Each month, the International Price Program (IPP) at the Bureau of Labor Statistics collects a sample of approximately 

14,000 import and 8,300 export price quotes from roughly 4,000 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 2018 to December 2018. 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. 

The values of the estimated percent changes show the average percent changes on a monthly, quarterly, and annual basis. 

Table 1. Percent change and standard error for the Import Price Indexes, for one, three, and 12-month intervals, 

January-December 2018 

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 0.8 0.18 1.9 0.16 3.4 0.24

February 0.2 0.11 1.2 0.11 3.3 0.19

March ‐0.2 0.05 0.8 0.14 3.3 0.21

April 0.5 0.07 0.5 0.12 3.5 0.15

May 0.9 0.05 1.1 0.07 4.5 0.15

June 0.0 0.08 1.3 0.13 4.7 0.16

July ‐0.1 0.07 0.8 0.16 4.8 0.19

August ‐0.4 0.06  ‐0.5 0.17 3.8 0.24

September 0.1 0.04  ‐0.4 0.13 3.1 0.23

October 0.5 0.07 0.2 0.10 3.4 0.24

November ‐1.6 0.10  ‐1.1 0.11 0.7 0.18

December ‐1.4 0.11  ‐2.6 0.14  ‐0.9 0.23

Variance Estimates for Price Changes in the  

Import and Export Price Indexes

January‐December 2018

Helen McCulley and David Mead
Table 2. Percent change and standard error for the Export Price Indexes, for one, three, and 12-month intervals, 

January-December 2018 

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 0.7 0.11 1.3 0.13 3.3 0.27

February 0.2 0.15 1.0 0.16 3.1 0.36

March 0.4 0.05 1.3 0.16 3.4 0.30

April 0.5 0.07 1.0 0.12 3.7 0.21

May 0.7 0.06 1.6 0.12 5.0 0.25

June 0.2 0.07 1.3 0.10 5.3 0.27

July ‐0.5 0.14 0.4 0.17 4.3 0.19

August ‐0.1 0.04  ‐0.4 0.15 3.6 0.17

September 0.0 0.06  ‐0.5 0.18 2.7 0.24

October 0.5 0.06 0.4 0.07 3.1 0.21

November ‐0.8 0.12  ‐0.3 0.15 1.8 0.27

December ‐0.6 0.09  ‐0.9 0.15 1.1 0.27

The standard error, the square root of the estimated 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 0.7 percent for export prices in May, as seen in Table 2. The standard 

error for exports in May was 0.06. 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 0.7 percent plus or minus 0.12, or between 

0.58 percent and 0.82 percent. 

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. 

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. 
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, 

where 

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

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

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

wi,0 = pi,0 qi,0, or the total revenue generated by item i in base period 0; 

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

ܴܵܶ௧ ൌ ∑೔ ௥೔,೟௪೔,బ

∑೔ ௥೔,೟షభ௪೔,బ

, 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. 

These weighted item price relatives are combined across establishments and aggregated to the lowest level stratum 

indexes as 

ܲ௛,௧ ൌ

∑ ∑∑ ௪ೖ,೟௪ೕ,೟௪೔,೟൬

೛೔,೟

೛೔,బ ೖ ೔ೕ ൰

∑ ∑∑೔ೕೖ ௪ೖ,೟௪ೕ,೟௪೔,೟

, 

where 

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

wk,t = the weight of detailed product category k, within stratum h; 

wj,t = the weight of establishment j, within detailed product category k; 

wi,t = the weight of item i, within establishment j and detailed product category k; and 

௣೔,೟

௣೔,బ

 = the price relative of item i, from period t, to base period 0. 
The weights used for these lowest level stratum indexes are derived from sampling frame trade dollar values, divided by 

the corresponding probabilities of selection determined by the sample design. 

At the next level of aggregation, child strata level indexes are aggregated to their corresponding parent stratum level 

indexes. A child stratum index is simply one level of 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 

ܲு,௧ ൌ ∑೓ ௪೓,೟௉೓,೟

∑೓ ௪೓,೟

, 

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. 

Replication and Variance Estimation 

A modified bootstrap method, applying rescaled sampling weights, is used to produce 150 replicate index set estimates 

from 150 simulated item set samples. Item set replicates 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) or a probability selection. The replicate resampling 

method takes this into consideration by first partitioning the selected items within each sampling stratum m into those 

items that resulted from certainty establishment selections and those items resulting from probability establishment 

selections. The item set resulting from establishment certainty 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; 

ܵ ൌ ራ ܵ௠

ே

௠ୀଵ

ൌ ራ ቌራܵ௠೛

ଷ

௣ୀଵ ቍ

ே

௠ୀଵ

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

for items selected from probability sampling classification groups within certainty establishments, and p=3 for items 

selected from certainty sampling classification groups within certainty establishments. 
Each bootstrap sampling, b, selects ݊௠೛

௕ units within each partition of each sampling stratum 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 

௜,௝,೛௠ݓ

௕ ൌ ൞

൭௜ ,௝,௠೛ݓ

݊௠೛

௕ ൅ 1

݊௠೛

௕ ൱ ݀௠೛,௝

௕ for ݊௠೛

൐ 1

೛௠݊ for௜ ,௝,௠೛ݓ

ൌ 1ൢ

where 

௜,௝,೛௠ݓ

௕ = the bth replicate item weight for item i, within establishment j and sampling stratum partition mp; 

ݓ೛௠,௝, = ௜the standard item weight for item i, within establishment j and sampling stratum partition mp; and 

݀௠೛,௝

௕ = the number of times establishment j, within partition p of sampling stratum m, is selected in bootstrap 

sample b. 

In the rare instances that	݊௠೛

ൌ 1, a simple random sample of items within that establishment is selected. If only one item 

exists under this establishment singleton, that item is chosen with certainty. 

For each of the 150 bootstrap samples, chained indexes of the desired length are calculated at all levels of aggregation 

using these modified item weights, original probabilities of selection, trade dollar values, and collected price data. For 

variance 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.