consumption expenditures may be important enough to influence the conclusions of empirical work.

Are our data relevant to the theory? Is the economic model really in error? Should

research be directed towards alternative models of economic behavior or is data

itself not suitable to validate existing models?

We will address these issues presenting three sets of results. Firstly, new evidence on the evolution of consumption inequality for the United States in the

last twenty years is presented using data from the DS. Consumption inequality for

the United States has recently received much attention amongst researchers, since

according to IS data it does not appear to have grown much during a period characterized by a marked increase in income inequality. This result has also generated

5
discussion on the appropriate measure of economic well-being in the evaluation of

inequality.

Secondly, the amount of misreporting of total non-durable expenditure characterizing the IS is discussed exploiting information from the DS. Since the two

surveys refer to different samples, individual level consumption cannot be straightforwardly defined from combined IS and DS data. For this reason, we will mainly

focus on figures for mean expenditure on different commodities for cohorts of people

identified by their year of birth or, equivalently, by their age in a base year.

Finally, the available information from the IS and the DS allows us to examine

to what extent tests of the permanent income (life cycle) hypothesis are sensitive to

the choice of consumption measure. Our motivating example is the work by Deaton

and Paxson (1994), where they examine the evolution over time of the variance of

total non-durable consumption at cohort level as a test for the original formulation

of permanent income theory (Hall 1978).

The main findings of this paper can be summarized as follows. Firstly, the

pattern of consumption inequality obtained from DS data is very different from

the pattern largely discussed in the literature using IS data (see for example the

discussion in Krueger and Perri 2001). DS inequality seems to grow over time, and

particularly over the last ten years when there is no evidence of increasing inequality

using IS data. A correction procedure is proposed to reconcile the evidence from

the CEX and to characterized the most likely pattern of consumption inequality.

This issue is further developed by Attanasio et al. (2003).

Secondly, we show that the quality of IS information has worsened with respect

to frequently purchased smaller items, housekeeping supplies and personal care

products and services (that is those items the DS is designed for). This decline is

particularly accentuated for the last ten years. On the other hand, we show that DS

data quality has improved also for all those non-durable items which presumably

are better described using IS data.

Finally, we point out that data quality issues in the IS should deserve more

attention in the evaluation of alternative constructs of economic behavior. We

emphasize this point showing that the permanent income model as formulated by

Hall (1978) cannot be rejected combining IS and DS data to improve the quality of

reported consumption. The same test applied to IS data over the period covered

by our analysis leads to different conclusions.

The remaining of this paper is organized as follows. Section 2 describes the

two surveys and compares descriptive statistics of household characteristics already

found to be relevant for data quality in previous studies of expenditure surveys.

Section 3 discusses the economic model used as a motivating example. Section 4

presents a puzzle implied by the comparison of means and inequality indicators

of non-durable expenditure exploiting information from the two surveys. Section

5 analyzes such discrepancies looking at the contribution of different non-durable

goods. Also, the identification restrictions needed to combine IS and DS information

are presented. Section 6 presents results on the reporting errors affecting nondurable components. Section 7 discusses the testing procedure to validate the

permanent income model using combined information from the IS and the DS.

Results from this test are presented in Section 8. Some more technical comments

on data collection issues and data problems are discussed in the Appendix.

6
2 DATA

The main characteristics of the two survey components from the CEX are summarized in what follows. In particular, Section 2.1 describes diary and recall questionnaires. Section 2.2 discusses the extent to which the IS and the DS are comparable

with respect to sample designs, population coverage and information collected; also,

the definition of household total consumption is presented. Finally, Section 2.3 discusses the working sample considered in this paper. The reader interested in more

specific details on the survey methodology in the CEX is referred to Bureau of

Labor Statistics (2002).

2.1 The consumer expenditure surveys

The CEX is currently the only micro-level data set reporting comprehensive measures of consumption expenditures for a large cross-section of households in the

United States. Essentially, sample consumer units are households (literally, “all

members of a particular housing unit who are related by blood, marriage, adoption, or some other legal arrangement”, Bureau of Labor Statistics 2002), whose

buying habits provide the basis for revising weights and associated pricing samples

for the Consumer Price Index. Only one person responds for the whole consumer

unit, typically the most knowledgeable of expenditures in the family.

The survey consists of two separate components, each of them with its own

questionnaire addressing a different sample. The IS sample is selected on a rotating

panel basis targeted at 5000 units each quarter; DS data refer to repeated cross

sections of households (around 4500 per year) interviewed over a two-week period.

Response rates for the two components are reasonably good (around 80 percent).

In the IS, households are interviewed about their expenditures every three

months over five consecutive quarters. After the last interview households are

dropped and replaced by a new unit, so that - by design - 20 percent of the sample

is tossed out every quarter. Expenditure information is collected in the second

through the fifth interview; one month recall expenditures are asked in the first

interview only for bounding purposes. The percentage of households completing all

five interviews is about 75 percent, with single persons more likely to attrit.

Households are retrospectively asked for their usual expenditure via two major

questions. The first type of question asks for the weekly/monthly purchase directly

for each reported expenditure; the exact wording is “What has been your usual

weekly/monthly expense for ... in the last quarter?”. For non-durable goods households are asked to report their usual weekly expenditure only for tobacco products

and for food and non-alcoholic beverages consumed at home. The expenditure on

the latter category is obtained as the difference between the usual weekly total

expenditure at grocery stores or supermarkets and how much of this amount was

for non-food items (specified as ‘paper products, detergents, home cleaning supplies,

pet foods, and alcoholic beverages’). Expenditures on alcoholic beverages and food

away from home (but not food consumed on vacation) are referred to the usual

monthly amount.

The second type of question asks for expenses in the last quarter by a detailed

collection of expenditures on a list of separate goods (referred to clothing, food

consumed on vacation and entertainments). In either case, recall data are collected

by a trained interviewer asking questions and providing examples of items in each

category.

7
The DS is instead a cross-section of consumer units asked to self-report their

daily purchases for two consecutive one-week periods by means of product-oriented

diaries. Each diary is organized by day of purchase and by broad classifications of

goods and services. Respondents are assisted by printed cues and - whether it is

needed - by interviewers at pick-up. The percentage of households completing both

diaries is about 92 percent. Interestingly, the DS also collects recall information for

food and non-alcoholic beverages consumed at home as in the IS, implying that for

this category of consumption recall and diary information is available for the same

households.

Both IS and DS collect information on a very large set of household characteristics (demographics, work-related variables, education and race) as well as on

income and assets (using a twelve-month recall period). Income figures refer to total

before-tax family income in the last year. This information is subject to top-coding

in the CEX, but only for a small proportion of household in the two samples (less

than 1 percent in any year). However, income and assets data are known to be not

as reliable as the expenditure data: the amount of incomplete income reporters is

about 20 percent in the two surveys and missing values are currently not imputed.

For this reason many applications in the literature have combined consumption

information from the CEX to income information from complementary data sets

(see, for example, Lusardi 1996 and Blundell et al. 2002).

The two survey components are based on a common sampling frame: the 1980

Census for those households sampled in the 1980s and the 1990 Census for households sampled in the 1990s. Sample designs differ only in terms of frequency and

over sampling of DS households during the peak shopping period of Christmas and

New Year holidays. The BLS constructs weights to control for systematic aspects of

the sampling, to post-stratify by region, home ownership, household size and race

and to compensate for under coverage (relative to the Census-based estimates) of

people by age, race and gender.

2.2 On the comparability of the two surveys

As far as consumption is concerned, the BLS follows the standard international

procedure of exploiting both information from recall questions for more durable

items bought in the quarter prior to the interview and diary-based records of purchases carried out within a two-week period. In fact, as discussed above, the IS

and the DS are explicitly designed to collect different types of goods and services

and neither survey is expected to represent all aspects of consumption.

Since aggregate consumption is obtained from integrated survey data, the question then arises of which survey component provides more accurate information on

different items. When data are available from both surveys, the procedure followed

by the BLS determines the most reliable survey by comparing CEX figures to those

from other data sources - typically from the Personal Consumption Expenditures

(see Branch and Jayasuriya 1997 and McCarthy et al. 2002). Of course, a potentially interesting question is the extent to which external sources provide an

accurate representation of total consumption (Slesnick 2001).

However, some expenditure items are collected only by either the IS or the

DS. The IS excludes expenditures on housekeeping supplies (e.g. postage stamps),

personal care products and non-prescription drugs, which are instead collected in

8
the DS.2 On the other hand, the DS excludes expenditures incurred by members

while away from home overnight or longer and information on reimbursements (such

as for medical care costs or automobile repairs), which are collected in the IS.

Throughout the analysis, only figures for expenditure on non-durable goods and

services will be considered.3 The expenditure categories considered have been defined so that IS and DS definitions are comparable and consistent over time. The

definition of non-durable expenditure closely follows the one already given by Attanasio and Weber (1995): food and non-alcoholic beverages (both at home and

away from home), alcoholic beverages, tobacco and expenditures on other nondurable goods such as heating fuel, public and private transports (including gasoline), services and semi-durables (defined by clothing and footwear). In particular,

expenditure on health, education (which can be considered as an investment in

human capital) and mortgage/rent payments are excluded. To improve the comparability of the two surveys, DS figures for total consumption are considered only

after 1986 (see the Appendix for a discussion of data collection problems in the

CEX affecting the definition of non-durable expenditures).

Changes in survey instruments characterize the CEX over the time covered by

our analysis. For example, a new diary form with more categories and expanded

use of cues for respondents was introduced in 1991, based on results from earlier

field and laboratory studies. Moreover, changes in the wording of the IS question on

food consumption heavily affect the mean of reported expenditure on this category

over time (data have been adjusted to solve for this problem; see the Appendix for

more details).

Only expenditure figures for the month preceding the interview are considered

for the IS sample, thus leaving four observations for each household (one observation for each interview).4 Monthly expenditure in the DS is defined as 26/12 = 2.16

times the expenditure observed over two weeks, assuming equally complete reporting.

2.3 The working sample

The CEX has a long history: the first survey was ran in 1917-18. Before the new

ongoing structure initiated by the BLS in 1980, previous surveys were conducted

only every 10 to 12 years. The 1980 survey is the first year providing consumption

information on a continual yearly basis. Differences in the survey methodology

between the surveys conducted after 1980 and those conducted before result in

visible inconsistencies when compared to the national accounts (see Slesnick 2001).

The information exploited in this paper covers twenty years of data from the IS

and the DS between 1982 and 2001. However, public use tapes permit to integrate

data on non-durable consumption from both surveys only after 1986, since only

selected expenditure and income data from the DS were published before then.

In what follows, the family head is conventionally fixed to be the male in all

H/W families (representing the 56 percent and 53 percent of the whole sample for

2This expenditures contribute about 5 to 15 percent of total monthly expenditures in our data.

3Assuming preference separability between durables and non-durables, expenditure on nondurable goods and services is a relevant consumption measure. 4It has been found that expenditures for many items are reported more frequently for this

month than for earlier months (see Silberstein and Jacobs 1989). This could obviously mean

a partial recollection of past events (mainly less important purchases) increasing with longer

reference period and/or a telescoping effect for the month nearest to the interview.

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Table 1: T-statistics from propensity score estimates; dependent variable 1=Interview, 0

Variable 1982-85 1986-89 1990-92 1993-95 1996-9Proportion of components 18

−

Proportion of components 64+ 2.22 -3.15 Proportion of children 0

− 3

Proportion of children 4

− 7 1.94

Proportion of children 8

− 12 -3.00 2.86

Proportion of children 13

− 18 3.43

Age of the reference person 2.11

Dummy for retired head -6.13 -11.94 -13.01 -8.38 -4.90Weeks worked per year -6.00 -10.73 -12.04 -9.34 -8.20 Total amount of income before taxes 10.03 -7.78 9.10 6.35 Total amount of income before taxes squared -4.25 -4.01 -6.12 -3.58 Dummy for Midwest region 2.57Dummy for South region 4.15 2.39Dummy for West region Husband and wife (H/W) only -2.93 -4.24 -2.64H/W, own children only, oldest child 0

− 5 -6.45 -2.65 -3.67 -2.06

H/W, own children only, oldest child 6

− 17 -2.75 -2.16 -2.04 -2.56 -3.16

H/W, own children only, oldest child over 18 -2.62 -2.41 All other H/W households -2.96One parent (male) at least one child 0

− 18

Single personsDummy for Black -2.31Dummy for American Indian -2.37 1.95Dummy for Asian or Pacific Islander -2.14 2.08High School Graduate -2.95 -5.01 -1.98College dropout -2.61 -3.63 -2.26At least College graduate -3.15 -4.75 -3.30

10
IS and DS data, respectively). Furthermore, only households headed by individuals

aged at least 23 and no more than 73 and not self-employed are considered. These

restrictions leave us with a sample of 190.080 and 46.244 households over the considered period of time, for IS and DS data respectively. Sample sizes by year and

a detailed description of less important selection criteria and of data problems are

presented in the Appendix.5

Although the two surveys are designed to be representative of the same population, significant differences in the two samples are found along several dimensions

and with a different pattern over time. Table 1 presents t-statistics from a logistic

regression of the binary indicator IS/DS household over a set of variables including

work-related information and characteristics found to be relevant for data quality in

previous analysis of CEX data (Tucker 1992). Weighted results are presented, using

population weights from each survey. The specification adopted includes polynomial terms in the age of the reference person and in the proportion of children

and members within certain age bands (these terms are not reported because not

statistically significant).

For ease of exposition, only statistically significant differences at the 95 percent

confidence level or above are reported. Moreover, pooled results for the following six time periods are presented: 1982-85, 1986-89, 1990-92, 1993-95, 1996-98,

1999-2001. A negative (positive) value in the table should be interpreted as an

higher concentration of households with that characteristic in the DS sample (IS

sample, respectively) with respect to the other sample. The main difference in the

composition of the two surveys is confirmed to lie in the DS relative over sampling

of more educated households (particularly between 1986 and 1996), and generally

of H/W households with retired heads. The amount of weeks worked per year by

the reference person and the distribution of total family income are lower in the IS

sample. However, significant differences are found along several other dimensions

and with a different pattern over time.

3 THE MOTIVATING EXAMPLE

This section follows closely Deaton and Paxson (1994) to review the economic

reasoning that motivates our exercise. The simplest formulation of Permanent

Income Hypothesis (PIH) implies that, for any cohort of people born at the same

time, consumption inequality should grow with age (see, for example, Deaton 1992).

The conventional model of consumption under uncertainty assumes that, in each

period t, individuals maximize expectation of a time-separable utility function

U(ct) + 

T

s=t+1

δsU(cs),

subject to expectation of an intertemporal budget constraint

(ct − yt) + 

T

s=t+1

rs(cs − ys) = ωt.

Throughout this section U will denote an utility function invariant over the life

cycle, δt the individual’s rate of subjective preference, rt the real interest rate and

5Note that, due to the criteria used to define the working sample, we might observe systematic

movements in/out of the sample for IS households over their one-year interview period.

11
ωt the accumulated wealth al time t. The length of the life cycle is T, while ct and

yt represents log real consumption and income in each period.

The first order conditions to solve this problem imply that marginal utility of

consumption obeys the following equation

U

(ct) = βtU(ct−1) + νt,

where νt is a shock to consumption resulting from new information at time t and

βt = δt/rt (see Hall, 1978). Otherwise stated, the last expression implies that only

the actual level of consumption is informative to predict future values of consumption. In particular, future consumption is independent of actual income and wealth

related variables given actual consumption.

The relationship between the evolution of consumption and the evolution of

marginal utility depends on the function U. If U

(ct) is approximately linear in ct

(that is if each subperiod’s utility function is quadratic up to discounting by the

rate of time preference δt), the intertemporal choice of consumption over the life

cycle is given by

ct = βtct−1 + υt.

The last expression has some testable implications that might be used to check

the validity of this set-up. First, since the lagged value of consumption incorporate

all information about consumers’ decision at time t, then consumption or any other

related variable (income, in particular) lagged more than one period shouldn’t have

any explanatory power to predict the current value of consumption. This can be

tested by looking at the coefficients in the regression of actual consumption on

lagged values of consumption and income (see Hall 1978).

Additional implications become available once we are willing to make assumptions on the evolution of the βt terms over time. To see that, consider the case

where βt = 1, as in the original model. Under this condition, individual’s variance

must increase over time (Deaton and Paxson 1994), since

V ar(ct) = V ar(ct−1) + V ar(υt). (1)

Secondly, since

ct

t = 1t

t

s=0

υs, (2)

individual’s consumption is a sample average of independent shocks and hence

asymptotically normal by means of the central limit theorem. Note, however, that

this result applies when t grows to infinity; since individuals have finite life-spans,

the distribution of consumption is expected to be normally distributed only amongst

older people (see Blundell and Lewbel 1999).

When βt varies over time, the implications of the model might be different.

The distribution of individual’s consumption at time t is more disperse than the

distribution of consumption at time t − 1 when βt is greater that one. If the rate

of interest is greater than the rate of time preference (i.e. if incentives to postpone

consumption dominate impatience), the distribution of individual’s consumption

can either concentrate or disperse depending on the variance of υt. On the other

hand, consumption in older age can be normally distributed only if βt is always

centered around one, to avoid having the sum of shocks in (2) be zero or infinity as

t grows to infinity.

12
Of course, the validity of the previous implications of the PIH rests on several

caveats that have been already pointed out in the literature (see the discussion in

Browning and Lusardi 1996 and the references therein).6 In the remaining of this

paper, we will build on Deaton and Paxson (1994) to investigate the validity of (1)

by looking at the cross-sectional dispersion of IS and DS non-durable consumption

within cohorts as they age.

4 EVIDENCE ON CONSUMPTION BEHAVIOR

A standard way to analyze the dynamic properties of consumption with repeated

cross-sections is to rely on cohort analysis. This section investigates the effects of

different data collection methodologies by comparing expenditures levels (Section

4.1) and expenditure variances (Section 4.2) from IS and DS data.

In what follows we will group IS and DS households into cohorts on the basis

of the year of birth of the reference person (defining six 10-year bands) and we

will produce some descriptive graphs for total non-durable consumption using average cohort techniques. We will focus on four cohorts of individuals born between

1930 and 1969; sample sizes for the IS and the DS are reported in the Appendix.

Throughout this paper we will tend to use ‘expenditure’ and ‘consumption’ as two

synonyms, since the distinction is not relevant in this context.7

Clearly household characteristics (occupation and economic activity of the head,

household composition, region of residence) affect the share of spending and the

quality of reporting own expenditures (Tucker 1992). Since differences in consumption across the two surveys might reflect differences in the composition of the

samples with respect to household characteristics, we re-weight DS households exploiting a weighting scheme based on the regressions in Table 1 (see Battistin et

al. 2003). Intuitively, the aim is to down-weigh (up-weigh) those households in

the IS sample exhibiting characteristics over represented (under represented) with

respect to the DS sample. Such a weighting scheme depends on the conditional

probability of observing those characteristics in the population represented by the

IS (the so-called propensity score), that is on the binary regressions reported in

Table 1. Under the assumption that sampling differences are adequately captured

by this weighting scheme, the remaining differences reflect solely the nature of the

instrument exploited in each survey (i.e. diary vs recall questions).

Note that the weighting scheme adopted leads the distribution of income and

household composition to be the same across the two samples over time; both these

variables could appreciably affect the shape of age profiles because of an increasing

dispersion of household size (and, as a consequence, available family income) as the

cohort ages.8

To summarize the evidence from this section, we find that the evolution of

consumption means and variances as cohorts age over the life-cycle turns out very

6For example, the evolution of within cohort inequality depends on people’s attitude toward

risk and on the mechanisms that are available for sharing risks between people and periods. 7The panel component of the IS survey is not exploited in this paper, that is quarterly observations for IS households are counted separately as if the four observations over the one year

interview referred to different households. 8It might be interesting considering how much robust our results are to variations in head’s

age definition (Deaton and Paxson, 2000). However, there is not particular reason to believe that

any bias arising from such problem affects the two instruments in a different way and/or with a

different sign.

13
different depending on the source we consider.9 In particular, although most of the

published research for the United States has noticed that using IS data consumption

inequality has not grown significantly over the last twenty years, we show that the

picture emerging using DS data is very different. Differences in the two surveys

with respect to this point are discussed in the remaining of this section. Section 5

below aims to reconcile the evidence from the IS and the DS and to characterize

the most likely pattern of consumption inequality.

4.1 Expenditure levels

Figure 2 presents total expenditure on non-durable consumption by cohort as obtained from raw IS and DS data. Each point in the graph represents mean expenditure of the cohort in a generic year over the period 1982-2001 (1986-2001 for the DS

survey). Family consumption is adjusted using an equivalence scale which depends

on the number of adults and children in various age ranges (see the Appendix for

more details).

There are several discrepancies between estimated profiles for the two surveys.

Perhaps the most striking feature is that DS cohorts consume less than IS cohorts,

consistently over time. Overall, DS data appear to pick up most of the trends in

total spending found in IS data. Non-durable consumption declines in the last part

of the life cycle in line with similar drops reported in other countries (the retirement

puzzle). However, this decline seems to be more pronounced for IS data.

A further point worth stressing is the sharp drop of the IS expenditure level in

the early 1990s. Apparently, differences in reported levels of consumption between

the two surveys converge to zero after this time, and both data sources suggest a

declining pattern of total consumption in the last ten years.

The implications of these findings for the whole population are presented in

Figure 3. Annual growth rates in the two surveys are positive until 1989, then the

two series present negative growth in real consumer spending (although this decline

occurs later in time for the DS). As already pointed out in the Section 1, the level

of expenditure from integrated IS and DS data is decreasing over time, particularly

in the 1990s. This finding strongly contradict the pattern of expenditure on nondurables as implied by looking at data from the national accounts (see Figure 1).

The observed pattern of consumption in the two surveys can be further analyzed

by breaking down total expenditure into the contribution of different categories of

non-durable consumption. The relationship between mean expenditures in the two

surveys varies a great deal considering different commodities. Thus, the overall

figure for total expenditure appears to be the aggregated outcome of a large number

of positive and negative mean differences on non-durable commodities. Figures

A.2-A.10 in the Appendix present cohort profiles for expenditure on the nine nondurable groups considered in this paper (as discussed in Section 2.2).

Trends in expenditure on food related items and transportation in IS and DS

data probably presents the most striking differences. IS households realized a sharp

decline in price-adjusted food expenditure over the 1990s, both at home and away

from home. This figure is not consistent with DS data about food at home expenditure, whose values present a less pronounced decline over time. Expenditure

for transportation exhibits different trends over time for the two surveys, with DS

figures increasing in the 1990s contrary to those from the IS.

9We removed the households with the highest and lowest 2% of expenditures in each year so

as to enhance robustness of results.

14
 Interview data Diary data

born 1960−69

6.3

6.4

6.5

6.6

6.7

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

6.3

6.4

6.5

6.6

6.7

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure 2: Mean of log family expenditure on non-durable goods by cohort (2001

dollars)

 Interview data Diary data

82 84 86 88 90 92 94 96 98 100 102

6.3

6.4

6.5

6.6

6.7

Figure 3: Mean of log family expenditure on non-durable goods (2001 dollars)

15
Expenditure budgets (that is, the expenditure on each commodity as percentage of expenditure on non-durables) follow the same pattern over time in the two

surveys, although they have different levels (see Battistin 2002). The percentage

of total expenditure attributable to clothing and footwear, tobacco and alcohol

is decreasing (particularly over the 1990s) and is compensated by an increase of

expenditure on housing and public services.

4.2 Expenditure inequality

While the pattern of income inequality in the United States during the last twenty

years is well documented, the evidence on the evolution of consumption inequality

is much less clear. Several researchers have pointed out a rise in consumption

inequality over the 1980s using IS data, both within age-cohorts and for the overall

population (see, amongst others, Deaton and Paxson 1994). However, during the

first half of the 1990s, inequality partially receded for consumer expenditures while

for income it continued to rise (see Johnson and Shipp 1995).10 For this reason, the

issue of what happened to consumption inequality during a period characterized by

marked increases in income inequality has recently received much attention. This

section shows that the inequality pattern emerging from the DS is different from

the one obtained using the IS.

We will at first discuss the evidence by cohort and then consider inequality for

the all population. Figure 4 presents the evolution of consumption inequality by

cohort both for IS and DS data exploiting all observations in each survey year. We

find inequality (defined as the variance of log monthly non-durable expenditure) to

be higher for DS as compared to IS data. This may be due to respondent issues, but

is definitely related to the interview time periods of the two surveys being different:

the shorter time period results in data with greater volatility (the DS reference

period is one week).

However, the information contained in each sample leads to contradictory results

with respect to the trend of inequality over time. As stated above, within cohort

inequality from the IS survey presents a mildly increasing pattern in the 1980s for

those born between 1930 and 1949, but there is no evidence of any increase after

1990. This result can be directly compared to the numbers reported in Deaton

and Paxson (1994) and Blundell et al. (2002) where IS information for the 1980s

is used. DS inequality is instead increasing over time uniformly for all cohorts.

According to what discussed in Section 3, raw information from the two surveys

leads to contradictory conclusions in the validation of the PIH.

To improve the readability of the information contained in Figure 4, Table 2

presents the values of the Gini coefficient for the same data. IS inequality remains

flat over time for all the considered cohorts, with the exception of the cohort defined

by heads born in 1940 − 49. Moreover, the cohort born in 1930 − 39presents a

mildly increasing pattern during the 1980s. Inequality seems to be most pronounced

exploiting DS data for all cohorts.11 The bump-shaped pattern before retirement

10Table A.3 in the Appendix presents values of the Gini coefficient for total family income

over the last twenty years, exploiting additional information from the Current Population Survey

(CPS) and the Panel Survey of Income Dynamics (PSID). The picture is consistent with the one

already reported in the literature, showing a general increase in income inequality over the time

covered by this analysis. 11The robustness of this result has been further investigated exploiting additional measures of

inequality selected from the generalized entropy family (see for example Shorrocks, 1982). The

resulting picture is consistent with the one presented here.

16
 Interview data Diary data

born 1960−69

.2

.3

.4

.5

.6

.7

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

.2

.3

.4

.5

.6

.7

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure 4: Variance of log family expenditure on non-durable goods by cohort (2001

dollars)

 Interview data Diary data

82 84 86 88 90 92 94 96 98 100 102

.2

.3

.4

.5

Figure 5: Variance of log family expenditure on non-durable goods (2001 dollars)

17
Table 2: Gini coefficient for total non-durable expenditure (2001 dollars)

Diary sample

year born 1930-39 born 1940-49 born 1950-59 born 1960-69 all

1986 0.34 0.31 0.31 0.31 0.31

1987 0.33 0.32 0.32 0.29 0.32

1988 0.34 0.32 0.31 0.34 0.33

1989 0.36 0.35 0.32 0.30 0.33

1990 0.35 0.34 0.31 0.32 0.33

1991 0.35 0.33 0.34 0.33 0.34

1992 0.37 0.33 0.32 0.33 0.34

1993 0.38 0.36 0.31 0.33 0.34

1994 0.39 0.35 0.33 0.34 0.35

1995 0.36 0.35 0.36 0.33 0.35

1996 0.37 0.34 0.34 0.34 0.34

1997 0.35 0.37 0.36 0.33 0.35

1998 0.36 0.38 0.34 0.34 0.35

1999 0.37 0.36 0.35 0.34 0.36

2000 0.38 0.37 0.35 0.36 0.36

2001 0.36 0.38 0.35 0.37 0.37

Interview sample

year born 1930-39 born 1940-49 born 1950-59 born 1960-69 all

1982 0.27 0.26 0.26 0.26

1983 0.27 0.26 0.27 0.28 0.27

1984 0.28 0.26 0.26 0.26 0.27

1985 0.28 0.26 0.27 0.27 0.27

1986 0.28 0.27 0.27 0.28 0.27

1987 0.28 0.27 0.26 0.25 0.27

1988 0.28 0.27 0.26 0.27 0.27

1989 0.29 0.27 0.26 0.26 0.27

1990 0.28 0.28 0.27 0.27 0.27

1991 0.30 0.27 0.26 0.26 0.28

1992 0.30 0.28 0.27 0.27 0.28

1993 0.30 0.28 0.27 0.26 0.28

1994 0.29 0.28 0.26 0.26 0.27

1995 0.30 0.28 0.28 0.27 0.27

1996 0.28 0.28 0.27 0.26 0.27

1997 0.31 0.28 0.28 0.27 0.28

1998 0.28 0.28 0.27 0.28 0.28

1999 0.31 0.29 0.27 0.28 0.29

2000 0.290.290.27 0.26 0.28

2001 0.290.290.28 0.27 0.28

18
for those born in 1930 − 39could reflect the effect of an increasing leisure time due

to the retirement age.

The corresponding indeces of consumption inequality in the population are presented in Figure 5 and in the last column of Table 2. The resulting pattern for

the IS survey is the one already documented in several other papers expoiting this

data source: see for example the graphs in Slesnick (2001; Chapter 6) or Krueger

and Perri (2001; Figure 1). The variance of log-consumption is 0.22 in 1982, 0.25

in 1993 and 0.25 in 2001. On the other hand, DS inequality presents a markedly

increasing pattern over time. A formal test reject the null hyphothesis of constant

inequality for the DS over time, while it fails to reject this hyphothesis using IS

information for the 1990s.

A first attempt to explain this difference is to describe how marginal changes

in expenditures for specific commodities can affect the inequality of total expenditure. This would required the identification of the contribution in overall inequality attributable to each group entering the definition of non-durable consumption.

The problem is related to an unique decomposition rule as suggested by Shorrocks

(1982), since the inequality contribution assigned to each source can vary arbitrarily depending on the choice of decomposition rule. Particularly important for our

purposes is the ability to meaningfully decompose the index into inequality between and within different commodities. The decomposition must be consistent, in

the sense that commodities’ contribution should add up to the overall amount of

inequality.

Table 3 reports non-durable commodities and their percentage contribution to

total inequality using the ‘natural’ decomposition rule

V ar(Y ) = 

j

Cov(Xj , Y ),

both for IS and DS data. The contribution of each commodity Xj to total inequality

is then expressed as the slope coefficient of the Engel regression of Xj on nondurable expenditure Y . Alternative procedures based on decompositions of the

Gini coefficient (see for example Garner, 1993) lead to the same result.12

The contribution of food at home and housing and public services is increasing

over time for both IS and DS data; the increasing weight of non-durable services for

IS inequality is not observed in the DS. The trend for the remaining figures is comparable across the two surveys, although different levels are observed. Food away

from home, alcohol and clothing are those commodities presenting a decreasing

contribution over time.

5 ACCOUNTING FOR INACCURACIES

Can IS and DS data be exploited together to derive a superior measure of nondurable consumption? The goal of this section is to address this issue by discussing

the nature of survey errors that are likely to affect the CEX. In fact, a natural

explanation for the different time pattern of means and variances in the two sur12However, under suitable constraints, it can be proved that there is an unique decomposition

rule for any inequality measure for which the proportion of inequality attributed to each commodity is the proportion obtained in the natural decomposition rule of the variance (Shorrocks,

1982).

19
Table 3: Factors contribution as percentage of total inequality

Interview 1982-85 1986-89 1990-92 1993-95 1996-98 Food and non-alcoholic beverages at home 10.62 10.37 11.17 10.43 10.87 Food and non-alcoholic beverages away 10.14 9.71 8.73 9.11 9.01 Alcoholic beverages (at home and away) 4.23 3.22 2.65 2.46 2.61 Non-durable goods and services 11.97 12.77 12.62 13.70 13.27 Housing and public services 7.72 7.88 9.14 10.22 11.39 Tobacco and smoking accessories 0.61 0.43 0.55 0.54 0.51 Clothing and footwear 16.96 18.35 17.66 15.21 13.85 Heating fuel, light and power 4.80 4.27 4.194.40 4.26 Transport (including gasoline) 32.95 33.00 33.30 33.93 34.24 Diary 1982-85 1986-89 1990-92 1993-95 1996-98 Food and non-alcoholic beverages at home 9.23 9.22 9.52 9.61 Food and non-alcoholic beverages away 11.85 10.71 9.57 9.37 Alcoholic beverages (at home and away) 2.58 2.10 1.83 1.62 Non-durable goods and services 18.25 18.5917.54 16.95 Housing and public services 9.91 10.80 12.19 13.25 Tobacco and smoking accessories 0.52 0.37 0.48 0.52 Clothing and footwear 15.54 15.21 13.78 12.11 

Heating fuel, light and power 8.92 8.75 7.98 8.08 Transport (including gasoline) 23.20 24.27 27.12 28.49

20
veys is the aggregate result of inaccuracies affecting the IS and DS reporting of

expenditures.

Lyberg et al. (1997), Bound et al. (2001) and Browning et al. (2002) review

data quality problems characterizing survey measurements. The main lesson from

their findings is that inaccuracies mainly come from those non-durable commodities

each survey is not targeted to: frequently purchased smaller items and services (IS)

and large expenditures occurring on a regular basis (DS). The aggregate effect of

these inaccuracies is likely to vary over time, because it depends on significative

changes in the structure of the two surveys (i.e. design and collecting strategies)

and on time-in-sample effects (i.e. people might change their disposition to answer

accurately or answer at all). As a matter of fact, there has been a deterioration

of the correspondence between CEX aggregates and PCE over the 1990s (see McCarthy et al. 2002).

Modelling the effects of survey errors generally requires strong assumptions. In

fact, although the error on a given variable is often assumed to be independent of

the true level of that and of all other variables, this assumption reflects convenience

rather than conviction. In what follows we will depart from any model-based assessment of the error. Instead, we will use information from the most reliable survey

as validation data to assess the quality of the other survey.

Internal validation data are to be preferred over validation data coming from

external surveys. In this sense, the two survey components from the CEX represent

a unique example for the United States. However, since the IS and the DS address

independent samples of households, diary and recall values are not observable for

the same survey households. The drawbacks of this design discussed in Section 6.

5.1 Nature and consequences of survey errors

Collection methodology differences between the two survey components of the CEX

certainly represent the main explanation for the evidence presented so far. While

the DS collects detailed disaggregated data and then sums these up to obtain total

spending, the IS asks a global retrospective question about totals. Differences in

levels and inequality indices might be determined by different expenditure estimates

on each commodity as a result of this aggregation.

Respondents’ partial recollection of past events (mainly related to less important

purchases) and/or telescoping effects for the month nearest to the interview are

factors likely to affect the accuracy of available information in the IS. In fact, recall

is certainly a complex cognitive process in the collection of consumer expenditures

data through household interviews. The important implications of forgetfulness and

telescoping in large-scale recall surveys have been largely discussed in the literature

(see, among others, Neter and Waksberg 1964). Moreover, measurement effects in

self vs proxy responses, differences in the interpretation of questions, inability or

unwillingness of respondents to provide full information or interviewers’ effect on

data collection might play a non-negligible role in determining the quality of IS

information.13

Under reporting of expenditures is likely to be limited in diary surveys, although

additional data collection effects might play an important role in the definition of

monthly aggregates. Household’s expenditures recorded during a limited period of

13Both surveys accept proxy responses from any eligible household member who is at least 16

years old if an adult is not available after a few attempts to contact that person.

21
time (two weeks) might give a misleading impression of its underlying consumption pattern over a longer period (a month). Since commodities entering total

non-durable consumption are purchased with a different frequency, zero recorded

expenditures might reflect preferences in the frequency of purchasing rather than

preferences in consumption behavior.

Besides, other conjectures on the sources of zero reported expenditures include

under reporting in an acknowledged purchase and item non-response (i.e. not reporting a purchase that was made). Several studies have shown that the negative

effect of poor quality information as the interview-time increases is bigger than the

positive effect due to respondent’s learning-by-doing process. For example, Turner

(1961) and Silberstein and Scott (1991) find that the average of reported expenditure using diary data decreases across day and week of participation, probably

reflecting under reporting related to a declining interest. Silberstein and Jacobs

(1989) find similar results with respect to the time-in-sample (i.e. the number of

cycles of participation) for the IS.14

An additional explanation for the evidence of the previous section are periodic

changes in the survey instruments over the years, both for the IS and the DS.

For example, a new diary form with more categories and expanded use of cues for

respondents has been introduced in the DS since 1991. The diary instrument for

the DS is respondent-filled, and reporting levels and accuracy are known to depend

on the adopted diary format (Tucker 1992). On the other hand, the definition of

expenditure categories collected in the IS has changed over time (notably for food,

as discussed in the Appendix) because of newly collected items or because some of

them have been supplanted by new ones.

It what follows, we will refer to measurement error as the difference between

the actual value of expenditure and the value reported by respondents. From what

discussed above, data collection effects and respondents’ partial recollection of past

events are likely to make such a difference not identically distributed across households and over time.15 Moreover, the assumption of classical measurement error

in survey measurements (i.e. zero mean error independent of the true unobserved

variable and of all other variables) has been largely criticized and is usually made

for ease of estimation rather than for any theoretical conviction (see Rodgers et al.

14Table A.4 in the Appendix presents reporting rates for IS and DS data, that is the proportion

of non-zero expenditures for a specific commodity. Year-to-year changes in this indicator provide

useful monitors of survey performance over time. The frequency of purchasing is generally lower

in the DS sample, with the only exception of expenditure on food away from home (which by

definition does not include expenditures on vacation). However, the overall pattern remains the

same uniformly over time, across samples and for each commodity; this we take as an evidence

that changes in consumption habits are well reflected in both the samples. 15Note that if the distribution of the measurement error is not stationary over time, we cannot

separately identify the effect of a real change in the inequality level from the effect induced by

variation in the quality of reporting. To give a flavor of such a problem, assume that the error

affecting reports of spending on commodity X is multiplicative and that its intensity is given

by a parameter σ (see Chesher and Schluter 2001). If we assume independence between X and

the reporting error process, a second-order approximation for the Gini coefficient of the errorcontaminated consumption is given by

GX + σ2 E[X2fX(x)]

E[X] ,

where fX and GX are the density and the Gini coefficient associated to X, respectively. It follows

that the ‘distance’ between the true and the observed Gini coefficient might be different over time

because of variations in σ or in the shape of expenditure distribution (indeed, the incidence of the

measurement error is not particularly high when the distribution of X is heavily right skewed).

22
1993, Torelli and Trivellato 1993 and Pischke 1995). In particular, Battistin et al.

(2003) provide some evidence for the case in which the magnitude of the measurement error is endogenously determined by the real amount of expenditure (with

higher expenditure levels associated with larger errors), so that the independence

assumption is no longer valid.

It follows that modelling measurement errors affecting IS and DS information

would require relatively strong assumptions. We instead depart from any modelbased approach and exploit information from the most reliable survey as a validation

source for the other survey.

Our procedure develops along the following lines. Since the IS and the DS are

explicitly designed to obtain reliable measures of expenditure on different commodities, the error affecting total non-durable expenditure results from those categories

of each survey component that are considered ‘less reliable’. Those commodities

either having regular periodic billing or involving major outlays easily recalled for

a period of three months or longer are better described using IS data. On the other

hand, those non-durable commodities referring to frequently purchased and smaller

items are presumably more reliable in the DS survey. The next section discusses

the validity of such an assumption by presenting the evidence from a broad range

of studies.

5.2 Choosing among alternative collection methodology

Throughout the analysis we will make the following assumptions.

Condition 1. Either IS or DS data identify correctly (i.e. report without measurement error) the actual spending on non-durable commodities.

Condition 2. We know which source (IS or DS) provides the actual amount of

spending on each commodity.

Condition 1 builds on the well-established conviction that diary and recall surveys

provide reliable consumption information on different commodities. Neither survey component alone can be exploited to get an accurate measure of non-durable

spending: it is the joint use of IS and DS information that leads to accurate totals.

Condition 2 defines the aggregation rule one should follow in pooling IS and DS

information, and it is therefore more debatable. As discussed earlier, the question of

which survey component provides more accurate estimates for non-durable items is

an issue of longstanding concern in the design of expenditure surveys (see Browning

et al. 2002). Reliability of expenditure data has been assessed either by examining

how aggregate spending on a certain commodity compares with aggregate spending

from national accounts (see for example Banks and Johnson 1998 and Slesnick

2001), or by looking at results from controlled experiments (see for example Winter

2002). The remaining of this section discusses the implications of these findings for

the aggregation rule used in this paper.

Even if potentially the sign of the bias in recall and diary data could be in both

directions depending on different commodities (over or under reporting of true

expenditures), the available evidence from several countries suggests that under

reporting is more likely to affect the great part of items in expenditure surveys.

Complete information on small expenditures is likely to be not always available

since the respondent may forget to report less important purchases below a certain

amount.

23
The magnitude of partial recollection of past events varies for different commodities exploiting recall and diary keeping methods. Those components having

regular periodic billing are more likely to be well reported by respondents in the

IS survey. Indeed, exploiting validation data from the national accounts IS expenditures for transports and fuel have been found to be reliable and heavily under

reported by diary data (Gieseman 1987).

Spending on alcoholic beverages and tobacco traditionally has been under reported in household surveys; some authors refer to this evidence as a puritan element in household data. Diaries were found to give more reliable information

about alcohol consumption than recall data (see Poikolainen and Kakkainen 1983

and Atkinson et al. 1990); comparisons of tobacco expenditures based on mean

squared error methods exploiting national accounts data suggest better quality

from recall data (Branch and Jayasuriya 1997).

Clothing is a category which requires fuller investigation. Several studies reveal

heterogeneity in results exploiting diary or recall information amongst goods within

this category. As expected, IS data seem to be more reliable for costly and salient

apparel items (with quite variable results exploiting different methods of source

selection), but DS data generally capture more apparel spending (Silberstein and

Scott 1991).

There is some evidence that diaries are the most reliable source to measure

purchases on food made away from home (Stanton and Tucci 1982). However, it is

also well documented that the quality of reporting is higher in the first week of the

diary (see Turner 1961 and Figure A.11 in the Appendix). On the other hand, as

explained in Section 2.1, IS information of food at home is derived as the difference

between the usual spending at grocery stores and how much of this amount was for

non-food items.

According to the evidence from the studies summarized in this section, Table 4

presents the aggregation rule we will follow to obtain aggregate values of total expenditure on non-durables. Branch and Jayasuriya (1997) discusses how consumer

expenditures from the two surveys are chosen by the BLS for publication. It is

worth noting that, although the level of aggregation considered in their paper is

finer than the one exploited here, the classification procedure suggested in Table 4

broadly reflects the one currently being used by the BLS. Not surprisingly, DS data

are exploited as the reference source for expenditures on grocery items and personal

care, entertainments and other services; IS data to identify expenditures on those

components having regular periodic billing or involving major outlays. In what

follows these two sets of commodities will be denoted by I and D, respectively.16

6 ERROR CORRECTION

The classification procedure discussed in the previous section provides a rule to

define a superior measure of total consumption by exploiting together IS and DS

information. Nevertheless, straightforward pooling cannot be implemented since

diary and recall expenditures are not observed for the same survey households.

The aim of this section is to formalize the restrictions presented in Table 4. Although by sampling design Conditions 1 and 2 do not allow us to fully identify the

16According to the classification procedure suggested, the ‘true’ unobserved value of expenditure

is a mixture of observed expenditures from the IS and the DS. Conditions 1 and 2 impose zero/one

restrictions on the weights of this mixture, so that either IS or IS expenditures are considered.

24
Table 4: Survey selection

Commodities in D Survey

Food and Non-Alcoholic Beverages at Home Diary

Food and Non-Alcoholic Beverages Away from Home Diary

Alcoholic Beverages (at home and away from home) Diary

Non-Durable Goods and Services Diary

Commodities in I Survey

Housing and Public Services Interview

Tobacco and Smoking Accessories Interview

Clothing, Footwear and Services Interview

Heating Fuel, Light and Power Interview

Transportation (including gasoline) Interview

distribution of total non-durable consumption, they are identifying restrictions concerning the first moment of this distribution. The identification of higher moments

(and percentiles) would require to model the relationship between true expenditures and reporting errors. However, the second moment of this distribution can

be bounded without imposing any restriction on the nature of the error by means

of the Cauchy-Schwartz inequality. To a certain extent, this turns out to be an

informative test for the relationship in (1) implied by the PIH.

Throughout the remaining of this paper we will focus on within cohort means

and variances. The issue of how IS and DS information can be jointly used to characterized the most likely pattern of consumption inequality for the entire population

is further elaborated in Attanasio et al. (2003).

6.1 A potential outcomes approach

Let

(Xr

j , Xdj )

be the two outcomes that result from being interviewed on commodity Xj via a

recall or a diary based questionnaire, respectively, and let X∗

j be the true expenditure on the same commodity (which is not observed). Commodities Xj ’s are those

that have been described in the previous section and reported in Table 4.

Clearly, the difference between Xr

j and Xdj is informative about the effect of

reporting expenditures exploiting recall rather than diary based questionnaires.

Since the two surveys refer to separate samples, an identification problem arises

from the fact that - by design - only one of these measurements is observed on each

household. In fact, they represent potential outcomes from using alternative survey

instruments to collect information on household consumption. The measurement

error on each commodity would be identified if we could observe the counterfactual expenditure for each household, that is what the same household would have

reported had it participated the other survey.

Let C∗ be total expenditure on all non-durable commodities Xj ’s

C∗ = C∗

1 + C∗2 ,

25
where C∗

1 and C∗2 represent expenditure on commodities in I and in D, respectively,

as defined by Table 4

C∗

1 = 

j∈I

X∗

j , C∗2 = 

j∈D

X∗

j .

Total expenditure is not observable to the analyst. Rather two error affected measurements of C∗ are observed, representing the aggregate expenditures that result

from using IS or DS data

Cr = Cr

1 + Cr2 ,

Cd = Cd

1 + Cd2 ,

where (Cr

1 , Cd1 ) and (Cr2 , Cd2 ) are IS and DS expenditures on commodities in I and

in D

Cr

1 = 

j∈I

Xr

j , Cr2 = 

j∈D

Xr

j ,

Cd

1 = 

j∈I

Xd

j , Cd2 = 

j∈D

Xd

j .

By means of Conditions 1 and 2, the measurement error affecting the aggregate

expenditures Cr and Cd depends on the measurement error on different subsets of

commodities entering total non-durable expenditure (i.e. those commodities either

in D or in I, respectively). That is, total expenditure from IS data can be written

as

Cr = C∗

1 + Cr2 ,

where the last expression follows since expenditures on commodities in I are not

affected by measurement error by assumption (Xr

j = X∗j , ∀j ∈ I). By analogy, it

follows that

Cd = Cd

1 + C∗2 ,

since Xd

j = X∗j , ∀j ∈ D. Accordingly, the error due to recall and diary interviews

can be written as

εr = Cr

2 − C∗2 , (3)

εd = Cd

1 − C∗1 , (4)

respectively. Any difference in the distribution of these errors over time is responsible for the different pattern of means and variances observed in raw IS and DS

data.

The main points arising from the last two expressions can be summarized as

follows. Firstly, the mean of the IS (DS) error can be written as a linear combination

of means of errors for commodities in D (I, respectively). Since error means on all

non-durable commodities are identifiable because of Conditions 1 and 2, the mean

values of (3) and (4) are identified by

E(εr) = E(Cr

2 ) − E(Cd2 ), (5)

E(εd) = E(Cd

1 ) − E(Cr1 ), (6)

26
born 1960−69

700

750

800

850

900

born 1950−59

born 1940−49

86 88 90 92 94 96 98 100102

700

750

800

850

900

born 1930−39

86 88 90 92 94 96 98 100102

Figure 6: Mean of family expenditure on non-durable goods by cohort after correction (2001 dollars)

respectively. Accordingly, Conditions 1 and 2 are identifying restrictions concerning

the first moment of the distribution of total non-durable consumption (see Section

6.2).

Secondly, recall and diary errors are likely to be correlated with C∗, since they

depend on a common set of commodities. This implies that the difference between

the real variance of consumption and the observed values of this variance in IS

and DS data depends on the variance of the terms in (3) and (4) and on their

correlation with C∗, which are not observable. Therefore the variance of C∗ cannot

be estimated without imposing additional restrictions on the error structure (see

Section 6.3).

6.2 Consumption levels

Differences in mean expenditure values for all commodities entering total nondurable expenditure have been already discussed in Section 4.1 (see also Figures

A.2-A.10 in the Appendix). By means of Conditions 1 and 2, they can be interpreted as the effect of collecting expenditure information using the less appropriate

methodology, depending on whether the considered commodity belongs to D or I.

Mean aggregate errors for IS and DS data (that is the quantities in (5) and (6),

respectively) are presented in Table 5, separately by cohort and over time. More

precisely, figures for IS and DS errors as proportion of total non-durable expenditure

are reported, that is

E(εr)/E(C∗) E(εd)/E(C∗)

27
Table 5: Survey errors as a proportion of total non-durable expenditure

year born 1960-69 born 1950-59 born 1940-49 born 1930-39 all

IS DS IS DS IS DS IS DS IS 

1986 -0.05 -0.12 -0.03 -0.10 -0.01 -0.10 -0.02 -0.11 -0.02 1987 0.02 -0.11 -0.02 -0.11 -0.03 -0.09 -0.01 -0.08 -0.01 1988 -0.05 -0.09 -0.06 -0.10 -0.05 -0.08 -0.04 -0.08 -0.05 1989 -0.07 -0.07 -0.05 -0.10 -0.05 -0.10 -0.07 -0.06 -0.06 1990 -0.02 -0.06 -0.07 -0.09 -0.06 -0.06 -0.07 -0.05 -0.06 1991 -0.03 -0.06 -0.08 -0.03 -0.07 -0.05 -0.06 -0.06 -0.06 1992 -0.09 -0.04 -0.08 -0.05 -0.05 -0.08 -0.03 -0.04 -0.07 1993 -0.07 -0.03 -0.05 -0.05 -0.06 -0.05 -0.06 -0.03 -0.06 1994 -0.06 -0.04 -0.05 -0.03 -0.03 -0.05 -0.04 0.01 -0.04 1995 -0.04 -0.06 -0.07 -0.03 -0.07 -0.02 -0.08 0.02 -0.06 1996 -0.05 -0.02 -0.06 -0.05 -0.07 -0.04 -0.08 0.02 -0.06 1997 -0.05 -0.02 -0.04 -0.03 -0.06 -0.01 -0.07 -0.06 -0.05 1998 -0.07 -0.06 -0.08 -0.01 -0.04 -0.08 -0.06 -0.02 -0.06 1999 -0.08 -0.03 -0.07 -0.06 -0.06 -0.06 -0.05 -0.04 -0.06 2000 -0.07 -0.05 -0.08 -0.02 -0.04 -0.04 -0.08 -0.05 -0.07 2001 -0.08 -0.05 -0.07 -0.01 -0.08 -0.02 -0.10 -0.06 -0.08 

28
respectively, where the mean of total expenditure is estimated by

E(C∗) = E(C∗

1 ) + E(C∗2 )

= E(Cr

1 ) + E(Cd2 ). (7)

Negative (positive) numbers in the table can be interpreted as the proportion of

expenditure under (over) reported in the IS or in the DS. Apparently, this proportion varies across cohorts, with younger cohorts more likely to have higher errors

than older cohorts, both in the IS and in the DS. The amount of misreporting has

an almost stationary distribution over time for the IS, while it varies for the DS.

In particular, the DS appears to become more accurate over time.17

Figure 6 presents estimated mean expenditures by cohort using (7) and can be

compared to the results presented in Figure 2. It is evident that, even after the

correction, consumption profiles are still sharply decreasing during the first half of

the 1990s.18

6.3 Consumption inequality

The goal of this section is to derive the analogue of Figure 4 once the effect of

reporting errors is accounted for. Throughout our analysis, we will consider figures

for the squared coefficient of variation of total expenditure instead of figures for the

variance of logs. The reason for this choice will be clear from what follows.

The variance of total expenditure can be expressed as a function of the variance

of commodities in D and I and between-group covariances

V ar(C∗) = V ar(C∗

1 ) + V ar(C∗2 )+2Cov(C∗1 , C∗2 ). (8)

The ratio of the previous quantity to the squared mean of total consumption represents a first order approximation for the variance of log consumption.19 Conditions

1 and 2 are identifying restrictions only for the first two terms of the previous

expression, using either IS or DS data. On the other hand, the covariance term

cannot be identified from the available information since by definition D∩I = ∅

17Given the restrictions imposed so far, the measurement error affecting the reporting of nondurable commodities cannot be further characterized. If we observed both recall and diary outcomes for commodity X on the same household, according to Condition 1 and Condition 2 the

error on that commodity would be (non-parametrically) identified. In fact, by writing

ξ = x − x∗,

we could identify the distribution of ξ by taking the difference between observed diary and recall

outcomes depending on the rule discussed in Table 4. In our case, although the first moment

of this distribution can be easily recovered, the identification of higher moments and percentiles

would require to know the structure of correlation between x and x∗, which is not observable. 18Following the same lines of the previous footnote, it can be shown that the distribution of C∗

is not identified. Additional restrictions are required. Alternatively, bounds on this distribution

could be derived exploiting Frechet’s or Markov’s inequalities. 19A Taylor expansion of lnC∗ in a neighborhood of its mean gives

lnC∗ = lnE(C∗) + C∗ − E(C∗)

E(C∗) − [C∗ − E(C∗)]22E(C∗)2 + error,

so that the variance of logs is approximated by taking the squared coefficient of variation

V ar(C∗)

E(C∗)2 = CV (C∗)

2  V ar(lnC∗).

29
so that true expenditures on commodities in D and I cannot be observed for the

same individuals.

Two measurements of the covariance between C∗

1 and C∗2 are actually observable

in our data. The first one is derived by taking the covariance between commodities

in D and commodities in I from the IS; the second one is derived from the DS.

Figure 7 presents the squared coefficient of variation obtained when the unknown

covariance is estimated by using IS or DS information and the mean of total consumption is estimated by (7). Apparently, both IS and DS figures suggest increasing

within cohort inequality over time.

However, without additional assumptions neither observed covariance is a consistent estimator for the covariance of interest. In fact, using (3) and (4) the

observed covariance in the IS sample can be written as

Cov(Cr

1 , Cr2 ) = Cov(C∗1 , Cr2 )

= Cov(C∗

1 , C∗2 ) + Cov(C∗1 , εr), (9)

and, by analogy, the observed covariance in the DS sample is

Cov(Cd

1 , Cd2 ) = Cov(Cd1 , C∗2 )

= Cov(C∗

1 , C∗2 ) + Cov(C∗2 , εd). (10)

Figure 8 presents the evolution of IS and DS covariances over time and separately

by cohort. The difference between the two lines of each panel reflects the difference

between the last terms in (9) and in (10). These two terms are not identified

from observed data. Overall there is evidence that DS covariances are higher and

present different trends than IS covariances. A formal test for the equality of

observed covariances in IS and DS data rejects this hypothesis over the time period

1986-2001. Similarly, the inequality indices presented in Figure 7 are found to be

statistically different (but both statistically increasing).

7 BOUNDS ON INEQUALITY

7.1 Definitions

As discussed in the last section, additional structure is required to identify the

covariance term in (8). For example, Attanasio et al. (2003) consider alternative

identification strategies based on the specification of a demand system for C∗

1 and

C∗

2 . In this paper we take a different route and we investigate whether informative

bounds on the within cohort variance can be derived from fairly general assumptions

on IS and DS errors.

The strategy we will take consists of three different steps. We will at first use

Cauchy-Schwartz bounds resulting from the following equality

|Cov(C∗

1 , C∗2 )| = |ρ∗|



V ar(C∗

1 )V ar(C∗2 ), (11)

where ρ∗ represents the Pearson correlation index between C∗

1 and C∗2 . The wellknown relationship

|ρ∗| ≤ 1 (12)

30
 covariances from Interview data covariances from Diary data

born 1960−69

.2

.25

.3

.35

.4

born 1950−59

born 1940−49

86 88 90 92 94 96 98 100 102

.2

.25

.3

.35

.4

born 1930−39

86 88 90 92 94 96 98 100 102

Figure 7: Squared coefficient of variation of family expenditure on non-durable

goods by cohort using observed covariances (2001 dollars)

 Interview data Diary data

born 1960−69

10000

20000

30000

40000

50000

born 1950−59

born 1940−49

86 88 90 92 94 96 98 100 102

10000

20000

30000

40000

50000

born 1930−39

86 88 90 92 94 96 98 100 102

Figure 8: Observed covariances by cohort (2001 dollars)

31
defines bounds for the within cohort covariance over time, since the product of the

variances in (11) is identifiable using the restrictions in Table 4.

However, although the interval [−1, +1] is often considered the reference range

for evaluating the strength of empirical correlation, it is well known that the upper

and the lower bounds can only be achieved if the marginal distributions of C∗

1

and C∗

2 are linearly related (indeed, the Pearson coefficient detects only linear

correlation!). If the two distributions cannot be linearly related, the maximum

attainable absolute correlation coefficient is lower than one (see for example the

discussion by Shih and Huang 1992).

More formally, let F∗

1 and F∗2 be the cumulative distribution functions of C∗1

and C∗

2 , respectively, and let the corresponding inverse be defined as

F∗−1

j (t) = inf{x : F∗

j (x) ≥ t}, j = 1, 2

for 0 ≤ t ≤ 1. Then the correlation between C∗

1 and C∗2 is bounded by

corr(F∗−1

1 (W), F∗−12 (1 − W)) ≤ ρ∗ ≤ corr(F∗−11 (W), F∗−12 (W)), (13)

where W is a random variable distributed uniformly between zero and one. Note

that, if C∗

1 can be written as a linear function of C∗2 , then the previous expression

collapses to the Cauchy-Schwartz bound in (12).20 Bounds in (13) can be estimated

by taking the correlation between percentiles of the empirical distributions of C∗

1

and C∗

2 which, by means of Conditions 1 and 2, are identified. They represent the

tightest bounds on the correlation coefficient attainable without imposing further

restrictions on the correlation of interest.

Bounds for the within cohort inequality resulting from this procedure are presented in Figure 9. To assess the importance of sampling variability, figures from

bootstrapped 95% confidence intervals will be discussed throughout this section.

As expected, results are not informative on the pattern of within cohort inequality

over time.

Bounds in (13) can be improved by looking at the partial correlation between

C∗

1 and C∗2 once a set of regressors Z is controlled for. The expression in (8) can be

written as the sum of the following within-group and between-group components

V ar(C∗) = EZV ar(C∗|Z) + V arZE(C∗|Z),

which are defined as

V arZE(C∗|Z) = V arZ{E(C∗

1 |Z) + E(C∗2 |Z)},

EZV ar(C∗|Z) = EZ{V ar(C∗

1 |Z) + V ar(C∗2 |Z)+2Cov(C∗1 , C∗2 |Z)},

where Z represents a set of family characteristics common across the two surveys.

The first component is identified by assumption, since it represents the variance

across groups defined by Z of total mean expenditure as it results from (7). The

second component is not identified because it contains the within group covariance

between C∗

1 and C∗2 .

This covariance can be bounded along the same lines of what discussed above.

To fix ideas, in what follows Z will represent total family income and households

will be split into five mutually exclusive groups defined by income deciles (20% or

20Shih and Huang (1992) provide several examples where the bounds implied by (13) are very

different from the ‘natural’ bounds implied by (12).

32
born 1960−69

0

.1

.2

.3

.4

.5

.6

born 1950−59

born 1940−49

86 88 90 92 94 96 98 100 102

0

.1

.2

.3

.4

.5

.6

born 1930−39

86 88 90 92 94 96 98 100 102

Figure 9: Squared coefficient of variation of family expenditure on non-durable

goods by cohort (2001 dollars) - bounds using (13)

less, 20-40%, 40-60%, 60-80%, 80% or more). The within group correlation between

C∗

1 and C∗2 is then bounded using the analogue of (13) once income is controlled for.

However, the resulting bounds are still not informative on the pattern of inequality

over time and therefore are not presented here.21

Tighter bonds on these covariances can be derived moving from the following

evidence. By using the relationships in (9) and (10), observed IS and DS covariances

can be first differenced over time to get

∆Cov(Cr

1 , Cr2 |Z)=∆Cov(C∗1 , C∗2 |Z)+∆Cov(C∗1 , εr|Z),

∆Cov(Cd

1 , Cd2 |Z)=∆Cov(C∗1 , C∗2 |Z)+∆Cov(C∗2 , εd|Z).

Table 6 presents results from a bootstrapped test at the 95% confidence level for

the null hypothesis

∆Cov(Cr

1 , Cr2 |Z) − ∆Cov(Cd1 , Cd2 |Z)=0.

For the sake of completeness, results are reported also for the test referred to

the covariance in levels. Numbers in the table refer to the proportion of income

groups for which the hypothesis is rejected, separately by cohort and over time.

Accordingly, 0.20 means that for one out of the five income groups considered the

null hypothesis is rejected, 0.40 means that for two out of five groups the null

hypothesis is rejected, and so on.

Although IS and DS covariances appear to have different levels even controlling

for Z (see the top panel of the table), overall results support the hypothesis of

stationary difference between observed IS and DS covariances over time once Z is

21Similar results are found by controlling for a richer set of regressors. Results are available on

request.

33
Table 6: Percentage of groups for which the null hypothesis is rejected

H0 : Cov(Cr

1 , Cr2 |Z) − Cov(Cd1 , Cd2 |Z)=0

year born 1960-69 born 1950-59 born 1940-49 born 1930-39

1986 0.40 0.40

1987 0.60 0.40

1988 0.20 0.20 0.20 0.20

1989 0.20 0.40 0.40

1990 0.20 0.20 0.20 0.40

1991 0.80 0.40

1992 0.20 0.60 0.20

1993 0.20 0.40 0.20 0.40

1994 0.60 0.60 0.20 0.80

1995 0.60 0.80 0.20 0.20

1996 1.00 0.40 0.20 0.40

1997 0.40 0.40 0.60 0.20

1998 0.20 1.00 0.40 0.20

1999 0.80 1.00 0.40 0.20

2000 1.00 0.80 0.60 0.20

2001 1.00 0.60 0.80 0.20

H0 : ∆Cov(Cr

1 , Cr2 |Z) − ∆Cov(Cd1 , Cd2 |Z)=0

year born 1960-69 born 1950-59 born 1940-49 born 1930-39

1987 0.20

1988

1989

1990

1991

1992

1993

1994 0.20

1995

1996

1997

1998

1999

2000

2001 0.20

34
born 1960−69

0

.1

.2

.3

.4

.5

born 1950−59

born 1940−49

86 88 90 92 94 96 98 100 102

0

.1

.2

.3

.4

.5

born 1930−39

86 88 90 92 94 96 98 100 102

Figure 10: Squared coefficient of variation of family expenditure on non-durable

goods by cohort (2001 dollars) - bounds using (13) and Condition 3

controlled for. This represents a necessary condition for the following assumption

to be satisfied.

Condition 3. ∆Cov(C∗

2 , εd|Z) = 0 and ∆Cov(C∗1 , εr|Z) = 0.

The previous assumption together with the bounds implied by (13) restricts the

range of values of the partial covariance between C∗

1 and C∗2 given Z. Although the

level of the covariance of interest is unknown, the series of its changes over time

turns out to be identified by the series of variations in IS and DS data. Condition

3 defines bounds on the inequality index over time. Technical details on how these

bounds are derived are presented in the Appendix.

Results are reported in Figure 10. Although Condition 3 turns out to be very

powerful in tightening the natural bounds implied by (13), it is not fully informative

on the pattern of within cohort inequality over time. Apparently, inequality grows

over time for the cohort of those born in 1930-39 and for those born in 1930-39

before 1993 (that is before the retirement age). Inequality for the two remaining

cohorts (in particular for those born in 1950-59) is not statistically increasing at

the confidence level considered.

7.2 Discussion

A sensitivity analysis of these results with respect to the true value of the correlation coefficient ρ∗(Z) = corr(C∗

1 , C∗2 |Z) is presented in Figures 11-13. More

precisely, the same procedure described in the previous section is derived exploiting the additional assumption that the correlation between C∗

1 and C∗2 net of Z is

lower than 0.90, 0.85 and 0.80 in absolute value.

35
born 1960−69

0

.1

.2

.3

.4

.5

born 1950−59

born 1940−49

86 88 90 92 94 96 98 100 102

0

.1

.2

.3

.4

.5

born 1930−39

86 88 90 92 94 96 98 100 102

Figure 11: Squared coefficient of variation of family expenditure on non-durable

goods by cohort (2001 dollars) - bounds using (13), Condition 3 and assuming

|ρ∗| ≤ 0.90

By means of a grid search on the support of ρ∗(Z) the following question is

addressed: what is the maximum correlation required such that the inequality

index for the two ‘central’ cohorts in Figure 10 is statistically increasing over time?

Apparently, a maximum value of |ρ∗(Z)| between 0.85 and 0.90 is enough to detect

increasing inequality for those born in 1940-49, while 0.80 is required for those born

in 1950-59.

As discussed above, the range of possible values of ρ∗(Z) implied by (13) is

different from the interval [−1, +1]. Table 7 presents the distribution over time

of the minimum and the maximum values attainable by this correlation by cohort

and income group as a result of (13) once Z is controlled for. Figures for the 5th,

50th and 95th percentiles of the minimum and maximum value of ρ∗(Z) between

1986 and 2001 are reported. The median of the minimum correlation attainable is

always below -0.90, thus the restrictions exploited in Figures 11-13 rule out high

and positive values of ρ∗(Z).

Table 8 presents the distribution over time of the observed values of ρ∗(Z) for

IS and DS data. Apparently both IS and DS data suggest that the strength of

empirical correlation net of Z is lower than the values considered to derive bounds

in Figures 11-13.

8 CONCLUSIONS

This paper has discussed how to account for reporting errors affecting diary-based

and recall-based data on non-durable consumption. In fact, it is likely that not

all the commodities defining non-durable consumption are well reported exploiting

36
Table 7: Bounds implied by (13)

income group born 1960-69 born 1950-59 born 1940-49 born 19

min max min max min max min 

0-20% 5% -0.88 0.91 -0.87 0.98 -0.91 0.92 -0.86 

50% -0.81 0.99 -0.81 0.99 -0.78 0.99 -0.76 95% -0.70 1.00 -0.76 1.00 -0.65 1.00 -0.68 

20-40% 5% -0.93 0.96 -0.94 0.98 -0.90 0.98 -0.92 

50% -0.89 0.99 -0.90 0.99 -0.85 0.99 -0.82 95% -0.77 1.00 -0.83 1.00 -0.78 1.00 -0.67 

40-60% 5% -0.95 0.97 -0.96 0.98 -0.95 0.98 -0.93 

50% -0.90 0.99 -0.90 0.99 -0.90 0.99 -0.85 95% -0.81 1.00 -0.87 1.00 -0.84 1.00 -0.76 

60-80% 5% -0.94 0.97 -0.96 0.97 -0.95 0.96 -0.92 

50% -0.89 0.99 -0.91 0.99 -0.88 0.99 -0.86 95% -0.83 1.00 -0.87 1.00 -0.81 1.00 -0.76 

80-100% 5% -0.95 0.95 -0.89 0.98 -0.91 0.97 -0.89 

50% -0.87 0.99 -0.85 0.99 -0.83 0.99 -0.81 95% -0.82 1.00 -0.79 1.00 -0.78 1.00 -0.65 

37
Table 8: Observed correlations

income group born 1960-69 born 1950-59 born 1940-49 born 19

IS DS IS DS IS DS IS 

0-20% 5% 0.34 0.26 0.390.46 0.41 0.51 0.42 

50% 0.42 0.51 0.46 0.61 0.48 0.56 0.50 95% 0.51 0.68 0.52 0.71 0.57 0.66 0.63 

20-40% 5% 0.23 0.22 0.35 0.390.28 0.28 0.26 

50% 0.390.50 0.44 0.48 0.42 0.53 0.38 95% 0.49 0.73 0.48 0.60 0.54 0.63 0.45 

40-60% 5% 0.11 0.190.28 0.38 0.23 0.21 0.26 

50% 0.30 0.43 0.34 0.46 0.36 0.490.37 95% 0.41 0.58 0.40 0.61 0.44 0.57 0.45 

60-80% 5% 0.14 0.25 0.190.33 0.23 0.27 0.17 

50% 0.27 0.42 0.34 0.46 0.32 0.43 0.31 95% 0.35 0.51 0.38 0.58 0.44 0.67 0.35 

80-100% 5% 0.18 0.18 0.23 0.25 0.23 0.28 0.26 

50% 0.290.37 0.33 0.40 0.30 0.46 0.3995% 0.40 0.55 0.42 0.56 0.39 0.55 0.46 

38
born 1960−69

0

.1

.2

.3

.4

.5

born 1950−59

born 1940−49

86 88 90 92 94 96 98 100 102

0

.1

.2

.3

.4

.5

born 1930−39

86 88 90 92 94 96 98 100 102

Figure 12: Squared coefficient of variation of family expenditure on non-durable

goods by cohort (2001 dollars) - bounds using (13), Condition 3 and assuming

|ρ∗| ≤ 0.85

born 1960−69

0

.1

.2

.3

.4

.5

born 1950−59

born 1940−49

86 88 90 92 94 96 98 100 102

0

.1

.2

.3

.4

.5

born 1930−39

86 88 90 92 94 96 98 100 102

Figure 13: Squared coefficient of variation of family expenditure on non-durable

goods by cohort (2001 dollars) - bounds using (13), Condition 3 and assuming

|ρ∗| ≤ 0.80

39
only one of these two survey methodologies. Expenditures on frequently purchased,

smaller items are presumably more accurate using diaries while recall data are more

appropriate for large expenditures or expenditures occurring on a regular basis.

It turns out that neither diary nor recall-based data alone provide a reliable

aggregate measure of total expenditure on non-durables. One might argue that for

any practical purpose these alternative data collection strategies lead to consistent

results in the estimation of economic models of consumption behavior. Unfortunately, evidence from the literature suggests that data themselves might not be

suitable to validate existing models and that conclusions are strongly related to the

information being used.

This issue has been addressed by looking at micro data from two independent

samples of households from the Consumer Expenditure Survey for the Unites States.

This survey represent a unique source of data because it consists of diary an recall information collected on the same set of items, although it refers to separate

samples of households. The integration of these datasets presents the problem of

determining the appropriate survey component from which to select expenditure

items. On the basis on evidence reported in a number of previous studies, we have

split the set of commodities entering non-durable consumption into two groups indicating which one of the two survey methodologies (diary or recall) leads to more

accurate data quality.

The comparison of the two surveys based on the overlap in coverage of expenditures offers insights for the effects of collection modes on data quality. Estimating

the true expenditure level on frequently purchased items for the recall sample (and

on bulky items in the diary sample) can be seen as a problem of inferring counterfactuals: what is the counterfactual diary (recall) expenditure measure for recall

(diary) respondents?

We have shown that the pictures emerging from the two surveys are very different, both with respect to mean expenditure and, more importantly, with respect

to indices of inequality. More precisely, while the differences in mean expenditure

are roughly constant over time, consumption inequality presents different levels and

trends in the two surveys. We have reconciled this evidence using integrated diary

and recall information to characterize the most likely pattern of consumption for

cohorts of individuals defined by their age.

This procedure allows us (i) to define an improved measure for mean and variance of non-durable expenditure over the 1990s and (ii) to characterize the measurement error affecting the commodities whose quality is doubtful according to

other studies in the literature. The implications of our findings for the estimation

of inequality indices have been discussed, with an application to permanent income

models. In particular, we have shown that using diary and recall data to improve

the quality of household consumption the permanent income hypothesis cannot be

rejected.

One general lesson can be deducted from this paper. Surveys are large measurement machines where many points might influence the ultimate error distribution.

The collection of consumption data may be better structured as a set of overlapping questionnaires asking about only a small subset of consumption (as for the

Family Expenditure Survey for the United Kingdom). In this way, more accurate information on different components of consumption would be obtained by the

most appropriate survey methodology. Moreover, information from diary and recall

questions collected for the same individuals would permit to shed more light into

the effects of collection modes on the quality of available information. The evidence

40
on this point currently available in the literature is very much limited to a small

groups of commodities (mainly food).

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44
TECHNICAL APPENDIX

Bounds on Cov(C∗

1 , C∗2 |Z) using Condition 3 presented in Section 7 are derived in

what follows. Let

ηt = Cov(C∗

1 , C∗2 |Z)

be the quantity of interest and let

ηt = ηt − ηt−1

be its changes over time. It is assumed that ηt can vary between known values lt

and ut defined by the Cauchy-Schwartz bounds. Moreover, ηt is also identified by

observed changes in IS and DS covariances if Condition 3 holds true.

Let

st = {...,ηt − ηt − ηt−1, ηt − ηt, ηt, ηt + ηt+1, ηt + ηt+1 + ηt+2,... }

be the the time series of covariances given ηt. In light of Condition 3, such a

sequence would be fully determined if ηt were observed.

Let

St = {st : lt ≤ ηt ≤ ut}

the set of sequences st consistent with the assumption that ηt is bounded between

lt and ut. The set

B = 

t

St

defines all the sequences that jointly satisfy the Cauchy-Schwartz bounds over time

and are consistent with the time series of observed changes ηt. This set represents

the basis to derive the bounds exploited in Figures 10-13.

I
DATA APPENDIX

This appendix describes the selection criteria used to derived the working sample for

the analysis presented in this paper. It also discusses how methodological changes

in the definition of consumption categories over time are handled.

Sample information from the the two survey components of the CEX is used for

the period 1982-2001. All those households satisfying at least one of the following

criteria

1. living in rural areas

2. with single females

3. residing in student housing

4. whose head is self-employed

5. whose head is aged below 22 and above 74

6. whose total expenditure on food (at home at away from home) is zero

are not considered in the final sample. Non-urban households are excluded to make

information comparable over time, since they were discontinuously sampled before

1984. Additionally, households

7. not completing the diary for two weeks

8. not belonging to the cohorts analyzed in the paper

are excluded. Sample sizes for each cohort are given in Table A.1. Throughout the

analysis, the family head is conventionally fixed to be the male in all H/W families

(representing 56 percent and 53 percent of the whole sample for IS and DS data,

respectively). Households presenting null expenditure on total food (both at home

and away from home) are dropped from the analysis (less than 1 percent in each

sample). These restrictions leave us with a sample of 190.080 and 46.244 units, for

IS and DS data respectively.

Because of the small within-quarter variation in reported IS expenditures (less

than 2 percent of the total variation in our sample), monthly consumption figures

for IS households refer to the month before the interview. Monthly expenditures

in the DS are instead defined as 26/12 = 2.16 times the expenditure observed over

two weeks, thus assuming equally complete reporting. All expenditure data have

been seasonally adjusted, by taking residuals from regressions on zero-sum monthly

dummies. Real expenditures are obtained using the the Current Price Index (CPI)

published by the BLS. Although CPI bias has been of considerable concern to policy

makers and researchers in recent years (see, for example, Costa 2001), we do not

deal with this issue in the paper.

Expenditures of different family types are adjusted onto a comparable basis

using an equivalence scale which depends on the number of adults and children in

various age ranges. Specifically, the scale assigns weight 1 to the first adult and

0.670 to all remaining adults in the household. Children are weighted 0.233 if aged

3 or below, 0.333 if aged 4 to 7, 0.400 if aged 8 to 12 and 0.533 if aged 13 to 18.

Sensitivity of main findings to alternative equivalence scales was investigated, and

results were found to be qualitatively similar.

II
Table A.1: Sample sizes

Diary sample

year born 1960-69 born 1950-59 born 1940-49 born 1930-39 Totals

1986 257 864 675 419 2,215

1987 383 849 633 466 2,331

1988 345 756 515 374 1,990

1989 422 738 603 412 2,175

1990 497 809 578 459 2,343

1991 574 808 571 396 2,349

1992 603 744 555 352 2,254

1993 624 726 527 370 2,247

1994 560 663 476 295 1,994

1995 542 587 444 265 1,838

1996 688 758 504 328 2,278

1997 722 740 579 411 2,452

1998 674 751 543 347 2,315

1999 985 997 716 481 3,179

2000 1,021 931 722 457 3,131

2001 1,044 950 691 418 3,103

Interview sample

year born 1960-69 born 1950-59 born 1940-49 born 1930-39 Totals

1982 2,881 2,883 1,979 7,743

1983 242 3,226 2,804 1,901 8,173

1984 435 3,059 2,577 1,729 7,800

1985 572 2,765 2,201 1,587 7,125

1986 989 3,498 2,690 2,023 9,200

1987 1,217 3,376 2,609 1,995 9,197

1988 1,436 2,852 2,411 1,611 8,310

1989 1,724 2,768 2,412 1,590 8,494

1990 1,943 2,904 2,309 1,472 8,628

1991 2,030 2,862 2,181 1,568 8,641

1992 2,334 2,869 1,978 1,467 8,648

1993 2,424 2,899 2,159 1,424 8,906

1994 2,380 2,869 2,132 1,384 8,765

1995 2,133 2,526 1,849 1,213 7,721

1996 2,954 3,244 2,380 1,541 10,119

1997 3,088 3,363 2,347 1,585 10,383

1998 3,043 3,221 2,268 1,691 10,223

1999 4,331 4,493 3,147 2,232 14,203

2000 4,393 4,381 3,259 2,216 14,249

2001 4,314 4,099 3,207 1,932 13,552

III
Table A.2: Definitions of expenditure categories

Commodities in D Diary Interview

Food and Non-Alcoholic Beverages at Home 83-87

Food and Non-Alcoholic Beverages Away from Home

Alcoholic Beverages (at home and away from home)

Non-Durable Goods and Services 82-85

Newspapers and Magazines

Non-durable Entertainment Expenses

Housekeeping Services

Personal Care

Commodities in I Diary Interview

Housing and Public Services 82-85

Home Maintenance Services

Public Utilities

Miscellaneous Home Services

Tobacco and Smoking Accessories

Clothing, Footwear and Services

Clothing, Footwear

Services

Heating Fuel, Light and Power 82-85

Transportation (including gasoline) 82-85

Fuel for Transportation

Transportation Equipment Maintenance and Repair

Public Transportation

Vehicle Rental and Misc. Transportation Expenses

IV
80.1 82.1 84.1 86.1 88.1 90.1 92.1 94.1 96.1 98.1 100.1

5.7

5.8

5.9

6

6.1

Figure A.1: Logs of monthly Food at Home expenditure in 2000 dollars - Interview

Survey

Detailed expenditure categories included in total non-durable consumption are

reported in Table A.2. The top panel refers to those commodities labelled in the

text as D, while the bottom panel to commodities in I. The second and the third

columns report the time intervals for which expenditure on non-durable commodities cannot be directly derived from public use tapes because of definitional problems, separately for IS and DS data. Accordingly, expenditures referring to “Housing and Public Services” and “Non-durable Services” have been introduced in the

DS only after 1986, with the exception of very few items for “Home Maintenance

Services” and “Non-durable Entertainment Expenses”. Similarly, information on

“Fuel” and “Transportation” expenses is not available from public tapes between

1982 and 1985.

As for IS data, the time series of food at home expenditure presents discontinuities introduced by changes in survey design in 1982 and 1987. In fact, recall

information on food was derived in different ways over the years and the average

spending appears to be heavily affected by the exact wording, as shown in Figure

A.1. Spikes downwards in 1982 and upwards in 1988 are the effect of asking respondents for usual spending on a monthly basis or on a weekly basis. In 1980-1981

the question was on usual weekly expenditure on food over the past three months.

In 1982-1987 the question was on how often and how much was spent in food over

the previous month. In 1988, the 1980-1981 question was resumed.

Discontinuities in levels are accounted for by running a regression of real food

expenditure on a quadratic time trend and a time dummy for the period 1982-

1987, a polynomial in family income and additional household controls. Reported

expenditure are then scaled up for all households between 1982 and 1987 by the

same amount. Accordingly, the underlying assumption made is that changes in

the wording of food-related questions only affect the level of reporting, but not the

V
variance.

Because of the definitional problems summarized in Table A.2, it follows that

total expenditure figures for IS and DS data are not fully comparable over time.

For this reason, figures reported in the paper for total spending on non-durables

exploit 16 years of data for the DS (i.e. 1986-2001) and 20 years for the IS (i.e. 1982-

2001). The aggregation rule suggested in Table 4 is therefore partially implemented

over time. As a first approximation, only aggregate values of IS expenditures are

considered before 1986, while integrated information for the DS is used after then.

The comparison of mean expenditure for the commodities defined in Table A.2

is presented in Figures A.2-A.10. A detailed description of the items used to define

the categories of non-durable consumption can be downloaded at

http://www.stat.unipd.it/∼erich/papers.html,

separately for IS and DS data.

Diary and recall information is available from the DS on expenditure for food

items. As noted earlier, food and non-alcoholic beverages is the only commodity

entering non-durable expenditure for which recall and diary-based measurements

on the same household are available. The information on the ‘usual’ spending

is collected for each household at the beginning of the (two-week) diary period.

Therefore, its accuracy is presumably not influenced by how respondents learn

about own expenditures during the interview.

Figure A.11 presents the difference between weekly expenditure derived from

diaries (as the sum of detailed food data) and weekly ‘usual’ expenditure for food

and non-alcoholic beverages at grocery stores given by respondents. As already

pointed out by several papers using similar data sources, diary expenditure levels

in the first week tend to be higher that those for the second week. However,

information from recall data leads to higher values of reported expenditures.1

It is worth noting that the difference between diary and recall figures conditional on the expenditure month (to control for seasonal effects) is always negative

for the period of time covered by this analysis, with values generally decreasing in

absolute value over time. There is a mild effect of the interview month on the magnitude of diary under reporting, since households interviewed in December usually

present values closer to zero. This evidence supports the idea that recall questions

overstate the real spending on food-related items possibly because households’ reporting includes more than just food-related goods. Indeed the recall question

about food expenditure in the IS is derived subtracting to the usual amount spent

at the grocery store the usual amount on non-food items.

1We found a very similar pattern comparing recall and diary information from the National

Food Survey for the United Kingdom. Results are available on request.

VI
 Interview data Diary data

born 1960−69

100

150

200

250

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

100

150

200

250

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure A.2: Mean of family expenditure on food and non-alcoholic beverages at

home (2000 dollars)

 Interview data Diary data

born 1960−69

50

100

150

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

50

100

150

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure A.3: Mean of family expenditure on food and non-alcoholic beverages away

from home (2000 dollars)

VII
 Interview data Diary data

born 1960−69

10

20

30

40

50

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

10

20

30

40

50

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure A.4: Mean of family expenditure on alcoholic beverages (2000 dollars)

 Interview data Diary data

born 1960−69

50

100

150

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

50

100

150

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure A.5: Mean of family expenditure on non-durable goods and services (2000

dollars)

VIII
 Interview data Diary data

born 1960−69

40

60

80

100

120

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

40

60

80

100

120

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure A.6: Mean of family expenditure on housing and public services (2000 dollars)

 Interview data Diary data

born 1960−69

5

10

15

20

25

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

5

10

15

20

25

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure A.7: Mean of family expenditure on tobacco (2000 dollars)

IX
 Interview data Diary data

born 1960−69

40

60

80

100

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

40

60

80

100

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure A.8: Mean of family expenditure on clothing, footwear and services (2000

dollars)

 Interview data Diary data

born 1960−69

40

50

60

70

80

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

40

50

60

70

80

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure A.9: Mean of family expenditure on heating fuel, light and power (2000

dollars)

X
 Interview data Diary data

born 1960−69

100

150

200

born 1950−59

born 1940−49

82 84 86 88 90 92 94 96 98 100102

100

150

200

born 1930−39

82 84 86 88 90 92 94 96 98 100102

Figure A.10: Mean of family expenditure on transportation (2000 dollars)

 at grocery stores 1st week diary

 2nd week diary

born 1960−69

20

40

60

80

100

born 1950−59

born 1940−49

86 88 90 92 94 96 98 100 102

20

40

60

80

100

born 1930−39

86 88 90 92 94 96 98 100 102

Figure A.11: Median weekly expenditure on food and non-alcoholic beverages at

home (2000 dollars): usual expenditure at grocery stores and expenditure by diary

week

XI
Table A.3: Gini coefficient for total family income

born 1930-39 born 1940-49 born 1950-59 born 1960-69 

year CEX CPS PSID CEX CPS PSID CEX CPS PSID CEX CPS PSID 

1982 0.34 0.31 0.31 1983 0.34 0.31 0.32 0.30 1984 0.36 0.32 0.32 0.32 1985 0.34 0.32 0.31 0.34 1986 0.32 0.32 0.32 0.32 1987 0.35 0.32 0.31 0.31 1988 0.37 0.32 0.30 0.32 1989 0.37 0.32 0.31 0.31 1990 0.36 0.33 0.31 0.32 1991 0.38 0.32 0.32 0.32 1992 0.41 0.34 0.33 0.34 1993 0.39 0.36 0.33 0.33 1994 0.41 0.35 0.34 0.34 1995 0.40 0.34 0.35 0.34 1996 0.44 0.38 0.34 0.32 1997 0.43 0.37 0.34 0.32 1998 0.42 0.37 0.35 0.34 1999 0.46 0.38 0.35 0.36 2000 0.44 0.38 0.36 0.34 2001 0.43 0.38 0.35 0.34 source for CPS data: http://www.census.gov/hhes/income/histinc/ie1.html

XII
Table A.4: Percentage of consumers: Pr(X > 0|Interview) and Pr(X > 0|Diary)

Interview 1982-85 1986-89 1990-92 1993-95 1996-98 Food and non-alcoholic beverages at home 1.00 1.00 1.00 1.00 1.00 Food and non-alcoholic beverages away 0.890.90 0.88 0.88 0.87 

Alcoholic beverages (at home and away) 0.67 0.63 0.57 0.56 0.51 Non-durable goods and services 0.97 0.97 0.97 0.97 0.95 Housing and public services 0.97 0.97 0.98 0.98 0.98 Tobacco and smoking accessories 0.50 0.44 0.390.36 0.34 Clothing and footwear 0.88 0.87 0.85 0.82 0.78 Heating fuel, light and power 0.88 0.89 0.91 0.92 0.93 Transport (including gasoline) 0.98 0.98 0.98 0.98 0.98 

Diary 1982-85 1986-89 1990-92 1993-95 1996-98 Food and non-alcoholic beverages at home 0.98 0.99 0.98 0.98 Food and non-alcoholic beverages away 0.94 0.92 0.890.90 

Alcoholic beverages (at home and away) 0.590.53 0.50 0.45 Non-durable goods and services 0.94 0.94 0.92 0.91 Housing and public services 0.66 0.65 0.64 0.63 Tobacco and smoking accessories 0.43 0.38 0.34 0.33 Clothing and footwear 0.73 0.73 0.70 0.67 Heating fuel, light and power 0.46 0.51 0.46 0.46 Transport (including gasoline) 0.96 0.94 0.93 0.94 

XIII