-1-

MEASURING EMPLOYMENT FROM BIRTHS AND DEATHS IN THE

CURRENT EMPLOYMENT STATISTICS SURVEY

Diem-Tran Kratzke, Heidi Shierholz, Steve Woodruff, Bureau of Labor Statistics.

Steve Woodruff, BLS, 2 Massachusetts Ave, N.E. Washington, DC 20212

KEY WORDS: Seemingly Unrelated Regression,

Generalized Least Squares, Mixed Estimation.

Any opinions expressed in this paper are those of the

authors and do not constitute policy of the Bureau of

Labor Statistics. The authors thank Alan Dorfman of the

Bureau of Labor Statistics for his review and comments.

1. Overview of the Current Employment Statistics

Survey

The Current Employment Statistics (CES) survey is a

nationwide monthly payroll survey of close to 400,000

business establishments. It provides current estimates of

employment, hours, and earnings in industry and area

detail for the 50 states, the District of Columbia, Puerto

Rico, and the Virgin Islands. The Bureau of Labor

Statistics (BLS) is currently redesigning the CES survey.

The research presented here is one component of the

redesign.

The CES sampling frame is called the Business

Establishment List (BEL). BEL is a list of seven million

establishments whose primary source is the quarterly

contribution reports that employers file with their state

Unemployment Insurance (UI) agencies. Quarterly BEL

data are available to statisticians and economists at BLS

approximately nine months after the end of a quarter.

The sampling frame is updated yearly to account for

changes such as new businesses starting up (births) and

existing businesses going out-of-business (deaths). Births

that occur in between frame updates can be problematic;

it is very difficult and expensive to maintain a

comprehensive and timely sampling frame for new births.

New deaths also pose a problem; though it should be

possible to directly measure them from the sample, in

practice it can be very hard to distinguish between deaths

and nonrespondents because of the vast sample size and

short data collection period of the CES.

This paper focuses on the cost effective and reliable

measurement of the gain of employment from births that

have not yet been added to the frame (birth employment)

and the loss of employment from deaths that have not yet

been removed from the frame (death employment).

2. Research Approach

The continuous portion of the universe consists of all

units that are not births, and their employment is called

continuous employment. The total employment figure

that the CES survey measures is continuous employment

plus birth employment. Our approach is not to measure

the birth employment directly, but rather to predict the net

employment, which is the difference between birth

employment and death employment. Since birth

employment is equal to death employment plus net

employment, it follows that total employment is equal to

the sum of continuous employment, death employment,

and net employment. In estimating total employment,

continuous employment and death employment are both

estimated from the CES sample. Death employment is

estimated by retaining all dead UI accounts in the sample

and imputing their employment with the same imputing

technique used for non-respondents. Net employment is

predicted using a statistical model.

Operationally, this approach is advantageous; a sample

unit is imputed whether it is a non-respondent or a death,

so it is unnecessary to distinguish between the two.

This paper discusses methods for modeling net

employment and examines the ability of net models to

predict net employment.

3. Data

Four historical CES data frames were constructed for a

simulation study of the CES redesign. These frames were

used for the research of this paper. They are called

Frame90, Frame91, Frame92, and Frame93. Frame(yy)

consists of all UI accounts that existed at March ‘(yy-1)

plus all new UI accounts that occurred between April

‘(yy-1) and March ‘(yy).

The employment estimates from April of one year to

September of the next year constitute a set of estimates.

Four sets of estimates that correspond to the four data

frames mentioned above were produced: April ’90 to

September ’91; April ’91 to September ’92; April ’92 to

September ’93; and April ’93 to September ’94. For a

given estimation period, the estimates do not reflect any

gain in employment due to new businesses that have

occurred since the beginning of that period. For example,

the May ’90 estimate does not reflect any gain in

employment due to businesses that were born in April ‘90

or May ’90.

The above file structure reflects the fact that the CES

frame has historically been updated when the first quarter

(January through March) of BEL data becomes available

each year. In this study, we assume that the data becomes

available in October. At this time, revised estimates are

computed for April through September, using an updated

sample. The sample that was used for the first round of

estimates is updated to include a sample of births and to

exclude any deaths that have occurred since April of

previous year until March of the current year. In
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addition, the March population employment is used to

ratio adjust the revised estimates in a process called

benchmarking. March is referred to as the benchmark

month and the employment in March is called the

benchmark employment. Benchmarking from the March

employment is also used for estimates from October of

the current year through September of the following year,

at which time a new benchmark employment will be

available.

In order to simulate the net employment, four files of

births and four files of deaths were constructed for the

same periods that estimates were produced. That is, file 1

consists of births (or deaths) that occur between April ’90

and September ‘91, file 2 consists of births (or deaths)

that occur between April ’91 and September ‘92, and so

forth.

A birth file consists of UI accounts that have zero

employment in the twelve months preceding the file

period, and at least one month of positive employment in

the file period. Birth dates were assigned to each birth

unit as the first month with positive employment.

A death file consists of UI accounts that have at least one

month of positive employment in the twelve months

preceding the file period, zero employment for the last

month of the file period and zero employment for the

three months following the file period. The requirement

of an additional three months of zero employment is given

in order to minimize the chance of defining seasonal

businesses that may have zero employment for some

months of the year as deaths. Death dates were assigned

to each death unit as the first month of lasting zero

employment.

In our study, data from the first two files of births and

deaths were used to model the net employment, and the

parameters from the models are used to predict net

employment for the last twelve months of the last file.

This timing reflects the availability of data for modeling

during the actual CES production.

4. Model of Net Employment

Net employment is calculated as the difference between

cumulative birth employment and cumulative death

employment. At any month on a file, cumulative birth

employment is defined as the sum of employment at that

month of all births since the benchmark month, and

cumulative death employment is defined as the sum of the

last positive employment of all deaths since the

benchmark month. Net employment at month t is

represented by Nt, where t = 0 at the benchmark month.

a. Explanatory Variables

Two explanatory variables are used in the model. The

first is the change in the industry employment level since

the benchmark. For month t, it is defined as the

difference between the employment at month t as

estimated by the sample and the benchmark employment.

It is denoted by Zt. The second explanatory variable is

the net employment for the previous year in the same

month, or Nt

p

. This variable makes use of the fact that

there is a high correlation between employment in the

same month from one year to the next.

b. Seemingly Unrelated Regression

When applied to CES data, classical normal linear

regression models do not incorporate all available

knowledge about the regression equations and the

variables involved. They ignore the fact that errors in a

regression equation for an industry are correlated with the

errors in the regression equation for the total (all

industries combined). A system of equations that has a

property of this type is referred to as a system of

seemingly unrelated regression (SUR) equations.

Generalized least squares (GLS) parameter estimates for a

system of SUR equations can be derived that take into

account the mutually correlated regression errors in

different equations. In what follows, such GLS parameter

estimates are derived using Zt Nt

P

 and as independent

variables. The application to any combination of

independent variables is straightforward.

The net predictor described below is based on partitioning

a set or aggregate (AG) of business establishments into

sub-aggregates (SAGs) and allowing the aggregate level

data to lend strength to the sub-aggregates. The

aggregate used for the research is the national total and

the sub-aggregates the industry divisions. When industry

information is available, establishments are assigned to

one of the eight industries: Mining, Construction,

Manufacturing, Wholesale, Retail, Transportation and

Public Utilities (TPU), Finance Insurance and Real

Estates (FIRE), and Services. Otherwise, they are

assigned to the unclassified industry. In the study, a SAG

refers to a classified industry at the national level and the

AG is the national total. Here, the unclassified industry

will not be included in the SUR system, even though a

linear regression could be applied to these establishments.

Let the subscript j refer to a SAG , (1≤j≤8), u refer to the

unclassified industry, and s refer to the AG. Then

Nts Ntj N

j

= + tu

=

∑

1

8

 is the national total net employment at

time t. 

Let ( )

p Xtj Ztj Ntj = , and ( ) j 1 j 2 j b′ = a , a . For SAG j, the

linear model then takes the form

Ntj = Xtjbj + tj e for 1 ≤ j ≤ 8

with E tj (e ) = 0 and Var tj at Bj(e ) = s

2

and for the unclassified industry, we have

Ntu = Xtubu + tu e
-3-

with E tu (e ) = 0 and Var tu at Bu(e ) = s

2

.

For fixed t, the {et1,et 2,.....,etw ,etu}are uncorrelated, B

denotes the benchmark employment, at is an increasing

function of t (to be further discussed below), and

t = 1,2,...,T (note that, for this problem, T = 18).

We have Nts Ntj N

j

= + tu

=

∑

1

8

 and Xts Xtj X

j

= + tu

=

∑

1

8

which imply

Nts Xtj X

j

j tu u tj

j

= + + + tu

= =

∑ ∑

1

8

1

8

b b ( e e ) .

Following the above linear regression form, let Nts also

be given by Nts = Xtsbs + ts e . Then by equating these two

expressions for Nts , we have:

e b b b b e e ts tj

j

j s tu u s tj

j

= X − + X − + + tu

= =

∑ ∑

1

8

1

8

( ) ( ) ( )

and this implies

E ts (e ) = Xtj X

j

j s tu u s

=

∑ − + −

1

8

(b b ) (b b )

and Var ts at Bj B a B

j

u t s

(e ) = s ( + ) = s

=

∑

2

1

8

2

.

It is reasonable to assume that E ts (e ) is negligible

compared to Xtsbs. It is therefore set to zero in our

models.

The above equations can be written as:

N. j = X. jbj + . je ,

and N.s = X.sbs + .se ,

where N. j and N.s are (T × 1) vectors,

X. j and X.s are (T × 2) matrices, bj and b are s(2 ×1)

vectors, and e e .( ) j and 

.s

 are T × 1 vectors.

Let A = {att '}= {Cov(Ntj , Nt′j)}. Then A is a (T × T)

matrix with the following form:

A

a a a a

a a a a a

a a a a a a

a a a a a a

a a

a a

a a a a a a

T T

T T

=













































− −

−

1 1 1 1

1 2 2 2 2

1 2 3 3 3 3

1 2 3 4 4 4

3 4

1 1

1 2 3 4 1

. . .

. .

.

.

. . . . .

. . . . .

.

Given A, let E(e.se.

′

s

) = s Bs A

2

, and

 

E

B A j i

j i

j i

( )

,

,

. .

e e

s

′ =

=

≠







2

0

j

 when 

 when 

.

To incorporate the correlation of regression errors

between SAGs within the AG and the whole AG

( E(e.se.′

j

) = s Bj A

2

), the system of SUR equations is

compressed into one equation. In the following matrix

expression, 0 denotes a (T × 2) matrix of zeros.

N

N

N

N

N

X

X

X

X

X

.s s s s

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1

2

4

8

1

2

3

8

1

2

3

1

2

3

8

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0





































=









































































+



















b

b

b

b

b

e

e

e

e

e



















.

Let the compact version of this expression be

N = Xβ + ε,

where N, β, X, and ε are the associated matrices in the

above system of equation.

The covariance matrix of ε is:

=





















































∑ s ⊗

2

1 2 3 8

1 1

2 2

3 3

7

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0

0 0

0 0 0 0 0 0

B B B B B

B B

B B

B B

B

B B

A

s

w w

. . ..

. ..

. ..

. ..

. . . ..

. . . . . . .

. . . . .

,

where ⊗ is the Kronecker product.

The least squares estimate of β is

b = ( ′ ) ′

− − − X Σ X X Σ N1 1 1

 (4b.1).

c. Mixed Estimation

Since we are only interested in estimating β in order to

use it to predict future net employment, it proved useful

to apply a regression technique called mixed estimation

from Belsley, Kuh, and Welsch (1980). The effect of this

additional structure is to place stochastic constraints on

the components of β so that the coefficient of the Ztj is

given extra importance. Recall that Ztj estimates the

change in employment since the benchmark month. This

variable is thus an indicator of current economic

movement, which we want the regression equation to

capture. It is given extra emphasis in hopes that it will

contribute to more accurate employment estimates in

times of an unsually strong or weak economy.
-4-

Let the stochastic constraint that we wish to force on β be

given by the linear model c=Rβ + ξ where E(ξ)=0 and

Var(ξ)= Σ1. R is a matrix with the same number of

columns as β and each row is a constraint on β. In our

case, it is logical to constrain each pair of components in

the {bs,b1,....b8} separately. Thus R has 9 rows with

each row being a constraint on the two components of a

vector of regression coefficients from this set. The

quantity c is a 9x1 vector of constants, and ξ is a random

vector that is uncorrelated with ε.

The estimation proceeds by augmenting N, X, and ε to

give the constrained linear model used to estimate β:

N

c

X

R











 =











 +











b 

e

x

 , where Var

e

x











 =













Σ

Σ

0

0 1

.

The mixed estimate of β is

b ( ) ( ) M = X ′ X + R′ R X′ N + R′ c

− − − − − Σ Σ Σ Σ1

1

1 1 1

1

1

(4c.1).

The following discussion gives the reason for the choice

of R, c, and Σ1. If we regress the Ztj on the net

employment, the regression coefficient is roughly around

.15. If we regress Nt

p

 on the net employment, the

coefficient is roughly 1. Thus both (.15)Ztj Ntj

p

and

estimate Ntj . If we were to take a weighted average of

these two estimates -- as in composite estimation -- then

with the right weights we would have an improved

estimate of Ntj . Write this composite as:

(1− a)(.15)Ztj + aNtj

p

. Then let b=(1-a)(.15). If a is

chosen optimally then b and a should roughly equal

a j1 a j2and (the components of bj) respectively. The

result is the restriction aj1 + (.15)aj 2 =.15 for all j.

Thus in the equation c=Rβ + ξ, we have R=I⊗(1,.15)

where I is the 9×9 identity matrix and ⊗ is the Kronecker

product, c is the column vector with .15 in each position,

and Σ1 is the diagonal matrix with .00004 along the

diagonal. The covariance matrix Σ1 forces a relatively

small variation from the purely deterministic constraint,

c=Rβ.

d. Parameter Estimation

The empirical results following the stochastic description

of the process that generates the net employment showed

that it is appropriate to set a

t t

t =

( +1)

2

. The vector of

regression coefficients is then only a function of only one

unknown parameter,s

2

. Initially set s

2

= 1 and estimate

β according to equation (4b.1) or (4c.1). This initial

estimate of b , b , is then used to obtain improved

estimates of s

2

, which are estimated in the “usual” way

from the squared residuals.

Thus the new estimate of s

2

 is

,

s

b

2

2

1

18 9

=













 −



























 ∑  x

N X

B a

jt jt j

j t

j t

, where the

summation is over the SAGs and the 18 months of the

modeling file and bj

 is the jth SAG component of b .

5. Empirical Studies

Recall that models are built using data from the first two

birth and death files. Four different regression models are

presented here. Each model regresses the net

employment on the change in employment since

benchmark month, Zt, and the previous year’s net

employment, Nt

p

. The four different models are:

a. Ordinary Least Square (OLS) – The regression is run

separately for each industry and for the national total,

with uncorrelated errors that have zero expectation and

constant variance.

b. Seemingly Unrelated Regression 1 (SUR1) – This

model follows the variance structure presented in Section

4b, with the exception that the off-diagnonal elements of

∑ are all zeros, indicating that E s j( ) . .

'

e e = 0 .

c. Seemingly Unrelated Regression 2 (SUR2) – This

model follows the SUR system as described in Section 4b.

d. Seemingly Unrelated Regression 3 (SUR3) – This

model follows the SUR system as described in Section 4b

along with the mixed estimation described in Section 4c.

Net employment was predicted for the last twelve months

of file 4. The prediction error is defined as the difference

between the true net employment and the predicted net

employment. The absolute relative prediction error

(ARPE) averaged over the twelve months was computed

as:

ARPE

N N

N

t t

t t

=

−

=

1

12∑

1

12

.

The absolute relative prediction error was also computed

for the month of March, which is of particular concern

because it is the benchmark month.

ARPE March

N N

N

( ) =

6 − 6

6

.

In the above equations, t=1,2,...,12 correspond to the

months October ’93 through September ’94.

The smaller the value for ARPE and ARPE( March) , the

better the model is predicting the net employment.

Values of ARPE and ARPE( March) that are greater
-5-

than one indicate that using the model is actually

detrimental. In these cases, the estimate of total

employment would be more accurate if the sample

estimate alone were used.

6. Results

The following tables show ARPE and ARPE( March) for

the four models at the total and industry level.

Table I

ARPE

OLS SUR1 SUR2 SUR3

Total Private 0.273 0.455 0.570 0.272

Mining 1.046 1.382 3.031 3.686

Construction 0.510 1.226 0.809 0.579

Manufacturing 10.125 9.300 1.697 1.901

TPU 0.732 0.809 0.419 0.811

Wholesale 5.015 1.147 1.677 3.125

Retail 0.124 0.110 0.530 0.232

FIRE 2.307 2.687 1.139 1.341

Services 0.181 0.245 0.539 0.240

Table II

ARPE( March)

OLS SUR1 SUR2 SUR3

Total Private 0.292 0.463 0.568 0.263

Mining 1.001 1.200 1.888 2.141

Construction 0.546 1.531 0.909 0.712

Manufacturing 71.842 67.109 11.358 13.503

TPU 0.647 0.862 0.113 0.752

Wholesale 1.880 0.483 0.162 0.296

Retail 0.087 0.067 0.561 0.254

FIRE 2.189 2.574 1.082 1.231

Services 0.092 0.162 0.483 0.146

From the two tables, we see that for Mining,

Manufacturing, and FIRE, none of the models was helpful

in predicting the net employment. For Wholesale Trade,

the three SUR models were helpful in March, but not on

average. For the four remaining industries, OLS is the

best model for Construction and Services, SUR1 is the

best model for Retail Trade, and SUR2 is the best model

for TPU. The model SUR3 is best for Total Private only.

In three out of the four industries where modeling is

helpful, using the mixed estimation technique along with

the seemingly unrelated regression is better then using the

seemingly unrelated regression alone. However, the

simplest model, OLS, is the best or second best model for

all the industries where modeling is helpful.

The following plots show the true net employment and

the net employment as predicted by the OLS model. The

graphs also show a reference line at zero, which

demonstrates whether or not the model is helpful by

comparing it to using no model at all (or equivalently, by

predicting that the net employment is zero, and therefore,

assuming that birth and death employment are roughly

equal).

Net Employment in Total Private at the National Level

-200

0

200

400

600

800

1000

1200

1400

oct

novdec

jan

feb

mar

apr

may

jun

jul

aug

sep

October 1993 - September 1994

Net Employment (in Thousands)

True

Model Prediction

No model

Net Employment in Mining at the National Level

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

oct

novdec

jan

feb

mar

apr

may

jun

jul

aug

sep

October 1993 - September 1994

Net Employment (in Thousands)

True

Model Prediction

No model

Net Employment in Construction at the National Level

-50

0

50

100

150

200

oct

nov

dec

jan

feb

mar

apr

may

jun

jul

aug

sep

October 1993 - September 1994

Net Employment (in Thousands)

True

Model Prediction

No model
-6-

Net Employment in Manufacturing at the National Level

-60

-40

-20

0

20

40

oct

novdec

jan

feb

mar

apr

may

jun

jul

aug

sep

October 1993 - September 1994

Net Employment (in Thousands)

True

Model Prediction

No model

Net Employment in TPU at the National Level

-10

0

10

20

30

40

50

oct

novdec

jan

feb

mar

apr

may

jun

jul

aug

sep

October 1993 - September 1994

Net Employment (in Thousands)

True

Model Prediction

No model

Net Employment in Wholesale Trade at the National Level

-20

-10

0

10

20

30

40

oct

novdec

jan

feb

mar

apr

may

jun

jul

aug

sep

October 1993 - September 1994

Net Employment (in Thousands)

True

Model Prediction

No model

Net Employment in Retail Trade at the National Level

-50

0

50

100

150

200

250

300

350

400

450

oct

novdec

jan

feb

mar

apr

may

jun

jul

aug

sep

October 1993 - September 1994

Net Employment (in Thousands)

True

Model Prediction

No model

Net Employment in FIRE at the National Level

-40

-30

-20

-10

0

10

20

30

40

oct

novdec

jan

feb

mar

apr

may

jun

jul

aug

sep

October 1993 - September 1994

Net Employment (in Thousands)

True

Model Prediction

No model

Net Employment in Services at the National Level

-50

0

50

100

150

200

250

300

350

400

450

500

550

600

650

oct

novdec

jan

feb

mar

apr

may

jun

jul

aug

sep

October 1993 - September 1994

Net Employment (in Thousands)

True

Model Prediction

No model

7. Conclusions

In the major industry divisions of Mining, Manufacturing,

Wholesale Trade, and FIRE, it was found that a net

model should not be used, that instead the sample

estimate alone should be used.

In Construction, TPU, Retail Trade, and Services, it was

found that the OLS model with employment since the

benchmark month and the previous year’s net

employment as explanatory variables should be used to

predict net employment. Though it is not the best model

in all the industries, the OLS model outperforms the SUR

models across industries. The OLS model also has the

advantage that it is simpler in concept and is easier to

implement.

8. Future Research
-7-

The modeling for this paper was conducted at the national

level. Modeling at the state level is a logical extension.

The research presented here was conducted using

simulated data. The next step in evaluating the net

modeling approach will be to test it in production in order

to make an assessment of how the approach works under

the real conditions of the CES.

The behavior of the net employment in all industries

should be monitored over time in order to verify or revise

the conclusion that modeling will be beneficial only in

Construction, TPU, Retail Trade, and Services.

9. References

Shail B., Stamas G., Brick M., (1997), “Sample Redesign

for the Current Employment Statistics Survey,” ASA

Proceedings of the Survey Research Methods Section.

Belsley D. A., Kuh E., Welsch R. E. (1980), Regression

Diagnostics, John Wiley & Sons, Inc.

Kmenta, J. (1971), Elements of Econometrics, The

Macmillan Company, New York.

Rao C. R. (1973), Linear Statistical Inference and Its

Applications, John Wiley & Sons.

Werking, G. (1997), “Overview of the CES Redesign,”

ASA Proceedings of the Survey Research Methods

Section.