Full text of Selected Techniques of Seasonal Adjustment : A Revision of Selected Techniques of Seasonal Adjustment Published June 1962

View original document

The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.

Row

SELECTED TECHNIQUES
OF
SEASONAL ADJUSTMENT

Research Department

Federal Reserve Bank of Atlanta

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

May

1963

Atlanta, Georgia

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

SELECTED TECHNIQUES

OF
SEASONAL ADJUSTMENT

A Revision of Selected Technique
of Seasonal Adjustment
Published June 1962

Compiled by
W. M. Davis
Senior Economist
and
Elizabeth Long
Technical Assistant

Research Department
Federal Reserve Bank of Atlanta
Atlanta, Georgia
May 1963

TABLE OF CONTENTS
Page

Preface

Computational Steps of Selected Methods of Seasonal Adjustment ............................. 1
Bureau of the Census Seasonal Adjustment Technique (Method II) ........................... 8
Listing of Tables Prepared by Census Method II ......................................................16
New Tables in Census Method II ............................................................................................ 19

Bureau of the Census Seasonal Adjustment Technique (The X-9
Version of Census Method II) .................................................................................. .. ..................... 21
Listing of Tables Prepared by X-9 Version of Census Method II ................. 26
Bureau of the Census Seasonal Adjustment Technique (The X-10
Version of Census Method II) .......................................................................................................... 27

The Seasonal Adjustment Method of the Bureau of Labor Statistics ...................... 34
Listing of Tables Prepared by Bureau of Labor Statistics Method ............ 42
The Regression Method of Deutsche Bundesbank ...................................................................... 44

Bureau of the Census Seasonal Adjustment Technique (Method III) .........................45
Listing of Tables Prepared by Census Method III ................................................... 47
Bibliography ............................................................................................................................................................ 49

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

PREFACE

This publication contains basic descriptions of various techniques
of seasonal adjustment that are in common use.

It was originally pre

pared for the Federal Reserve System Seminar on Seasonal Adjustment held
in Washington on June 5-6, 1962, and only a few minor changes have been
made since that time.

It is hoped that this compendium of techniques

will prove useful to the technician and non-technician alike.
In the last few years, there has been a renewed interest in seasonal
adjustment of time series.

Census Method II has been employed by users

of seasonally adjusted data for quite sometime.

This method adapted to

the electronic computer the essence of the ratio-to-trend procedure, which
the Federal Reserve System had used for many years.

The availability of a

computer program greatly facilitated seasonal analysis and led to a great
expansion in the number and types of series for which seasonal factors were

computed.

As the number grew, inadequacies of existing methods came to

light.
These shortcomings of existing methods of seasonal adjustment have led
to several developments in the last three or four years.

The Bureau of

Labor Statistics has designed a different version of the iterative,
ratio-to-trend procedure.

Many users have seriously studied the regression

technique of seasonal adjustment, especially, as currently used by the
Deutsche Bundesbank. The Bureau of the Census, moreover, has developed
the X-9 and X-10 versions of Census Method II, and work is proceeding

on Census Method III.

In addition, basic research on seasonal methods

is now being conducted by the Federal Reserve System, the Bureau of the

Budget, and other institutions.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

COMPUTATIONAL STEPS OF SELECTED METHODS OF SEASONAL ADJUSTMENT

CENSUS METHOD II

X-9 VERSION OF CENSUS
METHOD II

X-1O VERSION OF CENSUS
METHOD II

BUREAU OF LABOR
STATISTICS METHOD

REGRESSION METHOD OF
DEUTSCHE BUNDESBANK

Preliminary Computation
of Seasonally Adjusted
Series

First Iteration

Basic Method

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

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

*1 Develop a centered
12-month moving
average (MA) of
original.
Six
values at each end
are computed by a
series of steps.
Preliminary estimate of trend-cycle
(TC).

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

*2 Compute ratio of
original to centered
12-month MA. First
approximation of
seasonal-irregular
(SI).

2 Basic analysis based
on following additive
relationship:
ansstn + Pn + En
Original values—trend
values + seasonal
component + residual
component.

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

3 Other symbols:
a'=seasonal values
(regression values)
a"«residual values
(a-a1)
a*=seasonally adjusted
values

2 Compute ratio of
original to average
of preceding and
following months.

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

*A step used in derivation of seasonal factors.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

r—4

CENSUS METHOD II____

X-9 VERSION OF CENSUS
METHOD II

X-1O VERSION OF CENSUS
METHOD II

BUREAU OF LABOR
STATISTICS METHOD

*4 Force total to 1200

*4 Center 12-month MA.

REGRESSION METHOD OF
DEUTSCHE BUNDESBANK
*4 Standard regression
equation used for
each month:
a'=tB + A, where B—
slope; A—Y intercept:

’at - Et** Z a
Etr-nEt1
-nA

zt
*5 Calculate ratio of
original to centered
12-month MA.

*6 Omit step 6 in Census
Method 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-term
MA for each month
of data in step 5.
To 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.

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

step used in derivation of seasonal factors.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

*5 Compute seasonally
adjusted series.

*5 With A and B values,
compute seasonal
values (a1) for each
original value:
a’~tB + A

*6 Divide seasonally
adjusted series by
12-month MA of
original.
First
approximation of
irregular (I) with
s ome TC.

*6 Graphical check of
computations.
Using
(t) on the X axis and
(a) on the Y axis,
plot the values and
the regression line.
Visual proof of the
correct computation
of A and B.

Second Iteration

*7 Smooth I ratios in
step 6 by 7-month
WMA after extending I for three
months at each end.
Measure of residual
TC. Multiply resuiting values by
TC of step 1.

*8 Compute ratio of
original to TC of
previous step.
Second approximation of SI.

*7 Compute residual
values (a") by com
paring regression
values (a') with
original values (a):
a"=a-a’.
If a" is
greater than 0, there
is superseasonal
present; if a" is
less than 0, there
is subseasonal; if
0, there is only
purely seasonal.
ro

CENSUS METHOD II

X-9 VERSION OF CENSUS
X-10 VERSION OF CENSUS
METHOD II
MFTUOn TT

BUREAU OF LABOR
STATISTICS METHOD

c Replace extremes as
follows:
(1) Ratio
falling first in
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, simi
lar to beginning.

c Replace extremes as
follows:
(1) Ratio
falling first in
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, simi
lar to beginning,

d Six missing ratios
(due to step 4) at
beginning are sup
plied by extending
first available
ratios for corre
sponding months back
to initial month of
series. Six missing
at end supplied
similarly.

d For each month,
compute a 3-term
MA. Missing values
supplied for first
year-average of
first three ratios;
similar for end.

d For each month,
compute a 7-term MA
of ratios in
step 6c. Missing
ratios supplied in
first 3 years by
averaging first 3
years available.
Similar for last
years. MA values
computed by using
these estimates.

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.

e For each month,
compute the average,
*14 Compute ratio of
without regard to
original to TC of
sign, of year-toprevious step.
year percent changes
Third approxima
in MA of step 6d.
tion of SI.

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

*A step used in derivation

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

*9 For each calendar
month, compute
5-term WMA of SI
ratios. Unforced
seasonals, second
approximation.
*10 Force total to
1200.

*11 Compute seasonally
adjusted series.
*12 Divide seasonally
adjusted series by
TC of step 7.
Second approxima
tion of I.

Third Iteration

*13 Smooth I ratios in
step 12 by 7-month
WMA after extending
I for 3 months at
each end. Multiply
smoothed I by TC
in step 7. Result
is final TC unless
extreme values are
present.

REGRESSION METHOD OF
DEUTSCHE 3UNDESBANK

*8 Compute seasonally
adjusted values (a*)
by adding residual
values (a'1) to
corresponding trend
values : a*?=t + a".

9 To decompose time
series:
(seasonal) p=a' - t;
(residual) E-—a’~a - a'.

Refinement of Trend
Translation.
*10 Test whether trend
is "true," i.e.,
whether seasonal
fluctuations around
trend are distorted.

*11 If refinement is
necessary, improved
trend values (t^)
are obtained by
smoothing trend
values (t):

Where Ao and Bo are
original regression
equation coeffi
cients .

of seasonal factors.

CENSUS METHOD II

f Compute 3-term of
3-term MA of ratios
in 6e for each month.
Supply missing values
at each end. The re
sults are preliminary
seasonal factors.

X-9 VERSION OF CENSUS
METHOD II

X-10 VERSION OF CENSUS
METHOD II

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

f For each month,
divide step 6d into
step 6c. Estimate
of I.

*15 For each calendar
month, compute
5-term WMA of SI
in step 14. Un
forced seasonal,
third approxima
tion .

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 prelimi
nary seasonal
factors.

g For each month,
compute average,
without regard to
sign, of year-toyear percent
changes in I.

*16 Force total to
1200.
These are
final factors un
less extreme
values are present.

*A step used in derivation of seasonal factors.

REGRESSION METHOD OF
DEUTSCHE BUNDESBANK

*12 New improved trend
values (t|) are
then used as basis
of a refined corre
lation between the
trend and original
values.
Calculation
techniques for
various values are
the same as in the
Basic Method.

*17 Compute seasonally adjusted series.

h For each month,
compute ratio of
step 6g to step
6e.
Designated
as Moving Seasona
lity Ratios.
i For each month,
depending upon the
size of the ratio
in step 6h, MA of
ratios yielded by
step 6c is com
puted using the
term indicated in
the table at the
end of this method
(page 6). Missing
ratios supplied.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

BUREAU OF LABOR
STATISTICS METHOD

*18 Divide seasonally adjusted series by TC of
step 13. Final I unless extreme values are
present.
Fourth Iteration

*19 Test for extreme values and, if found, re
place with substitute values. Tests involve
developing, smoothing, and analyzing irre
gular component to determine whether values
fall outside + 2.8 sigma limits. Replacement
are calculated by multiplying TC by S for
a given month.

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

CENSUS METHOD II

X-9 VERSION OF CENSUS
METHOD II

X-10 VERSION OF CENSUS
METHOD II

j Compute a centered
12-month MA of 6i.
For missing values,
repeat first avail
able ratio six
times. Similar for
end. Divide into
step bi.
k For each month,
compute a 3-term
MA. For missing
values, use value
in step 6j corre
sponding to the
month missing.

BUREAU OF LABOR STATISTICS METHOD
Sixth Iteration
Steps 7-12 of Second Iteration are repeated
still using replacement values as original
values.

Seventh Iteration
Steps 13-18 of Third Iteration are repeated.
After derivation of final measures, original
values are replaced and final seasonally
adjusted series is derived.

1 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.

*7 Compute preliminary
seasonally adjusted
series.

*7 Same as Census II.

Final Seasonally Adjusted Series

Final Seasonally Ad
justed Series

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

*8 Same as Census II.

*9 Compute ratio of orig
inal to 15-month WMA.

*9 Same as Census II.

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

*10 Same as Census II

*10 Omit from Census II.

*A step used in
https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

derivation

of seasonal factors.

CENSUS METHOD II

X-9 VERSION OF CENSUS
METHOD II

X-10 VERSION OF CENSUS METHOD II

*11 Compute month-to-month
percent changes of step
10 and average without re
gard to sign. Measures
average amplitude of I.

*11 Same as Census II.

*11 Omit from Census II.

*12 Identify extreme values in
step 9 and replace in the
same manner as explained
in steps 6a-6c above.
Force total to 1200.

*12 Omit step 12 in Census
II and apply steps
6a-6c above to the re
sults of step 10.

*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.

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

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

*13 See step 12 above.

*14 Project seasonals in step
13 above 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.

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

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

*A step used in derivation of

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

seasonal factors.

Average of 6c Values

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

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

2
2
3
3
O'

CENSUS METHOD II
15 Compute seasonally adjusted series.

16 Compute ratio of final seasonally adjusted series to average of preceding and following month as test of
residual seasonal.
17 Develop an uncentered 12-month MA of seasonally adjusted series.
18 Compute ratio of uncentered 12-month MA of final series to 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)

20 Compute 15-month WMA of final series--measure of cyclical component (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.
22 Compute ratio of final series to 15-month WMA of final series.
percentage changes in I.

Yields estimate of I.

Calculate month-to-month

23 Derive mean of percentage changes in original (0), irregular (I), cyclical (C), seasonal (S), and seasonally
adjusted (CI).

24 Using averages in step 23, calculate:

I/C, I/S, S/C, I/O, C/0, S/0.

25 Compute ratio of I/C with percentage changes taken 2, 3, 4, and 5 months apart. The interval corresponding to
the last I/C ratio that is less than 1.00 is the "number of months for cyclical dominance, (MCD).” 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.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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 sample "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, i.e,, a two-month moving
average of the twelve-month moving average is computed. This operation
places the moving-average values at raid-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.

*"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, Paris, France, Organization for Economic Cooperation
and Development, 1961, pp. 391-98,
This description is the same as that pub
lished in Electronic Computers and Business Indicators, Occasional Paper
No. 57, New York, National Bureau of Economic Research, 1957, pp. 248-52

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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,
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. Moving average values for the last
two years are obtained in a similar manner.
b.

For each month, compute Lwo-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).

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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

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 preliminary seasonally adjusted series (Table 7).

II.

Computation of Final Seasonally
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 signif
icant seasonal factor to suppress. The weighted fifteen-month moving
average is much more flexible then 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 (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 ir
regular 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.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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."

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 preceding
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 (Table 10).

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 years’ 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 years' 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

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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).
g.

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 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 three years preceding the first year
available. This is equivalent to weighting the first four
years' centered ratios by 6/15, 6/15, 2/15, and 1/15, re
spectively, 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, 6/30, 6/30,
4/30, and 2/30, respectively, to obtain the third year's
moving average value. The missing values at the end are ob
tained 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 of the trend
between the factor for that year and the preceding year.
If
X?=seasonal adjustment factor for year n, then Xn 4- 1 is esti
mated by the equation Xn + 1 - 3X„ . p
2

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 1943, 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 identified 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).

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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 the adjusted and the unadjusted series for any twelve-month
period (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).

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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, C; seasonal, S’; 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 component 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 7/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).

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

28.

The average duration of run, i.e., 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, New
York, National Bureau of Economic Research, 1957, p. 252.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

Listing of Tables Prepared by Census Method II*
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

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/weighted 15-month moving average
Stable-seasonal adjustment factors
Stable-seasonally adjusted series
Centered ratios, original/weighted 15-month moving average
Final seasonally adjusted factors, 3x5-month moving average
Estimated seasonal factors, one year ahead
Final seasonally adjusted series
Ratios, final adjusted to preceding and following
Averages
Uncentered 12-month moving average, final adjusted series
Ratios, 12-month moving averages, final adjusted series to original
Ratios, each month to preceding January, final adjusted series
Weighted 15-month moving average of final adjusted series
Percent change from preceding month, original
Percent change from preceding month, seasonal
Percent change from preceding month, final adjusted series
Percent change from preceding month, seasonal-irregular ratios
Irregular component
Percent change from preceding month, irregular
Percent change from preceding month, cyclical
Moving average with term of MCD, of final seasonally adjusted series
Irregular, cyclical and seasonal components, their relationship
and average duration of run
Ratios, 12-month moving average of irregular and cyclical amplitudes

*"Actual Sample Univac Print-out for Private Non-farm Dwelling Units Started,
1951-56,” Electronic Computers and Business Indicators, Occasional Paper
No. 57, New York, National Bureau of Economic Research, 1957, pp. 253-57.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

Listing of Tables Prepared by Census Method II
Long Program - Complete Shiskin Method*

Table Number
1
2
3
4
5
6
7
8

9
10
11
12
13
14
15
16
17
18

20
21

23
24

25
26
27
28

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 'djustment factors
Preliminary seasonally adjusted series
Weighted 15-month moving average of preliminary seasonally
adjusted series
Ratios of original to weighted 15-month moving average
Percent change from preceding month, original
Percent change from preceding month, seasonal-irregular ratios
Modified ratios of original/weighted 15-month moving average
Centered ratios of original/weighted 15-month moving average
Final seasonally adjusted factors, 3x3-month moving average
Estimated seasonal factors, one year ahead
Final seasonally adjusted series
Percent change from preceding month, seasonal
Percent change from preceding month, final seasonally adjusted
series
Ratios, final seasonally adjusted series to preceding and
following
Averages
Uncentered 12-month moving average of final seasonally adjusted
series
Ratios of 12-month moving average--final seasonally adjusted
series to original
Ratios, each month to preceding January--final seasonally
adjusted series
Weighted 15-month moving average of final seasonally adjusted
series
Irregular component
Percent change from preceding month, irregular
Percent change from preceding month, cyclical
Ratios, 12-month averages of irregular and cyclical amplitudes
Moving average with term of MCD, of final seasonally adjusted series
Irregular, cyclical, and seasonal components, their relationship
and average duration of run

<4-K 1401 Program, Seasonal Adjustment of Monthly Time Series (Shiskin-Census II
Method), Long program-Complete Shiskin Method, Research Department, Federal
Reserve Bank of Philadelphia, December 1961.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

Listing of Tables Prepared by Census Method II*
Short Program - Shiskin Method*
Table Number

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

Point Chart

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, seasonal-irregular ratios
Modified ratios of original/weighted 15-month moving average
Centered ratios of original/weighted 15-month moving average
Final seasonally adjusted factors, 3x3-month moving average
Estimated seasonal factors one year ahead
Final seasonally adjusted series
X indicates seasonally adjusted
0 indicates unadjusted

*4-K 1401 Program, Seasonal Adjustment of Monthly Time Series (Shiskin-Census
II Method), Short Program with Point-Chart, Research Department, Federal
Reserve Bank of Philadelphia, December 1961.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

New Tables In Census Method II *

Since 1957, when the description of Census Method II was published
in Electronic Computers and Business Indicators, "three new sets of
tables have been added to provide additional information about the be
havior of each series. The first set of tables shows by stages how the
months for cyclical dominance, or MCD, curves are adjusted for amplitude.
The series shown in the final Table (26C) of this set is adjusted for
seasonality, irregularity and amplitude.
Such series add another level
to the adjustment process and facilitate comparisions of the cyclical
timing and pattern of different series.

Tables 26A, 26B and 26C are entitled Symmetrical Percentage Change
from Preceding Month in Short-Term Moving Average, Standardized Symmetrical
Percentage Change from Preceding Month in Short-Term Moving Average, and
Standardized Short-Term Moving Average Index, respectively. Table 26A
shows the month-to-month percentage changes in the short-term moving average
of period equal to the MCD. These percentage changes are computed by the
formula 200 (B-A)/(B+A), where A is the moving average value for the first
month and B is the moving average value for the second month. Table 26B
is obtained by dividing each value of Table 26A by the average (without re
gard to sign) of all the values of Table 26A.
This average value is shown
after the title of Table 26B. Table 26C is obtained by setting the value
for the first month for which the moving average is available at 100 and
by obtaining values for subsequent months by application of the formula
B'=A'(200+r)/(200-r), where A' is the Table 26C value for the month pre
ceding the desired month and where r is the standardized, modified rate of
change (shown in Table 26B) for the desired month.
The second set of tables shows the original observations adjusted for
large irregularities (Table 29) and then adjusted for seasonality (Table 30,
see Note at end of this description). If the modified original observations
are run through the program instead of the original observations, the whole
series of computations will not be affected by large irregularities, and
better measures of the seasonal, cyclical, and irregular components and a
better seasonally adjusted series may be expected.
Table 29 is obtained by multiplying the original observations (Table 1)
by the ratio of the modified seasonal-irregular ratios (Table 10) to the un
modified seasonal-irregular ratios (Table 9).
Table 30 shows these modified
original observations adjusted for seasonality. This table is computed by’
dividing the modified original observations (Table 29) by the final seasonal
adjustment factors (Table 12).

In addition a third set of tables will be computed upon request.
They
show the seasonal factors when converted to percentages of the yearly acti
vity. Each of the monthly seasonal factors is divided by 12 to make up

*"Test and Revisions of U.S. Census Methods," Seasonal Adjustment on Electronic
Computers, Paris, France, Organization for Economic Cooperation and
Development, 1961, pp. 138-39,

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

Table 12A (Monthly Seasonal Factors as Percent of Year), and appropriate
values from Table 12A are added to obtain Table 12B, Quarterly Seasonal
Factors as Percent of Year (e.g., first quarter values in Table 12B are
sums of January, February and March values of Table 12A, etc). The sura
of the percentages in each row (year) equals approximately 100.

These tables indicate the percentage of the year’s activity that may
be ’expected’ each month or each quarter as a result of the seasonal
factor alone. These tables may be helpful in distributing an annual fore
cast by month; however, in such a use an allowance should be made for the
cycle and trend within the year. Other uses in which test tables may be
helpful include distributing annual budgets seasonally and supervising
monthly expenditures of annual appropriations."
NOTE*: "The modified original observations are shown in a new table (print
out Table 29) and the modified original observations adjusted for seasonality
are shown in still another new table (print-out Table 30). Table 30 is
computed by dividing the modified original observations (Table 29) by the
final seasonal adjustment factors (print-out Table 12). A program which
recognizes no extremes is also available through SAG (Seasonal Adjustment
Generator program).
If this program is applied to modified original ob
servations, 'purer1 seasonal factors and trend-cycle curves, and a better
adjusted series will be obtained."

*"Test and Revisions of U.S. Census Methods," Seasonal Adjustment on Electronic
Computers, Paris, France, Organization for Economic Cooperation and
Development, 1961, p. 110.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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

Fit a five-terra 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 average”).

*”Specifications for the X-9 Version of the Census Method II Seasonal Adjustment
Program,” Bureau of the Census, Office of Chief Economic Statistician,
March 6, 1962.
**This description is the same as that published in Electronic Computers and
Business Indicators, Occasional Paper No. 57, New York, National Bureau
of Economic Research, 1957, pp. 248-52.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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 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 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 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.
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.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

23
- 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

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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
ODtained in a similar way (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
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
Xn + 1 is estimated by the equation Xp + I - 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: 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 co.thly amplitude of the irregular component is 4.0 or larger as

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

originally specified. Tables 6A and IOC are the Moving Seasonality Ratios
described in the specifications for X-10.
In X-9, these ratios do not
play a role in the selection of the seasonal factor curves; however, they
are useful as a descriptive measure of the type of seasonality present in
each month.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

Listing of Tables Prepared by X-9 Version of Census Method II*

Table Number
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

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
Preliminary seasonal adjustment factors
Preliminary seasonally adjusted series
Weighted 15-month moving average of preliminary seasonally
adjusted series
Ratios of original to weighted 15-month moving average
Modified ratios of original to weighted 15-month moving average
Stable-seasonal adjustment factors
Stable-seasonally adjusted series
Centered ratios of original to weighted 15-month moving average
Final seasonally adjusted factors, 3x5-month moving average
Estimated seasonal factors, one year ahead
Final seasonally adjusted series
Ratios of final seasonally adjusted series to preceding and
following
Averages of ratios
Uncentered 12-month moving average of final seasonally adjusted
series
Ratios of 12-month moving averages--final seasonally adjusted
series to original
Ratios of each month to preceding January--final seasonally ad
justed series
Weighted 15-month moving average of final seasonally adjusted
series
Percent change from preceding month, original series
Percent change from preceding month, final seasonally adjusted
factors
Percent change from preceding month, final seasonally adjusted
series
Percent change from preceding month, seasonal-irregular ratios
Irregular component
Percent change from preceding month, irregular component
Percent change from preceding month, cyclical component
Moving average with term of MCD, of final seasonally adjusted series
Irregular, cyclical, and seasonal components, their relationship
and average duration of run
Ratios of 12-month moving average of irregular and cyclical
amplitudes

*1410 Program, Actual Print-out for Non-manufacturing Employment in Georgia,
Washington, D. C., Board of Goversnors of the Federal Reserve System,
1962.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

BUREAU OF THE CENSUS SEASONAL
ADJUSTMENT TECHNIQUE

The X-10 Version of Census
Method II*

This procedure** replaces steps 6, 10, 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 Variations," Seasonal Adjustment on Electronic Computers.***

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 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 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, 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

*"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.

**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). The 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.
***This description is the same as that published in Electronic Computers and
Business Indicators, Occasional Paper No. 57, New York, National Bureau
of Economic Research, 1957, pp. 248-52.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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 the 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 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 Sy.

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_
the change in the irregular component and is referred to as Iy.

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

For each month, depending upon the si?e of the moving seasona
lity ratio computed in step 6h, an average of the modified
ratios yielded by step 6c is computed, as specified in the
table on the next page. 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 pre
ceding 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.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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").

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.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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 she year-to-year percentage changes in the moving aver
age values of step lid. This average is an estimate of the

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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

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 Ty.

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

Moving
seasonality
ratio step llh

Average of 11c
values

“No. oFTTeginnihg or ending
11c values average to
extend the moving 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

0 to 1.49

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

—

The values obtained in this step are printed out (Table 10D,
"Final 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
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 llj. 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 llj 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 Xn + 1 is estimated by the equation Xn + 1 ~ 3Xn ~

_ 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

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

"Stable-Seasonal Factors" and "Stable-Seasonally 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 speci
fied.
3. It is suggested that two additional features be incorporated into
the X-10 program. The first is to allow the parameters of the moving sea
sonality ratios in steps 6i and Hi to be changed easily if additional re
search suggests their change or if individual users want to modify the
program to their own needs. The second is to program an option which allows
the moving average used for each month to be specified in advance by the
user instead of being selected on the basis of the moving seasonality ratio;
this step will reduce revisions when series are brought up to date.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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 Number" reference
preceding a description refers to the table in the print-out provided by
the electronic computer program (IBM 650 basic installation). 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 begin six months later than the original
series. However, the difference has been reduced to three

*"BLS Seasonal Factor Method," 1960 Proceedings of the Business and Economic
Statistics Section, Washington, D. C. American Statistical Association,
1960, pp. 8-11.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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

Table 102:

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).

Table 103:

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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

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 104:

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.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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. Hie 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 orig
inal 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 (fina1 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

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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 irregular (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 +2.8(5" and are designed
to provide a probability of about 50 percent that all
"good" values will fall inside the limits.
The 2.8<f
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
encountered, a discriminating test is needed that will
detect the non-belonging observation without rejecting

too many acceptable values. The 2.8d is the point
in the distribution which will, in 50 percent of
the cases, reject no values; in the other 50 per
cent, it will reject one or more (usually one)
values.
Initially, different sigma limits based
on the length of the series were calculated. How
ever, since our computer program handled series of
from 6-12 years, and the limits varied only 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 de
signated 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 orig
inal 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 unforced seasonal (table 503) is multiplied by an ad
justment 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
irregulars (table 505) are smoothed in the same way described
for table 201. The smoothed series of irregulars is multiplied by

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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 of 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 adjust
ment 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
original values are divided 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 component. For the partial year at each
end of the series, the seasonal factors of the adjacent year
are used.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

*Table 708:

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,

*Table 709:

Original series.
values.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

This is the monthly series of original

Listing of Tables Prepared by Bureau of Labor Statistics Method*
Table Number

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 approximation (extremes replaced)
Irregular, first approximation (extremes replaced)

601
602
603
604
605

Moving average, modified once (extremes replaced)
Seasonal-irregulars, 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 (extremes replaced)
Seasonally adjusted series

709

Original series

*"Tables Resulting from BLS Computer Program for Securing Seasonal Factors,
1960 Proceedings of the Business and Economic Statistics Section,
Abe Rothman, American Statistical Association, Washington 25, D. C.,
p. 11.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

Format for All Tables, Except Table 407
Prepared by Bureau of Labor Statistics Method*

Series

Year

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov.

Dec.

Format for Table 407*

407 Extreme Values - Tests and Replacement Values

Series

Month

Year

Irregular

Lower

Upper

Original

Trend

Seasonal

Replacement

*1960 Proceedings of the Business and Economic Statistics Section, Abe Rothman,
American Statistical Association, Washington, 25. D. C., p. 11.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

THE REGRESSION METHOD
OF DEUTSCHE BUNDESBANK

The regression method as adapted by the Deutsche Bundesbank develops
seasonal measures by a regression between the actual data and a moving
average of actual data.
The Deutsche Bundesbank reports that reliable
factors can be calculated from a relatively short series. The following
quotation gives the rationale of this technique:

"The 'classical methods’ of seasonal adjustment (e.g., adjustment
by means of a seasonal index or the link relative method) cannot produce
useful results in analyzing statistical time series unless the course of
the seasonal movements is proportional to the general trend or the ratio
of monthly values to the values of the corresponding month of the pre
ceding year remains almost constant.
The seasonal fluctuations of many statistical time series, however,
are not proportional to the trend. For this reason, results of seasonal
adjustment by means of the existing procedures are frequently rather un
satisfactory. The fact that under certain conditions useful results have
been obtained by such methods should not obscure the truth that all these
methods contain a basic methodical error.
If the various kinds of seasonal behaviour are to be computed by
means of one method, this can be achieved only by a comprehensive
mathematical/statistical approach applicable to proportional, additive,
or any other seasonal behaviour (e.g., proportional and additive mixed).
Since all linear relations between two variables can be covered by a re
gression equation including one multiplicative (proportional) and one
additive parameter, the idea suggests itself to solve the problem of
seasonal fluctuations in analyzing statistical time series by means of
the correlation calculus in that the seasonal movement is conceived as
a correlated connection between series value and trend value. On the
basis of the correlation calculus, seasonal adjustments cannot only be
made in the case of purely proportional or purely additive behaviour, but
also in all other cases where the seasonal movements are composed of an
additive and a proportional component.
It goes without saying that these
components are the result of numerous single factors."*

*"The Problem," Application of the Regression Method to the Analysis of
Statistical Time Series, Frankfurt (Main), Germany, Deutsche Bundes
bank, 1959, p. 1.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

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

Method III will have further refinements and also utilize somewhat
different methods of dealing with our principal problems. The principal
improvement, however, will be the increased generality, flexibility, and
adaptability of the new program. The new program will be constructed by
utilizing a family of subroutines instead of a fixed set of consecutive
instructions. For example, one subroutine of this program will consist
of a generalized trend-cycle moving average with a period from 3 to 45
months and any weight pattern. Thus different moving averages can be
used for series with different characteristics. The program will also
be prepared in such a way that it will be easy to make fairly large
modifications by replacing one subroutine by an entirely different one
or by adding a new subroutine. Additional tests and controls will be
built into the program to determine what are the appropriate moving
averages, control limits, and weights to use for the particular series
being run. Because of its adaptability, this new program will be an
even more powerful tool for experimental work than SAG**.

The Census Bureau has recently expanded its studies of mathematical and
statistical methods which could be applied to the decomposition of economic
time series. When improved or additional techniques are developed by these
studies, they will be used in Method III.
The new program will be prepared for our new computers, the Univac 1105's.
These computers have about the same speed and capacity as the large scale
IBM machines-- the 704 and 709.
These machines compute much faster than our
Univac I's. Since a 10-year series will probably require about one minute's
running time on the 1105, it will matter little whether the short or long
run is requested. Consequently, the distinction between the long run and
the short run will probably be eliminated.
It may also be possible to reduce
the costs.

Some of the major improvements under consideration for Method III are
summarized below:

A working or trading-day adjustment based upon the internal
evidence of the series.

A wider range of moving averages to measure the trend-cycle
factor, selected on the basis of the I/C ratio.

A third iteration based upon original observations modified
for extreme irregularities.

*"Census Method III," Tests and Revisions of Bureau of the Census Methods of
Seasonal Adjustments, Technical Paper No. 5, Washington, D.C., U.S.
Bureau of the Census, 1961, p. 32.

**Seasonal Adjustment Generator program.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

A wider range of moving averages to measure the seasonal factor,
selected for individual months on the basis of the I’/S’ ratio.

An iterative technique for identifying extreme S-I ratios; i.e.,
a set of control limits and trend line would be computed without
the preliminary extremes affecting the computations.

An alternative method of identifying extremes based upon the
distribution of the complete series of irregular factors.

A moving amplitude adjustment.

Additional summary measures including CI/O, annual values for
I, C, S, quadratic means of the irregular, cyclical, and seasonal
factors and measures of the seasonal and irregular factors for
each of the 12 months (i.e., I’, S').

A technique for adjusting first for additive seasonality and
subsequently for moving multiplicative seasonality (if this
proves feasible).

10.

To the extent that our research warrants it, statistical tests
of significance will be added at appropriate sections of the
program.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

Listing of Tables Prepared by Census Method III*
Table Number

1
2

3.
4

5
6

7
8

9
10
10A
10B
11

12
12A
12B

13
14

15
16
17

18
19
20
21

22
23
24
25
26

Title of Table

Original series--to be adjusted
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
Preliminary seasonal adjustment factors
Preliminary seasonally adjusted series
Weighted 15-month moving average of preliminary seasonally
adjusted series
Ratios of original to weighted 15-month moving average
Modified ratios of original to weighted 15-month moving
average
Stable-seasonal adjustment factors
Stable-seasonally adjusted series
Centered ratios of original to weighted 15-month moving
average
Final seasonally adjusted factors, 3x5-month moving averages
Estimated seasonal factors, one year ahead
Monthly seasonal factors as percent of year
Estimated seasonal factors, one year ahead
Quarterly seasonal factors as percent of year
Estimated seasonal factors, one year ahead
Final seasonally adjusted series
Ratios of final seasonally adjusted series to preceding and
following
Averages
Uncentered 12-month moving average of final seasonally adjusted
series
Ratios of 12-month moving averages--final seasonally adjusted
series to original
Ratios of each month to preceding January--final seasonally
adjusted series
Weighted 15-month moving average of final seasonally adjusted
series
Percent change from preceding month, original series
Percent change from preceding month, seasonal
Percent change from preceding month, final seasonally adjusted
series
Percent change from preceding month, seasonal-irregular ratios
Irregular component
Percent change from preceding month, irregular component
Percent change from preceding month, cyclical component
Moving average with term of MCD, of final seasonally adjusted series

*”Sample Print-out Tables of the X-3 Program--Number of Private Nonfarm
Dwelling Units Started, United States, 1939-1959," Tests and Revisions
of Bureau of the Census Methods of Seasonal Adjustments, Technical
Paper No. 5, Washington, D. C., U.S. Bureau of the Census, 1961, pp. 33-51.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

Table Number

26A
26B
26C
27
28
29
30

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

Title of Table

Symmetrical percent change from preceding month in table 26
Standard symmetrical percent change from preceding month in
table 26
Standard short-term moving average index
Irregular, cyclical, and seasonal components, their relation
ship and average duration of run
Ratios of 12-month moving averages of irregular and cyclical
amplitudes
Modified original observations (T| x T^q/Tq)
Modified seasonally adjusted series (T.Z9/T.12)

BIBLIOGRAPHY

Census Method II

Electronic Computers and Business Indicators, Occasional Paper No. 57,
New York, National Bureau of Economic Research, 1957, pp. 248-57.
Seasonal Adjustment on Electronic Computers, Paris, France, Organization
for Economic Cooperation and Development, 1961, pp. 110, 138-39, and
391-98.
4-K 1401 Program, Seasonal Adjustment of Monthly Time Series (ShiskinCensus II Method), Long program-Complete Shiskin Method, Philadelphia,
Research Department, Federal Reserve Bank, December 1961.

4-K 1401 Program, Seasonal Adjustment of Monthly Time Series (ShiskinCensus II Method), Short program with Point-Chart, Philadelphia,
Research Department, Federal Reserve Bank, December 1961.

The X-9 Version of Census Method II

’’Specifications for the X-9 Version of the Census Method II Seasonal Ad
justment Program," Washington, D. C., Bureau of the Census, Office of
Chief Economic Statistician, March 6, 1962.
1410 Program, Actual Print-out for Non-manufacturing Employment in
Georgia, Washington, D. C., Board of Governors of the Federal Re
serve System, 1962.

The X-10 Version of Census Method II

"Specifications for the X-10 Version of the Census Method II Seasonal Ad
justment Program," Washington, D.C., Bureau of the Census, Office of
Chief Economic Statistician, March 6, 1962.

The Seasonal Adjustment Method of the Bureau of
Labor Statistics

1960 Proceedings of the Business and Economic Statistics Section, Wash
ington, D. C., American Statistical Association, 1960, pp. 8-11.

The Regression Method of Deutsche Bundesbank

"Application of the Regression Method to the Analysis of Statistical Time
Series,"Frankfurt (Main), Germany, Deutsche Bundesbank, 1959.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

"Experience in the Application of Regression Computing to the Seasonal
Adjustment of Statistical Time-Series," reprint from: Monthly Reports
of the Deutsche Bundesbank, Vol. 13, No. 8, Frankfurt (Main), Germany,
Deutsche Bundesbank, August 1961, pp. 19 et. seq.

"The Practice of Seasonal Adjustment with Regression Equations," Frankfurt
(Main), Germany, Deutsche Bundesbank, 1960.

Census Method III

Tests and Revisions of Bureau of the Census Methods of Seasonal Adjustments,
Technical Paper No. 5, Washington, D. C., U. S. Bureau of the Census,
1961,pp. 32-51.

https://fraser.stlouisfed.org
Federal Reserve Bank of St. Louis

Full text of Selected Techniques of Seasonal Adjustment : A Revision of Selected Techniques of Seasonal Adjustment Published June 1962

FRASER