Implementing a Reconciliation and Balancing Model in the 

U.S. Industry Accounts*

Dylan G. Rassier†

Thomas F. Howells III 

Edward T. Morgan 

Nicholas R. Empey 

Conrad E. Roesch 

U.S. Department of Commerce 

Bureau of Economic Analysis 

Industry Accounts 

Washington, DC 

September, 2007 

 

*

 We thank Baoline Chen for contributing motivation and knowledge to this project through her study to reconcile and 

balance the 1997 industry accounts (Chen, 2006). We also thank Karen Horowitz, Douglas Meade, Mark Planting, and 

George Smith for providing early direction, leadership, and advice to the project. 

This paper has been prepared for presentation at the 2007 Federal Committee on Statistical Methodology Research 

Conference in Crystal City, VA, November 5 – 7, 2007. We thank Baoline Chen, Sumiye Okubo, George Smith, Erich 

Strassner, Mary Streitwieser, and Bob Yuskavage for helpful comments. All errors and omissions are our own. 

The views expressed in this paper are solely those of the authors and not necessarily those of the U.S. Bureau of Economic 

Analysis or the U.S. Department of Commerce. 

†

 Corresponding author: Dylan Rassier, U.S. Bureau of Economic Analysis, Industry Accounts (BE-40), 1441 L St. NW, 

Washington DC 20230; e-mail dylan.rassier@bea.gov; phone 202-606-9575; fax 202-606-5366. 
1. Introduction 

In a series of papers beginning with Stone et al. (1942), Stone (1961, 1968, 1970, 1975, 1976, and 1984) advocates a 

generalized least squares framework to improve the economic and statistical accuracy of independent estimates of national 

income and expenditures based on relative reliabilities of underlying data used to construct the estimates. Researchers have 

subsequently revised the Stone method to facilitate its implementation (Byron, 1978; van der Ploeg, 1982, 1984), but federal 

agencies responsible for producing national economic accounts have generally not implemented the method. Two reasons 

for this have been a lack of technology that is typically required to solve the complex systems of equations faced by federal 

agencies and a lack of information regarding the relative reliabilities of underlying data used to construct national accounting 

estimates. While some agencies have resolved the latter challenge with subjective measures of relative reliabilities 

(Mantegazza and Pisani, 2000; Moyer et al., 2004a, 2004b), a lack of adequate technology has until recently continued to 

prevent the implementation of the Stone method (Nicolardi, 2000; Tuke and Aldin, 2004). In a recent study, Chen (2006) 

addresses both challenges by building an empirical model based on the Stone method and incorporating statistical measures 

of relative reliability in the model to reconcile and balance estimates in the 1997 U.S. industry accounts. 

As part of the U.S. Bureau of Economic Analysis’ (BEA’s) integration initiative (Yuskavage, 2000; Moyer et al., 2004a, 

2004b; Lawson et al., 2006), the Industry Accounts Directorate is building an empirical model based on the Stone method 

and Chen (2006) in order to reconcile the gross operating surplus component of value-added from the 2002 expenditurebased benchmark input-output accounts and the 2002 income-based gross domestic product (GDP)-by-industry accounts.1

 In 

addition to reconciling gross operating surplus, the model provides a tool for balancing the 2002 benchmark use table.2

The objective of the reconciliation is to use information regarding the relative reliabilities of underlying data that are used to 

estimate intermediate inputs in the benchmark use table and gross operating surplus in the GDP-by-industry accounts in a 

balanced input-output framework in order to improve intermediate input estimates and gross operating surplus estimates in 

both products. In the benchmark use table, gross operating surplus is derived as a residual between the industry distributions

of gross output and intermediate inputs, compensation, and taxes. Intermediate inputs in the benchmark use table are derived 

from external source data and BEA adjustment methodologies.3

 In the GDP-by-industry accounts, gross operating surplus is 

derived from the industry distributions of the gross domestic income (GDI) components of the national income and product 

accounts as well as BEA adjustment methodologies. Similar to gross operating surplus in the benchmark use table, 

intermediate inputs in the GDP-by-industry accounts are derived as a residual between the industry distributions of gross 

output and value-added. Results from current work will be published in the upcoming 2002 benchmark use table.4

A reconciliation model based on the Stone method has five advantages over past models used by the Industry Accounts 

Directorate. First, the Stone method yields a model that is transparent. The method is widely researched and familiar to 

national economic accounting agencies as well as national statistical agencies. In addition, models based on the Stone 

method are easy to describe to users. Second, the Stone method is built on a firm statistical foundation. In particular, the 

method uses information on the relative reliabilities of initial underlying data from which estimates are derived to make 

adjustments to the initial estimates in a balanced accounting framework. Thus, a model based on the Stone method yields a 

set of fully balanced accounts with adjustments that reflect the reliability of the initial data (Chen, 2006). Third, the least 

squares framework of the Stone method guarantees that adjustments made to initial estimates are as small as necessary to 

remove discrepancies between estimates and to satisfy the accounting constraints of the model. In this way, the method 

 

1

 Note that the term GDP typically refers to expenditure- or production-based estimates. However, estimates for the GDPby-industry accounts are derived from the industry distribution of the income-based gross domestic income estimates in the 

national income and product accounts. Thus, contrary to the typical reference, GDP-by-industry accounts are truly income 

based. 

2

 The U.S. benchmark input-output accounts are composed of several tables, including the make table, the use table, and 

supplementary tables (Lawson et al., 2002). This paper focuses on the benchmark use table, which shows commodities 

consumed by industries and final users as well as value-added contributed by industries. The benchmark make table has been 

balanced separately using a different methodology. The supplementary tables are outside the scope of this paper. 

3

 Likewise, gross output, compensation, and taxes in the benchmark use table are derived from external source data and BEA 

adjustment methodologies. However, the reconciliation model only adjusts intermediate inputs and gross operating surplus, 

so this paper focuses only on the external source data and BEA adjustment methodologies used to derive intermediate inputs 

and gross operating surplus. 

4

 In contrast, a prior reconciliation based on a different model set a best level for value-added in the 1997 benchmark use 

table for use in the GDP-by-industry accounts (Moyer et al., 2004a, 2004b). 
yields final estimates that are consistent with the economic concepts upon which the accounts are built. Fourth, the Stone 

method yields a model that is flexible (Chen, 2006). The model can be implemented with aggregated data or disaggregated 

data. The objective function can be easily altered to include the reliability-weighted squared differences of additional 

variables. Constraints can be easily removed or imposed. Finally, the Stone method yields a model that is replicable. If no 

changes are made to data that are introduced to the model, the model yields a duplicate set of results. Alternatively, updated

data can be introduced to the model without requiring any substantial changes to the model. 

This paper presents work by the Industry Accounts Directorate of the BEA to develop and implement an empirical model 

based on the Stone method and Chen (2006) to reconcile gross operating surplus estimates in the 2002 benchmark inputoutput accounts and the 2002 GDP-by-industry accounts as well as to balance the benchmark use table. In this spirit, the 

paper presents the empirical model and briefly describes the technology used to solve the model. Work during development 

of the model shows that reconciling and balancing a large system with disaggregated data is computationally feasible and 

efficient in pursuit of an economically accurate and reliable benchmark use table. 

Section 2 of the paper discusses the benchmark input-output accounts, focusing specifically on the use table, and section 3 

discusses the GDP-by-industry accounts. In addition to providing overviews of the accounts, each section discusses features 

of external source data and BEA adjustment methodologies that are relevant for the reconciliation. Section 4 presents the

empirical model as well as a discussion of the construction of reliability measures in the absence of a complete variancecovariance matrix required by theory to obtain an efficient estimator. Section 5 briefly describes the technology used to solve 

the model, and section 6 presents a set of preliminary results from the model. Section 7 concludes. 

2. The U.S. Benchmark Input-Output Accounts 

The benchmark input-output accounts are prepared quinquennially and provide a comprehensive picture of the flows of 

goods and services across all consuming industries and final use categories. The accounts are presented in a series of tables. 

The use table incorporates a complete, balanced set of economic statistics and presents a full accounting of industry and final

use transactions. In addition, the use table provides gross output estimates for industries and commodities, value-added

estimates by industry, a cross check for the variety of data used to estimate the national income and product accounts, and 

estimates of final consumption for the national income and product accounts that incorporate the best data available and are 

in balance with industry inputs and outputs. 

Chart 1 provides an overview of the structure of the benchmark use table. The matrix in the upper left corner of the use table

is the matrix of intermediate inputs, which shows commodities purchased by industries for the production of goods and 

services in those industries.5

 The matrix below the intermediate inputs is the matrix of value-added components, which 

includes compensation, taxes on imports less subsidies, and gross operating surplus. The matrix to the right of the 

intermediate inputs is the matrix of final use categories, which includes personal consumption expenditures, private 

equipment and software, structures, changes in private inventories, government expenditures, exports, and imports. Finally, 

the bottom row and the far right column show the gross output for industries and commodities, respectively, which come 

from the balanced benchmark make table. 

Intermediate input estimates in the benchmark use table are derived using expense data from the Census Bureau, data from 

non-Census sources, and BEA adjustment methodologies. Approximately 71 percent of the value of intermediate inputs 

comes from Census expense data, 22 percent of the value comes from non-Census data, and 7 percent of the value comes 

from BEA adjustment methodologies. 

Expense data from Census for 2002 are more comprehensive and improved than were available for previous benchmark use 

tables in the manufacturing sectors (NAICS 31 – 33) and the service sectors (NAICS 42 – 81).6

 Within the manufacturing 

 

5

 The publication level of detail in the benchmark use table includes approximately 400 industries and 400 commodities. The 

working level of detail includes approximately 1,000 industries and 1,000 commodities. Commodities are further 

disaggregated into approximately 9,000 items. Industry analysts estimate transactions for items purchased by industries for 

the production of goods and services in those industries. The model is implemented with the item level of detail. 

6

 However, intermediate input estimates continue to be derived entirely or partially from non-Census data and BEA 

adjustment methodologies for the following sectors: agriculture, forestry, fishing, and hunting (NAICS 11), utilities (NAICS 

22), transportation and warehousing (NAICS 48-49), information (NAICS 51), finance and insurance (NAICS 52), real estate 

(NAICS 53), and education (NAICS 61). 
sectors, nineteen categories of expenses are available from the 2002 Economic Census.7

 Within the service sectors, nineteen 

categories of expenses are available from the 2002 Business Expenses Survey (BES).8

 Given this depth of coverage in the 

manufacturing sectors and service sectors, initial gross operating surplus estimates in the benchmark use table are much 

improved in those sectors both before and after the reconciliation because gross operating surplus is calculated as a residual: 

industry gross output less intermediate inputs, compensation, and taxes. 

Expense categories provide a basis for each industry’s input category control estimates in the benchmark data.9

 However, 

adjustments to the input category control estimates similar to adjustments that are made to derive each industry’s estimate of 

gross output are required in order to maintain equality among industry inputs and outputs. In particular, adjustments are 

made for non-employer expenses, misreporting and non-filing, and auxiliary services.10

The Economic Census only covers establishments with employees and payroll. Thus, BEA makes an adjustment to capture 

outputs and inputs for non-employer establishments. Census provides receipts for non-employers derived from 

administrative data, which are used to estimate the gross output of non-employers.11 For each industry, the ratio of inputs to 

outputs is assumed to be the same for employers and non-employers. Thus, expenses for non-employers are estimated in a 

given industry by distributing total receipts for non-employers based on the ratio of inputs to outputs for employers in the 

industry, with the residual being distributed to gross operating surplus. Given the reasonable underlying assumption and the 

reliable source data for non-employers, the adjustment for non-employer expenses is considered to be more reliable than 

other adjustments in the benchmark data but less reliable than data from the Economic Census. 

Unlike non-employer establishments, small employers are included in estimates published for the Economic Census. 

However, similar to non-employers, Census derives estimates for small employers from administrative data. Administrative 

data for non-employers typically include individual income tax returns, and administrative data for small employers typically 

include individual, partnership, and corporate income tax returns. Given these administrative data, BEA adjusts estimates for 

non-employers and small employers for misreporting and non-filing.12 A misreport results when a tax return is filed with 

incomplete or incorrect information. Non-filing results when a business or individual who earns income fails entirely to file a 

return. An adjustment for misreporting is based on data from two Internal Revenue Service (IRS) programs: the Taxpayer 

Compliance Measurement Program (TCMP) and the TCMP-Information Return Program (TCMP-IRP).13 The most recent 

 

7

 Expense categories from the 2002 Economic Census for manufacturing sectors include the following: (1) accounting, 

auditing, and bookkeeping, (2) advertising, (3) communications, (4) computer services, (5) contract labor, (6) depreciation, 

(7) electricity, (8) fuels, (9) legal services, (10) management, consulting, and administrative services, (11) materials 

consumed, (12) other expenses, (13) payroll and fringe benefits, (14) refuse removal, (15) re-sales, (16) rental of buildings 

and other structures, (17) rental of machinery and equipment, (18) repairs and maintenance of buildings and/or machinery, 

and (19) taxes and license fees. 

8

 Expense categories from the 2002 BES for service sectors include the following: (1) accounting, auditing, and 

bookkeeping, (2) advertising, (3) communications, (4) contract labor, (5) data processing and other purchased computer 

services, (6) depreciation and amortization charges, (7) insurance, (8) lease and rental payments, (9) legal services, (10) 

packaging and containers, (11) printing, (12) management, consulting, and administrative services, (13) materials and 

supplies not for resale, (14) other expenses, (15) payroll and fringe benefits, (16) repair and maintenance, (17) taxes and 

license fees, (18) transportation, shipping, and warehouse services, and (19) utilities. In addition to payroll from the 2002 

BES, payroll is available from the 2002 Economic Census, which is considered more reliable than the 2002 BES. Thus, the 

benchmark data include payroll from the 2002 Economic Census. 

9

 Input category control estimates are used to guide transactions for industries’ purchases of items for intermediate uses. 

10 While other adjustments are made to source data for intermediate inputs, these three adjustments capture the majority of 

the dollar value of adjustments made in the benchmark data. 

11 Census provides receipts for non-employers at a more aggregated level than receipts for employer establishments. To 

distribute receipts for non-employers to the appropriate industry level, BEA uses a ratio of receipts for small-employer 

establishments by industry to total receipts for all small employers from the Economic Census. However, these adjustments 

do not affect the data for the reconciliation because gross output is fixed in the reconciliation model. 

12 According to Internal Revenue Service compliance studies, three components contribute to a tax gap between the amount 

taxpayers should pay and the amount taxpayers actually pay in a timely manner: non-filed returns, underreported income, 

and underpaid taxes (Brown and Mazur, 2003). 

13 Discontinued in the early 1990s, the TCMP was an audit program designed to study compliance patterns and levels of 

misreporting among sole proprietors from a stratified random sample of individual tax returns. The TCMP-IRP was designed 
year for which data are available from a TCMP study is 1988. An adjustment for non-filers is based on data from an exactmatch study conducted by Census.14 The most recent year for which data are available from an exact-match study at the time 

the 2002 benchmark use table is being composed and the 2002 GDP-by-industry accounts were published was 1996. In 

addition, data from the TCMP, TCMP-IRP, and exact-match studies are provided at a more aggregated level than the industry 

level in the benchmark use table. Given the infrequencies with which the TCMP, TCMP-IRP, and exact-match studies are 

conducted and the need to approximate the industry distribution of the adjustments, adjustments based on data from the 

studies are considered less reliable than other adjustments. 

BEA makes an adjustment for auxiliary services in order to allocate sector-level expense data to industries served by an 

auxiliary.15 For a given auxiliary, Census provides expense data tabulated for the sector served by the auxiliary. For each 

industry, the ratio of industry-level payroll to sector-level payroll is assumed to be the same as industry-level expense to

sector-level expense. The adjustment draws upon this assumption and industry-level payroll data to allocate the sector-level 

expense data to industries served. Additions to intermediate inputs in a given industry served are offset by reductions to 

gross operating surplus in that industry. The adjustment for auxiliary services is considered more reliable than the 

misreporting and non-filing adjustment but less reliable than the adjustment for non-employer expenses because of the source 

data and the relatively strong assumption regarding payroll and other expenses. 

Before gross operating surplus in the benchmark use table is reconciled with the GDP-by-industry accounts, all benchmark 

estimates are finalized and reviewed to ensure the data submitted to the model are as complete as possible and consistent with 

the economic concepts upon which the benchmark use table is built. In particular, total gross industry output must equal total

gross commodity output, controls for intermediate inputs and value-added components must equal industry gross output, and 

commodity gross output must be as fully allocated as possible to intermediate inputs. Likewise, the commodity distributions 

of final use categories are reconciled with the national income and product accounts, Census-based compensation and tax 

estimates are scaled to match total compensation and taxes, respectively, in the national income and product accounts. In 

addition, estimates for intermediate inputs, trade margins, and transportation costs must be non-negative. Furthermore, inputoutput coefficients and the ratio of value-added by industry to total GDP are examined for unreasonable changes from those 

in the 1997 benchmark use table. Finally, the industry distribution of initial residual gross operating surplus estimates from

the benchmark data are compared with the industry distribution of gross operating surplus estimates from the GDP-byindustry accounts, and total value-added from the GDP-by-industry accounts is compared to total final uses as reconciled 

with the national income and product accounts. 

3. The U.S. Gross Domestic Product-by-Industry Accounts 

The GDP-by-industry accounts focus on the industry shares of value-added to total GDP as well as contributions of industries 

to GDP growth and provide an annual time series of estimates for gross output, intermediate inputs, and value-added by 

industry (Moyer et al., 2004a). Intermediate inputs in the GDP-by-industry accounts are derived as a residual between the 

industry estimates of extrapolated gross output and value-added. Gross output in the GDP-by-industry accounts is calculated 

using annual survey data to extrapolate best-level gross output detail from the make table in the most recent benchmark 

input-output accounts. Best-level estimates of the compensation and tax components of value-added in the GDP-by-industry 

accounts are derived from the GDI components of the national income and product accounts, and these best-level estimates 

are used directly. Gross operating surplus in the GDP-by-industry accounts is derived by extrapolating best-level estimates

of gross operating surplus from the most recent benchmark use table using the sum of annual GDI-based components of gross 

operating surplus as an indicator.16 In addition to being utilized as annual indicators, the GDI-based estimates of gross 

 

to compare the results of the TCMP audits to information returns (e.g., W-2, 1099-INT, etc.) filed with the IRS in order to 

capture misreporting that TCMP auditors failed to find. 

14 An exact-match study compares records from the Current Population Survey (CPS) to records from the IRS in order to 

identify and estimate non-filed income for individuals who report income in the CPS but do not file a return with the IRS. 

15 An auxiliary is an establishment that provides to other establishments services that may not be part of the economic

activities of which the serviced establishment is primarily engaged. 

16 Best-level estimates of gross operating surplus from the most recent benchmark use table are derived from a prior 

reconciliation to set best-level value-added (Moyer et al., 2004a, 2004b). In particular, value-added in the GDP-by-industry 

accounts was built from GDI components and BEA adjustment methodologies and reconciled with value-added from the 

1997 benchmark use table. The resulting best-level value-added includes best-level estimates of compensation, taxes, and 

gross operating surplus, which are extrapolated in the GDP-by-industry accounts. In order to fit the sum of best-level 

estimates of compensation, taxes, and gross operating surplus with best-level value-added, estimates of gross operating 
operating surplus are used as inputs for the reconciliation model. Thus, estimates of 2002 gross operating surplus used in the

reconciliation model are different from estimates of gross operating surplus published in the 2002 GDP-by-industry accounts. 

GDI-based estimates of gross operating surplus for private, non-farm industries are derived using data from the IRS’s 

Statistics of Income Division (SOI), data from non-SOI sources, and BEA adjustment methodologies.17 Approximately 47 

percent of private, non-farm industries’ gross operating surplus comes from the SOI data, 14 percent comes from the 

misreporting and non-filing adjustment, 22 percent comes from concept and coverage adjustments, and 17 percent comes 

from non-SOI data and related adjustments.18 In addition, the SOI data are collected and classified on a company basis. 

BEA makes an adjustment to convert these company-based data to an establishment basis, which is the production unit used 

for both the benchmark input-output accounts and the GDP-by-industry accounts. This adjustment shifts gross operating 

surplus from one industry to another with no impact on total gross operating surplus. Company-based and establishmentbased industry estimates differ by an absolute average of 7.7 percent. 

Similar to the benchmark input-output accounts, BEA also makes a misreporting and non-filing adjustment for non-corporate 

business income tax-based source data in the GDP-by-industry accounts based on data from the TCMP, TCMP-IRP, and 

exact-match studies. Thus, the misreporting and non-filing adjustment is considered less reliable than other adjustments. 

The same assessment regarding the relative reliability of the adjustment is made in both the benchmark data and the GDP-byindustry data. 

Concept adjustments are a collection of adjustments designed to convert tax accounting-based concepts in the SOI data to 

economic accounting-based concepts consistent with national accounts. Concept adjustments include the removal of capital 

gains and dividends from business income. Coverage adjustments are designed to include the activities of entities that 

contribute to gross domestic product but are not required to file a return with the IRS. Coverage adjustments include adding 

income earned by Federal Reserve banks and imputing net income for owner-occupied housing. While concept and coverage 

adjustments are considered more reliable overall than the misreporting and non-filing adjustment, reliability varies by 

specific adjustment. Adjustments based on administrative data are considered second in reliability only to Economic Census 

data, and adjustments based on survey data are considered third in reliability of this type of adjustment. 

The company-establishment adjustment is limited to three of the income components of gross operating surplus derived from 

corporate income tax data: corporate profits before tax, corporate capital consumption allowance, and corporate net interest. 

The adjustment is made using a matrix of employment data from Census that relates employment by industry on a company 

basis and employment by industry on an establishment basis. By assuming that profits before tax, capital consumption 

allowance, and net interest are the same per employee for all establishments performing the same activity, regardless of the 

company to which that establishment belongs, the matrix converts the industry distribution of each income component from a 

company basis to an establishment basis. An employment matrix based on Economic Census data is used to adjust the 

corporate income tax data available for the reconciliation. Due to the weakness of the assumptions that underlie this 

adjustment methodology, the company-establishment adjustment is considered less reliable than most concept and coverage 

adjustments but more reliable than the misreporting and non-filing adjustment. 

Similar to the benchmark estimates, gross operating surplus estimates from the GDP-by-industry accounts are reviewed and 

finalized before they are submitted to the reconciliation model. In particular, gross operating surplus levels and the ratio of 

value-added by industry to total GDP is compared with corresponding measures in the benchmark use table. In addition, the 

initial GDI-based gross operating surplus estimates are reviewed for reasonability within the context of an annual time series.

 

surplus were adjusted as necessary. Thus, best-level estimates of gross operating surplus from the most recent benchmark 

use table are a residual determined from best-level estimates of compensation, taxes, and value-added. Gross operating 

surplus is composed of the following components: corporate profits before tax, corporate capital consumption allowance, 

corporate net interest, corporate inventory valuation adjustment, business transfer payments, proprietors’ income, noncorporate capital consumption allowance, non-corporate net interest, proprietors’ inventory valuation adjustment, rent and 

royalties, rental income of persons, and surpluses of government enterprises. 

17 Gross operating surplus estimates for farm and general government industries and the owner-occupied housing portion of 

gross operating surplus estimates in the real estate industry are not reconciled using the model presented in this paper. Thus, 

a detailed discussion of the data and methodologies used to prepare estimates for these industries is outside the scope of this

paper. 

18 NIPA tables 7.13, 7.14, 7.16, and 7.17 provide additional detail on the relationship between income components as 

reported by the IRS and as published by BEA. 
4. Reconciling Gross Operating Surplus in a Balanced Input-Output Framework

The objective is to reconcile in a balanced input-output framework the expenditure-based residual gross operating surplus 

estimates by industry in the 2002 benchmark use table with the gross operating surplus estimates derived from income-based 

external source data and BEA adjustment methodologies in the 2002 GDP-by-industry accounts. Based on an adaptation of 

Chen (2006), the objective is satisfied with a constrained optimization model that minimizes the reliability-weighted sum of 

squared differences between initial and final estimates subject to a system of accounting constraints.19

4.1. Empirical Model 

Estimates of expenditure-based final use categories in the benchmark use table are considered highly reliable and are not 

allowed to adjust.20 In addition, industry gross output, commodity gross output, and the compensation and tax components 

of value-added, all in the benchmark use table and all derived from the 2002 Economic Census, are considered to be highly 

reliable and are not allowed to adjust.21 Thus, the objective function in the model is constructed to adjust intermediate input 

estimates from the benchmark data and gross operating surplus estimates from the GDP-by-industry data. In addition to 

identifying fixed elements of the benchmark use table, the system of constraints incorporates the accounting identities upon 

which the table is built.22 In this way, the model provides a tool for balancing the benchmark use table in addition to 

improving intermediate input estimates and gross operating surplus estimates in the table. 

More formally, let z and v denote intermediate inputs and value-added, respectively. Let w

z

 and w

v

 denote variance measures 

for initial estimates of intermediate inputs and value-added, respectively. Let x, y, Y

E

, and Y

I

 denote gross output, final uses, 

aggregate expenditure-based GDP, and aggregate income-based GDI, respectively. Subscript i indexes industries, k indexes 

commodities, d indexes final use categories (including imports and exports), and h indexes the components of value-added 

(i.e., compensation, taxes, and gross operating surplus).23 Superscript 0 denotes an initial value. 

The model is described mathematically as follows: 

Min S {z, v} = ∑∑ ∑∑ −

+

−

i h

v

hi

hi hi

i k

z

ki

ki ki

w

v v

w

z z

2

,

0

, ,

2

,

0

, ,

( ) ( )

 (4.1.1) 

 

 

19 Chen (2006) builds an empirical model based on the Stone method and incorporates statistical measures of relative 

reliability in the model to reconcile and balance estimates in the 1997 U.S. Industry Accounts. While Chen’s (2006) model 

adjusts gross output, intermediate inputs, and all components of value-added, the model being implemented by the Industry 

Accounts Directorate only allows intermediate inputs and the gross operating surplus component of value-added to adjust. 

20 Fixler and Grimm (2005) show that expenditure-based GDP estimates are reliable over time based on the small magnitude 

of revisions. 

21 The compensation and tax components of value-added are scaled to match the totals in the national income and product 

accounts. 

22 In addition to the stated fixed elements of the benchmark use table, gross operating surplus estimates for farm and general 

government industries and the owner-occupied housing portion of gross operating surplus estimates in the real estate industry 

are fixed because they are reconciled using a different methodology. 

23 The Industry Accounts Directorate maintains multiple indices of industries and commodities. The most aggregated 

industry index comprises 65 industries, and the most aggregated commodity index comprises 69 commodities. The most 

disaggregated industry and commodity indices comprise nearly 1,000 industries and commodities, respectively. Each of the 

most aggregated industries and commodities comprise a mutually exclusive subset of the most disaggregated industries and 

commodities. Furthermore, commodities are composed of items. The benchmark database contains data for the most 

disaggregated industries and items. The GDP-by-industry database contains data for the most aggregated industries and no 

data for commodities. Because the reconciliation and balancing model is implemented with the most disaggregated industry 

and item data, the model requires separate indices and constraints to identify the most disaggregated industries included in 

the most aggregated industries. However, this strategy does not affect the specified model. Thus, for simplicity, this 

exposition focuses on a single index for industries and commodities. 
Subject to: 

 

 

 

 

 

 

0

0 ∑ , −∑ , =

d

dk

d

dk

y y ∀ k (4.1.2) 

0

0

xi − xi = ∀ i (4.1.3) 

0

0

xk − xk = ∀ k (4.1.4) 

0

0

vi,compensation − vi,compensation = ∀ i (4.1.5) 

0

0

vi,taxes − vi,taxes = ∀ i (4.1.6) 

0 −∑ ∑ , − , =

k h

i ki hi

x z v ∀ i (4.1.7) 

0 −∑ , −∑ , =

d

dk

i

k ik

x z y ∀ k (4.1.8) 

0 −∑∑ , =

i h

hi

E

Y v (4.1.9)

Initial conditions: 

0 ∑∑ , − =

E

k d

y dk Y (4.1.10)

E

i i k

∑ ∑ xi − ∑z ki ≠ Y

0

,

0

 (4.1.11)

−∑ ≠∑

h

hi

k

i ki

x z v

0

,

0

,

0

 ∀ i (4.1.12) 

−∑ ≠∑

i

ik

d

k dk

x y z

0

,

0

,

0

 ∀ k (4.1.13) 

0

0 0, ∑ , − =I

gos

i

v gosi Y (4.1.14)

∑ =∑

k

k

i

i

x x (4.1.15)

I E

Y ≠ Y

0,

 (4.1.16)

Constraints (4.1.2) to (4.1.6) identify fixed values for final use categories, industry gross output, commodity gross output, 

compensation, and taxes, respectively.24 Equation (4.1.7) constrains the sum of final estimates for intermediate inputs used 

 

24 Industry gross output and commodity gross output in the benchmark use table come from the balanced benchmark make 

table. Given the constraints in equations (4.1.3) and (4.1.4), the distribution of commodities produced by industries (i.e., the 

benchmark make table) is not affected by the model. 
by an industry and value-added for the industry to equal gross output for the industry. Equation (4.1.8) constrains the sum of

final estimates for a commodity used by industries as an intermediate input and sold for final uses to equal gross output for 

the commodity. Constraint (4.1.9) indicates that final total value-added equals aggregate expenditure-based GDP. 

 

The initial condition in equation (4.1.10) indicates that aggregate expenditure-based GDP is set exogenously by the total 

commodity distribution of final uses determined through the reconciliation with the national income and product accounts. 

Equation (4.1.11) indicates that total value-added is not initially equal to aggregate expenditure-based GDP. Equation 

(4.1.12) indicates that the sum of initial estimates for intermediate inputs used by an industry and value-added for the 

industry do not equal gross output for the industry. Equation (4.1.13) indicates that the sum of initial estimates for a 

commodity used by industries as an intermediate input and sold for final uses do not equal gross output for the commodity. 

The condition in equation (4.1.14) indicates that initial estimates of the gross operating surplus component of value-added 

come from the income-based GDP-by-industry data. Equation (4.1.15) identifies the initial equality between total industry 

gross output and total commodity gross output in the benchmark use table.25 The initial condition in equation (4.1.16) 

identifies the statistical discrepancy between aggregate expenditure-based GDP and aggregate income-based GDI.26

4.2. Reliability Weights 

The optimal estimator requires a variance-covariance matrix to weight the adjustments made to the initial estimates. In 

practice, the variance-covariance matrix for initial estimates comprising the benchmark use table and the GDP-by-industry 

accounts is unavailable. However, information regarding the relative reliabilities of initial estimates is used to construct a

variance measure for each estimate in the benchmark data and the GDP-by-industry data. In addition, given the lack of 

information regarding correlations among the initial estimates, covariance measures are assumed to be zero. While this 

assumption results in an estimator that is less than efficient, the inefficiency is less than may be introduced if the correlations 

are incorrectly determined (Byron, 1978; Nicolardi, 2000). 

The noted variance measures are constructed using coefficients of variation, where available or internally-assigned reliability

indicators where coefficients of variation are unavailable.27 Recall that estimates in the benchmark use table and GDP-byindustry accounts are derived from two sources of information: 1) external source data and 2) BEA adjustment 

methodologies. External source data are subject to sampling and non-sampling errors. Coefficients of variation measure 

sampling errors and are available for source data provided to BEA by SOI and Census.28 BEA adjustment methodologies are 

designed and implemented to correct non-sampling errors in external source data. However, the adjustments are themselves 

subject to non-sampling errors. Thus, BEA analysts assign reliability indicators to individual adjustment records based on a 

rubric required to ensure a consistent application of the reliability indicators across industries and adjustments common to 

both the benchmark use table and the GDP-by-industry accounts.29

Table 1 contains a summary of the general rubric used by analysts to assign reliability indicators to individual records.30 As 

shown in Table 1, internally-assigned reliability indicators range from 1 to 5 in one-unit increments, with 1 indicating the 

 

25 Recall that industry gross output and commodity gross output in the benchmark use table come from the balanced 

benchmark make table. 

26 At the most aggregated industry level and commodity level (i.e., 65 industries and 69 commodities), the model contains 

4,550 variables to be solved and 135 constraints to be satisfied. At the most disaggregated industry level and item level (i.e., 

987 industries and 8,910 items), the model contains 8,795,157 variables to be solved and 9,963 constraints to be satisfied. 

However, at the most disaggregated level, the intermediate input portion of the benchmark use table is not fully populated. 

Thus, the number of variables to be solved decreases accordingly. 

27 Coefficients of variation are defined as the standard error of an estimate divided by the estimate. 

28 In particular, coefficients of variation are available from SOI for all estimates. Because sampling errors apply to surveys 

and not censuses, coefficients of variation are available from Census for the expense data from the Business Expenses Survey 

and purchased services from the Annual Survey of Manufactures portion of the Census of Manufactures. Purchased services 

include the following expense categories: (1) accounting, auditing, and bookkeeping, (2) advertising, (3) communications, 

(4) computer services, (5) legal services, (6) management, consulting, and administrative services, (7) other expenses, (8) 

refuse removal, (9) repairs and maintenance of buildings and/or machinery, and (10) taxes and license fees. 

29 In addition, using the noted rubric, BEA analysts assign reliability indicators to all detailed estimates in the benchmark 

data, including those derived from external source data, regardless of the availability of coefficients of variation. 

30 In addition to reviewing estimates from the benchmark data and GDP-by-industry data before the data are submitted to the 

model, internally-assigned reliability indicators are reviewed for consistency of application across industries, type of 

adjustment, and data set (i.e., benchmark data and GDP-by-industry data) prior to implementing the model. 
most reliable estimates (e.g., Census-based estimates), 5 indicating the least reliable estimates (e.g., misreporting 

adjustments), and intermediate values indicating intermediately reliable estimates (e.g., concept adjustments). In this way, 

internally-assigned reliability indicators provide an indication of the percentage error or relative reliability of the related

estimate, similar to coefficients of variation. 

Table 2 contains distributions of internally-assigned reliability indicators and coefficients of variation for all industries and 

ten randomly chosen and sorted industries.31 The figures shown are the percentages of the value of intermediate inputs and 

gross operating surplus in the benchmark data and the GDP-by-industry data, respectively. No coefficients of variation are 

available for Census estimates, so there is generally a smaller percentage of benchmark data with coefficients of variation. 

As shown in Table 2, coefficients of variation are available for approximately 23 percent of the total intermediate input value

in the benchmark data and 47 percent of the total gross operating surplus value in the GDP-by-industry data. However, this 

result does not hold for industries A and I. In addition, internally-assigned reliability indicators of one are assigned only to 

Census estimates with no adjustments. Thus, there are no GDP-by-industry data with an internally-assigned reliability 

indicator of one. Internally-assigned reliability indicators of two and three are assigned to almost 50 percent of the total 

intermediate input value in the benchmark data and 40 percent of the total gross operating surplus value in the GDP-byindustry data. However, this result varies considerable across industries. Finally, internally-assigned reliability indicators of 

five are assigned to a higher percentage value of gross operating surplus value in the GDP-by-industry data because the 

misreporting adjustment plays a larger role in those data. This result holds for all industries. 

Coefficients of variation range from 0 to 1 with a theoretically infinite number of intermediate values.32 In order to use 

internally-assigned reliability indicators where coefficients of variation are unavailable, a function is applied to internallyassigned reliability indicators to generate an equivalent scale of reliability across all indicators of reliability (i.e., internallyassigned reliability indicators and coefficients of variation). The function maps internally-assigned reliability indicators, 

denoted θ, according to an exponential parameter, denoted α, as follows: 

α

θ

θ ⎟

⎠

⎞

⎜

⎝

⎛ −

=

4

1

f )( . (4.2.1) 

If α = 1 , internally-assigned reliability indicators are mapped linearly from 0 to 1 in increments of 0.25. If α > 1 , internallyassigned reliability indicators are mapped from 0 to 1, assuming that less-reliable estimates are exponentially less reliable 

than more reliable estimates. If α < 1, internally-assigned reliability indicators are mapped from 0 to 1, assuming that 

reliability decreases at a diminishing rate.33 In the end, a reliability indicator exists for each individual record and includes a 

coefficient of variation, where available or an internally-assigned reliability indicator mapped by equation (4.2.1) where a 

coefficient of variation is unavailable.34 The reliability indicators range from 0 to 1, with 0 indicating the most reliable 

estimates and 1 indicating the least reliable estimates.35

Upon generating a reliability indicator for each individual record, a variance measure is constructed for each initial estimate

by squaring the product of the initial estimate and the related reliability indicator, denoted CV. The following equations 

summarize the construction of a variance measure, w

z

 or w

v

, for each initial estimate of intermediate inputs, z

0

, and valueadded, v

0

, in equation (4.1.1): 

20 w (CV z )z

= × (4.2.2)

 

31 Industries are randomly chosen and sorted in order to disguise the identity of unpublished industry estimates in subsequent 

figures with results. 

32 Coefficients of variation greater than 1 are limited to 1. In the GDP-by-industry data, 1.8% of the coefficients of variation

are greater than 1. In the benchmark input-output data, .05% of the coefficients of variation are greater than 1. 

33 Evidence has not been generated regarding an appropriate value for α. Thus, given a lack of any evidence to support α ≠ 1, 

internally-assigned reliability indicators are mapped assuming α = 1. 

34 Given the use of fully disaggregated data in the model, this approach yields a variable with either a coefficient of variation

or a mapped internally-assigned reliability indicator for each related estimate. 

35 This approach assumes that the best estimates with internally-assigned reliability indicators are as good as the best 

estimates with coefficients of variation, and the worst estimates with internally-assigned reliability indicators are as bad as

the worst estimates with coefficients of variation. 
 

and 

20 w (CV v )v

= × . (4.2.3) 

Thus, according to equation (4.1.1), adjustments in the model are weighted by the reciprocal of the variance measure. 

5. Technology 

As noted in the introduction, recent technology makes implementation of the reconciliation and balancing model possible. In 

particular, the reconciliation model is solved using the CPLEX solver in the Generalized Algebraic Modeling System 

(GAMS). GAMS is a flexible optimization software package designed to handle large, complex mathematical programming 

problems. CPLEX is a GAMS solver with solution algorithms for linear, quadratically constrained, and mixed-integer 

problems. In particular, CPLEX includes simplex and barrier algorithms. The CPLEX solver automatically chooses the 

optimal combination of algorithms to efficiently solve the particular model specified. Alternatively, GAMS also allows users 

to adjust tuning parameters in order to set algorithmic options. The reconciliation model is solved with the combination of 

algorithms chosen by GAMS. 

6. Preliminary Results

Preliminary results from the reconciliation are shown for ten industries in Table 3. To disguise unpublished estimates, the 

sample of industries is randomly chosen and sorted. In addition, all estimates have been scaled by the smallest estimate. 

Given the specifications of the model, the reconciliation results are generally consistent with prior expectations. For 

industries with relatively small differences between initial benchmark and GDI gross operating surplus estimates (e.g., 

industries D, F, G, and J), reconciled estimates are generally close to both initial estimates. For industries with relatively

large differences between initial benchmark and GDI gross operating surplus estimates (e.g., A, B, C, E, H, and I), the 

reconciliation favors benchmark estimates in some industries (e.g., industries A, B, C, H, and I) and GDI estimates in other 

industries (e.g., industry E). This is a result of the relative reliabilities of estimates in those industries as demonstrated by the 

smaller relative variance in the data favored by the reconciliation.36 For example, initial benchmark and GDI estimates for 

industry A are 7,156 and 10,172, respectively, and the relative variances of the initial benchmark and GDI estimates are 207 

and 3,109, respectively. Reconciled gross operating surplus for industry A is 7,222, which favors the benchmark data and is 

consistent with the smaller relative variance in the benchmark data. Likewise, initial benchmark and GDI estimates for 

industry E are 10,259 and 16,124, respectively, and the relative variances of the initial benchmark and GDI estimates are 

20,071 and 11,740, respectively. Reconciled gross operating surplus for industry E is 12,634, which favors the GDP-byindustry data and is consistent with the smaller relative variance in the GDP-by-industry data. 

In some industries (e.g., industries C, F, and J), reconciled estimates lie just outside the range of initial benchmark and GDI

estimates. Recall that the initial estimates are adjusted based on relative reliabilities of other estimates in a given industry as 

well as estimates in other industries. In addition, there is no constraint specified in the model to yield a value between the

two initial estimates. Thus, in order for the industry accounting constraints to be satisfied, the reconciled estimates fall 

outside the range of initial estimates. 

An important measure of industry performance is calculated as industry value-added as a percentage share of total GDP. 

Since gross operating surplus is a component of value-added, Table 4 compares this measure before and after the 

reconciliation and also compares the percentage shares after the reconciliation to the percentage shares from 1997. The 

percentage shares before and after the reconciliation are similar in most industries (e.g., industries A, B, C, F, G, and J). 

However, there are some industries in which the percentage shares before and after the reconciliation are not similar (e.g., 

industries D, E, H, and I). In all industries except industry E, the percentage shares after the reconciliation are similar to the 

percentage shares from 1997. Because gross operating surplus measures profits, a higher percentage share in industry E 

reflects higher profits in that industry. 

 

36 A relative variance is calculated as the aggregate variance for an industry divided by the aggregate estimate in that 

industry. 
To evaluate the balanced intermediate input matrix of the benchmark use table, tables 5 and 6 show a comparison of initial 

and balanced total industry intermediates and initial and balanced total commodity intermediates, respectively. In addition to

the percent change from initial to balanced totals, a column is included in each table for the initial under- or over-allocation 

of gross output calculated as a percent of gross output.37 For both industry intermediates and commodity intermediates, the 

percent change from initial totals to balanced totals is generally small. This is a result of small initial under- or overallocations. The largest percent change for both total industry intermediates and total commodity intermediates occurs in 

industry C, which also has the largest initial under-allocation. The percent change for total industry intermediates is 34.0 

percent, while the percent change for total commodity intermediates is 21.6 percent. The initial under-allocations for total 

industry intermediates and total commodity intermediates in industry C are 13.4 percent and 28.1 percent, respectively. In 

addition, industry J experiences a relatively large percent change for total industry intermediates of 8.3 percent with an initial 

under-allocation of 2.9 percent. Finally, industry G also experiences a relatively large percent change for total commodity 

intermediates of 12.8 percent with an initial under-allocation of 20.5 percent. 

In Table 5, the percent changes for total industry intermediates are positive for all industries. However, not all industries are 

under-allocated (e.g., industries B, G, H, and I). This result occurs because the change for gross operating surplus is negative 

in order to satisfy the constraint that intermediate inputs plus value-added in the industry are equal to gross output. In Table 

6, commodity J is the only over-allocated commodity, which is reflected by the negative change in the total. 

An important measure of industry technology is calculated as a ratio of total industry inputs to total industry output. Since 

total industry intermediates can change in the model, Table 7 compares input-output coefficients before and after balancing 

and also compares coefficients after balancing to coefficients from 1997. The coefficients generally increase slightly after 

balancing (e.g., A, B, C, D, E, F, G, and J), reflecting increases in total industry intermediates for all industries as shown in 

Table 5. The coefficients for industries H and I do not increase, reflecting the small increases in total industry intermediates 

for those industries. Likewise, the coefficient for industry C experiences a large increase, reflecting the large increase in total 

industry intermediates for that industry. Coefficients after balancing are similar to coefficients from 1997 for most industries 

(e.g., A, C, D, E, G, I, and J) but different for some industries (e.g., B, F, and H). Coefficients increase over time for 

industries A and J, indicating more inputs are required in 2002 to produce a unit of output in those industries, but decrease for 

all other industries, indicating fewer inputs are required in 2002 to produce a unit of output in those industries. Given the 

generally small changes in input-output coefficients before and after balancing, the changes in coefficients from 1997 to 2002 

are most likely explained by changes in technology as reflected in the initial estimates of industry intermediate inputs and 

gross output. 

7. Conclusions

This paper presents work by the Industry Accounts Directorate of the BEA to develop and implement a model based on a 

generalized least squares framework in order to reconcile gross operating surplus estimates in the 2002 benchmark inputoutput accounts use table and the 2002 GDP-by-industry accounts as well as to balance the benchmark use table. Work 

during development of the model and preliminary results show that the model is computationally feasible and efficient with a 

set of fully disaggregated data in pursuit of an economically accurate and reliable benchmark use table. 

Preliminary results are currently being reviewed and are generally as expected. While the benchmark data and the GDP-byindustry data are subject to review before the model is run, there are unexpected results from the model that help identify 

opportunities for improvements in the data. While the model yields significant technological improvements over past 

reconciliation and balancing methods, the review process highlights the fact that industry expert intervention during 

reconciliation and balancing is inevitable. In particular, intervention efforts focus on improvements to the data being 

introduced to the model. For example, intermediate inputs should be as complete as possible to ensure that under- and overallocated output are as small as possible before data are introduced to the model. 

 

37 Under- or over-allocation of gross output occurs because complete information regarding the use of intermediate inputs is 

unavailable. Thus, estimates regarding the use of intermediate inputs either fail to satisfy output or exceed gross output. In 

particular, under- or over-allocation of industry gross output refers to the amount by which intermediate inputs plus valueadded in a given industry fail to satisfy industry gross output or exceed industry gross output, respectively. Likewise, underor over-allocation of commodity gross output refers to the amount by which intermediate inputs plus final uses for a given 

commodity fail to satisfy gross output or exceed commodity gross output, respectively. 
As work is completed to implement the model, there are three areas to focus in the future. First, the model can be adapted to 

other industry products, including the make table of the benchmark input-output accounts, the commodity flow table, and the 

annual input-output accounts. 

As a second area of focus for future work, there have been challenges encountered during implementation that programming 

alone cannot overcome. One such challenge has been working with a benchmark database that is configured for other 

reconciliation and balancing methodologies, which may be largely remedied by future database reconfigurations. Other 

challenges with the model may always exist, such as calibrating the degree of precision with which a model of this size is 

capable of reaching a solution. While expert intervention will always be required in balancing input-output tables, this model

provides a giant stepping stone to reduce routine interventions while at the same time improving estimate in the tables. 

Finally, given the important role of reliability indicators in the determination of final estimates, future work will focus on 

ways to effectively use coefficients of variation and internally-assigned reliability indicators together as well as ways to 

ensure the consistent application of internally-assigned reliability indicators. As one way to reduce dependence on internallyassigned reliability indicators, coefficients of variation should be collected from source data providers for expenses in 

industries not covered by Census. In addition, reliability information other than coefficients of variation may be available for 

some source data. For example, coefficients of variation are generally not available for the Economic Census, but 

information on imputation rates may provide a useful objective indication of reliability. 
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Chart 1: Overview of the Structure of the Benchmark Use Table 

Construction Manufacturing Trade Information Government Other Services

Total 

Intermediate 

Inputs

Personal 

Consumption 

Expenditures

Private 

Equipment and 

Software

Changes in 

Private 

Inventories Exports Imports Government GDP

Total 

Commodity 

Output

Construction

Manufacturing

Commodities Trade

Information

Government

Other Services

Total Intermediate 

Inputs

Compensation

Value-Added Taxes

Gross Operating 

Surplus

Total Value-Added

Total Industry Output

Industries Final Uses
Table 1: General Rubric for Assigning Reliability Indicators to Underlying Data

Reliability 

Indicator Source of Underlying Data 

1 • Economic Census data with no adjustments 

2 • Economic Census data with adjustments 

• Survey data with no adjustments 

• Concept and coverage adjustments based solely on administrative data 

• Non-employer adjustments 

3 • Survey data with adjustments 

• Trade association data 

• Concept and coverage adjustments based on survey data 

• Auxiliary service adjustments 

4 • Company-establishment conversion adjustment 

• Adjustments based on a combination of analyst judgment and external source data 

5 • Misreporting adjustments 

• Adjustments based solely on analyst judgment 
Table 2: Distribution of Reliability Indicators in the Benchmark and the GDP-by-Industry Data40

 

Industry Data 1 2 3 4 5 CV 

All Benchmark 13% 24% 20% 17% 3% 23% 

GDP-by-industry 0% 11% 27% 2% 13% 47% 

A Benchmark 0% 0% 10% 2% 3% 85% 

GDP-by-industry 0% 0% 6% 5% 21% 68% 

B Benchmark 3% 80% 6% 0% 0% 11% 

GDP-by-industry 0% 11% 8% 7% 15% 59% 

C Benchmark 80% 1% 18% 1% 0% 0% 

GDP-by-industry 0% 6% 2% 26% 11% 55% 

D Benchmark 0% 1% 99% 0% 0% 0% 

GDP-by-industry 0% 3% 9% 1% 10% 77% 

E Benchmark 0% 0% 5% 89% 6% 0% 

GDP-by-industry 0% 1% 1% 1% 33% 64% 

F Benchmark 0% 85% 5% 0% 0% 10% 

GDP-by-industry 0% 3% 16% 7% 22% 52% 

G Benchmark 0% 73% 6% 0% 0% 21% 

GDP-by-industry 0% 1% 7% 5% 12% 75% 

H Benchmark 5% 77% 9% 0% 1% 8% 

GDP-by-industry 0% 11% 1% 9% 4% 75% 

I Benchmark 0% 11% 3% 4% 0% 82% 

GDP-by-industry 0% 

 

0% 11% 2% 34% 53% 

J Benchmark 0% 0% 54% 4% 0% 42% 

GDP-by-industry 0% 3% 9% 2% 34% 52% 

 

40 Industries are randomly chosen and sorted. The figures shown are the percentages of the value of intermediate inputs and 

gross operating surplus in the benchmark and GDP-by-industry data, respectively. 
Table 3: Comparing Reconciled GOS to Initial Benchmark GOS and Initial GDP-by-Industry GOS41

Industry 

Initial 

Benchmark 

GOS 

Benchmark 

Relative 

Variance42

Initial 

GDP-byIndustry 

GOS 

GDP-byIndustry 

Relative 

Variance43

Reconciled 

GOS 

Benchmark 

Percent 

Change 

GDP-byIndustry 

Percent 

Change 

A 7,156 207 10,172 3,109 7,222 0.9% - 29.0% 

B 3,122 462 1,727 510 2,593 - 16.9% 50.1% 

C 1,000 31 1,456 585 978 - 2.2% - 32.8% 

D 14,859 7,951 12,964 1,784 13,848 - 6.8% 6.8% 

E 10,259 20,071 16,124 11,740 12,634 23.1% -21.6% 

F 1,166 472 1,187 784 1,148 - 1.5% - 3.3% 

G 1,877 580 1,332 389 1,626 - 13.4% 22.0% 

H 13,923 750 6,083 990 9,842 - 29.3% 61.8% 

I 16,915 117 11,323 16,758 15,464 - 8.6% 36.6% 

J 2,116 204 2,411 2,077 2,095 - 0.9% - 13.1% 

 

41 Results come from preliminary data. Industries are randomly chosen and sorted, and values are scaled by the smallest 

value in the sample. 

42 The benchmark relative variance is calculated by aggregating the variances for all intermediate input estimates in a given 

industry and dividing by the aggregate estimate in the industry. 

43 The GDP-by-industry relative variance is calculated by aggregating the variances for all gross operating surplus estimates 

in a given industry and dividing by the aggregate estimate in the industry. 
Table 4: Comparing Benchmark Industry Value-Added as a Percent of Total GDP44

Industry 

2002 Value-Added 

with Initial Benchmark GOS 

2002 Value-Added 

 with Reconciled GOS 

1997 Percent with 

Reconciled Value-Added45

A 3.1% 3.2% 3.2% 

B 0.8% 0.5% 0.6% 

C 0.3% 0.2% 0.3% 

D 4.1% 2.7% 2.4% 

E 5.7% 4.7% 3.7% 

F 0.5% 0.4% 0.5% 

G 0.8% 0.6% 0.8% 

H 3.1% 1.8% 1.9% 

I 5.9% 4.3% 4.1% 

J 0.9% 0.7% 0.6% 

 

44 Results come from preliminary data. Industries are randomly chosen and sorted. 

45 The percent for 1997 is based on a different reconciliation model that set a best level for value-added in the 1997 

benchmark use table for use in the GDP-by-industry accounts (Moyer et al., 2004a, 2004b). 
Table 5: Comparing Initial and Balanced Industry Intermediates46

Industry 

Initial 

Intermediate 

Total 

Balanced 

Intermediate 

Total 

Percent 

Change 

Initial 

Under- or OverAllocation 

A 11,433 11,750 2.8% 1.0% 

B 5,142 5,410 5.2% - 2.0% 

C 1,000 1,340 34.0% 13.4% 

D 14,567 15,483 6.3% 0.0% 

E 25,856 27,303 5.6% 1.3% 

F 5,129 5,486 7.0% 4.2% 

G 5,871 6,282 7.0% - 4.8% 

H 20,111 20,427 1.6% - 0.1% 

I 14,604 14,617 0.1% - 0.3% 

J 3,090 3,346 8.3% 2.9% 

 

46 Results come from preliminary data. Industries are randomly chosen and sorted, and values are scaled by the smallest 

value in the sample. The initial under- or over-allocation is calculated as a percentage of industry gross output (i.e., industry 

gross output less intermediate inputs and value-added divided by industry gross output). 
Table 6: Comparing Initial and Balanced Commodity Intermediates47

Commodity 

Initial 

Intermediate 

Total 

Balanced 

Intermediate 

Total 

Percent 

Change 

Initial 

Under- or OverAllocation 

A 1,000 1,005 0.5% 0.0% 

B 6,433 6,525 1.4% 1.4% 

C 1,812 2,203 21.6% 28.1% 

D 11,231 11,265 0.3% 0.1% 

E 7,033 7,281 3.5% 0.6% 

F 7,545 7,743 2.6% 0.8% 

G 6,575 7,418 12.8% 20.5% 

H 9,267 9,756 5.3% 6.7% 

I 35,927 36,465 1.5% 0.0% 

J 4,805 4,636 - 3.5% -1.6% 

 

47 Results come from preliminary data. Commodities are randomly chosen and sorted, and values are scaled by the smallest 

value in the sample. The initial under- or over-allocation is calculated as a percentage of commodity gross output (i.e., 

commodity gross output less intermediate inputs and final uses divided by commodity gross output). 
Table 7: Comparing Benchmark Input-Output Coefficients with Initial and Balanced Industry Intermediates48

Industry 

2002 Input-Output Coefficient 

with Initial Intermediates 

2002 Input-Output Coefficient 

with Balanced Intermediates 

1997 Input-Output Coefficient 

with Balanced Intermediates 

A 0.33 0.34 0.30 

B 0.51 0.54 0.65 

C 0.31 0.42 0.50 

D 0.42 0.45 0.45 

E 0.40 0.42 0.50 

F 0.56 0.60 0.71 

G 0.52 0.55 0.61 

H 0.54 0.54 0.74 

I 0.32 0.32 0.34 

J 0.38 0.41 0.35 

 

48 Results come from preliminary data. Industries are randomly chosen and sorted.