Monthly Labor Review September 2005 27

The Current Population Survey (CPS) is the

source of the Nation’s official estimates of

total employment and unemployment. The

CPS is a nationally representative, scientifically

selected monthly sample survey of approximately

60,000 households. The survey yields data that

are rich in demographic detail, including such

characteristics as age, sex, race, and Hispanic or

Latino ethnicity. Estimates from the survey are

published monthly in the BLS news release Employment Situation and in the BLS publication

Employment and Earnings.

In order to make the time-series data collected

from the CPS more useful to analysts and policymakers, the monthly data from the survey are

adjusted for seasonal fluctuations. As is well

known, the purpose of seasonally adjusting a

series is to remove seasonal fluctuations in the

data so that users can more easily observe fundamental changes in the level or trend of the series

that are associated with business cycle contractions and expansions. Approximately 116 time

series from the CPS are directly seasonally adjusted, and many more are indirectly seasonally

adjusted, as sums or ratios of the original 116.

There is, however, a source of spurious random

fluctuations in the CPS data that arises because

the CPS samples only a fraction—1 in 2,200, on

average, of the working-age population each

month: sampling error—the difference between

the survey estimates and the values that would be

produced by a complete census of the population.

Simultaneously removing both seasonality in

the data and noise due to sampling error can prove

Seasonal Adjustment Estimates

Model-based seasonally adjusted

estimates and sampling error

Estimating certain CPS series with a model

that filters out sampling error

may reduce volatility in the time series,

facilitating more meaningful trend analysis

Richard Tiller

and

Marisa Di Natale

mathematical

statistician on the

Statistical

Methods Staff,

and Marisa Di

Natale is an

economist

formerly in the

Division of Labor

Force Statistics,

Office of Current

Employment

Analysis, Bureau of

Labor Statistics.

quite challenging. The monthly estimates produced for the national aggregated series, such as

total employment and total unemployment, are

highly reliable relative to smaller, disaggregated

series. Many of the more detailed demographic

series, such as employment and unemployment for

blacks, are based on relatively small sample sizes,

so that survey error dominates movements in the

underlying level of the series. The standard error

for a (not seasonally adjusted) month-to-month

change based on the CPS can be quite high for some

of these series. For example, the standard error for

a change in the unemployment rate of adult black

males can be as large as 0.8 to 0.9 percentage point,

compared with 0.2 percentage point for the unemployment rate for all persons aged 16 years and

older. As a result, drawing meaningful conclusions

about trends or month-to-month changes is difficult, even after the data have been adjusted for

seasonal movements.

As an alternative to conventional seasonal adjustment, the study reported in this article applies

an experimental model-based method to selected

CPS demographic series. The method is designed

to remove the effects of sampling error, as well as

those of seasonality, from the series, thereby

making it easier to discern underlying trends in the

data.

Approaches to seasonal adjustment

The presence of large survey errors in the detailed

CPS series represents a major challenge to conventional methods of seasonal adjustment. Currently,

Richard Tiller is a
28 Monthly Labor Review September 2005

Seasonal Adjustment Estimates

the Bureau of Labor Statistics uses a seasonal adjustment

program called X-12-ARIMA to seasonally adjust its CPS series.

This program is based on the empirical moving-average

approach to seasonal adjustment.

An alternative that is gaining increasing attention is the

model-based approach to seasonal adjustment. A comparison

of the two approaches suggests that the model-based

approach provides much-needed flexibility in controlling for

the effects of sampling error. Such flexibility is not possible

with the conventional moving-average approach.

Conventional approaches to seasonal adjustment are based

on the classical decomposition of a time series, which assumes

that the series is composed of trend (or trend-cycle), seasonal,

and irregular components, in either an additive or a multiplicative

relationship. The first two components respectively account for

the long- and short-run systematic variation in the series. The

irregular component is a residual, usually assumed to be purely

random variation with a fixed variance.

Because the three components of the classical decomposition are not directly observable, they must be estimated in order to perform seasonal adjustment. The movingaverage method uses weighted moving averages of the

original data over a period of many years to produce a smooth

trend and a seasonal pattern. The estimated trend and

seasonal components are removed from the series, and the

residual is the irregular component. This approach makes no

attempt to define, in any formal statistical way, what is being

estimated, but rather applies a series of moving averages

directly to a series. While some of the moving averages are

chosen to satisfy a mathematical smoothness criterion, the

method was derived largely from empirical work with a wide

range of series.

By far the most successful application based on the

moving-average approach is the X-11 program,1

 which has

gone through several major revisions. The latest, enhanced

version is X-12-ARIMA.

2

 The original X-11 program, however,

remains at the core of X-12.

As an alternative to the moving-average approach, modelbased seasonal adjustment has been gaining increased

attention. The model-based approach specifies explicit

statistical models of the trend, seasonal, and irregular

components of the classical decomposition.3

 To seasonally

adjust the data, weighted moving averages of the observed

data actually are used in the model-based approach, but with

the important difference that they are derived directly from

the model. An essential characteristic of the approach is its

use of standard statistical procedures to estimate the unobserved components of the time series and to provide

associated statistical measures such as confidence intervals

and significance tests.

There is a large body of literature on the comparative

properties of the two approaches.4

 Each has its supporters

and critics. The major concerns with the model-based

approach are that it may be difficult to develop good models

for some series and that the model may fail occasionally when

new data become available. These concerns raise issues

about the robustness of the adjustment and the associated

statistical measures.

One major criticism of the moving-average approach is

that it lacks standard statistical measures. The absence of

standard errors for published seasonally adjusted data tends

to promote the mistaken impression that the final seasonally

adjusted values are exact rather than estimates. Moreover,

the lack of confidence intervals makes analysis of change in

the estimates and the location of turning points more difficult.

Still, supporters of the method argue that it is robust and

nonparametric; thus, its lack of an explicit statistical model is

viewed as an advantage. Even so, the absence of statistical

measures of reliability remains a major shortcoming.5

The model-based approach makes (testable) assumptions

about the underlying probability distribution generating the

data. Along with estimates of the model parameters, these

assumptions provide the means for constructing confidence

intervals and other statistical measures to quantify the

uncertainty in the estimates. Non-model-based estimates do

not, in general, afford a basis for producing measures of

uncertainty in the estimates.

Another criticism of the moving-average approach is that

it is not tailored to the specific properties of the series being

adjusted.6 In contrast, the model-based approach develops a

model on the basis of goodness-of-fit diagnostics. The resulting seasonal adjustment is based on the properties of the

series as represented by the model. In theory, under the

assumptions of the model, the seasonal adjustment is

“optimal” for the specific series. While the moving-average

method can make no such theoretical claim, its moving

averages were originally selected because they work well for

a very large number of series. This more generic approach

continues to work well in practice and may have an advantage

over the model-based approach when good models for seasonal adjustment cannot be developed.

Clearly, both approaches to seasonal adjustment have

their merits and limitations. Indeed, there have been a number

of studies of the relative performance of the two approaches,

but no general agreement as to how to interpret the results.

Perhaps a more balanced approach is to treat them as complementary tools for performing seasonal adjustment.7

Dealing with “noisy” CPS data series

The types of data series that are the focus of the study

discussed in this article—survey series with large sampling

errors—represent a class of series that presents special

problems for the moving-average approach to seasonal
Monthly Labor Review September 2005 29

adjustment. For these series, that approach (specifically, the

X-12 program) performs poorly, not because it has trouble

removing seasonality from the series, but because it cannot

adequately remove the effects of sampling error. The result is

a seasonally adjusted series that often is dominated by

sampling error, masking the underlying trend in the series.

This occurs because the moving averages are not tailored to

the specific properties of survey series with sampling errors.

What is implicitly assumed in applying these moving averages to survey data is that sampling error can be adequately

treated as part of the conventional irregular component,

which, conceptually, is a purely random series with a fixed

variance. The design of the survey, however, determines the

properties of the sampling errors, which may deviate in

important ways from typical irregular behavior. Moreover, for

well-designed probability samples, the characteristics of the

sampling errors are known, or at least, good estimates of

them can be obtained.

For the CPS, the standard errors provided along with the

point estimates from the survey routinely yield information

on the magnitude of the sampling errors. These standard

errors, however, are not constant and vary substantially over

time for some series, due to fluctuations in labor force levels,

redesigns of the survey, and changes in the sample size.

Another important characteristic of the CPS that is relevant to

seasonal adjustment is its panel structure, which generates

strong correlations of the sampling errors with their past

values. The CPS has a rotating panel whereby three-fourths

of the households are carried over from the previous sample

each month and one-half are carried over from the previous

year.8

These characteristics represent a serious challenge to

conventional approaches to seasonal adjustment. First, the

relatively large and changing magnitude of the survey error

will directly distort estimates of the trend and seasonal

components. In addition, because the survey error is correlated, the moving-average approach treats the induced correlations as if they were related to the trend. Consequently, too

much of the sampling error is absorbed into the estimated

trend and, to some extent, into the estimated seasonal

component, and not enough goes to the irregular component.

The standard model-based approach will do no better than

the moving-average approach, because it also is based on

the classical time-series decomposition, which ignores survey error as an important source of variation. The modelbased approach, however, has flexibility, which is not available with the moving-average approach. As mentioned earlier,

the latter, by virtue of its nonparametric structure, is a more

generic approach to seasonal adjustment, whereas the modelbased approach can be tailored to the idiosyncratic properties

of a particular series. From the model-based perspective,

survey error is just another unobserved component of the

time series for which a separate model can be specified, as is

done with the trend, seasonal, and irregular components of

the series. Then there is the further advantage, in modeling

survey error, of having external information from the survey

on the standard errors and correlations. This information can

be directly used to specify the parameters of the model. Of

course, measures of the survey error characteristics, such as

standard errors and autocorrelations, are estimates which are

themselves subject to errors that can adversely affect the

decomposition. For example, overestimates of the variances

could lead to “oversmoothing” the series. Still, given good

estimates of the survey error properties, a rather precise and

objective identification of the effects of survey error is

possible, resulting in a much cleaner separation of the

estimated trend and seasonal components from the

survey series than can be achieved by the moving-average

approach.

As described in the appendix to this article, research has

been conducted that uses a time-series model to seasonally

adjust some of the more volatile CPS series. For example, model

estimates have been produced for blacks 16 years and older,

for adult black men and women, and for black male and

female teens. The model producing these estimates has two

submodels as components: a “signal” model, which models

the true value of the specific demographic group; and a

“noise” model, which models the survey error associated

with that series. This model-based approach filters out the

survey error and the seasonal component of the series,

allowing the trend to be more cleanly separated from the error.

Using estimates from the model instead of from X-12 results in

smaller standard errors of the estimates and smoother

seasonally adjusted series. This type of model estimation

has been widely investigated in the context of “small-area”

estimation and has been used in the BLS Local Area

Unemployment Statistics program for more than 15 years to

estimate employment and unemployment for States and

selected metropolitan areas.9

Historical model-based estimates

There are two ways of processing time-series data to produce

model-based estimates. One, called the “forward filter,”

processes each observation as it first becomes available in

real time. Filtering is similar to the “concurrent” seasonal

adjustment method used to seasonally adjust new CPS data

series as they become available each month.10 The most

recent value of the time series is “filtered” by using all the

available data up to and including the latest month. Prior

months’ estimates are not revised as new data become

available.

The second approach to processing the data, called

“smoothing,” incorporates data before and after a particular
30 Monthly Labor Review September 2005

Seasonal Adjustment Estimates

observation. All the data are processed in batch mode, rather

than one observation at a time. During the smoothing process, estimates derived from filtering at each point in time are

revised to reflect information in all of the data. This approach

creates a much smoother seasonally adjusted series than

does filtering, although it does require revision of the entire

time series. (See the appendix for further discussion of

smoothing and filtering.)

Counterparts of smoothing and filtering also exist within

the conventional moving-average approach. For X-12, the

smoothest estimates are produced by symmetric moving

averages, which use an equal number of observations before

and after the time point being adjusted. Toward the end of

the series, where there are fewer observations after that point,

less desirable asymmetric moving averages must be used.

When the X-12 program is executed to seasonally adjust the

most recent observation, the result is referred to as the

concurrent estimate, which is the counterpart of the model’s

filtered estimate.

Charts 1 and 2 compare smoothed model-based and X-12

seasonally adjusted unemployment rates and employment levels

from January 1999 through December 2003 for blacks aged 16

and older.11 Chart 2 also includes the X-12 trend series, discussed

later. The X-12 seasonally adjusted series are published each

month in the BLS Employment Situation news release. The

graphs show that the model estimates create smoother seasonally adjusted lines or trend lines for both series than do the

corresponding X-12 seasonally adjusted series.

The conventional time-series decomposition represents the

observed series as being composed of trend, seasonal, and

irregular components. The seasonally adjusted series consists

of the trend and irregular components and thus is generally less

smooth than the trend. For the series modeled, the distinction

between the seasonally adjusted series and the trend series is

not important: once the estimated survey error is removed from

the series, very little residual irregular component is left. The

model-based seasonally adjusted series, therefore, is virtually

identical to the trend component. For X-12, the irregular

component is relatively large; thus, its seasonally adjusted series

is noticeably less smooth than its trend.

One advantage of the modeled estimates is evident from

the graphs: peaks and troughs in these series are more easily

identifiable, which is not the case with the X-12 seasonally

adjusted series. This makes identifying labor market turning

points for specific demographic groups easier. For instance,

in the black employment- and unemployment-rate series in

charts 1 and 2, the turning point clearly occurred in mid-2000.

X-12’s trend estimates, shown in chart 2, clearly involve

Chart1

Chart 1. Unemployment rates, blacks aged 16 years and older, seasonally adjusted, 1999—2003

6

7

8

9

10

11

12

6

7

8

9

10

11

12

Percent Percent

January

1999

July

1999

January

2000

July

2000

January

2001

July

2001

July

2002

January

2002

January

2003

July

2003

Model

X-12

Chart 1. Unemployment rates, blacks aged 16 years and older, seasonally adjusted, 1999—2003

Model

X-12
Monthly Labor Review September 2005 31

Chart2 & 3

Chart 2. Employment levels, blacks aged 16 years and older, seasonally adjusted, 1999—2003

14,600

14,700

14,800

14,900

15,000

15,100

15,200

15,300

14,600

14,700

14,800

14,900

15,000

15,100

15,200

15,300

Thousands Thousands

January

1999

July

1999

January

2000

July

2000

January

2001

July

2001

July

2002

January

2002

January

2003

July

2003

Model

X-12

Trend

Chart 3. Unemployment rates, blacks aged 16 years and older, concurrent, seasonally adjusted, 

 2003—04

8.5

9.0

9.5

10.0

10.5

11.0

11.5

12.0

8.5

9.0

9.5

10.0

10.5

11.0

11.5

12.0

Percent Percent

July

2004

January

2004

January

2003

July

2003

Model

X-12

Chart 3. Unemployment rates, blacks aged 16 years and older, concurrent, seasonally adjusted, 2003—04

Chart 2. Employment levels, blacks aged 16 years and older, seasonally adjusted, 1999—2003

X-12

trend

X-12

Model

Model

X-12
32 Monthly Labor Review September 2005

Seasonal Adjustment Estimates

further smoothing of the seasonally adjusted series, but this

does not help identify real turning points. Due to the effects

of the survey, the X-12 trend series displays spurious oscillations around the model-based trend series. These oscillations

could be misinterpreted as real turning points.

Real-time current-year estimates

Because the smoothed estimates use data from the entire

sample (January 1976–December 2004), they do not correspond to the estimates that would have been available in

2004 to data users in real time. As mentioned in the previous

section, the latest monthly estimate in a modeled series is a

filtered estimate that incorporates prior months’ data, as well

as data for the current month. Filtered estimates are not as

smooth as the “smoothed” series, because filtering cannot

incorporate data from the future, nor does it incorporate the

latest information into past months’ estimates. As a result,

the month-to-month changes in the filtered estimates are

much more volatile than the smoothed estimates, but still are

smoother than the X-12 estimates.

Charts 3 through 5 show forward-filter estimates of the

black unemployment rate and employment level, and of the

unemployment level for adult black women, for January 2003

through December 2004. In chart 3, it is clear why the modeled

data may be preferable to the X-12 estimates. The X-12 series

shows two rather large spikes in black unemployment in June

and October 2003. The June spike appears to be the larger of

the two. In contrast, the model series shows, not the June

spike, but rather a small, statistically insignificant rise,

followed by a flattening out of the rate and a slower decline

from October 2003. The model suggests that the rate peaked

at around 11.0 percentage points during the summer and early

fall of 2003, whereas X-12 gives a more confusing picture. The

employment level for blacks is shown in chart 4. The modeled

data indicate that the employment level bottomed out around

August 2003, whereas the X-12 series depicts the lowest level

of employment for this group during the first quarter of the

year. In chart 5, the unemployment level of adult black women

is shown. Although in both the modeled series and the X-12

series, the unemployment levels wind up being roughly the

same at the beginning and end of the period shown, the

modeled series is clearly less volatile over the year.

An important feature of the model-based approach, as

emphasized in the previous section, is that it provides

estimates of the standard errors for the point estimates,

Chart 4

Chart 4. Employment levels, blacks aged 16 years and older, concurrent, seasonally adjusted, 

2003—04

14,600

14,700

14,800

14,900

15,000

15,100

14,600

14,700

14,800

14,900

15,000

15,100

Thousands Thousands

July

2004

January

2004

January

2003

July

2003

Model

X-12

Model

Chart 4. Employment levels, blacks aged 16 years and older, concurrent, seasonally adjusted,

 2003—04

X-12
Monthly Labor Review September 2005 33

whereas X-12 does not. It is common practice for analysts to

use the standard errors for the not-seasonally-adjusted CPS

estimates as a proxy measure of error for the X-12 estimates.

While this practice is not strictly correct, recent research

suggests that these standard errors may be a reasonable

approximation (an overestimate of around 10 percent to 15

percent, sometimes less) for detailed series in which sampling

error is an important source of variation.12 The following

tabulation compares the model-based standard errors of

monthly change from the forward filter with the CPS proxy

measures and computes the minimum magnitude of monthly

change in the respective estimates required to achieve significance at the 90-percent level for the period from January

2000 to July 2004 (the figures shown are all mean values):

 Unemployment Employment Unemployment,

 rate, level, black adult

 blacks blacks women

 Standard error:

 CPS ........................ 0.51 101,271 49,261

 Model .................. .21 41,580 14,047

 Minimum change

required for

significance:

 CPS ....................... ±.84 ±166,591 ±81,034

 Model .................. ±.35 ±68,399 ±23,107

For the total black employment level, the minimum

magnitude of monthly change required for significance has

to be at least 167,000 with the CPS proxy measure—almost

two-and-a-half times the magnitude required for the model

estimates. For unemployed black adult women, the CPS proxy

measure requires a change of at least 81,000 in the X-12

seasonally adjusted estimates, or 3.5 times the change required in the model estimates. For the total black unemployment rate, the CPS proxy requires a change of more than

0.8 percentage point, compared with less than 0.4 percentage

point for the model. Thus, even though the CPS proxy may

overestimate the required monthly change in X-12 by up to 15

percent in relative terms, that still leaves major gains from using

the model estimates.

Research to this point suggests that the model-based approach to seasonal adjustment is a useful tool for helping to

discern trends in data series with large variances due to relatively

small sample sizes. The approach, however, is not necessarily

superior to the moving-average approach currently used by the

Bureau of Labor Statistics for adjusting aggregate series such

as total employment and unemployment when sampling error is

a much less important source of variation than it is for the detailed

demographic series. Research into the merits and limitations of

the model-based approach to smoothing time series and creating

estimates of variance is ongoing in order to evaluate possible

extensions of the method to other series.

Chart5

 Source and

 measure

Chart 5. Unemployment levels, adult black women, concurrent, seasonally adjusted, 2003—04

600

650

700

750

800

850

600

650

700

750

800

850

Thousands Thousands

July

2004

January

2004

January

2003

July

2003

Model

X-12

Chart 5. Unemployment levels, adult black women, concurrent, seasonally adjusted, 2003—04

X-12

Model
34 Monthly Labor Review September 2005

Seasonal Adjustment Estimates

APPENDIX: Methodology

To account for the special properties of the CPS series, a signal-plusnoise model is developed that combines a time-series model of the

“true” values and its unobserved components (trend, seasonal, and

irregular) with a model of the sampling errors. The latter is treated

as an additional unobserved component of the time series, with the

special advantage that its variance-covariance structure is objectively identified by design information.

This appendix discusses, in general terms, how the components

of the CPS are modeled, briefly considers the series that is directly

modeled in the text of this article, and examines the use of the

Kalman filter and smoother algorithms in the text to produce

seasonally adjusted estimates in both real and historical time.1

Component models of the CPS

For labor force series, seasonality is an important source of

variation. The nonseasonal, or trend, part of the series cannot be

observed directly; instead, trend values must be inferred solely

from observations of the aggregate series. Seasonal adjustment is a

Notes

1

 Julius Shiskin, Alan H. Young, and John C. Musgrave, The X-11

variant of the census method II seasonal adjustment program,

technical paper 15 (Bureau of Economic Analysis, 1967).

2

 David B. Findley, Brian C. Monsell, William R. Bell, Mark C.

Otto, and Bor-Chung Chen, “New Capabilities and Methods of the X12-ARIMA Seasonal-Adjustment Program,” Journal of Business and

Economic Statistics, April 1998, pp. 127–77 (with discussion).

3

 Examples are TRAMO-SEATS (see Victor Gomez and Agustin

Maravall, “Programs Seats and Tramo: Instructions for the User,”

Working Paper No. 9628 (Madrid, Bank of Spain, 1996)); and STAMP

(see Siem J. Koopman, Andrew C. Harvey, Jurgen A. Doornik, and

Neil Shephard, STAMP : Structural Time Series Analyser Modeller and

Predictor (London, Timberlake Consultants, 2000)).

4

 See, for example, William R. Bell and Steve C. Hillmer, “Issues

Involved with the Seasonal Adjustment of Economic Time Series,”

Journal of Business and Economic Statistics, October 1984, pp. 291–

320; Agustin Maravall, “Unobserved Components in Economic Time

Series,” Handbook of Applied Econometrics, ed. M. Hashem Pesaran,

and Mike Wickens (Oxford, U.K., Basil Blackwell, 1993), pp. 12–72;

and Findley, Monsell, Bell, Otto, and Chen, “New Capabilities and

Methods.”

5

 Recent research suggests that the absence of explicit statistical

models in x-11 may not prevent statistical inference from being carried

out after all. Stuart Scott and Danny Pfeffermann have experimented

with a promising approach to the development of standard error

measures for x-11 estimators. (See Stuart Scott and Danny

Pfeffermann, “Evaluation of Two Variance Methods for x-11

Seasonally Adjusted Series,” ASA Proceedings of the Joint Statistical

Meetings (Alexandria, VA, American Statistical Association, 2003),

pp. 3760–67.)

6 X-12 allows for some flexibility by automatically varying the

special case of the more general “signal-plus-noise” formulation of

an observed series as being composed of a number of unobserved

components, where the components of interest are referred to as

the signal and all other components are considered to be noise.

The goal is to “filter out” the signal from the noise when all that

can directly be observed is the data corrupted by noise. The

solution to this problem requires a model that specifies, either

implicitly or explicitly, the properties of each of the underlying

components. In the conventional seasonal adjustment problem, the

observed series, Yt

, is assumed to be measured without error and to

be composed of trend, Tt

, seasonal, St, and irregular, It, components

in additive form:

Multiplicative seasonality is handled by taking logarithms of the

data, fitting the model to the data, and then taking the antilogarithm

of the component estimates back to the original scale.

The signal is the nonseasonal component (trend plus irregular

 length of its filters on the basis of series properties.

7

 This view appears to be gaining momentum with the effort to

have X-12 and model-based seasonal adjustment as alternative options

in a single software program called X-12-ARIMA/SEATS. (See Brian C.

Monsell, John A. D. Aston, and Siem J. Koopman, “Toward X-13?” ASA

Proceedings of the Joint Statistical Meetings (Alexandria, VA, American

Statistical Association, 2003), pp. 1459–66.)

8

 As a result, if the sample, say, overestimated the true value in the

previous month, it is likely to continue overestimating in the current

and successive months. In other words, the behavior of the survey

errors is more similar to a trend with cumulative up-and-down

movements than to one with purely random irregular variation.

9

 For a discussion of these models, see R. Tiller, “Time series

modeling of sample survey data from the U.S. Current Population

Survey,” Journal of Official Statistics, June 1992, pp. 149–66.

10 For more on the switch to concurrent seasonal adjustment in the

CPS, see Richard B. Tiller and Thomas Evans, “Revision of Seasonally

Adjusted Labor Force Series in 2004,” Employment and Earnings,

January 2004, pp. 3–9; on the Internet at http://www.bls.gov/cps/

cpsrs2004.pdf.

11 Both the model-based and the X-12 series are based on data

extending back to January 1976. In chart 2, the estimates have been

adjusted to smooth out the effect of the introduction of new

population controls from Census 2000 in January of that year.

12 Stuart Scott, Michail Sverchkov, and Danny Pfeffermann,

“Variance Measures for Seasonally Adjusted Employment and

Employment Change,” ASA Proceedings of the Joint Statistical

Meetings (Alexandria, VA, American Statistical Association, 2004),

pp. 1328–35.

. YTSI t t tt =++
Monthly Labor Review September 2005 35

 component), and the noise includes the seasonal component (when

irregular fluctuations are relatively large, it may be more appropriate

to define the signal as just the trend and treat the irregular component

as part of the noise):

Following Andrew C. Harvey, the trend, seasonal, and irregular

components are specified as structural time-series models.2

 Each

component has an associated variance that determines its properties. The trend takes a local linear form, with its level and slope

varying randomly. A wide variety of patterns is a possibility,

depending on the relative magnitudes of the variances of the level

and slope. When the two variances are both zero, we have the

smoothest possible trend, which assumes a fixed linear form over

the entire observation period. A rapidly fluctuating trend results

when the variance of the level shift is much larger than the variance

of the slope.

The seasonal component also evolves over time according to its

variance. A seasonal pattern that is fixed from year to year implies

a zero variance, while patterns that change gradually imply a small

positive variance. Finally, the magnitude of the irregular component

depends directly on the size of its variance. A zero variance implies

the absence of irregular variation in the series. The four variances of

the structural model presented in the text of this article are unknown

parameters that will be estimated from the data once a complete

model is set up as described herein.

In the case of CPS data, there is a fourth component: the sampling

error et

, which arises because only a fraction of the total population

is sampled each month. This means that the true values Yt are no

longer directly observed, but instead, the survey estimates,

are what is observed. The definition of the signal is unchanged, but

now the noise includes survey error as well as seasonality; that is,

Since the presence of sampling error in the data can radically

alter the results of seasonal adjustment, a model is required that

takes the sampling error into account when it is an important source

of variation in the data. Unlike the situation with the other components, the properties of this component do not have to be estimated

from just the time series itself. Rather, direct information from the

survey microdata may be used to estimate the sampling error model

independently of the structural time-series model.

The Census Bureau routinely produces data to assess the reliability of national CPS statistics on an ongoing basis. The process

involves drawing a set of random subsamples, or replicates, from

the full sample surveyed each month, using the same principles of

selection as are used for the full sample, and applying the regular

CPS estimation procedures to the replicates that are drawn. Each

month, 160 replicates are produced for a large number of characteristics. The variability in the replicates provides the basis for

computing empirical variances and autocorrelations for the sampling

error. Variances are computed by fitting generalized variance

functions to the monthly replicate variances to smooth out volatility in the latter. Lag covariances are computed by averaging each

replicate lag covariance over time and normalizing for changes in

variance.

The empirical sampling-error variances and autocorrelation

estimates are then used to directly derive the parameters of the

survey error model, which is specified as an autoregression that

relates the current value of the survey error to its past values. Note

that the coefficients are determined from the sampling-error

autocorrelations, and the variances are adjusted to conform to the

survey variances. In this way, both the effect of the rotating panel

design and the magnitude of the sampling error are taken into

account.

The sampling-error model is combined with the time-series

model of the trend, seasonal, and irregular components, with the

latter variances estimated directly from the historical CPS series.

Thus, the basic properties of the overall model are tailored to the

empirical behavior of the CPS series.

Model fitting to CPS series

Black employment- and unemployment-level series for the

following four age-gender groups are directly seasonally adjusted,

and variance estimates are produced, with the models presented in

the text of the article:

• black male youths aged 16–19 years

• black female youths aged 16–19 years

• black men aged 20 years and older

• black women aged 20 years and older.

Seasonally adjusted total black employment and unemployment

levels are obtained indirectly by summing the appropriate model

estimates, and the unemployment rate is derived from the estimates

of these levels. Although seasonal adjustment of the totals is done

indirectly, the variances are produced by directly fitting the models

to the aggregate data.

The observation period of the study begins in January 1976 and

ends in December 2004. For each series directly modeled, a

preliminary model consists of the trend level, together with the

slope, seasonal, irregular, and survey error components. The model

is fine-tuned through diagnostic testing. Evaluating the test results

leads to decisions regarding (1) the need for a logarithmic transformation, (2) the presence of a trend slope, an irregular component,

and outliers of various forms, and (3) the composition of the

seasonal component. If necessary, impulse or step dummies can be

incorporated into the model to allow for exceptional temporary

shocks or even more permanent shifts in the series.

For all models, the estimated variance of the irregular component

was very close to zero. Therefore, the seasonally adjusted series

(trend plus irregular component) is virtually identical to the trend.

Thus, although there may be irregular variation in the true series,

it is empirically difficult to detect in the presence of the large

sampling-error variation.3

Real-time and historical estimates

Given estimates of the unknown parameters of the model presented

in this article, seasonal adjustment of the data may commence.

Because CPS data are generated each month, there are two ways to

approach estimation. One way is to make an estimate each time

new CPS data become available; the other way is to wait until data

, t tt yYe = +

. N Se t tt = +

Signal: Sig ; t tt = + T I

Noise: . N S t t =
36 Monthly Labor Review September 2005

Seasonal Adjustment Estimates

accumulate and then produce all the estimates at once. The first

approach occurs in real time when seasonal adjustment is performed

immediately after the latest data are available. (This approach is

sometimes referred to as concurrent seasonal adjustment.) The

second approach corresponds to the usual practice of revising the

entire series of seasonally adjusted estimates at the end of the year

once the latest data for the entire year become available. In the

time-series literature, the first approach is called filtering, the

second smoothing.

To produce seasonally adjusted estimates, the model is first

cast into the so-called state-space form, which allows the use of the

Kalman filter, a powerful algorithm for computing the estimates in

real time.

A state vector includes the values of all the components of the

model at a specific point in time. Two equations define the statespace form. First, there is the transition equation that specifies

how the state vector, Zt

, behaves over time. This equation takes a

simple form, with the current value of the state vector depending

only on its value in the previous period, plus Vt

, a vector containing

the random disturbances representing uncertainties in the dynamics

of the system. (The vector Vt

 is assumed to be normal and independently distributed.) Mathematically,

where Ft

 is a known transition matrix. This simple autoregressive

structure may appear to be very limiting, but in fact it encompasses

a wide variety of models by introducing artificial state variables.

The state variables, which are the signal and noise components,

are not directly observable. Instead, the information on the state of

the system is conveyed by the sample data yt

, which is related to

the state vector via the observation equation

This is the second of the two equations. Here, Ht, which has known

values, is called the observation matrix and sums up the components

of the state vector such that the observed datum is equal to the sum

of the signal (trend plus irregular component) and the noise (seasonal

plus sampling error components).

The task is to find the expected values of the state vector and its

variance, given the observed sample values. These are the “best”

estimates of the state vector, in the sense of minimizing the mean

square error (or, more precisely, the best linear unbiased predictors

of the state variables). The expected value is

and the variance is

where | ˆ

Zt t is the expected value of the state vector at time t, based

on all of the observed data up to time t.

The solution to finding a “best” estimate from noisy data as

they come in each period is given by the celebrated Kalman filter,4

which needs only the previous period’s estimates 1| 1 ˆZ t t − − and

1| 1 ˆPt t − − (which are based on all of the data up to time t – 1) and the

current observation yt

 to compute the “best” current-period

estimates and This is done in two steps: a prediction step

followed by an update step. First, the state vector is predicted for

time t by projecting forward its previous period’s estimate by means

of the aforementioned transition equation, where the disturbance terms,

v are set to their expected values of zero.

In fact, however, the disturbances are never exactly zero, and

the previous period’s estimates contain errors. Accordingly, in

the update step, the predictions are improved with the use of

information from the current sample values. The observation

equation provides the means for getting feedback from the sample

by forming a prediction of the new data value as the sum of the

signal and noise predictions, and comparing it with the actual

value

In this equation, the subscript t|t – 1 indicates that the prediction is

for time t, using data up to time t – 1 only.

Next, an overall prediction error

is computed. The error in predicting the latest yt

 value using data

only up to the previous period is the sum of the unobserved

prediction errors for the signal and noise. In the update step, the

signal and noise estimates are corrected in proportion to their

expected contribution to the total prediction error:

The subscript t|t indicates that information from time t has been

added to the estimate.

The weight gt, which varies between zero and unity, determines

how much of the prediction error is allocated to the signal and how

much to the noise. It is a function of the ratio of the variance in the

signal to the variance in the noise. As the noise variance gets larger,

gt gets smaller, and more weight is given to the previous period’s

prediction of the signal, because now the observed data at time t are

less reliable and therefore the sampling error for this observation is

more likely to account for the largest share of the overall prediction

error. The signal and noise weights on the prediction error add to

unity, ensuring that the updated estimated signal and noise add to

the observed sample value.

Each period, the Kalman filter makes a prediction of the

next observed value yt

, using only the most recent estimates of the

signal and noise. The filter calculates corrected estimates of these

components from the prediction error, which incorporates

information from the latest available data at time t. Because the

predicted values are based on all the data available up to time t – 1,

the corrected estimates for time t reflect all the available information

from both the historical and current values of the series. After each

prediction and update step, the prediction correction process is

repeated. This recursive nature is one of the appealing features of

the Kalman filter. A simple illustration follows.

Filtering is tailored to real-time processing of one observation at

a time as it first becomes available. Conceptually, this is analogous

to updating a running average of numbers by the

recursive formula

1 , Z FZ V t tt t = + −

Sig . t tt t t y HZ N = =+

|1 |1 |1 |1 ˆ ˆ ˆ ˆ Sig . tt t tt tt tt y HZ N − − −− = =+

|1 |1 |1 ( ) ( ) ˆ ˆ ˆ Sig Sig t tt t tt t tt yy NN − = − +− − −−

1 | 1 ˆ ˆ Sig Sig ; ˆ tt tt t t tt gy y − − = +− ⎡ ⎤ ⎣ ⎦

1 ( ) | 1 ˆ ˆ 1 . ˆ N N gyy tt tt − t t tt− = +− − ⎡ ⎤ ⎣ ⎦

1

1 1 , t ttt x xx t t −

⎛ ⎞ ⎛⎞ − = + ⎜ ⎟ ⎜⎟ ⎝ ⎠ ⎝⎠

( 1 | ) ˆ |,, , EZ y y Z t t tt K =

( 1 | ) Var | , , , ˆ Zy y P t t tt K =

: t y

ˆ , t V

| 1 ˆ , t t y −

| 1 ˆt t y −

| ˆ

Zt t |

ˆ . Pt t

1 2 ,,, t xt xx x K
Monthly Labor Review September 2005 37

which requires only the previous period’s estimate plus the current

observation, rather than by starting over each time with the formula

which requires all of the data. The significant point is that the

Kalman filter does no more work to process the last observation

than it does the first. The net result is an algorithm tailored to realtime applications, whereby data keep coming in and information

about the current value of the signal is needed immediately.

The Kalman filter, however, is not well suited to producing

historical estimates for a fixed set of data observations, because it is

designed to produce an estimate for the current period only and not

to revise any earlier estimates. Nonetheless, it is standard practice

to revise a seasonally adjusted estimate made at time t by using

information that had arisen subsequently, for the succeeding

observations up to the last observation at t = n are

bound to convey information about the trend and other components

that can supplement the information that was available at

time t. Clearly, the Kalman filter estimates for the observations at

the beginning of the time series will be extremely weak.

The retrospective improvement of the estimates by using ex

post information is achieved by a process conveniently described

as “smoothing.” This is a matter of revising each of the filter

estimates for a period running from t = 1 to t = n once the full set of

observations y1

,..., yn has become available. These “retrospective”

estimates are obtained from the “Kalman smoother,” which runs

the Kalman filter recursion backwards from t = n to t = 1 through the

earlier data, revising the estimates produced by filtering at each

time point. Smoothing is batch processing in the sense that it

operates on all of the data at once, in contrast to the Kalman filter

real-time processing of one observation at a time.

Not surprisingly, the estimates from the smoother typically

 look “smoother” than those from the filter. For historical analysis,

the smoothed estimates are superior (they have a smaller error

variance) to the filtered estimates. But it is important to note that,

because these smoothed estimates use data from the entire sample,

they do not correspond to estimates that would have been available

to data users in real time. In reality, smoothing would be done once

a year, in accordance with the BLS policy of revising official

seasonally adjusted estimates. Smoothing could be performed each

month, but there is an obvious disadvantage to publishing the

results, because all of the previous month’s estimates would be

revised each month, a practice that could be confusing to data users.

Notes to the appendix

1

 A more technical discussion is given in R. Tiller, “Seasonal

Adjustment of CPS Time Series with Large Survey Errors,” paper

presented at the Federal Economic Statistics Advisory Committee

Meeting, Washington, DC, December 13–14, 2001.

2 Andrew C. Harvey, Forecasting, Structural Time Series, Models

and the Kalman Filter (Cambridge, U.K., Cambridge University Press,

1990).

3 Tiller, “Seasonal Adjustment.”

4 Harvey, Forecasting.

( ) 1

1 , t t xx x t

= ++ L

1 2 , ,, tt n yy y + + K

1, , t y y K

1, , n y y K