Monthly Labor Review September 2005 3

Estimating Gross Flows

Estimating gross flows

consistent with stocks in the CPS

The basic gross-flow table formerly used

by the Bureau of Labor Statistics in examining

labor market flows was expanded and the resulting tables

were raked iteratively in order to produce labor market

flow statistics compatible with CPS stocks

Harley J. Frazis,

Edwin L. Robison,

Thomas D. Evans,

and

Martha A. Duff

Harley J. Frazis is an

economist, and

Edwin L. Robison,

Thomas D. Evans, and

Martha A. Duff are

matematical

statisticians, in the

Office of Employment

and Unemployment

Statistics, Bureau of

Labor Statistics.

The Current Population Survey (CPS) is

primarily a cross-sectional survey

designed to estimate the distribution of

labor force states—employed (E), unemployed

(U), or not in the labor force (N)—among the

population1 for a given month. However, the CPS

also can be used to examine the number of persons who change their labor force state between

months.

Gross-flow estimates describe the month-tomonth transitions from one labor force state to

another. The following 3 × 3 matrix gives an

example in which EU represents the number of

persons who were employed in the previous

month (May) and are unemployed in the current

month (June), and similarly for the other entries:

 Current month (June)

 E U N

Previous E EE EU EN

 month U UE UU UN

Gross-flow estimation is possible in the CPS

because households are interviewed for 4 consecutive months, are then rotated out of the survey

for 8 months, and are then interviewed for another 4

consecutive months. About three-fourths of the

sample households are in common across 2 consecutive months. Household records can be linked,

and month-to-month labor force transitions determined, for most persons in those households.

Gross-flow statistics from the CPS were published from 1948 until 1952. Publication was stopped

because there were clear discrepancies between

labor force changes derived from the flows and

labor force changes derived from the monthly stock

estimates. (The sources of these differences are

explained later.) Over the years, many analysts have

called for the Bureau of Labor Statistics (BLS) to

resume publishing gross flows. This article describes a new method of obtaining flow statistics

that are compatible with the monthly stock numbers.

Seasonal adjustment of gross-flow series also is

discussed.

Existing gross-flow data problems

The Census Bureau generates unpublished grossflow estimates as part of its monthly production of

CPS data. The current procedure used by the

Census Bureau to generate the tabulations each

month starts by matching respondents in the

current month to respondents in the previous

month; about 72 percent are matched. Next, the

sampling weights of the matched respondents are

adjusted so that weighted sample totals, by sex,

match known population totals. The adjusted

weights are then used to compute weighted estimates of labor force transition flows. The analysis

that follows focuses on two types of error inherent

in this procedure: classification error and margin

error.

(May) N NE NU NN
4 Monthly Labor Review September 2005

Estimating Gross Flows

Classification error. For a variety of reasons, some CPS

respondents may be classified into the wrong labor force state.

Errors in classifying the respondent can have large effects on

gross-flow calculations. In stock data, classification errors tend

to offset each other, whereas in flow data, errors tend to be additive. For example, if equal numbers of respondents are erroneously classified as employed when they are unemployed and as

unemployed when they are employed, stock data will be

unaffected, but both EU and UE flows will be increased.

Although research indicates that classification error may have

large effects on gross flows, the Bureau of Labor Statistics has

no current plans to publish classification-error-corrected flows.

While measurement error probabilities could be derived from

reinterview data, it is not entirely clear how such data should be

used. In their attempts to correct for classification error, John

Abowd and Arnold Zellner,2

 and James Poterba and Lawrence

Summers,3 used “reconciled” reinterview data, whereby the

interviewer attempted to establish a true labor force state in the

case of contradiction between the original survey and the

reinterview. Because of data quality problems, however, reconciled reinterview data are no longer being produced. Tin Chiu

Chua and Wayne Fuller4 used unreconciled reinterview data,

but doing so requires additional statistical assumptions. Moreover, the reinterview sample may not be representative of the

CPS sample as a whole, because response rates are lower than

they are for the CPS. (Currently, reinterview response rates are

approximately 80 percent, compared with 90–95 percent in the

CPS.)

Margin error. CPS stock estimates for a given month use responses

from all eight panels. Gross-flow estimates, by contrast, are restricted

to the six panels in months-in-sample (MIS) groups that continue

from one month to the next. These groups are MIS1–MIS3 and MIS5–

MIS7 in the previous month and become MIS2–MIS4 and MIS6–MIS8,

respectively, in the current month. Moreover, because the CPS does

not track persons who change their residence, and because

respondents may be absent or refuse to complete the survey in a

given month, not all persons in the survey can be matched, even in

the continuing months-in-sample.

It is natural to attempt to derive implied changes in stocks by

adding up gross flows. For example, one could derive an estimate

of the change in employment by adding the flows into employment (UE and NE) and subtracting the flows out of employment

(EU and EN). Consistently, the implied changes in stocks do not

match the changes in stocks estimated from the CPS as a whole.

The gross flows tend to show net flows out of the labor force, in

contrast to the stock numbers. For example, the following

tabulation shows both the average (not seasonally adjusted)

monthly changes in stocks directly from the CPS between

December 1994 and December 2004 and the changes in stocks

implied by adding up the flows calculated by the current procedure over the same period (numbers are in thousands):

 Change in—

 Not in

Method Employed Unemployed labor force

CPS ................................ 130 8 87

Current method ............. –203 –231 434

Whereas the stock numbers show large increases in employment

and a smaller increase in the number not in the labor force, adding

up the flows implies large decreases in both employment and

unemployment and a large increase in those not in the labor

force.

There are three sources of the discrepancy between published

estimates of changes in stocks and estimates from gross-flow

data. First, there is the problem of “nonidenticals”: respondents

who match from month to month may be systematically different

from those in the relevant MIS who do not match due to nonresponse, changing addresses, and so forth. Unmatched respondents are part of the stocks, but not the flows. Second is

the problem of rotation group effects: it is known that respondents’ answers to questions about their labor force status systematically differ by MIS. Third, the current gross-flow method does

not take into account changes in the population aged 16 years

and older.

The research preparatory to this article showed that rotation

group effects are the most important reason for the discrepancy

between stocks and flows. It has long been known that the labor

force state a respondent reports in the CPS is affected by MIS.

For example, in 2003, the average weighted percentage of

respondents who reported that they were not in the labor force

ranged from 32.8 percent in MIS1, to 33.4 percent in MIS2, to 34.3

percent in MIS8, with corresponding decreases in both the

percentage employed and the percentage unemployed.5

These rotation group effects have clear implications for gross

flows. Because respondents in MIS1 and MIS5 cannot be

matched to the previous month, the increase in respondents

reporting that they were not in the labor force for other MIS’s

implies that matched samples will show flows out of the labor

force even if there is no change in the stocks.

As mentioned, another source of margin discrepancies is that

the current gross-flow method does not account for population

growth and, more broadly, does not account for flows into and

out of the scope of the CPS. Abowd and Zellner pointed out that

the in-scope population for the CPS is not static.6 Thus, a

complete table of flows would include not only flows between

labor force states, but also flows from the labor force out of the

scope of the CPS (because of death, entry into the Armed

Forces, emigration, and the like) and flows from out-of-scope

states to a labor force state (for example, due to turning 16

and immigration). It is easy to deal with entering the scope of

the survey due to turning age 16, because 15-year-olds are in

the survey (and their birth dates are collected). Other flows

into and out of scope are more difficult to deal with, because

CPS
Monthly Labor Review September 2005 5

complete data do not exist within the CPS: persons exiting households are imperfectly tracked, and persons entering households

are not asked retrospective questions.7

A method for correcting margin error

The basic approach set forth in this article is to adjust the flows in the

matched CPS data by a method known as raking, so that they

correspond to the labor force stocks in both the current and the

previous month. The method accounts for flows into and out of the

scope of the CPS and thus corrects for all sources of margin discrepancies, so that implied changes in stocks derived from the flows

match changes in CPS stock estimates. As a basis for discussing this

method, table 1 expands upon the basic 3 × 3 gross-flow matrix

shown earlier, in order to deal with flows into and out of the scope of

the CPS.

Deaths refers to those individuals who were in scope in the

previous month, but had died by the current month. Just 16

refers to those individuals who just turned 16 in the current

month. The portions of the table set in roman type correspond

to the estimates of flows and stocks presented here, with P

denoting the previous month and C denoting the current month.

The remaining parts of the table consist of indirectly estimated

residual totals that are used to make the table “add up.”

As noted, the four boldface estimates JE, JU, JN, and JP in

the “Just 16” row can be computed directly from the CPS by

using the known ages of respondents in the previous and

current months, as well as their labor force status in the

current month. (Note that JP is simply the stock estimate of

those who just turned of age as of the current month.) Deaths

are reported in the CPS, but for various reasons are

undercounted by nearly half, so to estimate flows that are

out of scope due to death, a less direct approach needs to be

adopted. To get a more accurate estimate, average death rates

for each gender are derived from mortality tables published

annually by the National Center for Health Statistics.8 These

death rates are then applied to the CPS data to estimate total

deaths each month (DC in table 1). Finally, the deaths are allocated

among labor force states on the basis of those states’ average

allocation of deaths from historical CPS data, generating the

boldface estimates ED, UD, and ND in the “Deaths” column of

the table.

Three cells in table 1 are defined to be zero: those who would

be classified as inflows, but were immediate outflows due to

death, and those who would be defined simultaneously as other

inflows and other outflows.

Details

This section discusses in detail the computation of raked

tables. All cases presented refer to the table structure defined

in table 1. Gross-flow tables are computed for men and women

separately.

Margin adjustment step. Construct the stock labor force

estimates EP, UP, NP, and JP for the previous month, using the

previous month’s sampling weight, and construct the stock

labor force estimates EC, UC, and NC for the current month,

using that month’s sampling weight. Now construct the death

estimates ED, UD, and ND by first estimating the number of

deaths (the sum of ED, UD, and ND) by taking the previous

month’s total population and multiplying that by a death rate

(obtained from records on vital statistics) appropriate to each

sex. Next, distribute the total death estimate among the three

labor force estimates (ED, UD, and ND) on the basis of 3-year

EE EU EN ED EO EP

UE UU UN UD UO UP

NE NU NN ND NO NP

JE JU JN 0 JO JP

IE IU IN 0 0 IP

EC UC NC DC OC Total

Labor force status Other outflow

Deaths

Table 1.

Other

outflow

Row total E U N

Structure of basic gross flows

Row and column category

Previous

month

Labor force

status

Other inflow

Just 16

Column total

Inflow

E

U

N

NOTE: Shading indicates original 3 × 3 gross-flow matrix used as example early in text.

Current month
6 Monthly Labor Review September 2005

Estimating Gross Flows

average estimates of the proportion of deaths by labor force

classification. These estimates are obtained from the CPS.

The adjustments so far yield a total of P0 = (EP + UP + NP +

JP) – (ED + UD + ND) as a potential population in the scope of

the CPS during the current month. The actual in-scope population is P1 = EC + UC + NC. Usually, due to immigration, P1 will

be greater than P0. If so, then other inflow estimates (IE, IU,

and IN) are set by allocating the discrepancy (P1 – P0) to labor

force states in proportion to their shares of the current month's

population, and other outflow is set to zero. Some of the time

when CPS population controls are adjusted, P0 will be greater

than P1. In this case, other outflow estimates (EO, UO, NO, and

JO) are set by allocating (P0 – P1) in proportion to EC, UC, NC,

and JP, respectively (the totals by labor force state in the previous month, plus JP).

Matching step. Construct weighted counts of the 12 flow

cells (EE, EU, EN, UE, UU, UN, NE, NU, NN, JE, JU, and JN), using

the sampling weight for the current month for those individuals who had a labor force status in both the previous

and the current month. These totals are constructed for each

sex. The weighted flow counts will be too small by approximately

25 percent, because about 75 percent of the sample overlaps

from month to month. The initial iteration step presented next will

correct this undercount.

Iteration step. The table constructed in the matching step

is not entirely consistent, because not all the cells are guaranteed to add up to the appropriate row and column totals. In

order to obtain consistency, iterative raking is performed. The

death estimates (ED, UD, ND, and DC) and the inflows or

outflows are held fixed, while the remaining interior cells of

the tables are raked by iterative proportional fitting to ensure

additivity to the stock estimates in the row and column totals.

Final factor calculation. The procedure presented next

produces gross-flow microweights. Compute factors for each

of the 12 raked cells (EE, EU,..., JN) for each table (for men and

for women) by dividing the final estimate obtained in the

iteration step by the initial cell value (the weighted sample

count, using the current-month sampling weight, for those

individuals who are in the sample both months). When applied

to the current-month weights of matched individuals, the factors

yield gross-flow weights that enable the re-creation of the flows

for both men 16 years and older and women 16 years and older.

The computation of flows for demographic groups other than

those two will be facilitated by these weights, but will not be

perfectly consistent with CPS stock numbers.

Results

The foregoing procedure was implemented with data from 1990

through 2004. Table 2 shows a comparison of row percentages—

flows as a percentage of the population in a labor force state the

previous month—for the current procedure and for the procedure

just set forth.9 As can be seen, the average differences are fairly

small, 7 percent or less in magnitude. This is to be expected,

because the aim of the raking procedure is to revise the existing

flow data as little as possible while forcing compatibility with the

margins. Earlier, it was noted that rotation group bias caused the

current procedure to show more substantial flows out of the

labor force than could be reconciled with the stock data. Table 2

reflects this movement, because the new procedure slightly

reduces flows out of the labor force. In some months, there are

more substantial differences. For example, in June 1992, NU flows

were 3.7 percent of the previous month’s not-in-the-labor-force

stocks under the current procedure, but 4.2 percent under the

new procedure, a 12-percent increase. This discrepancy is associated with an unusually high increase in unemployment, from

9.4 million to 10.4 million. Chart 1 shows the time series of NU

flows. The June 1992 point in the adjusted series is a multiyear

high, whereas the point in the current series is not much different from other Junes.

Gross-flow row percentages, 1990-2004 CPS, different methods of estimation

EE 95.8 95.9 0 0 0 0

EU 1.4 1.4 4 4 –2 13

EN 2.8 2.7 –4 4 –11 6

UE 27.5 27.3 0 2 –5 4

UU 49.7 50.9 2 2 –2 6

UN 22.8 21.8 –4 4 –10 0

NE 4.7 4.9 4 4 –6 15

NU 2.5 2.7 7 7 1 16

NN 92.8 92.4 0 0 –1 0

Minimum percent

 increase

Maximum percent

increase

Average percent

increase,

adjusted method

divided by current

method

Average row

percentage,

current

method

Average row

 percentage,

adjusted method

divided by current

method

Average absolute

 percent change,

adjusted method

divided by current

method

 Flow

Table 2.
Monthly Labor Review September 2005 7

Seasonal adjustment

Seasonal adjustment is a natural way to further improve the

utility of gross flows, and it is also highly useful as an analysis tool to evaluate data quality and to uncover hidden or

hard-to-see characteristics. One major wrinkle in seasonally

adjusting gross-flow data is that the adjusted flow data need

to be made compatible with seasonally adjusted stocks. Accordingly, once the raked gross flows are seasonally adjusted,

they are raked a second time, this time to the total seasonally

adjusted stock numbers from the CPS. Final raked and seasonally adjusted flows were examined as part of the research for

this article, because it was not known exactly how reraking

would affect the flows. It is important that extraneous seasonality or any other systematic effects not be introduced and

that the seasonal factors remain reasonably stable from year

to year.

Seasonal adjustment is performed with the Census Bureau’s X-12-ARIMA program, the current standard at the Bureau

of Labor Statistics.10 Data from February 1990 to March 2004

were examined, with a missing data point in January 1994 due

to a survey redesign. Initially, the research analyzed flows for

a variety of demographic breaks by age, sex, and race. However, because these breaks resulted in smaller sample sizes

and flow estimates that were subject to greater sampling error,

it was determined that only flows for men 16 years and older

and women 16 years and older could be adequately seasonally

adjusted.

The 3 × 3 flows between the labor force states were seasonally

adjusted. The X-12-ARIMA procedure indicated that all of the

flows showed seasonality, although it was relatively weak for UN

flows for both men and women and UU flows for women. These

flows may be even more seasonal than is indicated here, but

sampling error is likely masking some seasonal properties,

because the diagnostic statistics indicating the presence of

seasonality increase when flows for men and women are added.

Flows with the largest seasonal factors are EU, EN, and NE, while

EE and NN factors are by far the smallest. In general, the

seasonally adjusted flows appear to have explainable seasonal

patterns.

A few of the flows show signs of seasonal factors that are

unstable from year to year. This instability can be expected,

because, clearly, some of the smaller flows are fairly noisy.

Plotting the seasonal factors for each month over the years of

the sample period showed no factors unstable enough to cause

serious concern.

An examination of the seasonally adjusted flow data revealed

another pattern. The CPS is fielded on the week containing the

19th of each month11 and asks questions referring to the week of

the 12th. Thus, CPS reference weeks are separated by either 4 or

5 weeks. The adjusted series revealed systematic differences in

the flows, depending on the distance between the reference

CHART 1

Chart 1. NU flows, 1990—2004, row percent

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Census Adjusted

Percent Percent

June 1992

January

1990

January

1992

January

1994

January

1996

January

1998

January

2000

January

2002

January

2004
8 Monthly Labor Review September 2005

Estimating Gross Flows

periods. Note that such differences are quite plausible: respondents have more opportunities to change their labor force status,

and thus end up in the off-diagonal cells, when reference periods

are 5 weeks apart. Population growth also implies that referenceweek intervals can affect levels of all the cells. Modeling these

calendar effects often results in smoother seasonally adjusted

series.12 In this case, modeling the calendar effects with dummy

variables resulted in seasonally adjusted series with no apparent

residual seasonality.

After seasonal adjustment, the flows for men and women are

raked to their respective published CPS seasonally adjusted

controls. The complete matrix for the flows, as shown in table 1,

has a row for 15-year-olds turning 16 (JE, JU, and JN). These

series are too small to adjust reliably, so, as reasonable

approximations, CPS seasonal factors for 16-to-19-year-old

employed, unemployed, and not-in-the-labor-force men and

women are applied to the cells corresponding to those categories

in the “Just 16” row of the matrix. Other cells for flows into and

out of the scope of the survey are assumed to be nonseasonal.

Once the seasonally adjusted flows were raked, they were

reexamined for differences with the unraked series, residual

seasonality, and stability of the seasonal factors. Basic statistics

for differences between the unraked seasonally adjusted flows

and the raked seasonally adjusted flows are presented as row

percentages (as in table 2) in table 3. The overall effect of raking

seasonally adjusted flows to seasonally adjusted stocks is less

than the effect of raking the flows that are not seasonally

adjusted, as is shown in table 2. The mean percentage difference

between raked and unraked seasonally adjusted flows is in all

cases less than 0.5 percent, and the mean absolute percentage

difference is in all cases less than 2 percent. None of the percentage differences is large; the largest differences between

raked and unraked flows were approximately 5 percent. Not

surprisingly, the two largest flows have by far the smallest

percentage differences.

The final check on the raked flows was to see whether seasonal

or calendar patterns were reintroduced by raking. A comparison

of the seasonally adjusted flows and the raked seasonally

adjusted flows showed some differences along this dimension.

Unfortunately, it appears that some of the calendar effects are

reintroduced into the final raked seasonally adjusted series, but

not enough to cause serious concern.

MORE THAN 50 YEARS AGO, the Bureau of Labor Statistics

stopped publishing series of labor market flows when it was

discovered that the published flows were incompatible with the

monthly labor force stock numbers. This article describes efforts

to produce labor market flow statistics from the CPS without the

incompatibilities that led to their curtailment. The basic grossflow table was expanded to estimate flows into and out of the

scope of the CPS. The resulting gross-flow tables were then raked

in an iterative process to match both the previous month’s and

the current month’s stock estimates. As part of this project, a

method was developed to seasonally adjust the flow series while

maintaining the flows’ compatibility with the seasonally adjusted

stocks. The Bureau of Labor Statistics is continuing to work on

this research, with the expectation of eventually resuming

publication of the gross flows.13

Seasonally adjusted gross-flow row percentages, 1990—2004 CPS, unraked and raked

EE 95.9 95.9 0 0 0 0

EU 1.4 1.4 0 1 –5 5

EN 2.7 2.7 0 1 –4 3

UE 27.4 27.4 0 1 –2 3

UU 50.8 50.8 0 1 –3 3

UN 21.8 21.8 0 1 –3 3

NE 4.9 4.9 0 1 –3 4

NU 2.7 2.7 0 1 –5 5

NN 92.4 92.5 0 0 0 0

Minimum percent

 increase

Maximum percent

increase

Average percent

increase,

adjusted divided

by raked

Average row

percentage,

seasonally

adjusted

Average row

 percentage,

seasonally

adjusted and

raked

Average absolute

 percent change,

adjusted divided by

raked

 Flow

Table 3.

Notes

ACKNOWLEDGMENT: The authors thank Fran Horvath of the Office

of Employment and Unemployment Statistics for his contributions to

the research preparatory to this article.

1

 “Population” refers to the U.S. civilian noninstitutional population

aged 16 and older.

2 John M. Abowd and Arnold Zellner, “Estimating Gross LaborForce Flows,” Journal of Business and Economic Statistics, July 1985,

pp. 254–83.

3

 James M. Poterba and Lawrence Summers, “Reporting Errors and

Labor Market Dynamics,” Econometrica, November 1986, pp. 1319–

38.

4

 Tin Chiu Chua and Wayne A. Fuller, “A Model for Multinomial
Monthly Labor Review September 2005 9

Response Error Applied to Labor Flows,” Journal of the American

Statistical Association, March 1987, pp. 46–51.

5

 Here is an example that shows how little difference matching

makes relative to the rotation group effect. In 1996, the average

monthly change in the percentage of the population that was employed

was 0.16 percent, the average change for the percentage unemployed

was –0.07 percent, and the average change for the percentage not in

the labor force was 0.09 percent. If, instead, one computes the same

figures, but this time using the changes in percentage from rotation

groups MIS1–3 and MIS5–7 in the first month to rotation groups MIS2–4

and MIS6–8, respectively, in the next month, whether the observations

do or do not match between months, the numbers are 0.02 percent for

the employed, –0.16 percent for the unemployed, and 0.13 percent

for those not in the labor force. (Due to rounding, the preceding

numbers may not add to zero.) If one now computes the same figures,

but using only matched observations from rotation groups MIS1–3 and

MIS5–7 in the first month and rotation groups MIS2–4 and MIS6–8 in the

next month, the numbers are 0.03 percent for the employed, –0.17

percent for the unemployed, and 0.14 percent for those not in the

labor force—very close to the unmatched figures, showing that failure

to match makes little additional difference after accounting for the

rotation group.

6

 Abowd and Zellner, “Estimating Gross Labor-Force Flows.”

7

 The CPS uses an address-based sample and does not attempt to

interview persons who moved out of the sampled address. If a

household member exits the household between interviews, a remaining

household member is asked the reason for the exit. If the entire

household is replaced, no such question is asked.

8

 National Center for Health Statistics, Vital Statistics of the United

States (Atlanta, Centers for Disease Control and Prevention, published

annually).

9

 Flows from April 1994 are missing in the current series; January

1994 also is excluded, due to linking problems associated with a redesign

of the survey.

10 X-12-ARIMA Reference Manual, version 0.2.10 (Census Bureau, 2002).

11 For those Decembers when this date would conflict with holidays,

the survey is fielded during the week that includes the 12th of the

month.

12 Stephanie Cano, Patricia M. Getz, Jurgen Kropf, Stuart Scott, and George

Stamas, “Adjusting for a Calendar Effect in Employment Time Series,”

Proceedings of the Survey Research Methods Section, 1996 (Alexandria, VA,

American Statistical Association, 1996), explain how this approach is similarly

implemented in seasonal adjustment for the BLS Current Employment Statistics

survey. For an in-depth discussion of calendar effects in the present case, see

Thomas D. Evans, “Analysis of Raking on Seasonally Adjusted Household

Gross Flows Data,” ASA Proceedings of the Joint Statistical Meetings, 2004,

(Alexandria, VA, American Statistical Association, 2004), pp. 1166-73.

13 The utility of the new gross-flow series to analysts is discussed in

the companion article to this one, “Analyzing CPS data using gross

flows,” by Randy Ilg, this issue, pp. 10–18.