EVALUATING FORMULA BIAS IN VARIOUS INDEXES USING

SIMULATIONS

June 20, 1996

Draft

ROBERT MCCLELLAND

Division of Price and Index

 Number Research

2 Massachusetts Ave., NE

Room 3105

Washington, D.C. 20212

INTERNET: MCCLELLAND_R@BLS.GOV

I would like to thank Ralph Bradley, Tim Erickson, John Greenlees, Ted

Jaditz, Brent Moulton, Marshall Reinsdorf and Keith Waehrer for useful

comments and suggestions. I would also like to thank Steve Verdon for his

research assistance and Elisa McClelland for her editing. All remaining

errors are, of course, my own. The views expressed here do not necessarily

reflect the policy of the Bureau of Labor Statistics or the views of other BLS

staff members.
1

INTRODUCTION

Current research on the consumer price index (CPI) has focused upon several

sources of potential bias. Reducing this bias is important because many

federal tax and expenditure programs are indexed to changes in the CPI.

This means that a change of 0.1 percent per year changes federal

expenditures and revenues by $600 million dollars in the first year. An

upward bias of 1.0 percent per year then represents a loss to the federal

government of approximately $6,000 million dollars.

In response to one of the problems described in Reinsdorf (1994), the Bureau

of Labor Statistics (BLS) recently implemented a method to reduce ‘formula

bias.’ This bias, also known as functional form bias and elementary index

bias, occurs when the BLS estimates a Laspeyres price index in which the

quantity weights are measured in a past ‘base period.’ Because the BLS

lacks information about these quantity weights, it estimates them by

dividing total expenditures on a good by its unit price. But because these

base period prices are themselves missing, they are imputed, causing a

problem known as ‘proxy’ bias. Imputing with the prices when they first

appear in the index leads to the special case known as formula bias, and its

correction involves using prices from some other time period.

In spite of the large sums of money involved, the size of this bias is not well

understood because its current estimates rely upon either educated

guesswork or strong parametric assumptions about the distribution of prices.

This reflects the fact that researchers face the same problem confronting the
2

BLS: without the base period prices, the Laspeyres index cannot be

constructed, and so we cannot know the size of the bias.

In this paper I evaluate the magnitude of the bias of several different price

indexes and provide measures of their closeness to the Laspeyres index

without making the strong parametric assumptions in previous work. This is

done by treating the price quotes collected by the BLS as a representative

population from which price quotes can be drawn and calculating the

population Laspeyres index. By sampling quotes from this population in the

same manner as the BLS samples from the actual population, I then

simulate price indexes and their distributions and compare their closeness to

the population Laspeyres index, including an estimate of the size of the bias

of the current BLS method. Because both the current method and the

method recently implemented by the BLS are simulated, the reduction in

bias from the new implementation is also estimated.

My results are: First, the estimate of bias size using the old imputation

method is lower than previous estimates, being about 0.20 point per annum

for the commodities and services component of the CPI.1 Second, the recently

implemented ‘seasoning’ method reduces the estimated bias to about -0.02.

Finally, estimates of other distance measures, such as root mean square error

(RMSE) and mean absolute bias, are also lower with the seasoning procedure

than with the previous index.

 1 Because this component excludes shelter, it represents about three-quarters of all

expenditures in the CPI’s market basket.
3

However, because the choice of a base period partially determines the level of

the population Laspeyres index, ceteris paribus, the distance of any index to

the Laspeyres varies with the base period. This implies that other indexes

may better approximate a Laspeyres index for some base period.

Nevertheless, the simulations suggest that an index using seasoning is closer

to the Laspeyres index than others considered here for reasonable definitions

of the base period.

A SIMPLE DESCRIPTION OF THE PROBLEM

To clarify the nature of the problem, Erickson (1995) considers several

simplified examples. Here one of those cases is generalized.2 The index to be

estimated is E(Pj,T/Pj.B)/E(Pj,S/Pj.B), where Pj,T is the price of good j in time T,

PT>0 and 0 < S < T.

This index does not resemble a standard Laspeyres index in two respects.

First, a Laspeyres index weights the prices in times T and S with quantities,

which can be calculated by dividing expenditures by prices. For this

example, we suppress the expenditures and weight prices with the inverse of

prices in period B, the ‘base period.’ In addition, in a standard Laspeyres

index, the base period would be period S, while here we assume it is earlier.

This follows BLS methodology, where period S is no earlier than the “link”

period L which itself is later than period B.

 2 I would like to thank Tim Erickson for several useful conversations regarding this

example. Other descriptions of the problem can be found in Reinsdorf (1994) and Erickson

(1995).
4

The index above can be consistently estimated with (n-1ΣjPj,T/Pj,B)/( n-1ΣjPj,S/Pj,B),

where n is the number of prices in the index. Because we do not actually

observe prices in the base period, we impute them using a proxy Pˆ i,B and form

the index (n-1ΣjPj,T/Pˆ i,B)/( n-1ΣjPj,S/Pˆ i,B). Assuming that {Pj,S, Pj,T, Pj,B, Pˆ i,B} is

independently and identically distributed with a nondegenerate distribution,

the ‘proxy’ bias can be written as the difference between the expected value of

the calculated index and the target:

(1) β = EP P

EP P

EP P

EP P

T B

S B

T B

S B

( / $ )

( / $ )

(/)

(/) − = EP P EP P EP P EP P

EP P EP P

TB SB TB SB

SB SB

( / $ )( / ) ( / )( / $ )

( / $ )( / )

− ,

where the subscripts denoting goods are suppressed. By noting that

E P

P

E P

P

E P

P

P

P

P

P

P

P

T

B

S

B

T

B

S

B

T

B

S

B $ $ cov $ , 



 



 



 



 = 



 



 − 



 





and that a similar equality holds for the second component of the numerator

in Equation (1), we may write the bias as

β =

cov , $ cov $ ,

( / $ )( / )

P

P

P

P

P

P

P

P

EP P EP P

T

B

S

B

T

B

S

B

SB SB





 



 − 



 





.

If these two covariances are equal, then the bias is zero. Selection of Pˆ B is

therefore a balancing act between one covariance and the other. Ceteris

paribus, as the correlation between the proxy and, say PS, increases, the first

covariance goes to zero while the second moves towards negative infinity.

BLS imputation procedure prior to January 1995 can be represented by Pˆ B =

ρPL, where ρ is a positive constant defined in the next section. The bias is

then

β =

cov , cov ,

( / )( / )

P

P

P

P

P

P

P

P

EP P EP P

T

B

S

L

T

L

S

B

SL SB





 



 − 



 





 .

Because period L will always be closer to S than to T, one might expect the
5

first covariance to be smaller in absolute magnitude than the second and the

second to be negative. The overall bias would then be positive.

Erickson considers the special case of formula bias, when S equals L. Then,

β =

− 



 

 cov , 

(/)

P

P

P

P

EP P

T

L

L

B

L B

 .

Erickson notes: “My intuition is that [the bias] will often be positive, because

for given PT and PB an increase in PL decreases PT/PL while increasing PL/PB.”

Moulton (1996) also points out that in this case larger variances in PL cause

larger biases by increasing the covariance between 1/PL and PL. If we

simplify even further by assuming that 1/PB is uncorrelated with PT and PL, β

reduces to E(PT/PL) - E(PT)/E(PL). Here the bias clearly exists and, by Jensen’s

inequality, must be positive.

THE PROBLEM AS IT EXISTS IN THE CPI

To relate the example in the previous section to the CPI, it is first necessary

to explain some aspects of the CPI’s construction. The BLS divides services

and commodities into expenditure classes (such as expenditure class 07: fish

and seafood), which are composed of several item strata (such as item

stratum 0702: fresh or frozen fish and seafood). Item strata are themselves

composed of one or more entry level items (ELI), such as ELI 07021:

Shellfish, excluding canned. Once a specific good to be priced -- such as one

pound of crab meat -- has been selected, a price quote for the good is collected

either monthly or bimonthly.

With these price quotes, indexes for each of the 207 item strata are calculated

in 44 geographic areas. For each item stratum, these 44 geographic indexes
6

are aggregated into a single national index. This national item stratum

index is then aggregated into the all-items CPI using expenditure weights

from the Consumer Expenditure Survey (CES). Every year, about 20 percent

of the price quotes in each population statistical unit (PSU). Each of the 44

geographic areas consists of one or more PSU, which are phased out and

replaced with price quotes on new items. In the ‘link period,’ the old sample

is used to compare prices with those of the previous month, while price

quotes in the new sample are collected so they can be compared with prices in

the following month. Thereafter, only price quotes in the new sample are

used.

To examine proxy bias, consider the calculation of the relative change in the

(modified) Laspeyres index of a specific item strata in a specific area. This is

the target that the BLS estimates when it calculates indexes for each item

stratum and area. Given a sample of n prices, the relative price change

between times T and T-1 is calculated as:

(2) R Q P

Q P T T

j jB jT n

j jB jT

, n

, ,

, ,

−

−

= ∑

∑ 1

1

,

where

Q EP jB jB jB , ,, = ,

and EjB is the total expenditures on good j in the base period B, the time

period during which the expenditures are made. Rather than occurring in a

single month, this period varies from one week to five years, depending upon

the item strata. In addition, the expenditures need to be multiplied by

several adjustment factors, such as a correction for the difference in the

definition of goods between the survey which collects expenditures and the

survey which collects the prices.
7

There are two separate problems complicating the calculation of Equation

(2). First, for each item stratum, the BLS collects only a small proportion of

all possible price quotes in a given area. To represent this, replace Ej,B

withEj,B≡Wj,B*Nj,B, with Nj,B being a random variable equal to unity if a

particular outlet and item is chosen and zero otherwise. The BLS sets the

probability of choosing a given outlet proportional to the outlet’s

expenditures. Within each outlet, the price of a specific good is ‘initiated’ by

being randomly selected in a similar manner. This ensures that the expected

value of Ej,B equals EjB. The variable Wj,B is the product of the adjustment

factors mentioned above a sample weight and total expenditures of the

sampled outlets ΣEj,B*Nj,B. The modified Laspeyres of such a sample is then:

(3) [ ]

[ ] $ ,

, ,,

, ,,

R

E PP

E PP T T

j jB jB jT n

j jB jB jT

n −

−

= ∑

∑ 1

1

.

Although the index in Equation (3) is a consistent estimator of the index in

Equation (2), it suffers from small sample bias βs; i.e., the expectation of the

index calculated on a small sample does not necessarily equal the index

calculated on the entire population. In spite of the fact that the numerator

and denominator in Equation (3) are each unbiased estimators of their

population counterparts, the ratio is not. Therefore, the right-hand side of

Equation (3) is not an unbiased estimator of its population counterpart.

While this bias is often assumed to be positive, this need not be the case.

The source of the second problem, proxy bias, occurs because we do not

observe any base period prices Pj,B. Prior to January 1995, the BLS imputed

these base period prices by deflating the link period prices by the index,

defined as ρ in the previous section. The left-hand side of Equation (2) would

then be estimated with the relative $RT T,

L

−1 :
8

(4) [ ] ( )

[ ] ( ) $ $

$ ,

, , ,,

, , ,,

R

E PR P

E PR P

L

T T

jB jL LB

L

j j T

n

jB jL LB

L

j j T

n −

−

= ∑

∑

1

1

.

For simplicity, assume from here on that all quotes in the area rotate

simultaneously (which occurs where a geographic area contains only one

PSU) so that $RL B,

L cancels out of Equation (4). In this case, Equation (4)

becomes

(5) [ ]

[ ] $ ,

, ,,

, ,,

R

E PP

E PP

L

T T

j jB jL jT

n

j jB jL jT

n −

−

= ∑

∑

1

1

.

Further, taking expectations and subtracting the right-hand side of Equation

(3) from the right-hand side of Equation (5) yields an expression analogous to

Equation (1). As in the simple example above, if Pj,L is more correlated with

Pj,T-1 than to Pi,T, then one might expect the bias to be positive.

The analog to the formula bias case considered in Erickson (1995) occurs

when T-1 is equal to L. Then:

( ) $ ( )

( ) , , ,,

,,,

,, ,

,

6 R

E PP

E PP

EP P

E

L

T L

j jB jL jT

n

j jB jL jL

n

j jB jT jL n

j j B

n

= ∑

∑

∑

∑ =

.

Again, intuition suggests that the bias will be positive. Higher variance in

Pj,L is also expected to cause a greater level of formula bias.

ALTERNATIVE SOLUTIONS

It should be noted that, due to small sample bias, none of the proposed

methods are unbiased. Indexes that impute base period prices will also

generally suffer from the proxy bias defined in Equation (1). In addition,

because we can only observe expenditures in the base period, desirable

indexes, such as superlative indexes, cannot be estimated. However, it may
9

be possible to select an index that is closer to the modified Laspeyres than an

index that imputes with the link month.

One alternate imputation method involves using the period in which price

quotes are first ‘initiated’ and the prices collected. This initiation period

occurs approximately four months prior to the link month. Because the price

quotes are collected at this time, the initiation period price can be used for

imputation.3 

This approach will be most effective in reducing bias if prices become

increasingly uncorrelated as the time period between them increases. It also

renders the worst case scenario in Erickson (1995) impossible. Using the

initiation period prices Pj,I instead of the link month period prices results in

the index:

(7) $ ( )

( ) ,

, ,,

, ,,

R

E PP

E PP T T

I j jB jI jT n

j jB jI jT

− n −

= ∑

∑ 1

1

.

A virtually identical procedure to using the initiation period is to collect the

price quotes at the link month, but wait several months before actually

linking in the new quotes. This procedure is termed seasoning and the BLS

has used it on food at home item strata since January 1995.4 The BLS has

recently extended this method to non-food items.

 3 Armknecht, Moulton and Stewart (1995) describe this method and several others, such as

the geometric mean, as alternative methods to alleviate the formula bias for the food at

home component of the CPI. See also Moulton and Smedley (1995) for simulations of the

current method, the Dutot and the Geometric means indexes.

4 Food at home items are those numbered below 1900. See Table A1 for a listing of all items

included in this study. In addition, quotes in several apparel ELI’s are seasoned for either
10

It is also possible to use some other procedure to impute the base period

prices. For example, imputing the base period with any non-zero constant

yields the weighted ratio of expectations (Dutot) relative:

(8) $ ,

, ,

, ,

R

E P

E P

r j jB jT n

j jB jT

T T − n = ∑

∑ −

1

1

.

Because the expenditure weights are not divided by prices, this index will

give too large a weight to high-priced items. This will be an especially severe

problem for groupings of widely disparate goods such as item stratum

number 6001, ‘sports vehicles, including bicycles’ which can include boats

with prices exceeding $300,000 as well as bicycles with prices of $50.

A final imputation procedure would be to estimate the base period prices

using a hedonic equation:

(9) $ ( $ )

( $ ) , , ,,

, ,,

R

E PP

E PP

H

T T

j jB jB jT

n

j jB jB jT

n −

−

= ∑

∑

1

1

,

where Pˆ j,B is the estimated price from a hedonic regression. The major

problem with this method is the difficulty in correctly specifying hedonic

equations for the hundreds of item strata that are used to make up the CPI.5 

It also requires that the item possesses useful and identifiable characteristics

with which prices can be estimated.6 

An alternative to imputing the base period price is to use the geometric

means (geomeans or Jevons) relative

 

eight or four months. In the simulations below, all of the quotes in any strata with such an

ELI are seasoned for eight months.

5 For example, the single market model of Kahn and Lang (1988) requires correct

specification of the functional form for model identification.

6 For these reasons hedonic regressions are currently used only for quality adjustment on a

small set of items in the CPI.
11

(10) [ ] $ ,

, ,

, ,

$ $

R PP G

jT jT

E E

j

n T T

jB jB

− = ∏ −

∑

1 1 .

It has often been noted that the geometric means relative is lower than the

Laspeyres relative because the Laspeyres is a true cost-of-living index when

the elasticity of substitution among goods is zero, while the geometric means

relative is a true cost-of-living index when the elasticity of substitution

among goods is unity.

This statement only applies when the base month is equal to the link month.

As the base month prices become increasingly uncorrelated with the prices in

times T and T-1, this condition no longer holds. When the base month prices

are entirely uncorrelated with the prices in times T and T-1, the modified

Laspeyres index equals the ratio of the arithmetic average prices. In this

case, the modified Laspeyres index and the geometric means index are

simply different measures of the central tendency of the distribution of

prices. The modified Laspeyres and the geometric means will then only

differ to the extent that the variance of prices changes over time.7 

SIMULATING THE INDEXES

The primary problem with comparing the performance of the above

alternatives is that we cannot observe the base month prices. If we could, we

wouldn’t need to impute. This problem is approached here by using price

quotes collected by the BLS nationally and arbitrarily designate specific

months as the base month, initiation month and link month. After assuming

that these quotes are the population for a representative area I constructed

 7 I would like to thank Marshall Reinsdorf and Brent Moulton for several useful

conversations regarding this point.
12

the Laspeyres index, the alternative methods described in Equations (4), (7),

(8) and (10), and an index that seasons the food components but uses link

month imputation elsewhere.8 Further, the distribution of the indexes are

simulated by sampling these quotes randomly 10,000 times with the

probability of selection proportional to expenditures. Several measures of the

closeness of the alternative methods to the Laspeyres were then applied and

rough estimates of the magnitude of the small-sample bias were made.

These Monte Carlo simulations were performed using 162 of the item strata

in the commodities and services section of the CPI’s market basket, which

represents approximately three-quarters of expenditures.

The link period is January 1994, and the initiation period is set to be

September 1993.9 As mentioned above, use of the initiation period method is

also equivalent to seasoning the quotes for four months. The sample

Laspeyres index uses a base period of January 1993 for monthly item strata

and bimonthly strata price quotes collected in odd months. To form the

population Laspeyres index, the base period price is defined as the average

price in one of two time periods. Base period price 1 is the average price from

July 1992 to January 1993.10 This is as similar to the base period for

expenditures as the data limitation to months after June 1992 will allow. In

this sense, it is the base period definition most consistent with forming the

 8 The mixture of link and seasoning methods represents the seasoning of food items that

started in January 1995 and the eight month seasoning of some apparel ELI’s started in

1991. Because of the complexity in performing hedonic regressions, this method is not

included.

9 As described in the Appendix, quotes from even-numbered months of bimonthly item

stratum were assumed to be odd-numbered.

10 The months prior to 1993 became available too late for use in the Monte Carlo

simulations. Therefore, all Monte Carlo simulations of the Laspeyres index use a base period

of January 1993. As described in the Appendix, by construction, any quote not available in

July 1992 was available no later than January 1993.
13

quantity weights in Equation (2). Base period price 2 is the average price for

the year prior to the initiation period September 1993, which is the recall

period used when initiating prices.

If a quote was missing in the initiation period or link period, it was imputed

using the quote in the first available month. Missing quotes in other months

were imputed using the last available quote and inflating by the index

formed without the quote. All of the indexes calculated compare short-run

price changes from one time period to the next. The long-run relatives from

January 1994 to September 1994 were calculated by estimating the monthly

relatives and then multiplying them together. Because actual expenditures

on an outlet were unavailable, the outlet expenditure used in the weights

and sampling was taken to be the average expenditure on all outlets in the

area. For more details on the construction of the data set, see the Appendix.

Table A1 reports the means of the sample indexes for the 162 separate item

strata.

RESULTS

Figure 1 depicts the proxy bias for the various short-run population indexes

from January to September 1994. Because this is the formula bias in

Equation (6), there is a large bias with the link month method in the link

month. The bias of the link month method then tapers off.

As mentioned above, this decline can occur under conditions that favor the

seasoning method. In the January/February period, the link/seasoning

index, which seasons food items and uses link month imputation on others,

reduces the bias by more than 85 percent. Seasoning the non-food items

reduces the bias in this month an additional 12 percent. In the following
14

months, however, there is relatively little bias reduction to seasoning of food

items and a much greater proportionate reduction to seasoning the non-food

items. While the link month method always has a positive bias, the

seasoning month method starts with a positive bias but falls below zero in

the latter months.

Figure 2 describes the cumulative effects of this bias. This shows that most

of the benefit from seasoning food comes in the first comparison. After that,

the link/season index tracks the link month imputation method. The

seasoning method, in contrast, starts with a small positive bias and appears

to have a shallow negative trend. Given sufficiently strong seasonality, this

bias would continue to be negative until one year after the link, when it turn

positive. In contrast, the Dutot method has a negative bias that trends lower

over time. The geometric means index also appears to underestimate the

population Laspeyres, with the difference growing over time.

Although not designed specifically to measure small sample bias, this study

can provide some evidence. These results should be interpreted with caution

because of the small number of months examined. In theory, the link month

imputation method should have no small sample bias when the quotes first

link. For subsequent short run comparisons, bias is possible, although the

cumulative bias should remain quite close to zero.11 In contrast, the

seasoning method, the Dutot method and the geomeans method should all

have some degree of small sample bias.

 11 The existence of missing quotes can cause the bias to vary from zero.
15

Figure 3 reports the small sample biases for the various short-run indexes.

In the link month, the link month method has an extremely small bias:

0.008. The remaining methods have some degree of bias, but only of about

0.05. For subsequent months, the bias of the link month method is larger,

but always below that of other methods. In contrast, the geomeans and the

Dutot always have the greatest level of small sample bias.

The cumulative bias is shown in Figure 4. The link month bias starts near

zero and rises to approximately 0.04 before declining below zero. Because it

should equal zero asympotically if no quotes are missing, the bias should

hover around zero for indexes beyond September. The bias in the seasoning

method, while not equal to zero, peaks below 0.15 and appears to be

declining towards zero at the end of the sample, where it equals about 0.07.

To estimate the size of proxy bias in the commodities and services section of

the CPI, I compare the population Laspeyres index with the various

population alternatives. The raw bias estimates are divided by five because

about twenty percent of the quotes rotate every year, and there is probably

little if any sample bias after the first year.

It is clear that the two base periods result in fundamentally the same

estimate.12 Table 1 lists both the estimate of the bias using the link month

method of about 0.20 per year, as compared with some previous estimates of

0.50 per year. The intermediate link/seasoning method appears to cut the

bias about in half, while the pure seasoning method seems to have

eliminated most of the additional bias.

 12 Moulton (1996) multiplies the figures in table one by 0.75 to represent the fact that

housing, which represents about twenty five percent of the index, is omitted.
16

Table 2 lists the results from applying several criteria for evaluating the

ability of sample indexes to estimate a population Laspeyres index that uses

base period price 1. In order to measure how precisely the current

methodology estimates the Laspeyres index, I use the root mean square error

(RMSE) criteria and the related mean absolute error criteria. Because the

mean bias has received the most attention, this criteria is also selected.

Because this number includes small sample bias and is not adjusted for the

percent of quotes rotating per year, it cannot be interpreted as an estimate of

the actual bias. Finally, the very conservative maximum absolute error

criteria and the frequency with which an index is within 0.5 percent of the

target are included.

As the table shows, the RMSE using the new imputation method is below

both the link/seasoning methodology and the link month method. The mean

absolute error is also lower for the seasoning method than for other methods.

In addition, the proportion of simulations with which the index is within 0.5

of the Laspeyres index is greater using the new method.

On the other hand, the maximum error of the seasoning method exceeds that

of the Dutot and the geomeans, implying that imputing the base period

generates occasional large outliers. In Table 3, the results from using the

second base period are listed. These results are qualitatively similar to those

of Table 2.

However, the above results depend upon the definition of the base period. To

show this, the RMSE of the indexes as a function of the base period of the

population Laspeyres index were recalculated. These base periods ran from
17

July 1992 to August 1993.13 The results are shown in Figure 3, where the

most noticeable feature is the effect of seasonality.

For all methods not using link-month imputation, the RMSE peaks with a

base period of February 1992 and falls away after that. For the link month

imputation method, RMSE is minimized at that period and rises elsewhere.

Because the link month is either January or February 1993, this suggests

that seasonality is the source of this pattern. Comparing the RMSE’s of July

1992 with July 1993 also suggests seasonality. For these two months, the

RMSE of the link month imputation method is equal to or lower than that for

the seasoning method. For any other month, the seasoning method is

preferred.

To examine the effects of base period selection upon the mean bias, a similar

experiment were performed. The results are described in Figure 4. The bias

for each method is minimized when the base period is set to January or

February 1992. For the link-month imputation method, the bias is very

nearly zero for February 1992. In addition, bias for the seasoning/imputation

method is near zero when the base period is September or October of 1992,

one year prior to the initiation month. Again, seasonality appears to be the

cause.

CONCLUSION

In this paper I provide estimates of the proxy bias of several price indexes

and provide measures of their closeness to the Laspeyres index. By using

 13 Odd bimonthly quotes use either the same month as the monthly quotes, if it is an odd

month, or one month earlier, if it is even. Even bimonthly quotes are one month later than

the odd bi-monthly quotes.
18

Monte Carlo simulations with BLS data, these estimates are made without

the strong parametric assumptions of previous work. The results show that

the recently implemented seasoning procedure more closely estimates the

Laspeyres index than previous BLS methods, using two definitions of the

base period. I also estimate the proxy bias using the old imputation method

at about 0.20 point. The recently implemented “seasoning procedure” reduces

the estimated bias to about -0.02, although it is difficult to attach a sign to

this number because of fluctuations in the bias over time.

Because these results are sensitive to the definition of the base period, some

care must be used in interpreting the results. In particular, the effects of

seasonality need to be considered. Nevertheless, for reasonable definitions of

the base period, the results suggest that the seasoning procedure is much

closer to the Laspeyres index than the older imputation procedure.
19

Table 1

Net Bias Estimates With Two Laspeyres Indexes

Imputation Base Period Price One Base Period Price Two

 Method Per Rotation Per Year Per Rotation Per Year

Link Month 1.048 0.210 0.989 0.198

Link/Season 0.622 0.124 0.563 0.113

Seasoning -0.074 -0.015 -0.133 -0.027

Geomeans -0.265 -0.053 -0.324 -0.065

Dutot -0.455 -0.091 -0.514 -0.103

Base Period Price One is the average price from July 1992 to January 1993.

Base Period Price Two is the average price from September 1992 to August

1993. Approximately 20 percent of the quotes rotate per year.

Table 2

Comparison of Estimation Methods of Long Run Relatives, Using the

Population Laspeyres (With Base Period Price One) as the Target

Sample

Index

Root Mean

Square

Error

Mean

Absolute

Error

Mean

Error

Maximum

Absolute

Error

Proportion

within .5 of

Laspeyres

Link Month 1.181 1.041 1.032 5.378 0.170

Link/Season 0.854 0.693 0.636 5.160 0.409

Seasoning 0.514 0.404 -0.001 3.010 0.683

Geomeans 0.536 0.426 -0.114 2.966 0.651

Dutot 0.622 0.499 -0.240 2.713 0.572

The Link/Season index seasons for all food items and imputes with the link

month for all non-food items.

Table 3

Comparison of Estimation Methods of Long Run Relatives, Using the

Population Laspeyres (With Base Period Price Two) as the Target

Sample

Index

Root Mean

Square

Error

Mean

Absolute

Error

Mean

Error

Maximum

Absolute

Error

Proportion

within .5 of

Laspeyres

Link Month 1.130 0.985 0.973 5.319 0.199

Link/Season 0.811 0.650 0.577 5.101 0.448

Seasoning 0.517 0.409 -0.060 2.951 0.672

Geomeans 0.552 0.441 -0.173 2.907 0.630

Dutot 0.647 0.521 -0.299 2.654 0.550

The Link/Season index seasons for all food items and imputes with the link

month for all non-food items.
20

Figure 1

Proxy Bias of Short-Run Population Indexes When Estimating A Population

Laspeyres Index (With Base Period Price One)

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Comparison Period

 Figure 2

Proxy Bias of Cumulative Population Indexes When Estimating A Population

Laspeyres Index (With Base Period Price One)

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Season

Geomeans

Dutot

Comparison Period

Bias
21

Figure 3

Small Sample Bias of Short-Run Indexes

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Bias

Comparison Period

Figure 4

Small Sample Bias of Cumulative Indexes

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22

Figure 5

RMSE of Indexes as a Function of the Base Period of the Population

Laspeyres Index

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Jul-93 May-93 Mar-93 Jan-93 Nov-92 Sep-92 Jul-92

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Base Period

RMSE

Figure 4

Mean Bias of Indexes as a Function of the Base Period of the Population

Laspeyres Index

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Base Period
23

APPENDIX

DATA SET CONSTRUCTION

The prices used in this analysis are the prices of items in the commodities

and services section that are used in calculating the CPI. Item classes in

which the average number of quotes was less than two were deleted --

calculated by dividing the total number of quotes by 44, the total number of

geographic areas for which indexes are created-- as were strata with an

unusually large number of missing prices or with anomalous price behavior.

The total weight of these three classes in the CPI is less than 0.5 percent.

The number of remaining item strata is 162.

Any quote that failed to exist continuously from January 1993 to November

1994 or for which the number of quotes in the area was unknown was

deleted. Because approximately 20 percent of the sample rotates every year

this left about 60 percent of the original sample. If the price for a given quote

was missing for every month from January to November of 1994, then the

quote was deleted. Quotes with missing prices were not otherwise deleted.

The total number of price quotes remaining was 48,390. This forms the

population from which quotes were drawn, either all at once to form the

population indexes, or several at time in the Monte Carlo simulations.

For the 82 item groups where prices are collected monthly, the ‘short run’

index is calculated from one month to the next, starting in January and
24

ending in September 1994. For the 83 groups where prices were collected

bimonthly, even-month indexes run from February through August and oddmonth indexes run from January through September. The even-numbered

months were treated as odd-month quotes (i.e., quotes from February were

combined with the January quotes.) There were also several large

metropolitan areas that collect the bimonthly items on a monthly basis. Oddmonth quotes of this type were used and even-month quotes were ignored.

The outlet expenditures used for the expenditure weights were taken to be

the average expenditures in the area from which a price quote was taken.

All the adjustment factors in the weights, such as the percent of POPS

correction, were used intact. This procedure inevitably resulted in the

expenditure information being older than would be true in the actual Bureau

procedure. By construction, the simulated link month of January 1994 is

after the month the price was actually linked, which must have been prior to

January 1993 to avoid being deleted. This problem exists to the extent that

relative outlet expenditures varied from the month in which the survey was

actually conducted to the month in which it should have been conducted in

the simulations.

To reflect Bureau procedure, ‘long run’ indexes between January and August

1994 were formed by multiplying ‘short run’ indexes together. If there were

no missing prices, this process yields exactly the same value as calculating

the long-run index directly. As with BLS policy, if at least one quote existed
25

in both periods of the index, then missing values were imputed using the

index formed by the existing quotes. This procedure creates an index slightly

different from the directly calculated index.

The long run quotes were aggregated using the relative importance weights

for December 1994. These weights are the product of the index in a given

period and the item’s expenditure share estimated from the 1982-4 Consumer

Expenditure Survey. While this method is slightly inaccurate, the

inaccuracy is only a function of how the relative inflation rates across goods

has changed from January to December 1994.

For the Monte Carlo simulations, the average number of quotes were drawn

from the population 10,000 times, using the definition of average described

above. The total number of simulations computed for each index formula was

then 10,000*(82*8 + 80*4)= 9,760,000. The probability of an outlet being

selected was proportional to its expenditures, as defined above.

Official Bureau procedure is to select randomly one entry level item (ELI) to

represent the item strata of goods and then randomly select outlets from that

ELI proportional to their expenditures. Here, I do not separate the quotes by

ELI, but simply sample outlet/quote combinations from the entire item

stratum.

If a short run index was missing because all of the values were missing, a

randomly selected index for the month from another simulation was
26

substituted. This is reflective of Bureau procedure, where missing indexes

are imputed using the index for a different area or the index for a different

class of goods in the same area. To check for the sensitivity of the results to

this procedure, all the index formulas were calculated on the item classes in

which there were no missing indexes. The resulting aggregate indexes were

essentially identical to those that substituted for missing indexes.
27

Table A1

Mean Long Run Relative For Simulations Using Mean Number of Quotes

Imputation

Item

Strata Name Laspeyres

Link

Month Season Dutot Geomeans

101 Flour and prepared flour mixes 102.52 103.20 102.35 102.18 101.51

102 Cereal 101.06 102.05 101.18 100.93 100.97

103 Rice, pasta, and cornmeal 102.95 103.68 102.64 102.23 101.69

201 White bread 103.82 104.87 103.69 103.75 103.70

202 Fresh other bread, biscuits rolls 101.78 102.05 101.65 101.42 102.86

204 Cookies, fresh cakes, and cupcakes 99.79 100.16 99.70 99.75 100.91

206 Other bakery products 107.77 109.25 106.46 107.37 105.33

301 Ground beef other than canned 96.97 97.59 96.76 96.64 97.31

302 Chuck roast 99.43 102.19 98.92 98.90 99.22

303 Round roast 98.03 100.28 97.42 97.30 97.44

304 Other beef and veal 99.12 100.34 99.21 98.90 99.05

305 Round steak 98.12 100.56 98.39 98.03 98.22

306 Sirloin steak 100.46 102.74 100.62 100.42 100.23

401 Bacon 98.41 99.80 98.74 98.32 98.64

402 Pork chops 101.29 103.19 101.74 101.44 101.34

403 Ham 106.65 109.33 106.00 106.65 105.38

404 Other pork, including sausage 100.23 101.26 99.98 99.58 99.66

501 Other meats 100.57 101.82 100.39 100.43 100.18

601 Fresh whole chicken 100.03 101.76 99.64 99.68 99.51

602 Fresh and frozen chicken parts 100.56 102.37 101.07 100.28 99.10

603 Other poultry 105.82 106.47 105.96 105.14 104.77

701 Canned fish and seafood 100.78 101.33 100.56 100.60 100.29

702 Fresh and frozen fish and seafood 100.26 101.27 99.07 99.16 99.80

801 Eggs 95.68 96.63 95.46 95.36 95.63

901 Fresh whole milk 98.31 98.52 98.32 98.22 98.34

902 Other fresh milk and cream 99.23 99.73 98.99 98.97 98.97

1001 Other dairy products incl. butter 100.01 100.93 100.22 100.10 100.20

1002 Cheese 100.41 100.95 100.39 100.29 100.61

1004 Ice cream and related products 102.78 104.61 102.42 102.80 101.60

1101 Apples 110.17 111.83 107.81 107.45 107.48

1102 Bananas 104.95 111.43 103.88 105.64 106.02

1103 Oranges, including tangerines 157.10 171.58 160.79 153.27 154.95

1104 Other fresh fruits 102.35 111.16 90.02 92.74 90.24

1201 Potatoes 100.94 103.76 99.93 98.45 100.15

1202 Lettuce 119.71 132.73 119.59 120.65 119.21

1203 Tomatoes 65.04 69.31 65.49 64.64 66.39

1204 Other fresh vegetables 90.38 96.15 88.37 88.76 90.06

1301 Fruit juices and frozen fruit 96.91 98.29 97.21 97.15 96.98
28

Table A1

Mean Long Run Relative For Simulations Using Mean Number of Quotes

Imputation

Item

Strata Name Laspeyres

Link

Month Season Dutot Geomeans

1303 Canned and dried fruits 98.88 99.42 98.80 98.69 99.10

1401 Frozen vegetables 101.68 101.95 101.38 101.33 100.92

1402 Other processed vegetables 104.35 105.75 104.21 104.31 103.36

1501 Sweets, including candy 100.38 100.60 100.45 100.22 100.55

1502 Sugar and artificial sweeteners 100.49 100.80 100.40 100.33 101.29

1601 Fats and oils 103.18 104.08 103.39 103.12 102.23

1701 Carbonated drinks 99.05 100.46 99.09 98.97 99.03

1703 Coffee 158.72 159.36 158.47 155.99 146.38

1705 Other noncarbonated drinks 98.37 98.63 98.37 97.97 97.11

1801 Canned and packaged soup 103.69 103.93 103.62 103.38 103.07

1802 Frozen prepared food 101.80 102.94 101.72 101.86 101.57

1803 Snacks 103.09 104.77 102.57 102.93 103.20

1804 Seasoning condiments sauces, spices 103.80 104.61 103.16 103.46 104.90

1806 Miscellaneous prepared food,

including baby food

102.21 102.73 102.16 102.07 101.93

1901 Lunch Away from Home 101.09 101.13 101.08 101.01 100.95

1902 Dinner Away from Home 101.12 101.15 101.14 101.06 101.51

1903 Other meals and snacks Away from

Home

101.09 101.29 101.10 100.97 100.93

2001 Beer and ale at home 99.95 100.12 99.85 99.88 99.79

2002 Distilled spirits at home 100.41 100.60 100.39 100.42 100.39

2003 Wine at home 97.56 98.18 97.65 97.66 97.33

2005 Alcoholic beverages away from home 100.54 100.70 100.66 100.42 101.07

2102 Lodging while out of town 104.96 106.13 100.13 102.02 97.48

2103 Lodging while at school 103.65 103.76 103.72 103.66 103.34

2401 Material supplies and equipment for

home

98.15 98.62 97.99 98.17 98.27

2501 Fuel oil 97.82 97.86 97.72 97.75 97.94

2502 Other household fuel commodities 97.20 97.22 97.09 97.10 96.01

2601 Electricity 106.86 107.22 105.74 106.46 105.76

2602 Utility (piped) gas 96.90 97.09 96.51 96.49 97.14

2701 Telephone local charges 99.82 99.81 99.81 99.76 99.84

2702 Water and sewerage maintenance 102.10 102.08 102.08 101.95 102.76

2703 Cable television 97.08 97.62 97.55 97.15 97.07

2704 Refuse collection 102.77 102.78 102.79 102.52 102.01

2705 Telephone interstate toll calls 105.89 105.96 105.93 105.80 103.28

2706 Telephone intrastate toll calls 99.84 99.85 99.85 99.79 99.52

2801 Textile House furnishings 98.84 100.07 98.68 98.41 100.33

2901 Bedroom furniture 100.41 100.69 100.34 99.84 102.19
29

Table A1

Mean Long Run Relative For Simulations Using Mean Number of Quotes

Imputation

Item

Strata Name Laspeyres

Link

Month Season Dutot Geomeans

2902 Sofas 101.14 101.33 101.33 100.92 101.42

2903 Living room chairs and tables 99.99 100.11 99.93 99.09 99.78

2904 Other furniture 101.70 101.88 101.58 101.18 103.39

3001 Refrigerators and home freezers 99.89 99.81 99.77 99.76 99.71

3002 Laundry equipment 102.91 102.93 102.92 102.80 102.84

3003 Stoves, ovens, dishwashers and air

conditioners

99.12 100.40 100.35 100.06 100.33

3101 Televisions 98.10 98.28 98.05 98.00 98.05

3103 Audio products 99.21 99.04 99.04 98.62 95.88

3201 Floor and window covering infants'

laundry,

98.88 98.53 98.75 98.08 99.26

3202 Clocks, lamps, and decor items 97.95 98.55 97.58 97.81 98.04

3203 Tableware serving pieces, and

nonelectric kitchenware

99.50 99.92 99.76 99.26 100.29

3204 Lawn equipment power tools, etc. 100.15 100.05 99.95 99.75 99.90

3205 Sewing, floor cleaning small

kitchen...

96.63 95.72 96.38 94.26 94.70

3206 Indoor plants and fresh cut flowers 100.31 101.40 101.06 100.78 101.50

3301 Laundry and cleaning products

including soap

100.72 101.15 100.78 100.69 100.59

3303 Household paper products and

stationary supplies

99.96 100.47 99.77 99.74 99.63

3305 Other household lawn, and garden

supplies

101.53 102.48 102.05 101.68 101.88

3401 Postage 100.00 100.00 100.00 100.00 100.00

3404 Gardening, other household services 101.06 101.08 101.05 101.00 101.65

3601 Men's suits, sport coats, coats, and 104.31 106.26 103.60 103.99 106.14

3603 Men's furnishing and special clothing 103.30 106.04 103.25 103.92 104.87

3604 Men's shirts 96.04 97.17 95.40 94.43 93.91

3605 Men's dungaree jeans, and trousers 97.59 99.90 98.04 97.61 97.61

3701 Boys apparel 98.27 100.55 97.68 97.37 96.82

3801 Women's coats and jackets 96.89 101.45 96.50 95.70 97.48

3802 Women's dresses 104.50 116.03 106.84 105.97 101.72

3803 Women's separated and sportswear 99.53 106.27 98.83 99.43 100.65

3804 Women's underwear nightwear

hosiery, and accessories

99.26 100.75 98.28 97.92 101.44

3805 Women's suits 99.70 110.38 100.77 99.03 99.21

3901 Girls apparel 93.66 98.50 94.50 93.56 93.44

4001 Men's footwear 99.18 100.46 98.91 98.79 98.71
30

Table A1

Mean Long Run Relative For Simulations Using Mean Number of Quotes

Imputation

Item

Strata Name Laspeyres

Link

Month Season Dutot Geomeans

4002 Boys' and girls' footwear 97.05 99.55 97.78 97.83 97.78

4003 Women's footwear 99.75 102.38 99.34 98.66 99.69

4101 Infants and toddlers apparel 103.51 103.35 102.98 101.94 101.14

4201 Sewing materials, notions and

luggage

102.89 103.23 101.58 101.03 101.88

4302 Jewelry 103.16 106.75 104.31 103.09 101.52

4401 Other apparel services 101.31 101.32 101.31 101.21 101.06

4402 Laundry and dry cleaning 102.05 102.03 102.04 101.88 102.09

4501 New cars 100.45 100.51 100.44 100.38 99.74

4502 New trucks 102.70 102.75 102.73 102.65 102.70

4503 New motorcycles 106.01 106.03 105.96 105.90 105.90

4701 Motor fuel 112.65 112.86 112.56 112.58 112.06

4702 Motor oil, coolant, and other products 103.04 103.22 103.11 102.90 103.22

4801 Tires 100.02 100.19 100.10 99.96 99.70

4802 Other automobile parts and

equipment

99.46 99.55 99.50 99.15 99.55

4901 Automobile body work 101.57 101.55 101.60 101.42 101.40

4902 Automobile drive train, brake, and

miscellaneous

101.89 102.05 101.98 101.81 102.11

4903 Automobile maintenance and

servicing

101.15 101.29 101.25 101.02 102.42

4904 Automobile power plant repair 102.02 102.11 102.04 101.97 102.06

5201 Automobile registration licensing

and inspection fees

100.15 100.14 100.14 100.14 100.18

5205 Other automobile fees 102.56 102.75 102.70 102.35 103.31

5301 Airline fares 97.32 97.87 96.51 96.05 97.57

5302 Other intercity public transportation 100.41 100.48 100.25 99.77 98.79

5303 Intracity public transportation 100.39 100.40 100.40 100.39 100.77

5401 Prescription Drugs 101.97 102.03 102.00 101.87 102.20

5502 Internal and respiratory over-thecounter drugs101.50 101.86 101.55 99.48 99.38

5503 Nonprescription medical equipment

and supplies

100.32 100.81 100.76 100.13 99.47

5601 Physicians' services 102.45 102.57 102.49 102.14 102.17

5602 Dental services 103.41 103.49 103.44 103.13 103.04

5603 Eye care 101.01 100.99 100.92 100.56 99.76

5604 Services by other medical

professionals

101.74 101.96 101.92 101.69 101.27

5701 Hospital room 103.13 103.20 103.12 102.95 102.41
31

Table A1

Mean Long Run Relative For Simulations Using Mean Number of Quotes

Imputation

Item

Strata Name Laspeyres

Link

Month Season Dutot Geomeans

5702 Other inpatient hospital services 104.16 104.17 104.02 103.78 103.68

5703 Outpatient services 105.84 105.86 105.75 105.31 105.24

5901 Newspapers 102.61 102.61 102.63 102.33 102.69

5902 Magazines periodicals, and books 102.10 102.20 102.15 101.90 102.68

6001 Sport vehicles including bicycles 98.94 98.99 99.00 98.79 100.03

6002 Other sporting goods 100.87 100.98 100.65 100.04 100.87

6101 Toys, hobbies, and music equipment 101.31 101.69 101.51 101.22 102.39

6102 Photography supplies and equipment 100.99 101.35 100.76 100.95 99.79

6103 Pet supplies and expense 100.83 101.08 100.84 100.44 100.21

6201 Club memberships 99.78 100.21 99.69 96.52 99.36

6202 Fees for participant sports, exclusive

club memberships

102.48 102.83 102.78 102.01 101.21

6203 Admissions 101.15 101.31 101.24 99.54 102.12

6205 Other entertainment services 101.14 101.23 101.13 101.08 101.18

6301 Tobacco products 101.82 101.98 101.96 101.62 100.32

6401 Other toilet goods and small personal

care

104.09 104.29 103.81 103.60 102.10

6403 Cosmetic bath, nail preparations

manicure.

98.04 98.06 98.00 97.83 98.76

6501 Beauty parlor services for females 101.76 101.87 101.81 101.46 101.88

6502 Haircuts and other barber shop

services for...

100.95 101.02 101.01 100.92 100.84

6601 School books and supplies, college 102.26 102.45 102.33 102.25 102.26

6701 College tuition 106.20 106.12 106.10 106.05 105.90

6702 Elementary and high school tuition 104.87 104.83 104.83 104.77 104.69

6703 Daycare and nursery school 102.93 102.95 102.94 102.91 103.15

6704 Tuition for technical/business/ and

other schools

104.45 104.45 104.44 104.42 104.13

6801 Legal service fees 100.94 100.88 100.91 100.83 100.42

6802 Personal financial services 100.70 100.71 100.71 100.70 100.83

6803 Funeral expenses 102.97 102.98 102.96 102.92 102.86
32

REFERENCES

Armknecht P. Moulton B. and K. Stewart: “Improvements to the Food at Home, Shelter,

and Prescription Drug Indexes in the U.S. Consumer Price Index”, BLS Working Paper

263, February 1995

Erickson, T.: “Effects of Mismeasuring Base Period Prices When Estimating the Laspeyres

Index: some Idealized Cases”, memo, November 1995

Kahn, S. and K. Lang: “Efficient Estimation of Structural Hedonic Systems”,

International Economic Review, v29 #1, February 1988

Moulton, B. and K. Smedley: “A Comparison of Estimators for Elementary Aggregates of

the CPI”, manuscript, June 1995

Moulton, B.: “Estimation of Elementary Indexes of the Consumer Price Index”,

manuscript, May 1996

Reinsdorf, M.: “Price Dispersion, Seller Substitution and the U.S. CPI”, BLS Working

Paper 252, March 1994