EVALUATION OF COMPOSITE ESTIMATION METHODS FOR COST WEIGHTS IN THE CPI

David C. Swanson, Sharon K. Hauge, Mary Lynn Schmidt, U.S. Bureau of Labor Statistics

David C. Swanson, 2 Mass. Ave., NE, Room 3655, Washington, DC 20212

Key Words: composite estimation, sample size, mean

squared error

Any opinions expressed in this paper are those of the

authors and do not constitute policy of the U.S.

Bureau of Labor Statistics.

Introduction

The Consumer Price Index (CPI) uses data from the

Consumer Expenditure Survey to weight individual

area/item price indexes in order to aggregate them to

higher-level price indexes. The accuracy of the

weights of local area indexes is improved through a

process called composite estimation, in which

strength is borrowed from a larger geographic area.

Larger geographic areas have similar expenditure

patterns to those of local areas, but the patterns are

more stable because they come from a larger sample.

In this paper we describe the method of composite

estimation used in the CPI’s 1998 Revision, and then

summarize research to improve the composite

estimation process conducted over the past ten years

at the U.S. Bureau of Labor Statistics.

Background

The CPI is one of the most important economic

indicators available in the United States. It measures

the change in prices of goods and services that are

purchased by American consumers. The goods and

services currently represented in the CPI are those

purchased by urban Americans during the 3-year

period 1993-95.

The samples and weights used to compute the CPI

are updated periodically to reflect changing

consumer expenditure patterns and changing

demographics. In the past these updates occurred

approximately every ten years, but in the future they

will occur more often. The weights come from

another survey conducted by the U.S. Bureau of

Labor Statistics called the Consumer Expenditure

Survey, and are updated every time the CPI is

revised.

To produce the CPI, price data are collected every

month from a random sample of stores in thirtyeight geographic areas across the United States.

These geographic areas are called index areas.

Examples of index areas are the Boston metropolitan

area, the St. Louis metropolitan area, and the San

Francisco metropolitan area. Within each of the 38

index areas price data are collected for 211 item

categories called item strata. Together the 211 item

strata cover all consumer purchases. Examples of

item strata are Bananas, Women’s Dresses, and

Electricity.

Multiplying the number of index areas by the

number of item strata gives us 8,018 (= 38 × 211)

different area/item combinations for which data need

to be collected. Price indexes are calculated for each

one of these 8,018 area/item combinations, and then

the indexes are aggregated to form higher-level price

indexes using expenditure estimates from the

Consumer Expenditure Survey as their weights.

Because of small sample sizes within individual

index areas, composite estimation is used to increase

the accuracy of the weights.

Current Method of Composite Estimation

In the CPI’s 1998 revision composite estimation of

Consumer Expenditure Survey data was performed

in the following way:

First, expenditure estimates were obtained from the

Consumer Expenditure Survey for the 3-year period

1993-95 for each of the 8,018 area/item

combinations. Then each item stratum’s relative

importance was computed. Relative importance is

the total expenditure made on a particular item

stratum divided by the total expenditure made on all

item strata. In other words, it is the proportion of

total consumer expenditures made on a specific item

stratum.

For example, according to the Consumer

Expenditure Survey, consumers in the Boston

metropolitan area spent $463 million on women’s

dresses during the 3-year period 1993-95, and $224

billion on all items combined. Dividing $463

million by $224 billion gives a relative importance of

0.0021, meaning that consumers in the Boston
metropolitan area spent 0.21 percent of their total

expenditures on women’s dresses.

Relative importances were computed by index area,

and also by major area. For the purpose of

composite estimation the United States was divided

into 8 major areas, which are the 4 Census regions

(Northeast, Midwest, South, West) cross-classified

by the sets of self-representing and non-selfrepresenting index areas.

After computing relative importances for each item

stratum within each index area and major area, the

composite estimation process simply involved

replacing every index area relative importance by a

weighted average of the index area and major area

relative importances. The weighted average is

expected to be a more accurate estimate of the index

area’s true relative importance because the index

area and major area have similar expenditure

patterns, but the major area has a larger sample size.

For example, in the Boston metropolitan area

consumers spent 0.21 percent of their total

expenditures on women’s dresses. In all selfrepresenting areas in the Northeast Region they

spent 0.32 percent on women’s dresses. In the

composite estimation process Boston’s estimate of

0.0021 is replaced by a weighted average of the two

estimates: 0.0021 is replaced by α0.0032 +

(1−α)0.0021 where 0 ≤ α ≤ 1. Of course the entire

Northeast Region has a larger sample size than the

Boston metropolitan area alone, so the weighted

average is expected to have less variability than

Boston’s original estimate.

Mathematically, let x be an item stratum’s relative

importance estimate for a major area, and y be its

estimate for an individual index area within that

major area. Then in composite estimation y is

replaced by y* = αx + (1−α)y, where 0 ≤ α ≤ 1.

This is a weighted average of the index area and

major area relative importance estimates. The

weight α is chosen to be the number between 0 and 1

that minimizes the mean squared error (MSE) of y*.

Although MSEs are not exactly the same as

variances, they are very similar. To give the basic

idea of how the formula for α is derived, the

following shows how to find the value of α that

minimizes the variance of y*:

V(y*) = V(αx + (1−α)y)

= α

2V(x) + (1−α)2V(y) + 2α(1−α)Cov(x,y)

= α

2[V(x) + V(y) − 2Cov(x,y)] −

2α[V(y) − Cov(x,y)] + V(y)

= α

2V(y−x) − 2α[V(y) − Cov(x,y)] + V(y)

Then to minimize the variance of y*, the derivative

is taken with respect to α and set equal to 0:

dα

dV(y )

*

 = 2αV(y−x) − 2 [V(y) − Cov(x,y)]

0

d

dV(y )

*

=

α

 implies α = 

V(y x)

V(y) Cov(x, y)

−

−

This shows how to obtain the value of α that

minimizes the variance of y*. When MSE is

minimized the formula becomes the following:

α = 

E[ (y x) ]

V(y) Cov(x,y)

2

−

−

When we look at all 8,018 area/item combinations,

the median reduction in root mean squared error

( MSE , or RMSE) achieved by this method of

composite estimation was 14 percent, so the method

was fairly effective. It was also relatively easy to

implement.

Research to Improve Composite Estimation

Although the current method of composite

estimation worked fairly well, ten years ago the CPI

program started to look for ways to improve the

process even further. This led to a research program

in which a number of alternative methods and

models were developed. They are briefly described

below.

First is a nearest neighbor method in which a

weighted average of the relative importances from

the index area and the index areas geographically

closest to it is used. After that, a multivariate

generalization of the current method is described,

followed by several Bayesian models, and then

finally a method in which the inputs to the current

model are improved.

For more details on these methods, see the articles

listed in the bibliography at the end of this paper.

Nearest Neighbor Method

This is the same as the current method, but with one

slight change: instead of compositing an index area

with its major area, the index area is composited

with the index areas that are geographically closest
to it. For example, Baltimore is in the South region,

which includes many index areas that are far from

Baltimore, such as Miami, New Orleans, Houston,

and Dallas. Baltimore’s nearest neighbors, however,

are mostly in the Northeast region, such as

Philadelphia and New York. In the nearest neighbor

method, instead of compositing Baltimore with the

set of self-representing index areas in the South,

Baltimore is composited with the set of index areas

that are geographically closest to it. An index area’s

expenditure patterns are expected to be closer to its

nearest neighbors than they are to those of distant

index areas that just happen to be assigned to the

same region of the country.

Here is an example. Baltimore’s 2 nearest neighbors

are Washington, DC and Philadelphia. The total

expenditures on women’s dresses and all items

combined during the 3-year period 1993-95 are

shown below. According to the Consumer

Expenditure Survey, people in the Baltimore

metropolitan area spent $230 million on women’s

dresses, and $80,395 million on all items. Thus

women’s dresses represented 0.29 percent of total

expenditures in Baltimore. Similar numbers are

shown for Washington, DC, Philadelphia, and all 3

areas combined.

Total Expenditures, 1993-95 ($ millions)

Index Area

Women’s

Dresses All Items

Relative

Importance

Baltimore 230 80,395 .0029

Washington, DC 557 180,951 .0031

Philadelphia 718 197,160 .0036

Total 1,505 458,506 .0033

The composite estimate of relative importance for

Baltimore is then computed as α0.0033 +

(1−α)0.0029, where 0 ≤ α ≤ 1. The number α is

computed the same way as in the current method –

by taking the derivative of the MSE with respect to

α, setting the derivative equal to 0, and then solving

for α. In this example α turned out to be 0.59.

When α=0.59 the composite estimate becomes

0.0031, which means that the new expenditure

estimate for women’s dresses in Baltimore is 0.0031

× $80,395 million = $249 million. Thus $249

million is the new (and presumably more accurate)

estimate of total expenditures on women’s dresses in

the Baltimore metropolitan area for the 1993-95

period.

In the example above Baltimore was composited

with its 2 nearest neighbors. Of course it could have

been composited with its 3 nearest neighbors, its 4

nearest neighbors, and so on. A graph comparing

the median reduction in RMSE ( MSE ) for item

stratum/index area relative importances against the

number of neighbors used is shown below.

The graph shows that the optimal number of

neighbors to use is 8. When 8 neighbors are used,

the median reduction in RMSE is 15 percent, which

is 1 percentage point better than the current method.

A Multivariate Generalization of the Current

Method

When we look at the formula for α in the current

method of composite estimation, it is clear that it

takes into consideration covariances between index

areas, but ignores covariances between item strata.

In 1992 a multivariate procedure similar to the

current method was developed in which covariances

between different item strata were taken into

consideration as well. This generalization of the

current method naturally produced superior results.

The median reduction in RMSE was 14 percent for

the current method, 15 percent for the nearest

neighbor method, and 18 percent for this

multivariate method. Thus the multivariate method

worked a little better than the current method.

The problem with the multivariate method was that

it required calculating some matrix inverses, and

whenever one or more of the item stratum/index area

combinations had no reported expenditures in it, the

matrices could not be inverted. In those situations

the method broke down. As a result more research

was needed.

8%

10%

12%

14%

16%

0 2 4 6 8 10 12 14 16

NUMBER OF NEIGHBORS
Bayesian Models

Eight different Bayesian models were proposed to

improve the direct survey estimates of expenditure.

Two of them are described here.

In 1992 a linear empirical Bayes model was

proposed to estimate the true expenditure. The basic

idea of the model was that the true expenditure can

be modeled as a linear combination of other

variables, θ = β + ε

T

x = β + β + + β + ε n n

x x K x 1 1 2 2,

where θ is the true expenditure, the ix ’s are some

exogenous variables, the βi’s are coefficients of

those variables, and ε is a random error term with

mean 0.

For example, the total expenditure on women’s

dresses might be modeled as a linear function of the

number of women living in the index area

( θ = β + β + β + ε 1 1 2 2 3 3

x x x , where 1x is the number of

women aged 15-24, 2x is the number of women aged

25-65, and 3x is the number of women over 65); or

the total expenditure on new cars might be modeled

as a linear function of the number of families in

various income categories ( θ = β + β + β + ε 1 1 2 2 3 3x x x ,

where 1x is the number of families with incomes

between $0 and $30,000, 2x is the number of families

with incomes between $30,001 and $75,000, and

3

x is the number of families with incomes over

$75,000).

Then under certain Bayesian assumptions the true

expenditure was estimated to be a weighted average

of the direct survey estimator and the linear model of

it:

θ = α + − α β

T

(1 )x

ˆ y , where 

V( ) ( | )

( )

θ + θ

θ

α =

V y

V

Of course this model assumes that we have some

outside variables that produce good estimates of

expenditure. However, the problem with the model

is that finding such variables can be difficult, and

using them to produce official expenditure estimates

is rather difficult to justify at a survey organization

such as the Bureau of Labor Statistics. As a result,

we did not pursue this model any further.

In the same year a hierarchical Bayes model was

proposed to estimate expenditures. Let θkj be the

true total expenditure for item stratum k in index

area j, letYkj be the direct survey estimator of the total

expenditure, and let T

j j Kj [ , , ] θ = θ1 K θ and

T

Yj Y j YKj [ , , ] = 1 K be vectors of those quantities. Also

let us make the following assumptions:

(i) Conditional on the true expenditure θ j, the Yj’s

are independent with | ~ ( , ) Yj θ j N θ j Vj

(ii) Conditional on Kμ ∈ R and 0 < r0 < ∞ , the θ j

’s

are independent with | , ~ ( , )

1

0 0

r N r I j

−

θ μ μ

(iii) The prior density function of μ and 0r is given

by ( , ) exp( )

2 0

1

1

0 0

2

1

r r er

f

π μ ∝ −

−

(iv) The values of Vj, f, and e are assumed to be

known.

Under this model, the posterior distribution of

T T

J

T

1

θ = [θ ,K,θ ] given T T

J

T

1 Y = [Y ,K,Y ] and 0

r is a

multivariate normal distribution with mean vector

T T

J

T E( |Y,r ) [Q , ,Q ] θ 0 = 1 K , where Qj = (I −VjWj

)Yj +

∑ ∑=

−

=

















m

j

j j

m

j

VjWj Wj W Y

1

1

1

, and where 1 1

0

( )

− − W = V + r I j j

.

The hierarchical Bayes estimator of θ was then

obtained using the familiar iterated formula for

expectations, θ = θ =

∧

E( | Y) [ ( | , ) | ] E E θ Y r0 Y , where

the expectation is computed using the posterior

density of 0r given Y, found to be:

∑ ∏ ∑ = =

−

=

∝ − +

m

j

j j

T

j

m

j

j

m

j

g r Y Wj W er Y W Y

1

2 0

1

1 1

( 0| ) exp[ {

2

1

2

1

( ) }]

1

1

1 1

∑ ∑ ∑=

−

= =

















−

m

j

j j

m

j

j

T

m

j

WjYj W W Y

Regardless of the model’s effectiveness, it is clear

that the model is rather hard to describe to nonstatisticians, which makes it is hard to convince

them of its value. It is also rather complicated to

implement in a production environment. As a result,

we did not pursue this model any further either.

In addition to the two models described above, six

other Bayesian models were proposed. All of them

were considered to be overly complex, both in terms

of the ability to explain them to non-statisticians,

and in terms of the ability to implement them in a

production environment. As a result, none of them

was pursued any further.

Current Research

As we mentioned before, the current method of

composite estimation involves replacing an item

stratum’s relative importance for a particular index
area with a weighted average of the index area and

major area relative importance estimates. That is, y

is replaced by y* = αx + (1−α)y, where y is the index

area estimate, x is the major area estimate, and

E[ (y x) ]

V(y) Cov(x, y)

2

−

−

α = .

Recently a proposal was made to use the same

model, but with an improved estimate of α. The

method of improving the estimate of α is simply to

improve the estimates of V(y), Cov(x,y), and

E[(y x) ]

2

− that go into its formula, which in turn

improves the estimate of α. Specifically, the

recommendation is to use stratified estimators of

V(y), Cov(x,y), and E[(y x) ]

2

− to improve their

stability.

Currently we are evaluating this method. So far we

have identified and tested over a dozen variables that

were conjectured to be effective in stratifying the

families in the Consumer Expenditure Survey. The

variables consisted of several expenditure/income

variables, several demographic variables, and several

geographic variables.

After stratifying the families with these variables, we

used the following formula to measure how much of

the total variance in the Consumer Expenditure

Survey was explained by each variable:

∑∑∑

∑∑∑

∈

∈

−

−

= −

i s f s

if i

i s f s

if is

RI RI

RI RI

R 2

2

2

( )

( )

1

where RIif is the relative importance of item

category i for family f, RIis is the relative importance

of item category i for all families in stratum s, and

RIiis the relative importance of item category i for

all families in the Consumer Expenditure Survey’s

sample. The numerator of this ratio is the stratified

variance, and the denominator is the un-stratified

variance. One minus the ratio is the proportion of

total variance “explained” by the stratification

variables, which is commonly called R-squared.

Stratifying Variable 2 R

Total family expenditures 9.4%

Total family pre-tax income 7.8%

Age 4.0%

Family Type 3.8%

Education 2.2%

Family size 2.0%

Race 0.7%

Tenure (owner/renter) 1.9%

Type of segment 1.2%

Degree urban 1.1%

The table above shows that expenditure/income

variables explained the greatest amount of variance,

followed by demographic variables, and then

geographic variables. This means that stratifying by

expenditure/income variables will probably improve

the stability of α by the greatest amount, followed by

demographic variables, and then geographic

variables.

After measuring the stratification effect of individual

variables, we also looked at whether stratifying by

more than one variable would have a greater effect.

For example, we looked at whether age and

education together would produce better results than

either variable by itself. We found that additional

variables did not have a noticeable effect, so in the

future we will be testing this method by stratifying

with just one variable.

Summary

The current method of composite estimation works

fairly well. It is easy to implement, and successfully

reduces RMSE by a median amount of 14 percent.

Some of the other methods reduce RMSE a little

more, but they are also harder to implement, and

sometimes break down. All eight Bayesian models

proposed are somewhat complex, difficult to describe

to non-statisticians, difficult to convince nonstatisticians of their value, and difficult to implement

in a production environment. Our current research

focuses on returning to the current method of

composite estimation and improving the stability of

the weight α that goes into it. Further improvement

of the current method looks promising.

References

Cohen, M. P. and Sommers, J. P. (1984).

“Evaluation of Methods of composite Estimation of

Cost Weights for the CPI”, Proceedings, Section on

Business and Economic Statistics, American

Statistical Association, pp. 466-471.

Datta, G. S., and Lahiri, P. (1992). “Robust

Hierarchical Bayes Estimation of Small Area

Characteristics in Presence of Covariates,” Technical
Report, University of Georgia, Department of

Statistics.

Ghosh, M., and Sohn, S. Y. (1991). “An Empirical

Bayes Approach towards Composite Estimation of

Consumer Expenditure,” Technical Report,

University of Florida, Department of Statistics.

Lahiri, P. (1992). “Estimation of Consumer

Expenditures for Small Areas – the Hierarchical

Bayes Approach.” Technical Report, University of

Nebraska – Lincoln, Division of Statistics.

Lahiri, P., and Wang, W. (1992). “A Multivariate

Procedure Towards Composite Estimation of

Consumer Expenditure for the U.S. Consumer Price

Index Numbers”. Survey Methodology, Statistics

Canada, vol. 18, no. 2, 279-292.

Lahiri, P. (1994). “Estimation of Consumer

Expenditure: A Linear Empirical Bayes Approach.”

Technical Report, University of Nebraska – Lincoln,

Division of Statistics.

U.S. Bureau of Labor Statistics, BLS Handbook of

Methods (1997), Washington, DC, U.S. Government

Printing Office.