Current Population Survey (CPS) 

 Technical Documentation 

 January 2010

Methodology for Seasonally Adjusting 

National Household Labor Force Series 

with Revisions for 2010 

Richard B. Tiller and Thomas D. Evans

hort-run movements in labor force time series are strongly influenced by seasonality—

periodic fluctuations associated with recurring calendar-related events such as weather, 

holidays, and the opening and closing of schools. Seasonal adjustment eliminates the 

influence of these fluctuations and makes it easier for users to observe fundamental changes in 

the level of the series, particularly changes associated with general economic expansions and 

contractions. 

While seasonal adjustment is feasible only if the seasonal effects are reasonably stable with 

respect to timing, direction, and magnitude, these effects are not necessarily fixed, but often 

evolve over time. The evolving patterns are estimated by X-12-ARIMA, a procedure based on 

moving averages, or “filters,” that successively average a shifting timespan of data, thereby 

providing estimates of seasonal factors that change in a smooth fashion from year to year. 

For observations in the middle of the series, a set of symmetric moving averages with fixed 

weights produces final seasonally adjusted estimates. A filter is symmetric if it is centered around 

the time point being adjusted, with an equal amount of data preceding and following that point. 

Standard seasonal adjustment options imply a symmetric filter that uses from 6 to 10 years of 

original data to produce a final seasonally adjusted estimate. Obviously, this final adjustment can 

be made only where there are enough data beyond the time point in question to adjust with the 

symmetric filter. 

To seasonally adjust recent data, shorter filters with less desirable properties must be used. 

These filters are referred to as asymmetric, because they use fewer observations after the 

reference point than preceding it. The weights for such filters vary with the number of 

observations that are available beyond the time point for which estimates are to be adjusted. 

Seasonally adjusted data for the current year are produced with a technique known as concurrent 

adjustment. Under this practice, the current month’s seasonally adjusted estimate is computed 

with the use of all relevant original data up to and including data for the current month. 

Revisions to estimates for previous months, however, are postponed until the end of the year. 

Revisions to a seasonally adjusted estimate for a given time point continue until enough future 

observations become available to use the symmetric weights. This effectively means waiting up 

to 5 years for a final adjustment when standard options are used. 

At the end of each calendar year, BLS reestimates the seasonal factors for the Current 

Population Survey (CPS) series by including another full year of data in the estimation process. 

On the basis of this annual reestimation, BLS revises the historical seasonally adjusted data for 

the previous 5 years. As a result, each year’s data are generally subject to five revisions before 

 Richard B. Tiller and Thomas D. Evans are mathematical statisticians on the Statistical Methods Staff, Office 

of Employment and Unemployment Statistics, Bureau of Labor Statistics. Telephones: (202) 691-6370 (Tiller) and 

691-6354 (Evans); e-mail: Tiller.Richard@bls.gov; Evans.Thomas@bls.gov. 

S
2 

the values are considered final. The fifth and final revisions to data for the earliest of the 5 years 

are usually quite small, while the first-time revisions to data for the most recent years are 

generally much larger. For the major aggregate labor force series, however, the first-time 

revisions rarely alter the essential trends observed in the initial estimates. 

Changes in 2010 

Revisions to 2009 estimates 

This year’s revisions incorporate data through December 2009 and provide revised estimates for 

January 2005 through November 2009 for all previous seasonally adjusted labor force series. 

Currently, there are 144 series that are directly seasonally adjusted and many more that are 

derived from these series. (See the subsequent section titled “Aggregation procedures.”) 

An important criterion for evaluating alternative methods of seasonal adjustment is how 

close initial estimates are to the results of subsequent revisions. Users of seasonally adjusted data 

are often most interested in current information. Thus, it is desirable that the initial seasonally 

adjusted estimates be as close as possible to the improved estimates made after more data 

become available. Even though the revisions currently being released for the 2009 seasonally 

adjusted data are not final, the first revisions are usually the largest and often indicate the 

direction of subsequent revisions. 

Table 1 shows the civilian unemployment rates for January to November 2009, as first 

computed and as revised. There are a few revisions to the rates with most coming early in the 

year. Note that December 2009 is the final month of the series and has no revision. 

Extension of seasonal adjustment to additional series 

Beginning with the release of data for January 2010, seasonal adjustment will be extended to 16 

series not previously adjusted. Ten of the series will be produced monthly and six quarterly. The 

monthly series are: 

1) Average hours, total at work, all industries 

2) Average hours, persons who usually work full time, all industries 

3) Average hours, total at work, nonagricultural industries 

4) Average hours, persons who usually work full time, nonagricultural industries 

5) Average hours, total at work, agriculture and related industries 

6) Average hours, persons who usually work full time, agriculture and related industries 

7) Average hours, at work in nonagricultural industries, wage and salary workers 

8) Average hours, at work in nonagricultural industries, wage and salary workers, persons 

who usually work full time 

9) Unemployment level—permanent job losers (table A-8 of “The Employment Situation” 

news release) 

10) Unemployment level—persons who completed temporary jobs (table A-8 of “The 

Employment Situation” news release) 

The quarterly series are from table 1 of the “Usual Weekly Earnings of Wage and Salary 

Workers” news release: 
3 

1) Median usual weekly earnings (second quartile), employed full time, wage and salary 

workers 

2) Median usual weekly earnings (second quartile), employed full time, wage and salary 

workers, men 

3) Median usual weekly earnings (second quartile), employed full time, wage and salary 

workers, women 

4) Employed full time, wage and salary workers 

5) Employed full time, wage and salary workers, men 

6) Employed full time, wage and salary workers, women 

The total “Employed full time, wage and salary workers” quarterly series is indirectly 

seasonally adjusted by summing its respective seasonally adjusted series for men and women. 

(See the subsequent section titled “Aggregation procedures” for a brief discussion on indirect 

seasonal adjustment.) 

Indirect seasonal adjustment of two other seasonally adjusted series has been modified. 

Previously, the “Job losers and persons who completed temporary jobs” series from table A-8 of 

“The Employment Situation” news release was directly seasonally adjusted. It is now the sum of 

the seasonally adjusted “On temporary layoff” and “Not on temporary layoff” series from the 

same table. The “Not on temporary layoff” series used to be adjusted by subtracting the “On 

temporary layoff” series from “Job losers and persons who completed temporary jobs.” The 

series is now a sum of “Permanent job losers” and “Persons who completed temporary jobs.” 

The series on total job losers, not on temporary layoff, and permanent job losers, and persons 

who completed temporary jobs will appear in table A-8 of “The Employment Situation” news 

release. The quarterly earnings series will appear in the “Usual Weekly Earnings of Wage and 

Salary Workers” news release. The other series on hours at work will be available through the 

BLS website in the Labstat database. Historical series for all newly added seasonally adjusted 

series will also be available through the BLS website. Note that the earnings series begin in 

1979; the hours at work series start in 1976 except for the usually full-time series that start in 

1994; and the “Permanent job losers” and “Persons who completed temporary jobs” series begin 

in 1994. 

Adjustment Methods and Procedures 

Beginning in 2003, BLS adopted the use of X-12-ARIMA as the official seasonal adjustment 

procedure for CPS labor force series, replacing the X-11-ARIMA program that had been used 

since 1980. Both X-12- and X-11-ARIMA are based on earlier versions of the widely used X-11 

method developed at the U.S. Census Bureau in the 1960s.1

 Statistics Canada added to X-11 the 

ability to extend the time series with forward and backward extrapolations from Auto-Regressive 

 1

 For a detailed discussion of X-12-ARIMA, see David F. Findley, Brian C. Monsell, William R. Bell, Mark C. 

Otto, and Bor-Chung Chen, “New Capabilities and Methods of the X-12-ARIMA Seasonal Adjustment Program,” 

Journal of Business and Economics Statistics, April 1998, pp. 127-152. For documentation on X-11-ARIMA, see 

Estela Bee Dagum, The X-11 ARIMA Seasonal Adjustment Method, catalogue no. 12-564E (Ottawa, Statistics 

Canada, January 1983). The X-11 method is described in Julius Shiskin, Alan Young, and John Musgrave, “The X11 Variant of the Census Method II Seasonal Adjustment Program,” Technical Paper no. 15 (Bureau of the Census, 

1967). 
4 

Table 1. Seasonally adjusted unemployment rates in 2009 and change due to revision

January ......................................... 7.6 7.7 0.1

February ....................................... 8.1 8.2 .1

March .......................................... 8.5 8.6 .1

April ............................................. 8.9 8.9 .0

May ............................................. 9.4 9.4 .0

June .............................................. 9.5 9.5 .0

July .............................................. 9.4 9.4 .0

August .......................................... 9.7 9.7 .0

September .................................... 9.8 9.8 .0

October ........................................ 10.2 10.1 -.1

November .................................... 10.0 10.0 .0

Month As revised Change As first

computed 

Integrated Moving Average (ARIMA) models, prior to seasonal adjustment. The X-11 algorithm 

for seasonal adjustment is then applied to the extended series. The use of forward and backward 

extensions results in initial seasonal adjustments that are subject to smaller revisions, on average, 

when the adjusted data are revised after future data become available. 

Developed at the U.S. Census Bureau, the X-12-ARIMA program includes all of the 

capabilities of the X-11-ARIMA program, while introducing new, major enhancements. These 

enhancements fall into three basic categories: (1) Enhanced ARIMA model selection and 

estimation, (2) detection and estimation of outlier, trading day, and holiday effects, and (3) new 

postadjustment diagnostics. 

For the majority of labor force series that are seasonally adjusted by BLS, the main steps of 

the seasonal adjustment process proceed in the following order: 

• Times series modeling—a REGARIMA model (a combined regression and ARIMA 

model) is developed to account for the normal evolutionary behavior of the time series 

and to control for outliers and other special external effects that may exist in the series; 

• Prior adjustments—given an adequate REGARIMA model, the series is modified by 

prior adjustments for external effects estimated from the regression part of the model and 

extrapolated forward 12 months by the ARIMA part of the model; 

• X-11 decomposition—the modified and extrapolated series is decomposed into trendcycle, seasonal, and irregular components by means of a series of moving averages, 

developed in the X-11 part of the program, to produce seasonal factors for implementing 

seasonal adjustment; and 

• Evaluation—a battery of diagnostic tests is produced to evaluate the quality of the final 

seasonal adjustment. 
5 

For two CPS series, the seasonal adjustment process begins with special user-defined prior 

adjustments for Easter effects. (See subsequent section titled “Aggregation procedures.”) 

Time series modeling 

Time series models play an important role in seasonal adjustment. They are used to identify and 

correct the series for aberrant observations and other external effects, as well as to extend the 

original series with backcasts and forecasts so that less asymmetric filters can be used at the 

beginning and end of the series. 

ARIMA models2

 are designed to make forecasts of a time series based on only its past 

values. While these models can represent a wide class of evolving time series patterns, they do 

not account for the presence of occasional outliers and other special external effects. An outlier 

represents a sudden break in the normal evolutionary behavior of a time series. Ignoring the 

existence of outliers may lead to serious distortions in the seasonally adjusted series. 

A common form of outlier that presents a special problem for seasonal adjustment is an 

abrupt shift in the level of the data that may be either transitory or permanent. Three types of 

outliers are usually distinguished: (1) An additive change that affects only a single observation, 

(2) a temporary change having an effect that diminishes to zero over several periods, and (3) a 

level shift or a break in the trend of the data, which represents a permanent increase or decrease 

in the underlying level of the series. These three main types of outliers, as well as other types of 

external effects, may be handled by the times series modeling component of X-12. This is done 

by adding to the ARIMA model appropriately defined regression variables based on intervention 

analysis originally proposed by George E.P. Box and George C. Tiao.3

The combined regression and ARIMA model is referred to as a REGARIMA model and is 

represented by 

Y XZ t tt = β + , 

where, Yt is the original series or a logarithmic transformation of it, Xt is a set of fixed regression 

variables, β represents the regression coefficients, and Zt is a standard seasonal ARIMA model 

described by the notation (p,d,q)(P,D,Q), where p is the number of regular (nonseasonal) 

autoregressive parameters, d is the number of regular differences, q is the number of regular 

moving average parameters, P is the number of seasonal autoregressive parameters, D is the 

number of seasonal differences, and Q is the number of seasonal moving average parameters. 

While the ARIMA model can be very complicated theoretically, in practice it takes a 

parsimonious form involving only a few estimated parameters. (See table 2.) There are welldeveloped methods for determining the number and types of parameters and the degree of 

differencing appropriate for a given series. 

With respect to specifying the regression component to control for outliers, X-12 offers two 

approaches. First, major external events, such as breaks in trend, are usually associated with 

known events. In such cases, the user has sufficient prior information to specify special 

regression variables to estimate and control for these effects. 

 2

 For a more detailed discussion of ARIMA models, refer to George E.P. Box and Gwilym M. Jenkins, Time 

Series Analysis: Forecasting and Control (San Francisco, Holden Day, 1970); and Sir Maurice Kendall and J. Keith 

Ord, Time Series (New York, University Press, 1990). 3

 George E.P. Box and George C. Tiao, “Intervention Analysis with Applications to Economic and 

Environmental Problems,” Journal of the American Statistical Association, vol. 70, no. 353, 1975, pp. 71-79. 
6 

Table 2. REGARIMA models used for the eight major civilian labor force components

Total employment:

Men, 20 years and over ............... (0, 1, 2) (0, 1, 1) LOG 1976

Women, 20 years and over ......... (0, 1, 0) (0, 1, 1) LOG 1976

Men, 16 to 19 years ..................... (3, 1, 0) (0, 1, 1) LOG 1976

Women, 16 to 19 years ................ (0, 1, 1) (0, 1, 1) LOG 1976

Total unemployment:

Men, 20 years and over ............... (0, 1, 3) (0, 1, 1) LOG 1990

Women, 20 years and over ......... (1, 1, 0) (0, 1, 1) LOG 1990

Men, 16 to 19 years ..................... (0, 1, 1) (0, 1, 1) LOG 1976

Women, 16 to 19 years ................ (0, 1, 1) (0, 1, 1) LOG 1976

Series Model Transformation Series start 

date

Second, because it is rare that there is sufficient prior information to locate and identify all of the 

aberrant observations that may exist in a time series, as a second approach to specifying the 

regression component, REGARIMA offers automatic outlier detection based on work by I. 

Chang, G.C. Tiao, and C. Chen.4 This approach is especially useful when a large number of 

series must be processed. Of course, both approaches may be combined so that readily available 

prior information can be used directly, while unknown substantial outliers may still be 

discovered. 

Model adequacy and length of series. The preference is to use relatively long series in fitting 

times series models, but with some qualifications. Sometimes, the relevance of data in the distant 

past to seasonal adjustment is questionable. The implied X-11 moving average does not use 

much more than 5 years of data before and after the central observation being adjusted. Using a 

sliding span of 10 years in length, never revising back more than 5 years at any point, is 

sufficient to produce final revised seasonal factors. 

Even though the X-12 filters have limited memory, there are reasons for using longer series. 

First, for homogenous time series, the more data used to identify and estimate a model, the more 

likely it is that the model will represent the structure of the data well and the more accurate the 

parameter estimates will be. The exact amount of data needed for time series modeling depends 

on the properties of the series involved. Arbitrarily truncating the series, however, may lead to 

more frequent changes in model identification and to large changes in estimated parameters, 

which in turn may lead to larger-than-necessary revisions in forecasts. 

Second, although level shifts and other types of outliers tend to occur more often in longer 

series, X-12 has the capability of automatically controlling for these effects. 

Third, some very useful diagnostics available in X-12 typically require a minimum of 11 

years of data and, in some cases, as much as 14 years of data. 

 4

 I. Chang, G.C. Tiao, and C. Chen, “Estimation of Timer Series Parameters in the Presence of Outliers,” 

Techonometrics, 1988, pp. 193-204. 
7 

Fourth, attempting to fit longer series often provides useful insights into the properties of the 

series, including their overall quality and the effects of major changes in survey design. 

On the basis of the foregoing considerations, REGARIMA models are initially estimated for 

series beginning in 1976, where data series of that length are available. Extensive use is made of 

intervention analysis to estimate the magnitude of known breaks in CPS series and of automatic 

outlier detection to identify and correct for the presence of additional aberrant observations. 

Once a model is estimated, it is evaluated in terms of its adequacy for seasonal adjustment 

purposes. The criteria essentially require a model to fit the series well (there should be no 

systematic patterns in the residuals) and to have low average forecasting errors for the last 3 

years of observed data. When there is a tradeoff between the length of the series and the 

adequacy of the model, a shorter series is selected. In this case, the identification of the model is 

not changed with the addition of new data, unless the model fails diagnostic testing. 

Acceptable REGARIMA models have been developed for all of the labor force series that 

are directly adjusted. For each of the eight major civilian labor force components, table 2 

presents the form of the ARIMA part of the model, the transformation selected, and the starting 

date of the series used to fit the model. 

Concurrent seasonal adjustment 

Concurrent seasonal adjustment of national labor force data began at BLS with the release of 

estimates for December 2003 in January 2004. This practice replaced the projected-factor 

method, which updated seasonal factors only twice a year. Under the latter procedure, projected 

seasonal factors were used to seasonally adjust the new original data as they were collected. At 

midyear, the historical series were updated with data for January through June, and the seasonal 

adjustment program was rerun to produce projected seasonal factors for July through December 

of the current year. 

With concurrent seasonal adjustment, the seasonal adjustment program is rerun each month 

as the latest CPS data become available. The seasonal factors for the most recent month are 

produced by applying a set of moving averages to the entire data set, extended by extrapolations, 

including data for the current month. While all previous-month seasonally adjusted estimates are 

recalculated in this process, BLS policy is not to revise previous months’ official seasonally 

adjusted estimates as new data become available during the year. Instead, revisions are 

introduced for the most recent 5 years of data at the end of each year. 

Numerous studies, including that discussed in a 1987 paper on the CPS labor force series,5

have indicated that concurrent adjustment generally produces initial seasonally adjusted 

estimates requiring smaller revisions than do estimates produced with the projected-factor 

method. Revisions to data for previous months also may produce gains in accuracy, especially 

when the original data are themselves regularly revised on a monthly basis. Publishing numerous 

revisions during the year, however, can confuse data users. 

The case for revisions to previous-month seasonally adjusted estimates is less compelling 

for CPS series, because the original sample data normally are not revised. Moreover, an 

empirical investigation indicated that there were no substantial gains in estimating month-tomonth changes by introducing revisions to the data for the previous month. For example, it was 

found that if previous-month revisions were made to the labor force series, the overall 

 5

 George R. Methee and Robert J. McIntire, “An Evaluation of Concurrent Seasonal Adjustment for the Major 

Labor Force Series,” in Proceedings of the Business and Economic Statistics Section (Alexandria, VA, American 

Statistical Association, 1987). 
8 

unemployment rate would be different in only 2 months between January 2001 and November 

2002, in each case by only one-tenth of a percentage point. (More detailed information about this 

study is available from the authors upon request.) 

Prior adjustments 

Prior adjustments are adjustments made to the original data prior to seasonal adjustment. Their 

purpose is to correct the original series for atypical observations and other external effects that 

otherwise would seriously distort the estimates of the seasonal factors. Prior adjustment factors 

are subtracted from or used as divisors for the original series, depending on whether the seasonal 

adjustment is additive or multiplicative. 

Prior adjustment factors for CPS series may be based on special user-defined adjustments or 

handled more formally with REGARIMA modeling. Most of the prior adjustment factors for the 

labor force series are estimated directly from REGARIMA. 

Level shifts. The most common type of outlier that occurs in CPS series is the permanent level 

shift. Most such shifts have been due to noneconomic methodological changes related to 

revisions in population controls and to major modifications to the CPS design.6

 One notable 

economic level shift was due to the 2001 terrorist attacks. 

Population estimates extrapolated from the latest decennial census are used in the secondstage estimation procedure to control CPS sample estimates to more accurate levels. These 

intercensal population estimates are regularly revised every 10 years to reflect the latest census 

data and, less frequently, on other occasions. 

During the 1990s, three major breaks occurred in the intercensal population estimates. 

Population controls based on the 1990 census, adjusted for the estimated undercount, were 

introduced into the CPS series in 1994 and, in 1996, were extended back to 1990. In January 

1997 and again in January 1999, the population controls were revised to reflect updated 

information on international migration. 

Population revisions, which reflected the results of Census 2000, were introduced with the 

release of data for January 2003 and were extended back to data beginning in January 2000. 

Since 2003, population controls also are updated in January to reflect new estimates of 

international migration in the postcensus period. 

In 1994, major changes to the CPS were introduced, including a redesigned and automated 

questionnaire and revisions to some of the labor force concepts and definitions. In January 2003, 

new industry and occupational classifications also were introduced and were extended back to 

data beginning in 2000. 

To test for the possibility that revisions to the population controls had important effects on 

those CPS series with large numerical revisions in 1990, 1997, 1999, or 2000, as well as to test 

for effects due to the 1994 redesign, each REGARIMA model was modified to include 

 6

 For further discussion of these changes, see the following articles in printed issues of Employment and 

Earnings: “Revisions in the Current Population Survey Effective January 1994,” in the February 1994 issue; 

“Revisions in Household Survey Data Effective February 1996,” in the March 1996 issue; “Revisions in the Current 

Population Survey Effective January 1997,” in the February 1997 issue; “Revision of Seasonally Adjusted Labor 

Force Series,” in the January 1998 issue; “Revisions in the Current Population Survey Effective January 1999,” in 

the February 1999 issue; “New Seasonal Adjustment Factors for Household Data Series,” in the July 1999 issue; and 

“Revisions to the Current Population Survey Effective in January 2003,” in the February 2003 issue, available on the 

BLS Web site at http://www.bls.gov/cps/rvcps03.pdf. 
9 

Table 3. Prior adjustment factors for the eight major civilian labor force components

Pre-1990 Pre-1994 Pre-2000

Total employment:

Men, 20 years and over ............. Multiplicative .993 .983

Women, 20 years and over ....... Multiplicative

Men, 16 to 19 years ................... Multiplicative .940 .957 .988

Women, 16 to 19 years .............. Multiplicative

Total unemployment:

Men, 20 years and over ............. Multiplicative

Women, 20 years and over ....... Multiplicative

Men, 16 to 19 years ................... Multiplicative

Women, 16 to 19 years .............. Multiplicative

Series Mode of adjustment Prior adjustment factors

intervention variables for those years. The coefficients for these variables provide estimates of 

the direction and magnitude of the intervention effects. 

Intervention effects for 2000 were necessary for selected employment series related 

primarily to Hispanic, adult, and agricultural categories. These effects mainly reflect increases in 

adult and Hispanic employment due to the introduction of Census 2000-based population 

controls and a decline in agricultural employment caused by the change in the industry 

classification system. (See “Revisions to the Current Population Survey Effective in January 

2003” in the February 2003 issue of Employment and Earnings or on the BLS Web site at 

http://www.bls.gov/cps/rvcps03.pdf.) 

For those series with significant intervention effects, the estimated level shifts were removed 

prior to seasonal adjustment, thereby providing a smooth link to the pre-1990, pre-1994, pre1997, pre-1999, and pre-2000 data. The resulting prior-adjusted series were then used to estimate 

the seasonal factors, which, in turn, were applied to the original series, without prior adjustment, 

to obtain the seasonally adjusted series. 

The prior adjustment factors used for all of the eight major civilian labor force component 

series are shown in table 3. Because all eight series are seasonally adjusted with the 

multiplicative mode, the prior adjustments also are multiplicative. That is, the original series is 

modified prior to seasonal adjustment by dividing it by its prior adjustment factor. 

Due to an unusual revision in the population controls in January 2000, the unadjusted 

employment level of black men 20 years or over had a strong upward shift in the first quarter of 

2000. This temporary effect is permanently removed from the seasonally adjusted series with the 

use of the REGARIMA model. 

Effect of September 2001. At the end of 2001, unemployed job losers were identified as having 

had substantial upward level shifts one month after the September 11, 2001, terrorist attacks on 

the World Trade Center in New York City. (See the seasonal adjustment article in the January 

2002 issue of Employment and Earnings for more details.) Also, four additional series related to 
10 

workers employed part time for economic reasons were identified as having substantial upward 

shifts at the time of the terrorist attacks. 

Calendar effects. Calendar effects are transitory level shifts in a series that result from calendar 

events such as moving holidays or the differing composition of weekdays in a month between 

years. These effects have different influences on the same month across years, thereby distorting 

the normal seasonal patterns for the given month. 

Two CPS series related to persons at work exhibit significant effects in their April data due 

to the timing of Easter: “Persons at work on part-time schedules for noneconomic reasons who 

usually work part time in all industries” and “persons at work on part-time schedules for 

noneconomic reasons who usually work part time in nonagricultural industries.” These series 

were seasonally adjusted with multiplicative models with a moving-holiday correction. A 

detailed discussion of the nature of the Easter effect in those series and of the procedure used to 

control for it was included in the January 1990 version of this article. 

The six nonagricultural “Average actual hours at work” seasonally adjusted series 

mentioned in the previous section titled “Extension of seasonal adjustment to additional series” 

also have prior adjustments for holidays. The holidays are Good Friday, Labor Day, Columbus 

Day, and Veterans Day. Whenever one of these holidays falls in the CPS reference week, the 

level of each series falls. A weighted regression variable is added to the REGARIMA models to 

estimate each holiday’s effect by series. Since the effects are periodic (the sequence of dates for 

each holiday is not stable in annual timing but follows a longer cycle), the weights are necessary 

to maintain the level of the series over time. The estimate for each holiday is permanently 

removed from the series so as not to distort the level of the trend. 

X-11 decomposition 

The X-11 method of seasonal adjustment contained within the X-12-ARIMA procedure assumes 

that the original series is composed of three components: trend-cycle, seasonal, and irregular. 

Depending on the relationship between the original series and each of the components, the mode 

of seasonal adjustment may be additive or multiplicative. Formal tests are conducted to 

determine the appropriate mode of adjustment. 

The multiplicative mode assumes that the absolute magnitudes of the components of the 

series are dependent on each other, which implies that the size of the seasonal component 

increases and decreases with the level of the series. With this mode, the monthly seasonal factors 

are ratios, with all positive values centered around unity. The seasonally adjusted series values 

are computed by dividing each month’s original value by the corresponding seasonal factor. 

In contrast, the additive mode assumes that the absolute magnitudes of the components of 

the series are independent of each other, which implies that the size of the seasonal component is 

independent of the level of the series. In this case, the seasonal factors represent positive or 

negative deviations from the original series and are centered around zero. The seasonally 

adjusted series values are computed by subtracting the corresponding seasonal factor from each 

month’s original value. 

Given an appropriate choice for the mode of adjustment, the prior-adjusted, forecasted series 

is seasonally adjusted by the X-11 component of X-12. X-11 applies a sequence of moving 

average and smoothing calculations to estimate the trend-cycle, seasonal, and irregular 

components. The method takes either a ratio-to- or difference-from-moving-average approach, 

depending on whether the multiplicative or additive model is used. For observations in the 
11 

middle of the series, a set of fixed symmetric moving averages (filters) is used to produce final 

estimates. The implied length of the final filter under standard options is 72 time points for the 3 

by 5 seasonal moving average or 120 time points for the 3 by 9 moving average. That is, to 

obtain a final seasonally adjusted estimate for a single time point requires up to 5 years of 

monthly data preceding and following that time point. For recent data, asymmetric filters, with 

less desirable properties than symmetric filters, must be used. 

All of the civilian labor force component series were adjusted with the multiplicative mode. 

In previous years, unemployed teenagers, nonagricultural employment, and some other series 

were additively adjusted. Formal testing for the mode of seasonal adjustment with REGARIMA 

resulted in the rejection of all additive adjustments in favor of multiplicative adjustments. 

Evaluation 

A series should be seasonally adjusted if three conditions are satisfied: the series is seasonal, the 

seasonal effects can be estimated reliably, and no residual seasonality is left in the adjusted 

series. A variety of diagnostic tools is available in X-12 to test for these conditions. Among these 

tools are the F test from the original X-11, the more extensive M and Q tests from X-11-ARIMA, 

and a set of tests first available in X-12. The latter include sliding-span diagnostics, frequencyspectrum estimates, and revision-history statistics. If diagnostic testing shows that any of the 

three conditions listed fails to hold for a given series, that series is deemed not suitable for 

seasonal adjustment. 

Aggregation procedures

BLS directly seasonally adjusts series on the basis of age, sex, industry, occupation, education, 

and other characteristics. BLS also provides seasonally adjusted totals, subtotals, and ratios of 

selected series. It is possible to seasonally adjust an aggregate series either directly or indirectly 

by seasonally adjusting its components and adding the results, or dividing in the case of ratios. 

Indirect and direct adjustments usually will not give identical results, because (1) seasonal 

patterns vary across series, (2) there are inherent nonlinearities in X-12, (3) many series are 

multiplicatively adjusted, and (4) some series are ratios. 

BLS uses indirect seasonal adjustment for most of the major labor force aggregates. Besides 

retaining, so far as possible, the essential accounting relationships, the indirect approach is 

needed because many of the aggregates include components having different seasonal and trend 

characteristics that sometimes require different modes of adjustment. 

Examples of indirectly seasonally adjusted series are the levels of total unemployment, 

employment, and the civilian labor force, as well as the unemployment rate for all civilian 

workers. These series are produced by the aggregation of some or all of the seasonally adjusted 

series for the eight major civilian labor force components. The seasonally adjusted level of total 

unemployment is the sum of the seasonally adjusted levels of unemployment for four age-sex 

groups: Men 16 to 19 years, women 16 to 19 years, men 20 years and over, and women 20 years 

and over. Likewise, seasonally adjusted civilian employment is the sum of employment in all 

industries for the same four age-sex groups. The seasonally adjusted civilian labor force is the 

sum of all eight components. The seasonally adjusted civilian unemployment rate is computed as 

the ratio of the total seasonally adjusted unemployment level to the total seasonally adjusted 

civilian labor force (expressed as a percentage). 

A problem with producing seasonally adjusted estimates for a series by aggregation is that 

seasonal adjustment factors cannot be directly computed for that series. Implicit seasonal 
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adjustment factors, however, can be calculated after the fact by taking the ratio of the unadjusted 

aggregate to the seasonally adjusted aggregate or, for additive implicit factors, the difference 

between those two aggregates. 

Availability of revised series 

These revisions replace the seasonally adjusted estimates previously published. Revised 

historical seasonally adjusted labor force data also are available in various forms on the BLS 

Internet at http://www.bls.gov, including ftp access at ftp://ftp.bls.gov/pub/special.requests/lf/

to all of the revised data. The seasonally adjusted data last published for 2004 and earlier years 

were not further revised.