Variance Estimates for Price Changes in the Import and Export Price Indexes 

January-December 2011 

Helen McCulley and David Mead 

This paper was revised on February 13, 2013, to correct an error in the formula in the final sentence. 

Each month, the International Price Program (IPP) at the Bureau of Labor Statistics 

collects a sample of approximately 16,000 import and 10,000 export price quotes from 

roughly 5,000 establishments. The number of prices from imported items is larger 

because of the larger volume of imports versus exports. The sample of import and export 

transaction prices is drawn from the complete universe of all items traded during a 

calendar year as measured by the Census Bureau. Sampling variability results whenever a 

sample is used rather than the complete universe. 

Variance Results 

The percent changes for overall import and overall export prices along with the standard 

error for those estimates was calculated for each of the 12 months running from January 

2011 to December 2011. Tables 1 and 2 display the estimated percent change for each 

index, as well as the standard error of that estimate, defined as the square root of the 

variance.1 The values of the estimated percent changes show the average percent changes 

on a monthly, quarterly, and annual basis. 

                                                      

1 The standard errors for the first six months of 2011 for the 12‐month percent changes are not available

because variance numbers were not calculated prior to July 2010.  

1
Table 1. Percent change and standard error for the Import Price Indexes, for one, 

three, and 12-month intervals, January-December 2011 

Month

All Commodities Import Price Index

One‐month Three‐month 12‐month

Percent

Change

Standard

Error

Percent

Change

Standard

Error

Percent

Change

Standard

Error

January 1.5 0.08 4.6 0.12 5.6

February 1.7 0.07 4.7 0.09 7.6

March 3.0 0.11 6.3 0.13 10.3

April 2.6 0.16 7.4 0.19 11.9

May 0.1 0.07 5.8 0.10 12.9

June ‐0.6 0.05 2.1 0.13 13.6

July 0.1 0.07 ‐0.3 0.17 13.7 0.10

August ‐0.4 0.05 ‐0.8 0.09 12.9 0.09

September ‐0.1 0.04 ‐0.4 0.07 12.7 0.08

October ‐0.4 0.09 ‐0.8 0.12 11.1 0.16

November 0.7 0.03 0.2 0.06 10.1 0.09

December 0.0 0.04 0.4 0.05 8.5 0.09

Table 2. Percent change and standard error for the Export Price Indexes, for one, 

three, and 12-month intervals, January-December 2011 

Month

All Commodities Export Price Index

One‐month Three‐month 12‐month

Percent

Change

Standard

Error

Percent

Change

Standard

Error

Percent

Change

Standard

Error

January 1.3 0.10 3.5 0.14 7.0

February 1.3 0.10 3.3 0.10 8.7

March 1.5 0.07 4.1 0.10 9.5

April 0.8 0.11 3.6 0.14 9.2

May 0.4 0.08 2.7 0.13 9.1

June 0.1 0.12 1.4 0.15 10.1

July ‐0.4 0.08 0.1 0.13 9.8 0.14

August 0.4 0.07 0.2 0.11 9.4 0.12

September 0.5 0.05 0.6 0.13 9.4 0.12

October ‐2.0 0.06 ‐1.0 0.11 6.3 0.11

November 0.1 0.06 ‐1.4 0.09 4.8 0.08

December ‐0.5 0.10 ‐2.4 0.12 3.6 0.15

2
The standard error of the estimated square root of the variance is a common measure used 

to derive confidence intervals for percent changes in the import and export price indexes. 

Confidence intervals can be used to determine if an index change is significantly different 

than zero. 

Take as an example the 1-month percent change of -0.5 percent for export prices in 

December, as seen in Chart 2. The standard error for exports in December was 0.10. So, 

deriving a confidence interval plus or minus two standard errors from the point estimate, 

assuming a normal distribution the true change would most likely be -0.5 percent plus or 

minus 0.10, or between -0.7 percent and -0.3 percent. 

Analyzing the Data 

One result from the data is that the relative standard error, defined as standard error 

divided by the percent change, decreases the longer the period in question. For example, 

looking at the index for overall import prices as seen in Chart 1, for the 1-month percent 

change in October, the standard error divided by the percent change (0.09/-0.4) is -0.23. 

For the 3-month and 12-month percent changes, the relative standard errors are -0.15 and 

0.01, respectively, meaning that relative to shorter intervals, the precision of the 

estimated percent change improves as the calculation interval lengthens. 

In the future, the variance tables will be expanded to include detailed product-level 

indexes. This will allow for greater analysis of how standard errors change as the data 

moves from disaggregated index categories to more aggregated categories. Analysis will 

also be possible comparing different index categories. For example, how standard errors 

vary between manufactured items and nonmanufactured items can be studied. 

3
Sources of Error 

There are different types of errors that are introduced when calculating the estimates of 

average price changes for imports and exports published by the Bureau of Labor 

Statistics. One way to look at measurements of error is the difference between sampling 

error and nonsampling error. Sampling error, what is reported on in the tables in this 

report, is the error resulting from drawing a sample of imported and exported items to and 

from the United States, rather than using the entire universe of trade. 

Nonsampling error can take a number of different forms. One form is misspecification 

error, which takes place if the universe of data from which the sample is being drawn 

does not correctly measure the actual population. For import and export prices indexes, 

this type of error could result if there are mistakes in the trade dollar value statistics 

measured by the Census Bureau. A second type of nonsampling error is nonresponse 

error. Each month, a subset of the items sampled by the Bureau of Labor Statistics does 

not have prices reported. This type of error results if the respondents and nonrespondents 

do not represent a similar cross section of the total universe. Another form of 

nonsampling error can be introduced from misreported prices, and is possible regardless 

of whether deriving the indexes from a sample or from the complete universe of data. 

One issue when deriving an estimate from a sample is the potential trade-off between 

variance and bias. Variance is a measure of how much the estimates derived from 

numerous samples differ from the true value of the estimate. Bias results if the expected 

value of the estimate is either higher or lower than the true value of what is being 

estimated. An estimate could have a high sampling variance and still be unbiased if the 

expected value of the estimate is equal to the true value. Likewise, an estimate may have 

small variation over numerous samples, yet be biased if the expected value of the 

estimate deviates from the true estimate. 

4
The Bureau of Labor Statistics strives to minimize both sampling and nonsampling error 

as much as possible. Sampling error is reduced by maintaining as many prices as possible 

to support an index given resource and company burden constraints. Nonsampling error 

is reduced by subjecting the data to careful review using automated checks and a staff of 

professional economists, as well as by employing methods to estimate missing 

observations. 

Sampling in the International Price Program 

Trade into and out of the United States is highly regulated and therefore highly 

documented. This allows the IPP to sample from a fairly complete and detailed frame. 

U.S. Customs and Border Protection provides the sampling frame for import 

merchandise, while the export merchandise frame is created from a combination of data 

collected by the Canada Border Services Agency, for exports to Canada, and from the 

Census Bureau for exports to the rest of the world. 

The import and export merchandise universes are divided into two panels each, with each 

product-based panel representing approximately half of the trade dollar value for its 

respective universe. One import and one export panel is sampled each year, resulting in a 

fully sampled universe every 2 years. Each panel is sampled using a 3-stage sample 

design. The first stage independently selects establishments within product-based 

sampling strata using systematic probability proportional to size, where the measure of 

size is the total trade dollar value for establishments within the sampling stratum. The 

second stage selects more highly detailed product categories, known as classification 

groups, within each establishment stratum combination selected during the first stage 

using the technique of systematic probability proportional to size with replacement. The 

final stage of sampling, which results in a unique item to price, occurs in the field and is a 

random selection technique with the probability of selection proportionate to field 

collected trade estimates. 

5
Index Calculations 

IPP calculates its indexes using a modified Laspeyres formula. Rather than calculate 

indexes relative to a base period, the IPP indexes are calculated relative to the previous 

period and are theoretically chained to the reweight period. For this reason, IPP’s indexes 

are often referred to as chained Laspeyres indexes. Explicitly, the index formula for the 

modified chained Laspeyres is derived from the classic formula as follows: 

ܮܴܶ௧ ൌ ൬

∑೔ ௣೔,೟௤೔,బ

∑೔ ௣೔,బ௤೔,బ൰ ሺ100ሻ ൌ ൭

∑ ೛೔,೟

೛೔,బ ೔ ௣೔,బ௤೔,బ

∑೔ ௣೔,బ௤೔,బ ൱ ሺ100ሻ ൌ ൬

∑೔ ௥೔,೟௪೔,బ

∑೔ ௪೔,బ ൰ ሺ100ሻ ൌ

൬ ∑೔ ௥೔,೟௪೔,బ

∑೔ ௥೔,೟షభ௪೔,బ൰ ൬∑೔ ௥೔,೟షభ௪೔,బ∑೔ ௪೔,బ ൰ ሺ100ሻ ൌ ൬ ∑೔ ௥೔,೟௪೔,బ∑೔ ௥೔,೟షభ௪೔,బ൰ ሺܮܴܶ௧ିଵሻ ൌ ሺܴܵܶ௧ሻሺܮܴܶ௧ିଵሻ, 

 0 in base period 0; 

 

ܴܵܶ௧ ൌ ೔ ೔,೟ ೔,బ

∑೔ ௥೔,೟షభ௪೔,బ

where 

LTRt = the long term relative of a collection of items at time t; 

pi,t = price of item i t at time ; 

qi,0 = quantity of item i in base period 0; 

wi,0 = pi,0 qi, , or the total revenue generated by item i

r p i,t = i,t i,0 / p , or the long term relative of item i t in period ; and 

∑ ௥ ௪ , or the short term relative of a collection of items at time t. 

Depending on the level of aggregation, the weights used during index aggregation are 

either trade dollar based or probability based. At the lowest level of aggregation, items 

are weighted by probability-based weights calculated monthly corresponding to detailed 

categories within establishments. 

6
These weighted item price relatives are combined across establishments and aggregated 

ܲ௛,௧ ൌ

∑ ∑∑ ௪ೖ,೟௪ೕ,೟௪೔,೟൬

೛೔,೟

೛೔,బ

to the lowest level stratum indexes as 

ೖ ೔ೕ ൰

∑ ∑∑ ೖ ೔ೕ ௪ೖ,೟௪ೕ,೟௪೔,೟

, 

where 

Ph,t = the price index for lowest level stratum h, at time t;

tum h; 

ry k; 

tegory k; 

wk,t = the weight of detailed product category k, within stra

wj,t = the weight of establishment j, within detailed product catego

wi,t = the weight of item i, within establishment j and detailed product ca

and 

௣೔,೟

௣೔,బ

= the price relative of item i, from period t, to base period 0. 

 used for these lowest level stratum indexes are derived from sampling frame 

t the next level of aggregation, child strata level indexes are aggregated to their 

vel of 

ܲு,௧ ൌ ∑೓ ௪೓,೟௉೓,೟

∑೓ ௪೓,೟

The weights

trade dollar values, divided by the corresponding probabilities of selection determined by 

the sample design. 

A

corresponding parent stratum level indexes. A child stratum index is simply one le

aggregation less than its parent stratum index. The weights used for this aggregation are 

based on Census Bureau trade dollar values for the base period. The aggregation formula

for these upper index levels is 

, 

where 

PH,t = the price index at period t, for upper level index H; 

wh,t = the weight at period t, for child index h; and 

Ph,t = the price index at period t, for child index h. 

7
Replication and Variance Estimation 

 modified bootstrap method, applying rescaled sampling weights, is used to produce 

es 

or a 

t 

ertainty 

ܵ ൌ ራ ܵ௠

ே

௠ୀଵ

ൌ ራ ቌራܵ௠೛

ଷ

௣ୀଵ ቍ

ே

௠ୀଵ

where N is the number of sampling strata, ݌ א ሼ1,2,3ሽ, with p=1 for items selected from 

ed from 

A

150 replicate index set estimates from 150 simulated item set samples. Item set replicat

are constructed according to IPP’s 3-stage sample design. At both of the first two stages 

of sampling, it is possible for a selection to be either a certainty selection (i.e. the 

probability of selection is greater than the iteratively calculated sampling interval) 

probability selection. The replicate resampling method takes this into consideration by 

first partitioning the selected items within each sampling stratum m into those items tha

resulted from certainty establishment selections and those items resulting from 

probability establishment selections. The item set resulting from establishment c

selections is further partitioned into two item sets: sampling classification group certainty

selections and sampling classification group probability selections. Thus, the set of all 

sampled items S is the union of these three partitions over all sampling stratum m; 

probability establishments, p=2 for items selected from probability sampling 

classification groups within certainty establishments, and p=3 for items select

certainty sampling classification groups within certainty establishments. 

8
9

ach bootstrap sampling, b, selects ݊௠೛

௕ units within each partition of each sampling 

tratum as follows: 

݊௠೛

௕ ൌ ቊ݊௠೛

െ 1 ݊௠೛൐ 1

1 ݊௠೛ൌ 1ቋ, 

where ݊௠೛is the number of units originally sampled in partition p of sampling stratum m. 

Bootstrap item weights are then calculated as 

݊௠

௕ ൅ 1

E

s

௜,௝,೛௠ݓ

௕ ൌ ൞ݓ೛௠,௝, ௜ቆ ೛

݊௠೛

௕ ቇ ݀௠೛,௝

௕ for ݊௠೛

൐ 1

೛௠݊ for௜ ,௝,௠೛ݓ

ൌ 1ൢ

where 

௜,௝,೛௠ݓ

௕ = the bth replicate item weight for item i, within establishment j and 

sampling stratum partition mp; 

 pling௜ ,௝,௠೛ݓ

݀௠೛,௝

௕ 

ootstrap sample b. 

In the rare in

establis r this establishment singleton, that 

item is chosen with certainty. 

gation using these modified item weights, original 

probabilities of selection, trade dollar values, and collected price data. For variance 

 = the standard item weight for item i, within establishment j and sam

stratum partition mp; and 

= the number of times establishment j, within partition p of sampling 

stratum m, is selected in b

stances that ݊௠೛

ൌ 1, a simple random sample of items within that 

hment is selected. If only one item exists unde

For each of the 150 bootstrap samples, chained indexes of the desired length are 

calculated at all levels of aggre

estimates, the variance is calculated across replicate percent change values for all

published indexes as ߥ ஻ൌ ଵ

ଵହ଴ ∑ ൫ߠ෠

௕ െ ߠ෠൯

ଶ ଵହ଴

௕ୀଵ where ߠ ෠is the full sample estimate.