A Flexible Mathematical Programming Model to Estimate Multiregional InputOutput Accounts

1
A Flexible Mathematical Programming Model to
Estimate Multiregional Input-Output Accounts

• Zhi Wang
• Bureau of Economic Analysis, US Department of
  Commerce
• Patrick Canning
• Economic Research Service, US Department of
  Agriculture

2
Presentation Outline

• Motivations and Objectives
• Literature Review and Basic Ideas
• Model Specifications and Major Properties
• Experiment Design
• Testing Results
• Conclusions and Limitations
• Apply to the U.S. -- the Large Dimension Case

3
Motivations

• There are tremendous disparities in economic
  development among different regions in large
  developing countries. Globalization has
  different impacts on the urban and coastal areas
  compared to the rural and inland less developed
  regions. How to maintain balanced development to
  reduce income inequalities between different
  parts of the country and achieve a high rate of
  economic growth simultaneously is one of the
  pressing policy challenges faced by governments
  in most large developing countries today.
• A major obstacle in conducting policy analysis
  for regional economic development is the lack of
  consistent, reliable regional data, especially
  data on interregional trade.

4
Motivations

• Despite decades of efforts, regional data
  analogous to national input-output accounts and
  international trade accounts remain unavailable,
  even for well defined sub-national regions in
  many developed countries. Regional economists
  have to develop various non-survey methods to
  estimate these data.
• In the past two decades, there have been well
  developed mathematical procedures to estimate
  unknown data based on limited prior information
  subject to a set of linear constraints
  (Constraint matrix balance), but they have not
  been widely used in regional economic analysis.

5
Objectives

• This research intend to bridge such gaps. We have
  three objectives
• develop and implement a formal model to
  estimate inter-regional, inter-industry
  transaction
• flows based on incomplete regional data
• To evaluate the models performance using
  real-world data
• To apply the model in a large dimension case
  using publicly available economic and
  transportation statistics describing the US
  economy in 1997

6
Literature Review Constrained matrix-balancing
problem

• It involves the computation of the best estimate
  of an unknown matrix from a given matrix, with
  some prior information to constrain the solution.
  It is a core mathematical structure in diverse
  applications
• Estimating input-output tables and inter-regional
  trade flows in regional science (todays
  presentation)
• Balancing of social/national accounts in
  economics (on going research)
• Estimating interregional migration in demography
  
• Analysis of voting patterns in political science
• Treatment of census data and estimation of
  contingency tables in statistics
• Estimation of transition probabilities in
  stochastic modeling
• Projection of traffic within telecommunication
  and transportation networks

7
Literature ReviewApplication to interregional
trade flows

• David F. Batten. The Interregional Linkages
  Between National and Regional Input-Output
  Models. International Regional Science Review,
  Vol.7, No. 1, pp. 53-67, 1982.
• David F. Batten and D. Martellato. Classical
  versus Modern Approaches to Interregional
  Input-output Analysis. Annals of Regional
  Sciences, 19 1-15, 1985.
• Golan, Amos, George Judge and Sherman Robinson
  Recovering Information from Incomplete or
  partial Multi-sectoral Economic Data. The Review
  of Economics and Statistics, 76(3),541-49,1994.
• Robinson, Sherman, Andrea Cattaneo, and Moataz
  El-Said. Updating and Estimating a Social
  Accounting Matrix Using Cross Entropy Methods.
  Economic System Research, 13(1), 47-64,2001
• Patrick Canning and Zhi Wang A Flexible
  Mathematical Programming Model to Estimate
  Interregional Input-Output Accounts.Journal of
  Regional Sciences, August, 2005.

8
Basic Ideas of the Framework

• To apply this methodology as a data
  reconciliation tool, three pieces of information
  are needed
• Initial estimates of the same economic variables
  from different sources (in national economic
  accounts, estimates of the same variables can
  often be obtained from income, expenditure or
  production data).
• An accounting framework and other constraints
  (demands have to equal supplies and components
  have to sum to totals, ).
• Reliability information on the initial estimates
  (standard error, ranking index, )

9
Problems of Proportional Adjustment
10
Problems of Proportional Adjustment(Cont.)
11
Notation Conventions

• Domestic deliveries
• Intermediate demand
• Gross output
• Value added
• Final demand
• Exports
• Imports

• Counterparts of national totals without
  superscripts
• Variables with a bar denote initial estimates for
  that variable
• An additional w before the variable indicates
  the reliability measure for that variable
• Superscripts denote regions, subscripts denote
  products

12
The Estimation Problems

• Given a nm2 non-negative array and a
  n2m non-negative array , determine a
  non-negative array D and a non-negative
  array Z that is close to D and Z such that
  accounting identities hold.
• In other words, modify a given set of prior
  inter-regional and inter-industrial transaction
  estimates according to the following objective
  function to satisfy known accounting constraints

13
Basic Accounting Identities in a National System
of Economic Regions
14
The Battern Interregional IO Model (1982)
15

Theoretical Properties

• Statistical interpretations underlying the model
  differ when different reliability weights are
  used
• The quadratic and entropy objective functions are
  equivalent in the neighborhood of initial
  estimates
• In all but the trivial case, posterior estimates
  derived from entropy or quadratic loss minimand
  will always be closer to the unknown, true values
  than the associated initial estimates.
• The choice of weights in the objective function
  has very important impacts on the estimation
  results

16
Why Balanced Estimates Better?

• Initial estimates
• W variance matrix of initial estimates ,
• A coefficient matrix of all linear constraints
  AD 0
• The BLUE
• D will always not worse than with equal or
  smaller variance

17
Empirical Advantages

• Flexibility
• This model permits a wider variety and volume of
  information to be brought to bear on the
  estimation process than is possible with scaling
  methods such as RAS
• Incorporation of data reliabilities in a systemic
  way
• The weights in the objective function reflect the
  relative reliability of a given set of priors.
  Entries with higher reliability should undergo
  less adjustment than entries with lower
  reliability.
• Smaller model dimension than Batten model
• Has NG2(N-1) less variables and the same
  constraints. This increases estimation efficiency
  and facilitates the computation process.

18
Testing Data Set

• Version 4 of the Global Trade Analysis Project
  (GTAP) database was first aggregated into a
  4-region, 10-sector data set.
• Then 3 of the 4 regions (the United States,
  European Union and Japan) were further aggregated
  into a single open economy which engages in both
  inter-regional trade among its 3 internal regions
  and international trade with rest of the world.
• The model was used to replicate the underlying
  inter-continental trade flows among Japan, EU and
  the United States as well as the individual
  countrys input-output account.

19
Experiment Design

• Experiment 1
• The three regions weighted average I-O flows
  and distorted inter-regional trade data in the
  GTAP were used as initial estimates.
• Experiment 2
• The region-specific I-O flows are assumed to be
  constant. The inter-regional shipments in the
  first experiment were re-estimated.
• Experiment 3
• The inter-regional shipments are known with
  certainty. The three regions weighted average
  I-O flows were used as priors to estimate the
  region-specific I-O flows.

20
Experiment Design (Cont.)

• Experiment 4
• David F. Battens model was used to estimate the
  inter-regional shipments and individual region
  I-O flows.
• Solutions from both models are compared with the
  true inter-regional trade and inter-sector I-O
  flow data in the GTAP data set.

21
Measures to evaluate test resultsMean absolute
percentage error (MAPE)
22
Estimate Results Mean absolute percentage error
of inter-regional tradePercent difference from
true trade data
23
Estimate Results (Cont.)Mean absolute percentage
error of inter-regional tradePercent difference
from true trade data
24
Estimate Results (Cont.)Mean absolute percentage
error of inter-sector flowsPercent difference
from true IO data
25
Observations

• In all experiments except the Batten model, most
  of the mean absolute percentage errors are about
  4-7 percent of the true data. In contrast,
  recovering the individual regions input-output
  flows from national averages values only had
  limited success.
• When there is no additional information that can
  be incorporated into the estimation framework, a
  more detailed model may not perform any better
  than a simpler model. However, the accuracy is
  improved by a more detailed model when more
  detailed data are available.

26
Observations(Cont.)

• The marginal accuracy gained from actual
  individual regional I-O flows is significant in
  estimating inter-regional trade flow using the
  IRIO model, but quite small in the MRIO version.
  In contrast, the marginal value of accurate
  inter-regional shipment data is rather small in
  estimating individual regional I-O coefficients
  under both versions of the model.
• However, caution is needed before developing a
  firm conclusion because the particular data set
  used to test the model in this paper may have
  skewed the results. Because the United States, EU
  and Japan are all large economies, their demand
  for intermediates are largely met by their own
  production.

27
Conclusions

• This paper developed a mathematical model to
  estimate inter-regional trade patterns and I-O
  accounts based on an inter-regional accounting
  framework and initial estimates of inter-regional
  shipments in a national system of economic
  regions.
• The model is quite flexible in its data
  requirements and has desirable theoretical and
  empirical properties.
• The model performed remarkably well in
  identifying the true patterns of inter-regional
  trade from highly distorted initial estimates of
  inter-regional shipments.

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Limitations

• Based on the data set aggregated from the GTAP
  data, tests show that the current model is
  limited in its ability to improve the IO
  transaction estimates of individual regions from
  national averages.
• Continuing research on the true underlying causes
  is needed to further enhance the models capacity
  as an estimating and reconciliation tool in
  building inter-regional production and trade
  accounts.

29
Estimating a U.S. Multiregional Input-Output
Account

• Apply the Mathematical Programming Model to a
  Large Dimension Case
• Data preparation for the United States

30
1997 Detailed Benchmark Input-Output Account 483
x 494
1997 Detailed Benchmark Input-Output Account 483
x 483
convert to C by C
expand farm sectors
USDA production, cost, and utilization
data, plus other ERS value added statistics
ERS expanded Detailed Input-Output Account 494 x
494
Economic, Ag, and Govt. Census, NASS,
ARMS APHIS, ERS Value added data
U.S. customs Data (SITC) by ports and detailed
data by customs districts (HS)
BLS-CES BEA St. Inc. tangible wealth
Census 2000 Gov GSA procurement ancillary
concordance optimization
concordance optimization
concordance optimization
31
State estimates of gross output, value added,and
wagebill
State estimates of imports and exports
State estimates of household and
govt. consumption investment

• Commodity flow survey
• USDA transportation stats
• Monopolistic comp. model

concordance aggregation optimization
concordance product mix
State-to-State flow estimates for goods and
services 51-regions, 94-sectors
State estimates of inter-sectoral transactions 51-
regions, 94-sectors
Unbalanced 51-region, 94-sector multi-regional
input-output account All initial data for the
final mathematical programming model
32
Specifications for Model to Reconcile Consumption
and Saving Statistics

• Data available from three different sources
  (observed statistics)
• Bureau of Labor Statistics (BLS) - Consumer
  Expenditure Survey
• rgexp0ik HHs expenditure by commodity and US
  regions
• szexp0is HHs expenditure by commodity and
  family size
• incexp0in HHs expenditure by commodity and
  family income groups
• rginc0k HHs disposable income by US regions
• szinc0s HHs disposable income by family size
• incinc0n HHs disposable income by family income
  groups
• Census Bureau - Population Census
• Number of HHs by size, income group and
  state
• Bureau of Economic Analysis (BEA) - Disposable
  income data
• sinc0r Disposable income by state

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Specifications of Model to Reconcile Consumption
and Saving Statistics (Cont.)

• Dimension of data
• i Commodity categories, i 1,2,,75
  
• s Households by family size, s 1,2,5
• n Households by income, n 1,2, 7
• r State
  r 1,2,51
• k Region k Northeast, Mid-West, South,
  West
• Unobservable statistics estimated from the model
• HHs expenditure by commodity and state
• A Two Stage Quadratic Programming Model to
  Reconcile the data

34
Estimates of Consumer Savings and Expenditures
by Commodities

• Disposable income by Region and Income Group from
  BLS survey is held constant in the adjustment
• Percentage adjustments are small for disposable
  income by state from BEA and for expenditure by
  family size from BLS survey.
• Percent adjustment from BEA state disposable
  income
• AL 1.621, AK -1.337, AZ 0.280, AR
  2.137, CA -0.894, CO -0.652, CT -1.093,
  DE -0.613,
• DC 0.733, FL 0.633, GA -0.066,
  HI -1.211, ID 0.894, IL -0.585, IN
  0.174, IA 0.685,
• KS 0.393, KY 1.711, LA 1.734,
  ME 1.135, MD -1.198, MA -0.725, MI
  -0.293, MN -0.605,
• MS 2.199, MO 0.916, MT 2.562, NE
  0.663, NV -0.397, NH -0.791, NJ -1.329,
  NM 1.914,
• NY -0.095, NC 0.636, ND 1.938,
  OH 0.348, OK 1.755, OR 0.369, PA
  0.500, RI 0.238,
• SC 0.980, SD 1.694, TN 1.167
  TX 0.195, UT -1.092, VT 0.444, VA
  -0.498, WA -0.431
• WV 2.928, WI -0.136, WY 1.067
• Percent adjustment from expenditure by family
  size from BLS survey
• size1
  size2 size3 size4 size5
• SAVE -6.185 -4.481 -4.085
  -4.541 -7.304

35
Specifications for Model to Fill the Missing
Value in Commodity Flow Survey Data

• Data available in CFS
• x0isr State to state shipment by commodity at 2
  digit SCTG level with missing values
• sx0ir Shipment of commodity i by state of origin
  at 3 digit SCTG (table 5)
• st0sr Total outbound shipment from state s to
  state r at 3 digit SCTG (table 7)
• dt0sr Total inbound shipment from state s to
  state r at three 3 SCTG (table 8)
• us0 i Total shipment of commodity i in the
  United States at 2 and 3 digit (US report table
  5)
• wsxir Variance of sx0ir (Appendix)
• wstsr Variance of st0sr (Appendix)
• wdtsr Variance of dt0sr (Appendix)
• dx0ir Shipment of commodity i by state of
  destination and x0isr at three digit SCTG are
  complete missing.
• First fill the missing x0isr at 2 digit SCTG
  level, then fill the missing values of dx0ir at
  3 digit SCTG level, finally fill the missing
  x0isr at 3 digit SCTG level

36
Service Sector Trade Flows

• Treyz, F., and J. Bumgardner. "Monopolistic
  Competition Estimates of Interregional Trade
  Flows in Services." In H. Kohno, P. Nijkamp, and
  J. Poot. eds. Regional Cohesion and Competition
  in the Age of Globalization. Edward Elgar, 2000.
• Monopolistic competitive service sector w/
  economies of scale technologies
• Consumer and producer demands for services are
  characterized by preference for variety, ala
  Dixit-Stiglitz (1977)
• Free entry and exit drives profits to zero and
  uniform firm sizes
• location of production and demand markets are
  pre-determined
• cif prices reflect market and non-market costs of
  overcoming distance
• Given location of production and demand, demand
  elasticity's, and distance costs, solve the the
  set of fob prices that minimize the cost of
  serving each market--this produces a unique
  spatial equilibrium in service trade flows

37
Specifications for Model to Fill the Missing
Value in Commodity Flow Survey Data