ESTIMATING PRICE RELATIVES FOR THE U.S. CONSUMER PRICE INDEX 

WHEN PRICED ITEMS ARE RESTRATIFIED 

Mary Lee Fuxa, U.S. Bureau of Labor Statistics 

2 Massachusetts Ave., N.E., Room 3655, Washington, D.C. 20212 

Any opinions expressed in this paper are those of the author 

and do not constitute policy of the Bureau of Labor 

Statistics. 

Key Words: Item Strata, Price Relative, 

Quote-Level Weights 

1. Introduction

Approximately every 10 years the fixed market basket 

of commodities and services supporting the Consumer 

Price Index is updated in a process referred to as a 

revision of the index. Samples of specific items in the 

fixed market basket are selected within a newly 

designed, stratified sample of geographic areas on a 

continuing rotation basis. The prices of the specific 

items are quoted every month by representatives of the 

Bureau of Labor Statistics. 

Commodities and services have 70% relative 

importance by weight in the index, and shelter has 30%. 

For the January 1998 revised index, the stratification of 

the items in the fixed market basket for commodities 

and services was extensively redefined. For the 36 

geographic areas that were not in the sample for the 

December 1997 index, new samples of specific items 

were selected for the 183 item strata. For the 51 

geographic areas continuing in the sample: 24% of the 

item strata were designated for new sampling; 33% did 

not change in definition; 43% changed but were not 

designated for new sampling due to budgetary 

constraints. 

This paper documents the adjustment in the monthly 

estimation of the index that is necessary for the 43% 

item strata until new samples are selected and rotated in 

for the 51 geographic areas. 

2. Method of Sampling and Formulation

 of the Index and its Components

The geographic areas are referred to as the primary 

sampling units (PSU’s). The PSU’s were selected to 

represent 38 index areas. A PSU is classified by the 

region it is in as Northeast, South, Midwest, or West. 

It is also classified by its population size as selfrepresenting or non-self-representing. A selfrepresenting PSU is large enough to be its own index 

area. An example is Tampa, Florida, and its 

surrounding area. For each region, there is an index 

area composed of non-self-representing PSU’s. The 

index area is divided into geographic strata, and a single 

PSU is selected from each geographic stratum. An 

example of a PSU selected to represent the index area 

for the South is Ocala, Florida, and its surrounding area. 

Indexes are computed at the item-stratum, index-area 

level each month. The indexes are weighted and 

aggregated to produce the all-items U.S. index. 

The sample of priced items for a combination of an 

item stratum and index area is composed of replicate 

samples. A self-representing PSU has 2 or 4 replicate 

samples selected for it. A non-self-representing PSU is 

one of the replicate samples for its index area, or a 

component of a replicate sample. Indexes are 

computed at the replicate level each month for the 

purpose of estimating index variance. 

For the first stage of sampling for an item stratum 

within a PSU, or replicate of the PSU, the item stratum 

is divided into item categories referred to as entry-level 

items (ELI’s). An example is the item stratum Cakes, 

Cupcakes, and Cookies. It is composed of the ELI 

Cakes and Cupcakes (Excluding Frozen) and the ELI 

Cookies. One or more ELI’s are selected with 

probability proportional to the expenditure they 

represent. At each stage of sampling, multiple 

selections of a sample unit are possible. A single 

selection of an ELI is referred to as an ELI hit. 

Each ELI has a corresponding sampling frame for 

outlets in the PSU that sell the ELI. A bakery in the 

food court of a shopping mall is an example of an outlet 

that sells the ELI Cookies. More than one ELI may 

share the same sampling frame. In the second stage of 

sampling, two or more outlets from a frame are selected 

with probability proportional to the expenditure they 

represent. A single selection of an outlet is referred to 

as an outlet hit. The outlets selected from a frame are 

matched to the selected ELI’s that correspond to the 

frame. 

For a given combination of an ELI hit and an outlet hit, 

a unique item within the outlet is selected with 

probability proportional to the expenditure it represents. 

An example of a unique item for the ELI Cookies is a 
 2 

1-lb. bag of chewy-style chocolate-chip cookies with 

walnuts, of a particular brand name. A single selection 

of a unique item is referred to as a quote. For each 

quote, the price of the unique item is quoted each 

month. 

An index t, 0 I , reflecting price change from month 0 

to the current month t , is calculated at the item-stratum 

replicate level by multiplying the index for the previous 

month ( ) t−1 , 0 I by the estimate of price change between 

t and t-1 . This one-month price change is referred to 

as the price relative Rt,( ) t−1 . 

 (1) 

() () ( ) , 1

1

, 0 1 ,0 , 1 −

=

= − − = ∏ k k

t

k

It I t Rt t R

For a replicate of a PSU that is self-representing, the 

estimator for the price relative Rt,( ) t−1 is: 

 (2) 

( ) ( )

( ) ( )

∑ ∑

∑ ∑

= =

−

= =











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

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









































































n n

a c i j

t

a

c c i j

n n

a c i j

t

a

c c i j

c

c

p

p

w

n n

p

p

w

n n

ELI 1 hit 1 quote ELI hit, quote , ,

1

, ,

ELI 1 hit 1 quote ELI hit, quote , , , ,

P

1

P

1

Where: n = sample size for ELI’s in the 

 item stratum 

 nc = sample size for outlets in the 

 frame corresponding to ELI c 

 Pc,i, j = probability that ELI c and 

 outlet i and unique item j are 

 selected given the item stratum 

 and PSU 

 

a c i j

t

a p

p

w

, ,

















 = measurement of expenditure 

 in month a for unique item j 

 in outlet i for ELI c which 

 has been updated by the ratio 

 of the price in month t to the 

 price in month a

The expected value of the numerator (or denominator) 

in formula 2 equals the total updated expenditure for 

the item stratum in the PSU during month t (or t-1),

where updating reflects the change in price of items 

purchased but not the change in quantity of items 

purchased. 

3. Bridging of Samples for the 1998 Revised Index

 and Adjustments in Price-Relative Estimation

For the 51 PSU’s continuing in the sample for the 

January 1998 revised index, the sample of quotes were 

mapped to codes for the newly defined item strata and 

ELI’s. Sample sizes were tested at the national level 

against the optimal sample sizes determined for the 

1998 sample design. A sample for an item stratum was 

defined as insufficient if the ratio of actual size to 

optimal size was less than .50. 

Based on this test, 24% of the item strata had 

insufficient samples and were designated for new 

sample selection. For the 76% item strata that had 

sufficient samples mapping to them, a process referred 

to as quote-level bridging was developed. The process 

involves: (1) coding of the quotes to retain their 

association with both the old and the new ELI 

definitions; (2) adjusting the quote-level weight for pt

and pt−1 shown in formula 2. 

Samples were bridged for 33% of the item strata where 

there was no change in the item-stratum or ELI 

definitions. For estimation, no adjustment of the quotelevel weights is necessary for these item strata. 

Samples were also bridged for 43% of the item strata 

where there were changes in definition for the itemstratum and/or ELI’s. Since the quotes in a bridged 

sample were not selected based on sampling stages 

determined by the new definitions, the expected sample 

proportion associated with a quote for a sampling stage, 

given the new definition, would not equal the original 

probability of selection. Thus, the expected value of 

the numerator in formula 2 would not equal the updated 

total expenditure for the item stratum in the PSU during 

month t . 

The following formulation explains these concepts 

more explicitly. Formula 3 is the basis for the current 

computation of the numerator of Rt,( ) t−1 . It can be 

shown that formula 3 is equal to the numerator of 

formula 2. 
 3 

 (3) 

( ) ( ) ( ) ∑ ∑ = = 













































N 

c 

r n

a c i j

t

a

c c c i j

c c c

p

p

w

n r n

r

1 quote 1

quote , , P , , 

Where: N = number of ELI’s in the item stratum

 rc = number of sample hits for ELI c .

 It is a random variable, such that: 

 r n c

c

N

=

∑ = 1

 

 Note: In practice, an adjustment is made in 

 formula 3. Instead of cnc r , the factor 

 ( )*

cnc r is used which has a value from 

 0 to cnc r . It is computed each month, 

 and its value depends on the number of 

 usable quotes for that month and other 

 contingencies (such as the duplication 

 of certain quotes to compensate for the 

 subsampling and dropping of quotes). 

 

The probability of selection associated with a quote is: 

P = Pr(c) Pr(i | c) Pr(j | c,i) c,i,j (4) 

Where: c = event that ELI c is selected 

 i = event that outlet i is selected 

 j = event that unique item j is 

 selected 

The expected sample proportion for ELI c is: 

E  = Pr ( ) c 

 





n

rc (5) 

The expected sample proportion for a unique item given 

ELI c is selected is:

 (6) 

E | c Pr (i | c) Pr (j| c,i) , , = 















c c

c i j

r n

r

Where: c i j r , , = number of quotes for unique 

 item j in outlet i for ELI c . 

 It is a random variable, such that: 

 c c

N

i

N

j

c i j r r n c c i

 1 1

, ,

,

∑∑ = = =

 Where: 

 Nc = number of outlets in the 

 frame corresponding to 

 ELI c 

 Nc,i = number of unique items 

 in outlet i for ELI c 

The adjustment of the quote-level weights for the 43% 

item strata is broken down into four levels of 

complexity. The bridged sample for an item stratum 

and PSU, or replicate of the PSU, is assigned to the 

lowest level of complexity that it qualifies for. 

Level 1, referred to as the 1987 method, involves item 

strata where the old ELI’s within an item stratum may 

be split apart and/or recombined into new ELI’s. 

However, the definition of the item stratum does not 

change. An example is the item stratum Rice, Pasta, 

and Cornmeal. The old ELI Rice and the old ELI 

Macaroni, Similar Products, and Cornmeal were 

combined into the new ELI Rice, Pasta, and Cornmeal. 

For estimation, a simple technique is used that was 

developed for the January 1987 revised index. The 

monthly computation of the factor ( )*

cnc r is based on 

quotes coded for the old ELI, not the new, and the 

expected sample proportion for the unique item equals 

the probability of selection for the unique item given 

the old ELI. Also, the factor rc / n does not change 

and is based on the old ELI. Thus, estimation is no 

different for a bridged sample than a newly selected 

sample, but the supporting sampling structure and 

coding system are different. 

Levels 2, 3, and 4 involve item strata where the old 

ELI’s in one or more old item strata may be split apart 

and/or recombined into new ELI’s. The newly defined 

ELI’s and the unchanged ELI’s are grouped into new 

item strata. An example is based on 2 old item strata: 

(1) White Bread, composed of the old ELI White Bread; 

and (2) Other Breads, Rolls, Biscuits, and Muffins, 

composed of 2 old ELI’s: (a) Bread Other Than White

and (b) Rolls, Biscuits, Muffins (Excluding Frozen). A 

new item stratum Bread is composed of the new ELI 

Bread, and a second new item stratum Fresh Biscuits, 

Rolls, and Muffins is composed of the new ELI Fresh 

Biscuits, Rolls, and Muffins. For estimation, a poststratification method is used. Substrata within a new 
 4 

item stratum are defined and, for the numerator (or 

denominator) of the price relative, the estimate of total 

updated expenditure in month t (or t-1) is the sum of 

the substratum estimates of updated expenditure in 

month t (or t-1). For the new item stratum Bread, the 

first substratum is White Bread and the second 

substratum is Bread Other Than White. Formula 7 

shows how formula 3 is adjusted for the poststratification method. 

 (7) 

( ) ∑∑∑ === 













































N 

s

N

c 

m

a s c i j

t

a

s s c s c i j

s c s s c

p

p

w

n m

r

1 1 1 qt

qt , , , , , ,

*

,

, ,

 P

Where: N = number of substrata s in the new item stratum 

 c = intersection of the old ELI 

 and the new item stratum 

 Ns = number of intersections c 

 in substratum s 

 ns = number of old-ELI sample hits 

 bridged to substratum s 

 s c r , = number of sample hits for 

 intersection c , such that: 

 s

N

c

rs c n s

∑ = =1

,

 Note: If intersection c is 

 smaller than the old 

 ELI, then s c r , may 

 equal 0 . Otherwise, 

 it will equal the original 

 number of sample hits 

 for the old ELI. 

 ms,c = number of quotes bridged to 

 intersection c 

 Ps,c,i, j = probability that intersection c 

 and outlet i and unique item j 

 are selected given substratum s

For level 2, referred to as stage-1 poststratification, a 

substratum is defined as the intersection of the old ELI 

and the new item stratum. In formula 7, each 

intersection c is its own substratum. For a bridged 

sample to qualify for estimation at level 2, each of the 

substrata for the new item stratum must be represented 

by at least one bridged quote. The probability of 

selection for intersection c given substratum s is equal 

to 1.00 and is incorporated into Ps,c,i, j . The sample 

proportion for intersection c given substratum s,

s c s r / n , , is equal to 1.00 . If the outlet sample for the 

old ELI is split apart when quotes for the old ELI are 

bridged to the new item stratum, then the probability of 

selection for an outlet given intersection c is adjusted 

and the probability is incorporated into Ps,c,i, j . 

Finally, the monthly computation of the factor ( )* ms,c

is based on quotes coded for intersection c, and, as a 

result, the sample proportion for a unique item is 

conditioned on intersection c . 

For level 3, referred to as stage-2 poststratification, a 

substratum is defined as the intersection of the old item 

stratum and the new item stratum. For a bridged 

sample to qualify for estimation at level 3, each of the 

substrata for the new item stratum must be represented 

by at least one bridged quote. The method of 

estimation is the same as it is for level 2, with the 

following exceptions: (1) A substratum s may contain 

more than one intersection c . (2) The probability of 

selection for intersection c is conditioned on substratum 

s and may be less than 1.00 . (3) The sample proportion 

for intersection c, s c s r / n , , is conditioned on 

substratum s and may be less than 1.00 . 

For level 4, referred to as stage-2 poststratification with 

adjustment, a substratum is defined the same as it is for 

level 3. However, at least one substratum is not 

represented by at least one bridged quote. The method 

of estimation is the same as it is for level 3 with the 

addition of an adjustment factor in the quote-level 

weight to account for missing substrata. This factor 

equals the total expenditure at time b for the new item 

stratum divided by the sum of the expenditures at time

b for the substrata having a bridged sample. (Time b is 

between month a, the sample-rotation reference month 

for the PSU, and January 1998.) The adjustment factor 

is important when estimating a higher level price 

relative where the summation in the numerator (or 

denominator) crosses over replicates and/or geographic 

strata. 
 5 

The adjustment of the quote-level weights for the 43% 

item strata is being implemented in 2 stages. 

Implementation of stage 1 was completed before the 

production of the January 1998 revised index. In stage 

1, the level-1 adjustments were made for the qualifying 

bridged samples and the level-2 adjustments were made 

for the samples qualifying for levels 2, 3, and 4. In 

stage 2, the level-3 and level-4 adjustments will be 

made. 

4. Empirical Findings

An analysis of the bridged samples and adjustment of 

quote-level weights was conducted for the PSU 

consisting of New York City. It is self-representing 

and has 4 replicate samples selected for each item 

stratum. For 180 of the 183 item strata, 2 replicate 

samples (A and B) were newly selected based on the 

new item-stratum and ELI definitions. The third and 

fourth replicate samples (C and D) were bridged. For 

the remaining 3 item strata, all 4 replicate samples were 

bridged. 

The analysis is based on the stage-1 implementation of 

the adjustment of the quote-level weights. For stage 1, 

the bridged samples assigned to the 3rd and 4th levels of 

adjustment are combined and coded as level 3. For 

these bridged samples, the level-2 adjustments were 

made with no adjustment factor added to account for 

missing substrata. 

At the time of the analysis, 8-month price relatives, 

reflecting price change from December 1997 to August 

1998, were available. Price relatives estimated from the 

replicate A and B newly selected samples (month a = 

May 1995) are compared against price relatives 

estimated from the replicate C and D bridged samples 

(month a = May 1994). The term control relative is 

defined as an A or B price relative, and the term test 

relative is defined as a C or D price relative. A control 

relative for an item stratum is coded the same level of 

adjustment as its paired test relative for the same item 

stratum, where relatives for replicates A and C are 

paired and relatives for replicates B and D are paired. 

Table 1 shows how the C and D replicate samples break 

down for the 183 item strata. The average number of 

active quotes per sample is 4.9, where active is defined 

as a combination of an ELI and outlet that was still 

existing in January 1998. 

Plot 1 shows the mean 8-month price relative for each 

level of adjustment for each replicate A, B, C, and D. 

The number of price relatives coded for each level of 

adjustment is listed by replicate under Plots 1 and 2. 

The number for replicates C and D is less than the 

number of samples shown in Table 1 since: (1) A and 

B replicate samples were not newly selected for 3 item 

strata and, as a result, control and test relatives could 

not be paired; (2) only nonimputed price relatives are 

used. An 8-month price relative is defined to be 

imputed if no quotes in its sample are usable in any 

month from January 1998 through August 1998. 

Plot 1 shows no notable difference in the mean 8-month 

price relative between the levels of adjustment or 

between the replicates. Likewise, no notable 

divergence of the set of test means from the set of 

control means is indicated. The divergence that appears 

at level 3 is fairly negligible relative to the standard 

deviations at level 3 in Plot 2, which is examined 

below. These results could indicate that estimates from 

the reweighted bridged samples are as adequate as the 

estimates from the newly selected samples. However, 

such a conclusion is premature given the limited time 

frame and geographic representation of the analysis. 

Plot 2 shows the standard deviation of the 8-month 

price relatives for each level of adjustment for each 

replicate A, B, C, and D. At level 0, a divergence of the 

set of test standard deviations from the set of control 

standard deviations is not obvious. If there is a 

divergence, it could indicate a sample-rotation effect. 

At level 2, a notable divergence does exist. This 

divergence could indicate that the stage-1 

poststratification method results in an estimate with 

lower variance when each substratum for a new item 

stratum is represented by at least one bridged quote. 

However, the divergence may fade over time. Also, it 

may not occur in other geographic areas, or it may 

depend on the particular groups of items in the analysis, 

or it may be due to a bridging effect other than 

reweighting. 

5. Conclusion

The poststratification method described in this paper 

provides a flexible tool when the reclassification of 

sampled items is necessary. Further study of both the 

immediate and long-term effects on the consumer price 

index and its component indexes is warranted. 
 6 

6. References

Bureau of Labor Statistics. BLS Handbook of Methods, 

September 1992, pp. 190-191. 

Casady, Robert J., Alan H. Dorfman, and Richard L. Valliant. 

“Sample Design and Estimation for the CPI,” BLS Note, July 

2, 1996. 

Lane, Walter F. “Changing the Item Structure of the 

Consumer Price Index,” Monthly Labor Review, December 

1996, pp. 18-25. 

 Table 1 

Type of C or D 

Replicate Sample 

Number of 

Samples &

Proportion 

Newly Selected 90 .24 

Bridged 

 Level of Adjustment

 0 No Change 

 1 1987 Method 

 2 Stage-1 Poststrat. 

 3 Stage-1 Poststrat. 

 No Adjustment Factor 

118 

32 

55 

57 

.32 

.09 

.15 

.16 

Designated as Bridged 

 but No Quotes to Bridge 14 .04 

All 366 1.00 

 

Plot 1: MEAN 8-MONTH RELATIVE FOR REPLICATES A, B, C, D BY LEVEL OF ADJUSTMENT * 

Plot 2: STANDARD DEVIATION FOR REPLICATES A, B, C, D BY LEVEL OF ADJUSTMENT * 

Number of Relatives: 

 A 46 13 13 23 

 B 43 13 15 27 

 C 46 13 13 23 

 D 43 13 15 27 

 178 52 56 100 

* A control relative is coded the same level of adjustment as its paired test relative, where 

 relatives for replicates A and C are paired and relatives for replicates B and D are paired.