metadata: authors: 0: author: Canning, Patrick
1: author: Wang,Zhi
count: World
docna: 0: docna: A flexible modeling framework to estimate
            interregional trade patterns and input-output accounts
docty: Policy Research Working Paper
subtopic: International Trade and Trade Rules,Trade and Services,Food Security,Transport Services,Trade and Multilateral Issues
teratopic: Education
abstracts: cdata!: This study implements and tests a
            mathematical programming model to estimate interregional,
            interindustry transaction flows in a national system of
            economic regions based on an interregional accounting
            framework and initial information of interregional
            shipments. A national input-output (IO) table, regional data
            on gross output, value-added, exports, imports and final
            demand at sector level are used as inputs to generate an
            interregional IO account that reconciles regional economic
            statistics and interregional transaction data. The model is
            tested using data from a multi-regional global input-output
            database and shows remarkable capacity to discover true
            interregional trade patterns from highly distorted initial estimates.
repnme: repnme: A flexible modeling framework to estimate
            interregional trade patterns and input-output accounts
display_title: A flexible modeling framework to
            estimate interregional trade patterns and input-output accounts
listing_relative_url: /research/2004/07/4986873/flexible-modeling-framework-estimate-interregional-trade-patterns-input-output-accounts
topicv3: Agriculture and Food Security m1326205 435,Intraregional Trade m1326248 1175,Transport m1326661 135,Trade m1326924 698,Trade in Services m1326928 780
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Education
no_of_pages: 32
author_keys: 0: author_key: 454781
model_extractions: # **A Flexible Modeling Framework to Estimate Interregional Trade Patterns and Input-Output Accounts**

Patrick Canning, Economic Research Service, US Department of Agriculture Zhi Wang, World Bank and City University of Hong Kong *

## **Abstract**

This study implements and tests a mathematical programming model to estimate interregional, interindustry transaction flows in a national system of economic regions based on an interregional accounting framework and initial information of interregional shipments. A national input-output (IO) table, regional data on gross output, value-added, exports, imports and final demand at sector level are used as inputs to generate an interregional IO account that reconciles regional economic statistics and interregional transaction data. The model is tested using data from a multi-regional global input-output database and shows remarkable capacity to discover true interregional trade patterns from highly distorted initial estimates.

### **JEL Classification Numbers** : R1, C67, C81

## World Bank Policy Research Working Paper 3359, July 2004

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors, or the countries they represent. Policy Research Working Papers are available online at http://econ.worldbank.org._

* Patrick Canning is a senior economist at the Economic Research Services, United State Department of Agriculture, 1800 M street, NW, Washington, D.C. 20036; email: <u>pcanning@ers.usda.gov. Zhi Wang is a consultant at the World Bank and adjunct Professor at</u> City University of Hong Kong. Corresponding address: 838 Summer Walk Drive, Gaithersburg, Maryland 20878. E-mail: zwang53@comcast.net. The authors thank the editor of this Journal and three anonymous referees for their valuable comments.

1
# **Executive Summary**

There are tremendous disparities in economic development across regions in large developing countries such as China, India, Indonesia and Brazil. Globalization may have different impact on urban and coast developed areas and rural and inland less developed regions. A major obstacle in conducting policy analysis for regional economic development under globalization is the lack of consistent, reliable regional data, especially data on interregional trade and interindustry transactions. This study implements and tests a mathematical programming model to estimate interregional, interindustry transaction flows in a national system of economic regions based on an interregional accounting framework and initial information on interregional shipments. A complete national input-outputtable plus regional sectoral data on gross output, value-added, exports, imports and final demand are used as inputs to generate an interregional input-output system that reconciles regional market data and interregional transactions. The model is tested on a four-region,10-sector example against data aggregated from a multi-regional global input-output database, and test results from seven experiments are evaluated against eight mean absolute percentage error indexes. The model has capacity to discover the true interregional trade pattern from highly distorted initial estimates. The paper also discusses some general guidelines for implementing the model for a large–dimension. multi-regional account based on real national and regional data.

2
# **1. INTRODUCTION**

A major obstacle in regional economic analysis and empirical economic geography is the lack of consistent, reliable regional data, especially data on interregional trade and interindustry transactions. Despite decades of efforts by regional economists, data analogous to national inputoutput accounts and international trade accounts, which have become increasingly available to the public today, still are generally not available even for well defined sub-national regions in many developed countries. Therefore, economists have had to develop various non-survey and semisurvey methods to estimate such data. In earlier years, quotient based, gravity based and regional purchase coefficient based non-survey methods were popular but lacked logical and theoretical structures, and so have been deemed as ‘deficiency methods’ (Jensen, 1990).

Since the 1980s, various constrained matrix-balancing procedures have become increasingly popular for estimating unknown data based on limited initial information subject to a set of linear constraints. Attempts have been made to estimate regional and interregional transactions in a unified national accounting system of economic regions. Batten (1982) extended earlier work by Wilson (1970)<sup>1</sup> and laid out an optimization model based on information theory and linkages between national and regional input-output accounts to simultaneously estimate interregional deliveries in both intermediate and final goods. Batten and Martellato (1985) establish a simple hierarchical relationship among five classical models associated with authors such as Isard, Chenery and Leontief that address interregional trade within an input-output system. They find those models could be reduced to a statistical estimation problem based on varying degrees of available interregional trade data and demonstrate that the net effect of additional data or additional theoretical assumptions is similar in reducing the number of unknown variables in the underdetermined estimation problems. They also demonstrate such estimation problems are best undertaken with a closed system, i.e., when all the geographic components of the national or state data are estimated simultaneously. Following this philosophy, 3
Byron et al (1993), Boosma and Oosterhaven (1992) and Trendle (1999) find evidence that the additional accounting constraints imposed by such a closed system are useful as a checking device on individual cell values and so improve estimation accuracy. Golan, Judge and Robinson (1994) further generalize such an estimation problem to an ill-posed, underdetermined, pure inverse problem that can be formulated in an optimization context that involves a nonlinear criterion function and certain adding up and consistency constraints. They also show that under such a framework, it is easy to take account of whatever initial information and data that exist through the specification of additional constraints. However, they do not pay attention to how such procedures could be used in a multi-regional context and thus the potential gain from implementing the procedure in a closed national system of economic regions.

Methods for matrix balancing can be classified into two broad classes -- bi-proportional scaling and mathematical programming. The scaling methods are based on the adjustments of the initial matrix to multiplying its rows and columns by positive constants until the matrix is balanced. It was developed by Stone and other members of the Cambridge Growth Project (Stone et al., 1963) and is usually known as RAS. The basic method was originally applied to known row and column totals but has been extended to cases where the totals themselves are not known with certainty (Senesen and Bates, 1988; Lahr, 2001). Mathematical programming methods are explicitly based on a constrained optimization framework, usually minimizing a penalty function, which measures the deviation of the balanced matrix from the initial matrix subject to a set of balance conditions.

Scaling methods such as RAS have been one of the most widely applied computational algorithms for the solution of constrained matrix balancing problems. They are simple, iterative, and require minimal programming effort to implement. However, as pointed out by van der Ploeg (1982), they are not straightforward to use when including more general linear restrictions and when allowing for different degrees of uncertainty in the initial estimates and restraints. They also lack a theoretical interpretation of the adjustment process. Those aspects are crucial for an 4
adjustment procedure to improve the information content of the balanced estimates rather than only adjusting the initial estimates mechanically. Mohr, Crown and Polenske (1987) discuss the problems encountered when the RAS procedure is used to adjust trade flow data. They point out that the special properties of interregional trade data increase the likelihood of non-convergence of the RAS procedure and propose a linear programming approach that incorporates exogenous information to override the unfeasibility of the RAS problem.

In recent years, more and more researchers have tended to formulate constrained matrix balancing problems as mathematical programming problems (var der Ploeg, 1988, Nagurney and Robinson, 1989, Bartholdy, 1991, Byron _et al_ ., 1993), with an objective function that forces "conservatism" on the process of rationalizing **X** from the initial estimate **X** . The theoretical foundation for the approach can be viewed from both the perspectives of mathematical statistics and information theory, and the solution of RAS is equivalent to constrained entropy minimization with fixed row and column totals, as shown by Bregman (1967) and McDougall (1999), and thus can be seen as a special case of the optimization methods<sup>2</sup> .

Another important advantage of mathematical programming models over scaling methods is their flexibility,. which allows a wide range of initial information to be used efficiently in the data adjustment process. Additional constraints can be easily imposed, such as allowing precise upper and lower bounds to be placed on unknown elements, inequality conditions, or incorporating an associated term in the objective function to penalize solution deviations from the initial row or column total estimates when they are not known with certainty. Therefore, it provides more flexibility to the matrix balancing procedure. This flexibility is very important in terms of improving the information content of the balanced estimates as showed by Robinson, Cattaneo and El-said (2001).

A Mathematical programming approach also permits one to routinely introduce relative degrees of reliability for initial estimates. The idea of including data reliability in matrix 5
balancing can be traced back over half a century to Richard Stone and his colleagues (1942) when they explored procedures for compiling national income accounts. Their ideas were formalized into a mathematical procedure to balance the system of accounts after assigning reliability weights to each entry in the system. The minimization of the sum of squares of the adjustments between initial entries and balanced entries in the system, weighted by the reliabilities or the reciprocal of the variances of the entries is carried out subject to linear (accounting) constraints. This approach had first been operationlized by Byron (1978) and applied to the System of National Accounts of the United Kingdom by van der Ploeg (1982, 1984). Zenios and his collaborators (1989) further extend this approach to balance a large social accounting matrix in a nonlinear network-programming framework. Robinson and his colleagues (2001) provide a way to handle measurement error in cross entropy minimization via an error-in-variables formulation. Although computational burden is no longer a problem today, the difficulty of estimating the error variances in a large data set by such approaches still remains unsolved.

The objectives of this paper are threefold. The firstis to develop and implement a formal model to estimate interregional, interindustry transaction flows in a national system of economic regions based on incomplete statistical information at the regional level. The second is to evaluate the model’s performance against data from the real world. And the third is to discuss the issues that arise when applying this modeling framework to estimate a multi-regional input-output account containing well-defined sub-regions.

The paper is organized as follows. Section 2 specifies the modeling framework and discusses its theoretical and empirical properties. Section 3 tests the model by using a four-region, ten-sector data set compiled from a global database documented in McDougall, Elbehri, and Trong (1998). Test results from seven experiments are evaluated against eight mean absolute percentage error indexes. Section 4 discusses some empirical issues involved in applying such a framework to data from a national statistical system. The paper ends with conclusions and direction for future research.

6
# **2. MATHEMATICAL PROGRAMING MODEL FOR ESTIMATING INTERRGIONAL TRADE AND INTERINDUSTRYTRANSACTION FLOWS**

Our model builds upon earlier work by Wilson (1970) and Batten (1982) with two important departures. First, it explicitly incorporates interregional trade flow information into both its accounting framework and initial estimates. We find this greatly enhances the accuracy of estimation results. Second, a multi-regional input-output (MRIO) account is estimated first, then extended to an interregional input-output (IRIO) account, which substantially reduces data requirements and the "dimension explosion" problem in real world applications.

Consider a national economy consisting of N sectors that are distributed over M regions. The sectors use each other’s products as inputs for their own production, which is in turn used up either in further production or by final users. Each region exports some of its products to other regions and some to other nations. They also import products from other regions and nations to meet their intermediate and final demand. Assuming a predetermined location of production that defines the structure of the national economic system of regions, the deliveries of goods and services between regions are determined by imbalances between supply and demand inside the different regions.

_r r r r r_ Denote<sup>_x_</sup> _i_<sup>,</sup><sup>_y_</sup> _i_<sup>,</sup><sup>_v_</sup> _i_<sup>,</sup><sup>_e_</sup> _i_<sup>, and</sup> _mi_ as sector i’s gross output, final demand (excluding exports), value-added, exports, and imports in region r respectively, and denote<sup>_x_</sup> _i_<sup>,</sup><sup>_y_</sup> _i_<sup>,</sup><sup>_v_</sup> _i_<sup>,</sup><sup>_e_</sup> _i_<sup>,</sup> _sr_ and _mi_ as their respective national counterparts. Also denote<sup>_d_</sup> _i_ as delivery of sector i’s product from region r to region s,<sup>_z_•</sup> _ijr_<sup>and</sup><sup>_z_</sup> _ij_<sup>as intermediate transactions from sector i to sector j in</sup> region r and the national level respectively.<sup>3</sup> All variables are measured in annual values. In such a static national system of economic regions, the following accounting identities must hold at each given year for all i ∈ N and s, r ∈ M.

7
Collectively, equations (1) to (9) define a multi-regional input-output account (MRIO). Such an account stops short of assigning specific intermediate or final uses for inter/intra regional product flows, but guarantees that these flows exactly meet all regional demands. The economic meanings for each of the nine equations are straightforward: equation (1) defines the sum of sector i’s intermediate and primary factor input equals the sector’s total output in each region. Equation (2) states the sum of each region’s intermediate and final demand must be met by deliveries from all regions (including from its own) within the nation plus imports from other nations. Equation 3 defines a region can only deliver to all regions within the nation and export to other nations what it produces, while equations (4) to (9) are simply the facts that sums of all the region’s economic activities within a nation must equal to the national totals.

8
Suppose statistics exist for each regional sector on the gross outputs and value added

> (<sup>_x_</sup> _ir_<sup>and</sup><sup>_v_</sup> _ir_<sup>), the origin of exports and destination of imports (</sup><sup>_e_</sup> _ir_<sup>and</sup> _mir_ ), and the final regional

demands (<sup>_y_</sup> _ir_<sup>). The MRIO estimation problem can be formally stated as follows:</sup> _sr Given a n_ × _m_ × _m non-negative array_ **D** _= {_<sup>_d_</sup> _i } and a n_ × _n_ × _m non-negative array_ **Z** _= {_<sup>_~~z~~_</sup> _ij_<sup>•</sup> _r_<sup>_}, determine a non-negative array_</sup><sup>**D**</sup><sup>_={d_</sup> _isr } and a non-negative array_ **Z** _= {_<sup>_z_•</sup> _ijr_<sup>_} that is_</sup> _close to_ **D** _and_ **Z** _such that equations (1) to (9) are satisfied_ **,** _where s_ ∈ _M denotes the shipping regions, r_ ∈ _M denotes the receiving regions, and i, j_ ∈ _N denotes the make and use sectors respectively_ **_._** In plain English, the estimation problem is to modify a given set of initial inter-regional and interindustrial transaction estimates to satisfy the above nine known accounting constraints. The mathematical programming model used to conduct the estimation employs an objective function that penalizes the deviations of the estimated array **D** and **Z** from the initial array **D** and **Z** . Two types of alternative functional forms could be used: (i) Quadratic function: (ii) Cross-entropy function (Harrigan & Buchanan, 1984, Golan et al., 1994): There are desirable theoretical properties of the above estimation framework that are well documented in the literature. Firstly, it is a separable nonlinear programming problem subject to linear constraints. The entropy function is motivated from information theory and is the objective function underlying the well-known RAS procedure with row and column totals known with certainty (Senesen and Bates, 1988). It measures the information surprise contained in **D** and **Z** 9
given the initial estimates **D** and **Z** . The quadratic penalty function is motivated by statistical arguments. There are different statistical interpretations underlying the model by choices of different reliability weights _wd isr_ and _wzij_<sup>•</sup> _r_ . When the weights are all equal to one, solution of this model gives a constrained least square estimator. When the initial estimates are taken as the weights, solution of the model gives a weighted constrained least square estimator, which is identical to the Friedlander-solution, and a good approximation of the RAS solution. When those weights are proportional to the variances of the initial estimates and the initial estimates are statistically independent (the variance and covariance matrix of **D** and **Z** are diagonal), the solution of the model yields best linear unbiased estimates of the true unknown matrix (Byron, 1978), which is identical to the Generalized Least Squares estimator if the weights are equal to the variance of initial estimates (Stone, 1984, van der Ploeg, 1984). Furthermore, as noted by Stone et al. (1942) and proven by Weale (1985), in cases where the error distributions of the initial estimates are normal, the solution also satisfies the maximum likelihood criteria.

Secondly, the quadratic and entropy objective functions are equivalent in the neighborhood of initial estimates, under a properly selected weighing scheme. By taking second order Taylor expansion of equation (11) at point _(_<sup>_d_</sup> _isr_ ,<sup>_~~z~~_</sup> _ij_<sup>•</sup> _r_<sup>_)_we have</sup> This is the quadratic function (10) plus a remainder term R. As long as the posterior estimates and the initial estimates are close and the initial estimates are used as reliability weights<sup>4</sup> , the term R will be small and the two objective functions can be regarded as approximating one another.

10
Thirdly, as proved by Harrigan (1990), in all but the trivial case, posterior estimates derived from entropy or quadratic loss minimand will always better approximate the unknown, true values than do the associated initial estimates. In this framework, information gain is interpreted as the imposition of additional valid constraints or the narrowing of bounds on existing constraints as long as the true but unknown values belong to the feasible solution set. This is because adding valid constraints or further restricting the feasible set through the narrowing of interval constraints cannot move the posterior estimates away from the true values, unless the additional constraints are non-binding (have no information value). Although the posterior estimates may not always be regarded as providing a "reasonable" approximation to the true value<sup>5</sup> , they are always better than the initial estimates in the sense the former is closer to the true value than the later, so long as the imposed constraints are true. In other words, the optimization process has the effect of reducing, or at least not increasing, the variance of the estimates. This property is simple to show by using matrix notation. Define **W** as the variance matrix of initial estimates **D** , **A** as the coefficient matrix of all linear constraints. The least squares solution (equivalent to the quadratic minimand as noted above) to the problem of adjusting **D** to **D** that satisfies the linear constraint, **A** • **D** = **0** can be written as: (13) **D** = ( **I** - **WA**<sup>T</sup> ( **AWA**<sup>T</sup> )<sup>-1</sup> **A** ) **D** Thus, (14) var( **D** ) = ( **I** - **WA**<sup>T</sup> ( **AWA**<sup>T</sup> )<sup>-1</sup> **A** ) **W** = **W** - **WA**<sup>T</sup> ( **AWA**<sup>T</sup> )<sup>-1</sup> **A** ) **W** Since **WA**<sup>T</sup> ( **AWA**<sup>T</sup> )<sup>-1</sup> **A** ) **W** is a positive semi-definite matrix, the variance of posterior estimates will always be less, or at least not greater than the variance of the initial estimates as long as **A** • **D**<sup>true</sup> = **0** holds. This is the fundamental reason why such an estimating framework will provide better posterior estimates. Imposing accounting relationship’s (1) to (9) will definitely improve, or at least not worsen the initial estimates, since we are sure from economics those constraints are identities and must be true for any national system of economic regions.

11
Finally, the choice of weights in the objective function has very important impacts on the estimation results. For instance, using the initial estimates as weights has the nice property that each entry of the array is adjusted in proportion to its magnitude in order to satisfy the accounting identities, and the variables cannot change sign and that large variables are adjusted more than small variables. However, the adjustment relates directly to the size of the initial estimates<sup>_d_</sup> _isr_ and<sup>_~~z~~_</sup> _ij_<sup>•</sup> _r_ ,<sup>and does not force the unreliable initial estimates to absorb the bulk of the required</sup> adjustment. Furthermore, only under the assumptions: (1) the initial estimates for different elements in the array are statistically independent, and (2) each error variance is proportional to the corresponding initial estimates, this commonly used weighing scheme (underlying RAS) can obtain best unbiased estimates, while those assumptions may not hold in many cases. Fortunately, the model is not restricted to use only a diagonal-weighing matrix such as the initial estimates. When a variance-covariance matrix of the initial estimates is available, it can be incorporate into the model by modifying the objective function as follows: The efficiency of the resulting posterior estimator will be further improved if the error structure of the initial estimates is available, because such a weighting scheme makes the adjustment independent of the size of the initial estimates. The larger the variance, the smaller its contribution to the objective function, and hence the lesser punishment for<sup>_d_</sup> _isr_ and<sup>_z_</sup> _ij_<sup>*</sup> _r_<sup>to move</sup> away from their initial estimates (only the relative, not the absolute size of the variance affects the solution). A small variance of the initial estimates indicates, other things equal, they are very reliable data and thus should not change by much, whilst a large variance of the initial estimates indicates unreliable data and will be adjusted considerably in the solution process. Therefore, this weighing scheme gives the best-unbiased estimates of the true, unknown inter-regional and interindustrial transaction value under the assumption that initial estimates for different elements in the array are statistically independent. Although there is no difficulty to solve such a nonlinear 12
programming problem like this today, the major problem is lack of data to estimate the variancecovariance matrix associate with the initial estimates.

Stone (1984) proposed to estimate the variance of<sup>_~~z~~_</sup> _ij_<sup>•</sup> _r_<sup>as var(</sup><sup>_~~z~~_</sup> _ij_<sup>•</sup> _r_<sup>) = (</sup> θ _ij_ * _r_<sup>_~~z~~_</sup> _ij_<sup>•</sup> _r_ )<sup>2</sup> , where θ _ij_ * _r_ is a subjectively determined reliability rating, expressing the percentage ratio of the standard error to<sup>•</sup> _r_<sup>. Weale (1989) had used time series information on accounting discrepancies to infer</sup><sup>_~~z~~_</sup> _ij_ data reliability. The similar methods can be used to derive variances associated with those initial estimates in our model.

Despite the difficulties in obtaining data for the best weighting scheme, advantages of such a model in estimating interregional trade flows and interindustry transactions are still obvious from an empirical perspective. Firstly, it is very flexible regarding the required know information. For example, it allows for the possibility that the state total of output, value-added, exports, imports and final demands are not known with certainty. In the real world, these regional statistics typically have substantial gaps and inconstancies with the national total. Incorporating associated terms similar to **D** and **Z** in the objective function to penalize solution deviations from the initial estimates from statistical sources allows the estimation of those regional totals, together with entries in the inter-regional delivery and inter-industrial transaction array. With the use of upper and lower bounds, this fact can also be modeled by specifying ranges rather than precise values for the linear constraints (1) - (3). In addition, the estimation of **D** or **Z** will be a special case of the framework when only one set of additional data is available.

Secondly, it permits a wider variety and volume of information to be brought into the estimation process. For example, the ability of introducing upper and/or lower bounds on those regional totals is one of the flexibilities not offered by commonly used scaling procedures such as _sr r_ RAS. The gradient of the entropy function tends to infinity as<sup>_d_</sup> _i_ and<sup>_z_•</sup> _ij_<sup>→ 0, and hence</sup> 13
restricts the value of the posterior estimates to nonnegative. This is a desirable property of estimating inter-regional trade data.<sup>6</sup> Thirdly, the weights in the objective function reflect the relative reliability of a given set of initial estimates. The interpretation of the reliability weights is straightforward. Other things equal, entries with higher reliability should be changed less than entries with a lower reliability. The choice of those weights is also very flexible. They will use the best available information to insure that reliable data in the initial estimates are not being modified by the optimization model as much as unreliable data. In practice, such reliability weights can be put into a second array that has the same dimension and structure as the initial estimates. The inverted variance-covariance matrix of the initial estimates is statistically interpreted as the best index of the reliability for the initial data.

Finally, solution of this estimation problem exactly provides the data needed to construct a so-called multi-regional input-output (MRIO) model (Miller and Blair, 1985, Isard, et al. 1998). This model was pioneered by professor Polenske and her associates at MIT in the 1970s (Polenske, 1980), and is still widely used in regional economic impact analysis today.

The above model could be easily extended to further allocate **Z** and **D** to distinguish intermediate and final delivery of goods and services within a national system of economic regions. The extended model will be similar in many aspects with the interregional accounting framework proposed by Batten (1982) two decades ago. However, as we will show later in this paper, it becomes more operational and provides better empirical estimation results on interregional shipments because of the explicit incorporation of interregional trade flow information into both the initial estimates and the accounting framework.

> To demonstrate, denote<sup>_z_</sup> _ijsr_ as intermediate inputs delivered from sector i in region s to

sector j in region r within a nation, and _yihsr_ as final goods and services delivered from sector i in _r r_ region s to type h final demand in region r. Further, denote _mij_ and _mih_ as imported (from other 14
nations) intermediate and final goods and services delivered to sector j or final demand type h in region r respectively.<sup>7</sup> Other notation regarding state gross output, intermediate inputs, valueadded, exports and imports are the same with the aggregated model. Then the accounting framework for the national system of economic regions can be defined as follows: Adding a quadratic penalty objective function, we have an extended model to estimate a detailed interregional input-output account based on the results from the earlier model.<sup>8</sup> This model has the theoretical and empirical properties similar to the earlier model, but with much higher details. The solution to (22), subject to constraints (16) to (21), provides a 15
complete set of data for a so-called inter-regional input-output (IRIO) model with imports endogenous (Miller and Blair, 1985, Isard, et al. 1998).

# **3. EMPIRICAL TEST OF THE MODEL AND EVALUATION MEASURES**

# **_The Testing Data Set_**

How does the model specified above perform when applied to data from the real world? In order to evaluate the models’ performance, a benchmark data set from the real world is needed. Because good interregional trade data are quite rare and very difficult to obtain in any country, a natural place to find such data sets is existing global production and trade databases such as the GTAP (Global Trade Analysis Project) database. For instance, version 4 GTAP database contains detailed bilateral trade, transportation, and individual country’s input-output data covering 45 countries and 50 sectors (McDougall, Elbehri, and Trong, 1998). For our particular purpose, version 4 GTAP database was first aggregated into a 4-region, 10-sector data set. Then three of the four regions (the United States, European Union and Japan) were further aggregated into a single open economy which engages in both interregional trade among its 3 internal regions and international trade with the rest of the world. We will use this partitioned data set as the benchmark for a hypothetical national economy, and attempt to use our model to replicate the underlying inter-continental trade flows among Japan, EU and the United Sates as well as the individual country’s input-output accounts.

# **_Experiment Design_**

In the first experiment, we do this without use of the region-specific input-output coefficients as the situation encountered in the real world, where only the national IO table is available to economists (it is the three region’s weighted average in our experiment and are defined as _zij_ • _r_ = ( _zij_ /( _x j_ − _v j_ ))× ( _x rj_ − _v rj_ ) to make full use of the known information). Initial estimates of interregional commodity flows are from the ‘true’ interregional trade data in the 16
GTAP database but was distorted by a normally distributed random error term with zero mean and the size of standard deviation as large as 5 times the “true” trade data. The solution from the model is compared with the benchmark data set for both the inter-regional shipment and intersector transaction flows.

In the second experiment, we use the region-specific input-output coefficients as constant in the model. We re-estimate the interregional shipment data as the first experiment, and compare the model solution with the benchmark data set for the inter-regional trade data only.

In the third experiment, we assume the interregional shipment pattern is known with certainty and we use the three region’s weighted average IO coefficients as initial estimates to estimate the region-specific input-output accounts.

In the fourth experiment, Batten’s model was used to estimate the interregional shipment and individual region’s IO flows. In the fifth to the seventh experiments, experiments 1-3 were repeated by using the extended model. Solutions from both models are compared with the “true” interregional trade and inter-sector IO flow data in the aggregated GTAP data set. The assumptions, initial estimates and expected model solution are summarized in table 1.

# **(** **_Insert Table 1 here_ )**

# **_Measures to Evaluate Test Results_**

Each experiment produces a different set of estimates, and it is desirable to know how much each set of estimates differs from the true, known data. However, it is difficult to use a single measure to compare the estimated results. Since there are so many dimensions in the model solution sets, a particular set of estimates may score well on one region or commodity but badly on others. It is meaningful to use several measures to gain more insight on the model performance in different experiments. Generally speaking, it is the proportionate errors and not the absolute errors that matter; therefore, the "mean absolute percentage error" with respect to the true data will be calculated for different commodity and regional aggregations. Consider the following aggregate index measure for intra/inter-regional trade flows: 17
(23) Alternating the removal of summations over i, s, and r in equation (23) produces MAPE estimates on shipments by commodities, shipping regions, and receiving regions respectively. For regional intermediate transactions, the aggregate MAPE index is defined as: Alternating the removal of summations over i, j, and r in equation (24) produces MAPE estimates on intermediate transactions by inputs, using sectors, and regions respectively. The model and all test experiments are implemented in GAMS and the complete GAMS program and related data set are available from the authors upon request.

# **_Test Results_**

Table 2 summarizes all the eight measurement indexes from the seven experiments listed in Table 1. The accuracy of the estimates is judged by their closeness to the true interregional trade and individual region’s input–output flows aggregated from the GTAP database.

# **_(Insert Table 2 here)_**

Generally speaking, the model has remarkable capacity to rediscover the true interregional trade flows from the highly distorted data. The estimated shipment data are very close to the true data, as judged by the eight MAPE measurements, in all testing experiments except the Batten model. Most of the mean absolute percentage errors are about 4-7 percent of the true data value, which implies the model has great potential in the application of estimating interregional trade flows. In contrast, recovering the individual region’s input-output flows from weighted average national values only obtained limited success, indicating national IO 18
coefficients in detailed sectors may be the best place to start in building regional IO accounts if there is no additional prior information on regional technology or cost structure available.<sup>9</sup> Comparing estimates from different test experiments, there are several interesting observations. First, when there is no additional information that could be incorporated into the estimation framework, a more detailed model may not perform better than a simpler model. Comparing results from Exp-1 and Exp-5, the more sophisticated extended model actually brings less accurate estimates overall because of increasing numbers of unknown variables without additional known data. However, as results in Experiments 2, 3, 6, and 7 show, the estimation accuracy does improve by a more detailed model when more useful data become available. Second, the marginal accuracy gained from actual individual regional IO flows is significant in estimating interregional trade flows using the extended model, but very small in the aggregate version. In contrast, the marginal value of accurate interregional shipment data is rather small in estimating individual regional IO coefficients under both versions of the model. Finally, Batten’s model performed poorly in interregional shipment estimation, but obtained similar estimates on individual regional IO flows as our model, providing further evidence that there _may_ be no high dependency between individual regional IO coefficients and interregional trade flows. However, this is not a firm conclusion because the particular data set used to test the model in this paper may be part of the problem. Since the United States, EU and Japan are all large economies, their intermediate demands are largely met by their own production. Therefore, the correlation between individual inter-industrial flow and inter-regional shipments may be particularly low.

The extended model only provides better estimates of interregional shipments when regional IO data are available, so the aggregate version of the model specified in this paper may be the best practitioner’s tool in estimating interregional trade flows because of the lack of subnational IO data in the real world. It demands less statistical information and has a smaller model dimension, which facilitates the implementation and computation process.<sup>10</sup> 19
# **4. IMPLICATIONS FOR APPLYING THE MODEL**

Results in the previous section offer some guidance for applying the framework outlined in this paper to real world statistics. It was found that initial estimates of regional commodity trade flows based on survey data with very high statistical variability are highly preferable (in the experiments) to a widely used non-survey approach for producing initial estimates.<sup>11</sup> This finding holds promise for opportunities to use other survey data to recover unobserved regional economic accounts. It was also found that solving an aggregate account (e.g., a MRIO or MR-SAM) as an intermediate step is at least as accurate (in the experiments) as producing a direct solution to an extended account (e.g., IRIO or IR-SAM) when superior data unique to the later are not widely available. This finding is useful when working with regional economic accounts of considerable sector and region details. Results also support the _product mix_ approach, whereby the most feasible sector detail for regional gross output estimates are used to derive weighted average national technical coefficients for more aggregated regional sectors.

Statistical systems vary by nation and no one-size fits all rules exist that tell us how to seamlessly employ every data-system to best advantage.<sup>12</sup> However, there are general guidelines for implementing the optimization framework presented in this paper to a large dimension multiregional account. To facilitate discussions of implementation, we assume that a detailed national account always exists and regional sector statistics are also available in a variety of details. Then the implementation process may be classified into three broad phases as discussed below.

# **_Develop Independent Estimates for Major Components of a Multi-regional Account_**

It has been stressed as far back as Wilson (1970) that information used to produce parameters and initial estimates of a regional economic system should be estimated independently. While this produces unbalanced initial accounts, it avoids introducing spurious information that can lead to biased estimates (McDougall, 1999). A useful approach is to partition the multi-regional account into components that coincide or are related to known statistical survey series published regularly in the nation under study.

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For the multi-regional IO account outlined in equations (1) to (9), the major components are gross regional output (<sup>_x_</sup> _ir_<sup>), final demand (</sup><sup>_y_</sup> _ir_<sup>), primary factor payments (</sup><sup>_v_</sup> _ir_<sup>), international</sup> trade (<sup>_e_</sup> _ir_<sup>and</sup> _mir_ ), inter-industry transactions (<sup>_z_•</sup> _ijr_<sup>) and inter/intra-regional trade flows (</sup><sup>_d_</sup> _isr_ ). In many cases, data for several of these components are available from a single major statistical survey series—for example, in the United States<sup>_x_</sup> _ir_<sup>and</sup><sup>_v_</sup> _ir_<sup>are available from an Economic</sup> Census conducted every five years. Other components, for example<sup>_y_</sup> _ir_<sup>, may themselves require</sup> multiple disparate data sources to compile. While the strategic groupings may differ by country, it is likely that for large dimension (N × M) multi-regional accounts, primary data for individual regional sectors become sparse.

When the best available data are not consistent to the model structure, it may be necessary to restructure the adding up requirements in the model to accommodate the data. For example, in equation’s (2) and (3) of our model, the accounting identities require data for international exports (<sup>_e_</sup> _ir_<sup>) and imports (</sup> _mir_ ) on an origin of movement and destination of use basis respectively. However, in many countries such as the United Sates, port of entry/exit data are far more reliable. Therefore, different formulation of the corresponding accounting identities should be used.

For certain elements of the multi-regional account, very often only a purely theoretical inference is available to produce informed guesses about the initial estimates. A common example is the information about service trade flows within and between regions. In using a theory-based alternative to data, a case must be made for a prevailing empirical model that calibrates the unobserved activities to some other statistics or available survey data.

# **_Determine Model Dimensions Based on Maximum Concordance among Different Components_**

In compiling different components of the multi-regional account, the volume and nature of data available for each component can greatly vary. Detailed and survey based data may be 21
obtained on, for example, gross regional output and incomes, but survey data on the inter/intraregional trade flows of this output may be far less detailed. Inter-industry transactions may only be available at the national level, and international trade data may be very detailed, but based on a different product classification system. The notion of conservatism, both in the information theoretic sense and in terms of computational burden, should be the primary guiding principal in reconciling this information.

Robinson et al. (2001) interpret conservatism by the rule of using ‘only, and all’ information in the estimation problem. Considering this rule in the present context, the fact that a component such as gross regional outputs are available from highly detailed and reliable statistics suggests all this information should be used. However, if the associated intra/inter-regional trade flow account has more general product aggregations than the output account, it appears that one is faced with an ‘only **or** all’ decision. Although the specific situation often guides the approach one takes, it is worth noting that there are usually many opportunities to introduce all information available into the estimation process.

In practice, conserving on computational burden may also become an issue. When employing a more general estimation framework such as the model presented in this paper, the use of iterative techniques that diminish computational burden may not be readily available.<sup>13</sup> Both computer hardware and software available to the researchers may become binding in many such instances. For example, access to special solvers or greater programming finesse becomes a more prominent issue when computational burdens grow tremendously as model dimension increase. In addition, while conventional personal computers have improved dramatically, limits on current 32-bit operating systems to manage sufficient memory on PC’s may become a binding constraint for very large models. Solutions to these issues can become expensive.

# **_Add Additional Constraints to Use All Available Information_**

The greatest opportunities to use all relevant information are in the form of additional binding linear constraints, beyond the adding up and consistency requirements, on any selected 22
groups of variables in the aggregate or extended model. Information deemed ‘superior’ and that is related to any group of elements in either the aggregate or extended accounts is a candidate for a linear constraint. Since both interregional and multi-regional economic accounts are comprehensive and detailed, there are many opportunities to introduce such constraints. A few general guidelines are notable.

Both the aggregate and extended accounts describe flows of payments and products in the form of a matrix with known adding-up and consistency requirements. Any information used to formulate new constraints—either equality or inequality linear constraints—can greatly diminish the feasible solution set of the calibration procedure. However, new constraints that are nonbinding add no information to the problem, but do increase the computational burdens.

Where and how information is used to formulate constraints depends on many factors. For example, the U.S. Government has published state measures of farm productivity that include estimates of purchased farm inputs by state for broad input categories. A pro-rated version of this data could form the basis for additional linear constraints for agricultural sector I-O flows in the model. Other restrictions could be designed to replicate certain highly reliable economic statistics that can be formed by special groupings of certain flow statistics contained in the account being estimated. Although such information must be carefully compiled, their incorporation in the form of constraints will improve the estimation accuracy greatly.

# **5. CONCLUSIONS AND DIRECTION FOR FUTURE RESEARCH**

This study constructed a mathematical programming model to estimate interregional trade patterns and input-output accounts based on an interregional accounting framework and initial estimates of interregional shipments in a national system of economic regions. The model is quite flexible in its data requirements and has desirable theoretical and empirical properties. An empirical test of the model using a 4-region, 10-sector example aggregated from a global trade database shows that the model performs remarkably well in discovering the true patterns of 23
interregional trade from highly distorted initial estimates on interregional shipments. It shows the model may have great potential in the estimation and reconciliation of interregional trade flow data, which often are the most elusive data to assemble. In addition, solutions from the aggregated model exactly provides the data needed for a MRIO model and the solution from the extended model exactly provide the data needed for an IRIO model. This will greatly reduce the data processing burden in such analysis. Therefore, application of the model will further facilitate quantitative economic analysis in regional sciences.

Lessons from the experiments in this study shaped our view on approaches for applying the model to real data from a particular nation’s statistics. A logical conclusion is that widely available and disparate survey data on the economy, including commodity flows data and incomplete geographic data, can effectively be used to substantially narrow the margins for error in obtaining feasible solutions to interregional input-output systems. It is also evident that data on region-to-region commodity flows represent a limiting factor in determining the optimal sector dimensions to be solved in the modeling framework.

However, there are important questions not yet answered by the current study. First, test results from the data set aggregated from GTAP also show that our model’s ability to improve the IO transaction estimates of individual regions from national averages may be limited. Continuing research on the real underlying causes and means of improvement are needed to further enhance the model’s capacity as an estimating and reconciliation tool in building interregional production and trade accounts. Second, the relative importance of regional sector output, value-added, exports, imports and final demand as model input in the accuracy of a model solution is also not analyzed, and could be addressed with minor changes of the current model. Third, the approach employed in this study draws primarily from regional science and constrained matrix balancing literatures. How insights from economic geography theory can help define a bounded solution needs to be explored. Finally, the robustness of the model’s performance should be further tested using other data sets.

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# **Footnotes** :

1. Wilson (1970) had suggested an entropy maximizing solution for a model which integrated gravity models and multi-regional input-output equations as constraints to estimate inter-regional commodity flows. However, his work did not clearly incorporate a complete system of national and regional input-output accounts as did in Batten (1982).

2. Using Monte Carlo simulation, Robinson, Cattaneo and El-said (2001) shows that when updating column coefficients of a Social Accounting Matrix (SAM) is the major concern, the cross entropy method appears superior, while if the focus is on the flows in the SAM, then the two methods are very close with the RAS performing slightly better.

3. The variables<sup>_d_</sup> _isr_ and<sup>_z_•</sup> _ijr_<sup>have no counterparts in Batten’s framework, reflecting important</sup> departures in the present approach.

4. The quadratic functional form has a numerical advantage in implementing the model. It is easier to solve than the entropy function in very large models because they can use software specifically designed for quadratic programming.

5. The minimand objective function reflects the principle that the 'distance' between the posterior and initial estimates should be minimized. What we would like is to minimize the 'distance' between the posterior estimates and the unknown true values. This 'distance' cannot be measured, but a good estimation procedure should have a desirable influence on it.

6. Zeros can become non-zeros and vice versa under a quadratic penalty function. However, a side effect for the cross entropy function is that if there are too many zeros in the initial estimates, the whole problem may become infeasible.

7. The assignment of an intermediate (j) or final demand use (h) of international imports has no counterpart in Batten’s notation since he makes no such assignments. Either approach is valid and would be dictated by the data available.

8. By incorporated the 6 accounting identities that the sum of all regions in the nation should equals their national totals defined in equations (4-9), the model could be solved independently without use of the earlier model, however, the dimension and data requirements of the model will be much larger than the aggregated model.

9. Following the product mix method outlined in Miller & Blair (1985), initial estimates of IO coefficients for each of the 10 aggregated industries are unique for each region. They are weighted averages of the 3-region detailed (50-industry) IO coefficients where the weights are the gross regional outputs of the relevant detailed industries. Experiment results show that a “product mix” approach improves the accuracy of the true regional IO flow estimates compared to an 25
approach that directly uses the 3-region average IO coefficients, although the differences are small in our particular model aggregation.

10. The aggregate model only has N(NM+M<sup>2</sup> +5M) variables and N(3M+N+5) constraints, while the extended model has (N<sup>2</sup> M + NHM)(M+1) variables and N(M<sup>2</sup> +NM+N+5) constraints. This is a much larger model, having NM<sup>2</sup> (N-1) + NM(HM-5) more variables and MN(M+N-3) additional constraints.

11. A random normal distortion of the ‘true’ trade data by an average of 400-percent was produced in the previous section to simulate a well designed but poorly sampled transportation survey of annual commodity flows.

12. Comprehensive studies by West (1990) and Lahr (2001) consider how to identify and use superior data in a regional accounting system context.

13. For example, by allowing both regional technical coefficients and intra/inter-regional flows to adjust, the optimal solution to the cross-entropy or quadratic formulations in section 2 must be jointly solved.

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