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Section on Survey Research Methods – JSM 2009

Sample Design and Estimation of Volumes and Trends
in the Use of Paper Checks and Electronic Payment
Methods in the United States
May X. Liu1, Geoffrey R. Gerdes1, Darrel W. Parke1
1

Federal Reserve Board, Washington, DC 20551

Abstract
The Federal Reserve System relies on surveys of banks to monitor the aggregate use of
paper checks and other major noncash payment methods. In recent surveys, the bank
population was stratified by type and by a universally available measure of size
correlated with payments, checkable deposits. For the estimation of, say, the number of
checks, the separate ratio estimator has many desirable features. However, questions
arose as to which and how many auxiliary variables should be used. Also, due to varied
and significant levels of item nonresponse and adding-up requirements, constrained
imputation methods were used for estimation which created special challenges for
constructing error measures using standard methods, e.g., multiple imputation. Despite
the difficulties, we find that the conclusion that check usage is declining relative to
electronic payment methods is robust.

Key Words: Ratio estimator, auxiliary variables, item nonresponse, imputation, sample
design

1. Background
An efficient payments system is important for the smooth functioning of the large and
complex U.S. economy. In the 20th century, the use of cash and checks were the
predominant methods of payment in the United States and paper checks accounted for the
majority of noncash payments. As the availability and use of technology has evolved,
payments by cards and other electronic methods have become increasingly common
among individuals, businesses, and governments. In addition, checks themselves are
increasingly being cleared electronically.
Over the last decade, the Federal Reserve has conducted several payments studies to
estimate changes in the aggregate number and value of check and electronic payments.
The aggregate number and value of checks need to be measured by surveying depository
institutions (banks) because check processing is not centralized in the same way as, for
example, card networks. Furthermore, there are a variety of ways that checks can be
processed, and the transformation of paper-based clearing to electronic image-based
clearing spurred by the “Check 21” law prompted the need to estimate not only changes
in the aggregate number and value of check payments, but also changes in the underlying
proportions of paper and electronic check clearing methods.1
Our check estimates are based on data collected from several voluntary bank surveys.
The recent surveys were conducted in 1996, 2001, 2004, 2006, and 2007, and were used
1

For information on the Check 21 law see
http://www.federalreserve.gov/paymentsystems/truncation/default.htm.

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Section on Survey Research Methods – JSM 2009

to estimate figures for the year that preceded each survey. 2 The surveys contained
questions on a core set of items regarding checks, and as new policy questions came to
the forefront, other questions were added or deleted to adapt to demand. In general, the
surveys have increased in complexity, and we have adopted new methods and analysis
over time.
The Federal Reserve conducted another bank check survey in 1979 and the Federal
Deposit Insurance Corporation conducted one in 1971. Over the years, estimates from the
bank surveys were combined with estimates from other Federal Reserve studies to
compute national estimates of noncash payments (Figure 1). The estimates showed that
checks peaked sometime around 1995 and declined since then. The most recent study
indicated that by 2006 the number of electronic payments was about twice the number of
check payments, or about two-thirds of all noncash payments.
Billions of payments
70
Electronic
60

Check

50
40
30
20
10
0
1971

1979

1995

2000

2003

2006

Figure 1: Noncash payments in the United States, selected year.

For simplicity and because checks are the main focus of our surveys, this article will
concentrate on checks. We will discuss the design of the 2007 bank survey, and consider
whether the separate ratio estimates for total checks can be improved through the use of
alternative auxiliary variables (covariates and stratification variables). We will also
discuss some issues we have encountered in dealing with item nonresponse, and how we
have used imputation to address them. Our analysis will show that a new stratification
variable may improve the estimates in future surveys, but does not suggest the
replacement of our traditional covariate. Imputation achieves further improvements for
the estimates. Finally point estimates among the different estimators we investigated
continue to support our findings about recent trends in checks and other noncash
payments.

2

Detailed reports on these and related surveys are available at
http://www.federalreserve.gov/paymentsystems/paymentsresearch.htm and
http://www.federalreserve.gov/boarddocs/RptCongress/check21/check21.pdf
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Section on Survey Research Methods – JSM 2009

2. Survey Design3
In the 2007 survey, the questionnaire collected data on checks, as well as ACH payments,
debit card payments, and ATM withdrawals.4 The survey period was March and April of
2007. During each of these two months, banks were asked to report the number and
dollar value of each payment type. For presentation purposes, the reported data are
annualized by multiplying the sum of the two months of data by six.
The population in 2007 comprised over 13,000 insured banks, broadly divided into the
categories of commercial banks, savings institutions, and credit unions. Affiliated banks
were treated as a single entity. These banks provide a variety of balance sheet and
income statement information on a periodic, usually quarterly basis. One balance sheet
item from these so-called “call reports” is the value of total checkable deposits, which we
call CHKD.
For simplicity of analysis, we concentrate on the commercial banks, which represent
about half of the bank population and which are responsible for the majority of check
payments. Because most checks are paid from checkable deposit accounts, CHKD has a
natural connection with the volume of paid checks.5 Over the years, CHKD has been
found to be highly correlated with the reported number and value of check payments
across banks. (See Figure 2 for an example.) Traditionally, CHKD has been used as our
size variable and covariate for the separate ratio estimator, as provided in public reports
such as Gerdes and Walton (2002), Gerdes, Liu, Parke, and Walton (2005), Board of
Governors (2007), and Gerdes (2008).
The population of banks is highly-skewed, as demonstrated in an empirical density plot
of CHKD for commercial banks (Figure 3). In the banking industry, most of the assets,
deposits, and other activities are controlled by a small number of very large banks. To
account for the skewness, we used a stratified random sampling approach in order to
achieve higher precision in the estimates of checks by using a separate ratio estimator
with CHKD as covariate.
We stratified the population by the value of CHKD as of September 2006. This was the
most current bank data available that would also allow enough time to prepare for data
collection in Spring of 2007. The largest banks, as determined by the value of CHKD,
and some banks known to have highly unusual check volumes, such as issuers of rebate
checks, were grouped in a certainty stratum, meaning that all were included in the
sample. The remaining banks were then stratified by CHKD. The strata boundaries were
chosen using the cum f method (Dalenius and Hodges 1959).

3

Here we discuss the most recent survey, but much of the discussion applied to the previous
surveys as well. In cases where differences between surveys are relevant, they will be mentioned.
4
A copy of the survey instrument from the 2007 survey is available starting on page 88 of
http://www.frbservices.org/files/communications/pdf/research/2007_depository_institutions_paym
ents_study.pdf
5
Checkable deposits are the only type of bank deposits against which an unlimited number of
payments may be made. Other types of accounts are limited to no more than six payments per
month.
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Section on Survey Research Methods – JSM 2009

22

20

18

16

14

12

10

8
6

8

10

12

14

16

18

20

Figure 2: Scatter plot of the log of the number of checks (y-axis) against the log of checkable
deposits (CHKD). Axes are in logs for display purposes.

Figure 3: Empirical density function of checkable deposits (CHKD) for the 2007 population of
commercial banks.

Based on experience with previous surveys, which had overall response rates higher than
50 percent, a stratified random sample of about 1,500 banks was chosen to produce
estimates with an expected precision of at least ±5 percent at a 95 percent level of

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confidence. We used a Neyman approach to allocate the sample for the noncertainty
portion of the population.
This combination of boundary selection and sample allocation was expected to minimize
the standard error of the estimated aggregate number of checks with a separate ratio
estimator for each size stratum.
By the time survey responses had been received, March 2007 financial data, including
CHKD had become available. Using those later data, the sample and population were
restratified. Strata changed because of changes in reported values of CHKD, and also
because of the entry and exit of some banks between the sampling date and the survey
period. The restratification allows us to group banks together that are more similar to
each other at the time of data collection, and better represent conditions at the time of the
survey.
A notable change resulting from the restratification was an adjustment to the largest size
stratum so that it would be a certainty stratum (that is, all members of the stratum must
have responded to the overall survey, although not necessarily to each item). Size
differences between the largest banks are greatest. Regrouping the largest banks into a
certainty stratum greatly reduces total variance because the finite population correction
factor, discussed in Section 3, becomes zero for that stratum.

3. Estimation Models
The traditional estimates for the population of commercial banks were made using
separate ratio estimators for each size stratum with CHKD as covariate and stratification
variable. In section 4, we will investigate several alternatives to this measure of bank
size.
Let yhi be the reported amount of the dependent variable of interest for the ith bank in
stratum h and let xhi be its covariate, either CHKD or another variable to be introduced
later, where h  1,..., L, i  1,..., nh , and L is the total number of strata while nh is the
number of respondents in stratum h. Then the ratio estimate for the population total Yˆh
of stratum h is given by the reported total multiplied by the ratio of the covariates in the
population to the covariates from the respondents:

y
X
Yˆh  rh X h  h X h  yh h ,
xh
xh
where xh 

nh

nh

i 1

i 1

 xhi and yh   yhi are the respondent total for the covariate and the

dependent variable, respectively, X h 

Nh

x
i 1

hi

is the population total of the covariate, and

N h is the total number of banks in the population.

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The estimated standard error for Yˆh is given by the following classical formula that
accounts for the uncertainty arising from sampling:

 N 2 (1  f h ) 2 
sh 
ˆYˆ  var(Yˆh )   h
h
nh



( y

where sh  [

hi

1/ 2

,

 rh xhi )2 / (nh 1)]1/2 , f h  nh / N h is the sampling fraction, and the

factor (1  f h ) is the correction for a finite population. We used an alternative version of
the variance discussed in Rao (1978), which accounts for the relative size of banks in the
stratum population and response:

var* (Yˆh )  ( X h / xh )2 var(Yˆh ).
Based on the separate ratio estimators, the estimated population total and associated
variances are the sum of the stratum total estimates Yˆ 

L

 Yˆ
h 1

estimates var * (Yˆ ) 

and stratum variance

h

L

 var (Yˆ ), respectively.
*

h 1

h

As we shall be comparing alternative covariates, we will also compare the univariate
estimators with multivariate versions discussed by Olkin (1958). In the multivariate
extension we assume p covariates X 1 , , X p . Without losing generality, we also assume
only one stratum. Then the multivariate ratio estimate of the population total is given by

Yˆ  wˆ 1r1 X 1    wˆ p rp X p ,

wˆ  ( wˆ 1 , , wˆ p ), and

where ri  y / xi for i  1,..., p,

p

 wˆ
i 1

p

 1 is a weighting

function. Weights that minimize the variance of the estimated population total are

eAˆ 1
wˆ  1 ,
eAˆ e '
with corresponding variance

N ( N  n) 1
,
n
eAˆ 1e '

var(Yˆ ) 

where e  (1, ,1)1 p and Aˆ  ( aˆij ) pxp with
n

aˆij 

 ( y  r x )( y  r x
t 1

t

i it

n 1
6
2289

t

j

jt

)
.

Section on Survey Research Methods – JSM 2009

4. Covariate and Stratification Variable Selection
Our traditional covariate CHKD can be used for other types of payments, but funds from
other types of accounts can also be used to make payments. In addition to checks,
electronic transfers can be initiated from checkable deposits using the automated
clearinghouse (ACH) system, debit card networks, automated teller machine (ATM)
networks, and other funds transfer systems. Banking regulations require that only a
limited number of withdrawals (six per month or per statement cycle) can be made from
other types of accounts such as savings and money market deposit accounts (MMDAs)
for payments.
Changes in the way that banks report their deposits over time has, however, led to an ever
increasing disconnection between our measured checkable deposits and the funds that are
used to pay checks. Since 1994, banks have increased the use of so-called retail sweep
programs. Retail sweep programs, which first appeared in January 1994, are designed to
reduce the required amount of funds banks must hold on reserve at the Federal Reserve.
In a retail sweep, banks move unused funds from checkable deposit accounts to special
purpose MMDA subaccounts and return them to the checkable deposits only as needed to
cover payments. This practice does not adversely impact the accountholder, but allows
the bank to reduce nonearning assets.
Over time, these retail sweeps have expanded, increasing the importance of the funds
held in MMDAs for check payments. We would prefer to obtain a direct measure of the
amount swept to sum with CHKD, but it is not available at the bank level. (We can
observe total MMDA but not that portion used in the sweep accounts.) Still, the sum of
CHKD and MMDA (CHKD+MMDA) might be useful as a covariate. As shown in
Figure 4, the sum of CHKD and the estimated aggregate amount of funds swept into
MMDAs was about twice CHKD in March and April of 2007, while CHKD+MMDA
was several times larger, and growing.
Banks’ increasing use of retail sweep programs suggested to us that CHKD+MMDA
might perform well as stratification variable and/or covariate. As MMDAs are used for
purposes other than sweep accounts it was unclear a priori whether CHKD+MMDA
would perform better than CHKD. In addition, we wanted to know how well other
measures of size might perform, because other measures of size could influence check
payments indirectly. Bank customers could, for example, move funds between CHKD
and other accounts on their own. Thus, we also considered the use of total deposits—the
sum of CHKD, MMDA, and other savings and time deposits—and total assets—a
traditional measure of bank size. To compare the alternatives, we stratified by the four
variables, and we combined each stratification variable with each covariate to estimate
the total number and total value of checks paid by commercial banks.

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4,500

4,000

Checkable Deposits

3,500

Checkable deposits + sweep estimate
Checkable Deposits + MMDAs

3,000

2,500

2,000

1,500

1,000

500

0
1993

1995

1997

1999

2001

2003

2005

2007

2009

Figure 4: Aggregate checkable deposits (CHKD), CHKD plus estimated amount swept into
money market deposit accounts (MMDAs), and CHKD plus MMDAs from 1993-2009, billions of
dollars. Amounts are not adjusted for inflation. Sources: Federal Reserve Bank of St. Louis and
Federal Reserve Board.

Because of changes in the rank of respondents caused by restratifications with different
variables, the size of the largest-bank certainty strata varied. For example, stratification
by CHKD had the largest certainty stratum (37 members), reflecting the original design
of the study. By comparison, stratification by MMDA+PCD included only 25 members.
Because of the finite population correction factor, these differences could bias
comparisons in favour of stratification variables that produced larger certainty strata. To
control for this, we reduced the size of the certainty stratum of all estimates to 25. (Of
course, membership in the certainty strata varied depending on the variable used.)
The estimates of the number and value of paid checks for commercial banks using
different combinations of stratification variables and covariates are shown in Table 1.
The table shows that the point estimates using alternative variables differ by no more than
6 percent from the traditional estimates (CHKD as covariate and stratification variable).
The relative differences between standard errors, however, are much larger, with the
largest differences exceeding 30 percent. None of the combinations clearly dominates.
Thus, the choice appears to be left to our judgement. Attempting to strike a balance in
minimizing the standard error for both the number and value estimates, we tentatively
prefer the estimates that use CHKD+MMDA as the stratification variable with CHKD as
covariate.
We also investigated the performance of bivariate (two covariate) ratio estimators using
several ways of pairing the four variables. The estimates of the number and value of paid
checks for commercial banks in Table 2 show the 5 (out of 8) covariate combinations that
appeared to perform best, each using the different stratification variables. Among the
bivariate estimates, none dominates, but we find that, as with the univariate models,
among the estimates CHKD+MMDA performs well as a stratification variable. These
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estimates show some improvement to the standard errors, but perhaps not enough to
abandon the simplicity of a univariate model. For example, if all microdata satisfy the
logical constraints, then so will the aggregates produced by the separate ratio estimators.
But with a bivariate ratio estimator, the aggregate estimates may not satisfy the adding-up
constraints unless one imposes additional constraints that the weights ( w ) are equal
across all estimates.
Table 1: Univariate ratio estimates of paid checks (number and value) with alternative
covariates and stratification variables for the commercial bank population.
Stratification
Variables
EST.
SE.
EST.
SE.
EST.
SE.
EST.
SE.

CHKD
CHKD +
MMDA
Total
Deposits
Total Assets

CHKD
# (mil)
$ (bil)
23,577
37,110
331
930
22,716
35,429
232
806
22,169
35,047
229
878
22,155
34,984
230
957

Covariates
CHKD+MMDA
Total Deposits
# (mil)
$ (bil)
# (mil)
$ (bil)
23,052
35,764
22,769
35,702
270
954
230
996
23,481
36,100
23,099
35,840
228
890
221
922
22,849
35,540
23,288
36,120
231
951
208
974
22,772
35,363
23,240
36,014
236
931
217
999

Total Assets
# (mil)
$ (bil)
22,852
35,588
255
1,204
23,152
35,706
251
1,126
23,224
35,949
289
1,159
23,379
35,994
235
1,211

Table 2: Bivariate (two covariate) ratio estimates of paid checks (number and value) with
alternative bi-variates and stratification variables for the commercial bank population.
Covariates
Stratification
Variables
CHKD
CHKD +
MMDA
Total
Deposits
Total
Assets

EST.
SE.
EST.
SE.
EST.
SE.
EST.
SE.

CHKD,
MMDA
# (mil)
$ (bil)
23,274
36,214
233
801
23,226
36,006
210
762
22,561
35,496
200
822
22,449
35,334
208
822

CHKD,
Total deposits
# (mil)
$ (bil)
23,039
36,091
215
805
22,913
35,793
196
743
22,946
35,961
190
806
22,886
35,959
195
839

CHKD,
Total Assets
# (mil)
$ (bil)
23,136
36,340
228
750
22,867
35,954
206
670
22,801
36,098
203
725
22,892
36,171
203
806

Total Deposits,
CHKD+MMDA
# (mil)
$ (bil)
22,817
35,692
227
931
23,233
35,859
205
852
23,205
36,013
204
909
23,148
35,650
214
924

Total Assets,
CHKD+MMDA
# (mil)
$ (bil)
22,871
35,780
241
923
23,317
35,939
213
826
23,164
36,035
216
871
23,146
35,698
223
920

5. Item Nonresponse and Imputation
Because the survey is voluntary and because some of the underlying categories of check
payments are difficult for some banks to report, there is fairly extensive item nonresponse
in the survey. At the same time, there is a hierarchy of subtotals and other relationships
leading to a variety of logical relationships that should be maintained in the aggregate
estimates for consistency. We required a rectangular dataset for studying a variety of
questions. To solve both of these problems we imputed missing items. In addition,
depending on the patterns of response, imputations that make use of the logical
relationships could improve estimates.

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Each respondent was asked to provide four figures (number and value for March and
April of 2007) per item. Each item had logical relationships with other items. For
example, number-value pairs should not have a zero amount accompanied by a nonzero
amount. Also, groups of subtotals should add up to totals. To illustrate the adding-up
constraints, Figure 3 provides a diagram with details of the variety of check clearing
methods appearing on the questionnaire. As shown in the chart, Paper Checks should be
the sum of Original Paper, Substitute, and Electronic Presentment; Truncation should be
the sum of Image Exchange, and MICR Presentment. Paper checks and Truncation
should add up to Inclearings. Finally, Inclearings and “On-Us” Checks should add up to
Payor Bank Checks or paid checks. We find it convenient to refer to totals as parents,
subtotals as children, and subtotals below children as grandchildren.
Payor Bank Checks

Inclearings

“On-Us” Checks

Truncation

Paper Checks
Original Paper
Substitute

Image Exchange

Electronic Presentment

MICR

Figure 3: Diagram showing an example of a hierarchy of adding-up constraints in the survey.

Since it was more common and easier for banks to report totals, for each incomplete
response we performed imputation in a hierarchical fashion by filling totals (or parents)
first, followed by children and then grandchildren. We used an EM algorithm-based
approach to impute each missing figure, where the missing figure was the predicted value
from a linear regression using data from respondents in the same stratum (Little and
Rubin, 2002). The regression models were univariate, where, for each missing item, the
regressor was chosen to be the reported variable with the closest relationship to the
missing value. After adjustments were made to ensure that logical relationships were not
violated, the imputed values produced on the final iteration of the EM algorithm were
used for estimation.
We applied a multiple imputation technique to account for any error from the imputation
model. On the final iteration, each fitted regression yielded a predicted value and an
associated standard deviation for the missing figure. To arrive at an imputed value for
the five datasets, a random deviate was added to the predicted value, drawn from a
normal distribution having a mean of zero and the standard deviation from the fitted
regression. This imputation procedure was repeated five times, each time using a newly
drawn deviate in the calculation, to create the five datasets. The variation among the
estimates calculated using the five datasets provided information about the uncertainty in
the overall estimate arising from the imputations and was used to compute standard
errors.
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Table 3 shows a comparison of univariate ratio estimates and standard errors of the
number and value of paid checks with only observed data and with both observed and
imputed data. The covariate in both cases was CHKD, and for both cases we stratified
two ways, one with CHKD and one with CHKD+MMDA. Imputation mattered little for
total number of paid checks because nearly all respondents reported this item. Standard
errors for the dollar value of paid checks, however, were reduced substantially. In
general, we believe this is because the imputation method uses the reported information
on the number of checks from each response to impute missing data. The standard errors
for several other survey items (children and grandchildren not shown here) also were
improved because of the use of close relationships with other reported figures within the
same observation.
Table 3: Ratio estimates of paid checks for commercial banks using imputed data
compared with estimates with non-imputed data (with CHKD as covariates). Nonimputed estimates are from Table 1.
Stratification Variables
CHKD
CHKD+MMDA

EST.
SE.
EST.
SE.

No Imputation
# (mil)
$ (bil)
23,577
37,110
331
930
22,716
35,429
232
806

Imputation
# (mil)
$ (bil)
23,573
37,471
330
451
22,722
36,043
231
362

6. Conclusions and Future Directions
This study showed that the quality of the estimates may be improved by using different
covariates and stratification variables. Regardless of the various potential stratification
variables, covariates, and imputation methods, our estimates still lead to the same general
results discussed in Section 1, and the conclusion that check usage is declining relative to
electronic payment methods is robust.
With the increasing use of retail sweep program by banks as well as the recent financial
turmoil, there should be more bank-to-bank variability in the data. So in the future study
planned for 2010, we will probably need to choose a larger sample size to get estimates
as reliable as before. Based on this work, we may use checkable deposits plus MMDA as
the stratification variable for selecting the sample. We will need to maintain consistency
to allow comparisons with estimates from previous years. When the newly collected data
become available, we will re-examine some of the issues that we have explored in our
current study.

Acknowledgements
The authors wish to thank Rich Oliver and Adrienne Wells of the Retail Payment Office
of the Federal Reserve Bank of Atlanta, David Stewart and Michael Argento of Global
Concepts, and Sam Slowinski and Jack Walton, now retired, of the Federal Reserve
Board. Many others provided exceptional assistance with our work over the years
including Samia Husain, Thomas Guerin, Namirembe Mukasa, Amin Rokni, Kathy
Wang, and Jaqueline Iwata.

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Board of Governors of the Federal Reserve System (2007), Report to the Congress on the Check
Clearing for the 21st Century Act of 2003.
Dalenius, T. and J. L. Hodges, Jr. (1959). “Minimum variance stratification,” Journal of the
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Gerdes, Geoffrey R. (2008), “Recent Payment Trends in the United States,” Federal Reserve
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Gerdes, Geoffrey R., Jack K. Walton II, May X. Liu, and Darrel W. Parke (2005), "Trends in the
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Gerdes, Geoffrey R. and Jack K. Walton II (2002), "The Use of Checks and Other Noncash
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