ADJUSTING FOR A CALENDAR EFFECT IN EMPLOYMENT TIME SERIES

Stephanie Cano, Patricia Getz, Jurgen Kropf, Stuart Scott and George Stamas, Bureau of Labor

Statistics

George Stamas, BLS, 2 Massachusetts Ave. NE, Room 4985, Washington, DC 20212

Key Words: Regression, Seasonal Adjustment

1. INTRODUCTION

The Current Employment Statistics (CES) program

is a survey of nearly 400,000 business establishments

nationwide, which provides monthly estimates of

nonfarm payroll jobs and the hours and earnings of

workers. The month-to-month movements in these

series are closely followed by policy makers and

forecasters as timely indicators of the overall strength

and direction of the nation's economy. Over-the-month

changes in CES series are nearly always analyzed on a

seasonally adjusted basis, but historically a calendarrelated limitation in the CES seasonal adjustment

process complicated the interpretation of month-tomonth movements in the series. While the CES survey

is referenced to a consistent concept, the pay period

including the 12th of each month, inconsistencies arose

in the seasonal adjustment process because there are

sometimes 4 and sometimes 5 weeks between the

weeks including the 12th in a given pair of months. In

highly seasonal industries, this variation can be an

important determinant of the magnitude of seasonal

hires or layoffs that have occurred at the time the

survey is taken, thereby complicating seasonal

adjustment.

 Standard seasonal adjustment methodology relies

heavily on the experience of the most recent three years

to determine the expected seasonal change in

employment for each month of the current year.

Because the previous methodology did not distinguish

between 4 and 5 week survey intervals, the accuracy of

the seasonal expectation depended on how well the

current year's survey interval corresponded with those

from the previous three years. All else the same, the

greatest potential for distortion when applying

projected factors occurred when the current month

being estimated had a 5-week interval but the three

years preceding it were all 4-week intervals, or

conversely, when the current month had a 4-week

interval but the three years preceding it were all 5-

week intervals. In the first case, for a month in which

seasonal growth typically occurs, the seasonal

expectation was too low and the seasonal employment

gain for the current month often exceeded the

expectation, thus exaggerating the strength of job

growth. In the second case, again for a month in

which seasonal growth typically occurs, the seasonal

expectation was too high and the resultant employment

series risked presenting a weaker employment picture

than genuinely existed. A recent example of this latter

situation occurred with April 1995: There were 4

weeks between the March and April surveys in 1995,

while each of the three preceding years had 5-week

intervals. When the April 1995 survey registered a

very weak employment gain, concerns were raised as to

whether this calendar effect had significantly

dampened the published over-the-month change.

 Recent availability of X-12-ARIMA seasonal

adjustment software developed by the Bureau of the

Census enabled the Bureau of Labor Statistics (BLS) to

conduct detailed research into refining the CES

seasonal adjustment process to remove this calendar

effect. Initial research on a calendar effect in CES

series was conducted at the Federal Reserve Board.

Researchers there utilized time series modeling

techniques to identify and remove the varying interval

effect before seasonally adjusting CES series. BLS

adopted the basic method used by the Federal Reserve

Board researchers, using a technique known in X-12-

ARIMA software as REGARIMA modeling to identify

the estimated size and significance of the calendar

effect. REGARIMA modeling combines standard

regression analysis with ARIMA modeling. The

REGARIMA models evaluate the variation in

employment levels attributable to 11 separate survey

interval variables, one specified for each month except

March, which always has a 4-week reporting interval

from February.

Results obtained from a joint chi-square test that

examines statistical significance across the 11 monthly

variables, as well as those from t-tests on individual

coefficients, provide evidence supporting the presence

of this interval effect in most CES employment series.

As expected, controlling for the 4- vs. 5-week calendar

effect produced smoother seasonally adjusted series, as

exaggerated movements caused by the less precise

capture of seasonal components were reduced by the X12-based procedure. The interval modeling may
The views expressed here are those of the authors and do not necessarily reflect those of the Bureau of Labor

Statistics.

introduce a little instability, as measured by revisions

between estimates for a given month derived from

different overlapping spans.

2. MODELING AND DIAGNOSTICS

2.1 The Model

Consider the multiplicative time series decomposition Y = T ⋅ S ⋅ I ⋅ P, where P denotes a prior

adjustment factor, and has the decomposition

P = PT PS PI⋅ ⋅ . In our application, the interval

effect is month-specific, and is estimated as the factor

PS. For seasonal adjustment, it is combined with the

seasonal factor, and the seasonally adjusted value

becomes

Y

S P

T I P P

S

T I

⋅

= ⋅ ⋅ ⋅ .

As with other interventions and calendar effects, we

use extended ARIMA models to estimate the interval

effect. We write

log yt (, ) jt j M

jt j

X B B a

t

−∑a −∑b = Ψ

12

where yt is the observed series, the Mj's represent the

month variables, the Xj's represent outliers or other

interventions, at represents the noise term and Ψ

denotes a seasonal ARIMA model. In other words,

after accounting for certain identifiable effects, the

series follows an ARIMA model. On the log scale, the

interval effect of month j at time t is

− =











a −

j

Mjt Mjt ,

 1 t = j(mod 12),5-week month

6 t = j(mod 12), 4-week month

0, otherwise 

,

. ,

Notice that the interval adjustment is sometimes

positive and sometimes negative. Since there are more

instances of 4-week intervals, the factor -0.6 helps

achieve a balance in these effects. This is analogous to

the property that the mean of the seasonally adjusted

series should be close to the mean of the unadjusted

series.

2.2 Model Estimation

The model is estimated using the REGARIMA

procedure in X-12-ARIMA software. A REGARIMA

model is a regression model whose error component is

assumed to follow an ARIMA model. X-12-ARIMA

estimates the REGARIMA model by iterating on two

steps until convergence: A generalized least squares

(GLS) estimate of the regression coefficients is

estimated, and then an ARIMA model is fit to the

regression errors using exact maximum likelihood.

The ARIMA model provides the covariance structure

for the next GLS.

The automatic model selection process of X-12-

ARIMA does not both perform outlier detection and

test final model estimation for each model considered

for selection. To this end, a modified automatic

selection procedure was developed to identify models

for estimation with the interval effect. The

modification resulted in more and better model fits, as

well as increased model selection consistency across

three time spans (59 vs. 11). In addition, the revised

model selection procedure provided more consistency

in the identification of particular observations as

outliers in successive estimates.

This research is based on the modified approach,

which tests five ARIMA models: (011)(011)s,

(012)(011)s, (022)(011)s, (210)(011)s and (110)(011)s.

The method screens each series for additive outliers

and level shifts, testing at the 3.5 sigma level, and

estimates replacement values. We do not currently test

for ramp effects. Models are evaluated on normality of

residuals, forecast error, and evidence of overdifferencing, similar to the X-12-ARIMA automatic

model selection. The model best meeting the criteria is

selected with a preference for the model that had been

selected in the previous span. If none of the five

ARIMA models meets the criteria for selection,

analysts review the autocorrelation functions and

attempt to find a model that meets the criteria. If all

modeling attempts fail, no model is used for seasonal

adjustment. Of 75 employment series directly adjusted

for CES, 74 could be conveniently analyzed. Due to

time limitations, implementation required the use of X12-ARIMA’s automatic model selection procedure, and

not the modified approach used in the research

discussed here.

3. SLIDING SPAN RESULTS

3.1 Analysis Across Overlapping Spans

The 74 employment series were estimated with the

X-12-ARIMA software for three different spans with

the specified REGARIMA model. The programs

included prior adjustments, primarily for strikes, that

have been identified and are used in current production

estimates. The spans analyzed were January 1983--
March 1993, January 1984--March 1994, and January

1985--March 1995. Results were reviewed across the

three spans for model selection consistency, individual

t-statistics and a joint test of significance for including

the monthly interval effect variables, the difference in

estimated levels between each span, and the

smoothness of the time series.

3.2 Tests for Significance of the Interval Effect

Variable

Chi-square and t-statistics were observed for testing

the significance of the joint contribution and

coefficients for each of the 11 monthly interval variables. Of the 74 series fitted with models using the

explanatory variables, 59 had t-statistics equal to or

greater than 2 for the last span estimated (January

1985--March 1995); 46 series had at least one month

with a t-statistic greater than 2 for all three spans. This

t-value corresponds roughly to a .05 test of

significance. These results are summarized by industry

division in Table 1.

Table 1. A summary of test statistics for inclusion of explanatory

variables across three spans.

Industry Number

of series

t>2

across

3 spans

Joint p

-value

< .10

TOTAL 74 46 41

TOTAL PRIVATE 68 40 36

Mining 4 2 2

Construction 3 2 2

Manufacturing 20 12 9

Transportation &

Utilities

9 5 6

Wholesale Trade 2 0 0

Retail Trade 8 6 6

Finance, Insurance &

Real Estate

7 4 2

Services* 15 9 9

Government 6 6 5**

* One series in services was not modeled with explanatory

variables.

** Two government series include an extra intervention variable

accounting for nonseasonal employment events.

Each month had an interval variable with a t-value

greater than 2 across all three spans in at least one of

the 74 industries. Those series with t-values of 2 or

more in less than three spans generally estimated the

interval effect in the same direction across all three

spans. While many of the series show a significant

interval effect in at least one month, some present an

inconsistent picture across spans. Notably, in SIC 15,

building construction, January has large t-values in the

first and last span but a small value in the second span.

Table 1 also provides results on the joint test for the

inclusion of all 11 interval variables given a specific

ARIMA component. For a total of 51 series, p-values

generated from the third span indicate rejecting the

hypothesis that the variables have no effect at the .10

level of significance. Across all spans, 41 series had

consistent rejecting p-values.

Individual and joint tests for inclusion of the

monthly regression variables suggest that about half of

the series should be corrected for the interval effect.

These tests are consistent for most series. Of the 41

series with a joint test significant at the .10 level across

all three spans, 34 had at least one month with a

significant t-value across all three spans. Given the

constraints of production under automated systems, we

preferred to treat all series identically, i.e., the

correction for the interval effect in all months.

Modeling results suggest that this may be feasible.

3.3 Smoothness

Individual series and aggregates of those series were

examined for evidence that the interval adjustment

actually represented an improvement in portraying the

underlying economic trend when compared to seasonal

adjustment without the interval adjustment. This was

done with calculations of revisions and smoothness

statistics. The smoothness of each series for each span

was calculated as the square root of the sum of the

squared first differences. Ratios were formed by

dividing the smoothness statistic with interval

adjustment by the statistic without interval adjustment,

so that a ratio less than 100% tells us that the adjusted

series is smoother. The distribution of 2-digit SICs

across ranges of smoothness ratios shown below

suggests that interval modeling helps reveal the

underlying trend in two-thirds to three-fourths of the

series, depending on the span, and rarely causes a

serious distortion of the series.

Table 2. Frequencies of 2-digit SIC Smoothness Ratios (%)

Spans

1st 2nd 3rd

<90 12 10 8

90-100 36 44 48

100-110 25 17 16

> 110 1 3 2

Smoothness statistics across the three estimation

spans are presented by division in Table 3. Evidence

supporting adjustment for the varying interval effect is

not equally compelling for all series, but with few

exceptions, the adjustment did not appear to do harm.

It was decided that the adjustment should be made to

all the series except for seven. Three of these are the

construction industries, one is in transportation and

public utilities, and three are in services. With the
exception of construction, the series are as smooth or

smoother with the adjustment for the interval effect.

The overall results illustrated in total nonfarm and

total private indicate a small difference in smoothness

in favor of removing the interval effect. The difference

in smoothness is greatest in retail trade, where 4 of 8

series appear to be substantially smoother across all

three spans. If the adjustment is made in only 68 of

the series the resulting aggregates are slightly

smoother. The improvement to totals results primarily

from the decision not to adjust construction series.

Table 3. Ratio of square roots of the sum of squared first

differences, the series with the interval effect removed to the

series with the effect present, times 100, by division

All Series Adjusted 68 Series Adjusted

Series/Span 1 2 3 1 2 3

Total 98.2 98.2 97.7 97.5 97.4 97.2

Private 98.0 98.6 97.9 97.3 97.6 97.3

 Mining 99.1 98.4 97.6 99.0 98.4 97.6

 Construction 104.3 107.5 106.8 100.0 100.0 100.0

 Manufacturing 97.8 99.1 97.9 97.8 99.1 97.9

 Transp. & Utils. 97.7 99.5 100.7 98.2 99.7 99.9

 Wholesale 99.0 99.8 100.4 99.0 99.8 100.4

 Retail 87.6 87.9 86.4 87.6 87.9 86.4

 Fin. Ins.& RE 96.7 97.7 99.8 96.7 97.7 99.8

 Services 98.8 99.1 99.3 98.7 99.0 99.1

Government 96.8 94.7 93.3 96.8 94.7 93.3

3.4 Revisions

Revisions in estimates of level and over-the-month

change were calculated across the three time spans in a

“sliding span” analysis for series with the adjustment

and without the adjustment. Median, 85th percentile

and maximum revisions were observed. Sliding spans

provide three estimates for each month from January

1985--March 1993 for most series and January 1988--

March 1993 for other series. At detailed and aggregate

levels, the evaluation of revisions was not positive

when comparing adjustments to all series to seasonal

adjustment without the interval adjustment. While the

series without adjustment for the interval effect appear

to be more stable, the differences are under .01

percentage points for total and total, private

employment series. Revision results could be a

symptom of estimating the interval effect with few

observations. Table 4 presents the median, 85th

percentile and maximum revision, expressed in percent, between spans in estimates of levels over all of

the possible months for each series.

 Table 5 presents median, 85th percentile and

maximum difference in percentage over-the-month

changes estimated for each of the different spans.

Again for total and total, private, differences in the

revisions between span estimates look similar whether

the interval effect is modeled or not. Series adjusted

for the effect tend to have larger revisions, but not

much larger.

Tables 4 and 5 show that excluding seven series

from interval effect modeling does very little to the

revisions in affected aggregate series, with the

exception of construction.

4. EFFECTS ON MONTHLY EMPLOYMENT ANALYSIS

At the total nonfarm level the difference in over-themonth change between the currently published

seasonally adjusted series and the X-12 based

experimental series ranged from zero up to ±100,000;

about two-thirds of the differences were in the 20,000

to 70,000 range. (See Table 6.) The more significant

impacts generally occurred in highly seasonal months

and industries, i.e., the ones in which having one more

week translated into a significantly larger number of

hirings or layoffs than expected. Effects were

especially concentrated in July and August for

manufacturing and local government, and in December

and January for retail trade.

Another way to view the improvement from using

the new methodology is comparison of the over-themonth changes reported for a given month under the

old and new methods, for cases when the current

month has a different survey interval from the three

years immediately preceding it. April poses such a

case in 1995; it had a 4-week interval while the Aprils

of 1992 - 1994 all had 5-week intervals. In this case,

the previous method produced a seasonal expectation

which was too high for the 4-week situation that

existed in April 1995. Because April is a strong

seasonal upturn month, the potential for distortion was

relatively high. The published seasonally adjusted

over-the-month change based on the previous

procedure was +8,000. With the new method the

estimated over-the-month change would have been

+97,000, a better representation of the underlying

economic movement.

Another recent example occurred with November

1995 which had a 5-week interval but was preceded by

three Novembers with 4-week intervals. At the total

nonfarm level, November exhibits a slightly upward

seasonal expectation; the previous method produced an

expectation of a smaller seasonal increase than

appropriate for a 5 week interval. When November

1995 employment came in stronger than the 4-week

dominated factors expected, the job gain estimated for

the month, 212,000, was somewhat exaggerated.

Under the new method, the over-the-month job gain

was estimated at 167,000.
 Like all time series techniques including seasonal

adjustment itself, the 4-5 week survey interval effect

modeling assumes a predictable continuation of

historical patterns and relationships. Thus an event

like the onset of a recession after a prolonged period of

growth can be problematic for both projected seasonal

factors and interval effect factors. For example, when

general merchandise stores (SIC 53) is modeled for the

interval effect, an extra (5th) week in December is

currently associated with a larger increase in hiring

than a 4-week interval. If this industry went into

decline, the 5th week might actually translate into an

opposite effect, more layoffs and a larger net

employment decline. In this case, the modeled interval

effect would be in the wrong direction. A comparison

of series that result from applying projected factors of

each method to estimates of employment through the

July 1990--March 1991 recession suggests that this is a

problem with or without modeling the calendar effect.

The issue deserves further research.

Table 4. Quantiles for percent revisions in estimates of levels under 3 modeling approaches

All Series Adjusted 68 Series Adjusted No Effect Adjustment

Series Median 85%ile Max Median 85%ile Max Median 85%ile Max

Total 0.014 0.036 0.068 0.014 0.031 0.072 0.012 0.024 0.065

Private 0.017 0.038 0.077 0.017 0.039 0.082 0.014 0.034 0.071

 Mining 0.089 0.194 0.362 0.089 0.194 0.362 0.047 0.176 0.440

 Construction 0.107 0.246 0.586 0.068 0.224 0.723 0.068 0.224 0.723

 Manufacturing 0.034 0.065 0.184 0.034 0.065 0.184 0.023 0.077 0.194

 Transp. & Utils. 0.052 0.087 0.148 0.047 0.096 0.170 0.021 0.060 0.178

 Wholesale 0.021 0.040 0.129 0.021 0.040 0.129 0.014 0.032 0.089

 Retail 0.047 0.091 0.165 0.047 0.091 0.165 0.028 0.069 0.130

 Fin. Ins.& RE 0.025 0.054 0.100 0.025 0.054 0.100 0.019 0.041 0.082

 Services 0.031 0.066 0.126 0.028 0.064 0.134 0.024 0.048 0.115

Government 0.045 0.084 0.137 0.045 0.084 0.137 0.031 0.074 0.137

Table 5. Quantiles for percent revisions in estimates of over-the-month changes under 3 modeling approaches

All Series Adjusted 68 Series Adjusted No Effect Adjustment

Series Median 85%ile Max Median 85%ile Max Median 85%ile Max

Total 0.019 0.035 0.072 0.020 0.032 0.059 0.019 0.035 0.096

Private 0.024 0.039 0.092 0.023 0.039 0.076 0.018 0.038 0.112

 Mining 0.103 0.162 0.280 0.103 0.162 0.280 0.060 0.159 0.310

 Construction 0.158 0.295 0.625 0.070 0.250 0.614 0.070 0.250 0.614

 Manufacturing 0.038 0.071 0.112 0.038 0.071 0.112 0.025 0.058 0.121

 Transp. & Utils. 0.047 0.098 0.172 0.046 0.106 0.178 0.024 0.068 0.184

 Wholesale 0.026 0.043 0.161 0.026 0.043 0.161 0.017 0.040 0.091

 Retail 0.045 0.080 0.240 0.045 0.080 0.240 0.031 0.060 0.197

 Fin. Ins.& RE 0.017 0.042 0.137 0.017 0.042 0.137 0.015 0.030 0.086

 Services 0.035 0.068 0.172 0.034 0.067 0.166 0.031 0.060 0.112

Government 0.048 0.095 0.185 0.048 0.095 0.185 0.030 0.073 0.141

5. CONCLUSIONS

The previous CES seasonal adjustment had a

calendar-related limitation which could affect the

accuracy of published over-the-month changes in

employment. Recently conducted research at BLS has

resulted in the development of a more refined seasonal

adjustment process which will mitigate the effects of
varying time intervals between surveys and provide

improved measurement of underlying economic trends.

REGARIMA modeling techniques are used to account

for the interval effect with month-specific explanatory

variables. Smoothness statistics, t-statistics and joint

tests for inclusion of the explanatory variables provide

supporting evidence for the majority of published

employment series. BLS implemented the new X-12-

ARIMA-based seasonal adjustment procedures with

release of May 1996 data.

REFERENCES

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

adjustment Method, Catalogue 12-546E, Statistics

Canada

Findley, D. F., B. C. Monsell, W. R. Bell, M. C. Otto,

and B.-C. Chen (1996), New Capabilities and Methods

of the X-12-ARIMA Seasonal Adjustment Program,

JBES (forthcoming)

X-12-ARIMA Reference Manual, Beta Version 1.0

(1996) Bureau of the Census

Acknowledgments

We are grateful to Bill Cleveland and Bill Wascher of

the Federal Reserve for advice on methods and to

David Findley and Brian Monsell of the Bureau of the

Census for early use and support of X-12-ARIMA.

Table 6. CES Total NonFarm Employment (in thousands), Seasonally Adjusted Over-The-Month Changes,

experimental series (X-12 w/ interval effect model) vs. current published series

JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC

1988 experimental 443 276 217 238 *353 282 128 *253 276 277 *295

published 470 286 262 183 362 241 152 240 281 330 303

difference -27 -10 -45 55 -9 41 -24 13 -5 -53 -8

1989 experimental **312 *290 200 206 69 115 117 51 169 116 *267 138

published 343 230 190 170 89 146 68 87 174 66 322 107

difference -31 60 10 36 -20 -31 49 -36 -5 50 -55 31

1990 experimental 221 350 201 46 252 100 -151 *-182 -76 -157 -219 -99

published 282 320 190 5 285 134 -213 -140 -74 -190 -185 -133

difference -61 30 11 41 -33 -34 62 -42 -2 33 -34 34

1991 experimental -258 -230 -149 -191 *-37 **59 -50 67 15 11 -128 19

published -183 -268 -179 -215 29 16 -74 112 17 -17 -86 -8

difference -75 38 30 24 -66 43 24 -45 -2 28 -42 27

1992 experimental *-9 **-35 66 *156 187 32 *68 136 66 *146 **175 170

published -58 -3 94 191 143 30 134 69 75 208 95 176

difference 49 -32 -28 -35 44 2 -66 67 -9 -62 80 -6

1993 experimental 246 333 -54 288 296 149 235 215 *245 337 234 *258

published 203 362 -19 311 270 128 307 155 246 322 236 299

difference 43 -29 -35 -23 26 21 -72 60 -1 15 -2 -41

1994 experimental 194 331 473 270 303 *268 300 308 252 200 457 171

published 207 275 511 276 275 305 228 339 252 173 492 197

difference -13 56 -38 -6 28 -37 72 -31 0 27 -35 -26

1995 experimental 187 *318 167 **97 -107 252 114 226 100 66 *167 246

published 186 313 179 8 -62 299 28 263 94 68 212 145

difference 1 5 -12 89 -45 -47 86 -37 6 -2 -45 101

1996 experimental -186 653 109 22 5 week intervals are shaded

published -146 631 178 2 * = a 5-week interval preceded by three 4-week intervals

difference -40 22 -69 20 ** = a 4-week interval preceded by three 5-week intervals