Effect of response errors on parameter estimates of models of savings behavior

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Faculty Working Papers

\

Effect of Response Errors on Parameter
Estimates of Models of Savings Behavior
Robert Ferber and Lucy Chao Lee
University of Illinois

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

FACULTY WORKING PAPERS
College of Commerce and Business Administration
University of Illinois at Urbana- Champaign
June 18, 1971

Effect of Response Errors on Parameter
Estimates of Models of Savings Behavior
Robert Ferber and Lucy Chao Lee
University of Illinois

No. 17

Effect of Response Errors on Parameter
Estimates of Itodels of Savings Behavior

Robert Ferber and Lucy Chao Lee

A considerable body of evidence has accumulated indicating* that
substantial errors exist in the reporting of asset and debt holdings in
consumer financial surveys. The characteristics of these errors vary from
one asset or debt to another, being more pronounced for more sensitive
holdings, such as savings accounts, common stock and personal debt, and
being of lesser importance for holdings such as life insurance, real
estate and installment credit. Overall, however, the evidence indicates
that nonreporting of ownership may be substantial, that very small holdings
may be overstated and vary large holdings understated, and that those who
refuse information on their holdings are likely to hold mere of that asset
or debt than would be expected on the basis of the usual averaging process.

Although current research may lead to means of detecting and correcting
such errors, tha fact remains that all of otir past and currant consumer
financial survey data are subject to these errors. In view of the

^-For example, s?e Lansing, J. B. , Ginsburg, G. P., and Br?.aten, K. ,
An Investigation of Rer.pjp.Fe Error, University of Illinois, Bureau of
Economic and Eusiness Research, Studies in Concumer Savings, No. 2, 1961;
Ferber, R. , The Relir.bil.lty of Consular Reports of Financial Assets
and Debts, University of Illinois, Bureau of Eccromic and Buriness Research,
Studies in Consular Savings, No. 5, 1566; Ferber, R. , Fcrcythe, J.,
Guthrie, H. W. , Maynes, E. S. , "Validation of Consumer Financial Characteris-
tics: Common Stocks',' Journal of the American Statistical Association.

June 1969, pp. 415-22; , "Validation of a National Survey of

Consumer Financial Characteristics: Savings Accounts, "Review of Economics
and Statistics, November 1969, pp. 436-44.

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widespread interest in ascertaining and measuring the determinants of
consumer savings behavior, it x«>uld, therefore, seem of critical
importance to evaluate the effects of these errors on analytical studies
of this type.

These effects are explored in this paper. More specifically, its
objective is to assess the effects of errors in asset and debt variables
on the estimates of parameters of models of consumer portfolios. This
is done in a two-stage process. First, estimates are made of the magni-
tudes of the bias in estimates of the parameters of consumer portfolio
models, using data from a validation study permitting relatively accurate
determinations of the magnitude of response and nonresponse errors in
the variables.

This first stage involves initially the formulation of alternative
hypotheses on the determinants of consumer portfolios. These hypotheses
are transformed into structural relations as a basis for the estimation
of parameters. The parameters of the models are then estimated in two
ways, one way by using the data on consumer portfolios as reported in the
surveys and the other way after adjusting these variables for response and
nonresponse errors in the data, based on the validation information. The
latter adjustment is a rather tricky one, because the validation data
provide only partial information on the errors in the variables , so that
additional inferences of the nature of the error in the nonvalidated
component of the variables have to be made. To obtain some idea of the
sensitivity of the estimates of the parameters to these inferences,
these estimates are made under alternative assumptions of these errors.

-3-

This first, econometric, approach yields rather narrow results,
providing estimates of the effect of these errors on a particular type of
sample. To obtain a more general idea of the nature of these effects,
a simulation approach is used next. This approach involves five distinct
steps. First, as before, certain structural hypotheses and corresponding
portfolio functions are formulated. Second, based on the results obtained
with the prior econometric approach, assumptions are made of the true
values of the parameters of the models. Third, error properties are
attributed to the portfolio variables based on the validation information
from the sample used in the prior econometric approach as well as from
previous validation studies of consumer financial behavior. Fourth,
the same validation sample used in the first stage is run through the
error process, and sets of observations are generated making use of the
error properties postulated at the prior stage. These sets of observations
are generated by Monte Carlo methods 150 different times, to yield some
idea of the range of variables obtained with these error properties.

Finally, estimates are obtained of the parameters in each of these
simulations and compared with the true values postulated in the second
stage. The distributions of the parameter estimates around the true values
provide a fairly comprehensive picture of the effect of these response
and nonresponse errors, at least for the types of models postulated in
this study.

The theoretical aspects of the effect of errors in variables has
been well covered in the literature, in addition to a few empirical studies

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of this question. A brief summary of this literature is provided in the
following section, which serves as a framework for the present paper.
Section 3 presents alternative formulations of models of savings behavior
and describes the data ur;ed in this study. The results of the econo-
metric approach are presented in Section 4 and that of the simulation
approach in Section 5. The concluding section then summarises the results
and discusses their implications both for the study of savings behavior
and with regard to the methodological issue of the effect of errors
and variables on parameter estimates.

2. Review of Relevant Literature

The great majority of studies of consumer savings have Iodised on the
flow aspects rather than on the stock of savings, which is not surprising
in view of the much greater availability of data of the former type.
Nevertheless, an increasing amount cf data has begun to be available in
recent years en household financial assets , and these data have served
as a basis for a number cf studies on the determinants of there asset
holdings. Although thr: cross-section studies of this questi'ii are of
primary interest, it seems desirable to refer briefly to some of the more
recent time series studies because of their relevance to ot^o of the
principal aspects of a model of consumer portfolios. This is the question
whether income or -ssets, or both, are most relevant to the determination
of holdings of a financial asset.

The bulk of the evidence appears to point rather strongly toward some
measure of vrealth (usually net worth) rather than income as a more likely

-5-

primary determinant. Thus, in a study of factors influencing liquid asset
holdings in England, Lydall found that new worth was far more important
than the level of income.2 Similar results were obtained by Meltzer as
well as by Bronfenbrenner and Mayer. In a still more recent study,
Hamburger found that wealth was consistently more important than income
in a number of single equation models of the influence of various factors
on the demand for four financial assets — marketable bonds, time and
savings deposits at commercial banks, life insurance reserves, and savings
accounts at credit unions, savings and loan associations and mutual
savings banks.

A study by Feige might also be cited in which he found that demand
deposits, as well as time deposits of commercial banks, and savings and
loan associations, were strongly influenced by an estimate of "permanent
personal income."5 However, the income variable reflected a weighted

2

Harold Lydall, "The Life Cycle in Income, Saving and Asset Owner-
ship." Ecoacg-ntrica. Vol. 23, April 1955, Pages 131-50.

JA. H. Meltser, "The Demand for Money: The Evidence from the Time
Series," Journal of Political Economy, Vol. 61, June 1963, Pages 219-246;
M. Bronfenbrenner and T. Irayer, "Liquidity Functions in the American
Economy," Eccnometrica, Vol. 28, October, 1960, Pages 810-834.

M. J. Hanburgar, "Household Demand for Financial Assets," Econometrica,
Vol. 36, January, 1968, Pages 97-118.

5E. L. Feige, The Demand for Liquid Assets: A Temporal-Cross
Section Analysis, Englewood Cliffs, New Jersey: Prentice Hall, 1964.

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average of present and past values of personal income and is, therefore,

closer in concept to a net worth variable than to a current income

variable. Partly for the latter reason, no measure of wealth was used in

this study.

The crcss-scction studies have teen relatively few and of a somewhat

varied nature because they necessarily had to be molded to fit the

particular set of data. Thus, Watts and Tobin ran a series of multiple

regressions on the 1950 BLS consumer expenditures data using as

dependent various stock variables (mortgage debt, installment debt, cash

balances and insurance) and also corresponding flow variables. They found

a variety of socio-economic and demographic character is tics to influence

these dependent variables, among which was income. However, net worth

was not available and therefore was not included in the study. It might

be noted that housing level, a variable that w^s included a.id that might

be considered as a proxy for permanent income and fcr net wealth, was

highly significant ir. almost all cases.

In another cress-section study, using data from the. Cc.nsurer Savings

Project, Claycamp found that wealth, in the form of total assets,

dominated income as a deter: air ant of the proportion of assets held in

liquid form as we: 1 as the proportion held in variable-dollar form

(meaning assets whose valve fluctuates with changes in prices).

6h. W. Watts and James Tcbin, "Consumer Expenditures and the Capital
Account," in Irwin Friend and Robert Jones, Editors, Proceedings of the
Conference on Consumption and Saving, Vol. 2, Philadelphia: University of
Pennsylvania Press, 1960, Pages 1-48.

7H. J. Claycamp, Jr. The Composition of Consvmer Savings Portfolios.
Urbana, Illinois: University of Illinois Bureau of Economic and Business
Research, Studies in Consumer Savings, No. 3, 1963.

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In another study, Crockett and Friend found that regressions of
different asset items on net worth gave about the same result as regressions
of these holdings on disposable income, using data from the 1962 Federal
Reserve Survey of Financial Characteristics of Consumers. Later in the
same study, results from a University of Michigan Survey Research Center
panel of consumers for 1959-61 indicated that both income and net worth
were significant in determining the flow into, and stocks of, particular
assets, varying with the asset. As in the Watts and Tobin study, however,
the income term was invariably a maasure of long run, or "normal," income,
which again might be construed as a proxy for rat worth. Also, the
bulk of this analysis focused on asset flows rather than on stocks.

Dorothy Projector found that equity in an asset at the beginning
of the period, and also occasionally a net worth variable, were more
important than disposable income in determining saving in the form of a

publicly traded stock, checking accounts, savings accounts, and investment

9
assets. Invariably, equity in the particular asset at the beginning

of the period was the dominant variable.

Also pertinent are various studies made by Rreinin with Survey

Research Center Data on the factors influencing ownership of liquid assets,

8jean Crockett and Irwin Friend, "Consumer Investment Behavior,"
in Robert Ferber, Editor, Determinants of Investment Behavior. New
York: National Bureau of Research, 1967, pages 15-123.

^Dorothy S. Projector, Survey of Changes in Family Finances,
Washington, D. C. : Board of Governors of the Federal Reserve System, 1968.

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life insurance and common stock. A variety of socio-economic factors
were found relevant, and stock ownership was found also to be influenced
by liquid assets . 10

3. Alternative I lode Is

Based on the previous review of the literature, three alternative
models were formulated for use in this experiment. In all three cases,
assets are subdivided into three categories in accordance with the valida-
tion information that is available. These categories are savings accounts
and savings certificates (S) , common stock (C) , and all other assets
and debt (L).

The three models represent alternative hypotheses on the
determinants of these three forms of asset holdings. The models are
as follows :

Model A

This model assumes that each of the three asset holdings is
dependent on the other asset holdings in accordance with the following
hypothesis. Savings accounts and common stock are jointly dependent on
each other and on total assets (T) , while other asset holdings are
dependent on savings accounts and on common stock. In addition, all
three categories of assets are influenced by a set of family character-
istics (Z) , which are treated as exogenous. The model is formulated

1"m. E. Kreinin, J. B. Lansing and J. N. Morgan, "Analysis of
Life Insurance Premiums," Vol. 39, Feb., 1957, Pages 46-54; "Factors
Associated with Stock Ownership," Review of Economics and Statistics,
February, 1961, Vol. 43, Pages 76-80.

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in linear arithmetic terms as follows i

1. S = aQ + a^ + a2Ti + a^ + i^

2« ci = bo + blSl + b2Ti + b3Zi + v-

3. L± = cQ + c-^ + C2Q± + c3zi + wi

4. S± + C± + L± = T±

Model B

The second model is partially recursive in that it assumes savings
accounts are determined first as a function of total asset holdings and
of family characteristics. Holdings of common stock and of other assets
are then assumed to be determined by savings accounts holdings and by total
assets as well as by family characteristics. The common stock function
is hence the same as in Model A. This hypothesis is in line with the
general advice given by personal finance and money management people,
that families just starting out should try to build up assets in the form
of savings accounts (as well as life insurance) for reserves before
accumulating other assets .

The exact form of the model is as follows :

1. S± = aQ + a2T± + a3Zi + u.

2. C± = bQ + h-fii + b2Ti + b3Zi + v±

3. L± = cQ + c-j^ + c2Ti + C3Z± + w±

A. S± + C± + L± = T±

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Hodel C

This model differs primarily from the preceding models in treating
as dependent the proportion of assets in a particular form rather than
the absolute amount. In other words, the endogenous variables are the
proportion of total assets in savings accounts (S/T) , the proportion of
total assets in common stock (C/T) and the proportion of total assets
in other forms (L/T).

The basic hypothesis is similar to the preceding model, namely,
that a family initially determines what proportion of its total assets
should be in savings accounts, based on its total assets and its socio-
economic characteristics. It then determines what proportion of its total
assets should be in the form of common stock as a function of the prior
determined proportion of its assets in the form of savings accounts, its
total assets and its socioeconomic characteristics. The proportion
of assets in other forms is obtained as a residual.
The exact form of the model is:

1. (S/T)i = a0 + a2T± + ajl± + u±

2. (C/T)i = bQ + bx (S/T^+b^ + b3Z± + Vi

3. (L/T)i = 1 - (S/T)1 - (C/T)±

The Data

The empirical part of this study is based on a combination of two
surveys carried out to validate reports of savings accounts and of common
stock holdings in the Federal Reserve 1963 Survey of Financial

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Characteristics (SFC). The SFC itself was a nationwide probability
survey designed to obtain information on the complete financial position
of families in this country, with over-representation from the high-
income groups. The two validation studies were carried out toward
the end of the SFC, using identical questionnaires and interview pro-
cedures and with the same interviewing organization (the U. S. Bureau
of the Census). Unlike the SFC, however, the validation surveys were
restricted geographically and, by their nature, contained only owners
of that particular asset.

It is clear, therefore, that the data used in this study do not
represent a cross section of the U.S. population. Rather, these data
constitute a very special sample for part of which savings account
holdings are known and for the other part, stockholdings are known. In
each case, however, the nature of the validation process precludes
complete knowledge of either savings accounts or common stock for a parti-
cular family, so that adjustments have to be made for that part of the
asset holding which was not validated and possibly not reported correctly.
These adjustments, to be described shortly, introduce an additional source
of error in the data. However, judging from the validation results
presented in previous studies, there is little doubt that the resulting

l*-For a more complete description of these studies and for summaries
of the results, see Robert Ferber, John Forsythe, Harold Guthrie, E.
Scott Maynes, "Validation of a National Survey of Consumer Financial
Characteristics: Common Stock," Journal of the American Statistical

Association, June, 1969, Pages 415-32; , "Validation of a

National Survey of Consumer Financial Characteristics: Savings Accounts,"
The Review of Economics and Statistics, November, 1969, Pages 436-444.

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data are far more accurate with regard to family holdings of savings
accounts and of common stock than of any other set of data that might be
used for this purpose.

The types of adjustments made in these data and the manner in which
they were made may be summarized as follows :

1. To adjust for errors in reported savings account balances or
of stockholdings in the appropriate validation sample, the
following rule was applied:
a. If V > 0, and T > V :

T = VI x (T - VJ + V

A — R R ]

VR

If V > 0, and T - V ;
R R R

V

VI " VR

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+ V
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If V = 0, and T > 0:

R R

TA = (TR + V + VI

where T = adjusted total stocks or savings accounts
A

T = reported total stocks or savings accounts
R

V = institution record (amount) of validated stocks or

savings accounts

V = reported part of stocks or savings accounts for
R

validation (amount)
N a reported total number of batches of all stocks or all

savings accounts
N = reported number of batches of stocks or savings

accounts for validation

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2. To adjust for nonreportinft of the particular holding in each of
the validation samples, regressions were run using the adjusted
total holding as the dependent variable (after adjusting the
data from the preceding step) with a number of socioeconomic
characteristics as independent. Separate regressions were ob-
tained for common stock and for savings accounts. One of the
independent variables in each case was the validated part of
that holding. In both cases the fit was quite good, R2

being .43 for the common stock regression and .37 for the
savings account regression, with a large number of indepen-
dent variables significant (particularly occupation, race,
family size, income, and the validated part of that asset)'.

For each of the sample members , its characteristics were sub-
stituted into the regression and an estimate obtained of the
adjusted total holdings of that asset. The equation estimate
was accepted except if the estimate was less than the validated
figure for that sample member, in which case the validated figure
was used as the total.

3. A further adjustment was made to spot nonreporting owners of
savings accounts in the common stock sample, and nonreporting
owners of common stock in the savings account sample. This was
done by estimating the relative frequency of nonreporting of
each asset by each income class. The same frequency of nonreport-
ing was attributed to the comparable income class in the other

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sample, which produced an estimate of the number of non-
reporters in each income class in that validation sample.
There were 28 estimated nonreporters for common stock and 13
estimated nonreporters of savings accounts.

To identify the specific sample members considered to be non-
reporting owners in each validation sample, discriminant
functions were derived for reporting of savings accounts and
of stock for each sample, in each case the dependent variable
being a 1-0 (reporter-nonreporter) variable. The independent
variables were a variety of family characteristics including
income, occupation, marital status, education, size of city,
race and family size. A number of these variables were
statistically significant at the .05 level, as was the
coefficient of determination, but the overall goodness of
fit was not high, namely, an R of .10 for the stockholding
function and .0 5 for the savings account function. Never-
theless, these functions were used on the nonvalidation sample
to pinpoint nonreporters, namely, as those people with the
highest estimated values of the function in each case on the
presumption that these people were owners and should have
reported their ownership.^

12lt might be argued that the nonreporters should have been
selected from the opposite end of the distribution, on the basis that
low values of the dependent variables denoted nonreporters (but note
that these people are also more likely to be nonxnmers) • In any event,
this approach was tested empirically and led to imputed ownership amounts
for the nonvalidation sample which, when aggregated, yielded average
amounts not reported far below the amounts indicated by the validation
sample.

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Once these nonreporting owners were pinpointed, estimates
had to be made of the amounts the presumed nonowners owned in
those particular assets. This was done by obtaining two
additional least squares regressions, one regressing for the
stock validation sample of the total savings accounts balances
reported by the reporters as a function of socioeconomic
characteristics; and the other for the savings account sample,
the amount of stock holdings reported by the reporters as a

function also of socioeconomic characteristics. In these

2
instances, the goodness of fit was much better, namely, R

of .19 for savings accounts and .31 for stock holdings. A

number of independent variables were significant at the

.05 level, particularly income class, age of head, race and

family size. Estimated amounts were accordingly obtained by

substituting the characteristics of the presumed nonreporting

owners into the appropriate function one at a time. It

should be noted, however, that no adjustment could be made

for reporting errors in this sample.

4. No adjustments were made for reporting or nonreporting in other

assets, as there was no basis for doing so.

Effect of the Adjustments

A general idea of the effect of the adjustments in the asset

variables is provided in Table 1. Not surprisingly, the table

indicates that the average holdings per sample household were increased

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substantially , by roughly 50 percent for savings accounts, by 22 percent
for common stock holdings and hardly at all for other assets. As a result
the average total asset holdings per sample household increased by 12
percent. Correspondingly, the proportion of total assets held in the
form of savings accounts rises from 8.4 percent to 11.1 percent, while the
proportion of total assets held in the form of common and preferred stock
rises from 31.5 percent to 34.2 percent.

These increases are to be expected in view of the fact that the
validation findings had indicated substantial nonreporting as well as
reporting errors in the direction of underestimation for savings accounts,
somewhat lesser reporting errors for common stocks, while the other validation
studies had indicated low reporting errors for other assets. Since the
adjustment procedures were designed to correct the data for these errors,
as described in the previous section, substantial increases in asset
holdings were only to be expected.

More or less paralleling these increases in average holding are the
increases in the variances of these holdings. Because of the highly skewed
nature of these holdings, the standard deviations exceed the means sub-
stantially for each type of asset. Not surprisingly, the adjustments serve
to increase the standard deviations somewhat more though the coefficient of
variation goes down for every variable but other assets.

4. Results of the Econometric Approach

The parameters of the three models shown on pages 8-9 were estimated
both by ordinary least squares and by three-stage least squares. The

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-17-

Table 1

Effect of Adjustments in Asset Variables on
Their Ileans and Standard Deviation

Variable Mean Standard Deviation

Unad.j . Ad j . Unadj . Adj .

Savings accounts $ 11,298 $ 16,917 $ 18,991 $ 24,107

Stocks 42,699 51,808 145,271 153,661

Other Assets 83,271 83,485 299,268 304,956

Total Assets 137,268 152,210 395,476 407,315

Savings accounts /total assets
Stocks/total assets
Number of observations

8.4%

11.1%

4.8%

5.9%

31.5%

34.2%

36.6%

38.6%

1,182

1,135

1,182

1,135

\b

-18-

latter estimation procedure is much more efficient than ordinary least
squares but may be more sensitive to specification error. The two sets of
results taken together should provide some idea, however, of the extent
to which differences in estimates of the parameters caused by errors in
the data may be affected further by the estimation procedures.

The ordinary least squares estimates of the beta coefficients of the
equations of Model A before and after adjustment for errors in the asset
variables are presented in Table 2. The socioeconomic characteristics,
the vector Z, used in all three equations are the same, namely, the varia-
bles listed in the table following the three asset variables at the top.
The selection of these variables was governed partly by data availability
and partly by the findings of previous studies regarding what character-
istics appear to be related to household savings in one form or another.

As is evident from this table, differences betxreen the two sets of
estimates are more of degree than of anything else. For the savings

function, the goodness of fit declines after the asset variables have been

13
adjusted, although five variables are significant at the .05 level after

the adjustment as compared to three prior to the adjustment. Host signi-
ficant perhaps is the change in the sign of the coefficient of the common

l^For comparability with the 3SLS estimates, A & U statistics are
given in addition to P.2, t:hc">~h the main reference in Tables 2-4 is to R .
A is defined as the average absolute value of the residuals while U,
Theil's measure of forecast accuracy is:

WZi y±2/N+ £• yf I 11

A

where y. is the function estimate for the ith observation and y is the
corresponding observed value.

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-20-

stock variable from a significant positive value to a nonsignificant
negative value almost of the same magnitude (and which would have been
judged significant at the .09 level). At the same time, the adjusted
data indicate significance for marital status and for service workers and
assign appreciably greater importance to the effect of the presence of a
male head and of older people on increasing savings account balances.

The effect of the adjusted data seems to be more pronounced on the
estimates of the parameters of the common stock function, but less so
on the estimates of the parameters of the function for other assets. In
the case of the common stock function, the adjustments serve to increase
the number of coefficients significant at the .05 level from four to
seven. In particular, the adjustments highlight the significance of
nonwhites as a factor reducing common stock ownership, and of the presence
of a male head and an older head for increasing stock ownership. The
function also ascribes much greater importance to education in affecting
positively) the amount of stock owned.

The effects of the adjustment seem least pronounced on the estimates
of the parameters of the function for other assets. In the case of this
function, the goodness of fit is increased hardly at all while the number
of variables significant at the .05 level declines from six to five.

The results for Model B are very similar to those obtained for
Model A, as can be seen from Table 3. This is not especially surprising
since the models are similar to each other (the common stock equation
is in fact the same). Elimination of the common stock variable from the
savings account function seem to have virtually no effect, as would be

•".-1-

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-21-

expected from the relatively little importance attached to the coefficient
of this variable in Model A. Nevertheless, the adjusted data do seem
to have brought about much larger differences in the estimates of the
parameters of the savings functions of Model B than of Model A. Thus,
the effect of total assets is reduced substantially. On the other hand,
the effects of the other significant variables are much more pronounced,
particularly the now significant effect of marital status and of service
occupations.

Highlighting the changed effect of the total assets variable is that
the elasticity of savings account balances relative to total assets
declines from 3.1 before the data adjustments to 1.4 after the adjustments.

Though still elastic, the effect of total assets on savings accounts

14
has been reduced by more than half.

In the case of the function for other assets, the estimates of the
parameters seem to have been affected dramatically not only by the
adjustments in the data but also by the addition of the variable for
total assets. The latter variable clearly dominates the relationship, as
is evidenced by the increase in the goodness of fit of this function from
an R of approximately .30 before the inclusion of this variable to an
R of about .90 with its inclusion. The error adjustments in the asset
variables increase the goodness of fit only slightly but increase sub-
stantially the number of variables significant at the .05 level — from
5 to 8. Presence of a male head and older age of head now have signif-
icant negative effect on holdings of other assets while nonwhite race
has a negative effect.

14The corresponding elasticities for the savings function of Model A
are 2.5 before adjustment of the data and 1.9 after the adjustments.

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-22-

Table 3

Estimated Beta Coefficients of Equations of Model B
Before and After Adjustment for Errors in Asset Variables

Savings

accounts

Other

assets

Variable

Before

After

Before

After

Total Assets

.258**

.016**

.973**

.968**

Savings accounts

-.082**

-.062**

Harried, spouse present

-.071

-.103*

.023

.028

Separated or widowed

.009

-.025

-.003

-.021

Self employed

.045

.030

.044*

.042*

Salaried professional

-.064

-.100

.040

.041

Clerical or sales

-.034

-.055

.022

.023

Craftsmen, kindred worker

-.057

-.065

.025

.027

Service worker

-.067

-.085*

-.007

-.004

Laborer

-.013

-.013

.011

.004

Retired

-.067

-.058

-.004

-.022

City size*

.028

.030

-.007

-.019

City size2

.055

-.005

.005

-.005

City size^

.011

.051

-.019

-.030

City size5

.005

-.018

-.004

-.003

Hale head

.146**

.158**

-.017

-.036*

Race white

.010

.017

.054**

.068**

Race nonwhite

-.016

-.018

.019

.026*

Education of head

.051

.004

-.033**

-.051**

Age of head

.166**

.179**

-.020

-.033*

Family size

-.060

-.059

.022

.038

No. of children under 18

-.040

-.030

-.011

-.025

R2 (adj.)

.137

.106

.900

.900

A

10.64

13.77

39.34

43.81

U

.491

.467

.155

.154

//Common stock function is the same as in Model A (Table 2).

*Significant at .05 level
**Significant at .01 level

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-23-

Perhaps the most important changes brought about by the adjustment
for the errors in the asset variables are apparent in the estimates of
the parameters of the two equations of Model C, as shown in Table 4.
In both cases, the goodness of fit is increased substantially, R rising
from .17 to .22 for the savings account function and from .24 to .39 for
the common stock function. The number of variables significant at the
.05 level, however, is hardly changed. The effect of total assets on
the savings account ratio is substantially more negative, as is also true
for marital status and education of family head.

In the case of the common stock function, the error adjustments
produce nonsignificance of the coefficient of the total assets variable
while increasing substantially the negative importance of the savings
account ratio variable, of race and of family size. The adjustment also
serves to remove the significance of the service and laboring occupations
of heads of families, while at the same time highlighting strong positive
effects due to age of head and to the number of children under 18. In
this case it seems clear that any inferences regarding the effects of the
other assets variable as well as of socioeconomic characteristic on the
proportion of a family's total holdings in the form of common stock would
be substantially different depending on which set of data were used.

It might be noted that the substitution of income for total assets,
which was tested in a couple of cases , yielded much poorer goodness of

fit. Thus, in the case of the common stock function of Model A, sub-

2

stitution of income for assets yielded an overall value of .251 for R

as compared to .575 when total assets was used.

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To summarize, the principal effects of the adjustment for the errors
in the asset variables would seem to be some improvement in the goodness
of fit and, more important, realignment of the importance of the asset
variables in relation to each other and increased importance of a number of
socioeconomic variables previously not judged significant. In view of
these adjustments, the question naturally arises of the extent to which the
particular adjustment process employed has predetermined these results.

There is little question that such predetermination is inherent in
the adjustment process. At the same time, there is also little question
that nonreporting of the financial data as well as reporting errors are
related to socioeconomic characteristics. Thus, nonreporting of savings
accounts tends to be higher among older people and among those with higher
incomes. Under the circumstances, adjustments to correct for these errors
serve in effect to eliminate part of the noise in the data and to restore
regularities which should have been there in the first place.

To be sure, there is always the possibility that the adjustments
may have gone too far and have introduced irregularities which are not
really present. Such a possibility cannot be eliminated simply from these
results alone. It is worth noting, however, that the principal effect of
these adjustments seems to have been on the common stock and the other-
assets functions, both assets for which the amount and nature of the

15Robert Ferber, op. cit., Chapter 4.

-26-
adjustments were less than for savings accounts. Ilore information about
the effect of these adjustments is provided in the following section,
which describes the simulation undertaken to explore in further detail
the effects of such errors on the estimates of the parameters of these
models .

In broad outline the results of the three-stage least squares estimates
are the same as those just reported for the ordinary least squares estimates
although appreciable differences are apparent in some of the individual
functions. Thus, as is evident from the 3SLS of Model A in Table 5, the
goodness of fit is improved primarily for the common stock function after
adjustment for errors in the asset variables. The number of coefficients
significant after adjustment is the same as the number before adjustment
for two of the functions and is slightly higher for the savings account
function. Also, the adjustment process serves to highlight the impor-
tance of common stock as a determining variable both in the savings accout
function and in the function for other assets and, more generally, seems
to accentuate the influence of the variables that are significant even
before adjustment.

Rather surprisingly, however, total assets does not seem to be important
at all in the common stock function in the 3SLS estimates whereas it is
by far the most important variable in the OLS functions.

In the case of Model B, all three functions exhibit improvement in
goodness of fit at least in terms of U, after adjustment for errors in the
asset variables. At the same time, the common stock and other assets
functions of this model show a very substantial improvement in goodness of

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Table 5

Estimated 3SLS Beta Coefficients of Equations of Model A
Before and After Adjustment for Errors in Asset Variables

Variable

Savings

accounts

Common

stock

Other assets

Before

After

Before

After

Before

After

Total Assets

.500**

.373**

-.024

.468

Savings accounts

2.859

1.591

3.106**

3.142**

Common stock

-.306**

-.276**

.169

.557**

Harried

-.087

-.115*

.152

.108

.254*

.398*

Separated or widowed

.003

.073

— — —

— — —

Self-employed

.086**

.087**

-.316*

-.170

-.231*

- . 311*

Salaried professional

-.001

-.033

-.039

.039

-.047

.098

Clerical or sales

-.028

.008

.024

-.014

Draftsman, kindred worker

-.016

.009

.045

-.007

Service worker

-.037

-.042

.169

.126

.156

.110

Laborer

-.009

.009

.046

-.039

Retired

-.012

-.015

.115

.115

.088

-.066

City sizel
City size2

M mt

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.170**
.265**

.133**
-.016

City size3
City size^

::::

— — — —

__—

„—_

.124**
.039**

.220**
-.065**

Hale head

.142**

.174**

-.355

-.193

-.446**

-.577**

Race white

-.150**

-.176**

—

Race nonwhite

-.036

-.035

Education of head

.019

. 100**

Age of head

. 174**

.200**

-.436

-.251

-.532**

-.657**

Family size

-.105**

-.096**

.240

.102

.329**

.322*

A

10.88

L3.89

234.9 154.1

541.0 576.2

U

.503

.468

.706

.593

.733

.742

*Significant at .05 level
**Significant at .01 level

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-28-

fit in terms of both U and A compared to the corresponding functions of
Model A. ° Besides the fact that the total assets variable is highly
significant in all three functions, both before and after error adjustment,
more variables are significant after adjustment in both the common stock
and other assets functions.

As compared to the OLS estimates, the main differences are that now
many different variables are significant. In the far majority of cases,
however, the direction of the relationship is the same.

The effect of the adjustment for errors on Model C appears mixed
(Table 7) because on one basis, the statistic A, the goodness of fit worsens,
whereas on another basis, Theil's U, the goodness of fit improves. However,
since Model C entails a pronounced change in the unit of the dependent variable
as compared to the previous models, it x^ould not seem unreasonable to
select U as the better basis for comparison, especially with the other models.
On this basis, the goodness of fit for the savings account function after
adjustment seems to be the lowest of any of the savings accounts functions in
any of the models and about the same as by the OLS method of fit. The
goodness of fit of the common stock function is also appreciably lower after
adjustment than before, and in this sense presents a very similar result
to those obtained for the common stock function of Model B.

16lt will be recalled that the same improvement was evident for the
other assets function by OLS but not for the common stock function which,
by definition, is the same in the two models. This would seem to be a very
good example of the greater efficiency of the 3SLS estimating procedure.

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-29-

Table 6

Estimated 3SLS Beta Coefficients of Equations of Ilodel B
Before and After Adjustment for Errors in Asset Variables

Variable

Savings accounts
Before After

Common stock
Before After

Other assets
Before After

Total assets

Savings accounts

Common stock

Harried

Separated or divorced

Self-employed

Salaried professional

Clerical or sales

Craftsman or kindred

Service worker

Laborer

Retired

City sizel

City size^

City size3

City size5

Male head

Race white

Race nonwhite

Education of head

Age of head

Family size

A
U

259** .155**

.891** .788**
-.611 -.403

1.069** 1.014**
-.445 -.369

-.065

-.101

-.093

-.100

.011

.018

.003

.021

.102**

.107**

.013

-.026

. 101**

. 104**

-.046

-.062

.062**

.085**

-.024

-.037

.032

.047**

-.020

-.043

.044*

.057**

-.048

-.047

-.018

-.004

.005

.012

.010

.019

.050

.058

.021

.004

.139** .178**

.114 .118

.030 .024

-.109** -.136**

.050** .057**

-.034 -.045*

.026* .033**

.050* .101**

-.056** -.053**

.149** .194**

.126 .132

.025 .034

-.096** -.087**

-.091 -.070

-.021 -.003

10.65 13.85

67.31 56.97

77.79 74.38

.494 .470

.434 .359

.219 .205

*Significant at .05 level
**Significant at .01 level

I

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-30

The number of significant coefficients is appreciably higher for
both functions of "odel C after error adjustment. Indeed, for both equations
in this model the large majority of coefficients were statistically significant
after adjustment, the only model or method of fit for which this was true.

All things considered, therefore, the results of the 3SLS estimation
procedure supports that of the OLS estimates in improvement in the goodness
of fit and in highlighting relationships previously judged not significant.

Results of the Simulation Study

The simulation study was carried out using only a fifth of the original
sample because of the larger size of that sample (1,135) and because it was
felt more important to obtain more simulations on a smaller sample than
fewer simulations on a larger sample. Accordingly, after arranging
the observations in numerical order every fifth observation was selected,
yielding a sample of 226 observations. With this sample, 150 simulations
were planned (50 simulations for each of the three models), a figure that
could be accomodated with the available computer resource? and which was
felt to be large enough to yield reasonably stable results.

Using the adjusted data for this smaller sample, as explained in the
section on methodology, estimates were obtained of the parameters of the
equations of each of the three models using alternately single equation
least squares and three-stage least squares. For the purposes of the
simulation, these estimates serve as the "true"population values.

', <

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-31-

Table 7

Estimated 3SLS Beta Coefficients of Equations of Model C
Before and After Adjustment for Errors in Asset Variables

Variable

Savings assets/total assets
Before After

Connon stocks/total assets
Before After

Total assets

Savings accts/total assets

-.094** -.132**

Married

-.179**

-.211**

Separation or divorced

-.096**

-.103**

Self employed

-.077

-.082**

Salaried professional

-.066

-.094**

Clerical or sales

-.057

-.086**

Craftsman or kindred

Service workei

Laborer

.022

.077**

Retired

-.010

-.045

City size *

.142**

.204**

City size *

City size 3

-.003

.091**

City size 5

Ilale head

.029

.057

Race white

Race nonwhite

-.024

-.052

Education of head

-.133**

-.097**

Age of head

.032

.059

Family size

-.178**

-.123**

A

.158

.189

U

.431

.375

137**

-.078*

089

-.182

086

-.127*

172**

-.145**

182**

-.183**

111*

-.127**

189**

-.201**

105**

-.100**

100**

-.050

067

-.009

-.066*

-.065*

009

-.029

020

.033

040

-.132**

023

.070*

195**

.234**

214**

.221**

094

-.128**

168

.171

431

.369

*Significant at .05 level
**Significant at .01 level

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-32-

Next, the simulation itself was carried out on the unadjusted data
for the same 226 observations. The procedure for simulating the unad-
justed asset information for each sample was as follows:

1. An estimate had already been obtained of the proportion of non-
reporting owners of common stock, and of savings accounts,

as described in steps two and three of the section on data
adjustment. This yielded point estimates of 31.3 percent of
the families not reporting common stock and 38.5 percent of the
families not reporting savings accounts.

2. Adjustments for nonreporting of assets other than savings
accounts and common stock were made on the basis of what is
known about nonreporting errors of these other assets.
Based on those remits, the average number of ncoreporters of
these other assets was taken as half of the number- ot nonrepoxters
of common stock.

3. These estimated proportions were assumed to be normally
distributed with the mean being the point estimate and with the
variance that ccnputed using that point estimate. Using this

l^The main source for such information is the summary results of the
validation studies conducted as part of the Consumer Savings Project of
the Inter-University Committee for Research on Consumer Behavior. These
results pertain to demand deposits, personal loans, auto loans, and farm
assets. The results are summarized in Robert Ferber, The Reliability of
Consumer Reports of Financial Assets and Debts, University of Illinois
Bureau of Economic and Business Research, Studies in Consumer Savings,
No. 6, 1966.

-33-
assumption, the numbers of nonreporters of each of these
three assets for each of the 150 samples were generated with
the aid of standardized random normal variants.

4. The specific nonreporters of each asset in each of the 150

samples was generated according to a uniform distribution,

18
in the absence of any other information.

5. To simulate errors in the asset holdings that were reported,
three log normal distributions were used — one for the ratio
of reported total savings account balances to adjusted total
balances in savings accounts, one for the ratio of reported
holdings of common stock to adjusted total holdings of common
stock, and one for the ratio of reported other assets to adjusted
other assets (assuming a mean of 1.0 and a variance of .3,

based on findings of other studies noted elsewhere in this paper).
A log normal distribution was used because this was felt to be
typical of most economic variables. The holdings of all the sam-
ple members were adjusted for each of the 150 samples by adjusting
this ratio with the aid of standardized log normal random
variables.

6. Using the 150 samples generated in this manner, estimates of
the parameters of the equations of each of the three models were

^Contrary to what might be anticipated, the limited work that has
been done provides no support for hypothesizing that nonreporters of one
asset are more likely than reporters to be nonreporters of another asset.

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-34-
generated by three-stage least squares. These estimates were
compared with the "true" values of the same parameters by
computing for each parameter estimate the mean of the estimates ,
the variance of the estimates and the mean square error. In
addition, for each fitted equation two measures of the adequacy
of the fit were computed, namely, A, and Theil's measure of
forecast accuracy, U, as noted previously.
The results of the simulation are quite surprising and are exempli-
fied to a large extent by the summary statistics in Table 8 pertaining
to the OLS estimate for Model A. The summary statistics shown in this
table for each parameter estimate of each of the three equations are the
value of the true parameter (Column 2), the average of the 50 estimates
obtained from the simulation (Column 3), the ratio of the latter to the
former (Column 4), the proportion of times the 95% symmetrical confidence
interval of the estimate contains the true parameter (Column 5), the average
width (range) of this confidence interval (Column 6), the average lower
bound (Column 7) and the average upper bound (Column 8) of this interval,
and the coefficient of variation of the parameter estimate (Column 9).

The surprising nature of the results is perhaps best highlighted by
the following capsule overview:

1. The average of the parameter estimates, even after 50 replica-
tions, does not come very close to the parameter, with some
exceptions (Column 4) . Only one quarter of these average of the
estimates were within 10 percent, while nearly half of the

-35-

averages (21 of 47) deviated from the parameter by more than
20 percent; 9 of these 47 averages were in error by over 50 per-
cent.

2. Nevertheless, the 95 percent confidence interval contained the
true parameter almost invariably, with a few notable exceptions
(Column 5). Thus this interval contained the parameter value more
than 90 percent of the time for 40 of the 47 parameters in the
model. (Note, however, that most of the key financial variables
are among the exceptions — the confidence intervals for five of
the six financial variables excluded the true parameter value
anywhere from 36 percent to 64 percent of the time.)

3. Why could the parameter estimates differ so appreciably from the
parameter estimates and still invariably encompass the true values
in their 95 percent confidence intervals? The answer is in
Columns 6-8, where we see that with rare exceptions the width of
the confidence interval is so large as to be virtually meaningles.-!.
Not only is the range as a rule many times the size of the parameter
estimate but it tends to cover both negative and positive values.
Hence, not only is there hardly any indication of the magnitude

of the parameter but the significance of the variable, and the
direction of any such effect, is in considerable doubt; this is
true of 41 of the 47 variables.

4. In other words, the variances of the parameter estimates tend
to be tremendous relative to the estimates themselves.

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-39-

This is reflected in the very high coefficients of variation of
these estimates (Column 9). Only one of the 47 coefficients of
variation is less than 10 percent, only four more are less than
50 percent, while 29 of 47 exceed 100 percent. Note that four
of the six financial variables have coefficients of variation
of only about 50 percent or less, and for these variables the
95 percent confidence intervals are as likely as not to miss
the true parameter.

5. Highly unstable estimates are most likely to characterize the
"other assets" equation and least likely the savings accounts
equation. Thus, though the bases are small, the proportion of
coefficients of variation exceeding 100 percent is 78 percent
for the former equation, 59 percent for the common stock
equation, and only 24 percent for the savings accounts equation.

6. By specific variables no clear pattern is apparent in the
reliability of the estimates except the scale phenomenon that
coefficients with higher absolute values tend to have lower
coefficients of variation. In other words, a variable estimate
with a very large confidence interval in one equation may or
may not have a very large confidence interval in another
equation, but it is difficult to generalize on the basis of
such a small sample.

All things considered, then, the picture is one of extreme insta-
bility of the parameter estimates brought about by the errors introduced
into the data. As a result, the 95 percent symmetrical confidence

-40-

intervals have the rather odd property of high reliability in the sense
of including the true parameter and at the same time of being meaningless
because the intervals are too large to have any substantive value.

The extent to irfiich the same results are borne out by the other
models and by the 3SLS estimates is shown in Table 9. On the whole, the
results are much the same as before. The parameter estimates for Model C
by three stage least squares appear to be somewhat closer to the true
values than the estimates obtained from the other models, but the gain
in accuracy is not substantial. Even for this model, nearly one-third of
the average values of the estimates after 50 simulations deviate from
the true figure by 20 percent or more.

Also, as before, for more than 80 percent of the parameters and
for each of the models, the 95 percent symmetrical confidence interval
includes the parameter 90 percent of the time or more. At the same
time, between 70 and 90 percent of the confidence intervals cover both
plus and minus values (the exact percentage verying with the model and
the method of fit) so that there is as a rule no reliability as to either
the significance or the direction of the effect of a particular variable.

i. i:..iaii. ■-:

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-41-

Table 9

Summary Statistics

on Reliab

lility of

Simulation Model

Estimates

Value

OLS estimates

3

SLS estimates

Statistic

Model A

Model B

Model C

Model A Model B

Model C

Frequency p .

Within 10%

12

6

9

11

10

13

within 1

given percent

10-19%

14

12

10

9

6

11

of p.

20-49%

12

16

10

8

15

6

50% or more

9

8

6

19

11

5

Total

47

42

35

47

42

35

Pet. of

85%

79%

100%

75%

81%

100%

parameters for

which 95%

confidence

interval includes

parameter 90% of

the time

Pet. of average
95% confidence
intervals covering
both plus and
minus values

87%

86%

72%

85%

90%

74%

Size distribu-
tion of average
coefficients of
variation

Under 10%

1

2

0

0

0

0

10-49%

4

5

11

6

4

8

50-99%

13

9

14

9

9

14

100% or more

29

26

10

32

29

13

Total

47

42

35

47

42

35

-42-

The reason is again brought out when we consider the size distri-
bution of the average coefficients of variation, shown at the bottom
of Table 9. For Models A and B, regardless of the method of fit, the
far majority of the standard deviations of the regression coefficients
exceed the estimates of the coefficients themselves. Here too, Model
C turns in a better performance, by either method of fit, with most of
the standard deviations of the regression coefficients being less than
their standard errors. Indeed, in the case of the least squares
estimates for this model nearly one-third of the standard deviations
of the coefficients are less than half the size of the coefficients
themselves, the best showing of all the models in the methods of fit.

The fact remains, however, that even with Model C one would
hesitate to impute much reliability to the results. The inescapable
conclusion is that the effect of the noise (reporting errors) in the
financial variables are such as to render highly questionable any
estimates of parameters that might be obtained with such data.

It might be noted also that Model C also contains the best acting
equations both in terms of stability of the coefficients of varia-
tion and in terms of closeness of approximation of the parameter
estimates to the parameters themselves. In the latter sense, 15 of

the 35 average parameter estimates are uithin 20 percent of the true
value by the 3SLS method of fit and 13 by the OLS method. The
standard deviations of the parameter estimates of this equation
were less than the estimates themselves 16 of 19 times for the OLS
method of fit and 14 of 19 times for the 3SLS method of fit.

-43-

Conclusions

The results of this study might appear at first to be contradictory.
Thus, the first part of the study, the econometric approach, suggests
that adjustment for reporting errors in the financial variables, at
least in so far as possible, brings about some improvement in the
goodness of fit of different equations of a model and the highlights
the significance of variables not otherwise significant.

At the same time, the results of the simulation approach suggest
that introduction of errors into what are taken to be a relatively
error-free set of data leads to parameter estimates that differ sub-
stantially from the "true" parameters, and to confidence intervals
that are meaningless for all practical purposes . In other words ,
the data produce a very high degree of instability in the parameter
estimates.

Are these two sets of results inconsistent? Not at all. This
becomes apparent if we compare the percentage deviation of the
parameter estimates from the econometric approach after adjustment
for errors in the variable:; with the "before" estimates, taking the
"after" estimates as the supposedly true values. Dividing the
"before" estimates by the- "&£ ter" estir^tes yr;.eld a set of
ratios comparable to those shown in Column 4 of Table 8 for the
simulation estimates. As an example, reproduced here are the ratios
of the "before" to the "after" OLS estimates of Model A (Table 2)
in conjunction with the ratios for the same variables from the
simulation for the same equation from Table 8.

-44-

Econometric

model estimates:

Simulation estimates:

Variable

"Before'VAfter"

b*/^

Constant

11.29

1.12

Total assets

.95

1.68

Common stock

1.00

4.30

Harried

.63

.94

Self-employed

2.16

1.33

Salaried

.54

.97

Services

.81

.83

Laborer

.79

.79

Retired

1.24

1.68

Male head

.88

.97

Age of head

.88

.83

Family size

.89

.83

"45"

It is rather striking that the two sets of ratios are of the same
order of general magnitude, except that the ratios of the parameter
estimates from the econometric approach appear to be much more
volatile than those from the simulation approach. This is only to be
expected because it should be recalled that the parameter estimates
from the econometric approach are single estimates, x^hile from the <?
simulation approach they represent ratios of an average over 50
simulations. Even so, many of the ratios are very similar and, with
only one exception, are also in the same direction.

This tabulation, plus others prepared from the other equations
and other models, indicate strongly that the results of the two
approaches tend to complement rather than conflict with each other.

It therefore seems clear that response errors of the magnitudes
that appear to exist in financial data can distort seriously not only
the means and variances of the corresponding univariate distributions
but in addition estimates of the parameters of econometric models not
only of these variables but of many other "variables included in the
model. The problem is clearly a most serious one. Where response
errors of these magnitudes exist, parameters estimates have to be
treated with a great deal of caution. Indeed, there may be no substi-
tute to putting a great deal of additional effort into getting better
data, partly through better data collection methods and partly through
the use of such supplementary methods as validation techniques.
Fortunately, most types of economic data do not seem to contain the

j:""l V

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sJ -.».••..

■I,i. : ;•,

-46-

large response errors characteristic of consumer saving data, so
possibly parameter estimates for models of other types of consumer
behavior are less subject to distortion from this source. This is
a question v/hich remains to be investigated.

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