Journal of Economic and Social Measurement 22 (1996) 

103-128 IOS Press

Do expenditures explain income? A study of variables for 

income imputation

Geoffrey D. Paulin

U.S. Bureau of Labor Statistics, 2 Massachusetts Avenue NE 113985, Washington, D.C, 20212. USA 

Phone:(203) 606-6900; Fax: (202) 606-7006

David L. Ferraro US. Bureau of Census. Room 3784-3. Washington, D.C. 

20233. USA

Income data in the U.S Consumer Expenditure Survey are subject to nonresponse. Model-based 

imputation is being explored to diminish missing data problems. Since income is an important 

variable to predicting expenditures, might expenditures be useful in predicting incomes? Incomes 

from wages and salaries and self-employment are modeled. These results are compared to regressions 

on demographic characteristics alone. Although each expenditure category adds to the predictive 

power of the model, total expenditures adds the most.

1. Introduction

Income is one of the most important variables in studies of consumer spending patterns. 

Not only is it an important predictor of expenditures, it has also been used to predict the 

probability of events as diverse as the consumption of wine [2] and the purchase of a 

home [4, 8]. Yet, nonresponse to income questions is a common problem in househ old 

surveys. The U.S. Consumer Expenditure Survey (CE) is no exception.

Many studies have been conducted to ascertain the best way to fill in the blanks For 

example, Lillard et al. [12] and David et al. [6] examine methods for imputing income in 

the Current Population Survey. Eltinge and Yansaneh [7] pursue weighting adjustment as a 

method to estimate mean consumer income in the CE. Paulin and Sweet [16] experiment 

with model building to estimate wage and salary income for individual nonrespondents in 

the CE. None of these studies directly uses expenditure data to impute income.

Given the problems associated with using a partial data set, and the fact that income is 

so often used to predict expenditures, two questions come to mind: Do expenditures 

explain income? If so, should they be used that way? This study examines the predictive 

power of similar equations, one without expenditures and the others including different 

expenditure variables. These equations are used to predict labor income (wage and salary, 

self-employment income) for single persons. The predictive powers of these equations are 

compared using various statistical techniques.

0747-966J96/58.00 ©1996 - IOS Press .411 rights reserved
2. The survey

Sponsored by the U.S. Bureau of Labor Statistics (BLS) and collected under contract by 

the U.S. Bureau of the Census, the CE is the only major U.S. government survey to collect 

detailed expenditure and demographic data from families. The information is collected 

from participating consumer units

1

in a series of five quarterly interviews. Income data are 

collected in the second and fifth of these interviews.

Currently, consumer units are divided into two groups: "complete" and "incomplete" 

income reporters, depending on the respondent's answers t o income questions2. Although 

85% of consumer units are classified as complete income reporters, even these families do 

not always provide a complete accounting of all types of income. As a result, these 

classifications do not completely correct for the problems caused by missing data. For 

example, many groups are shown on average to spend more than their reported incomes, 

even though only complete reporters are used to define income classes. It is hoped that 

imputing data to replace missing income values will improve the quality of the published 

CE data.

3. Prediction issues

3.1. Data issues

Based on the terminology of Little and Rubin [13], the data are assumed to be missing at 

random (MAR). If the data are MAR, the propensity to respond may be related to any 

number of demographic characteristics (such as age, number of earners, and other factors), 

but it is not related directly to level of income. Support for the MAR assumption is 

described in other recent work using CE data [15, 16].

3.2. The sample

Because there are questions to be addressed about whether expenditures should be used 

at all in the imputation process, preliminary investigation should be as simple as possible, 

consistent with achieving useful results. For this reason only single-member consumer 

units are studied. In this way the earner and the decision maker are the same person, and 

there are no questions about infra-household

1

Consumer units (the basic unit of comparison in the CE) are defined as a single person either living alone or 

sharing a household with others from whom the single person is financially independent; two or more members of a 

household related by blood, marriage, adoption, or other legal arrangement; or two or more persons living together 

who share responsibility for at least 2 out of 3 major types of expenses - food, housing, and other expenses.

2

For more information about sources of income collected and the definitions of complete and incomplete 

reporters, see Paulin and Ferraro [15, pp. 30-31).
resource sharing or other interactions among persons for which to account when 

incorporating expenditures into the models. Further, singles comprise a large enough 

portion of the sample (29% of consumer units interviewed in 1992) to constitute a 

sufficiently important group to study.

The sample is further restricted to consumer units in their second interview who said 

that most of their earnings during the past year had come from work at a particular 

occupation, either self-employed or working for wage and salary income, as opposed to 

pension or other supplemental income. Those who report that most of their earnings are 

from a wage and salary occupation are defined as salaried singles, even if they have some 

self-employment or other income; similarly, those who report that most of their earnings 

are from self-employment are classified as self-employed singles, even if they report wage 

and salary or other income. Additionally, only "valid" reporters of income are included. 

Salaried singles are included as valid report ers if the respondent reports positive wage and 

salary income. Self-employed singles are included if the respondent reports no negative 

self-employment income either from business or farm, and if neither source of income 

(business or farm) has an invalid response (refusal or "do not know").

The salaried singles were interviewed between 1988 and 1990. (These years are chosen 

because they are the internal data to which Census had access under an interagency 

agreement.) The self-employed data are from interviews taking place between 1988 

through 1992 in order to achieve a large enough sample for study. In theory. the sample 

could be broadened by including single-earner families, and adding independent variables 

to the models to control for factors such as number and age of children, marital status of 

the earner, and interactions between expenditures and family size, age of oldest child, and 

other characteristics. However, Chow tests [10] show that single-earner families and single 

persons are distinct groups, regardless of type of income earned.

The total sample size for the salaried singles is 2,247. The total sample size for the 

self-employed singles is 207. In the regression results described below, a small number of 

consumer units have a missing value for one of the independent variables (length of 

interview). Therefore, the sample size for the regressions is 2,207 for single salaried 

consumer units, and 202 for the self-employed.

3.3. Selecting expenditure categories

Choosing appropriate expenditure variables presents a challenge. Those with an 

identifiable Engel curve (i.e., expenditures as a function of income) are the best candidates. 

The stronger the relationship, the more obvious the shape of the Engel curve should be, 

and the more useful the expenditure information becomes in predicting income. (The shape 

can be postulated from scatter plots; the strength of the relationship can be gauged by 

examining t-statistics or performing
standard specification tests.) Perhaps the most obvious candidate is total expenditures, 

since these data should clearly be related to major sources of income. However, other 

expenditures may also be significantly correlated with income. It is even possible that some 

subcategories of expenditures may be better predictors than total expenditures. For 

example, the Engel curve for a specific expenditure may be estimated with a lower variance 

than the Engel curve for total expenditures.

But many expenditure categories have disadvantages that total expenditures do not. For 

example, virtually all consumer units have some value reported for total expenditures, but 

not every consumer unit incurs every type of expenditure. Therefore, it is important that if 

specific expenditures are chosen, there should be few non-purchasers; or, if there are a 

substantial number of zeros reported, it is important that those zeros be meaningful in the 

present context. That is, if almost no one under a certain income ever purchases a certain 

item, and almost everyone with more than the critical amount makes a purchase, then the 

zero expenditure may yield useful information. But if purchases of the item are naturally 

lumpy over time regardless of income (e.g., automobile purchases), then the lack of an 

expenditure is not a meaningful indicator of level of income.

Of further interest is whether the income elasticity of a particular expenditure (i.e., the 

percent change in the expenditure due to a one percent increase in income) might play a 

role in predicting income. For example, items with a low income elast icity (i.e., less than 

one) may help predict wage and salary income, being that these incomes are relatively 

stable, whereas items with a high elasticity (i.e., greater than one) may better predict the 

more transitory self-employment incomes, since high elasticity items are by definition more 

sensitive to changes in income. Based on Logit results (to predict the probability of 

reporting the expenditure given income and other characteristics), graphical analysis (i.e., 

plots of level of expenditure on income), and statistical comparisons, three candidates are 

selected. These are: food at home, shelter and utilities, and telephone services. However, 

the endogeneity issue becomes more complicated if individual expenditure categories are 

used for imputation instead of total expenditures. For example, some endogeneity may exist 

for all data users if total expenditures are used. But if food at home is chosen for use in 

imputation, researchers analyzing housing demand will have little concern with 

endogeneity whereas researchers analyzing food demand will have a greater concern. To 

help address these issues, some compromise candidates are proposed, based on Paulin

[14]. In this paper, the author examines the relationship between housing tenure choice 

(i.e., whether to own or tent one's home) and expenditures. Two categories he studies are 

basic goods and services (i.e., the sum of food at home, shelter and utilities,, and apparel) 

and recreation and related expenditures (i.e., the sum of entertainment, food away from 

home, vacation and other housing, and reading). These groups are chosen as candidates 

here because they each have few zeros, the endogeneity
problem is lessened with summed expenditures, and a priori, they should have different 

income elasticities (basic goods should be low and recreation should be high). Reading is 

not included in the present definition of recreation and related expenses because its value is 

too small and infrequently reported to be useful in predicting incomes.

In each case, expenditures are deflated by the level of the Consumer Price Index (CPI) 

for all goods and services for the month in which the interview takes place. This puts 

expenditures in real dollar terms, because the CPI measures the rate of changes in prices. It 

is important to control for price changes in some way because multiple years of data are 

used in each sample, and because expenditures, if used ultimately in imputation, will 

change as prices do; the division by CPI puts expenditures in real dollar terms. Incomes are 

also deflated by the CPI before being included in the regressions; thus, real dollar incomes 

are predicted from the models using real dollar expenditures.

3.4. Demographic variables

Other independent variables are also included in the models to predict income, as 

outlined in Appendix B. These include demographic variables identifying age, education 

(including a dummy variable for current student status), race, and sex of the respondent; 

dummy variables for location (region of the country and urban/rural area) and tenure 

(owner or renter) of housing.

In addition to the continuous demographic variables (age, age squared, and education), 

interaction terms are included (age*education and age squared*education). These 

interactions, which are found to be import ant in Paulin and Sweet [16], also have 

significant explanatory power in the present models, at least for the salaried singles (Table 

8). It should also be pointed out that although the education variable is continuous, and 

reflects the number of years of education of the respondent, the maximum value that is 

available for this variable is 18, or at least two years of graduate school.

Housing tenure is included because results of the Interview survey routinely show that 

homeowners report higher incomes than renters. Additionally, a dummy variable is 

included describing whether the person, if a homeowner, still owes for a mortgage. The 

interaction of this dummy variable and the level of the expenditure variable is used in each 

regression. Paulin [14] finds that owners with and without mortgages differ frequently in 

expenditure pattern, even when income and other characteristics are controlled, probably 

because a mortgage, once negotiated, is a fixed cost for the consumer. A person with the 

same level of income could choose to save the amount that would have gone toward a 

mortgage, or spend it on goods and services other than housing. The interaction term helps 

to account for these differences.
3.5. Labor-related variables

Attributes of the respondent's occupation (dummy variables describing type of 

occupation, whether other forms of labor income are also earned, and other variables 

describing number of hours per year worked) are included in each model. Perhaps the most 

interesting of the independent variables are those describing hours per year worked. When 

viewing scatter plots of labor income by hours per year worked (HOURYEAR), changes in 

the relationship are observed at different points. Although this is particularly we of 

self-employment income, in each case there is a spike at 2,080 hours (40 hours per week, 

52 weeks per year), and a noticeable slope change for those working more than 2,080 hours 

per year. For self-employed persons the slope appears to be close to zero. To account for 

these discontinuities, dummy variables and interactive terms are used. For those who work 

exactly 2,080 hours, the variable FULLTIME equals one; for all others, it equals zero. For 

those who work more than 2,080 hours, the variable OVERTIME equals one; for all others, 

it equals zero. Finally, OVERTIME is interacted with HOURYEAR to form OTSLOPE 

(i.e., OTSLOPE = OVERTIME * HOURYEAR). This allows the change in slope for those 

who work overtime to be measured.

Somewhat related to these issues is that of whether the respondent works two jobs, as 

may be the case when the respondent reports more than one source of labor income (i.e., 

wage and salary income and self-employment income). David et al. [6] note that their 

model predicts wage and salary income better for individuals for whom wages and salaries 

are the only source of income than for those who also report self-employment income, 

because hours per year is the total worked for both sources of income [6, p. 32]. However, 

it may not be the case that the person concurrently works for both sources, but may have 

changed jobs at some point in the last year. In either event, there may be other hidden 

relationships between the income sources for a person who has both. For example, a person 

whose main income is self-employment may earn supplemental salaried income if business 

is slow. Or a salaried earner may choose to work fewer hours away from home if some 

self-employment work at home is available. To make some attempt to control for these 

complicated possibilities, a dummy variable is included in each regression if the earner 

reports receipt of both types of labor income.

3.6. Survey attribute variables

Variables describing survey attributes including length of interview (continuous); 

quarter of the year in which the interview took place (binary); and for the self-employed, 

whether the interview occurred after 1990 (binary) are included for several reasons. 

Persons with longer interviews may have more expenditures or income information to 

report than those with shorter interviews. Incomes may
also be better reported during certain quarters of the year due to their proximity to the tax 

season. For the self-employed a dummy variable RECESS is included for those who are 

interviewed in 1991 or 1992. This variable serves a dual purpose. According to Survey of 

Current Business [18], Gross Domestic Product, when measured in constant 1987 dollars, 

experienced two consecutive quarters of negative growth in the last quarter of 1990 and the 

first quarter of 1991 [18, p. 3]. Since the respondent is asked to recall income for the 

previous year, each respondent interviewed between January 1991 and December 1992 is 

asked to recall. income for which at least one full quarter falls during the period of negative 

economic growth. At the same time, this variable controls for the difference in sample 

periods between the self-employed and salaried singles, since salaried singles are included 

for 1988-90 only. Table 9 shows that the coefficient for RECESS is negative but not 

statistically significant, suggesting the weakening economy had a small negative effect on 

average self-employment income, but this result is not conclusive.

3.7. Transformations

Neither income data nor expenditure data are often found to have a normal distribution. 

Many authors [6, 9] use the log of income in their models to approximate normality. 

Although log transformations have some desirable properties, it is not clear that the log 

transformation is optimal for approximating normality, nor that the best results are obtained 

by predicting log of income. Paulin and Sweet [16] find that the best results are obtained 

using a more general transformation is described by Box and Cox [3]. The formula for such 

a transformation is:

3

Paulin and Sweet do not divide income by the CPI before transforming, but this division is not expected to have 

a large effect on the normality of the distribution because the CPI changes slowly and steadily over the period under 

study.
3.8. Weighting

The regressions are weighted to reflect the population and to account for sample design 

effect.

3.9. Multicollinearity

Usually, when the goal is to impute a variable, multicollinearity in the model stage is not 

a serious problem. The reason is that it is the predicted outcome, and not any individual 

parameter estimate, that is of interest. However, in the present case, it is important to know 

whether expenditures are highly collinear with demographic characteristics for 

implementation. If expenditures are perfectly explained by the other independent v ariables, 

then it is more efficient to include only expenditures in the model. On the other hand, if 

processing is more complicated when expenditures are used, it may be more efficient to use 

only demographics in the model. Kennedy [10, p. 1811 suggests that if the R2 from the 

regression of income on expenditures and other demographic characteristics exceeds the R2 

for the regression of expenditures on the other demographic characteristics, 

multicollinearity is not a serious problem. When expenditures are r egressed on 

characteristics, the largest R2 is 0.5204 (for total expenditures for the self-employed). Since 

this value is smaller than the smallest R2 for income regressed on an expenditure and other 

characteristics (0.6167 for self-employed food at home model), multicollinearity is not 

serious. Table 2 summarizes results of the regressions of expenditures on demographic 

characteristics.
4. Results .

4.1. Predictive power of expenditures

A simple test of the power of each expenditure is to compare t he R2values for each 

regression to find the largest value. For the salaried singles, when income is regressed 

solely on demographic characteristics (i.e., expenditures are excluded from the model) the 

resulting R2is 0.6498. When food at home is added, the value increases slightly to 0.6512. 

When total expenditures are added, the value increases to 0.7091. Similar results are 

obtained from the self-employed singles. Without expenditures, the R2is 0.6042. When 

food at home is added to the model, the value increases to 0.6167. When total expenditures 

are added, the value increases to 0.7070. The R2values for all models are shown in Table 3.

It is interesting that whether wage and salary or self-employment income is examined, 

the order of increase in R2for each expenditure is the same. That is, in each case food at 

home adds the least to R2, which becomes succeedingly larger in the regressions for 

telephone services, recreation and related expenditures, shelter and utilities, basic goods 

and services, and finally total expenditures.
9.2. Mean square error comparisons for actual income

The models described above are designed to predict transformed incomes. But the real 

goal of imputation is to predict actual income values. How well do expenditures predict 

actual income? In order to answer this question, a comparison of mean square errors 

(MSEs) for each expenditure category is proposed. In this test, the transformed value of 

income is untransformed in the following way:

where Y is observed income, n is the number of observations of Y'.

Although "untransforming" the dependent variable in this way in theory can cause bias 

in the error term [16], in, practice the bias is found to be of little consequence, as shown 

later (Table 7). Table 4 summarizes the MSEs for the equations with and without 

expenditures.

Once again, the results of the MSE test are similar for both salaried and self-employed 

singles. The largest MSE (and therefore the least tight fit) is found for the model in which 

no expenditures are included. The variables, in descending order, of MSE are: food at 

home, telephone services, recreation and related expenditures, shelter and utilities, basic 

goods and services, and total expenditures; the same order is seen when R2 values for the 

models using transformed variables are compared (Table 3).

Additionally, the Wilcoxon rank-sum test [I l] is a distribution-free test used to ascertain 

whether or not the sum of squared errors (SSE) of the model with no expenditures is equal 

to the SSEs of the various models with expenditures. For
both the salaried and self-employed models, only the total expenditures models are found to 

have an SSE that is different in a statistically significant way from the model using no 

expenditures. Although the test results are stronger for the salaried singles (significant at 

the 1% confidence level) than for the self-employed singles (significant at the 10% 

confidence level), none of the other models tested display differences that are significant at 

any level even close to an accepted level of statistical confidence, as shown in Table 5. 

(The Z-variable is the standardized W-test statistic.)

4.3. Comparisons of means and standard errors

Another way to determine which models are most useful is to compare the means an d 

standard errors of the predicted incomes to the actual incomes to see which models produce 

the closest results. (For accuracy in comparisons, consumer units for which no length of 

interview is recorded are omitted before the mean and standard error of th e actual income 

values are calculated, since these observations are also omitted from the regressions.) The 

income data shown in Table 6 are for the untransformed values. The standard errors of each 

mean are shown in parentheses below the mean. All statist ics in Table 6 are unweighted.

Table 6 indicates that the standard errors of the means of values predicted from the 

models are significantly lower than the observed values. However, Little and Rubin [ 13, p. 

61] suggest that before the imputation process is considered complete, a random residual 

value should be added to the value predicted by the regression model to reflect the 

uncertainty in the predicted value. Before calculating the means shown in Table 7, residuals 

are added to each predicted observation using the following formula:
This new predicted value is then untransformed as described earlier. Using the overall 

MSE in this way is simplistic; however, use of a more appropriate method [5] is beyond the 

scope of this paper.

An even more striking anomoly with the distribution of the transformed predicted values 

(Table 6) is that the mean of the predicted values is much lower than the mean of the actual 

values. This bias in the mean of the transformed values is largely eliminated when the 

residuals are added to the predicted values before they are untransformed (Table 7). 

Because all predicted means are well within the 95% confidence interval associated with 

the observed means, no further statistical testing is performed.
5. Summary, future work, conclusions

Although some work has been done to adjust for nonresponse to income questions in the 

U.S. Consumer Expenditure Survey, no other work has been completed in which 

expenditures are used to impute income. Because expenditure data are expected to be 

correlated with income, and because these data are unique to the CE, this study examines 

which expenditures yield the most benefit in predicting incomes.

Single persons who have earned most of their income in the last year, either from wages 

and salaries or self-employment, are studied. Several expenditure categories are also chosen 

for analysis. Incomes and expenditures are normalized using Box-Cox transformations, 

which also correct for heteroscedasticity. Transformed expenditures are regressed on other

demographic characteristics before being included in the income prediction model to 

ascertain whether multicollinearity is a substantive problem. Results of a model using only 

demographic characteristics to predict incomes are compared to results of models that also 

include expenditures. For each type of income expenditures add to the explanatory power 

of the regression as measured by R2. Food at home expenditures add the least to predictive 

power; total expenditures add the most.

If expenditures are to be used to impute income, total expenditures emerge as the best 

choice in every method tested here. For researchers using CE data the problem of 

endogeneity is less for total expenditures than for more specific expenditure categories. 

Total expenditures also add the most (about 7% for the salaried and 10% for the 

self-employed) to the R2value of these regressions. Finally, regressions using total 

expenditures have the lowest mean square error, as proven with the Wilcoxon test.

However, this study only addresses single persons. The relationship of expenditures to 

income becomes more complex as family size, and particularly number of earners, 

increases. These relationships warrant fuller examination before expenditures can be 

recommended for use in imputation.

Appendix A: A preliminary experiment with income shares

According to Bannock et al. [1], Engel's original proposition of 1857 is that as incomes 

increase, the proportion of income spent on food diminishes [1, p. 140J. Because shares of 

other goods and services may also vary with level of income, it is worthwhile to test some 

relationships. However, because income is endogenous, it is necessary to use total 

expenditures as a proxy in the prediction of the share. Thus, the dependent variable in the 

model becomes the untransformed level of the specific expenditure (i.e., total expenditure, 

food at home, etc.) divided by the untransformed income from the appropriate source 

(wage and salary or self-employment income), or the income shares. The independent 

variables include the demographic characteristics and transformed total expenditures.
Surprisingly, none of the shares tested are very useful in predicting income. For the 

salaried singles the models all have extremely low R2values - 0.02 or less in each case. 

The models also predict negative shares for more than one-fourth of the sample regardless 

of the model. The coefficient on total expenditures is not statistically significant in any of 

the models tested; therefore, it is not surprising that the results change little when total 

expenditures is removed to compare models using only demographic characteristics to 

predict shares. Part of the problem is that so many respondents report extremely large 

income shares. For example, the average value of total expenditures divided by wage and 

salary income is 4.77, meaning that total expenditures are 477% of wage and salary income 

on average. The most extreme observation is greater than 1200 (or 120,000%), but there are 

several observations exceeding even 100 (10,000%), so it is not the case that one outlier is 

causing the problem.

In light of these results, a second experiment, this time using budget shares, is 

undertaken. A budget share is defined as a specific expenditure (e.g., shelter and utilities) 

divided by total expenditures. Now, levels of incomes (not shares) are regressed on budget 

shares and other characteristics. Budget shares are used as independent variables for two 

reasons. First, income is the variable to be predicted; hence it cannot be on both sides of the 

equation in any way. Second, total expenditures can be used as a proxy for permanent 

income, as described earlier. In these equations the level of income remains transformed, 

but neither the shares nor their components (specific or total expenditures) are transformed 

in any way.

Appendix B: Labor variables

Although other authors cited in the text have used labor-related variables in their studies 

of Consumer Expenditure Survey income data, none has used variables
as detailed as those described in the text. Because hours per year worked and other similar 

variables are undoubtedly strong predictors of income, it is useful to examine their roles in 

the models.

For salaried singles only FULLTIME is not statistically significant (Table 8). The 

parameter estimate for HOURYEAR has a t-statistic of 22.0, higher than any other variable 

tested except total expenditures. The coefficients for OVERTIME and OTSLOPE also have 

large t-values, ranging from 8.0 to 10.0 depending on the model considered. The 

coefficients for HOURYEAR and OVERTIME are positive, while the coefficient for 

OTSLOPE is negative. This implies that those who work more than full-time receive some 

extra base pay, such as a bonus, but receive a lower return to overtime hours worked than to 

regular hours worked. (An F-test shows the hypothesis that the sum of the coefficients for 

HOURYEAR and OTSLOPE is zero can be rejected with 95% confidence in most cases 

and more than 90% confidence in all cases.) But caution must be used in interpreting t his 

result because the workers included earn wage and salary income. A person who earns a 

high salary but whose overtime hours vary (e.g., a lawyer, doctor, or accountant) may 

indeed receive a lower "effective" wage (i.e., salary divided by hours per year) than 

someone working fewer hours in another occupation. Unfortunately, with CE data there is 

no way to distinguish between wage earners and salary earners to test these ideas.

For the self-employed these variables are also significant (Table 9), though generally 

with smaller t-values than for the salaried singles (Table 8). However, the parameter 

estimate for FULLTIME is negative, implying that those who work exactly 2,080 hours per 

year earn less, if all else is equal, than those who do not. For those who work more than 

full-time this result is not so surprising; they may be aggressively seeking business or once 

again may be in high paid professions that require many overtime hours. For those who 

work less than full-time, the result is puzzling - those who work less are predicted to have 

higher incomes (all else being equal). This is perhaps because working full-time (i.e., 40 

hours per week, 52 weeks per year) is common for so many workers. Indeed, 21 % of the 

self-employed singles sampled work exactly 2,080 hours per year. There is a large potential 

for variation at this point, since some persons are starting new businesses, some are 

established, some are having slow periods, and so forth, but all are working 2,080 hours per 

year. It is possible that those with new or slow businesses are pulling down the mean 

income for the full-timers compared to par and overtime workers.

The test of the hypothesis that reporting a second source of labor income indicates a 

lower than average value for the primary source of income yields results (Tables 8 and 9) 

that are inconclusive due to a lack of statistical significance for both salaried and 

self-employed singles. However, even if a strong evidence of a relationship were found to 

exist, the model does not account for the type of relationship that may exist. For example, 

although the sign of the parameter estimate for the selfemployed is consistently negative, 

providing weak evidence that the self-employed
singles use wage and salary income to supplement lower earnings, or vice versa. it also 

may be true that the dual-source self-employed singles have left a salaried position to start 

a business, and therefore report income from wages and salaries and also lower 

self-employed earnings than their more established counterparts.

Appendix C: Variable description

Dependent variables

BOXSELF: Sum of self-employment income (business or farm), divided by CPI for 

month of interview, and subjected to Box-Cox transformation.

BOXWGSAL: Wage and salary income, divided by CPI for month of interview. and 

subjected to Box-Cox transformation.

Expenditure variables

Note: All expenditure variables are divided by CPI for month of interview, and subjected 

to Box-Cox transformation.

BOXEXP: Total expenditures:

BOXFOODH: Food at home.

BOXSHELU: Shelter (rent or owned dwelling expenditures for primary home) and 

utilities.

BOXTELE: Telephone services.

BOXBASIC: Basic goods and services (food at home, shelter and utilities, apparel and 

services).

BOXRLFUN: Recreation and related expenditures (entertainment, food away from home, 

lodging away from home).

Other independent variables

AGE: Age of the respondent.

AGESQ: Squared age of the reference person.

EDUCLEVL: Educational attainment of the respondent, with 0 being no schooling and 18 

being at least 2 years of graduate school.

AGEEDUC: AGE * EDUCLEVL.

AGESQED: AGESQ * EDUCLEVL.

TM-INTER: Length of interview in minutes.

HOURYEAR: Number of hours per year worked.

FULLTIME: Dummy variable; equals one if HOURYEAR equals 2,080.

OVERTIME: Dummy variable; equals one if HOURYEAR exceeds 2,080.

OTSLOPE: OVERTIME * HOURYEAR.

OCCUPATIONAL CLASSES: TECHSALE: Respondent is in technical/sales work. 

PRECPROD: Respondent is in precision/production work. OPERATOR: Respondent is 

an operative or machinist.
SERVICES: Respondent is in service work. 

Control group is managers and professionals.

OTHLBINC: Dummy variable; indicates secondary source of labor income.

BLACK: Respondent is black.

FEMALE: Respondent is female.

BLACKFEM: BLACK * FEMALE.

STUDENT: Respondent is enrolled in college full- or part-time.

REGION OF RESIDENCE: NOREAST/MIDWEST/WEST: Indicate region in which 

consumer unit is located. Control group is located in Southern region.

RURAL: Consumer unit is located in a rural area.

RENTER: Respondent rents primary dwelling.

OWNOMORT: Respondent owns primary dwelling outright (i.e., no mortgage).

NOMBEXP: OWNOMORT * BOXEXP. Note: This variable is redefined and renamed as appropriate. For example, when food at home expenditures are 

included in the income prediction models, this variable is called 

"NOMRTFDH", and equals OWNOMORT times BOXFOODH.

SEASON OF INTERVIEW: QUARTER2/QUARTER3/QUARTER4: Indicate in which 

part of the year the interview takes place. Control group is the first quarter of the year 

(January, February, or March).

RECESS: Interview took place in 1991 or 1992. (Appears in self-employed regressions 

only.)







Acknowledgements

The authors gratefully acknowledge Donald B. Rubin (Chairman, Department of 

Statistics, Harvard University) and David Brownstone (Department of Economics. 

University of California) for discussing an earlier version of this paper at the Joint 

Statistical Meetings of the American Statistical Association (Toronto, Ontario, Canada, 

August 1994); Charles H. Alexander, Jr. (U.S. Bureau of the Census) for his help and 

support throughout the writing of this paper; and Ralph Bradley (U.S. Bureau of Labor 

Statistics) for his insightful comments, especially on the final drafts of the paper.

References

[1] G. Bannock, R. Baxter and R. Rees, A Dictionary of Economics, Penguin Books Ltd., Harmondsworth, 

Middlesex, England, 1972.

[2] JR. Blaylock and W.N. Blisard, Wine consumption by U.S. men, Applied Economics, 24 May, 1993, 

64.5451.

[3] G.E.P. Box and D.R. Cox, An analysis of transformations, J. R. Stat Soc., Series B. 1964. 211-243.

[4] D. Brownstone and P. Englund, The demand for housing in Sweden: equilibrium choice of tenure and type of 

dwelling, J. Urban Econ. 29(3) (1991), 267-281.
[5] N. Chand and C.H. Alexander, Unpublished results of recent preliminary experiments using data with a log 

transformation. For a description of their methodology, see: Chand and Alexander, Imputing income for an 

N-person consumer unit, 1994 Proceedings of the Section on Survey Research Methods, American 

Statistical Association, Alexandria, Virginia, 199-1, pp. 412-.116..

(6] M. David, R.J.A. Little, ME. Samuhel and R.K. Triest, Alternative methods for CPS income imputation, J. 

Am. Star. Assoc. (Applications) 81(393) (1986), 29-41.

[7] J.L. Eltinge and 1.5. Yansaneh, Weighting adjustments for income nonresponse in the U.S. Consumer 

Expenditure Survey. Technical Report No. 202, Texas A&M University. Department of Statistics, 1993.

[8] R. Gillingham and R. Hagemann, Cross-sectional estimation of a simultaneous model of tenure choice and 

housing services demand, J. Urban Econ. 14(1) (1983), 16-39.

[9] J.S. Greenlees, W.S. Reece and K.D. Zieschang, Imputation of missing values when the probability of 

response depends on the variable being imputed, J. Am. Star. Assoc. 77(378) (1982), 251-261.

[10] P. Kennedy. A Guide to Econometrics, 3rd ed., MIT Press, Cambridge, 1992.

[11] E.L. Lehmann, Nonparametrics: Statistical Methods Based nn Ranks, Holden-Day, San Francisco, 1975.

[12] L. Lillard, J.P. Smith and F. Welch, What do we really know about wages? The importance of nonreporting 

and Census imputation, J. Polit. Econ. 94(3) (1986), 489-506.

[13] R.J.A. Little and D.B. Rubin, Statistical Analysis with Missing Dana, John Wiley & Sons, New York, 1987.

[14] G.D. Paulin, A comparison of consumer expenditures by housing tenure, J. Cons. Affairs 29(1) (Summer, 

1995), 164-198.

[IS] G.D. Paulin and D.L. Ferraro, Imputing income in the Consumer Expenditure Survey, Mon. Labor Rev., 

December, 1994, 23-31.

[16] G.D. Paulin and E.M. Sweet, Modeling income in the U.S. Consumer Expenditure Survey, J. O/]: Star. 

(forthcoming).

(17] S. Scott and D.J. Rope, Distributions and transformations for family expenditures, 1993 Proceedings of the 

Section on Social Statistics, American Statistical Association, 1993, 741-746.

[18] Survey of Current Business, United States Department of Commerce, Economics and Statistics 

Administration, Bureau of Economic Analysis, 71(12) (December, 1991), Table 1.2, p. 3.