Aggregation of Dependently Adjusted Seasonally Adjusted Series

The Bureau of Labor Statistics is changing the way aggregation weights for seasonally

adjusted Consumer Price Index data are calculated, effective January 2002. At that time,

the expenditure weights used to calculate the Consumer Price Index (CPI) will be

updated and the seasonally adjusted Consumer Price Indexes for dependently adjusted

series will be revised from January 1987 through December 2001. The revised

seasonally adjusted series will use the aggregation weights used for the unadjusted series.

These aggregation weights are slightly different from the seasonal aggregation weights

originally used. The month to month index movement will be the same except for the

months in which weight revisions were introduced (December 1986 to January 1987 and

December 1997 to January 1998).

Beginning in January 2002, the CPI will use the same aggregation weights for unadjusted

and seasonally adjusted data. This will simplify the processing of seasonally adjusted

data. In addition, the index level of the seasonally adjusted data will more closely track

the level of the unadjusted data.

The revised data will be available through the CPI home page, https://www.bls.gov/cpi/ on

Friday, February 15, 2002.

Aggregation Using Standard Aggregation Weights

In January 2002, the formula for calculating dependently seasonally adjusted aggregate

series will be changed to

I p

i I

i p i t

I t

aggwt

aggwt SIX

SIX

,

, ,

,

∑( * )

∈ =

where

SIX = seasonally adjusted series

i = component series

I = aggregate series

aggwt = standard aggregation weight

t = current collection period

p = expenditure base

When standard aggregation weights are used and the weights are revised, the month to

month change across the weight revision will include a price change effect and a weight

change effect:

I p

i I

i p i t

I p

i I

i p i t

I p

i I

i p i t

I p

i I

i p i t

I p

i I

i p i t

I p

i I

i p i t

I t aggwt

I t aggwt

aggwt

aggwt SIX

aggwt

aggwt SIX

aggwt

aggwt SIX

aggwt

aggwt SIX

aggwt

aggwt SIX

aggwt

aggwt SIX

SIX

SIX

,

, ,

,

, ,

,

, ,

,

, , 1

,

, ,

,

, , 1

, ,

, 1,

( * )

( * )

*

( * )

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( * )

( * )

∑

∑

∑

∑

∑

∑

∈

′

∈

′

′

∈

′

′

∈

′ +

∈

′

∈

′ +

+

= =

 price change effect weight change effect
The expenditure weights used in calculating the Consumer Price Index are updated

periodically. Updates occurred in 1940, 1953, 1964, 1978, 1987, and 1998. The next

weight update will occur in January 2002; future updates will occur every 2 years, in

2004, 2006, and so on.

In order to provide continuous unadjusted series, the standard aggregation weights in the

CPI are set in such a way that the first index for the series calculated using the new

weights is equal to the index for the last month calculated using the old weights; this

overlap month is called the pivot or link month. The pivot month for the last weight

update was December 1997; the new weights were introduced in January 1998.

For seasonally adjusted aggregate series, constructed by aggregating seasonally adjusted

component indexes, there are technically two pivot month indexes. One is calculated

using the old weighting structure, and one is calculated using the new weights. The

difference between these two pivot month indexes, which occurs because of the different

weighting applied to the component seasonal factors, is known as the weight effect.

When the published pivot month index (old weights) is compared to the following month

index (new weights) the resulting change will include both price effects and weight

effects.

The purpose of seasonally adjusted data is to highlight the underlying price trend. The

inclusion of significant weight effects in seasonally adjusted data can distort the view of

the underlying trend. There are several methods that can be used to minimize the weight

effect; the CPI currently uses a specially calculated seasonal aggregation weight. With

weight updates occurring every 10 or more years, the weight effect was significant. The

expenditure weights will be updated frequently in the future; we expect that the weight

effect at the time of each update will be negligible.

Based on current data, the median weight effect at the 1998 Revision was 0.04 percent.

The largest weight effect for a continuous series was 0.48 percent, reflecting 11 years of

change in spending patterns. With biennial weight revisions, we expect the weight effect

to be less than 0.1 percent, and significantly less for most series.

Processing

The advantage to using standard aggregation weights is simplification of our procedures

for calculating aggregate seasonally adjusted data. Calculation of seasonal aggregation

weights is complicated by revisions of seasonally adjusted data. Seasonally adjusted data

are subject to revision for 5 years. Seasonal aggregation weights are recalculated each

year until the data are declared final. With weight updates occurring every 2 years, the

process for calculating and revising seasonal aggregation weights grows in complexity.

In addition to calculating seasonal aggregation weights for the current weight structure,

revised seasonal aggregation weights would need to be calculated for the 2 previous

weight sets.

Elimination of a separate seasonal aggregation weight greatly simplifies processing.

Aggregation Using Seasonal Aggregation Weights

The current formula for calculating dependent seasonally adjusted indexes using a

seasonal aggregation weight is

I p

i I

i p i t

I t

saggwt

aggwt SIX

SIX

,

, ,

,

∑( * )

∈

=
where saggwt is the seasonal aggregation weight.

For aggregate series, the saggwt is calculated as follows:

I v

i I

i v i v

I v

i I

i v i v i v

I v

i I

i v i v

I p

SIX

aggwt SIX

SIX

aggwt IX sf

SIX

E sf

saggwt

,

, ,

,

, , ,

,

, ,

,

∑ ∑( * ) ∑( * )

∈ ∈ ∈

≡ = = 

where Ei,v = Expenditures from the Consumer Expenditure Survey

sfi,v = seasonal factor for directly adjusted series

v = pivot month

The seasonally adjusted expenditures for the pivot period are divided by the seasonally

adjusted index for the aggregate for the pivot period to get the new seasonal aggregation

weight. When seasonal aggregation weights are used, the month to month change across

a weight revision will show only the price change effect.

One disadvantage of using the seasonal aggregation weight is that the index levels for the

unadjusted and seasonally adjusted series will not always be similar. When a new series

is introduced, the pivot month seasonally adjusted index is arbitrarily set equal to 100. If

the series is at a low point in its seasonal cycle at the pivot month, the seasonally adjusted

series will consistently track at a lower level than the unadjusted series. The opposite is

true for a series at a seasonal cycle peak.

Month-to-Month Changes

It can be shown that for any time period when the weights are constant, the month-tomonth percent change will be the same whether seasonal aggregation weights or standard

aggregation weights are used.

I t aggwt

I t aggwt

I p

i I

i p i t

I p

i I

i p i t

I p

I p

i I

i p i t

i I

i p i t

i I

i p i t

i I

i p i t

I p

i I

i p i t

I p

i I

i p i t

I t saggwt

I t saggwt

SIX

SIX

aggwt

aggwt SIX

aggwt

aggwt SIX

aggwt

aggwt

aggwt SIX

aggwt SIX

aggwt SIX

aggwt SIX

saggwt

aggwt SIX

saggwt

aggwt SIX

SIX

SIX

, ,

, 1,

,

, ,

,

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,

,

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, , 1

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, 1,

( * )

( * )

*

( * )

( * )

( * )

( * )

( * )

( * )

+

∈

∈

+

∈

∈

+

∈

∈

+

∈

∈

+

+

= =

= = =

∑

∑

∑

∑

∑

∑

∑

∑

Both the standard aggregation weight and the seasonal aggregation weight are constants.

The standard aggregation weight is based on expenditure data and is constant once

calculated. The seasonal aggregation weight is dependent on the seasonally adjusted

index for the pivot month. The seasonal aggregation weight is revised each year until the
pivot month index is final. At a given point in time, though, the seasonal aggregation

weight is constant, and the same weight is used for all months after the pivot month.

Advantages and Disadvantages

As indicated earlier, elimination of a separate seasonal aggregation weight greatly

simplifies processing. In addition, the index level of the seasonally adjusted data will

more closely track the level of the unadjusted data. The primary disadvantage of using

the standard aggregation weights for aggregation of unadjusted and seasonally adjusted

data is that the month-to-month seasonally adjusted index percent change at the revision

will have weight effects as well as price effects. With frequent weight updates, this

concern is eliminated. In the past, the weight effects at a revision were significant.

Weight revisions occurred approximately once a decade. It was felt that minimizing the

weight effect would give the best representation of economic movements. The January

2002 revision takes place only 4 years after the last revision, in January 1998. Beginning

with the January 2004 revision, the weights will be revised every other year. Thus, the

importance of the weight effect is declining. Both methods result in the same relative

month-to-month movement when the timeframe does not include a weight revision.

The advantages of the simplified processing afforded by using standard aggregation

weights outweigh the disadvantage of showing a negligible weight effect when the

expenditure weights are updated. In addition, the index level of the seasonally adjusted

data will more closely track the level of the unadjusted data.

In order to provide consistent seasonally adjusted series, historical dependently adjusted

series will be revised to the appropriate level by applying level adjustment factors.i The

level adjustment factors, which will be applied to data for January 1987 through

December 1996, are presented in table 1 and table 2. Data for January 1997 through

December 2001 will be revised and aggregated using the standard aggregation weights.

Because the pivot month (December 1997) indexes are subject to revision, estimates of

the level adjustment for January 1997 through December 2001 are not available in

advance. The revised data will be available on Friday, February 15, 2002 through the CPI

home page: https://www.bls.gov/cpi/
Table 1. Consumer Price Index for All Urban Consumers (CPI-U), U.S. city average:

Level adjustment factors for use with seasonally adjusted data for January 1987 through

December 1996.

Item Code Item Title Level Adjustment Factor

SA0 All items 0.99932659526447

SA0E Energy 1.00350571809222

SA0L1 All items less food 1.00032069654291

SA0L12 All items less food and shelter 1.00062005796949

SA0L12E All items less food, shelter, and energy 0.99890160108245

SA0L12E4 All items less food, shelter, energy, and used cars and trucks 0.99969338109949

SA0L1E All items less food and energy 0.99862946847454

SA0L2 All items less shelter 0.99975903320569

SA0L5 All items less medical care 1.00083330902630

SA0LE All items less energy 0.99879094018791

SA311 Apparel less footwear 1.00132214674933

SAA Apparel 0.99995324550230

SAA2 Women’s and girls’ apparel 1.00043267902463

SAC Commodities 0.99981421715880

SACE Energy commodities 1.00099461925798

SACL1 Commodities less food 0.99867857080338

SACL11 Commodities less food and beverages 0.99963937523226

SACL1E Commodities less food and energy commodities 0.99988575763120

SACL1E4 Commodities less food, energy, and used cars and trucks 1.00119169627147

SAF Food and beverages 0.99972034675630

SAF1 Food 0.99868585267151

SAF11 Food at home 0.99990526538189

SAF112 Meats, poultry, fish, and eggs 1.00017927084735

SAF1121 Meats, poultry, and fish 0.99968580695353

SAF11211 Meats 1.00029150982754

SAF113 Fruits and vegetables 0.99989404268686

SAF1131 Fresh fruits and vegetables 1.00286856973450

SAF115 Other food at home 1.00045455985688

SAG Other goods and services 0.99645615635526

SAH Housing 1.00072227985618

SAH1 Shelter 1.00163859180900

SAH2 Fuels and utilities 1.00017196702959

SAH21 Fuels 1.00717345558590

SAH3 Household furnishings and operations 1.00054457710847

SAM Medical care 0.99898550137250

SAS Services 0.99967847389687

SAS367 Other services 0.99708969720805

SAS4 Transportation services 1.00033267531316

SASLE Services less energy services 0.99949903212504

SAT Transportation 0.99948728584425

SAT1 Private transportation 0.99856994633031

SEAC Women’s apparel 1.00013481541914

SEFK Fresh fruits 0.99856411046612

SEHF Gas (piped) and electricity 1.00599001717135
Table 2. Consumer Price Index for Urban Wage Earners and Clerical Workers (CPI-W),

U.S. city average: Level adjustment factors for use with seasonally adjusted data for

January 1987 through December 1996.

Item Code Item Title Level Adjustment Factor

SA0 All items 0.99986976295709

SA0E Energy 1.00239424663013

SA0L1 All items less food 1.00132140627495

SA0L1E All items less food and energy 0.99855667920074

SA0L2 All items less shelter 0.99987406497296

SA0L5 All items less medical care 1.00043259245613

SA0LE All items less energy 0.99918541082466

SA311 Apparel less footwear 1.00082806932476

SAA Apparel 1.00046429995875

SAA2 Women's and girls' apparel 0.99830926603719

SAC Commodities 0.99971844608330

SACE Energy commodities 1.00049021139203

SACL1 Commodities less food 0.99896306607373

SACL11 Commodities less food and beverages 1.00065887489629

SACL1E Commodities less food and energy commodities 0.99950036434787

SAF Food and beverages 0.99896327975145

SAF1 Food 0.99914464838781

SAF11 Food at home 0.99957910152453

SAF112 Meats, poultry, fish, and eggs 1.00032954489545

SAF1121 Meats, poultry, and fish 1.00024319048413

SAF11211 Meats 1.00070449887252

SAF113 Fruits and vegetables 0.99765340605007

SAF1131 Fresh fruits and vegetables 0.99232814442297

SAF115 Other food at home 1.00065445411741

SAG Other goods and services 0.99812850299715

SAH Housing 1.00112851501844

SAH1 Shelter 1.00072786448314

SAH2 Fuels and utilities 1.00106064678852

SAH21 Fuels 1.00825918692716

SAH3 Household furnishings and operations 1.00066000976400

SAM Medical care 0.99950521056747

SAS Services 1.00002220193720

SAS367 Other services 0.99759052510255

SAS4 Transportation services 1.00003146958059

SASLE Services less energy services 0.99855912477519

SAT Transportation 1.00006820851772

SAT1 Private transportation 0.99955073963680

SEAC Women's apparel 0.99824205799826

SEFK Fresh fruits 1.00096742067240

SEHF Gas (piped) and electricity 1.00660608543668

 

ii

Both the aggregation weight (aggwt) and the seasonal aggregation weight (saggwt) are constants. Aggwt

is based on expenditure data and is constant once calculated; saggwt is dependent on the pivot month
 

seasonally adjusted index, and is constant once that index has been declared final. (December 1986

seasonally adjusted indexes were final in February 1991.)

I p

i I

i p i t

saggwt I t

saggwt

aggwt SIX

SIX

,

, ,

, ,

∑( * )

∈

=

I p

i I

i p i t

aggwt I t

aggwt

aggwt SIX

SIX

,

, ,

, ,

∑( * )

∈

=

I p

I p

i I

i p i t

I p

I p

i I

i p i t

I t saggwt

I t aggwt

aggwt

saggwt

aggwt SIX

saggwt

aggwt

aggwt SIX

SIX

SIX

,

,

, ,

,

,

, ,

, ,

, ,

( * )

*

( * )

= =

∑

∑

∈

∈

I p

I p

I t aggwt I t saggwt aggwt

saggwt

SIX SIX

,

,

, , , , = *