ESTIMATORS FOR AVERAGE HOURLY EARNINGS AND AVERAGE WEEKLY HOURS FOR THE CURRENT

EMPLOYMENT STATISTICS SURVEY

S. West, D. Kratzke, and P. Grden, Bureau of Labor Statistics

Sandra A. West, 2 Massachusetts Ave. N.E., Washington, D.C. 20212

KEY WORDS: Ratio Estimators, Regression, HorvitzThompson, Composite, Monthly Change

Any opinions expressed in this paper are those of the

authors and do not constitute policy of the Bureau of

Labor Statistics.

I. Introduction

In this paper the results of a theoretical and empirical

investigation of different estimators for average hourly

earnings, average weekly hours, and the respective

monthly changes are presented. The study considers

imputation under both situations of low and high non

response rates.

The investigations began in connection with the

revision program for the Bureau of Labor Statistics’

Current Employment Statistics (CES) survey. This is a

longitudinal survey of approximately 400,000

establishments that provides monthly estimates of,

among other parameters, total employment, average

weekly hours and average hourly earnings for

production workers, for industry groups and for total

private industry. In this program, population

employment counts are obtained once a year from

Unemployment Insurance administrative records. The

CES frame is updated annually, based on population

information up to the first quarter of the year. At this

time, estimation from the second quarter forward is

revised, post-stratified to the March population

employment. March is referred to as the "benchmark

month," and the March population employment is

called "benchmark employment." Note that population

counts for the number of production workers, their

hours, and their earnings are not available.

The ten estimators considered along with some of their

theoretical properties are presented in Section II. In

Section III the details of the empirical investigation are

explained, and the recommendations are given in

Section IV.

II. Theoretical Investigation

1. Definitions and notation

Before discussing the estimators the following notation

are needed.

PW i

k

( ) = a random variable denoting the number of

production workers for establishment i

during month k,

PR i

k

( ) = a random variable denoting the weekly

payroll for production workers for

establishment i during month k,

WH i

k

( ) = a random variable denoting the weekly

production worker hours for establishment

i during month k,

Y i

k

( ) = a random variable denoting all

employment for establishment i at month

k,

w i

k

( ) = the sampling weight associated with

establishment i at month k,

Sk = the set of establishments in the sample at

time k, for k=0,1,2,..., and k= 0 denotes

the benchmark month for employment,

S S k −1 k = S S k −1 k set of establishments that

responded in both time periods k-1 and k,

(referred to as a matched sample)

N = number of establishments in the

population

(It will be assumed that the number of

establishments in the population is fixed

from month to month.)

Let A denote a subset of the population, such as an

industry class. We are interested in an estimator for

the average weekly hours and average hourly earnings

for the set A during time k. This is defined as,

Average weekly hours for A:

AWHk( A) =

WH i

PW i

k

i A

k

i A

( )

( )

∈

∈

∑

∑

 =

WH A

PW A

k

k

( )

( )

Average hourly earnings for A:

AHEk(A) = 

PR i

WH i

k

i A

k

i A

( )

( )

∈

∈

∑

∑

 =

PR A

WH A

k

k

( )

( )

(1)

2. Estimators

The investigation for average weekly hours, AWH, and

average hourly earnings, AHE, differs from the

employment estimation in two main areas. One, in

employment estimation, we estimate a population total,

whereas here we are estimating a ratio of totals.

Second, in employment we have a true population

count, a benchmark at certain time intervals. For our

variables, which are total weekly payroll, total weekly
hours and total number of production workers there are

no benchmarks. However, since our variables and total

employment are highly correlated, it may be possible to

model our variables on total employment and make use

of the benchmark employment.

Nine estimators are developed and compared with the

current estimator. The nine estimators can be

categorized into one of two types. One type uses

information only from the current time period and is

the ratio of two Horvitz-Thompson estimators for total.

The second considers more than the current time

period, and is a regression type estimator for all

months but the benchmark month. The benchmark

month is estimated either by modeling or by a HorvitzThompson type estimator.

We will start with the first type, which contains only

one estimator, the ratio of two Horvitz -Thompson

estimators for total. That is:

AWHk

(A) = 

WH i w i

PW i w i

k

i S A

k

k

i S A

k

( ) ( )

( ) ( )

( )

( )

∈

∈

∑

∑

and (EST1)

AHEk

(A) = 

PR i w i

WH i w i

k

i S A

k

k

i S A

k

( ) ( )

( ) ( )

( )

( )

∈

∈

∑

∑

where S(A) represents a random sample from A.

Since it is known that payroll, hours, and production

workers from the previous time period are each

correlated with the corresponding variables in the

current time period, this estimator does not make use

of all the information available, including benchmark

employment.

All the estimators in the second type are developed

from a modeling point of view, even though they can

be obtained from a strictly probabilistic view point.

With the use of models the assumptions are clearly

visible, and if appropriate data sets are available the

models can be tested. In fact, in the early eighties,

West (1982), we were exploring models for the

employment variable using the universe data base. We

found that the most promising model was the simple

proportional regression model:

E Y i Y y y i

k k k k k

( ( )| ) ( ) −1 = −1 = b −1

(2)

cov( ( ), ( )| )

( )

Y i Y j Y y

Y i if i j

if i j

k k k k

k

− −

−

= =

=

≠







1 1

2

s 1

0 

 

For a variable that does not have a benchmark, such as

payroll, PR i

k

( ) , or hours, WH i

k

( ) , it is possible to

add to the above two equations a link with the

benchmark employment. Thus for the benchmark

month, payroll is studied under the following model:

E PR i Y y y i PR ( ( )| ) ( ) 0 0 = 0 = b 0

(3)

cov( ( ), ( )| )

( )

PR i PR j Y y

Y i if i j

if i j

0 0 0 0

2

0

= =

=

≠







s

0 

 

This relation is a way of representing the assumption

that payroll is roughly proportional to employment.

The same model is considered for relating production

workers and hours to benchmark employment. Using

these relationships, the new variables are all estimated

for the benchmark month. For example, an estimator

for total payroll at the benchmark month is:

PR (A)

PR Y (A), 0 = b 0

where (4)

b ( ) ( ) ( ) ( )

PR

j S j S

= pr j w j y j w j

∈ ∈

∑ 0 0 ∑ 0 0

0 0

For the subsequent months:

PR A PR A

pr j w j pr j w j

for k

k k k

k k

j S S

k k

j S S

k

k k k k

( ) ( ),

( ) ( ) ( ) ( )

, , ,

=

=

=

−

∈ ∩

−

∈ ∩

−

− −

∑ ∑

b

b

1

1 1

1 1

1 2 3

where (5)

Note that the estimator in (5) has the same form as the

current employment estimator (link relative estimator)

except here the regression coefficient reflects the

sample design by the incorporation of the sampling

weights.

Similar formulas exist for hours, WHk(A) , and

production workers, PWk( A) . Thus estimators for

average weekly hours, AWHk(A) , and average hourly

earnings, AHEk

(A) , are:

AWHk(A) =WHk(A) PWk( A)

and (EST2)

AHEk(A) = PRk(A) WHk(A)

for k = 1,2,3,..

AWH (A)

0 =WH ( A) PW (A)0 0

AHE (A)

0 = PR (A) WH (A)0 0

where for time period 0 the totals are estimated as in

equation (4).

The next two estimators differ from EST2 only for time

period, k =0. For the third estimator, EST3, for k =0,
the ratio of Horvitz-Thompson estimators is used,

rather than using the model that relates the new

variables to employment, as in (4).

Note that although EST2 and EST3 are defined

differently for k =0, they can produce the same result.

Since in EST2 both the numerator and denominator

are functions of benchmark employment, it is possible

for employment to cancel leaving simply the ratio of

two Horvitz-Thompson estimators as in EST3. This

happens when the “modeling cell” (formation of b ) is

the same as the publication cell (the point where the

ratio is formed). This was avoided as much as possible

by selecting the modeling cells as small as possible.

Thus, the estimates for the larger cells were obtained

by estimating the numerator and denominator

separately and then taking the ratio at the higher level.

The next estimator, EST4, differs from EST2 only in

the alternative estimation procedures for k =0. EST4

uses a prediction approach which resulted in one of the

top estimators in earlier studies, West(83,84).

Specifically, the numerator and denominator of each

estimator is written as the sum of the sampled units

plus the sum of the non sampled units. In this case the

effect of employment will not cancel out.

Recall that in estimator, EST2, the movement from one

month to the next is done by moving the numerator

and denominator and then forming the ratio. In the

next estimator, EST5, the movement is done on the

ratio. If in the underlying model, the conditional

variance is proportional to the prior months value

squared, then the resulting estimator is the sum of the

ratios, as opposed to the EST2 which is the ratio of

sums. EST6 is the current estimator, which is an ad

hoc estimator that has produced reasonable results, for

the most part, over the years, but has no statistical

theory underlying it. For details on these estimators

see West, Kratzke, Grden, (1997).

The next two estimators are link relative type

estimators, only now the current month is linked to the

0

th month, rather than to the prior month as in the case

of EST2 through EST5. For lack of space the formulas

will not be given, but they will be summarized.

EST7 and EST8: These two estimators are similar to

the link relative except “linking” is to the 0th month

rather than the (k-1)th month. EST7 is most like the

link relative in that it does not use imputation and uses

matched samples. EST8 uses imputation and does not

require matched samples Both these estimators use

benchmark employment in modeling for the 0th month

estimators. (Note that the totals in EST8 are of the

same form as the proposed employment estimator.)

EST9 and EST10: For both these estimators it is

assumed that total payroll for production workers is

known for the 0th month. EST9 is similar to EST8,

and EST10 is similar to EST2, with the difference

being that payroll in the 0th month does not have to be

modeled.

Note that only EST1, EST8 and EST9 use an explicit

imputation method, which will be discussed in the next

section. In the situation of 100% response rate EST2 =

EST4 = EST7 = EST8.

III. Empirical investigation

1. Population

We used 13 months, from March 1994 through March

1995, of national CES data, which is the only data base

that has wages for production workers. A frame was

created which consisted of CES reporters with private

ownership and with non-missing data for the variables

of interest: number of production workers, weekly

payroll, weekly hours, and employment. This frame

consists of 173,434 establishments.

2. Design of the sample

A national sample (26,308) was taken from the frame

according to the selection weights specified for the

employment simulation study. The allocation was

done by Allocation Industry Cell/Size.

3. Nonrespondents and imputation methods

We are concerned only with the first closing monthly

estimates. The expected response rate for

establishments making up the first estimates was

suggested as 80%. However the current response rate

is more around 60%. We decided to consider both

scenarios. Note that we are concerned only with unit

nonresponse. From previous investigations, it is clear

that the nonrespondents are not missing at random, so

we devised the following scheme to select the

nonrespondents.

Each observation on the frame has a closing code

which indicates the time in which the establishment

reported to the survey. The closing codes have values

from 1 to 4, which indicate the month that the data

were reported after the reference month. Initially, in

the sample the nonrespondents are considered to be the

reporters with closing codes 3 or 4. Across industries

and time, there is a range from 3 to 17 percent of

reporters whose closing codes are 3 or 4, which

actually gives us a better than 80% response rate

overall. In order to see how the best two estimators

and the current estimator withstand high nonresponse

rates, the study was also done with nonrespondents

considered to be the reporters with closing codes 2, 3

or 4. Across industries and time, there is a range from

14 to 47 percent of reporters whose closing codes are 2,

3 or 4, which actually gives us a fairly low overall

response rate.
Explicit imputation is done only for estimators, EST1,

EST8, and EST9. The imputation methods are as

follows: for the first month essentially a mean

imputation is used, and a ratio imputation for the

subsequent months. For a subsequent month, the

imputed value for nonrespondent j is obtained by

multiplying the previous month’s value for

establishment j by the ratio of the sum of values of

current month to sum of values of previous month in a

matched sample. Matched samples are done by

industry/size class. This method had top ratings in

earlier studies, West, Butani, Witt (1991), using

population data for wage and employment. In that

study it turned out that whether the imputation was

done on wage or wage over employment the results

were similar. In our method, it is assumed that an

editing procedure will be run after all data are imputed.

This is a good procedure to follow for any imputation

method that is used.

4. Comparison at the 3 digit SIC level of the first

five estimators

EST1 through EST5 were tested on the construction

and durable goods industries at the 3-digit SIC level.

The two most promising estimators were EST1 and

EST2.

5. Comparison of EST1, EST2, and the current

estimator, EST6, when publishing estimates for the

major industry divisions, and for total private

industry.

For each estimator, estimates were made for the nine

major industry divisions, and for total private industry

for thirteen months for level and twelve months for

monthly change. The error is defined as the estimate

minus the true. We looked at the sums of errors and

absolute errors (as well as relative errors). The three

estimators were compared in a number of ways. Since

they led us to the same conclusion we will just exhibit

the summary tables for the absolute errors. The

absolute errors for each major industry estimate are

summed over the thirteen (or twelve) months and the

nine major industries. The results are shown in Table

I. for both small and large nonresponse rates. Here we

see the pattern: EST1 is best for level and EST2 is

best for change The absolute error for total private

industry is summed over the months and is shown in

Table II. Here in the higher nonresponse rate the

pattern emerges again: EST1 best for level, EST2 best

for change.

6. Observations

a. The size of the response rate has less of an effect on

EST1 than EST2 when it relates to level, but more of

an effect for change. EST6 is not as good as either of

the other two.

b. As theory would predict, EST2 did very well for

estimating the monthly change, quite a bit better than

EST1. Since EST2 makes use of last month’s

information and is based on a matched sample, it

should do better for change. However EST2 did not do

as well as EST1 for level. What is particularly

troubling about the EST2 level estimate is that it is

made up of a ratio, where both the numerator and

denominator are each formed by ratio adjusting an

initial value, which is not the true value (as in the

employment estimator). The estimate is the ratio of

two parts where each part is made up of an initial

estimate multiplied by a string of ratios, the length of

which, depends on the distance from the “benchmark”.

The properties of the variance of this estimator is not

clear. Even if the variance of the estimators for the

numerator and denominator increased the further away

you were from the benchmark month, the variance of

the ratio would not necessarily be monotone.

c. Between the top two estimators, the HorvitzThompson estimator, EST1, with ratio imputation for

the nonrespondents is the best choice if level is the

most important issue. Note that using ratio imputation

is essentially using EST2 for the nonrespondents. On

the other hand if change is the major issue than EST2

would be a better choice. However EST1 and EST2 do

not differ by much, and in light of the objection with

EST2 stated previously, EST1 with ratio imputation,

would be the preferred method at this stage. However,

we will look at the additional estimators in the next

section and estimate the variances and bias of the top

two estimators.

d. In EST2, the model based estimators, Link

Relative, in the numerator and in the denominator

were better than the corresponding Horvitz-Thompson

estimators in EST3. Specifically, for the ratio, say

AWE = PR / WH, most often the Link Relative

estimator did better at estimating PR and WH than did

the Horvitz-Thompson. However. for the ratio, AWE,

the opposite was true, Horvitz-Thompson did better

than Link Relative. In West et al (1997), it is shown

theoretically under what conditions the numerator and

denominator each can be better estimated by a specific

estimator over an alternative estimator, but for the

resulting ratio the reverse is true.

7. Comparison of additional estimators

Estimators EST7 through EST10 were added to the

empirical study for the major industry estimates with

the result that they did not do as well as EST1 and

EST2. As expected, EST7 and EST8, which link to

the 0th month which is not a benchmark, did not do as

well as EST2, which links to the prior month. This
was especially true for monthly change, which is the

most important parameter.

One estimator, EST10, is worth mentioning; it has the

same form as the link relative estimator, EST2, except

for the benchmark month, where production worker

payroll, PR, is assumed known. Note that EST9 and

EST10 can not be applied in practice, since we do not

have production worker wages for the population. The

closest that exists is on the data base that gives us the

population employment for the 0th month, which also

has a quarterly wage for all employees. With this in

mind, we decided to see how well we could do if in fact

we did have the true wages for production workers at

the benchmark month. Note we still do not have true

total hours. Thus, for average hourly earnings for the

benchmark month we have true total payroll for the

numerator and an estimate from our model for total

hours. Average weekly hours would not change at all.

In Table III., EST1, EST2, EST8, EST7, and this

hypothetical estimator, EST10, are compared. For

each estimator, estimates were made for the nine major

industry divisions, and for total private industry for

thirteen months for level and twelve months for

monthly change. In Table III. the absolute errors are

summed over the thirteen (or twelve) months for each

industry.

We arrive at the same conclusion as before that EST1

does well for level and EST2 does well for change.

What may seem surprising is that knowing true

production worker payroll for benchmark month did

not do better than our original way of modeling on

benchmark employment. Reflecting on this, it is not

surprising. When both numerator and denominator are

highly correlated random variables we do better than

with a fixed number and a random variable. Note that

the estimators that require matched samples, EST2,

EST7, and EST10, all do better for change than the

estimators that use imputation, EST1 and EST8.

8. Mean Square Error Estimation

One last comparison between EST1 and EST2 was

made. A mean square error was computed over thirty

samples from which we could obtain estimates of

variance and bias. Specifically, thirty samples were

randomly drawn from our population. For each

sample, estimates were computed for the four

parameters using EST1 and EST2. For each estimator,

the mean square error, MSE, was computed for each

parameter over the thirty samples. The mean square

error was computed for each month and the average

formed over the thirteen (or twelve) months. The

results, shown in Table IV., support the same

conclusion as before. Note that the results in these

Tables are computed using all the establishments that

had a closing code of 2, 3 or 4 as the nonrespondents.

IV. Recommendations

EST1, which is the ratio of Horvitz-Thompson

estimators, is best for level, and EST2, which is the

ratio of Link Relative estimators, is best for change.

Since EST1 and EST2 do not differ by much, our

recommendation would be to use EST1 with ratio

imputation. The main problem with EST2 is that we

are starting out at the “benchmark” month with an

estimate and we are continually ratio adjusting it from

one month to the next, so that if we have a bad

estimate initially the level will continue to be bad for

subsequent months. This is not a problem with

monthly change. Thus, until we have a benchmark for

AWH and AHE, we recommend EST1 over EST2, or

any EST2 derivative, such as the ratio to 0th month

(which is EST2 with 100% response rate). EST1 could

be thought of as a composite of the Horvitz-Thompson

and Link Relative estimators, since the imputation

used for the Horvitz-Thompson is the same model

underlying the Link Relative estimiator.

References

Bureau of Labor Statistics (1989). Current

Employment Statistics State Operating Manual.

U.S. Department of Labor, Washington, D.C.

Butani, Harter, Wolter, (1997) “Estimation Procedures

for the Bureau of Labor Statistics Current

Employment Statistics Program” Proceedings

of the Section on Survey Research Methods,

American Statistical Association

Werking, George, (1997), “Overview of the CES

Redesign Research,” Proceedings of the Section

on Survey Research Methods, American

Statistical Association.

West, Sandra A., Kratzke, Tran, Grden, Paul, (1997)

“Estimators for Ratios of Totals in the Current

Employment Statistics Survey.” Bureau of

Labor Statistics Statistical Note.

West, Sandra A., Butani, Shail, and Witt, Michael.

(1991), "Alternative Imputation Methods for

Employment, Wage and Ratio of Wage to

Employment Data," Proceedings of the 78th

Indian Science Congress, India.

West, Sandra A. (1984), "A Comparison of Estimators

for the Variance of Regression-Type Estimators

in a Finite Population," Proceedings of the

Section on Survey Research Methods, American

Statistical Association, 170-175.

West, Sandra A. (1983), "A Comparison of Different

Ratio and Regression type Estimators for the

Total of a Finite Population," Proceedings of

the Section on Survey Research Methods,

American Statistical Association, 388-393.
West, Sandra A. (1982), “Linear Models for all

Employment Data” Bureau of Labor Statistics

report dated September 29, 1982.
Table I. Sum Of Absolute Errors For Major Industry Estimates Over Time And Major Industries

Small NonResponse Rate Larger NonResponse Rate

EST1 EST2 EST6 EST1 EST2 EST6

Level AWH 12.229 15.676 15.871 22.450 40.803 40.708

Level AHE 07.350 10.533 08.927 8.993 18.028 16.876

Change AWH 10.923 08.857 10.841 29.366 20.489 26.951

Change AHE 03.576 02.854 05.073 06.070 04.229 07.994

Table II. Sum Over Time Of Absolute Errors For Total Industry Level

Small NonResponse Rate Larger NonResponse Rate

EST1 EST2 EST6 EST1 EST2 EST6

Level AWH 0.414 0.363 0.950 0.709 0.919 2.431

Level AHE 0.287 0.269 0.391 0.429 1.098 1.573

Change AWH 0.382 0.321 0.618 1.005 0.554 2.460

Change AHE 0.135 0.089 0.348 0.258 0.146 1.414

Table III. Sum of monthly absolute errors for major industry estimates over time.

LEVEL - Average Hourly Earnings

INDUSTRY EST1 EST2 EST8 EST7 EST10

 FIRE 0.76 0.36 1.20 0.59 5.71

 Construction 0.68 1.13 0.80 1.23 0.76

 Durable 0.95 3.27 2.63 3.50 0.71

 Mining 2.70 9.72 2.50 4.05 3.29

 Nondurable 0.75 0.48 0.36 0.50 2.39

 Retail 0.45 0.52 0.59 0.74 1.25

 Services 0.50 0.79 0.85 0.91 0.55

 Transportation 1.03 1.03 1.15 1.00 1.15

 Wholesale 1.16 0.74 1.28 0.95 1.24

CHANGE - Average Hourly Earnings

INDUSTRY EST1 EST2 EST8 EST7 EST10

 FIRE 0.32 0.26 0.30 0.46 0.26

 Construction 0.52 0.42 0.52 0.46 0.41

 Durable 0.56 0.28 0.57 0.19 0.25

 Mining 1.34 1.30 1.47 1.77 1.72

 Nondurable 0.46 0.14 0.47 0.24 0.14

 Retail 0.49 0.31 0.51 0.41 0.29

 Services 0.22 0.15 0.24 0.28 0.16

 Transportation 1.30 0.85 1.31 0.84 0.82

 Wholesale 0.86 0.51 0.88 0.60 0.51

Table IV. Average MSE, MSE , and Average Bias, B , for EST1 and EST2

(Computed for each month using 30 sample replicates and then averaged over the 3/94 to 3/95 time period)

 EST1 EST2

EARNINGS LEVEL CHANGE LEVEL CHANGE

INDUSTRY MSE B MSE B MSE B MSE B

 FIRE 0.008 0.021 0.003 -0.005 0.008 0.008 0.001 0.000

 construction 0.011 -0.011 0.004 0.010 0.016 -0.043 0.004 0.008

 Durable 0.009 0.018 0.010 -0.030 0.069 0.258 0.001 -0.005

 Mining 0.071 -0.088 0.042 -0.051 0.131 -0.254 0.020 -0.034

 Nondurable 0.005 0.016 0.004 -0.016 0.004 -0.012 0.000 -0.004

 Retail 0.003 -0.014 0.003 0.015 0.002 -0.016 0.001 0.002
 Services 0.002 -0.002 0.001 0.003 0.004 0.043 0.000 -0.003

 Transportation 0.018 0.008 0.018 -0.008 0.017 0.003 0.012 0.008

 Wholesale 0.014 0.030 0.005 -0.011 0.039 0.139 0.003 -0.003

Total Private 0.002 0.005 0.002 -0.012 0.007 0.082 0.000 -0.001