Estimating the Level of 

Underreporting of Expenditures 

among Expenditure Reporters: 

A Micro-Level Latent Class Analysis 

Clyde Tucker, Bureau of Labor Statistics 

Paul Biemer, Research Triangle Institute 

Brian Meekins, Bureau of Labor Statistics 

Jennifer Shields, 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 demographic 

characteristics as well as 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. 

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 

In this section, we describe the data sets that will 

be analyzed in the study and some key 

operational definitions. 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 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. Finally, methodology 

for estimating missing expenditures was 

outlined. 

4. New Design 

Unlike previous work, these authors chose a 

micro-level approach incorporating measures 

specific to a given interview. 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. At 

this point we also allowed the latent variable to 

have four classes. Our goal was to develop a 

latent variable (preferably ordered) that indicated 

the quality of responses, such that poor reporters 

could be easily identified. The following 

indicators were explored: 

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. Model Selection 

Models were estimated using lEM. Model 

selection is 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 

values 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 (See Table 1 for and 

example). 

In addition to model fit, we used the ordered 

models as a guide. If the indicators of the 

unrestricted model were aligned to the latent 

classes in a similar manner as the restricted 

model, then the unrestricted model showed 

promise, at least in its interpretability. In cases 

where they did not and the objective fit statistics 

were not good, the models were immediately 

discarded. We were looking for a final model 

that was able to line up indicators with the latent 

classes in a logical way, while not sacrificing 

model fit. 

Boundary values were also of considerable 

influence in selecting models, especially in the 

case of restricted models. The use of ordinal or 

inequality restrictions can lead to a number of 

boundary issues, especially if the models fit 

poorly. These models were discarded if it was 

obvious that the ordinal constraints were the 

source of the boundary problem, regardless of fit 

statistics (although they were usually quite poor). 

If this was not obvious, the models were re-run 

with a priori specified starting points on the 

conditional probabilities so that they were 

“pushed-off” of the boundary. 

Table 2 shows a portion of one of the tables for 

the purposes of example used to examine 

differentiation between latent classes. The first 

three columns show the values of the indicators. 

The final three columns show the probability of a 

respondent falling into that group of indicators 

given each value of the latent class, or X. For 

example, approximately 76% of respondents 

with a value of “1” on each of the three 

indicators are classified as a “1” on the latent 

variable. 

If values are higher for one latent class than the 

other two, we can say that the indicators are able 

to differentiate between them, if the values are 

similar, then we would not be able to draw this 

conclusion. 

6. Best Model 

Using these methods a “best” model was 

selected. The model uses four indicators to 

define a three class latent variable. This model is 

not ordinal, and has adequate fit statistics: 

 BIC = -86.01, Dissimilarity index = 0.02, L2 = 

233.96, p=0.00. 

 The indicators are: 

1. Number of contacts 

2. Missing income question 

3. Type and frequency of records used 

4. Length of interview 

Each of these variables is thought to be related to 

the amount of effort expended by the respondent. 

After this final model was identified, we again 

used the probability of being in each latent class 

given a combination of indicators to assign each 

combination to a latent class, using the plurality 

rule. We then returned to the CE data and 

assigned each respondent to that latent class 

which corresponded to their characteristics. 

Expenditure means were then found for each 

“latent” class. 

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 (See Table 3). This may suggest 

that those in the high underreporting class 

failed to report more of their expenditures than 

did those in the other two classes. 

Table 1: Probability of Indicator A Given 

Latent Variable X 

A X P(A|X) 

1 1 .0794 

2 1 .9206 

1 2 .8119 

2 2 .1881 

1 3 .1217 

2 3 .8783 

Table 2: The Probability of Latent Variable X 

Given Indicators A, B, and C 

 X 

A3 B3 C3 1 2 3 

1 1 1 0.7638 0.0013 0.2349 

1 1 2 0.6370 0.1903 0.1727 

1 1 3 0.1089 0.8232 0.0679 

1 2 1 0.7020 0.0021 0.2959 

1 2 2 0.5297 0.2735 0.1967 

1 2 3 0.0671 0.8757 0.0573 

1 3 1 0.5617 0.0014 0.4369 

. . . . . . 

. . . . . . 

. . . . . . 

3 3 3 0.0233 0.6542 0.3225 
After examining how each indicator related to 

the latent classes, we found that those in the 

latent class with the least underreporting: 

1. Were not more or less likely to have a 

certain number of contacts, 

2. Were more likely to use more records 

frequently 

3. Were more likely to have longer 

interviews and 

4. Were more likely not to have missing 

data on the income question 

Expenditure means for respondents assigned to 

each latent class confirmed this finding (See 

Table 4). 

7. Discussion 

This paper is ony a preliminary step toward a 

final measure of underreporting in the CEIS. A 

more thorough analysis of micro-level models 

will be completed, and a final model will be 

selected. Demographic analyses using the latent 

variable as a dependent variable will be 

examined to identify the charactersitics of 

members of each underreporting classs. Besides 

looking at mean total expenditure, the means for 

each latent class by commodity category will be 

compared to evaluate the discriminatory power 

of the latent variable for different types of 

expenditures. 

At the final stage, the results from this microlevel analysis will be merged with the earlier 

aggregate level analysis to evaluate the overall 

underrepoting of expenditures in the CEIS. At 

that point, the contributions to underreporting 

from both nonreporters and incomplete reporters 

will be estimated for each commodity category 

and compared to external estimates of 

underreporting.

References 

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Screening Questions in the CE Interview 

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Table 3: Mean Expenditure by Latent Class 

Latent Class N. Expenditure 

Mean St. Dev 

Low 

Underreporting 11,507 10,625.18 112,8021

Moderate 

Underreporting 18,963 8,137.10 845,866

High 

Underreporting 12,525 6,948.97 904,372

Table 4: Indicator Mean by Latent Class 

Latent Class Indicator Indicator 

Mean*

Contact 2.025

Interview Length 1.069

Type of Records 2.11

Low 

Underreporting 

Missing Income 2.80

Contact 1.91

Interview Length 1.00

Type of Records 1.53

Moderate 

Underreporting 

Missing Income 1.90

Contact 2.08

Interview Length 1.64

Type of Records 1.25

High 

Underreporting 

Missing Income 1.43

*Coding for indicators provided in Appendix A. 
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Appendix A 

Coding for indicator variables used in final 

model: 

Amount of contact 

 1 “0-2 contacts” 

 2 “3-5 contacts and missing” 

 3 “6 + contacts” 

Interview Length 

 1 “ < 45 minutes” 

 2 “45 <= minutes < 90” 

 3 “>=90” 

Type of records 

 1 “Almost never or never use of 

 records” 

 2 “Single type of record and/or 

 mostly or occasionally used 

 records” 

 3 “Multiple types of records, 

 almost always or always” 

Missing on income 

 1 “Income not missing” 

 2 “Income missing”