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SELECTED TECHNIQUES
OF
SEASONAL ADJUSTMENT

33
F3cl

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Federal Reserve Bank of St. Louis

Seminar on

Seasonal Adjustment

Federal Reserve System
Washington, D C.

June 5-6, 1962

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n■ *
33

CONTENTS

Computational Steps of Selected Methods of Seasonal Adjustment

Bureau of the Census Seasonal Adjustment Technique (Method II)

X-9 Version of Census Method II

X-10 Version of Census Method II

Seasonal Adjustment Method of the Bureau of Labor Statistics

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Prepared By:
Research Department
Federal Reserve Bank of Atlanta

18469

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COMPUTATIONAL STEPS OF SELECTED METHODS OF SEASONAL ADJUSTMENT

The tables appearing on the following four pages were compiled
from published descriptions as follows:
Census Method II, Seasonal Adjustment on Electronic Com
puters, Organization for Economic Cooperation and
Development, pp. 389-398 and Electronic Computers and
Business Indicators, Occasional Paper 57, National
Bureau of Economic Research, Inc, pp. 248-257.

X-9 and X-10 Versions of Census II, mimeographed material
from Office of Chief Economic Statistician, Bureau of
the Census.
BLS Method, ”BLS Seasonal Factor Method,” Abe Rothman,
1960 Proceedings of the Business and Economic Sta
tistics .Section, American Statistical Association,
pp. 2-12.

Regression Method, ’’The Practice of Seasonal Adjustment
with Regression Equations,” Deutsche Bundesbank,
Frankfurt, October, 1960.

BASIC CALCULATION STEPS USED ST SEIECTED METHODS CP SEASONAL ADJUSTMENT

CENSUS METHOD II

X-9 VERSION OP CENSUS
METHOD II

Computation of Preliminary
Seasonally Adjusted Series

*1. Adjustment for trading
days is optional. If
used, daily averages
became original data,

*1, Steps 1-5 are the same
as Census Method II.

X-1O VERSION OP CENSUS
METHOD H

Computation of Preliminary
Seasonally Adjusted Series
*1, Steps 1-5 are the same
as Census Method II.

2. Compute ratio of ori
ginal to average of
preceding and following
months,

BUREAU OF LABOR
STATISTICS METHOD
First Iteration

*1, Develop a centered
12-month moving average
(MA) of original. Six
values at each end are
computed by a series of
steps. Preliminary estlmate of trend-cycle (TC),
*2, Compute ratio of original
to centered 12-month MA.
First approximation of
seasonal-irregular (SI),

*3. Develop an uncentered
12-month moving average
(MA) of original,

*3, For each calendar month,
compute a 5-term weighted
moving average (WMA) of
SI ratios in step 2. Un
forced seasonals, first
approximation,

*4, Center 12-month MA.
*5, Calculate ratio of ori
ginal to centered
12-month MA.

REGRESSION METHOD OP
DEUTSCHE BUNDESBANK
Basic Method

*1, Develop an uncentered
12-month MA of original
(a). This is used to
represent trend (t).

2, Basic analysis based on
following additive re
lationship:
an»tn + Pn + En
Original values-trend
values + seasonal com
ponent + residual com
ponent.
3, Other symbols:
a’-seasonal values (re
gression values)
a"«residual values (a-a>)
a*»seasonally adjusted
values.

*4, Force total to 1200,

*6, Identify extreme values
of step 5 and replace
with more representative
ones as follows:

*6. Omit step 6 in Census II
and substitute the
following:

a. Compute 5-term MA for
each month. To get MA
for first two years,
average the first two
ratios available, MA
for last two years are
obtained similarly.

a. Compute a 5-terni MA for
each month of data in
step 5*
get MA for
first two years, repeat
MA of third year, MA
for last two years are
obtained similarly.

b. For each month, compute
2-sigma control limits
about 5-term MA. All
ratios falling outside
limits are extreme.

Replace extremes as
follows: (1) Ratio
falling first of series,
average of first three
ratios; (2) Ratio falling
in middle, average ex
treme ratio and preceding
and following ones; (3)
Ratio falling at end,
average extreme and two
preceding ratios.

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c. Replace extremes as
follows: (1) Ratio
falling first of series
average of second, third,
and fourth ratios; (2)
Falling second, average
of first, third, and
fourth ratios; (3)
Falling middle, average
two preceding and two
following; (4) Falling
next to last or last,
similar to beginning.

*5, Compute seasonally ad
justed series,

*4. Standard regression equa
tion used for each month:.
a»«tB + A, where B slope;
A»Y intercept: ,

*6, Divide seasonally adjusted
series by 12-month MA of
original. First approxi
ft =
~
mation of irregular (I)
&
with some TC.
*5. With A and B values, com
pute seasonal values (a’)
Second Iteration
for each original values:
a’etB + A.
*7, Smooth!ratios in step6by
7 -month WMA after extending *6. Graphical checking of com
b. For each month, compute
putations, Using (t) on
2-sigma control limits
I for three months at each
the X axis and (a) on the
about 5-term MA. All
end. Measure of residual
TC. Multiply resulting
ratios falling outside
Y axis, plot the values
values by TC of step 1.
limits are extreme.
and the regression line.
Visual proof of the correct
*8. Compute ratio of original
c. Replace extremes as
computation of A and B.
follows: (1) Ratio
to TC of previous step.
falling first of
Second approximation of
*7. Compute residual values
(a”) by comparing re
series, average of
SI.
gression values (a*) with
second, third and
fourth ratios; (2)
original values (a):
*9. For each calendar month,
a"=a-a*. If a" greater
Falling second, aver
compute 5-term WMA of SI
age of first, third,
ratios. Unforced seathan 0, there is superseasonal present; if a"
and fourth ratios;
sonals, second approxi
(3) Falling middle,
mation,
is less than 0, there is
average two preceding
subseasonal; if 0, there is
*10, Force total to 1200,
and two following;
is only purely seasonal.
(4) Falling next to
last or last, similar
to beginning.
a. Compute a 5-termMAfor
each month of data in
step 5»
get MA for
first two years, repeat
MA of third year. MA
for last two years are
obtained similarly,

CENSUS METHOD II
*6* (Continued)

1-9 VERSION OP CENSUS
METHOD II
*6. (Continued)

X-1O VERSION OP CENSUS
- METHOD II

♦6

d. Six missing ratios
(due to step 4) at
beginning are supplied
by extending first
available ratios for
corresponding months
back to Initial month
of series. Six
missing at end suppli
ed similarly.

d. For each month, com
pute a 3-term MA.
Missing values suppli
ed for first year-aver
age first three ratios}
similar for end.

e. Force total to 1200.

e. Compute a centered
12-month MA. Missing
values—repeat first
available ratio six
times. Similar for
end. Divide Into
step 6d.

(Continued)
d. For each month, com
pute a 7-term HA of
ratios In step fie.
Missing ratios suppli
ed In first 3 years by
averaging first three
years available. Simi
lar for last years.
MA values computed by
using these estimates.
e.

For each month, com
pute a 3-term MA.
Missing values-use
value in step 6e
corresponding to the
month missing.

g. Six factors missing
at end (due to step 4)
are obtained by using
the factor for the
same month of the first
or last available year.
These are preliminary
seasonal factors.

f. Compute 3-term of 3f.
term MA of ratios In fie
for each month. Supply
missing values at each
end. The results are pre
liminary seasonal factors.

(Step 7 follows on the next page)

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BUREAU OF LABOR
STATISTICS METHOD
*11. Compute seasonally ad
justed series.

*12. Divide seasonally ad
justed series by TC of
step 7. Second approxi
mation of I.
Third Iteration

REGRESSION METHOD OP
DEUTSCHE BUNDESBANK

*8. Compute seasonally ad
justed values (a*) by
adding residual values
(a") to corresponding
trend values: a*«t + a”.
9. To decompose time series:
(seasonal) p-a’ - t;
(residual) E~a"«a - a’.

*13. Smooth I ratios In step
Refinement of Trend Trans
lation.
12 by 7-month WMA after
extending I for 3
*10. Test whether trend Is
For each month, compute
months at each end.
’’true," i.e., whether
the average, without
Multiply smoothed I by
TC in step 7. Result
seasonal fluctuations
regard to sign, of
Is final TC unless ex
year-to-year percent
around trend are dis
treme values are present.
changes In MA of step
torted.
fid.
*14. Compute ratio of ori
*11. If refinement Is
ginal to TC of previous
necessary, improved
trend values (tj) are
step. Third approxi
For each month, divide
step 6d Into step fic*
mation of SI.
obtained by smoothing
trend values (t):
Estimate of I.
♦15. For each calendar month,
compute 5-term WMA of
So
SI In step 14. Unforced
Where Ao and Bo are ori
seasonal, third approxi
ginal regression equa
mation.
For each month, com
tion coefficients.
pute average, without
*lfi. Force total to 1200.
regard to sign, of
*12. New Improved trend values
(ti) are then used as
These are final factors
year-to-year percent
changes In I.
unless extreme values
basis of a refined corre
are present.
lation between the trend
For each month, com
and original values.
pute ratio of step fig
*17. Compute seasonally ad
Calculation techniques
to step 6e. Designated
justed series.
for various values are
as Moving Seasonality
the same as in the Basic
*18. Divide seasonally ad
Ratios.
Method.
justed series by TC of
step 13. Final I unless
For each month, de
Regression Method
extreme values are pre
Concluded
pending upon siae of
ratio in step fih, MA of
sent.
ratios yielded by step
6c Is computed using
Fourth Iteration
term indicated in the
table at the end of the *19. Test for extreme values
Instructions. Missing
and, If found, replace
ratios supplied.
with substitute values.
Tests involve developing,
Compute a centered
smoothing and analysing
12-month MA of fil.
Irregular component to
Missing values—repeat
determine whether values
first available ratio
fall outside +2.8 sigma
six times. Similar for
limits. Replacements
end. Divide Into step
are calculated by multi
61.
plying TC by S for a given
month.

CENSUS METHOD II

X-9 VERSION OP CENSUS
METHOD II

X-10 VERSION OF CENSUS
METHOD II
*6. (Continued)

*7. Compute preliminary sea
sonally adjusted series.
Final Seasonally Adjusted
Series

*7. Same as Census II.
Final Seasonally Adjusted
Series

Fifth Iteration

k. For each month, com
pute a 3-term MA.
Missing values—use
value in step 6j corre
sponding to the month
missing.

Repeat basic steps 1-6 of
First Iteration using re
placements for extreme
values. This Iteration
develops first approximation
to time series components.

l. Six factors missing at
end (due to step 4) are
obtained by using the
factor for the same
month of the first or
last available year.
These are preliminary
seasonal factors.

Sixth Iteration

•7. Same as Census II

Final Seasonally Adjusted
Series

*8. Develop a 15-month WMA
of preliminary seasonally
adjusted series supplying
missing values.

*8. Same as Census II.

*9. Compute ratio of original
to 15-aonth WMA

*9. Same as Census II

*9. Same as Census II.

*10. Compute ratio of pre
liminary seasonally ad
justed series to its
15-month WMA.

*10. Same as Census II.

*10. Omit from Census II.

*11. Compute month-to-month
percent changes of
step 10 and average
without regard to sign.
Measures average ampli
tude of I.

*11. Same as Census II.

*11. Omit from Census II.

*8. Same as Census II.

Steps 7-12 of Second Itera
tion are repeated still
using replacement values as
original values.

Seventh Iteration
Steps 13-18 of Third Itera
tion are repeated. After
derivation of final measures,
replacement values are re
placed with original values
and final seasonally adjusted
series is derived.

Bureau of Labor
Statistics Method
Concluded

*12. Identify extreme values
*12. Omit step 12 in Census
in step 9 and replace
II and apply steps
6a-6c above to the re
in the same manner as
explained in steps
sults of step 10.
6a-6c above. Force total
total to 1200.

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BUREAU OF LABOR
STATISTICS METHOD

*12. Omit steps 12 and 13 in
Census II and substitute
as explained above in
steps 6a-6k using the
results in step 9 above.
These are final seasonal
factors.

- 4 -

X-10 VERSION OP CENSUS
METHOD II

X-9 VERSION OF CENSUS METHOD II

CENSUS METHOD II
*13. Final seasonal factors are derived as follows: If
irregular in step 11 averages under 2, use a 3-term MA
of a 3-term MAj If I is 2 or more, use a 3-term MA of a
5-term MA. Missing values at each end are supplied.

*13. Pinal seasonal factors are derived as follows: If I in *13. See step 12.
step 11 averages under 2, use 3-term MA; if I is 2 or.
more, compute a 5-terra MA. Missing values at each end
are supplied. Then perform steps be and 6f. These are
final seasonal factors.

*14. Project seasonals in step 13 for year ahead on basis of
the seasonal factors for the last two years.

*14. The remaining steps are identical to the steps in
Census II.

*14. The remaining steps are
Identical to the steps
in Census II.

13. Compute seasonally adjusted series.
16. Compute ratio of final seasonally adjusted series to
average of preceding and following month as test of
residual seasonal.

TERM OP MOVING AVERAGE FOR DIFFERENT
SEASONALITY RATIOS IN X-10. SEE STEP 6i.

17. Develop an uncentered 12-month MA of seasonally ad
justed series.
18. Compute ratio of uncentered 12-month MA of final series
te similar average of original series to provide test
for bias.
19. Calculate ratio of each month to the preceding January
in final series as test for residual seasonal of more
than a month’s duration.

Measures of Irregular (I), Cyclical (C), Seasonal (S)

Moving Seasonality
Ratio step 6h

0-1.49
1.50- 2.49
2.50- 4.49
4.50- 6.49
6.50- 8.49
8.50 and over

Average of 6 c Values

None (Leave 6c values
unchanged)
3-term moving average
5-term moving average
9-term moving average
15-tena moving average
Arithmetic average of
all 60 values

Number of beginning or
ending 6 c values averaged
to extend MA

2
2
3
3

20. Compute 15-month WMA of final series-yields (C).
21. Compute month-to-month percentage changes in original
(0), seasonal factors (S), final seasonally adjusted
series (CI), cyclical (C), and ratio of original to
12-month WMA.

An asterisk (♦) identifies a step involved in derivation of seasonal factors.

22. Compute ratio of final series to 15-month WMA of final
Series. Yields estimate of I. Calculate month-to-month
percentage changes in I.
23. Derive mean of percentage changes in original fo), irre
gular Cl). cyclical (C), seasonal C5), and seasonally
adjusted (Cl).
24. Uslng_averages in step 23, calculate: "I/C, I/T,

Vo; c/o; S/0".

~S/C~,

25. Compute ratio of VS”with percentage changes taken 2,
3, 4, and 5 months apart. The interval corresponding to
the last
ratio that is less than 1.00 is the "number
of months for cyclical dominance." Calculate MA of final
series, using this number as its period.

26. Derive average duration of run for CI, I, C, and CI,
smoothed in step 25 •
27. Compute, without regard to sign, ratio of 12-month MA of

month-to-month percent changes in I to 12-month MA of
month-to-month percent changes in C.
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RESEARCH DEPARTMENT
FEDERAL RESERVE BANK OF ATLANTA
MAY, 1962

BUREAU OF THE CENSUS SEASONAL
ADJUSTMENT TECHNIQUE*
(METHOD II)

Computation of Preliminary Seasonally
Adjusted Series

Original observations. Where an adjustment for the number of working
or trading days is made, these figures are shown after adjustment and
all subsequent computations are based on these adjusted figures (Table
I of sanple "print-out”).

Ratios of the original observations for each month to the average of
the original observations for the preceding and following months are
computed. Arithmetic means of these ratios for each month are given
at the bottom of the table (Table 2).

A twelve-month moving average of the original series is computed. This
curve provides a measure of the trend-cycle component of the series.
It also provides annual averages of the original series (Table 3).

The twelve-month moving average is centered—that is, a two-month
moving average of the twelve-month moving average is computed.
This
operation places the moving average values at mid-months.
The first
value of the centered moving average is placed at the seventh month of
the original series. Thus six moving average values will be missing
at the beginning and at the end of the series (Table 4).

Ratios of the original observations to the centered twelve-month moving
average are computed. This computation results in a series which shows
primarily the seasonal and irregular components of the original series
(Table 5).

This step will provide a method for identifying extreme items among the
ratios computed by step 5, substituting more representative ratios for
these extreme ratios and fitting smooth curves to all ratios for each
month.

Fit a five-term moving average to the ratios for each month.
This results in the loss of moving average values for the
first two and the last two years for which ratios are avail
able. To obtain moving averages for the first two years,
use the average of the first two ratios as the estimated
value of the ratio for each of the two years preceding the
first year available. This is equivalent to weighting the
first three years' ratios by 2/5, 2/5, and 1/5, respectively,

*"A Description of the United States Bureau of the Census Method of Adjustment
of Series of Monthly Data for Seasonal Variations," Seasonal Adjustment o
Electronic Computers, pp. 391-398. This description is the same as cone
tained in Electronic Computers and Business Indicators, Occasional Paper
No. 57, National Bureau of Economic Research, 1957, pp. 248-257.

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- 2 (CM II)

to obtain the first year’s moving average value, and to
weighting the first four years’ ratios by 3/10, 2/10,
2/10, and 2/10, respectively, to obtain the second year’s
moving average value. Moving average values for the last
two years are obtained in a similar manner.
b.

For each month, compute two-sigma control limits about
the five-term moving average line. All ratios falling
outside these limits are designated as extreme.

Replace extreme ratios as follows: for an extreme ratio
falling at the first point in the series, substitute the
average of the first three ratios of the series; for an
extreme ratio falling in the middle of the series, sub
stitute the average of the extreme ratio and the pre
ceding and following ratios; for an extreme ratio falling
at the end of the series, substitute the average of the
extreme ratio and the two preceding ratios.

The six missing ratios at
supplied by extending the
corresponding months back
The six missing ratios at

For each year, center the twelve ratios (i.e., adjust the
twelve ratios so that their sum will be 1,200) by division
of the twelve items by their arithmetic mean. If the ini
tial year is incomplete, use as the ratio for any missing
month the value of the average ratio for the same month in
the next two years in centering the initial year's ratios.
Treat the terminal year’s ratios in a similar manner.

For each month, compute a three-term moving average of a
three-term moving average of the centered ratios yielded
by step 6e, above. This will result in the loss of two
moving average values at the beginning and two at the end.
To obtain the values missing at the beginning, use the
average of the first two centered ratios as the estimated
value of the centered ratio for each of the two years pre
ceding the first year available. This is equivalent to
weighting the first three years’ centered ratios by 9/18,
7/18, and 2/18, respectively, to obtain the first year’s
moving average value, and to weighting the first four
years’ centered ratios by 5/18, 7/18, 4/18, and 2/18, re
spectively, to obtain the second year’s moving average
value. The missing values at the end are obtained in a
similar way. The values of these twelve curves constitute
the preliminary seasonal adjustment factors (Table 6).

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the beginning of the series are
first available ratios for the
to the initial month of the series.
the end are supplied similarly.

- 3 (CM II)

These seasonal factors are divided into the corresponding figures of the
original series, month by month; i.e., the seasonal factor for January,
1947, is divided into the original observation for January, 1947; the
factor for January, 1948, is divided into the original observation for
January, 1948. Similarly, the factor for February, 1947, is divided in
to the original observation for February, 1947; the factor for February,
1948, into the original observation for February, 1948; and so on. This
yields the preliminary seasonally adjusted series (Table 7).

II,

Computation of Final Sesonally
Adjusted Series

Compute a weighted fifteen-month moving average (Spencer’s fifteen-term
formula) of the preliminary seasonally adjusted series. The weights are
as follows: -3/320, -6/320, -5/320, 3/320, 21/320, 46/320, 67/320,
74/320, 67/320, 46/320, 21/320, 3/320, -5/320, -6/320, -3/320. This is
equivalent to a weighted five-month moving average (weights are -3/4,
3/4, 1, 3/4, -3/4) of a five-month moving average, of a four month moving
average, of a four-month moving average of the data.
To obtain values for the beginning points of this curve, use the aver
age of the first four values of the preliminary seasonally adjusted series
as the estimated value of this series for each of the seven months pre
ceding the first month available. The values for the end are supplied
similarly.
The preliminary seasonally adjusted series contains the cyclical,
trend, and irregular components of the series with only a trace of the
seasonal component. The weighted fifteen-month moving average can be
used in place of a twelve-month moving average because there is no signi
ficant seasonal factor to suppress. The weighted fifteen-month moving
average is much more flexible than a twleve-month moving average and will,
therefore, provide a better measure of the trend-cycle component; it is
also much smoother than a simple five-month moving average (Table 8).

Ratios of the original observations to the weighted fifteen-month moving
average are computed (Table 9).

10.

Compute the ratios of the preliminary seasonally adjusted series (step 7)
to its weighted fifteen-month moving average (step 8). Month-to-month
changes in these ratios are computed and averaged without regard to sign.
This yields a preliminary measure of the average amplitude of the irre
gular component.

11.

This step will provide a method for identifying extreme items among the
ratios computed by step 9, substituting more representative ratios for
these extreme ratios, and fitting smooth curves to all ratios for each
month.

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- 4 (CM II)

Fit a five-term moving average to the ratios for each month.
This results in the loss of moving average values for the
first two and the last two years. To obtain moving averages
for the first two years, use the average of the first two
ratios as the estimated value of the ratio for each of the
two years preceding the first year available. This is equi
valent to weighting the first three years’ ratios by 2/5,
2/5, and 1/5, respectively, to obtain the first year’s moving
average value, and to weighting the first four years’ ratios
by 3/10, 3/10, 2/10, and 2/10, respectively, to obtain the
second year’s moving average value. The moving average values
for the last two years are obtained in a similar manner.

For each month, compute two-sigma control limits about the
five-term moving average line. All ratios falling outside
these limits are designated as "extreme."

For each year center the twelve ratios (i.e., adjust the
twelve ratios so that their sum will be 1,200) by division
of the twelve items by their arithmetic mean.
If the ini
tial year is incomplete, use as the ratio for any missing
month the value of the average ratio for the same month in
the next two years in centering the initial year’s ratios.
Treat the terminal year's ratios in a similar manner (Table
11).

If the average irregular amplitude, computed in step 10 above, is under 2, use step Ilf; if it is 2 or more, use
step llg.

For each month compute a three-term moving average of a
three-term moving average of the centered ratios yielded by
step lid, above. This will result in the loss of two moving
average values at the beginning and two at the end. To ob
tain the values missing at the beginning, use the average of
the first two centered ratios as the estimated value of the
centered ratio for each of the two years preceding the first
year available. This is equivalent to weighting the first
three year's centered ratios by 9/18, 7/18, and 2/18, re
spectively, to obtain the first year’s moving average value,
and to weighting the first four year’s centered ratios by
5/18, 7/18, 4/18, and 2/18, respectively, to obtain the
second year’s moving average value. The missing values at

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- 5 -

(CM II)
the end are obtained in a similar way. These smoothed ratios
constitute the final seasonal adjustment factors. This series
is identified later by the symbol S (Table 12).

For each month compute a three-term moving average of a fiveterm moving average of the centered ratios yielded by step
lid, above. This will result in the loss of three moving
average values at the beginning and three at the end. To ob
tain the values missing at the beginning, use the average of
the first two centered ratios as the estimated value of the
centered ratio for each of the three years preceding the first
year available. This is equivalent to weighting the first
four year’s centered ratios by 6/15, 6/15, 2/15, and 1/15,
respectively, to obtain the first year’s moving average value;
to weighting the first five years' centered ratios by 9/30,
9/30, 6/30, 4/30, and 2/30, respectively, to obtain the
second year’s moving average value; and to weighting the
first six years' centered ratios by 5/30, 7/30, 6/30j 6/30, 4/30,
and 2/30, respectively, to obtain the third year's moving
average value. The missing values at the end are obtained
in a similar way. These smoothed ratios constitute the final
seasonal adjustment factors. This series is later identified
by symbol S (Table 12).

Estimates of the seasonal factors one year ahead are given
at the bottom of Table 12. These estimates are made by adding
to the seasonal factor for the end year, one-half the trend
between the factor for that year and the preceding year. If
X=seasonal adjusment factor for year N, then X^ +
is esti
mated by the equation

12.

These seasonal factors are divided into the corresponding figures of the
original series, month by month; i.e., the seasonal factor for January,
1947, is divided into the original observation for January, 1947; the
factor for January, 1948, is divided into the original observation for
January, 1948. Similarly, the factor for February, 1947, is divided into
the original observation for February, 1947; the factor for February,
1948, into the original observation for February, 1948; and so on. This
yields the final seasonally adjusted series. This series is later iden
tified by the symbol CI (Table 13).

13.

The ratios of the final seasonally adjusted series to the averages of
the final seasonally adjusted series for the preceding and following
months are computed. This is a rough test for residual seasonality,
similar to that made on the original observations described in step 2
above. Arithmetic means of these ratios for each month are given at
the bottom of the table (Table 14).

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14.

Compute an uncentered twelve-month moving average of the final season
ally adjusted series. This step is required to carry out the test de
scribed in step 15.
It also provides annual averages of the seasonally
adjusted series (Table 15).

15.

Compute ratios of the uncentered twelve-month moving average of the
standard seasonally adjusted series to the uncentered twelve-month
moving average of the original series. This is a test of the effect
of the seasonal adjustment on the level of the series, showing whether
the adjustment has resulted in significant differences between the
level of adjusted and the unadjusted series for any twelve-month per
iod (Table 16).

16.

Using the final seasonally adjusted series, compute the ratio of the
value of each month, from February through the following January, to
that of the preceding January. Such a table of ratios will disclose
repetitive patterns in successive years of more than one month's dura
tion (Table 17).

III.

Measures of the Irregular, Cyclical
and Seasonal Components

17.

Compute a weighted fifteen-month moving average (Spencer's fifteen-term
formula) of the final seasonally adjusted series. The weights are as
follows: -3/320, -6/320, -5/320, 3/320, 21/320, 46/320, 67/320, 74/320,
67/320, 46/320, 21/320, 3/320, -5/320, -6/320, -3/320. This is equivalent
to a weighted five-month moving average (weights are -3/4, 3/4, 1, 3/4,
-3/4), of a five-month moving average, of a four-month moving average, of
a four-month moving average of the data.
To obtain values for the beginning points of this curve, use the aver
age of the first four values of the final seasonally adjusted series as
the estimated value of this series for each of the seven months preceding
the first month available. The values for the end are supplied similarly.
The final seasonally adjusted series contains the cyclical, trend, and
irregular components of the series. The weighted fifteen-month moving
average can be used in place of a twelve-month moving average because
there is no seasonal factor to suppress. The weighted fifteen-month
moving average is much more flexible than a twelve-month moving average
and will therefore provide a better measure of the trend-cycle component;
it is also much smoother than a simple five-month moving average, and It
fits the data about as closely as does the five-month moving average.
This series is identified by the symbol C (Table 18).

18.

Compute the month-to-month percentage changes in the original series
(Table 19).

19..

Compute the month-to-month percentage changes in the final seasonal ad
justment factors (Table 20).

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20.

Compute the month-to-month percentage changes in the final seasonally
adjusted series (Table 21).

21.

Compute the month-to-month percentage changes in the ratios (step 9)
of the original observations to the weighted fifteen-month moving aver
age (Table 22).

22.

Compute the ratios of the final seasonally adjusted series (step 12)
to its weighted fifteen-month moving average (step 17). This provides
a measure of the irregular component of the series. This series is
identified by the symbol I (Table 23).

23.

Compute the month-to-month percentage changes in the irregular component
(Table 24).

24.

Compute the month-to-month percentage changes in the weighted fifteenmonth moving average of the final seasonally adjusted series (Table 25).

25.

Compute the average, without regard to sign, of the percentage changes
in steps 18, 19, 20, 23, and 24. This operation yields measures of the
average monthly amplitude of the original series, the seasonal component,
the seasonally adjusted series, the irregular component, and the cyclical
component, respectively. The symbols used to represent these averages
are original, 0; irregular, T; cyclical, (2; seasonal, ?»; and seasonally
adjusted, CI (Table 27).

26.

Compute the following ratios of the average monthly amplitudes of
step 25:

Irregular component to cyclical component (I/C);

Irregular component to seasonal component (I/S);

Seasonal component to cyclical component (S/C);

Irregular component to original series (I/O);

Cyclical con^onent to original series (C/0);

Seasonal component to original series (S/0);

See Table 27.

27.

Compute the ratio of the average monthly amplitude of the irregular to
the cyclical components when percentage changes are taken between entries
two, three, four, and five months apart (Table 27).
The interval corresponding to the last T/C ratio that is less than
1.00 is designated as "Number of Months for Cyclical Dominance,” and
a moving average of the seasonally adjusted data is computed, using
this interval as its period (Table 26).

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28.

The average duration of run, that is, the average number of months the
series moves before changing direction, is computed for the following:
a.

Seasonally adjusted series;

Irregular component;

Cyclical component;

Seasonally adjusted series smoothed by moving average with
period as given by number of months for cyclical dominance;

See Table 27.
29.

Compute the ratios of a) the twelve-month moving average of the monthto-month percentage changes in the irregular component (step 23) to
b) the twelve-month moving average of the month-to-month percentage
changes in the cyclical component (step 24).
In the computation of
these moving averages, the signs of the percentage changes are dis
regarded (Table 28).

IV,

30.

Notes**

Where the average monthly amplitude of the irregular component is
4.0 or larger (on the basis of the preliminary seasonally adjusted
series) and for special purposes, two additional tables are computed
and inserted between Tables 10 and 11. In the first one, the stable
adjustment factors are computed by averaging the modified ratios of
step 11c for each month and then centering the average so that their
sum will be 1,200. In the second table, these stable factors are
divided into the corresponding values of the original data, yielding
a seasonally adjusted series based on a constant seasonal pattern.
These two additional tables do not affect the computations in any*
other tables.

^Electronic Computers and Business Indicators, Occasional Paper No. 57, National
Bureau of Economic Research, 1957, p. 252.

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Listing of Tables Prepared by Census II Method
Table Number
1
2

3
4
5
9
10

11
12

13
14

15
16
17
18:
19
20
21
22
23
24
25
26
27
28

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Title of Table

Original series
Ratios of original to preceding and following
Averages of ratios
Uncentered 12-month moving average of original
Centered 12-month moving average of original
Ratios of original to 12-month moving average
Ratios of original to weighted 15-month moving average
Modified ratios, original/WTD 15-month moving average
Stable-seasonal adjustment factors
Stable-seasonal adjusted series
Centered ratios, original/WTD 15-month moving average
Final seasonal adjusted factors, 3*5-month moving averages
Estimated seasonal factors one year ahead
Final seasonally adjusted series
Ratios, final adjusted to preceding and following
Averages
Uncentered 12-month moving average, final adjustment
Ratios, 12-month moving average, final adjustment to original
Ratios, each month to preceding January, final adjustment
Weighted 15-month moving average of final adjustment
Percent change from preceding month, original
Percent change from preceding month, seasonal
Percent change from preceding month, final adjustment
Percent change from preceding month, S-I ratios
Irregular component
Percent change from preceding month, irregular
Percent change from preceding month, cyclical
2-month moving average, final adjusted series
I, C, & S components, their relations, & average duration of run
Ratios, 12-month moving averages of irregular and cycle amplitudes

- 10 (CM II)

of Tables Prepared under the 1401 Version of Census II
Method (Philadelphia Program)
e ;

1
2

3
4
5
6
7
8
9
10
11
12
13
14

15
16
17
18
19
20
21
22
23
24
25
26
27
28

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Title of Table

Original series
Ratios of original to preceding and following
Averages
Uncentered 12-month moving average of original
Centered 12-month moving average of original
Ratio of original to 12-month moving average
Preliminary seasonal factors
Preliminary adjusted series
Weighted 15-month moving average of preliminary series
Ratios of original to weighted 15-month moving average
Percent change from preceding month, original
Percent change from preceding month, S-I ratios
Modified ratios of original/WTD, 15-month moving average
Centered ratios of original/WTD, 15-month moving average
Final seasonal adjusted factors, 3*3-month moving averages
Estimated seasonal factors one year ahead
Final seasonally adjusted series
Percent change from preceding month, seasonal
Percent change from preceding month, final adjustment
Ratios, final adjustment to preceding and following
Averages
Uncentered 12-month moving average, final adjustment
Ratios, 12-month moving averages, final adjustment to original
Ratios, each month to preceding January, final adjustment
Weighted 15-month moving average of final series
Irregular component
Percent change from preceding month, irregular
Percent change from preceding month, cyclical
Ratios, 12-month moving averages of irregular and cycle amplitudes
2-month moving average, final adjusted series
I, C, & S components, their relations, and average duration of run

BUREAU OF THE CENSUS SEASONAL
ADJUSTMENT TECHNIQUE

The X-9 Version of Census
Method II*

This procedure replaces steps 6 and 11 of Census Method II as described
in the foregoing description, "A Description of the United States Bureau of
the Census Method of Adjustment of series of Monthly Data for Seasonal Vari
ations," Seasonal Adjustment on Electronic Computers.
6.

Fit a five-term moving average to the ratios for each month.
This results in the loss of moving average values for the
first two and the last two years for which ratios are avail
able. To obtain moving average values for the first two
years, repeat the moving average value of the third year.
This is equivalent to weighting the first five years’
ratios by 1/5, 1/5, 1/5, 1/5, and 1/5 to obtain the first
and second years' moving average values. Moving average
values for the last two years are obtained in a similar
manner.

For each month, compute two-sigma control limits about the
five-term moving average line. All ratios falling outside
these limits are designated as extreme.

Replace extreme ratios for each month as follows: for an
extreme ratio falling at the first point in the series,
substitute the average of the second, third, and fourth
ratios; for an extreme ratio falling at the second point
of the series, substitute the average of the first, third,
and fourth ratios; for an extreme ratio falling in the
middle of the series, substitute the average of the two pre
ceding and two following ratios; for an extreme ratio falling
at the next to last or last point, follow a procedure similar
to that for the beginning of the series (Table 5A, "Modified
Ratios, Original/12-month Moving Average4’).

^’Specifications for the X-9 Version of Census II Method Seasonal Adjustment
Program," Bureau of the Census, Office of Chief Economic Statistician,
March 6, 1962.

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- 2 (X-9)

For each month, compute a three-term moving average of the
modified ratios yielded by step 6c. This results in the
loss of moving average values for the first and last years
for which ratios are available. To obtain the moving aver
age value for the first year, use the average of the first
three ratios as the estimated value for the ratio preceding
the first year available. This is equivalent to weighting
the first three years' ratios by 4/9, 4/9, and 1/9, re
spectively, to obtain the first year’s moving average value.
The missing value at the end is obtained in a similar way
(Table 6B, "Preliminary Uncentered Seasonal Factors"),

For the entire series, compute a centered twelve-month
moving average (a two-term of a twelve-term moving average)
of the preliminary uncentered seasonal factors yielded by
step 6d (Table 60, "Preliminary Centering Factors"). For
the six missing values at the beginning of the centered
twelve-month moving average, repeat the first available
value six times. The six missing values at the end are ob
tained in a similar way. The values computed in step 6d
are divided by these values (Table 6D, "Preliminary Centered
Seasonal Factors").

For each month, compute a three-term moving average of the
preliminary centered seasonal factors yielded by step 6e.
This results in the loss of moving average values for the
first and last years. To obtain the moving average value
for the first year, use the first 6e value as an estimated
value for the year preceding the first year for which a
value is available. This is equivalent to weighting the
first two years' values by 2/3 and 1/3, respectively, to
obtain the first year's moving average value. The missing
value at the end is obtained in a similar way.
To obtain the six factors missing at the beginning of
the series (due to the use of the twelve-term moving average
in step 4), repeat the factor from the same month of the
first available year. Fill in the six missing factors at
the end of the series in a similar way (Table 6B, "Preliminary
Seasonal Factors").

Continue with step 7 of "A Description of the United States Bureau of
the Census Method of Adjustment of series of Monthly Data for Seasonal
Variations," Seasonal Adjustment on Electronic Computers.
11.

This step will provide a method for identifying extreme items among
the ratios computed by step 9, substituting more representative ratios
for these extreme ratios, and fitting smooth curves to all ratios for
each month.

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- 3 (X-9)

Fit a five-term moving average to the ratios for each month.
This results in the loss of moving average values for the
first two and the last two years for which ratios are avail
able. To obtain moving average values for the first two
years, repeat the moving average value of the third year.
This is equivalent to weighting the first five years* ratios
by 1/5, 1/5, 1/5, 1/5, and 1/5 to obtain the first and
second years’ moving average values. Moving average values
for the last two years are obtained in a similar manner.

For each month, compute two-sigma control limits about the
five-term moving average line. All ratios falling outside
these limits are designated as extreme.

Replace extreme ratios for each month as follows: for an
extreme ratio falling at the first point in the series,
substitute the average of the second, third, and fourth
ratios; for an extreme ratio falling at the second point
of the series, substitute the average of the first, third
and fourth ratios; for an extreme ratio falling in the
middle of the series, substitute the average of the two
preceding and two following ratios; for an extreme ratio
falling in the next to last or last point, follow a pro
cedure similar to that for the beginning of the series
(Table 10, "Modified Ratios, Original/Weighted 15-Month
Moving Average").

If the average irregular amplitude, computed in step 10
above, is under 2, use step lie; if it is 2 or more, use
step Ilf.

For each month, compute a three-term moving average of the
modified ratios yielded by step 11c. This results in the
loss of moving average values for the first and last years
for which ratios are available. To obtain the moving aver
age value for the first year, use the average of the first
three ratios as the estimated value for the ratio preceding
the first year available. This is equivalent to weighting
the first three years’ ratios by 4/9, 4/9, and 1/9,
respectively, to obtain the first year’s moving average
value. The missing value at the end is obtained in a simi
lar way (Table 10D, "Final Uncentered Seasonal Factors").

For each month, compute a five-term moving average of the
modified ratios yielded by step 11c. This results in the
loss of moving average values for the first two and last
two years for which ratios are available. To obtain
moving average values for the first two years, use the
average of the first four ratios as the estimated value
for the ratios for each of the two years preceding the
first year available. This is equivalent to weighting

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_ 4 (X-9)

the first four years’ ratios by 6/20, 6/20, 6/20, and 2/20,
respectively, to obtain the first year’s moving average
value and to weighting the first four years’ ratios by
5/20, 5/20, 5/20, and 5/20 to obtain the second year’s
moving average value. The missing values at the end are
obtained in a similar way (Table 10D, "Final Uricentered
Seasonal Factors").
g#

For the entire series, compute a centered twelve-month
moving average (a two-term of a twelve-term moving average)
of the final uncentered seasonal factors Yielded by step
lie or Ilf (Table 10E, "Final Centering Factors"). For the
six missing values at the beginning of the centered twelvemonth moving average, repeat the first available value six
times. The six missing values at the end are obtained in a
similar way. The values computed in step lie or Ilf are
divided by these values (Table 11, "Final Centered Seasonal
Factors").

For each month, compute a three-term moving average of the
final centered seasonal factors yielded by step llg. This
results in the loss of moving average values for the first
and last years. To obtain the moving average value for
the first year, use the first llg value as an estimated
value for the year preceding the first year for which a
value is available. This is equivalent to weighting the
first two years’ values by 2/3 and 1/3, respectively, to
obtain the first year’s moving average value. The missing
value at the end is obtained in a similar way (Table 12,
"Final Seasonal Factors").

Estimates of the seasonal factors one year ahead are given
at the bottom of Table 12. These estimates are made by
adding to the seasonal factor for the end year, one-half
the trend between the factor for that year and the preceding
year.
If X=seasonal adjustment factor for year n, then
Xjj + 1 is estimated by the equation Xj^ + 1 « 3Xh - Xn - 1.

Continue with step 12 of "A Description of the United States Bureau of
the Census Method of Adjustment of series of Monthly Data for Seasonal
Variations,” Seasonal Adjustment on Electronic Computers.

NOTE: In these specifications, no description is given for Tables 6A, 10A,
10B, and 10C. In the Census Bureau's printout, Tables 10A and 10B are the
"Stable-Seasonal Factors" and "Stable-Seasonal Adjusted Series" described
in step 30 of Occasional Paper No. 57. They are printed out regardless of
the size of the irregular component, not only when the average monthly ampli
tude of the irregular component is 4.0 or larger as originally specified.
Tables 6A and 10C are the Moving Seasonality Ratios described in the specificiations for X-10. In X-9, these ratios do not play a role in the

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- 5 (X-9)

selection of the seasonal factor curves; however, they are useful as a
descriptive measure of the type of seasonality present in each month.

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BUREAU OF THE CENSUS SEASONAL
ADJUSTMENT TECHNIQUE

The X-10 Version of Census
Method II*

This procedure** replaces steps 6, 10, 1.1, of Census Method II as
described in the foregoing description, "A Description of the United States
Bureau of the Census Method of Adjustment of Series of Monthly Data for
Seasonal Variations,11 Seasonal Adjustment on Electronic Computers.

Fit a five-term moving average to the ratios for each month.
This results in the loss of moving average values for the
first two and last two years for which ratios are available.
To obtain moving average values for the first two years, re
peat the moving average value of the third year. This is
equivalent to weighting the first five years’ ratios by 1/5,
1/5, 1/5, 1/5, and 1/5 to obtain the first and second year’s
moving average values. Moving average values for the last
two years are obtained in a similar manner.

For each month, compute two-sigma control limits about the
five-term moving average line. All ratios falling outside
these limits are designated as extreme.

Replace extreme ratios for each month as follows: for an
extreme ratio falling at the first point in the series, sub
stitute the average of the second, third and fourth ratios;
for an extreme ratio falling at the second point of the
series, substitute the average of the first, third, and
fourth ratios; for an extreme ratio falling in the middle of
the series, substitute the average of the two preceding and
two following ratios; for an extreme ratio falling at the
next to last or last point, follow a procedure similar to
that for the beginning of the series (Table 5A, "Modified
Ratios, Original/12-Month Moving Average").

For each month, compute a seven-term moving average of the
modified ratios yielded by step 6c. This results in the
loss of moving average values for the first three and the
last three years for which ratios are available. To obtain

*”Specifications for the X-10 Version of the Census Method II Seasonal Adjust
ment Program,” Bureau of the Census, Office of Chief Economic Statistician,
March 6, 1962.

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- 2 (X-10)

moving average values for the first three years, use the
average of the first three ratios as the estimated value
for the ratios for each of the three years preceding the
first year available. Then the moving average values for
the first three years are computed by including these
estimated ratios in the moving average (see part (1) of
Note at end of specifications). The missing values at
the end are obtained in a similar way.

For each month, compute the average, without regard to
sign, of the year-to-year percentage changes in the
moving average values of step 6d. This average is an
estimate of the change in the seasonal component for a
particular month and is referred to as 5y.

For each month, divide the moving average values in step
6d into the modified ratios from step 6c. The resulting
series is an estimate of the irregular component.

For each month, compute the average, without regard to
sign, of the year-to-year percentage changes in the
irregular component yielded by step 6f. This average is
an estimate of thexchange in the irregular component and
is referred to as Iy.

For each month, compute the ratio of the 6g value to the
6e value, Ty/Sy. These ratios are designated Moving Sea
sonality Ratios (Table 6A, “Moving Seasonality Ratios").

For each month, depending upon the size of the moving
seasonality ratio computed in step 6h, an average of the
modified ratios yielded by step 6c is computed, as speci
fied in the table below. When a moving average is selected
and computed, there is a loss of moving average values at
the beginning and end. The number of values lost depends
upon the length of the moving average selected. To obtain
the moving average values at the beginning, a specified
number of beginning ratios are averaged to obtain estimated
ratios for the years preceding the first available ratio.
Then the moving average values for the first years are
computed by including these estimated ratios in the average.
The number of ratios to be averaged, in order to obtain the
estimated ratios, is shown in the last column of the table
(See part (1) of Note at end of specifications). The moving
average values missing at the end are obtained in a similar
way.

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Moving
seasonality
ratio step 6h

Average of 6c
values

No. of beginning or ending
6c values average to
extend the moving average

0 to 1.49

None (Leave 6c values
unchanged.)

—

1.50 to 2.49

3-term moving average

2.50 to 4.49

5-term moving average

4.50 to 6.49

9-term moving average

6.50 to 8.49

15-term moving average

8.50 and over

Arithmetic average of
all 6c values

—

The values obtained in this step are printed out (Table 6B, ’’Pre
liminary Uncentered Seasonal Factors").
j.

For the entire series, compute a centered twelve-month moving
average (a two-term of a twelve-term moving average) of the pre
liminary uncentered seasonal factors yielded by step 6i (Table
6C, "Preliminary Centering Factors"). For the six missing
values at the beginning of the centered twelve-month moving
average, repeat the first available value six times. The six
missing values at the end are obtained in a similar way. The
values computed in step 6i are divided by these values (Table
6D, "Preliminary Centered Seasonal Factors").

For each month, compute a three-term moving average of the pre
liminary centered seasonal factors yielded by step 6j. This
results in the loss of moving average values for the first and
last years. To obtain the moving average value for the first
year, use the first 6j value as an estimated value for the year
preceding the first year for which a value is available. This
is equivalent to weighting the first two years’ values by 2/3
and 1/3, respectively, to obtain the first year’s moving average
value. The missing value at the end is obtained in a similar way.
To obtain the six factors missing at the beginning of the series
(due to the use of the twelve-term moving average in step 4), re
peat the factor from the same month of the first available year.
Fill in the six missing factors at the end of the series in a
similar way (Table 6E, "Preliminary Seasonal Factors").

Continue with step 7 of "A Description of the United States Bureau of
the Census Method of Adjustment of Series of Monthly Data for Seasonal
Variations," Seasonal Adjustment on Electronic Computers.

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- 4 (X-10)

10.

Delete step 10.

11.

This step will provide a method for identifying extreme items among
the ratios computed by step 9, substituting more representative
ratios for these extreme ratios, and fitting smooth curves to all
ratios for each month.

Fit a five-term moving average to the ratios for each month.
This results in the loss of moving average values for the
first two and the last two years for which ratios are avail
able. To obtain moving average values for the first two
years, repeat the moving average value of the third year.
This is equivalent to weighting the first five years’ ratios
by 1/5, 1/5, 1/5, 1/5, and 1/5 to obtain the first and
second years' moving average values. Moving average values
for the last two years are obtained in a similar manner.

For each month, compute two-sigma control limits about the
five-term moving average line. All ratios falling outside
these limits are designated as extreme.

Replace extreme ratios for each month as follows: For an
extreme ratio falling at the first point in the series,
substitute the average of the second, third, and fourth
ratios; for an extreme ratio falling at the second point
of the series, substitute the average of the first, third
and fourth ratios; for an extreme ratio falling in the
middle of the series, substitute the average of the two
preceding and two following ratios; for an extreme ratio
falling at the next to last or last point, follow a pro
cedure similar to that for the beginning of the series
(Table 10, ’’Modified Ratios, Original/Weighted 15-Month
Moving Average").

For each month, compute a seven-term moving average of the
modified ratios yielded by step 11c. This results in the
loss of moving average values for the first three and the
last three years for which ratios are available. To obtain
moving average values for the first three years, use the
average of the first three ratios as the estimated values
for the ratios for each of the three years preceding the
first year available. Then the moving average values for
the first three years are computed by including these esti
mated ratios in the moving average (See part 1 of Note at
end of specifications). The missing values at the end are
obtained in a similar way.

For each month, compute the average, without regard to sign,
of the year-to-year percentage changes in the moving aver
age values of step lid. This average is an estimate of the

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- 5 (X-10)

change in the seasonal component for a particular month and
is referred to as Ky.

For each month, divide the moving average values in step lid
into the modified ratios from step 11c. The resulting series
is an estimate of the irregular component.

For each month, compute the average, without regard to sign,
of the year-to-year percentage changes in the irregular
component series yielded by step Ilf. This average is an
estimate of the change in the irregular component and is re
ferred to as Iy.

For each month, compute the ratio of the llg value to the
lie values, Iy/Sy. These ratios are designated Moving
Seasonality Ratios (Table 10G, ”Moving Seasonality Ratios”).

For each month, depending upon the size of the moving sea
sonality ratio computed in step llh, an average of the
modified ratios yielded by step 11c is computed as specified
in the table below. When a moving average is selected and
computed, there is a loss of moving average values at the
beginning and end. The number of values lost depends upon
the length of the moving average selected. To obtain the
moving average values at the beginning, a specified number
of beginning ratios are averaged to obtain estimated ratios
for the years preceding the first available ratio. Then
the moving average values for the first years are computed
by including these estimated ratios in the average. The
number of ratios to be averaged, in order to obtain the esti
mated ratios, is shown in the last column of the table (See
part 1 of Note at end of specifications). The moving aver
age values missing at the end are obtained in similar way.

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Moving
seasonality
ratio step llh
0 to 1.49

Average of 11c
values

TIo. "of beginning or ending
11c values average to
extend the roving average

None (Leave 11c values
unchanged.)

1.50 to 2.49

3-term moving average

2.50 to 4.49

5-term moving average

4.50 to 6.49

9-term moving average

6.50 to 8.49

15-term moving average

8.50 and over

Arithmetic average of
all 11c values

- 6 (X-10)

The values obtained in this step are printed out (Table 10D,
"Final Uncentered Seasonal Factors").

For the entire series, compute a centered twelve-month moving
average (a two-term of a twelve-term moving average) of the
final uncentered seasonal factors yielded by step Hi (Table
10E, "Final Centering Factors"). For the six missing values
at the beginning of the centered twelve-month moving average,
repeat the first available value six times. The six missing
values at the end are obtained in a similar way. The values
computed in step lli are divided by these values (Table 11,
"Final Centered Seasonal Factors").

For each month, compute a three-term moving average of the
final centered seasonal factors yielded by step 11j. This
results in the loss of moving average values for the first
and last years. To obtain the moving average value for the
first year, use the first 11j value as an estimated value
for the year preceding the first year for which a value is
available. This is equivalent to weighting the first two
years’ values by 2/3 and 1/3, respectively, to obtain the
first year’s moving average value. The missing value at the
end is obtained in a similar way (Table 12, "Final Seasonal
Factors").

Estimates of the seasonal factors one year ahead are given
at the bottom of Table 12. These estimates are made by
adding to the seasonal factor for the end year, one-half
the trend between the factor for that year and the pre
ceding year.
If X^—seasonal adjustment factor for year n,
then Xh + 1 is estimated by the equation Xn + 1 “ 3Xn - Xn - 1.

Continue with step 12 of "A Description of the United States Bureau of
the Census Method of Adjustment of series of Monthly Data for Seasonal
Variations," Seasonal Adjustment on Electronic Computers.
NOTE: 1. No implicit weights are given for steps 6d, 6i, lid, or lli, as
are given for steps 6a, 6k, etc., because when the series is shorter than
the moving average, the weights vary with the length of the series. The
original Method II was programmed to accept series with a minimum of 72
months (six years) of data. For the 15-term moving average, different sets
of weights are required for 14, 13,............ 6-year series; for the 9-term,
sets for 8, 7, and 6-year series are required; and for the 7-term, sets
for 6-year series are needed. The purpose in using a 15-term moving aver
age with a series as short as six years is that it is a convenient way to
fit a straight line within the framework of the method.
2. In these specifications, no description is given for Tables 10A
and 10B. In the Census Bureau's printout, Tables 10A and 10B are the

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’’Stable-Seasonal Factors" and "Stable-Seasonal Adjusted Series" described
in step 30 of Occasional Paper No. 57. They are printed out regardless of
the size of the irregular component, not only when the average monthly
amplitude of the irregular component is 4.0 or larger, as originally specified

**The technique for selecting the seasonal factor curves on the basis of the
moving seasonality ratios, which is incorporated in X-10, was developed
by Stephen N. Marris, Head of Statistics Division of the Organization
for Economic Cooperation and Development, Paris, France, and is described
in Seasonal Adjustment on Electronic Computers, pages 257-309, OECD (Paris
1961). Hie Bureau of the Census and the Organization for Economic
Cooperation and Development have cooperated in further theoretical and
empirical development of this technique during the past two years. The
X-10 program differs slightly from that described in the OECD paper.

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THE SEASONAL ADJUSTMENT
METHOD OF THE BUREAU OF
LABOR STATISTICS*

Detailed Listing of Steps

The following steps describe the method used by the Bureau of Labor
Statistics in developing seasonal factors. The “Table No." reference pre
ceding a description refers to the table in the print-out provided by the
electronic computer** program. The BLS method may involve four or seven
iterations, depending on extreme values detected in the original data.
The table numbers have been assigned so that the first digit indicates the
iteration; the third digit identifies the type of information contained in
the table as follows:

Table X01

always refers to trend-cycle

X02

to seasonal-irregular ratios

X03

to unforced seasonals

X04

to forced seasonals

X05

to irregular movements

X07

to extreme values

X08

to deseasonalized original values

X09

to original data

The computer program used with the BLS method permits selection of
either a complete or partial record (print-out) of the values developed.
The partial record includes the final trend-cycle, seasonal, and irregular
components, the detected extreme original values and their substituted
values, the deseasonalized series, and the centered 12-month moving average
Tables included in the partial record (short print-out) are identified by
an asterisk immediately preceding the table number. The complete record
(long print-out) includes all the tables shown.
*Table 101:

12-month moving average. This is a centered moving average
of the original values (table 709), developed as a first
approximation to the trend-cycle component. A centered
moving average would being six months later than the original
series. However, the difference has been reduced to three

*“The BLS Factor Method," Abe Rothman, Proceedings of the 1960 American
Statistical Association, pp. 8-11,
**I.B.M. 650 basic installation.

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- 2 (BLS)

months by the following series of steps.
(Corresponding
operations are applied at the end of the series. All
operations in the entire procedure are symmetical with
respect to the time scale.)

Table 102:

Table 103:

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Seasonal-irregulars are computed as described
for table 102. These seasonal-irregulars be
gin with the following January, the first
month for which the 12-month moving average is
available.

Unforced seasonals are computed as described
for table 103. These begin with January.

Forced seasonals are computed as described
for table 104. These begin with January.

A seasonally adjusted series is computed by
dividing the original values (table 709) by
the forced seasonal factors (step c). For
the first six months of the original series,
the seasonal factor for the same month of
the following year is used. The adjusted
series begins with July.

The average of the first three seasonally ad
justed values (those for July, August, and
September) is multiplied by the seasonal
factors (step c) for April, May, and June of
the following year to provide synthetic ori
ginal values for the three months preceding
the beginning of the original series. The
centered 12-month moving average of this ex
tended original series is printed as table
101.

Seasonal-irregular, first approximation. The original values
(table 709) are divided by their 12-month moving average (table
101).
Unforced seasonal, first approximation. For each calendar
month, the seasonal-irregular ratios (table 102) are arranged
by year and a weighted average is secured. The weights ,30,
.30, .20, .10, .10 are applied to the first five seasonalirregulars.
(the underline weight is applied to the term
(year) whose seasonal is being computed.) For the second
term, the weights .24, ,26, .20, .16, .14 are applied to
the same first five values. For the third and all subsequent
terms up to the last two, the weights .17, .20, .26, .20,
.17 are applied to a centered group of five years. The

- 3 (BLS)

next-to-last term applies weights of .14, .16, .20, .26,
.24 to the last five values. The last term applies weights
of .10, .10, .20, .30, .30, to these same five end values.
The weights for the central term are a compromise between
a pattern with uniform weights (.20) and one with weights
associated with a 3 x 3 moving average (.11, .22, .33, .22,
.11). The actual pattern is very close to the average of
these two patterns but is a little flatter in shape.
Table 104j

Forced seasonal, first approximation. Each unforced seasonal
(table 103) is multiplied by an adjustment factor which is the
ratio of 1200 to the sum of the unforced seasonals in the whole
calendar year. This makes the average of the seasonal factors
equal to 100.

Table 105:

Irregular, first approximation. A seasonally adjusted series
is computed by dividing the original values (table 709) by the
forced seasonal factors (table 104). This, in turn, is divided
by the 12-month moving average (table 101) to produce an esti
mate of the irregular component which also includes some resi
dual trend-cycle. For the partial year at each end of the
series, the seasonal factors of the adjacent year are used.

Table 201:

Moving average, modified.once. The irregulars (table 105) are
extended three months at each end by tapering the first and
last values to 100 percent. The extended series of irregulars,
arranged in normal time sequence, is then smoothed by a weighted
7-month moving average to remove the irregular part and leave
only the residual trend-cycle. The weighting pattern used,
.090, .127, .183, .200, .183, .127, .090, is the average of a
pattern with equal weights (.143) and a pattern associated with a
3-term of a 3-term of a 3-term (3x3x3) moving average (1, 3, 6, 7,
6, 3, 1 equal to .037, .111, .222, .259, .222, .111, .037).

Table 202.

Seasonal irregular, second approximation. The original values
(table 709) are divided by the improved estimate of trend cycle
(table 201).

Table 203;

Unforced seasonal, second approximation. This is a weighted
5-term moving average of the seasonal-irregulars (table 202)
for each calendar month, using the same weights as for table
103.

Table 204;

Forced seasonal, second approximation. Each unforced seasonal
(table 203) is multiplied by an adjustment factor which is the
ratio of 1200 to the sum of the unforced seasonals in the whole
calendar year.

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Table 205:

Irregular, second approximation. A seasonally adjusted series
is computed by dividing the original values (table 709) by the
forced seasonal factors (table 204). This is in turn divided
by the trend-cycle (table 201) to estimate the irregular com
ponent. For the partial year at each end of the series, the
seasonal factors of the adjacent year are used.

*Table 301:

Moving average, modified twice (final trend if no extremes).
The irregulars (table 205) are smoothed in the same way de
scribed for table 201. The smoothed series of irregulars is
multiplied by the previous estimate of trend-cycle (table 201)
to produce table 301 as an improved estimate. This table gives
the final trend-cycle component if there are no extreme values
(revealed in next iteration).

Table 302:

Seasonal-irregular, third approximation. The original values
(table 709) are divided by the latest estimate of trend-cycle
(table 301).

Table 303:

Unforced seasonal, third approximation. This is a weighted
5-term moving average of the seasonal-irregulars (table 302)
for each calendar month, using the same weights as for table
103.

*Table 304:

Forced seasonal, third approximation (final if no extremes).
Each unforced seasonal (table 303) is multiplied by an adjust
ment factor which is the ratio of 1200 to the sum of the un
forced seasonals in the whole calendar year. This table
gives the final seasonal component if there are no extreme
values.

*Table 305:

Irregular, third approximation (final if no extremes). A
seasonally adjusted series is computed by dividing the ori
ginal values (table 709) by the forced seasonal factors
(table 304). This, in turn, is divided by the trend-cycle
(table 301) to yield the Irregular component. This table
gives the final irregular component if there are no extreme
values. For the partial year at each end of the series, the
seasonal factors of the adjacent year are used.

*Table 308:

Seasonally adjusted series (final if no extremes). The original
values (table 709) are divided by the forced seasonal factors
(table 304). For the partial year at each end of the series,
the seasonal factors are taken from the corresponding months of
the adjacent year.

*Table 407:

Extreme values - tests and replacement values.
This table con
tains the results of the series of steps designed to determine
whether the series contains any extreme values.
If any are
found, the procedure provides replacement values.
If no extreme

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-5 (BLS)

values are found, tables 407 through 708 are omitted.
test for extreme values includes the following steps:

The

The irregulars (table 305) are smoothed in the same
way described for table 201, except that the central
weight is zero instead of .200. The "mid-zero"
weight pattern provides a trend-cycle which minimizes
the effect of an extreme value on the test criterion.

The smoothed series of irregulars (step a) is multi
plied by the latest trend-cycle (table 301) to pro
duce the test trend-cycle. These values are uniformly
20 percent too low, because the weights used in step
a add to only .800.

The original values (table 709) are divided by the
test trend-cycle (step b) to yield test seasonalirregulars, which are uniformly 25 percent too high.

The test seasonal-irregulars (step c) are smoothed
by a weighted 5-term moving average for each calendar
month to produce test seasonals, using the following
"mid-zero" weights. For the first year, the weights
are 0, .43, .29, .14,' .14. For the second year they
are .32, 0, .27, .22, .19. For the third and subse
quent years up to the last two, they are .23, .27, 0,
.27, .23. For the next-to-last year, they are .19,
.22, .27, 0, .32. For the last year they are .14,
.14, .29, .43, 0.
(These weights are proportional to
those for table 103 except that the target year always
receives zero weight.) The test seasonals, like the
test seasonal-irregulars, are uniformly 25 percent too
high.

The test seasonal-irregulars (step c) are divided by
the test seasonals (step d) to produce test irregulars.

The mean and standard deviation are computed for the
entire (all months of all complete calendar years)
distribution of test irregulars (step e). Control
limits are set at the mean +2.86* and are designed
to provide a probability of about 50 percent that all
"good" values will fall inside the limits. The 2.86*
and the 50 percent probability are based on the assump
tion that all values in the original series are "good"
and belong to the series. However, since an original
value not really belonging in the series may be en
countered, a discriminating.test is needed which will
detect the non-belonging observation without rejecting

- 6 (BLS)

too many acceptable values. The 2.8<f is the point in
the distribution which will, in 50 percent of the
cases, reject no values; in the other 50 percent, it
will reject one or more (usually one) values.
Initially,
different sigma limits were calculated based on the
length of the series. However, since our computer pro
gram handled series of from 6-12 years, and the limits
varied by .2 sigma, the single limit of 2.8 sigma was
considered close enough for 6-12 year series.
g.

Particular months whose test irregulars (step e) fall
outside the control limits (step f) are designated as
extreme and are listed in table 407. The replacement
value for each extreme value is obtained by multiplying
the test trend-cycle (step b) by the test seasonal (step
d). This provides a value whose extreme irregularity
has been removed.

*Table 501:

12-month moving average (extremes replaced). The set of ori
ginal values (table 709) is modified by substituting for each
extreme value the replacement value given in table 407. Table
501 is a centered moving average of these modified original
values with extensions at the ends of series computed the same
way as for table 101.

Table 502:

Seasonal-irregular, first approximation (extremes replaced).
The modified original values are divided by the 12-month
moving average (table 501).

Table 503:

Unforced seasonal, first approximation (extremes replaced).
This is a weighted 5-term moving average of the seasonalirregulars (table 502) for each calendar month, using the same
weights as for table 103.

Table 504:

Forced seasonal, first approximation (extremes replaced). Each un
forced seasonal (table 503) is multiplied by an adjustment
factor which is the ratio of 1200 to the sum of the unforced
seasonals in the whole calendar year.

Table 505:

Irregular, first approximation (extremes replaced). A sea
sonally adjusted series is computed by dividing the modified
original values by the forced seasonal factors (table 504).
This is in turn divided by the trend-cycle (table 501) to
estimate the irregular component. For the partial year at
each end of the series, the seasonal factors of the adjacent
year are used.

Table 601:

Moving average, modified once (extremes replaced). The irre
gulars (table 505) are smoothed in the same way described for
table 201. The smoothed series of irregulars is multiplied by

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the previous estimate of trend-cycle (table 501) to produce
table 601 as an improved estimate.
Table 602:

Seasonal-irregulars, second approximation (extremes replaced).
The modified original values are divided by the latest estimate
of trend-cycle (table 601).

Table 603:

Unforced seasonal, second approximation (extremes replaced).
This is a weighted 5-term moving average o£ the seasonalirregulars (table 602) for each calendar month, using the
same weights as for table 103.

Table 604:

Forced seasonal, second approximation (extremes replaced).
Each unforced seasonal (table 603) is multiplied by an adjustment factor which is the ratio of 1200 to the sum of the un
forced seasonals in the whole calendar year.

Table 605:

Irregular, second approximation (extremes replaced). A sea
sonally adjusted series is computed by dividing the modified
original values by the forced seasonal factors (table 604)
This is in turn divided by the trend-cycle (table 601) to esti
mate the irregular component. For the partial year at each end
of the series, the seasonal factors of the adjacent year are
used.

★Table 701:

Final trend-cycle (extremes replaced). The irregulars (table
605) are smoothed in the same way described for table 201.
The smoothed series of irregulars is multiplied by the previous
estimate of trend-cycle (table 601) to produce this final esti
mate.

Table 702:

Final seasonal-irregular (extremes replaced). The modified ori
ginal values are divi.ded by the final trend-cycle (table 701).

Table 703:

Final unforced seasonal (extremes replaced). This is a weighted
5-term moving average of the final seasonal-irregulars (table
702) for each calendar month, using the same weights as for,
table 103.

★Table 704:

Final seasonal (extremes replaced). Each unforced seasonal (table
703) is multiplied by an adjustment factor which is the ratio of
1200 to the sum of the unforced seasonals in the whole calendar
year.

★Table 705:

Final irregular (extremes replaced). A seasonally adjusted
series is computed by dividing the actual original values
(table 709) by the final seasonal factors (table 704). This is,
in turn divided by the final trend-cycle (table 701) to yield
the final irregular comjponent. For the partial year at each
end of the series, the seasonal factors of the adjacent year
are used.

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Seasonally adjusted series. The original values (table 709)
are divided by the final seasonal factors (table 704).
For the partial year at each end of the series, the seasonal
factors are taken from the corresponding months of the adja
cent year.
Original series.
values.

This is the monthly series of original

Listing of Tables Prepared
Title of Table

101
102
103
104
105

12-month moving average
Seasonal irregular, first approximation
Unforced seasonal, first approximation
Forced seasonal, first approximation
Irregular, first approximation

201
202
203
204
205

Moving average modified once
Seasonal irregular, second approximation
Unforced seasonal, second approximation
Forced seasonal, second approximation
Irregular, second approximation

301
302
303
304
305
308

Moving average modified twice *final trend if no extremes*
Seasonal irregular, third approximation
Unforced seasonal, third approximation
Forced seasonal, third approximation *final if no extremes*
Irregular, third approximation *final if no extremes*
Seasonally adjusted series *final if no extremes*

407

Extreme values - tests and replacement values

501
502
503
504
505

12-month moving average *extremes replaced*
Seasonal irregular, first approximation *extremes replaced*
Unforced seasonal, first approximation *extremes replaced*
Forced seasonal, first approximationn *extremes replaced*
Irregular, first approximation *extremes replaced*

601
602
603
604
605

Moving average modified once *extremes replaced*
Seasonal irregular, second approximation *extremes replaced*'
Unforced seasonal, second approximation *extremes replaced*
Forced seasonal, second approximation *extremes replaced*
Irregular, second approximation *extremes replaced*

701
702
703
704
705
708

Final trend cycle *extremes replaced*
Final seasonal irregular *extremes replaced*
Final unforced seasonal *extremes replaced*
Final seasonal,*extremes replaced*
Final irregular *extrernes replaced*
Seasonally adjusted series

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Original series

Full text of Selected Techniques of Seasonal Adjustment

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