Estimating the Level of 

Underreporting of Expenditures 

among Expenditure Reporters: 

A Further Micro-Level Latent Class 

Analysis (August 2005) 

Clyde Tucker, Bureau of Labor Statistics

Paul Biemer, Research Triangle Institute 

Brian Meekins, Bureau of Labor Statistics 

1. Introduction 

This paper makes estimates of the level of 

underreporting of consumer expenditures. The 

paper examines reporting in particular 

commodity categories and attempts only to make 

estimates of underreporting among those that 

report at least one expenditure in the category. 

The measure of the level of underreporting in a 

category by a particular responding unit is based 

on latent class analysis using characteristics of 

the respondent’s reporting behavior. It is 

assumed that the level of underreporting is 

similar within a particular subpopulation defined 

by these characteristics. A demographic analysis 

will identify the types of responding units that 

have various levels of underreporting. 

Latent class analysis, a theory for detecting 

unobserved variables, was developed by Paul 

Lazarsfeld (1950). According to Lazarsfeld, an 

unobservable variable (such as underreporting) 

could be constructed by taking into account the 

interrelationships between observed variables. 

The mathematics underlying this theory were 

extended by Lazarsfeld and Henry (1968) and 

Goodman (1974). 

The paper begins with an introduction to the 

Consumer Expenditure Interview Survey (CEIS) 

sponsored by the Bureau of Labor Statistics 

(BLS) and conducted by the Census Bureau. 

Previous related work by the authors in this area 

is summarized and the design of this particular 

study is outlined. The following section presents 

the analytical results, and a final section is 

devoted to the discussion of the results and the 

consideration of additional avenues for research. 

2. CEIS 

The data used in this study consists of interviews 

collected in six years of the CEIS: 1996 through 

2002. Each survey was designed to collect 

information on up to 95 percent of total 

household expenditures. We define a consumer

unit (CU) as the members of a household who 

are related and/or pool their incomes to make 

joint expenditure decisions. In the CEIS, CU’s 

are interviewed once every three months for five 

consecutive quarters to obtain the expenditures 

for 12 consecutive months. The initial interview 

for a CU is used as a bounding interview and 

these data are not used in the estimation. The 

survey is designed to collect data on major items 

of expense which respondents can be expected to 

recall for three months or longer. New panels are 

initiated every quarter of the year so that each 

quarter, 20 percent of the CU’s are being 

interviewed for the first time. Only CU’s 

completing and reporting an expense in wave 2 

are used in this analysis, for a total of 14,877 

respondents. 

3. Previous Work 

For panel surveys such as the CEIS, a related 

statistical method referred to as Markov latent 

class analysis (MLCA) is available, which 

essentially relaxes the requirement that the 

replicate measurements pertain to the same 

point. Thus, this method of analysis is feasible 

for analyzing repeated measurements of the same 

units at different time points available in panel 

surveys. MLCA requires a minimum of three 

measurements of the same units, as would be the 

case for a panel survey where units are 

interviewed on three occasions. The MLCA 

model then specifies parameters for both the 

period-to-period changes in the status of the item 

as well as the measurement error associated with 

measuring those changes. 

Previous work by the authors used MLCA to 

make aggregate estimates of underreporting in a 

category only by respondents reporting no 

expenditures in that category. Biemer (2000) 

applied the MLCA methodology to the CEIS in 

order to determine whether useful information on 

the magnitudes and correlates of screening 

question reporting error can be extracted directly 

from the CEIS panel data. Biemer and Tucker 

(2001) extended the earlier analysis using data 

from four consecutive quarters of the CEIS by 

considering CU’s that were interviewed four 

consecutive times beginning in the first quarter 

of 1996 and ending in the last quarter of 1998. 

This allowed the authors to consider a widerrange of models including second-order Markov 

models. First order Markov models assume that 

a purchase or non-purchase at quarter q is 
affected only by quarter q-1 purchases or nonpurchases. A second order Markov model 

assumes that both quarters q-1 and q-2 affect 

purchasing behavior at quarter q. Their analysis 

provided evidence of second-order Markov 

effects and recommended that second-order 

terms be included in the models. 

In Tucker, Biemer, and Vermunt (2002), model 

estimates with both unweighted and weighted 

data were compared. The results indicated that 

few differences were found between the two; 

therefore, given the ease of use, unweighted data 

were used in these analyses. A thorough 

examination of all explanatory variables 

considered in the previous studies was 

undertaken, and a reduced set of the most 

powerful ones was identified. A new diagnostic 

technique was developed and used to evaluate 

the validity of the models. In 2003, Tucker, 

Biemer, and Meekins developed methodology 

for estimating the amount of the missing 

expenditures. 

Unlike the previous work, a micro-level 

approach incorporating measures specific to a 

given interview was used by Tucker, Biemer, 

Meekins, and Shields (2004) to examine 

underreporting for for total expenditures across 

approximately 20 commodity categories. In 

essence, a latent variable that adequately 

accounted for the shared varaince among a set of 

observed response error indicators was created. 

The observed varaibles were based on 

information collected from each CU during the 

interview. The latent variable was believed to be 

a better measure of underreporting than any of 

the observed varaibles taken individually. Each 

CU then was assigned to a particular class of the 

latent variable representing its hypothesized level 

of expendituire underreporting based on the 

CU’s values on the observed variables. See 

Tucker (1992) for an earlier empirical example.

We used only second wave data1

. We examined 

reporters of expenditures and ignored 

nonreporters. We wished to develop a model 

separate from covariates with only indicators of 

the quality of response. We began with the 

simplest identifiable model composed of three 

indicators (each with three classes) and a latent 

variable with three classes. From this point we 

ran all possible combinations of three indicators 

 

1

 Wave 2 data are used because wave 1 is a 

bounding interview. 

for a three class latent variable. The analysis 

was further extended by examining restricted 

models based on the hypothetical relationship of 

some of the indicators with the latent variable, 

thus ordering the latent classes in what we 

believed to be an interpretable manner. These 

“restricted” models were compared to the 

unrestricted models to aid in interpretability and 

choices of model fit. Some of the indicators are 

dichotomous. These were entered into the best 

three variable models along with other 

combinations to create four-indicator models. 

Our goal was to develop a latent variable 

(preferably ordered) that indicated the quality of 

responses, such that poor reporters could be 

easily identified. 

 

Models were estimated using lEM. Model 

selection was based on a number of objective 

and subjective measures. We primarily used the 

Bayesian Information Criteria (BIC), the L2 test 

statistic, and the dissimilarity index. However, 

for each model we examined the conditional 

probabilities of the latent variable given each 

value of each indicator. In this way we assessed 

the relative influence of each indicator and the 

degree to which an indicator effectively 

differentiated the respondents with respect to the 

classes of the latent variable 

Using these methods a “best” model was 

selected. Latent classes aligned with expenditure 

means as expected. Those with lower 

expenditure means had higher levels of 

underreporting. For example, those in the low 

underreporting class had a total expenditure 

mean of $10,625, while those in the high 

underreporting class had a mean of $6,948 

4. Current Design 

In this paper, we continue with a more in-depth 

exploration of micro-level measures of 

underreporting. Rather than considering just 

total expenditures, separate categories of 

expenditures are examined. A more refined set 

of indicators of underrepoting are utilized, and 

they are the following: 

1. Number of contacts the interviewer 

made to complete the interview 

2. The ratio of respondents to total number 

of household members 

3. The ratio of household members 

earning an income to the total number 

of household members 
4. Whether the household was missing a 

response on the income question 

5. The type and frequency of records used. 

This variable indicates whether a 

respondent used bills or their checkbook 

to answer questions, and how often they 

did so. 

6. The percent of data requiring 

imputation or allocation. 

7. The length of the interview 

8. A ratio of expenditures reported for the 

last month of the 3 month reporting 

period to the total expenditures for the 3 

months 

9. And a combination of type of record 

used and the length of the interview. 

5. Results 

The LCM table gives the model fits for the 

commodity groups examined. All commodities 

had acceptable fits based on the statistical 

diagnostics. The dissimilarity indexes were less 

than .05 in all cases. The BIC was negative in 

each case, and the absolute size of the BIC 

depended on the degrees of freedom in each 

model –the larger the degrees of freedom, the 

greater the BIC. 

For each commodity, cases were assigned values 

based on an appraisal of the loadings for each 

cell on three latent classes. These classes were 

labelled Poor Reporting (1), Fair Reporting, (2), 

and Good Reporting (3) based on theoretical 

expectations concerning the individual indicators 

in the models. The most influential indicators 

were number of contacts, missing income, length 

of the interview, and use of records. 

The substantive diagnostic chosen for evaluating 

the models was the mean expenditure of the 

commodity for each laent class. The expenditure 

means table indicates that, for Kid’s Clothing 

and Minor Vehicle Repairs, the expenditures are 

in the expected direction—increasing from Poor 

to Good. For Women’s Clothing, Men’s 

Clothing, and Kitchen Accessories, the Poor 

category, indeed, has the lowest mean; but the 

other two categories are indistinguishable. It is 

possible that a two-class model is more 

appropriate in these cases. The model for 

Furniture does not follow the expected pattern. 

Other models should be examined in this case. 

The means for Electricity also do not have the 

expected pattern, but earlier results indicated that 

electricity is reported very well by most 

respondents. 

The proportional odds model for Minor Vehicle 

Repairs presents an example of the factors most 

important for determining respondents’ scores on 

the latent variable. Odds ratios geater than 1.0 

indicate a tendency toward better reporting, and 

those less than 1.0 indicate the opposite 

tendency. Poorer reporting is most common for 

young people, single-person households, those 

with less income,and renters. 

6. Discussion 

These results provide some validation for the 

latent class approach to modeling measurement 

error. Other commodity categories as well as 

total expenditure will be examined. Adding a 

Markov component that takes advantage of 

expenditure patterns over the four quarters might 

lead to improvements in the models. Eventually, 

estimates from both nonreporters and 

underreporters will be combined to produce 

overall estimates of underreporting by 

commodity category and respondent 

characteristics. 

7. References 

Bassi, Franesca, Jacques A. Hagenaars, Marcel 

A. Croon and Jeroen Vermunt. (2000). 

“Estimating True Changes When Categorical 

Panel Data Are Affected by Uncorrelated and 

Correlated Classification Errors: An 

Application to Unemploymnet Data.” 

Sociological Methods & Research 29: 230-

268. 

Biemer, P.P. (2000). “An Application of Markov 

Latent Class Analysis for Evaluating the 

Screening Questions in the CE Interview 

Survey,” Technical Report Submitted to 

BLS, June 6, 2000. 

Biemer, P. P. and Tucker, C. (2001). “Estimation 

and Correction for Purchase Screening Errors 

in Expenditure Data: A Markov Latent Class 

Modeling Approach,” Proceedings of the 

International Statistics Institute, Seoul, 

Korea. 

Tucker, C. (1992). “The Estimation of 

Instrument Effects on Data Quality in the 

Consumer Expenditure Diary Survey.” 

Journal of Official Statistics. 8: 41-61. 

Tucker, C., Biemer, P., and Vermunt, J. (2002). 

“Estimation Error in Reports of Consumer 

Expenditures,” Proceedings of the ASA, 
Survey Research Methods Section, New 

York, NY. 

Goodman, L. A. (1974), "Exploratory Latent 

Structure Analysis Using Both Identifiable 

and Unidentifiable Models," Biometrika, 61, 

215-231. 

Langeheine, R. and Van der Pol, F. (2002). 

“Latent Markov Chains,” in Hagenaars, J. 

and McCutcheon, A. (eds.) Applied Latent 

Class Analysis, Cambridge University Press, 

Cambridge, UK 

Lazarsfeld, P.F. (1950). “The Logical and 

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II, Vol. 4, Measurement and Prediction. 

Princeton: Princeton University Press. 

Lazarsfeld, P.F. and Henry, N.W. (1968). Latent 

Structure Analysis. Boston: HoughtonMifflin. 

Tucker, C., Biemer, P., and Meekins, B. 

(2003),“Latent Class Modeling and 

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Reports,” Proceedings of the Section on 

Survey Research Methods, Washington, DC: 

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Tucker, C., Biemer, P., Meekins, B., and Shields, 

J. (2004), “Estimating the Level of 

Underreporting of Expenditures among 

Expenditure Reporters: A Micro-Level 

Latent Class Analysis,”Proceedings of the 

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Van de Pol, F. and de Leeuw, J. (1986). “A 

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Appendix 

Indicator Coding

• #contacts (1=0-2; 2=3-5; 3=6+)

• Resp/hh size (1= <.5; 2= .5+)

• Income missing (1=present; 2=missing)

• Records use (1=never; 2=single type or 

sometimes; 3=multiple types and always)

• Interview length (1= <45; 2=45-90; 3= 90+)

• Month3 expn/all (1= <.25; 2= .25-.5; 3= +.5)

• Combined records and length (1= poor; 2= fair; 

3=good) 

Demographic Coding

• CU size (1=1; 2=2; 3=3+)

• Age (1= 30<; 2= 30-49; 3=50+)

• Education (1=< H.S.; 2= H.S.+)

• Income rank (1= <=.25; 2=.25-.75 and 

missing; 3=+.75)

• Race (1= White; 2= Other)

• Tenure (1= renter; 2= owner)

• Urban (1= urban; 2= rural)

-119.7008 0.0288 Kitchen 

Accessories

-221.5239 0.0306 Minor Vehicle 

Repairs

Electricity -8.6450 0.0021

Furniture -239.6923 0.0327

Men’s Clothing -244.0034 0.0258

Women’s Clothing -23.1575 0.0221

Kid’s Clothing -7.4029 0.0040

BIC Diss Index

LCM Fit by Commodity
Value of Latent Class

2,934

52.51(a)

26,288

57.03(b)

4,377

198.87(b)

16,315

66.22(b)

842

107.04(a)

11,281

148.08(a)

12,001

59.62(b)

2

26,589

23.27(b)

9,864

39.47(a)

32,905

230.47(a)

23,437

117.25(a)

36,080

78.98(b)

21,316

99.00(b)

24,666

44.90(a)

1 = Poor

n 13,.475

Kitchen Mean 47.58(a)

Accessories

n 6,846

Minor Vehicle Mean 82.28(c)

Expenditures

n 5,716

Electricity Mean 223.30(c)

n 3,246

Furniture Mean 266.63(c)

n 6,076

Men’s Clothing Mean 105.46(a)

n 10,401

Women’s Mean 152.94(a)

Clothing

n 6,331

Kid’s Clothing Mean 71.09(c)

3 = Good

Expenditure Means by Latent Class

Max-rescaled R .0566 2

urban 1.104 <.0001

Tenure .909 <.0001

Race 1.056 <.0001

Inclass 2 .762 <.0001

Inclass 1 .776 <.0001

Educ .984 .1492

Age 2 1.033 .0293

Age 1 .862 <.0001

Famsize 2 1.063 <.0001

Famsize 1 .801 <.0001

PR(X2 Exp(b) )

Proportional Odds Model Results for Minor Vehicle