Seasonal Adjustment of Hybrid Economic Time Series

Stuart Scott, George Stamas, Thomas J. Sullivan, and Paul Chester, U.S. Bureau of Labor Statistics

Stuart Scott, 2 Massachusetts Ave., NE, Rm. 4915, Washington, DC 20212

Key Words: Employment statistics, revisions,

 X-11-ARIMA, smoothness

1. Introduction

State industry employment is estimated monthly

from the Current Employment Statistics survey, a

sample of about 380,000 employers, and seasonally

adjusted with X-11-ARIMA. An annual benchmarking

process revises estimates to reflect universe counts

available from administrative records of the

Unemployment Insurance (UI) programs of each state.

At any point in time, the current series consists of

universe data through the latest benchmark month

followed by sample data up to the current month. A

straightforward application of X-11-ARIMA to this

hybrid series gives projected seasonal factors which are

heavily influenced by the universe data, but which are

applied to sample data. Distortions can occur, because

the two data sources historically have displayed

different seasonal patterns.

Beginning with January 1994 data, the U.S.

Bureau of Labor Statistics (BLS) implemented an

alternative method that separately adjusts each part of

the series, an approach first carried out by Berger and

Phillips (1993). The decision to implement the

alternative method, which we refer to as the two-step

method, was based on the evaluation reported in this

paper.

The major users of the employment statistics

include the Federal Reserve Board, the President's

Council of Economic Advisors, the Joint Economic

Committee of Congress, and various other policyoriented groups. Where economic statistics are used

as the basis for their policy analysis, it is important that

the preliminary estimates be accurate and that the

economic information found in these data be

discernible. Highly variable economic series make the

interpretation of such data difficult. Furthermore,

large annual revisions to the data may also impact the

validity of policy analysis conducted on the original

estimates, as Berger and Phillips (1994) suggest.

Our analysis of the seasonal adjustment of state

industry employment statistics compares the two-step

method with the combined method formerly used

(Shipp and Sullivan, 1992), i.e., a basic application of

X-11-ARIMA to the hybrid series. Our findings are:

● there are meaningful differences between universe

and sample seasonal patterns,

● the two-step method produces smoother seasonal

adjustments, and

● the two-step method results in smaller revisions to

the seasonally adjusted data and one- and 12-month

change estimates.

We feel these attributes improve the economic

interpretation of the data.

2. The Current Employment Statistics Survey

The time series from the Current Employment

Statistics survey combine available universe

employment data with ratio estimates of sample

employment. For the period for which the

Unemployment Insurance (UI) data are available, the

universe value is the time series value, AEt (t= 1, 2, ...,

T) where AEt is the all employees count in month t

and month T represents the latest benchmark. In the

post-benchmark period ( t > T) , for which only sample

data are available, a ratio of the sample count in the

current month divided by the sample count in the

previous month is multiplied by the previous month's

employment estimate. Only "matched" reporters are

used, i.e., a sample unit’s values are used in the ratio

only if it reports in the two adjacent months. For k>0,

AET+k = AET

.

j

k

=

∏

1

ae

ae

T j

T j

+

+ −1

where aeT+j = the sample employee count in month

T+j summed over all matched units. Each year, in the

annual benchmark process, the value of T increases by

12 months as universe values replace sample values..

Statistics Canada’s X-11-ARIMA program

(Dagum, 1980) is applied to state industry employment

series as follows:

● The 1980 version of the program is used, with the

automatic option for ARIMA extrapolation.

● Data are adjusted either additively or

multiplicatively, depending on which form has better

diagnostics.

● Prior adjustments are made for strikes and other

atypical employment-related activity.

● Series are adjusted directly at the major industry

division levels and indirectly for higher aggregates, by

adding seasonally adjusted components.

● Twelve-month-ahead projected seasonal factors are

used.
3. Seasonal Differences between Universe and

Sample

In order to compare seasonality in sample and

universe data, a time series from each source is needed.

A sample link Lt is computed as the ratio of the sample

estimate for month t divided by the sample estimate for

month t-1. To form a sample series on a common

benchmark, say AEt

, the links can be applied forward

and backward, starting with

AET-1 = AET/LT , AET+1 = AET⋅ LT+1

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Figure 1. Mean Seasonal Factors, Texas

99

99.5

100

100.5

101

Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Sample Universe

Figure 1 compares eight-year average seasonal

factors for Texas for pure sample and universe data.

There are noticeable differences in the factors, with the

most pronounced difference in January. When the

universe factor is more extreme, it will tend to overadjust a sample estimate. For January, the universe

factor, based on larger universe declines, will turn a

typical sample decline into an increase. The combined

method, as seen in Figure 2, estimates the change in

January, 1993 as over 90,000. Summing up change in

January across states yields a change over 600,000.

That most of this change is spurious is supported by

the national January change estimate of about 180,000,

derived in a different way from the same data.

Table 1 exhibits mean absolute differences in

seasonal factors for eight states. January differences

range from 0.29 in New York to 0.58 in Texas. Other

months with appreciable differences in several states

are February and September.

4. Analysis

To evaluate the alternative seasonal adjustment

methods, data are examined for the eight states listed

in Table 1. By including one state from each BLS

region, geographic diversity is achieved and any

region-specific benchmarking activities can be at least

partially examined. Revision and smoothness statistics

are used.

Revisions

One measure used for evaluation is mean absolute

revision, used in many seasonal adjustment papers,

e.g., McKenzie (1984). Revision in one-month per cent

change is defined as

R

At

f At

f

At

f

At

i At

i

At

i = − −

−

⋅ − − −

−

⋅

() ()

( ) %

() ()

( )

1 %

1

100 1

1

100

where At denotes the seasonally adjusted value in

month t, (i) the initial value, and (f) the final value.

Revisions are also computed for seasonally adjusted

values and for 12-month % change.

Figure 2. One-Month Change, Texas, 1993

(in 1000's)

-20

0

20

40

60

80

100

Jan 93 Mar May Jul Sep Nov

Combined Two-Step

Table 1. Mean Absolute Difference in Seasonal Factors

State Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

Colorado 0.52 0.58 0.25 0.33 0.27 0.35 0.41 0.54 0.43 0.06 0.08 0.35

Florida 0.36 0.22 0.02 0.14 0.02 0.19 0.22 0.15 0.17 0.07 0.14 0.24

Massachusetts 0.53 0.49 0.30 0.23 0.08 0.32 0.29 0.40 0.58 0.02 0.00 0.12

Michigan 0.52 0.33 0.22 0.20 0.09 0.11 0.20 0.32 0.31 0.18 0.14 0.08

New York 0.29 0.20 0.04 0.16 0.01 0.16 0.04 0.03 0.14 0.04 0.16 0.22

Oregon 0.44 0.49 0.35 0.20 0.10 0.07 0.15 0.24 0.37 0.19 0.28 0.33

Pennsylvania 0.30 0.31 0.06 0.04 0.08 0.11 0.16 0.16 0.28 0.12 0.03 0.03

Texas 0.58 0.40 0.04 0.01 0.15 0.30 0.12 0.11 0.36 0.02 0.05 0.19
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F igure 3. E xperimental Des ign

Jan85Jan86Jan87Jan88Jan89Jan90Jan91Jan92Jan93Jan94Jan95

Round 4

Round 3

Round 2

Round 1

AAA AAA

AAA

Universe AAA AAA

AAA

S ample

Sliding span statistics developed by Findley et al

(1988) are not necessarily applicable, since they

measure stability in seasonal factors, seasonally

adjusted values or seasonally adjusted change. The

two-step method, by definition, implies advantages for

changes between the initial and final values of the

seasonal factor for a given month. Going from sample

to universe data with benchmarking affects the level of

the series, as well as the seasonality, contributing to a

departure from stability in seasonally adjusted values.

Revision statistics are also subject to this latter

problem. Our initial thinking was that focusing on

one-month change revision might largely eliminate the

effects of changes in level, but the revisions in level are

not constant across months. This leads us to attempt

below a decomposition of total revision into seasonal

and level components. Finally, revision statistics focus

on behavior at the end of the series, where the

difficulties lie in this application.

Figure 3 shows four spans of data used as input to

seasonal adjustment, intended to follow current

practice for official seasonal adjustment. For Round 1,

the input data are the years 1985-90. We would go

back further, except that sample data are readily

available only back to 3/85. Having earlier universe

data, for the combined method we actually use universe

data from 1/84-3/90 and sample data for the remainder

of 1990. For the two-step method, universe seasonal

adjustment uses input data 1/84-3/90 and sample

seasonal adjustment uses data 3/85-12/90. Seasonal

factors for each method are applied to sample values

for the last nine months of 1990 and for all of 1991,

the projection period. These factors come entirely from

the sample seasonal adjustment for the two-step

method and, of course, from the one seasonal

adjustment based on the hybrid series for the combined

method. Each successive round appends an additional

year of universe data, and then extends the series

through the last nine months of the year with sample

data. The sample periods from Rounds 1 and 2 are

used as the evaluation periods, and values from Round

4 are used as the “final” values. While the latter are

not truly final for these evaluation periods, they should

be reasonably close to final, especially for the sample

period 4/90-12/91 in Round 1.

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Figure 4. Total Revisions in One-Month % Change, 

Texas

-1.5

-1

-0.5

0

0.5

Apr 90 Jul 90 Oct 90 Jan 91 Apr 91 Jul 91 Oct 91

Combined Two-Step

One-month change revisions for Texas appear in

Figure 4. The two-step method has a smaller absolute

revision in 1/91 and in some other months. Still, after

studying the results, we felt these statistics were not

fully satisfying. As mentioned above, benchmarking

changes the levels in the series, so the revisions in

level contribute to the overall revisions. To separate

revision effects, we introduce the concept of a

"corrected" observed sample value, corrected in the

sense of correcting the level to be the universe level.

OS

* = (OU / SU) ⋅ SS

Dividing the observed universe value for a particular

month by a universe seasonal factor gives a pure

seasonally adjusted value. Multiplying by a sample

seasonal factor yields an observed sample value, but

now at the universe level. These values are computed

for the period 4/90-12/92. We use the best seasonal

factors available for this correction, namely factors

from Round 4. In the simplest case, adding and

subtracting O* gives the kind of decomposition we

seek for a seasonally adjusted value. We have

R = A4 - A1

 = (A4- O

S

*

1

) + ( ) * O O

S

− 1

1

 = RS + RL

the terms on the right being the seasonal revision and

the revision in level. Similarly, O* can be used to

obtain decompositions for one- and 12-month change.

We call the seasonal revision component the “adjusted”

revision, and focus most of our analysis on it. Figure 5

shows adjusted revisions for Round 1 for Texas. Now,

revisions are uniformly lower with the two-step

method. While some of the strongest results are for

Texas, the two-step method gives an improvement in

all eight states. Table 2 shows mean absolute adjusted
revisions for Round 1. In most states, the statistics are

about half or less with the two-step method. The table

shows that total revisions are also lower on average

(but to a lesser degree) in all states. Similar to results

for one-month change, the two-step method reduces

revisions to 12-month change estimates. Section 5

contains more on 12-month change.

Table 2. Mean Absolute Revisions to

One-Month % Change

 Total Adjusted

 State Combined TwoStepCombined TwoStep

Colorado 0.33 0.18 0.22 0.08

Florida 0.29 0.22 0.16 0.04

Massachusetts 0.33 0.21 0.19 0.10

Michigan 0.32 0.25 0.22 0.13

New York 0.17 0.15 0.10 0.06

Oregon 0.27 0.23 0.16 0.11

Pennsylvania 0.16 0.14 0.13 0.07

Texas 0.30 0.19 0.15 0.03

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Figure 5. Adjusted Revisions in One-Month % 

Change, Texas

-0.5

-0.3

-0.1

0.1

0.3

Apr 90 Jul 90 Oct 90 Jan 91 Apr 91 Jul 91 Oct 91

Combined Two-Step

One check on the revisions decomposition has

been carried out. Given that the level revision is not

intrinsically linked to a seasonal adjustment method,

this revision component has been compared for the two

methods. For Rounds 1 and 2, estimates for mean

absolute % revision in level, which ranges from .15%

to .23% across the states, never differ more than .01%

for the two methods. These positive results support the

use of the adjusted revisions.

Smoothness

 Application of the two-step method to the Texas data

in Figure 2 shows a marked improvement in the

smoothness of the seasonally adjusted series. The large

upward spike (90,900 employees) in the December

1992-January 1993 one-month change under the

combined method has been replaced by a less

significant level change under the two-step method

(2,300). More stability is seen in Figure 6 for onemonth % change in Texas in Round 1.

The departure from smoothness of the series,

measured by the sum of squared first differences, is

much less in the two-step method than the combined

method. Table 3 below contains the ratio x 100 (twostep divided by combined) for the eight states. The

relative smoothness was computed for the latter part of

the series for Rounds 1 and 2.

Table 3. Relative Smoothness of the

 Two-Step Method

State Round 1 Round 2

Colorado 61 94

Florida 82 73

Massachusetts 88 70

Michigan 91 104

New York 120 92

Oregon 79 79

Pennsylvania 88 94

Texas 65 65

5. Analysis of Twelve-Month Change

Recently, BLS began using seasonally adjusted

data for estimating 12-month change in all its

employment and unemployment statistics. Unadjusted

estimates at first seem natural and preferable for at

least two reasons:

● “by definition,” a 12-month change should

contain no seasonality;

● as an imperfect process, seasonal adjustment

may introduce error into the estimate.

Arguments for using adjusted estimates include:

● their use is consistent with the use of

seasonally adjusted values for other comparisons

across months;

● in presence of moving seasonality, unadjusted

estimates will contain residual seasonality.

In most situations, the differences are small. In this

setting, we can expect that the two-step method will

perform better, since residual seasonality from

unadjusted estimates is likely for certain months. For

Round 1, 12-month change estimates for 4/90-3/91 will

involve subtracting a universe value from a sample

value. For 3/90 and earlier months, the subtraction

involves two universe values; from 4/91 on, it involves

two sample values. Figure 7 shows Round 1 results for

1990-91; clearly, the two-step method is smoother.

Table 4 gives mean absolute revisions for the eight

states for the 12-month period 4/90-3/91 for Round 1.

For all states except
Michigan, the statistic is at least twice as large with

unadjusted estimates.

For Texas, January has adjusted revisions -.82%

and -.84%, the largest revisions for the unadjusted

data, compared to -.28% and -.02% with the two-step

method. The latter method has only three revisions out

of the 24 larger than 0.15%, while 17 revisions exceed

.15% for the unadjusted estimates. Another way of

expressing this result is that, with the two-step method,

the initial rounded estimate is rarely off by more than

0.1% while the unadjusted estimate differs by .2% or

more two-thirds of the time.

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Figure 6. One-Month % Change, Texas, 

Round 1

-0.4

-0.2

0

0.2

0.4

0.6

Jan

90

Apr

90

Jul

90

Oct

90

Jan

91

Apr

91

Jul

91

Oct

91

Combined Two-Step

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F igure 7. 12-Month Change, R ound 1

 (in 1000's)

50

90

130

170

210

250

Jan

90

Apr

90

Jul

90

Oct

90

Jan

91

Apr

91

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91

Oct

91

Observed T wo- S tep

Table 4. Mean Absolute Revisions to 12-Month

% Employment Change

Total Adjusted

State Unadjusted TwoStepUnadjusted TwoStep

Colorado 0.39 0.18 0.28 0.12

Florida 1.01 1.01 0.18 0.06

Massachusetts 0.54 0.31 0.31 0.08

Michigan 0.50 0.36 0.29 0.16

New York 0.32 0.26 0.13 0.05

Oregon 0.82 0.62 0.28 0.09

Pennsylvania 0.50 0.39 0.15 0.06

Texas 1.10 1.06 0.29 0.13

6. Issues and Limitations

With the two-step method, there is always one

calendar year with a mixture of universe and sample

seasonal factors. For Round 1, 1990 has universe

factors for the first three months and sample factors for

the rest of the year. Even with a standard application

of X-11, the seasonal factors need not average to 100,

unless that X-11 program option is selected, but, with

independent seasonal adjustments, the differences may

be greater. To check whether the two-step method

appreciably affects the level of the series, adjusted and

unadjusted monthly employment averages are

compared in Table 5.

Table 5. Unadjusted and Seasonally Adjusted

Monthly Averages Round 1, 1990 (in 1000’s)

State Unadjusted Combined Two-Step

Colorado 1517.8 1517.7 1519.4

Florida 5416.0 5416.1 5417.7

Massachusetts 2983.7 2983.7 2986.5

Michigan 3954.3 3954.6 3957.5

New York 8205.4 8206.4 8209.0

Oregon 1246.0 1245.9 1247.1

Pennsylvania 5171.2 5171.2 5175.2

Texas 7030.2 7030.0 7034.4

For both rounds, the difference from the unadjusted

average is less than 1,000 for the combined method,

while it always exceeds 1,000 for the two-step method.

For the two-step method, the maximum per cent

difference is .10% in Round 1 and .15% in Round 2.

The amounts for Texas are 4,300 and 9,800, or .06%

and .14%, respectively. Furthermore, the difference is

always in the same direction, with the two-step average

being higher, so there can possibly be some

accumulation across the states. For some detailed

industries, the differences are larger. Still, with

differences in the neighborhood of .10%, the

discrepancies for total employment are fairly small,

and they occur only in the just-concluded year, not the

year using projected factors.

A second question is whether comparisons across

the seam month, March for our data, are distorted.

Berger & Phillips (1994) present another version of the

two-step method intended to avoid such occurrences.

For t>T, the last seam month, they compute an

adjusted seasonal factor

SS

*(t)=k.

SS(t), with k=SU(T)/SS(T),

where SS and SU denote sample and universe seasonal

factors. This has the appealing property that successive

ratios of seasonal factors are natural ratios, i.e., ratios

of seasonal factors from one seasonal adjustment. On

the other hand, the factor k will consistently move the
seasonally adjusted series up or down, according as

k<1 or k>1. Figure 7 for Texas illustrates the bias in

the Berger-Phillips version of the two-step method.

Visually, the BLS values appear to “go through” the

data, while the Berger-Phillips values nearly all lie

above the unadjusted values. For all the months where

the Berger-Phillips values are above the unadjusted

values and the BLS values are below, the seasonal

factor has changed directions, i.e., moved from above

100 to below.

Examining monthly averages as above, their

variant gives differences over .50%; moreover, these

differences occur through both the previous year and

the projection year. We feel that the potential for

distortions across the seam deserves more study, but

that the bias in the Berger-Phillips formula is

unacceptable.

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Figure 8. Unadjusted and Seasonally Adjusted 

Employment, Round 1, Texas

7025

7125

7225

7325

Jan

91

Apr

91

Jul

91

Oct

91

Jan

92

Apr

92

Jul

92

Oct

92

Observed

BLS

FRB

Care is required in application of the two-step

method, due to state variations in data handling. In

our eight-state experiment, universe replacement was

controlled with the seam in March. With speedier

processing of universe data, most states now can use

universe data through June; others are able to make

September the seam month. Other special situations

arise which require monitoring and communication

with the states.

A different approach to the overall problem is to

predict universe values from sample values. Stamas,

Kratzke and Mueller (1993), for instance, formulate a

structural time series model for this purpose. This

would provide a more consistent seasonal pattern, and

eliminate the need for a two-step method. No

definitive results have been obtained yet. Which

seasonal pattern is more accurate is not known, but

that is not an issue with the two-step method.

A more fundamental solution is to examine

measurement error and coverage issues. A large

response analysis survey now in progress may shed

some light on error sources. The CES survey is

presently embarking on an effort to put in place a

sustainable probability design. Over time, projects

investigating both data sources can lead to program

improvements eliminating systematic differences.

Given the size of both programs, this will require many

years and substantial resources.

7. Conclusions

The important economic time series presented in

this paper are a hybrid of two data sources with

different characteristics, including different seasonal

patterns. The two-step seasonal adjustment method,

implemented in 1994, improves smoothness and

reduces revisions in seasonally adjusted statistics. In

particular, it largely eliminates spurious jumps in

January. The CES state program provides an example

where seasonally adjusted estimates of 12-month

change perform better than unadjusted estimates. The

two-step method is a short-term solution for the CES

state program. The preferred long-term solution is to

eliminate the systematic differences between the data

sources.

References

Berger, Franklin D. and Keith R. Phillips (1993),

"Reassessing Texas Employment Growth," The

Southwest Economy, July/August, Federal Reserve

Bank of Dallas, pp. 1-3

__________________ (1994), "The Disappearing

January Blip and Other State Employment

Mysteries," Federal Reserve Bank of Dallas Working

Paper 94-03

Dagum, Estela Bee (1980), The X-11-ARIMA Seasonal

Adjustment Method, Catalogue 12-546E, Statistics

Canada

Findley, David F., Brian Monsell, Mark Otto, William

Bell, and Marian Pugh (1988), “Towards X-12-

ARIMA,” Proceedings of the Bureau of the Census

Annual Research Conference, 591-622

McKenzie, Sandra K. (1984), “Concurrent Seasonal

Adjustment with Census X-11,” Journal of Business

& Economic Statistics 2, 235-249

Shipp, Kenneth and Thomas J. Sullivan, (1992),

"Using X-11-ARIMA to Seasonally Adjust Industry

Employment Data in the Current Employment

Statistics Program," Proceedings of the ASA Business

and Economic Statistics Section, 349-354

Stamas, George, Diem Tran Kratzke, and Kirk Mueller

(1993), “Improving Models Used to Estimate Bias in

Payroll Employment Estimate.” Proceedings of the

ASA Survey Methods Research Section, vol. 1, 498-

503.