ESTIMATING THE SAMPLING VARIANCE FOR ALTERNATIVE ESTIMATORS OF THE U. S.

CONSUMER PRICE INDEX

 Sylvia G. Leaver and Robert A. Cage, U.S. Bureau of Labor Statistics

Sylvia G. Leaver ,USBLS, 2 Massachusetts Ave., N.E., Room 3655, Washington, D.C. 20212

Any opinions expressed in this paper are those of the authors and do not constitute policy of the Bureau of Labor

Statistics.

KEY WORDS: Laspeyres, geometric mean,

superlative indices, Tornqvist, Fisher Ideal, Paasche,

stratified jackknife, VPLX

The official published United States Consumer

Price Index (CPI) is a modified Laspeyres index. The

Bureau of Labor Statistics has been investigating

alternative forms of basic level and aggregate index

estimators, and recently published a test series based

on a geometric mean basic index (BLS, 1997.) BLS

has also published index series based on 4 alternative

aggregation formulae, two of which are termed

superlative indices (Aizcorbe and Jackman, 1993.)

This report presents estimates of the sampling variance

for these alternative series.

In section one the estimators will be described. The

variance estimates for the basic level geometric mean

and the alternatively aggregated series will be given in

section two and three, respectively. In section four

comparisons between the stratified jackknife and

stratified random groups variance estimators for the

production Laspeyres series will be presented, and

section five will discuss conclusions and future

research directions.

1. Publication and Alternate Index Estimators

For a full discussion of the CPI the reader is

referred to Chapter 19 of the BLS Handbook of

Methods, (1992) and Leaver and Valliant (1995).

However, we will describe certain features of the CPI

pertinent to this study. The CPI is a modified

Laspeyres index, which is a ratio of the costs of

purchasing a set of items of fixed quality and quantity

in two different time periods. The CPI is calculated

monthly for the total US urban population for all

consumer items, and it is also estimated at other levels

defined by geographic area and item groups such as

food, shelter, and apparel.

Prices for the CPI are collected in 88 primary

sampling units (PSUs) in 85 geographic areas (the

New York area consists of 3 PSUs and the Los Angeles

area consists of 2 PSUs). Of these PSUs, 32 are selfrepresenting. The remaining 56 were selected

according to a stratified design in which one PSU was

selected from each stratum for each of 12 non-selfrepresenting index areas defined as medium-sized MSA’s,

small-sized MSA’s and urban, non-MSA’s in each of the four

Census regions (Northeast, Midwest, South, and West.)

The CPI is estimated at the item stratum-index area

level, although not all such indexes are published

every month. It is constructed in two stages. In the

first or basic level stage, the price index for an itemarea is updated every 1 or 2 months via a function of

sample price changes called a price relative. Let

I

ia

t

denote the index at time t, in item stratum i, area a,

relative to time period 0. Then

I

ia

t = RL I

ia

t t

ia

, −1 t −1

where RLia

t,t −1

denotes the price relative between times t

and t-1.

RLia

t,t −1

=











∑ 











∑ 

−

iaj

iaj t

j iaj c

iaj

iaj t

j iaj c

w

P

P

w

P

P

,

,

,

,

1

, where

P and w represent the price and sampling weight of

sample item j , and c represents the outlet frame

construction reference period for index area a.

 The index for higher level item I and area A

groupings is computed as a weighted sum of basic level

indices:

I

t

= ∑ ∑

i∈ ∈

ia

b

ia

t

a

r I , where

ia

b

r = the item-area relative importance or relative

consumer expenditure share, computed from the

Consumer Expenditure Survey for reference period b.

The first alternative form of the index we examine

is the basic level geometric mean index, published as

an experimental index series in April 1997. At the

basic level, this index is also a ratio of expenditure

estimates which are updated each month with a

geometrically averaged price relative. If IGia

t

denotes

the alternative index, called the geo-index, at time t,

relative to time period 0, then

IGia

t = RGia IG t t

ia

, −1 t−1
where RGia

t,t−1

denotes the one-period price relative

which is the weighted geometric mean of sample unit

price changes between times t and t-1.

 RGia

t,t−1

=















 ∏ =

′

∈ −

iaj

i

w

iaj t

iaj t j I

P

P

,

, 1

e

w

P

iaj P

iaj t

iaj t j I

ia

′











∑ 

∈ −

ln ,

, 1

here P represents the price and w¢ represents the quotelevel sampling weight of sample item j, normalized to

the same sample rotation base for all quotes in an itemindex area. Basic level geo-indices are aggregated in

the same manner as the Laspeyres index described

above.

BLS has also computed index series based on

alternative aggregations of yearly average basic level

production Laspeyres indices, relative to different base

years {b}:

{ I I

ia

t

ia

b

/ }, where I I

ia

t

ia

u

u year t month

year t month

= ∑

=

1

12 1

12 ,8612

,

,

, where the u’s

are in the same calendar year t. The formulae for the

different ways of aggregating the basic Laspeyres

indexes are given below:

Table 1: Alternative Index Aggregation Formulae

Laspeyres Lt,b r

I

I

i,a

b i,a

t

i,a

b

i,a

=

















∑

Paasche Pt,b =

=

















∑





















−1

ri,a

t

Ii,a

b

Ii,a

t

i,a

Fisher

Ideal

Ft,b = [Lt ,b×Pt ,b ]

1/2

Tornqvist

Tt,b

I

I

i,a

t

i,a

b

i a

r

i,a

b

r

i,a

t

=

















∏

+













,

1

2

1

2

=

∑ +





















































e

r

i,a

b

r

i,a

t i a

t

I

i a

b

I

i a

1

2

ln ,

, ,

Geometric

Mean

Aggregate Gt,b

I

I

i,a

t

i,a

b

i a

r

i,a

b

=

















∏

,

=

∑









































e

r

i,a

b i a

t

I

i a

b

I

i a

ln ,

, ,

where t indicates the current year, b indicates the base

year, and r

i,a

t

is the item-area relative importance

computed from the Consumer Expenditure Survey for

year t.

Diewert (1995) discusses the properties of the

different index forms. This paper will not discuss the

relative merits of the alternative index forms but will

only attempt to investigate their sampling variances.

Previous work in estimating the sampling variance

of the CPI has largely been devoted to the Laspeyres

estimator. Dippo and Wolter (1983) compared Taylor

series approximations to jackknifing. In a series of

papers, a hybrid random-groups-Taylor series approach

was used to estimate the sampling variance of the CPI

in Leaver (1990), Leaver et. al. (1991), and Leaver and

Swanson (1992). Leaver and Valliant (1995) compare

this hybrid estimator with a stratified random groups

estimator using VPLX software. Baskin and Leaver

(1996) explored variance estimation for the basic

geometric means estimator for the housing component

of the CPI. This paper builds on these previous studies

and is the first to provide standard error estimates for

the all items geo-index series and for alternatively

aggregated series and their differences.

2. Test Geo-Indices on Production Data

The Bureau is currently producing a test geo-index,

and recently published six years of back indices from

January 1991 through February 1997.

The test geo-index is not being calculated in the

same processing environment as the official Laspeyres

index. For this reason, a test Laspeyres index is being

computed simultaneously with the geo-index for

purposes of comparison.

 Estimates of variance for price change for both the

test geo-index and the test Laspeyres index were

produced using a stratified jackknife estimator

implemented in VPLX. For a description of the VPLX

software see Fay (1990). Details of this estimator are

given below.

 The stratified jackknife variance computations in

this application are based on a segmentation of the CPI

sample into 8 separate strata, one for each of 7 major

item groups for self-representing index areas, and one

for all items for non-self-representing index areas. The

sample for each of the 7 major item group strata

consists of 32 clusters, where each cluster is the sample

from one of 32 self-representing index areas. The

sample for the eighth stratum consists of 12 clusters,

where each cluster is the sample from one of 12 nonself-representing index areas.

 For the U.S. All Cities Index, VPLX constructs

replicate indices I

sm

u

 ( u = t or t - k ) for each of ns

clusters in each stratum s. This is done by deleting the
index for cluster (m) in stratum (s), and aggregating

the indices for the remaining clusters, while rescaling

their weights (relative importances) to produce an

estimate for the full stratum. These are then

aggregated with full sample estimates for the

remaining strata. The index corresponding to replicate

(sm) is

{ } { }

( )

' { }

' ' ' '

'

I r I r I sm

u

s S s

s m

b

s m

u

m M

sm

b

sm

u

s m M s m

= ∑ ∑ + ∑

∈ −

′

′∈

′

′∈ −

where {Ms-m} denotes the set of clusters for stratum s

omitting cluster m, and sm

b

r '

 is the rescaled relative

importance for the stratum-cluster sm′ in replicate

(sm). Replicate estimates of k-month price change

were derived by taking ratios of replicate indices:

( ) ( ) ( )

,

I I I sm

t t k

sm

t

sm

− t−k

=

The stratified jackknife estimator of the variance of

,

I

t t−k

 is given by

( )

( )

v I [ ]

n

n

I I

sjk

t t k s

s

sm

t t k t t k

m

n

s

S s

,

( )

, − − , −

= =

=

−

∑ ∑ −

1 2

1 1

where the first sum is over all strata in the sample.

Price change variances were calculated for both one

and twelve month periods, at the All Cities level for

January 1991 through December 1996. Graphs of 12-

month price change with two-standard error bands are

presented in Figures 1-3. Index change estimates for

the six-year period and their standard errors are given

for All Items-All Cities in Table 3. The graphs show

that the test indices are smoother and have smaller

variances than the corresponding production index.

The estimates of sampling variance for both test

Laspeyres and test geo-indices are very similar.

Differences in price change estimates between the

two test series and the production Laspeyres series and

their standard errors were also estimated. A graph of

the difference in cumulative price change between the

test Laspeyres and test geo-index series with twostandard error bands is given in Figure 4. Differences

in index change over the six-year period and their

standard errors for All Items and 7 major groups are

given in Table 4.

3. Higher Level Aggregates with Production Data

 Estimates of variance for alternatively aggregated

index series were also produced for 1987-1996 using a

stratified jackknife estimator. Stratum and cluster

definitions were the same as described for the basic

level geo-index. The replicate index aggregate

formulae for each alternative index are given below.

Price change variances were calculated for year to

year and cumulative index change at the All Items,

U.S. All Cities level. In general, the alternative indices

reflect smaller price changes than the corresponding

Laspeyres index. The estimates of sampling variance

for all alternative indices are very similar.

Table 2: Replicates for Alternative Index Aggregation

Formulae

Laspeyres ( )

{ }

{ }

( )

,

' { }

' ' '

,

'

,

'

L r I

r I

sm

u b

s S s

s m

b

s m

u b

m M

sm

b

sm

u b

m M m

s

s

= ∑ ∑

+ ∑

∈ −

′

′∈

′

′∈ −

Paasche

{ }

{ }

( )

,

' { }

' '

'

,

'

'

,

P

r

I

r

I

sm

u b

s S s

s m

u

s m

u b

m M s

sm

u

sm

u b

m M s m

=

∑











∑ 

















+ ∑

















































−

∈ − ′∈ ′

′∈ − ′

1

Fisher

Ideal

( ) [ ]

,

/

( )

,

( )

,

F sm L P

u b

sm

u b

sm

u b

= ×

1 2

Tornqvist

{ }

{ }

{ }

ln( )

( )

ln( )

( )

( )

,

' { }

' '

'

,

'

'

,

' { }

' '

'

,

'

'

,

T e

r

I

r

I

e

r

I

r

I

sm

u b

s S s

s m

b

s m

u b

m M s

sm

b

sm

u b

m M s m

s S s

s m

u

s m

u b

m M s

sm

u

sm

u b

m M s

=

∑











∑ 

+ ∑





























×

∑











∑ 

+

∈ −

′

′∈

′

′∈ −

∈ −

′

′∈

′

′∈

2

2

2

ln

2

ln

{ − }

∑



























 m

Geometric

Mean

Aggregate

( )

{ }

{ }

ln( )

( )

( )

,

' { }

' ' '

,

'

,

'

G e

r I

r I

sm

u b

s S s

s m

b

s m

u b

m M

sm

b

sm

u b

m M m

s

s

=

∑ ∑

+ ∑





















∈ −

′

′∈

′

′∈ −

ln

Of particular interest were differences between

alternative estimators in the “true index change,”

computed as the average index change occurring over

the year following a given index base year. Differences

in “true one-year index change” estimates between the

Laspeyres series and the four alternative index series,

and their standard errors were also estimated using the

same methodology and are given in Table 5.
Table 3. Six-Year Price Change and Standard Errors for All Items,

 U. S. All Cities CPI for Production Laspeyres, Test Laspeyres, and Test Geo-Index Estimators

Index Series

6-Year Percentage Price Change,

9012-9612

Standard Error

Production Laspeyres

All Items 18.4841 .3822

Test Laspeyres

All Items 17.8190 .4064

Test Geo-index

All Items 15.5384 .3898

Table 4: % Differences between Estimator Series and Their Standard Errors for All Items and

7 Major Groups, U.S. All Cities Average CPI, December 1990-December 1996

Series

Comparison

Test Laspeyres vs. Geo-index Production Laspeyres vs. Geo-index

Item Difference,

Six-Year

Price

Change

S. E.

6-Year

Average %

Difference,

Dec-Dec

Price Change

S. E.

Difference,

Six-Year

Price

Change

S. E.

6-Year

Average %

Difference,

Dec-Dec

Price Change

S. E.

All Items 2.2806 .1127 .3252 .0159 2.9458 .1220 .4186 .0174

Food 3.1918 .0866 .4598 .0120 3.1898 .0920 .4595 .0127

Housing .8516 .1822 .1222 .0260 2.4932 .2517 .3546 .0359

Apparel 7.8098 .3505 1.2919 .0548 7.7869 .3463 1.2883 .0542

Transportation 2.0359 .1064 .2995 .0158 2.0181 .1074 .2969 .0159

Medical Care 3.6469 .2361 .4511 .0289 3.6467 .2291 .4510 .0280

Entertainment 4.4196 .3600 .6258 .0523 4.3807 .3483 .6204 .0506

Other C&S 1.7078 .2153 .2151 .0272 1.6367 .2319 .2062 .0291

Table 5: % Differences between Paired Alternatively Aggregated Estimators of True One-Year

 Average Price Change and Their Standard Errors for All Items, U.S. All Cities , 1987-1995

Series

Comparison

Laspeyres vs.

Tornqvist

Laspeyres vs. Fisher

Ideal

Geomean vs.

Tornqvist

Paasche vs.

Tornqvist

Base Year,

Index Year

%

Difference,

True OneYear Price

Change

S.E. %

Difference,

True OneYear Price

Change

S.E. %

Difference,

True OneYear Price

Change

S.E. %

Difference,

True OneYear Price

Change

S.E.

8612, 1987 .1143 .0204 .1153 .0178 -.0489 .0203 -.1164 .0160

1987, 1988 .1309 .0185 .1302 .0183 -.0308 .0208 -.1297 .0183

1988, 1989 .0995 .0245 .1037 .0254 -.0547 .0213 -.1081 .0265

1989, 1990 .1485 .0337 .1443 .0327 -.0189 .0292 -.1403 .0318

1990, 1991 .1277 .0199 .1301 .0214 -.0259 .0176 -.1326 .0232

1991, 1992 .1319 .0175 .1332 .0184 -.0175 .0165 -.1346 .0195

1992, 1993 .0873 .0189 .0844 .0187 -.0488 .0183 -.0815 .0186

1993, 1994 .1025 .0161 .1045 .0159 -.0239 .0163 -.1066 .0161

1994, 1995 .1151 .0200 .1173 .0204 -.0188 .0197 -.1196 .0209

8-Year Avg

(1987-95)

.1179 .0086 .1185 .0087 -.0299 .0062 -.1191 .0088

4. Comparison of Variance Estimators

 Test geo-index and alternatively aggregated index

variances were computed using a stratified jackknife

estimator because within-index area replicate indices

were not available for these series. The stratified

jackknife estimates for the production Laspeyres

series in Leaver and Valliant (1995) and for shelter
geo-index series in Baskin and Leaver (1996) are

based on availability of within-index area

replication.

 The stratified jackknife estimator assumes equal

expected price change among the clusters within a

stratum, so the resulting variances should on the

average represent overestimates of the true variances

for items for which this assumption does not hold.

An example of this would be rent change, which

varies substantially between index areas.

 Graphs of the ratios of standard error estimates

computed using a stratified random groups estimator

(Leaver and Valliant, 1995) and stratified jackknife

estimators for the production Laspeyres series for all

items and apparel for January 1991 through

December 1996 are given in Figures 5 and 6.

5. Conclusions

 Regarding the test geo-index series and its test

and production Laspeyres counterparts, Table 3

indicates very small differences in variance estimates

among the three series. In terms of long term

change, it is quite clear that the two test series are

estimating different measures. The test geo-index

estimator produced a significantly lower price

change measure than the test Laspeyres index for

every major group over the six year study. This

difference is most remarkable in apparel, followed by

entertainment, medical care, and food.

The difference in price change between the test

Laspeyres and geo-index series has been regarded by

some as a measure of the lower-level substitution

bias inherent in the CPI. Previous BLS research has

estimated this difference to be around 0.3 to 0.5

percentage points per annum since 1990, and

approximately 0.25 since January 1996. Results here

indicate that the estimate is .3252 percentage points

per annum, over the six year period 1991-96, with a

95% confidence interval of (.294, .356).

 It is also quite clear from the comparisons in

Table 5 that the Fisher Ideal and Tornqvist series

are estimating different measures from the Laspeyres

aggregate. Differences in “true index change”, their

9 year average, and cumulative price change are

significant. In a similar manner the Paasche price

change estimator consistently underestimates the

Tornqvist series. The geomeans series also

underestimates the Tornqvist series, though not

significantly so with each yearly change. The 9-year

average percentage difference between the two

series, however, is significant.

The difference between the Laspeyres

aggregation and the Tornqvist aggregation has been

viewed as a measure of the upper-level substitution

bias in the CPI. Most estimates of this difference

cluster around 0.1 to 0.25. Results here indicate that

the measure of .1175 percentage points per annum

over the 9 year period 1987-95 with a 95%

confidence interval of (0.101, 0.134).

 Regarding variance estimators, the stratified

jackknife fairly consistently overestimates the

stratified random group variance for food,

transportation and housing, though these graphs are

not shown here due to space limitations. These

findings are not entirely surprising, since these

product groups are associated with larger local

market effects. It underestimates the stratified

random group estimator quite remarkably for apparel

and this warrants further investigation.

 The current research indicates the variance

estimates for the geo-index and alternative indices

behave similarly to the estimates for the Laspeyres

index. There is little evidence that the differences in

functional forms are producing different estimates of

variance.

6. Acknowledgments

The authors would like to thank Janet Williams,

Richard Valliant, Robert Baskin, Owen Shoemaker,

and John Greenlees for their careful reading of this

paper. The authors also thank Robert Fay for

providing invaluable assistance using the VPLX

software. The authors thank Brian Hedges and Janet

Williams for their support on this project.

7. References

Aizcorbe, A. and Jackman, P. (1993) “The

commodity substitution effect in CPI data, 1982-91,”

Monthly Labor Review, December 1993, pp. 25-33.

Baskin, R.M. and Leaver, S.G. (1996) “Estimating

the Sampling Variance for Alternative Forms of the

U.S. Consumer Price Index,” Proceedings of the

Survey Research Methods Section, American

Statistical Association, to appear

Bureau of Labor Statistics (1992) BLS Handbook of

Methods, Washington. DC: U.S. Government

Printing Office, pp. 176-235.

Bureau of Labor Statistics, (1997) "The

Experimental CPI using Geometric Means (CPI-UXG)," April 10, 1997 (Washington: Bureau of

Labor Statistics).
Diewert, W.E. (1995), “Axiomatic and Economic

Approaches to Elementary Price Indices,” University

of British Columbia Discussion Paper No. 95-01.

Dippo, C. S., and Wolter, K. M. (1983), "A

Comparison of Variance Estimators Using the

Taylor Series Approximation," Proceedings of the

Survey Research Methods Section, American

Statistical Association, pp. 113-119.

Fay, R. E., (1984), "Some Properties of Estimates of

Variance Based on Replication Methods,"

Proceedings of the Survey Research Methods

Section, American Statistical Association, pp. 495-

500.

Fay, R. E., (1990), "VPLX: Variance Estimates for

Complex Samples," Proceedings of the Survey

Research Methods Section, American Statistical

Association, pp. 266-271.

Leaver, S. G., (1990), "Estimating Variances for the

U.S. Consumer Price Index for 1978-1986,"

Proceedings of the Survey Research Methods

Section, American Statistical Association, pp. 290-

295.

Leaver, S. G., Johnstone, J. E., and Archer, K. P.

(1991), "Estimating Unconditional Variances for the

U.S. Consumer Price Index for 1978-1986,"

Proceedings of the Survey Research Methods

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