AGGREGATION & MODEL-BASED METHODS IN SEASONAL ADJUSTMENT OF LABOR FORCE SERIES

Stuart Scott and Peter Zadrozny, U.S. Bureau of Labor Statistics

Stuart Scott, 2 Massachusetts Avenue, NE, room 4915, Washington, DC 20212

Keywords: X-12-ARIMA, TRAMO/SEATS, direct

and indirect adjustment, unemployment rate

1. Introduction

One of the most closely-watched numbers

produced by the U.S. Bureau of Labor Statistics (BLS)

is the seasonally adjusted civilian unemployment rate.

It is computed from eight employment and four

unemployment series. These and other labor force

statistics are derived from the Current Population

Survey (CPS), a survey of close to 50,000 households.

A motivation for examining the aggregation issue is

that four of twelve components of the unemployment

rate are for teenagers and four are agricultural

employment. With the decline in size of the farm

sector and reductions in the CPS sample, it makes sense

to examine adjustment quality. This study revisits work

by Estela Dagum (1978) for the Levitan Commission.

An impetus for the use of models is the

development by Augustin Maravall and Victor Gomez

(1997) of the TRAMO/SEATS software for adjustment

based on seasonal ARIMA models. TRAMO/SEATS

offers advances in model identification and signal

extraction, making the use of models more practical,

and its availability makes these methods more

accessible to the public. Fischer and Planas (1998) for

Eurostat have worked extensively with the package.

2. Aggregation

As practiced, seasonal adjustment is nonlinear,

i.e., if A denotes the seasonal adjustment filter,

A(X + Y) ≠ A(X ) + A(Y) . X-11 starts with linear

filters, so a "pure" adjustment with its filters is linear.

Several factors can introduce nonlinearity: X-11

robustness features, ARIMA extrapolation, estimation

of trading day or other calendar effects, differing modes

of seasonal adjustment (additive, multiplicative). In

addition, our most important statistic, the

unemployment rate, is a ratio.

Even small departures from linearity can cause

problems. There's a story that one month major

components of the old Wholesale Price Index went up,

while the total went down. When Julius Shiskin, then

BLS commissioner, appeared before Congress, he was

asked whether or not the Bureau knew how to do

arithmetic. This led Shiskin to expand the use of

indirect adjustment, which provides for consistency.

With indirect adjustment (IA), the seasonally adjusted

aggregate is computed from seasonally adjusted

components (thus insuring consistency); with direct

adjustment (DA), the aggregate itself is seasonally

adjusted. For labor force series, there are so many

series of interest, based on demographics, geography,

job characteristic (e.g., industry, full-time vs. parttime), that it is not feasible to enforce complete

consistency across all published series. Currently, the

four series determined by gender and age (teenage,

adult) are seasonally adjusted directly for

unemployment, agricultural employment, and

nonagricultural employment, and all aggregate series

derived from these 12 series, including the civilian

unemployment rate, are seasonally adjusted indirectly.

This system is satisfying, because the seasonally

adjusted data are consistent for all these series, among

the most important products of the CPS. Table 1

shows codes used in the paper for the 12 basic

components and a few aggregates. (The first character

denotes age; the second, gender; and the last two, labor

force status).

The question that arises is whether this use of IA

sacrifices quality. This is largely an empirical question;

in principle, either method could be better, depending

on the application. Let's think about the case of only

two components.

Case 1. Components are seasonal, but "noisy."

Often the aggregate will be less noisy, so

DA may be better.

Case 2. Components are highly seasonal,

but with different patterns.

Table 1. Series codes

YFUN Unemployment, teenage females

YMUN Unemployment, teenage males

AFUN Unemployment, adult females

AMUN Unemployment, adult males

YFEA Agricultural employment, teenage females

YMEA Agricultural employment, teenage males

AFEA Agricultural employment, adult females

AMEA Agricultural employment, adult males

YFEN Nonag employment, teenage females

YMEN Nonag employment, teenage males

AFEN Nonag employment, adult females

AMEN Nonag employment, adult males

YUN Unemployment, teenagers

FUN Unemployment, females

MEA Agricultural employment, males

AEN Nonag employment, adults

etc.
The patterns may "wash out" somewhat in the

aggregate, so the combined pattern may be more

difficult to estimate. Here, IA may be better.

Assuming one wishes to set up an indirect adjustment,

Dagum (1978) gives four principles to consider in

selecting components.

(1) identifiable seasonality,

(2) simple pattern,

(3) pattern distinguishable from other

 components,

(4) meaningful economic entity.

The first is obvious. (2) is desirable, since a complex

pattern may be due to a lack of homogeneity of the

series, i.e., it may be the aggregate of series with

different patterns. If two potential components have a

similar seasonal pattern, then, according to (3), it may

make sense to combine them. Again, the last is fairly

obvious: it would not be desirable to form a component

from two series simply on the basis of a similar pattern,

rather than their economic sense. Such a series would

not be very usable. An in-depth discussion of these

principles is contained in Dagum's paper.

Dagum recommended reducing the number of

components for application of IA from 12 to 7. In

agreement with Dagum, this paper will show that in

some cases DA has better properties than IA, and that

IA outperforms DA for the civilian unemployment rate.

However, given the many aggregates that are derived

from the basic set of 12 series, it will argue against

combining any components in the current set.

A variety of statistics can be used to compare DA

and IA; among them are the Shiskin-Dagum quality

control statistics (M1-M11 and summary Q), sliding

span statististics, spectra, and revision statistics (cf.

Findley et al, 1998). In general, Q statistics tend to

favor DA. An initial look at spectra of seasonally

adjusted series fails to show appreciable differences.

Some use is made of revision statistics. Tables 2 and 3

contain sliding span statistics, the principle evaluation

tool. As discussed in Findley et al, for each month

common to two or more sliding spans, a maximum per

cent difference (MPD) is computed. It is desirable that

most MPD's be acceptably small. We focus on the 85th

percentile of MPD's for seasonal factors or seasonally

adjusted series, on the 60th percentile for month-tomonth change (change being more volatile), and on the

maximum. Only nine years of data, 1990-98, are used,

since pre-1990 data are not consistent with 1990 Census

data. This has limited us to applying sliding spans with

three 7-year spans.

Two of the smallest and noisiest series are YFEA

and YMEA, agricultural employment for teenage

females and males, respectively. Even so, the quality

control statistics support seasonal adjustment of YEA

with either method, with stable F statistics around 200,

M7's below 0.2, and Q's below 0.5; these are highly

seasonal series. From Table 2, the aggregate YEA has

substantially better statistics with DA. For seasonal

factors, the 85th percentile is 3.5 vs. 5.3, and the

maximum is 7.0 vs. 7.8; similarly, the statistics for

month-to-month change favor DA. From Table 3,

Table 2. Sliding Spans Statistics

Factors or adjusted series Month-to-month change

85th max 60th max

DA IA DA IA DA IA DA IA

YUN 1.6 1.2 3.9 4.2 1.2 1.4 5.0 5.6

AUN 1.2 0.8 2.8 1.3 0.7 0.7 3.2 1.7

FUN 0.9 0.9 2.9 1.5 0.8 1.0 3.7 2.0

MUN 1.0 1.1 1.8 1.9 0.8 0.8 2.1 2.1

YEA 3.5 5.3 7.0 7.8 2.5 4.5 7.8 12.1

AEA 1.0 0.6 2.1 1.4 0.7 0.7 1.9 1.8

FEA 1.4 1.7 3.6 4.2 1.2 1.1 3.8 4.4

MEA 1.1 0.8 2.4 1.4 0.6 0.6 2.0 1.6

YEN 0.7 0.7 1.3 1.6 0.5 0.6 1.5 1.5

AEN 0.1 0.1 0.2 0.2 0.1 0.1 0.2 0.2

FEN 0.1 0.1 0.3 0.3 0.1 0.1 0.3 0.3

MEN 0.1 0.2 0.3 0.3 0.1 0.1 0.2 0.3

UR 1.2 0.7 2.7 1.1 0.8 0.6 3.6 1.5
Table 3. Breakdown of Large MPD's

Series 2-3% 3-4% 4-5% >5%

YEA DA 18 11 4 5

IA 14 18 7 19

YUN DA 6 4 0 0

IA 5 1 3 0

FEA DA 6 4 0 0

IA 4 1 1 0

FUN DA 6 0 0 0

IA 0 0 0 0

with IA, 19 months (23%) even exceed a "liberal"

standard of 5% for differences in seasonal factors,

compared to 5 with DA. Further support for DA comes

from principle (3), the similarity of average seasonal

patterns for the components YFEA and YMEA in

Figure 1. Somewhat surprisingly, revision statistics are

lower for IA, with average absolute revision 1.2% for

1996 vs. 2.1% for DA.

The values in Tables 2 and 3 also favor DA for

YUN, teenage unemployment, but not by a very great

margin. Its components also have a fairly similar

average pattern, and the revision statistics favor DA.

Dagum also suggested combining AMEA and AFEA.

However, the sliding span and revision statistics are

smaller with IA.

So far, aggregates by age have been analyzed. An

aggregation scheme must take into account aggregates

by gender, which also receive attention from users.

Table 2 shows that IA performs as well or better for all

except FEA. Even in this case, DA has more MPD's

greater than 2, although IA has the largest MPD. A

similar seasonal pattern for components AFEA and

YFEA supports DA.

Looking at aggregates by both age and gender

brings out the difficulty of reducing components for

seasonal adjustment. Suppose we use YEA in place of

its two components. FEA can either be adjusted

directly or indirectly from YFEA and AFEA, but in

either case total agricultural employment based on

gender series may be inconsistent with the sum of

AMEA, AFEA, and YEA. Principle (3) is not

compelling when there is independent interest in the

components or aggregates formed from them. Some

pairs of components have different patterns. For

example, AMUN has a single January peak, while

AFUN, has peaks in both January and August. In

summary, for most aggregates, IA provides sliding span

(and revision) statistics which are as good or better than

DA. Given that both age and gender aggregates are

important, there appears to be no good case for

combining any of the current set of 12 components of

the unemployment rate.

As the foregoing analysis indicates, not all 12 of

the current components are equally strong series. Part

of the reason for both this study and Dagum's is the

declining size of the agricultural sector, and the fact that

CPS sample sizes are small for both agriculture and

teenagers. One reason for the split between farm and

nonfarm is that BLS's establishment employment

survey covers nonfarm only. This, in itself, is probably

not sufficient justification for using this employment

breakdown. For the demographic variable age,

alternatives could be considered. Dagum studies the

age split 16-24 and 25+. As the percentage of

individuals attending college increases, with a

corresponding delay in stable, year-round employment

for many, the 20-24 group may be more like the 16-19's

than the 25+ group. A series with this alternative

+

+

+

+

+

+

+

+

+

+

+

+

2 4 6 8 10 12

0.6 0.8 1.0 1.2 1.4 1.6

Fig. 1. Average Seasonal Factors

0

0

0

0

0

0

0

0

0

0

0

0

+ YMEA

0 YFEA

Time in months

4.5 5.0 5.5 6.0 6.5 7.0 7.5

Jan 90 Jan 92 Jan 94 Jan 96 Jan 98

Fig. 2. Unemployment Rate Trend

Direct

Indirect
"youth" classification should have lower variability

than the current one. Finally, some other variable, such

as race or ethnicity, may be important enough to merit

consideration for contributing to the formation of the

unemployment rate. The suggestions in this paragraph

are largely the province of the economists and data

users, and are beyond the scope of this paper. Given

that some of the current components are rather weak

series, however, it seems fitting to point out that

alternatives are possible.

What about the unemployment rate itself? At that

highly aggregated level, the statistics are quite good by

both methods. IA does have smaller sliding span

statistics in Table 2, and the graph of trends in Figure 2

does seem slightly smoother with IA. Thus, in addition

to providing consistency, IA appears to give a slightly

better seasonal adjustment for our most important

statistic, the civilian unemployment rate.

3. Comparisons using X-11 and model-based

 seasonal adjustment

For this paper, TRAMO/SEATS (T/S) and X-12-

ARIMA (X12), respectively, are used for model-based

and X-11 seasonal adjustments. Comparing underlying

methods has not been achieved by us, since X12 has

more features and diagnostics, and is better known to

us. What is done is to carry out an automatic seasonal

adjustment with T/S for a series, and, starting from the

T/S model, work with X12 to try to improve the

ARIMA modeling part. This "user-tinkering" includes

X12's transformation and outlier detection, examination

of graphs, and comparison of results for closely related

series. Thus, the aim of this section is not to gauge

superiority of method or software. Instead, we address

briefly the following questions:

(1) Are model estimation and outlier detection

similar with T/S and X12?

(2) Does user-tinkering (with X12) improve on

automatic adjustment (with T/S)?

To emphasize what is being compared, we use the

notation T/A for automatic TRAMO/SEATS and X/U

for X12 with user-tinkering. Table 4 summarizes

modeling results for the 12 components and a few

aggregates. T/S has a full-fledged model selection

procedure, to be incorporated into X12, which is why

we start with the T/A model. Both packages were

asked to choose between a log transformation and no

transformation, and to identify outliers. From the table,

there is agreement on the choice of transformation,

except for two series, YMUN and AFEA, where X12

chooses no transformation. T/A is geared very much to

the airline model, choosing it for all but two series,

YMUN and YEA. In these two cases, we accept the

T/A model. For four series, where T/A chooses

Table 4. T/A and X/U Modeling Results

Series TransformationModel qq q12No. of

outliers

Box-Ljung

p value

T/A X/U T/A X/U T/A X/U T/A X/U T/A X/U T/A X/U

YFUN L L 011 011 011 000 .78 .72 .99 - 0 0 .21 .16

YMUN L N 210 011 210 011 - - .77 .62 3 5 .94 .15

AFUN N N 011 011 011 011 .41 .23 .99 .77 0 3 .43 .38

AMUN L L 011 011 013 011 .31 -.28 .72 .75 0 2 .02 .34

YFEA L L 011 011 100 011 .49 - .93 .82 0 0 .14 .20

YMEA L L 011 011 300 000 .41 - .94 - 0 0 .09 .20

AFEA L N 011 011 011 011 .31 .22 .59 .55 1 3 .96 .50

AMEA L L 011 011 011 000 .39 .31 .92 - 0 2 .24 .34

YFEN N N 011 011 011 011 .40 .32 .91 .79 0 0 .69 .57

YMEN N N 011 011 011 011 .31 .17 .82 .67 0 1 .28 .48

AFEN L L 011 011 011 011 .37 .37 .53 .44 0 3 .05 .18

AMEN L L 011 011 110 011 .10 - .60 .57 0 1 .08 .15

YUN L L 011 011 011 000 .58 .52 .97 - 0 5 .19 .35

AUN L L 011 011 013 011 .16 -.25 .45 .42 0 4 .68 .86

YEA L L 100 011 100 011 - - .99 .97 0 0 .40 .23

AEA N N 011 011 011 011 .35 .35 .77 .77 0 0 .55 .37
an airline model, we use a deterministic seasonal, built

in for X12, but not T/S. In all four cases, the seasonal

MA parameter exceeds 0.90 with T/A, and is even

closer to 1 with X12. However, after comparing

results, keeping seasonal parameters very near one may

be advantageous. For another series, YEA, we accept

the airline model in X/U, even with .97 12 q = . For a

series such as AMUN, the rather low p-value for Q(24)

from the airline model has caused us to seek a betterfitting model.

T/A selects outliers for only two series. The

comparatively large number of outliers with X/U is

mostly due to examination of graphs and tables, i.e.,

user-tinkering. Sometimes, this leads to sets of outliers

which achieve the rather stringent standards of both

packages; often, one or more outliers is selected with

lower t-statistics. A possible advantage comes when

including an outlier allows a more parsimonious model

to have an acceptable fit. With AMEN, one outlier with

X/U improves a bit on the marginal fit with T/A, with

an equally parsimonious model. Probably, little harm is

done with marginal outliers, at least when they are

additive outliers; such values are also likely to be

replaced by X-11 in its iterations.

The table exhibits differences, even with a

common model, although these differences don't appear

to be important. For YFEN, small differences appear in

the estimation of both parameters and in the p-values.

For AEA, the parameter estimates agree, but there is a

difference in the p-values. Overall, more differences in

modeling appear than we expected.

As mentioned above, in a rare departure from the

airline model, T/A selects a (1 0 0) (0 1 1) model for

YEA, teenage agricultural employment. Notice that in

automatic mode, it selects no regular difference, a

decision corroborated by examination of residual

autocorrelation graphs. Average seasonal factors are

quite close for the two methods. Figure 3 compares

trends. Several countries are now publishing or doing

research on trend estimates. Some of the research in

this area involves the problem of false turning points.

Examining trends may be a useful diagnostic, even if

the trends are not published. Most people tend to prefer

a smoother, simpler trend. In fact, both these methods

seem to give pretty detailed trends, so smoother seems

better. Overall, we like T/A's trend a little better.

X/U's is smoother in the sense of being rounded. On

the other hand, most of the time, its peaks and troughs

are more extreme than T/A's. Here are a couple of

hypotheses arising from this graph, which we would

like to test further.

(1) T/S tends to give a more moderate trend,

because, with the canonical decomposition, it tries to

place as much as possible in the irregular.

(2) Locally X12's trend is smoother because its

robustness feature point by point assigns certain

variability to the irregular.

Figure 5 contains monthplots of the seasonal

factors. While the averages are very close, the withinmonth patterns are quite different. In X/U's view,

seasonal effects evolve. There is a fairly smooth curve

for each monthly subseries. According to T/A, seasonal

effects fluctuate. Maybe this latter view is better for

some series.

At the seasonal adjustment conference in

Bucharest, some comparisons of X12 & T/S used

defaults for both. That made for a "fair" comparison,

but in practice for important series such as these, we

Time in months

220 240 260 280

Jan 90 Jan 92 Jan 94 Jan 96 Jan 98

Fig. 3. YEA Trend

T/A

X/U

Time in months

2500 3000 3500 4000 4500

Jan 90 Jan 92 Jan 94 Jan 96 Jan 98

Fig. 4. AMUN Trend

T/A

X/U
may want to use some of the tools that are available.

Part of the reason for examining AMUN further is that

the Box-Ljung statistic is not too good for T/A.

Examining the residual autocorrelations leads fairly

easily to the 013 011 model. Secondly, an additive

outlier and a temporary change are included in the

model, both with marginal t-statistics by the standards

of both programs. Is there any difference? Figure 4

shows the trend graphs. They're quite close. However,

to use Estela Dagum's term, there are more ripples in

the T/A trend. The X/U trend appears preferable. The

sliding span statistics (not shown) are a bit smaller with

X/U. The monthplots (not shown) are similar; unlike

the previous example, both generate evolving seasonal

effects.

4. Summary

Indirect adjustment appears to perform as well or

better than direct adjustment for most of the labor force

aggregates studied, including the civilian

unemployment rate. Direct adjustment seems better for

teenage unemployment (YUN) and, especially, teenage

agricultural employment (YEA), but its use would

introduce inconsistency with gender aggregates, where

indirect adjustment also performs well. This argues

against any change in the method of adjustment with

the current set of components. With small samples for

the four components of teenage unemployment and

teenage farm employment, it would be desirable to

consider alternative components, such as a different age

breakdown.

For two series, comparisons have been made

between model-based and X-11 adjustments, with

automatic TRAMO/SEATS and user-augmented X-12-

ARIMA, respectively. Analysis of teenage agricultural

employment shows that with model-based adjustment

(1) seasonality fluctuates and (2) the trend has more

moderate peaks and troughs, which may be desirable

properties for some series. Analysis of adult male

unemployment shows modest gains applying X-11 with

user-tinkering.

Acknowledgments

Thanks to Catherine Hood for much help with

TRAMO/SEATS and to Bob McIntire, Tom Nardone,

and David Findley for information and useful

conversations.

The views expressed are those of the authors and

do not represent official positions of BLS.

References

Dagum, Estela Bee (1978), A Comparison and

Assessment of Seasonal Adjustment Methods for

Employment and Unemployment Statistics,

Background Paper No. 5, U.S. National Commission

on Employment and Unemployment Statistics,

Washington, D.C.

Findley, David F., Brian C. Monsell, William R. Bell,

Mark C. Otto, and Bor-Chung Chen (1999), X-12-

ARIMA Reference Manual, Version 0.2.3, Bureau of

the Census, Washington, DC.

Findley, David F., Brian C. Monsell, William R. Bell,

Mark C. Otto, and Bor-Chung Chen (1998), “New

Capabilities and Methods of the X-12-ARIMA

Seasonal Adjustment Program,” with discussion,

Journal of Business and Economic Statistics 16, 127-

177

Fischer, Bjorn and Christophe Planas (1998), "Large

Scale Fitting of ARIMA Models and Stylised Facts of

Economic Time Series," Eurostat Working Paper

9/1998/A/8, Eurostat

Gomez, Victor and Augustin Maravall (1997),

Programs TRAMO and SEATS: Instructions for the

User, Bank of Spain

Model-based

60 80 100 120 140 160 180

Jan Mar May Jul Sep Nov

X-11

60 80 100 120 140 160 180

Jan Mar May Jul Sep Nov

Fig. 5. Seasonal Factors for YEA