DEVELOPING IMPUTATION MODELS FOR THE SERVICES 

SECTORS PORTION OF THE ECONOMIC CENSUS 

Katherine J. Thompson and Quatracia Williams1

U.S. Census Bureau 

Room 3108-4, Washington, DC 20233/katherine.j.thompson@census.gov and quatracia.williams@census.gov

Introduction 

The editing software used by the Economic Census offers a variety of imputation options, many of which employ statistical 

models. In prior censuses, the services sectors portion of the Economic Census has relied on industry average imputation as 

its primary statistical imputation model. This ratio imputation method uses weighted least squares estimates for no-intercept 

simple linear regression models as imputation parameters. The weighted regression method currently used compensates for 

heteroscedasticity (unequal error variances). The industry average imputation method used by the services sectors portion of 

the Economic Census assumes the following weighted regression model for each basic data item Y: Yij = βi

 Xij + ξij, ξij ~ (0, 

Xij σ

2

), i.e., var(ξij) = σij

2

 = Xij σ

2

, where j indexes the establishments within industry i that satisfy the ratio edit li ≤ Yij/Xij ≤ ui

. 

Thompson and Sigman (1996) and Huang (1984) demonstrate the plausibility of this model for the services sectors data. The 

best linear unbiased estimator (B.L.U.E) of βi

 (the industry average) for this model is the weighted least square estimate with 

weight of 1/Xij, so that ∑ ∑ =∑ ∑ (Draper and Smith, 1981, p.111). 

j

ij

j

ij

j

ij ij

j

wij X Yijij/ w X Y / X

2

Although statistically sound, the industry average imputation method is limited. First, it assumes a strictly linear relationship 

between the independent and dependent variables, when the actual regression line may not go through the origin. Second, it 

assumes that the variability in the dependent variable can be best “explained” by one variable. A multivariate regression 

model could have more predictive power. 

This paper compares alternative methods of developing multiple regression models using data from the 1997 Economic 

Census. We consider three different approaches, two designed to obtain optimal estimates under conditions of 

heteroscedasticity and in the presence of outliers (weighted least squares regression and resistant regression) and one 

designed to obtain stable estimates in presence of multicollinearity (ridge regression). We discuss issues of model selection 

and validation in the presence of multicollinearity and heteroscedasticity. Finally, we make recommendations for developing 

imputation models for the services sectors portion of the Economic Census. 

Current Editing and Imputation Practices 

The services sectors portion of the Economic Census mails out over 400 different forms to over four million businesses. This 

portion of the Economic Census encompasses five trade areas: Retail Trade, Wholesale Trade, Services Industries, 

Transportation, Communication, and Utilities Industries (Utilities), and Finance, Insurance, and Real Estate (FIRE). All 

trade areas collect a core set of “basic data” items: annual payroll, 1st quarter payroll, employment, and receipts (or sales). In 

addition, tax-exempt Services industries collect operating expenses, and Wholesale Trade collects operating expenses and 

costs of purchases. 

Basic data items are automatically edited using the ratio edit module of a generalized editing and imputation subsystem, 

called Plain Vanilla (PV), which was developed for the 1997 Economic Census. A ratio edit compares the quotient of two 

highly correlated items to upper and lower bounds, known as tolerances. The ratio editing software employed by the 

Economic Census tests all ratio edits simultaneously and requires that imputed values satisfy all of these edits (Greenberg, 

1986). The software determines the minimum number of edit-failing items that must be deleted (replaced with imputed 

values) so that the edited questionnaire passes all edits. To maximize the efficiency of this editing software, the input items 

 

1The authors thank Samson Adeshiyan, Scot Dahl, and Richard Sigman for their useful comments on earlier versions of this 

manuscript. This paper reports research and analysis undertaken by Census Bureau Staff. It has undergone a more limited 

review than official Census Bureau publications. This report is released to inform interested parties and to encourage 

discussion. 

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must all be highly correlated. From a regression model-building perspective, this introduces the issue of multicollinearity. 

Thompson and Adeshiyan (2003) provide more detail on the PV ratio edit methodology. 

PV users specify the order in which edit-deleted items are imputed and the sequence of imputation models attempted for each 

item. For each edit-failing item, the program attempts each listed imputation model sequentially until obtaining a satisfactory 

imputation. The imputation region for each deleted item is calculated from the values of the remaining non-deleted items and 

their associated ratio edit bounds. When more than one item is deleted, the choice of imputation model can greatly affect the 

imputation region for all but the first deleted item. 

Many of the available PV imputation models are deterministic, including direct substitution or logical edits (e.g., substituting 

an associated sum of details for a total). There are, however, several available statistical model-based methods, including 

regression imputation. Of these statistical methods, industry average imputation is intuitively appealing. The statistical 

model follows logically from the form of the ratio edit: a no-intercept regression model where the edit’s numerator is the 

dependent variable, and the denominator is the independent variable. Moreover, industry average parameters are easy to 

interpret and verify, e.g., average annual payroll per employee. In this paper, we use the term “industry average” to denote 

simple linear no-intercept regression modeling, and “regression imputation” to refer to any other regression-based imputation 

model. 

Research Data 

We used 1997 Economic Census data for this evaluation and conducted each of the described analyses separately for each 

trade area. The five trade areas each classify their data slightly differently for editing and imputation purposes. For example, 

Wholesale Trade develops edit/imputation parameters for each type-of-operation classification within industry, whereas 

Services Industries uses industry-by-tax status to define edit/imputation parameters. We used the trade area’s edit/imputation

cells for our research, but for simplicity refer to each classification cell as an industry. We examined a total of 149 industries: 

19 for Wholesale Trade; 30 for FIRE; 30 for Utilities; 30 for Services; and 40 for Retail. For the Wholesale Trade research, 

we used all six-digit industries within one four digit NAICS industry code for our research. For the other trade areas, we 

selected a random sample of industries that contained at least 40 observations whose reported data satisfied all ratio edits (ten 

observations per independent variable). By restricting our model-building data to regions defined by the set of all ratio edits, 

we reduced the incidence of outliers and guaranteed that our prediction region would correspond to the imputation region 

used for the 2002 Economic Census data. 

Using 5-year old data may be sufficient for researching the viability of alternative regression imputation models, but we have 

no way of knowing how well the recommended 1997-data-models will predict 2002 data values. We are particularly 

concerned about production implementation of 1997-data regression models containing employment for two reasons. First, 

the effect of the between-census economy change could be negligible in models that contained only dollar value covariates 

(affecting all variables equally), but might not be negligible in models that contain both employment and dollar value 

covariates. Second, employment data is collected differently in 2002 from 1997: the 1997 Economic Census requested total 

employment (which included leased and direct employees), whereas the 2002 Economic Census collects payroll and 

employment data for direct and leased employees separately. 

Methodology 

Regression Parameters Estimation 

As mentioned above, the weighted least squares regression model used to obtain industry averages assumes that the error 

term variances increase with the independent variable. It is difficult to theoretically extend this simple weighted regression

model to multiple regression. Neter, Wasserman, and Kutner (1989, pp. 420-421) suggest grouping the model building data 

according to the fitted value of Y calculating the variance of the residuals for each group, and using this as each group’s 

weight. Chen and Shao (1993) propose constructing weights based solely on each group’s residuals. Our dependent variable 

distributions varied greatly by industry, making these approaches impossible in a generalized way. Instead, we used the 

following model 

,

ˆ

Yij = β1i X1ij + β2i X2ij + ...+ βpi Xpij + ξij, ξij ~ (0, X1ij σ

2

) (1)

Like the industry average model, the regression weight is given by 1/X1ij. We used different weighting variables for different 

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combinations of covariates. 

Weighted least squares estimation reduces heteroscedasticity by “down-weighting” observations with large expected

residuals. However, this approach assumes a particular data + error model, which may not be correct for all cases. For 

example, the assumed model may not adequately explain the presence and magnitude of outliers. Several of the current 

services sectors surveys as well as the manufacturing, mining, and construction sectors of the Economic Census use resistant 

regression (Mosteller and Tukey,1977, pp.356-361) to develop imputation parameters. This iterative procedure develops 

regression weights (biweights) that are smaller for observations with large residuals (generally outliers), assumes a particular 

data model with an unspecified error structure, and uses weighted least squares estimation to obtain coefficient estimates. 

When the data model contains no intercept, resistant regression yields the same coefficient estimates as the weighted least 

squares estimates for the model 

Yij = β1i X1ij + β2i X2ij + ...+ βpi Xpij + ξij, ξij ~ (0, wij

-1σ

2

) (2)

The biweight (wij) measures the “influence” of each observation on the estimation of model coefficients. The biweight is a 

function of the observed value, the predicted value from the k

th fit, and a measure of spread for all the residuals in the kth fit 

(Spk); wij is small for large residuals. Unlike (ordinary) weighted least squares regression, resistant regression down-weights 

observations with large observed residuals. We evaluated resistant regression with two different measures of spread: the 

median of the absolute value of the residuals (proposed by Mosteller and Tukey), and the mean of the absolute value of the 

residuals (used in the current survey’s programs). 

Regardless of regression weight, the majority of our fitted multiple regression models exhibited heteroscedasticity. Thus, the

parameter estimates from these weighted least squares regressions are still unbiased but do not have minimum variance. 

Using the different weighted regression methods reduced our models’ heteroscedasticity but did not address the problem of 

multicollinearity. We used the variance inflation factor (VIF) to measure multicollinearity of our full covariate models: a 

single VIF of 10 or an average VIF greater than the number of covariates indicates severe multicollinearity (Neter, 

Wasserman, and Kutner, 1989, pp. 408-410). The majority of our full covariate models for each data item had at least one 

VIF greater than 10. 

Ridge regression is a biased estimation technique designed to reduce multicollinearity (Neter, Wasserman, and Kutner, 1989, 

pp. 412-418). This method introduces a biasing constant c into the normal regression equations for correlation-transformed 

variables [The correlation transformation standardizes all covariates, helping control roundoff errors and making the 

regression coefficients units comparable]. Fitted ridge regression models always include an intercept term (coefficients 

calculated from correlation-transformed data are translated to the original scale). To choose c in our test industries, we tried 

to find one value of c per dependent variable (per trade area) for each unique set of covariates that yielded all VIFp ≤ 1 and 

also yielded similar parameter estimates for all greater values of c. Ridge regression has two major drawbacks. First, the 

choice of the biasing constant c is judgmental, and there is no guarantee that the same value of c will work “from one 

application to another” (e.g., 1997 and 2002 data for the same dependent variable and covariates). The second drawback is 

that ridge regression sums-of-squares estimates’ distributional properties are unknown and consequently cannot be used. 

Evaluation Statistics 

Because our data were highly multicollinear, we could not use estimates based on the weighted regression sums-of-squares to 

select the “best” method of parameter estimation or model selection. Moreover, there are no sums-of-squares diagnostic 

statistics for ridge regression. This restriction eliminates most of the “traditional” multiple regression diagnostics, such as 

adjusted-R2

 or F-tests. It also eliminates any form of automatic model selection procedure. Instead, we used cross-validation 

(Neter, Wasserman, and Kutner, 1989, pp. 466-468) and delete-a-group jackknife estimation (Rao, 1993) to obtain measures 

of model performance and t-statistics for individual parameter estimates, respectively. To perform cross-validation, we 

randomly split the establishments in each industry into two equal-sized sets: a model-building set and a validation set. Then

for each multiple regression model, we computed four alternative sets of regression parameters (one per regression method) 

from data in the model-building set and used these parameters to predict the same characteristic in the validation set. After 

this, we calculated three different statistics to measure prediction performance: 

Mean Absolute Error (MAE) for method m Y Y n 

n

i

im i

/ ∑ ˆ −

(DeGroot, 1987, pp. 209-211) where i indexes the industry within trade area, Y is the tabulated value obtained from im

ˆ

3 

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imputing data item Y in the validation data set using regression parameters obtained with method m (m = 1- 4: weighted least 

squares, resistant regression with Spk = mean of residuals, resistant regression with Spk = median of residuals, and ridge 

regression), and Yi is the tabulated value (from reported data in the validation data set) of the data item in industry i.

Mean Absolute Deviation (MAD) for method m and industry i i

n

j

ijm ij Y Y n

i

/ ∑ ˆ − 

(Nordholt, 1998) where j indexes the establishment within industry i and ni

 is the number of establishments in industry i .

MAD score for method m # of industries where MAD(m,i) is minimum of all four methods. For each model, this score 

identifies the method that produces the lowest MAD in the most industries. 

Mean Ratio of Predicted to True Value (MRPT) for method m Y Y n

n

i

im i

/ / ∑ ˆ

For delete-a-group jackknife estimation, we randomly split the establishments in each industry into 16 groups. Each delete-agroup jackknife replicate estimate was obtained by dropping one group k at a time, weighting the remaining observations by 

(16/15) and calculating the parameter estimates from the remaining 15 groups. Delete-a-group standard errors of each 

parameter estimate βpim are: 

[ ] 2

1

)

ˆ ˆ

) ( /)1 (

ˆ ˆ(

pimk pim

K

k

pim σ β = k − k ∑ β − β

=

. 

where p indexes the regression parameter. Within each trade area, we evaluated the average coefficient of variation of each 

parameter (for each multiple regression model) and the frequency of statistically significant parameters in each model by 

industry (using t-statistics at the 90% confidence level). 

Selecting Regression Methods (Parameter Estimation Methods) for Each Item 

Prior to model selection, we had to choose a parameter estimation method for each item, assuming that the “best” regression 

method within a trade area could differ by item, and the “best” regression method for a given item could differ by trade area. 

To predict annual payroll, we considered all permutations of available covariates (up to three covariates in Retail Trade, 

Utilities, FIRE and certain Services Industries; up to four covariates in Wholesale Trade (excluding purchases) and the 

remaining Services Industries. For the remaining data items’ models, we excluded 1st quarter payroll as an independent 

variable because of its perfect correlation with annual payroll (i.e., we did not include both items as independent variables in 

the same model). 

0

10000

20000

30000

40000

50000

60000

70000

123

Regression Model

Value of MAE

Weighted Least Squares Resistant Regression (Mean)

Resistant Regression (Median) Ridge Regression

4

To assess the performance of the four parameter 

estimation methods, we looked at all possible 

multiple regression models for an item within each 

trade area [Note: we considered only weighted least 

squares regression for the industry average 

imputation]. For each item (within trade area), we 

selected one parameter estimation method from the 

four after examining their respective MAE and MAD 

statistics. We believed that the estimation method 

should consistently yield the lowest MAE for a given 

dependent variable regardless of covariates and 

should generally yield the lowest MAD within trade 

area. Ideally, the mean ratio of the predicted to true 

value (MRPT) should also be close to one. 

Figure 1 illustrates this process, plotting MAE for all 

3- and 2-covariate models predicting annual payroll 

in our 30 FIRE industries [Note: we use only weighted least squares estimation – the production method – for the single 

covariate models]. Each point on the horizontal axis represents a multiple regression model (see Models 1- 4 in Table 1). 

Except for model 2, weighted least squares regression yields the lowest MAE (for model 2, the MAEs for the three “close” 

models are all within 150 units of one another). Frequency distributions of MAD/method by industry (MAD scores) show 

Figure 1: MAE for Multiple Regression Models Estimating Annual Payroll 

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0

2

4

6

8

10

12

123

Regression Model

Number of Industries

Weighted Least Squares Resistant Regression (Mean)

Resistant Regression (Median) Ridge Regression

4

the same pattern, as plotted in Figure 2, i.e., methods 

with low MAEs in Figure 1 are associated with high 

MAD scores in Figure 2. 

 

In all trade areas, the multiple regression models fit 

with weighted least squares generally had the lowest 

MAEs for all dollar value items (annual payroll, 

receipts, operating expenses, and purchases). Except 

for the Retail Trade and Utilities industries, ridge 

regression yielded the lowest MAE for the models 

predicting employment. 

Initially, we were hesitant to recommend ridge 

regression models based on 1997 data for modeling 

2002 employment. Choosing a biasing constant for 

ridge regression is a subjective procedure, even when 

f our ridge regression models for employment to the 

choice of biasing constant, within our Wholesale Trade industries, we refit the ridge regression employment models in with c

= 0.1, 0.2, 0.3, and 0.5 and found the same 

patterns of MAE (i.e., lowest from ridge 

regression of the four parameter estimation 

methods for all multiple regression models). Since our employment models introduce very small biasing constants, this 

insensitivity to c is not unreasonable. However, we were still concerned that the values of c obtained from our modelbuilding datasets might not be appropriate for other samples in the same prediction range. We were also concerned that we 

were simply not comparing enough multiple regression models within industry. Except for tax-exempt Services Industries 

and Wholesale Trade, we only had one available regression multiple regression model to compare to the industry averages. 

To provide more models and to assess the sensitivity of c to the model-building dataset, we repeated the cross-validation in 

many of our test industries. The results were quite consistent. 

the choice is based on diagnostic statistics. To assess the sensitivity o

Figure 2: MAD score for Multiple Regression Models Estimating Annual Payroll 

While ridge regression worked quite well for predicting employment, its performance was inconsistent when modeling dollar 

value items. We believe that ridge regression works so well for modeling employment because all of the available 

independent variables are dollar value items. Much of the multicollinearity is alleviated by the correlation-transformation, 

and the biasing constants used in the employment ridge regressions are quite small (compared to those used to model dollar 

variables). 

We were surprised by the poor performance of resistant regression with most of our data, given the current surveys’ excellent 

resistant regression results. However, the model-building data sets used by the current surveys include outliers, and the 

current surveys generally use simple-linear regression as imputation models. The census model-building data are restricted 

to observations that satisfy all of the ratio edits and consequently do not contain many outliers. 

Model Selection 

The MAE comparisons used for parameter estimation selection provided credible evidence of improved predictions over 

industry average predictions with many of our multiple regression models. The degree of improvement over industry average 

varied considerably by trade area and by item, however. Our subject-matter analyst experts were generally satisfied with the 

industry average imputation results and required proof of “substantial” improvements via multiple regression to justify the 

additional analyst-time in parameter estimation and data storage space. We developed the following procedure for obtaining 

reduced sets of multiple regression models for a data item/trade area: 

1. Using the statistics obtained with the recommended regression method (weighted least squares for all dollar value items 

and employment in the Retail and Utilities trade areas, ridge regression for employment in the FIRE, Wholesale and 

Services trade areas), rank the no-intercept regression models for the same data item by ascending MAE; 

2. Drop all models that do not show any improvement over each of the already implemented industry average imputations 

in terms of MAE; 

3. Use significance test results to further reduce the candidate set of reduced models (e.g., drop models that contain more 

than one independent variable whose parameter tests are rarely significant, especially if they do not exhibit marked 

5 

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improvements in MAE from other models); 

4. If the MAE between two or more models was quite close, examine MRPT to further reduce models; 

5. Again using MAE and significance test results, determine whether including an intercept term in the recommended beta 

models and industry average models improves their predictive power. 

Characteristics of the production software also affected our model selection procedure. The PV ratio module attempts all 

regression imputations in one consecutive block. Regression models cannot be interspersed with other types of imputation 

models, e.g., the user could not try a regression model, then a logical edit, then another regression model for the same 

variable. 

This example from the FIRE trade area illustrates our model selection procedure. The FIRE data had three available 

covariates for modeling annual payroll: employment (EMP), receipts (RCP), and 1st quarter payroll (QPR). FIRE currently 

uses three different industry 

average imputation models, 

one per covariate. Table 1 

presents the seven candidate 

no-intercept regression 

models, with their 

associated MAE rankings. 

Model 5, which uses the 

industry average from 1st

quarter payroll, has the lowest MAE. Models 6 and 7 – the other industry average models – have considerably higher MAE’s 

than many of the multiple regression models. So, any retained multiple regression models must have lower MAE than Model 

6. For this reason, we dropped Model 2, leaving three remaining candidate multiple regression models. Of these three, we 

eliminated Model 1 since it had higher MAE than those from Models 3 and 4 (and more covariates). Finally, we dropped 

Model 4 because its MAE was very close to the Model 3 MAE, and the MRPT was lower than the corresponding value for 

Model 3. This left us with four possible models (models 3, 5, 6, and 7). We then refit these models with an intercept and 

repeated the same analysis. Table 2 presents the MAE and MRPT for each of these models, again ranked by ascending 

MAE. 

Table 1: Candidate Models for Annual Payroll 

Model Model Rank Model Type MAE MRPT

1. APR = β1EMP + β2RCP + β3QPR 4 Multiple Regression 12872.77 .9643 

2. APR = β1EMP + β2RCP 6 Multiple Regression 32592.12 1.0266 

3. APR = β1QPR + β2EMP 2 Multiple Regression 10479.41 .9838 

4. APR = β1QPR + β2RCP 3 Multiple Regression 10759.53 .9614 

5. APR = β1QPR 1 Industry Average 8838.25 .9839 

6. APR = β1EMP 5 Industry Average 24924.56 .9958 

7. APR = β1RCP 7 Industry Average 65526.00 1.1275 

Of the four new withintercept models, only 

Model 3a appears 

promising, with slightly 

improved predictions over 

its corresponding nointercept model. However, 

the intercept term was 

significant in only four of 

the 30 test industries, so we 

decided not to recommend this imputation model. In fact, the intercept term was rarely significant in our models. This is not

unreasonable: for example, one would expect an establishment with no payroll to have no 1st quarter payroll, employment, or 

receipts. Ultimately, we recommended adding only multiple regression Model 3 to the suite of available imputation options. 

Table 2: Candidate Reduced Models for Annual Payroll With and Without Intercept Terms 

Model Model Rank Model Type MAE MRPT

3. APR = β1QPR + β2EMP 4 Multiple Regression 10479.41 .9838 

3a. APR = β0 + β1QPR + β2EMP 3 Multiple Regression 10215.30 .9864 

5. APR = β1QPR 1 Industry Average 8838.25 .9839 

5a. APR = β0 + β1QPR 2 Multiple Regression 8850.87 .9839 

6a. APR = β0 + β1EMP 5 Multiple Regression 23053.72 1.0114 

6. APR = β1EMP 6 Industry Average 24924.56 .9958 

7. APR = β1RCP 8 Industry Average 65526.00 1.1275 

7a. APR = β0 + β1RCP 7 Multiple Regression 62371.41 1.1035 

Table 3 presents the number of recommended multiple regression models in each trade area for each item. For models that 

do not predict annual payroll, we can substitute 1st quarter payroll for annual payroll in any recommended model, thereby 

increasing the number of available imputation options. 

At first glance, it may appear that we 

are recommending adding an excessive 

number of new imputation models for 

purchases. However, this item is 

traditionally very poorly reported and 

is very difficult to impute from a single 

covariate. Recall that Wholesale Trade 

industries collect six basic data items 

Table 3: Number of Recommended Multiple Regression Models 

Payroll Employment Sales 

Operating 

Expenses Purchases 

Wholesale 3 5 3 4 17 

FIRE 1 1 1 N/A N/A 

Services Tax exempt 4 3 2 3 N/A 

Non Tax Exempt 3 2 4 N/A N/A 

Retail 2 6 5 N/A N/A 

Utilities 4 6 2 N/A N/A 

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7 

(five excluding 1st quarter payroll), giving up to 41 possible imputation models. For the other items, the number of 

recommended additional imputation models is quite in line with the number of available covariates [Note that we did not

include purchases as a covariate in any Wholesale Trade model because of its poor reliability]. 

Conclusion

This paper presents the results of a comparison of alternative regression parameter estimation methods using data from the 

1997 Economic Census. Our model-building data are highly multicollinear and our fitted models exhibit heteroscedasticty. 

We present methods of model evaluation and model selection that are not dependent on sums-of-squares statistics. Thus, our 

evaluation methods can be applied to other (similar) situations. 

In theory, (weighted) least squares estimates from multicollinear regression models can be used for prediction, as long as the 

prediction region is in the same range as the model-building data. We attempted to enforce this by using the region defined 

by the ratio edit parameters as our prediction region. While this may be sufficient for our dollar value variables, we are not

sure that it will be true for employment. Furthermore, we are uncertain whether our recommended ridge regression models 

for employment will perform consistently with the 2002 Economic Census data because of the change in data collection as 

well as the subjectivity of the modeling procedure.

All of the trade areas represented by the services sectors calculate two sets of imputation parameters for the Economic 

Census: the first set are developed from the complete prior census data and the second set are developed from the current 

census data after sufficient cases have been processed. Rather than develop two sets of multiple regression parameters for 

production, we used the 1997 data for research, developing recommendations for trade-area specific regression models for 

each basic data item. We plan to use the available 2002 Economic Census data to validate the recommended models and to 

re-assess the proposed imputation hierarchy. Thus, the production regression parameters will be developed from 2002 data 

only, if in fact, our results hold up. 

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