ALTERNATIVE IMPUTATION METHODS FOR WAGE DATA

Sandra A. West, Shail Butani, and Michael Witt, Bureau of Labor Statistcs

441 G Street NW, GAO Bldg Room 2126, Washington DC 20212

1. Introduction

 In this paper the results of an empirical investigation of

different imputation methods for wage data and the ratio

of wage to employment data are presented. This study is

a sequel to the paper entitled "Alternative Imputation

Methods For Employment Data" (1989). Both projects

began in connection with a revision project for the Bureau

of Labor Statistics (BLS) program that maintains the BLS

Universe Data Base (UDB). The UDB is a sampling

frame of business establishments that is constructed from

the State ES-202 microdata files. The information used to

maintain this file is obtained from quarterly

unemployment insurance (UI) reports which each covered

employer is required to submit. These quarterly reports

contain, among other things, information on employment

for each month of the quarter, total quarterly wages, as

well as a standard industrial classification (SIC) code for

the establishment. Although the filing of the contribution

report is mandatory under the current UI laws, each

quarter there are always some reports that are filed late,

delinquent accounts, as well as returns with partial data.

 The goal of this project was to develop a single

imputation procedure for total quarterly wages of an

establishment that would work reasonably well for all SIC

groups within each state. The methods included

regression modeling, distribution modeling with

maximum likelihood estimators for the parameters,

multiple imputation, and standard procedures such as hot

decks, and mean value.

 The wage data used in this study are discussed in

Section 2. Section 3 presents the notation used and the

evaluation criteria that are used to compare the various

imputation methods. Section 4 provides a description of

the standard procedures such as: carryover method of

imputation, two mean imputation procedures, and two hot

deck procedures. In Section 5, eight regression models

for imputing wages are presented. One problem with a

"best" regression-based prediction method is that all

imputed values will fall on the estimated regression line

and therefore, will lead to biases in estimates that involve

the residual variance for nonrespondents. Simple

methods that attend to this problem draw random

residuals which are added to the model predictions.

Details of such methods are given in Section 6. In

Section 7, imputations are created under an explicit

Bayesian model and multiple imputations are developed

in Section 8. In a multiple imputation context, several

imputed values would be created for each missing value,

where ideally, uncertainty due to the imputation

procedure would be reflected. Section 9 compares the

results from the various imputation methods and

summarizes the findings of this study.

2. Data

 The data used for the wage study were the ES-202

microdata files obtained from the State Employment

Security Agency (SESA) of Wisconsin. Although results

of this project are needed for all the states, due to various

reasons it was not possible to obtain a sufficient amount

of data from any other state.

 Five consecutive calendar quarters of data, (January

1988 through March 1989), were selected and used to

impute wages for the latter four quarters. Four quarters

were considered so that fluctuations in total wages due to

seasonality could be incorporated into the analyses.

Because most of the imputation methods required a unit's

total wages from the previous quarter, data for five

quarters were needed.

 All the procedures were tested for three different SIC

groups of establishments. They are: SIC 16, Heavy

Construction Other Than Building Construction--

Contractors; SIC 37, Transportation Equipment; and SIC

50, Wholesale Trade--Durable Goods. Two of the three

SICs were chosen so that the results of this study could be

compared against the authors previous study (West, et

al, 1989).

 Intuitively, an establishment's total wages are highly

correlated with its total employment at any given point in

time. Consequently, a measure of size was created for

each establishment based on the establishment's oldest,

nonmissing monthly employment value beginning with

January 1988. This size measure was used to stratify the

data set by three different size partitions (Table I) in order

to examine the size class effect, if any, on the imputation

procedures.

 At the time of this study, there were no data available

that would indicate the establishments for which the

wages or employment were imputed by the SESA of

Wisconsin. Therefore, it was assumed that the missing

data mechanism is ignorable, and random sets of units

were chosen to represent the set of nonrespondents. To

examine the effects of nonresponse rates, two sets

consisting of 10% and 20% of the datafile were selected

and designated as the nonrespondents and imputation

procedures were examined using the remaining 90% and

80% , respectively.

 All of the imputation procedures were constructed using

an establishment's total quarterly wages, and in an

analogous fashion, the ratio of total quarterly wages to

total quarterly employment (i.e. mean quarterly wages).

The ratio of wages to employment was considered

because it was felt that the ratio would stabilize total

wages, which usually fluctuate across quarters much more

than employment. In both cases, however, the error
measures, defined in the next section, were computed for

total quarterly wages.

 The error measures for 90 different methods (Table II)

by three size class partitions (1, 3, and 8) were computed

for each of the three SICs and two response rates. This

was done separately for models based on total quarterly

wages and ratio of total quarterly wages to total

employment. Due to space limitations, the results using

all different combinations of these factors could not be

presented but will be briefly discussed in the conclusion

section.

3. Notation and Evaluation Criteria

 In this Section and in Sections 4 through 7, the

imputation procedures are discussed using total wage

data. The imputation procedures applied to mean wages

analogously follows by letting "Y" represent mean wages

as opposed to total wages.

Notation

Let the variables:

ESt,i = Establishment i, in quarter t

St,r,m = Set of respondents in domain of procedure m

St,nr,m=Set of nonrespondents in domain of procedure m

Yt,i = Reported quarterly wages of ESt,i

Yp

t,i,m = Predicted quarterly wages of ESt,i

Nt,r,m = Number of units in St,r,m

Nt,nr,m = Number of units in St,nr,m

Et,i,m = Error in the prediction = (Yp

t,i,m - Yt,i)

AEt,i,m = Absolute error in the prediction

 = ❘Yp

t,i,m - Yt,i❘

Evaluation Criteria

a. Relative Error (%):

 REm = {Σ Σ Σ Et,i,m / Σ Σ Σ Yt,i} * 100.00

size t i size t i

b. Relative Absolute Error (%):

 RAEm = {Σ Σ Σ AEt,i,m / Σ Σ Σ Yt,i} * 100.00

 size t i size t i

where i ε St,nr,m.

 Note that REm represents a macro level statistic that

indicates the effect that the imputation procedure has on

total quarterly wages, while RAEm is a micro level

statistic that indicates the effect on the unit's quarterly

wages. The corresponding mean unit errors per

nonrespondent were also computed but are not presented

in this paper due to space.

4. Standard Methods

CO: Carryover Method of Imputation

 Under this method, total quarterly wages of each

nonrespondent, ESt,j , is imputed as follows:

Yp

t,j,CO = Yt-1,j .

MN: Mean Imputation Method

 For any fixed SIC group, employment size class, and

quarter t and for all ESt,j ε St,nr,MN:

Yp

t,j,MN = Σ Yt,i / Nt,r,MN .

 i

 Thus Yp

t,j,MN is equal to the average of the total

quarterly wages of all respondents in the stratum.

MNL: Mean Of Log Wages

 This method is the same as the mean imputation method

stated above except log wages are substituted for wages.

HD1: Hot Deck Imputation Method - Random Selection

 For any fixed SIC group, employment size class, and

quarter t, the quarterly wages of ESt,j ε St,nr,HD1 is:

Yp

t,j,HD1 = Y*t,i

where Y*

t,i is the total quarterly wages of a randomly

selected respondent from St,r,HD1. Selection was done

independently within strata and with replacement.

HD2: Hot Deck Imputation Method - Nearest Neighbor

 The Nearest Neighbor hot deck method is desirable

because for any particular nonrespondent, it selects the

respondent that appears closest to the nonrespondent in an

ordered list, and substitutes the respondent's total

quarterly wages value for the nonrespondent's.

 Within any fixed SIC group, employment size class, and

for each quarter t, all ESt,i were ordered by Yt-1,i by Yt2,i by state. For this ordering procedure, missing values

for Yt-1,i and Yt-2,i, were considered -1.

 For all ESt,j ε St,nr,HD2 , let Y(1)t,i be the total quarterly

wages for the first establishment ES(1)t,i ε St,r,HD2 that

precedes ESt,j on the ordered list, and Y(2)t,k be the total

quarterly wages for the first establishment ES(2)t,k ε

St,r,HD2 that succeeds ESt,j on the ordered list. If

❘Y(1)t-1,i - Yt-1,j❘ < ❘Y(2)t-1,k - Yt-1,j❘,

then Yp

t,j,HD2 is set equal to Y(1)t,i. Otherwise, Yp

t,j,HD2

is set equal to Y(2)t,k.

5. Modeling Employment and Wages by Regression

Regression Models
 A common method for imputing missing values is via

least squares regression (Afifi and Elaskoff 1969). In

three papers on estimators for total employment (West

1982, 1983) and West, et al. (1989), it was discovered

that the most promising models for employment were the

proportional regression models. These models specify

that the expected employment for establishment i in the

t

th month, given the following vector of y values for

month

t-1:

Yt-1 = [Yt-1,1, Yt-1,2, ... Yt-1,n]

is proportional to the establishment's previous month's

employment, Yt-1,i. That is,

E(Yt,i ❘ Yt-1,i = yt-1,i) = βyt-1,i,

where β is some constant depending on t. In the

remaining sections, for clarity, the subscripts t and m are

suppressed in conjunction with the parameters σ, α and β.

 It was further assumed that the y's are conditionally

uncorrelated. That is,

νt,i if i = j

 cov(Yt,i,Yt,j ❘ Yt-1 = yt-1) = 

0 Otherwise

where νt,i represents the conditional variance of Yt,i,

which in general will depend on Yt-1,i. Choosing a

specific simple function to represent the variance νt,i

accurately is difficult. Fortunately, knowledge of the

precise form of νt,i is not essential (see Royal 1978).

 The model can be rewritten as:

Yt,i = βYt-1,i + εt,i

where E{εt,i} = 0, and

 νt,i if i = j

 E{εt,i,εt,j} =

 0 Otherwise

 In previous papers, νt,i = σ2Yt-1,i and νt,i = σ2 were

considered and it was found that the model:

Yt,i = βYt-1,i + εt,i

with νt,i = σ2Yt-1,i worked reasonably well for

employment data.

 Since this model with the above assumptions worked

well with employment data, it was decided to apply

variations of the same model with wage data. For the

current data set, the following eight models were

considered for total quarterly wages.

Model 1: Yt,i = α + βYt-1,i + εt,i

Model 2: Yt,i = βYt-1,i + εt,i

Model 3: Ln(Yt,i) = α + βLn(Yt-1,i) + εt,i

Model 4: Ln(Yt,i) = βLn(Yt-1,i) + εt,i

 Models 1 - 4 assume νt,i = σ2. Models 5 - 8 are

similar to models 1 - 4 respectively, except it is now

assumed that νt,i = σ2Yt-1,i for models 5 and 6, and νt,i = σ2Ln(Yt-1,i) for models 7 and 8.

 When the imputation procedure is based on a regression

model, m will be prefixed by RM. The regression model

parameters were estimated using the establishments in the

set St,r,m and an imputed value was calculated for those

establishments in the set St,nr,m. Note that in the case

when Yt,i denotes the ratio of wages to employment it is

assumed that employment is known. The model is

conditional on Yt-1,i and the employment at time t.

Example Using Model 8

Ln(Yt,i) = βLn(Yt-1,i) + εt,i with νt,i = σ2Ln(Yt-1,i)

and β is estimated as:

βp = Σ Ln(Yt,i) / Σ Ln(Yt-1,i) .

i ε St,r,RM8 i ε St,r,RM8

For any nonrespondent, ESt,j, in St,nr,RM8, the

establishment's predicted total wages at time t is:

Yp

t,j,RM8 = exp {βp Ln(Yt-1,j)} .

Adjustments for Models 4 and 8

 Considering models 4 and 8, if it is assumed that εt,i is

normally distributed then Yt,i has a lognormal

distribution with

Mean: exp{ βLn(Yt-1,i) + .5Var(εt,i)}

Variance:{exp[Var(εt,i)]-1}{exp[2βLn(Yt1,i)+Var(εt,i)]}.

 Therefore, an unbiased estimator of Yt,j is:

exp{ βLn(Yt-1,j) + .5Var(εt,j)} .

 As an estimate of Var(εt,j), the residual mean square

error, MSE, from the regression was used. The predicted

total wages for m = 4 and 8 are computed as:

Yp

t,j,RMmA1 = exp{ βpLn(Yt-1,j) + .5MSE}

 An alternative adjustment to the logarithmic regression

models was also tried. This adjustment was used by

David (1986), and led to the following unbiased

prediction of Yt,j for models 4 and 8:

exp{ βpZt-1,j + .5[Var(εt,j) + Z2

t-1,j]Var(βp)}

where Zt-1,i = Ln(Yt-1,i). Thus,
Yp

t,j,RMmA2 = exp{ βpZt-1,j + .5(MSE)(WGS)}

where WGS = 1 - {Ζ2

t-1,j / Σ Ζ2t-1,i} for m = 4

 i

and WGS = 1 - {Zt-1,j / Σ Zt-1,i} for m = 8.

 i

6. Adding Residuals to the Regression Models

 The methods discussed in the previous section could be

thought of as imputing for missing total quarterly wages

by using the mean of the predicted Yt (or ln(Yt))

distribution, conditional on the predictors, Yt-1 (or ln(Yt1)). As a result, the distribution of the imputed values has

a smaller variance than the distribution of the true values,

even if the assumptions of the model are valid. A simple

strategy of adjusting for this problem is to add random

errors to the predictive means; that is, select residuals

rest,k, with mean zero, to add to Yp

t,j,RMm (or the

predicted ln(Yt,jRMm)).

 In this project, it was decided to consider this

imputation procedure with the residuals, rest,k, equalling:

 1. A randomly selected respondent's residual using

model RMm (procedure denoted by RMmRS).

 2. A random normal deviate, from the distribution

with mean 0 and variance MSE *τj

, where τj takes on one

of three values defined below, using model RMm

(procedure denoted by RMmRGλ).

 τj = 1

= Ej = { (Nt,r,m)

-1 + A2

j

 / Σ A2

i

 }

 i

= Pj = 1 + Ej,

where for models 1 and 5

 Aj = Yt-1,j - ( Σ Yt-1,i)/ Nt,r,m)

 i

and for models 2 and 6

 Aj = Yt-1,j.

For the corresponding log models Yt-1,j is replaced

by

Ln (Yt-1,j). Note that the estimated variances, MSE * τj for τj

 = Ej and Pj are estimates of the variances of the

estimator of the mean of Yt,j, and a single new

observation Yp

t,j,m, respectively (Neter and Wasserman,

1974).

 For each of the eight models, residuals were added to

the model predictions by the above methods. For

example, using model 8 and the first method described

above, a prediction of Yt,j is:

Yp

t,j,RM8RS = exp{βp ln(Yt-1,j) + rest,k},

where rest,k is the residual from a randomly selected

respondent k; that is,

rest,k = ln(Yt,k) - βp ln(Yt-1,k).

 Using model 6 and the second method described above:

 Yp

t,j,RM6RGt = βpYt-1,j + sδj

,

where δj is a random number from a N(0,1) distribution

and s2 is equal to the MSE * τj.

7. Bayesian Model

 In creating imputed values under an explicit Bayesian

model, three formal tasks can be defined: modeling,

estimation and imputation. The modeling task chooses a

specific model for the data. The estimation task

formulates the posterior distribution of the parameters of

that model so that a random selection can be made from

it. The imputation task takes one random selection from

the posterior distribution of y missing, denoted by

Yt,BAY, by first drawing a parameter from the posterior

distribution obtained in the estimation task and then

drawing Yt,BAY from its conditional posterior

distribution given the drawn value of the parameter.

 For the modeling task, consider model 2 and Yt,i having

a N(βYt-1,i,σ2) distribution. This is the specification for

the conditional density f(Yt,i ❘ Yt-1,i, θ) where θ = (β,σ).

In order to complete the modeling task, the conventional

improper prior for θ, Probability(θ) proportional to a

constant, is assumed.

 For the estimation task, the posterior distribution of θ is

needed. Standard Bayesian calculations show that:

f(σ2 ❘ Yt,i) = (σp

1)2[n - 1] / χ2n-1

f(β ❘ σ2) = N(βp

1,σ2ν)

where

(σp

1)2 = Σ {Yt,i - βp1Yt-1,i}2 /(n-1) = MSE

 i

βp

1 = Σ Yt,iYt-1,i / Σ Y2t-1,i

 i i

ν = 1 / Σ Y2

t-1,i

 i

n = number of respondents.

 Since the posterior distribution of θ is in terms of

standard distributions, random draws can easily be

computed. The imputation task for this model is as

follows:

1. Estimate σ2 by a χ2

n-1 random variable, say h, and let

σ2

2 = (σp1)2(n-1)(h)

-1

2. Estimate β by drawing one independent N(0,1) variate,

say Zo, and let
β2 = βp

1 + σ2(n).5(Zo)

3. Let no be the number of values that are missing, that

is, the size of St,nr,BAY. Draw no values of Yt,BAY as

Yp

t,j,BAY = β2Yt-1,j + σ2Zj

 (7.1)

where the no normal deviates, Zj are drawn

independently.

Equation (7.1) can be rewritten as:

 (MSE).5(n-1).5

Yp

t,j,BAY=βp1Yt-1,j + ÄÄÄÄÄÄ [(ν)

.5ZoYt-1,j+Zj

].

 (h)

.5

8. Multiple Imputation

 Multiple imputation is the technique that replaces each

missing value with two or more acceptable values from a

distribution of possibilities. The idea was originally

proposed by Rubin. The main advantage of multiple

imputation is that the resultant imputed values will

account for sampling variability associated with the

particular nonresponse model.

 Multiple imputation was obtained from the Bayesian

method by repeating the above three steps five times.

The average of the five values was taken as the imputed

value.

 Multiple imputation was also obtained for the following

procedures: hot deck random selection; regression model

with randomly selected residuals; and regression model

with randomly generated residuals, N(0, MSE * τj) . For

all of the multiple imputation methods, error measures

were computed by using the average of five such repeated

imputations.

9. Comparison of Imputation Methods and

Conclusions

 Each imputation method was applied to an

establishment's total quarterly wages and to the ratio of

total quarterly wages to total quarterly employment (i.e.,

mean quarterly wages). In order to have comparability

between the two data types, the error measures,

Percentage Relative Error (%RE) and Percentage Relative

Absolute Error (%RAE) were based on total quarterly

wages. For both the data types, each imputation method

was applied to each of the three SICs by three sizes class

partitions and two response rates, and, accordingly, the

%RE and %RAE were computed for each combination.

Due to space limitations, the results are presented in

Table II only for SICs 16 and 37, and for the 80%

response rate. Data are presented only for data type total

quarterly wages for reasons stated below.

 For the three SICs considered, imputing total wages

based upon ratio of wages to employment faired about the

same as imputing wages based on total wages, in terms of

the two error measures. The knowledge of the

employment values did not yield smaller errors.

Additionally, the ratio of wages to employment assumes

that total employment is known for every establishment

on the file, which is generally not the case. In fact,

because of the nature of the U.I. reports, one of the

following occurs: (1) both the employment and wage data

are missing or (2) wages are provided and employment

data are missing. It is an extremely rare case when

employment data are provided but not the wages. Since

the effect of using an imputed employment value on

predicting total wages has not been analyzed, at this time

it is recommended that only total wages data be used for

imputation.

 Selecting the best imputation method based only on

total wage data type from the set of 90 methods

considered was difficult because one method of

imputation did not consistently and clearly yield the

smallest error measures. Consequently, in order to

determine the best method of imputing total wages for the

three SICs by two nonresponse patterns, it was decided to

consider only those methods that yielded less than 10 for

the |%RE| and less than 50 for the %RAE for any SIC

group by any size class partition. From this set of

imputation procedures and size class partitions, the subset

that overlapped across the three SICs and the two

response rates was retained. The resulting methods and

size class partitions that had the |%RE| less than 10 and

the %RAE less than 50 across the three SICs and two

response rates are listed in Table A.

Table A: Procedures with |%RE| < 10 & %RAE < 50

METHOD SIZE CLASS

1 3 8

Carryover x x x

Regression Model 4 x

Regression Model 8 x x

.5(MSE) Model 8 x x

.5(MSE)(WGS) Model 8 x x

Randomly Generated Normal Residual:

 S2 = MSE Model 8 x x

 S2 = MSE Model 8* x x

 S2 = MSE*P Model 8 x x

 S2 = MSE*P Model 8* x x

 S2 = MSE*E Model 4 x

 S2 = MSE*E Model 4* x x

 S2 = MSE*E Model 8 x x

 S2 = MSE*E Model 8* x x

ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ

* Indicates data presented are multiple imputation results.

 Model 8 with three and eight size class partitions also

dominated the list when the ratio of wages to employment

data were used. Note that the two basic models that are

among the contenders use the logarithm of wages as the

dependent variable. Most wage models in the literature,

such as David, et al. (1986) and Greenlees, et al. (1982)

are based on household surveys and have different

independent variables in the model, but the dependent

variable is generally the logarithm of wages.

 Because of the dominance of some form of model 8 and

the consistency of three and eight size class partitions,
some form of model 8 is preferred over model 4. Of all

the procedures involving model 8, the one with no

adjustment is preferred in the interest of simplicity. In

the above list, the three size class partition for the

regression model based procedures usually performed as

well as or better than the eight size class partition. Also,

since many State/SIC cells will have only a small number

of observations, it is recommended that three size classes

be employed if a regression model is selected.

 The above discussion limits the selection to either the

carryover method or to regression model 8 with three size

class partitions. Regression model 8 with three size class

partitions is recommended instead of the carryover

method, because the data used for this study were for

January 1988 through March 1989, a relatively stable

period economically. It is expected that the carryover

method will not perform as well during a period of large

economic growth or decline. Also a similar study

conducted last year for employment data recommended

the use of model 6, which is similar to model 8, the only

difference being model 6 uses raw data while model 8

uses the transformed data.

 Future work will include testing of both the carryover

method and model 8 with three size class partitions for

different SICs, States, and response patterns.

References

1. Afifi, A. A., and Elaskoff, R. M., (1969), "Missing

Observations in Multivariate Statistics III: Large Sample

Analysis of Simple Linear Regression", Journal of the

American Statistical Association, vol. 64, pp. 337-358.

2. David, M., Little, R., Samuel, M. and Triest, R.,

(1986), "Alternative Methods for CPS Income

Imputation", Journal of the American Statistical

Association, vol. 81, pp. 29-41.

3. Greenlees, W. S., Reece, J. S., and Zieschang, K. D.

(1982), "Imputation of Missing Values when the

Probability of Response Depends on the Variable Being

Imputed," Journal of the American Statistical

Association, vol. 77, pp. 251-261.

4. Little, R. J. A. and Rubin, D. B., (1987), Statistical

Analysis With Missing Data, John Wiley & Sons Inc.,

New York.

5. Neter, J., and Wasserman, W.,(1974), Applied Linear

Statistical Models, Richard D. Irwin, Inc., Homewood,

Illinois.

6. Royall, R. M. and Cumberland, W. G., (1978),

"Variance Estimation in Finite Population Sampling",

Journal of the American Statistical Association, vol. 73,

pp. 351-358.

7. Rubin, D., (1987), Multiple Imputation for

Nonresponse in Surveys, John Wiley and Sons Inc., New

York.

8. West, S. A., (1982), "Linear Models for Monthly All

Employment Data", Bureau of Labor Statistics Report.

9. West, S. A., (1983), "A Comparison of Different

Ratio and Regression Type Estimators for the Total of a

Finite Population", ASA Proceedings of the Section in

Survey Research Methods.

10. West, S., Butani, S., Witt, M., Adkins, C., (1989),

"Alternate Imputation Methods for Employment Data",

ASA Proceedings of the Section in Survey Research

Methods.

Table I: Establishment Size Class Definitions

Size class is determined by the establishment's oldest,

nonmissing employment during the time period: January

1988 to March 1989. The definition of one, three and

eight size classes are as follows (table entries indicate

number of employees):

ONE THREE EIGHT

0 and above 0 - 49 0 - 9;

100 - 249

 50 - 249 10 - 19; 250 - 499

250 + 20 - 49; 500 - 999

50 - 99; 1000 +

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