Handbook on Extended Supply and Use Tables and Extended Input‑Output Tables

5. Balancing methods applied to extended supply and use tables

Abstract

This chapter outlines strategies and practical implications in balancing extended supply and use tables in the absence of sufficient information. It begins with a discussion of automatic balancing procedures before turning to an examination of proportional scaling methods. Next, the chapter considers conditions for unique solutions. It also reviews other problematic scenarios. It concludes with a checklist for designing automatic balancing processes and a set of recommendations.

Introduction

This chapter outlines strategies and practical implications in balancing extended supply and use tables (ESUTs) in the absence of sufficient information. The main imbalances should be tackled manually, as this ensures robustness across varying data quality. For example, trade data are typically more reliable than intermediate use data. Other balancing methods should be used only when improving data sources is not feasible. The handling of conflicting data in ESUTs presents a significant challenge when balancing. This chapter, written by Juan M. Valderas-Jaramillo and José M. Rueda Cantuche (European Commission’s Joint Research Centre), describes various strategies in detail, including proportional and optimisation methods, to address the challenges and ensure meaningful outcomes. The chapter also discusses conditions for achieving unique solutions in non-negative matrices and identifies potential pitfalls. Note that theoretical and methodological aspects of extending supply and use table (SUT) frameworks need careful consideration, ensuring that all available information is synthesised efficiently.

Automatic balancing procedures

This section briefly describes automatic balancing techniques to focus the attention on proportional adjustment methods in the following section. United Nations (2018[1]) provides a good overview1 of the existing projection methods for SUTs that can, in principle, be used in the reconciliation of information from different data sources and in the balancing process of SUTs. There are three different ways to address the problem where data gaps for the interior elements of the matrices outnumber the external constraints in the form of, for example, row and column totals. These are:

1. constrained optimisation methods based on probability and information theory or on distance measures

2. proportional scaling methods, which can be one-sided or biproportional

3. model-based methods.

The first group of methods, in general, has in common the fact that they minimise some measure of distance between all elements of the two matrices, the prior and the resulting estimated matrix. There are many types of distance measures, such as absolute differences and square differences, among others. Alternatively, this group of methods also comprises objective functions that are based on the statistical concept of information loss, taken from the probability and information theory laid out by Kullback and Leibler (1951[2]). The measure typically represents a solution to the problem that provides minimum information loss. This group of methods can also incorporate constraints or weights in the objective function to account for different robustness in the data (United Nations, 2018[1]).

This concept is also associated with the second group, that of proportional scaling methods. Indeed, this balancing problem could also be expressed in terms of a simple biproportional2 iterative scaling method, which was the so-called RAS method used by Stone (1961[3]). This method prompted a lot of discussion because the elements of the matrices can be positive and negative (generalised RAS method), key examples include Günlük-Senesen and Bates (1988[4]); Junius and Oosterhaven (2003[5]); Huang, Kobayashi and Tanji (2008[6]); Lenzen, Wood and Gallego (2007[7]); and Lemelin (2009[8]). The literature also provides several refinements, such as those dealing with row and column totals with positive and negative elements, and non-preserving sign (Temurshoev, Miller and Bouwmeester, 2013[9]; Lenzen et al., 2014[10]) or allowing more flexibility to find a compromise with exogenous conflicting constraints (Lenzen, Gallego and Wood, 2009[11]). More recently, there have been further developments in multidimensional proportional balancing methods (Temursho, Oosterhaven and Cardenete, 2021[12]; Valderas-Jaramillo and Rueda Cantuche, 2021[13]), where the constraints are also sums of various parts of the interior matrices, on top of the row and column totals.

The third group of methods, model-based methods, uses input-output models to balance SUTs and input-output tables, for example, Leontief price and quantity models used by Snower (1990[14]); Beutel (2002[15]; Beutel, 2008[16]); Valderas-Jaramillo et al. (2019[17]); time series analysis covered by Wang et al. (2015[18]); and econometric methods used by Kratena and Zakarias (2004[19]).

This chapter focuses on the second group of methods because of their prevalent use in balancing SUTs and input-output tables at national statistical offices.

This chapter specifically focuses on the generalised RAS (GRAS) method, which seeks to find a unique solution to the balancing problem. Moreover, the SUT-RAS method is a particular case of the GRAS method for balancing SUTs all at once (Temursho, Oosterhaven and Cardenete, 2021[12]). However, the process is not entirely automatic, and the solution algorithm of RAS-type proportional methods (e.g. GRAS) is only equivalent to that of constrained optimisation problems if the problem is well-defined and a unique solution3 exists. Therefore, a crucial concern is to ascertain in advance the conditions that guarantee a unique solution and to establish some guidelines for this purpose. Elucidating these conditions and offering guidance are the main objectives of this chapter.4

Proportional scaling methods

Throughout this chapter, a stylised example will be used to guide the reader (Table 5.1). For a particular industry, integrating regular SUT data (e.g. industry and product output) with information from other sources (as described in Chapter 2) to delineate activities among, for instance, domestic multinational enterprises (MNEs), foreign MNEs and domestic non-MNEs, may lead to inconsistencies in the total output figures. From the regular SUTs, total intermediate use by industry 1 of product 1 is 8; for product 2, it is 12; and for value added, it is 10. Similarly, one only ascertains that the total output of industry 1 is 28 by examining the regular SUTs. Nevertheless, alternative data sources indicate that the distribution of industry output is 10 (for domestic MNEs), 12 (for foreign MNEs) and 6 (for domestic non-MNEs). Assuming a predetermined input structure for each type of enterprise (derived from ad hoc surveys, enterprises’ annual accounts, etc.), the challenge lies in determining a matrix of intermediate uses that satisfies the constraints imposed by the regular SUTs and other relevant data sources that disaggregate industries (the shaded cells).

Table 5.1. Illustrative example of an unbalanced extended supply and use table

	Industry 1
	Domestic MNE	Foreign MNE	Domestic non-MNE	Intermediate use
Product 1	1	2	5	8
Product 2	4	2	3	12
TLS	-1	2	-2	-2
Value added	6	1	2	10
Output	10	12	6

Note: MNE: multinational enterprise; TLS: taxes less subsidies on products. Predetermined constraints.

The GRAS method

United Nations (2018, pp. 491-494[1]) describes in detail the iterative process of the GRAS method with real numerical examples. This chapter presents succinctly how the GRAS solution algorithm applies to obtain a balanced matrix. This is applied to one specific industry (i.e. industry 1) but can easily be extended to others. It can even be used to automate the whole process. Therefore, each row or column total of Table 5.1 (the shaded cells) is treated independently, as a one-dimensional vector with positive and negative values. Following the GRAS method, for each total vector, a quadratic equation5 for k like the following needs to be solved:

where P is the sum of the positive elements of the vector, N the sum of the negative elements (in absolute value) and S the desired target value (i.e. shaded cells in Table 5.1). The solution for k would be given by:

In the first iteration, the column of domestic MNEs is already balanced, so there is no need to make any change to the column values. Hence, k = 1. For foreign MNEs, there are no negative elements in the column, so the solution would be reached by multiplying all column elements by the ratio 12/7 = 1.71. For instance, 2*12/7 = 3.43. However, there is a negative element in the column of domestic non-MNEs, where the solution for k would therefore be given by:

Table 5.2 and Table 5.3 show the results for the first iteration by columns.

Table 5.2. Unbalanced extended supply and use table with P, N and k

	Industry 1
	Domestic MNE	Foreign MNE	Domestic non-MNE	Intermediate use
Product 1	1	2	5	8
Product 2	4	2	3	12
TLS	-1	2	-2	-2
Value added	6	1	2	10
Output	10	12	6
Sum	10	7	8
P	11	7	10
-N	-1	0	-2
k	1.00	1.71	0.84
1/k	1.00	0.58	1.19

Note: MNE: multinational enterprise; TLS: taxes less subsidies on products. Predetermined constraints.

Table 5.3. Results after the first iteration (by columns)

	Industry 1
	Domestic MNE	Foreign MNE	Domestic non-MNE	Intermediate use	Sum	P	-N	k	1/k
Product 1	1	3.43	4.19	8	8.62	8.62	0.00	0.93	1.08
Product 2	4	3.43	2.52	12	9.94	9.94	0.00	1.21	0.83
TLS	-1	3.43	-2.39	-2	0.04	3.43	-3.39	0.74	1.34
Value added	6	1.71	1.68	10	9.39	9.39	0.00	1.06	0.94
Output	10	12	6
Sum	10.00	12.00	6.00
P	11.00	12.00	8.39
-N	-1.00	0.00	-2.39
k	1.00	1.00	1.00
1/k	1.00	1.00	1.00

Note: MNE: multinational enterprise; TLS: taxes less subsidies on products. Predetermined constraints.

As shown in Table 5.3, the sum of the column values matches the desired output totals exactly. However, the product totals (by each row) have not yet been balanced. Following the same strategy as outlined for the columns, Table 5.3 shows the values of k that would yield the same product totals that were defined as target. By applying these new k values row-wise, Table 5.4 shows the results obtained. At this point, the first iteration ends with a perfect match at the product level but with a mismatch at the industry level, as it was in the beginning.

Table 5.4. Results after the first iteration (by rows)

	Industry 1
	Domestic MNE	Foreign MNE	Domestic non-MNE	Intermediate use	Sum	P	-N	k	1/k
Product 1	0.93	3.18	3.89	8	8.00	8.00	0.00	1.00	1.00
Product 2	4.83	4.14	3.04	12	12.00	12.00	0.00	1.00	1.00
TLS	-1.34	2.55	-3.21	-2	-2.00	2.55	-4.55	1.00	1.00
Value added	6.39	1.83	1.79	10	10.00	10.00	0.00	1.00	1.00
Output	10	12	6
Sum	10.80	11.69	5.51
P	12.14	11.69	8.71
-N	-1.34	0.00	-3.21
k	0.94	1.03	1.04
1/k	1.06	0.97	0.96

Note: MNE: multinational enterprise; TLS: taxes less subsidies on products. Predetermined constraints.

Subsequent iterations can be run column-wise and row-wise until both industry output and product output totals match with the desired target. In our illustrative example, as shown in Table 5.5, after the fifth iteration, the maximum divergence is 9.07*10^-11.

Table 5.5. Results after the fifth iteration

	Industry 1
	Domestic MNE	Foreign MNE	Domestic non-MNE	Intermediate use	Sum	P	-N	k	1/k
Product 1	0.83	3.20	3.97	8	8.00	8.00	0.00	1.00	1.00
Product 2	4.50	4.30	3.20	12	12.00	12.00	0.00	1.00	1.00
TLS	-1.46	2.56	-3.10	-2	-2.00	2.56	-4.56	1.00	1.00
Value added	6.13	1.94	1.93	10	10.00	10.00	0.00	1.00	1.00
Output	10	12	6
Sum	10	12	6
P	11.46	12.00	9.10
-N	-1.46	0.00	-3.10
k	1.00	1.00	1.00
1/k	1.00	1.00	1.00

Note: MNE: multinational enterprise; TLS: taxes less subsidies on products. Predetermined constraints.

This proportional scaling algorithm can be replicated in as many directions of a hyper-matrix (with n dimensions) as needed, in a sequential way. Temursho, Oosterhaven and Cardenete (2021[12]) set out a particular case for an additional constraint (dimension) added to a matrix balancing problem with non‑overlapping elements and unity6 coefficients. Valderas-Jaramillo and Rueda-Cantuche (2021[13]) defined the general methodological framework for 3, 4 and n dimensions. KRAS (Lenzen, Gallego and Wood, 2009[11]) provides a framework allowing deviations from the target vectors to achieve balance, with unity coefficients.

However, balancing actual data is not straightforward and can introduce potential inconsistencies in the data, which may hinder the balancing method from reaching a solution. Such a scenario occurs when the method fails to converge to the intended target within a specified maximum deviation (threshold). The following section examines the conditions required for convergence in a matrix balancing problem. Understanding these conditions can assist ESUT compilers in pre-emptively resolving issues, thereby enhancing the likelihood of achieving convergence.

Conditions for unique solutions

This section elaborates on the conditions for unique solutions. It starts by considering non-negative matrices and then considers matrices with positive and negative elements.

Non-negative matrices

In the context of the RAS-type methods for matrix balancing, a pivotal reference is the necessary and sufficient conditions for convergence established by Bacharach (1965[20]), specifically for non-negative matrices. Bacharach (1965[20]) introduces the concept of “connectedness” to describe the structure of a matrix. A matrix is called disconnected if rows and columns can be ordered, in at least one way, so it can be expressed as a block-diagonal matrix, as in the following example. A matrix is called connected if it is not disconnected. Connectedness relates to the concept of sparsity, which refers to the number or proportion of zero entries in a matrix. While there is no universally accepted threshold to classify a matrix as sparse, generally a matrix with a higher proportion of zero7 entries is considered sparser.

Three different situations can be distinguished to illustrate Bacharach’s necessary and sufficient conditions:

1. disconnected matrix

2. connected matrix without zeros

3. connected matrix with zeros.

For the disconnected matrix one can use a simple 3x3 matrix as an example, such as:

In this example, the element at the intersection of the third row and third column is disconnected from the rest of the matrix. Consequently, matrix A can be divided into two independent submatrices, forming a block-diagonal structure. This configuration results in an infeasible problem that will not converge because the solitary element in the third row and column cannot simultaneously meet both the row and column targets. The same issue applies to the first diagonal block. Attempting to resolve this problem is futile because it is impossible to force an independent submatrix to have the same sum when adding the elements by rows (which totals 20) as when adding them by columns (which totals 23).

In conclusion, if a matrix is disconnected, a solution may not exist. However, the connected sub-blocks within the matrix can be examined. Then, convergence needs to be checked for all the connected sub‑blocks, as described hereafter.

Second, the connected matrix without zeros needs to be addressed. Table 5.1 shows an example of a connected matrix without zeros. This matrix does not contain any null elements (i.e. it is a full matrix) and is considered “connected” according to Bacharach’s terminology. Bacharach (1965[20]) proves8 that if a matrix is connected, then a solution of the RAS-type problem is unique if the sum of the row targets matches the sum of the column targets of the matrix. Therefore, the example in Table 5.1 is a well-posed problem, and a unique solution will exist in accordance with Bacharach’s conditions. Although it seems logical to think so, it is essential to verify that the sum of the row targets matches the sum of the column targets of the initial matrix. It may be the case that this condition would not naturally be met. Careful attention must be given to ensure that the newly added or modified rows and columns in the ESUT uphold this critical requirement. In the illustrative example, the sum of the row targets equals 28, as the sum of the column targets (i.e. the shaded cells).

Third, for a connected matrix with some zeros, a solution will exist if and only if for any sub-matrix of null values ${\mathbf{A}}_{{\mathit{I}}^{\mathrm{\text{'}}}\mathit{J}}=0$ that can be built reordering rows and columns, the following two conditions9 hold:

${\mathit{\iota }}^{\mathrm{\text{'}}}{\mathit{u}}_{{\mathit{I}}^{\mathrm{\text{'}}}}\le {\mathit{v}}_{{\mathit{J}}^{\mathrm{\text{'}}}}\mathit{\iota }$ (1)

${\mathit{\iota }}^{\mathrm{\text{'}}}{\mathit{u}}_{\mathit{I}}{\ge \mathit{v}}_{\mathit{J}}\mathit{\iota }$ (2)

for a matrix A such that:

$\left[\begin{array}{cc}{\mathit{A}}_{\mathit{I}\mathit{J}}& {\mathit{A}}_{\mathit{I}{\mathit{J}}^{\mathrm{\text{'}}}}\\ 0& {\mathit{A}}_{{\mathit{I}}^{\mathrm{\text{'}}}{\mathit{J}}^{\mathrm{\text{'}}}}\end{array}\right]$ $\left[\begin{array}{c}{\mathit{u}}_{\mathit{I}}\\ {\mathit{u}}_{{\mathit{I}}^{\mathrm{\text{'}}}}\end{array}\right]$.

$\left[\begin{array}{cc}{\mathit{v}}_{\mathit{J}}& {\mathit{v}}_{{\mathit{J}}^{\mathrm{\text{'}}}}\end{array}\right]$

Otherwise, the RAS-type problem is not feasible, and no solution exists. For the sake of simplicity, consider the following example of a connected matrix with zeros where there is no other way to reorder rows and columns to obtain a different null sub-matrix.

As shown in the example, equation (1) does not hold since ${\mathit{\iota }}^{\mathrm{\text{'}}}{\mathit{u}}_{{\mathit{I}}^{\mathrm{\text{'}}}}=10>{\mathit{v}}_{{\mathit{J}}^{\mathrm{\text{'}}}}\mathit{\iota }=7$ and equation (2) does not hold either, since ${\mathit{\iota }}^{\mathrm{\text{'}}}{\mathit{u}}_{\mathit{I}}{=\mathrm{ }20\mathrm{ }<\mathit{v}}_{\mathit{J}}\mathit{\iota }=23$. Therefore, no solution to this problem exists. Intuitively, considering that the third row of the matrix contains only one non-zero element, and the sum of that row must equal 10, one might think to change the value of a₃₃ from 2 to 10 to achieve the row’s target sum. However, this adjustment renders it impossible to attain the target sum for the third column, which already exceeded the desired total. A similar reasoning can be applied for the other set of rows and columns.

It is crucial to scrutinise the target values when dealing with a matrix that has a high level of sparsity to verify whether conditions (1) and (2) are met. If conditions (1) and (2) are not satisfied, the assumptions or the target values themselves must be re-evaluated to determine appropriate adjustments for the balancing problem.

Matrices with positive and negative elements

On certain occasions, the presence of negative elements can significantly alter the necessary conditions outlined by Bacharach (1965[20]), thereby introducing a degree of additional flexibility to the problem. Conversely, in other cases, the impact is the opposite. In other words, problems that were originally insolvable with non-negative matrices can become solvable when negative values are introduced. However, the reverse is also true: negative values can hinder the convergence of a solution. Furthermore, the inclusion of negative numbers is often associated with sign changes – whether in the aggregate totals or in the individual elements when comparing the initial and target matrices.

Apparently, there are no theorems that guarantee the convergence of general matrices that include both positive and negative elements (see Valderas-Jaramillo and Rueda-Cantuche (2021[13]) for a broader discussion on this issue). Experience suggests that for matrices with positive and negative elements, the greater the sparsity, the higher the likelihood of encountering convergence issues. Nonetheless, the conditions for the existence and uniqueness of a solution to a GRAS (generalised RAS) problem remain an open area of research.

Let us assume the following matrix as an illustrative example:

This matrix does not have a solution as can be easily proved. The first condition does not hold since ${\mathit{\iota }}^{\mathrm{\text{'}}}{\mathit{u}}_{{\mathit{I}}^{\mathrm{\text{'}}}}=5>{\mathit{v}}_{{\mathit{J}}^{\mathrm{\text{'}}}}\mathit{\iota }=4$; neither does the second condition, i.e. ${\mathit{\iota }}^{\mathrm{\text{'}}}{\mathit{u}}_{\mathit{I}}{=\mathrm{ }5\mathrm{ }<\mathit{v}}_{\mathit{J}}\mathit{\iota }=6\mathrm{ }.$ In other words, the element 𝑎₂₂ should be 5 (instead of 2) in order to match the target value (= 5). However, this would make it impossible to match the second column target, which is already lower (= 4). Assuming that 𝑎₁₂ is negative (= -1), then there would exist a solution such that:

The opposite case is when a matrix with a solution becomes infeasible due to a negative value. Assuming the following example:

This matrix fulfils the two conditions of Bacharach (1965[20]), i.e. ${\mathit{\iota }}^{\mathrm{\text{'}}}{\mathit{u}}_{{\mathit{I}}^{\mathrm{\text{'}}}}=5<{\mathit{v}}_{{\mathit{J}}^{\mathrm{\text{'}}}}\mathit{\iota }=6$ and ${\mathit{\iota }}^{\mathrm{\text{'}}}{\mathit{u}}_{\mathit{I}}{=\mathrm{ }5>\mathit{v}}_{\mathit{J}}\mathit{\iota }=4\mathrm{ }.$ The solution would be given by:

However, assuming now that 𝑎₁₂ is negative (= -1), then there would not be any solution to the new problem, even when Bacharach’s two conditions still hold. That is:

Analogously, 𝑎₂₂ should be 5 (instead of 2) in order to match the second row target value (= 5). However, this would make it impossible to match the second column target, which would be lower (= 4) due to the negative value.

Other possible problematic scenarios

Other potential scenarios can be proactively examined to detect issues of infeasibility and undesirable behaviours in matrices containing both positive and negative elements. These situations are generally straightforward to spot and may expose inconsistencies in the initial assumptions of the matrix and the dataset. It is both feasible and beneficial to develop algorithms capable of identifying these issues and implementing automatic solutions. This section discusses various scenarios, including zero (non-zero) targets with non-negative or non-positive (zero) elements.

Non-zero target with all elements null

The first problematic scenario is such that a null vector must match a non-zero target value, as shown in the example below.

Identifying a multiplying factor k using the GRAS algorithm is unfeasible because zeros will consistently remain as such. This situation is readily identifiable, and multiple potential solutions can be considered. The choice to have a full row or a full column of null values should be substantiated or otherwise abandoned, and a different row/column structure should be estimated. Alternatively, in the absence of information, a binary vector of zeros and ones can be created. By doing so, it must be established those elements for which one believes zeros must remain and ones otherwise. It is also possible that the target value should be null, which would imply setting the target to zero as a solution. In this respect, expert knowledge is key to defining either one or the other solutions to the infeasibility problem.

Zero target with all elements non-positive or non-negative

The second scenario entails a situation where a vector consisting solely of non-positive or non-negative values is required to align with a target value of zero:

This particular circumstance results in the nullification of the vector following the initial iteration. While it is conceivable that this may not affect the problem substantially, as previously observed, it enhances the sparsity of the prior matrix, potentially complicating the algorithm’s convergence. It is possible that the problem was initially well formulated; however, subsequent rounds after the first iteration may turn the problem into an ill-posed and infeasible one. This can be shown in the following example:

Although the Bacharach’s conditions (1) and (2) hold10 there may not be a solution since v₂ = 0 and the elements of the second column are all positive. After the first iteration (matching v), the initial matrix yields:

For the next iteration, the second column has been set to zero. Hence, the problem has turned infeasible because the first element a₁₁ cannot be 5 (to meet u₁ = 5) and at the same time meet v₁ = 4 if a₂₁ and a₃₁ are positive elements.

The solutions to this type of infeasibilities will depend on the context and assumptions inside the initial matrix and the target vectors that should be considered carefully by the ESUT compilers.

Zero target with positive, negative and null elements

The third scenario entails a situation where a vector consisting of positive and negative values is required to align with a target value of zero. In principle, nothing should be wrong in this particular situation (as long as negative values make sense).11 Otherwise, experts will have to evaluate what the zero target means (e.g. absence of activity) and whether that would entail setting all the elements to zero as a solution to the infeasibility problem.

This scenario, while not initially leading to infeasibilities, merits preventive examination to allow for informed decision making rather than deferring to the algorithm’s discretion. Furthermore, such situations may give rise to secondary complications. At times, all positive values may escalate due to the other targets, obligating the negative values to increase correspondingly to balance the positives. It is therefore advisable to set up alert controls in every iteration to detect large positive and large negative values compensating each other. Moreover, this situation can result in an unnecessary proliferation of iterations required for convergence or may even render the problem infeasible.

Negative (positive) target with non-negative (non-positive) elements

The preceding paragraphs addressed scenarios involving zeros, such as null vectors or those with zero targets. This section considers a different situation in which zeros are not relevant but there is a change in sign among the vector elements in the prior vector compared to the desired target. For instance, there may be a situation like the one described in the following example:

As the GRAS method is sign-preserving, this is a clear example of an infeasible situation since there is a non-negative vector as prior and a desired negative target. However, the GRAS algorithm may still produce a solution, although not necessarily the solution to the initial optimisation problem. Indeed, the GRAS algorithm is equivalent to solving an optimisation problem when the problem is well-defined – that is, when a solution exists and is unique. However, if the optimisation problem is ill-defined and lacks a feasible solution, the GRAS algorithm might still operate and potentially yield a plausible outcome. In such cases, the solution is not really a solution to the initial optimisation problem but to a different one. Consequently, any solution derived is not a result of the original optimisation problem,12 as discussed by Lenzen et al. (2014, p. 203[10]) and Valderas-Jaramillo and Rueda-Cantuche (2021, p. 1611[13]). Analogous conclusions come from situations where the vector has non-positive elements, and the target is positive.

There are three outstanding issues to consider in this scenario. First, the initial optimisation problem is infeasible. Second, the economic meaning of the negative target (or the negative values of the vector) has to be substantiated, as does the subsequent results of the GRAS algorithm, if necessary. Third, the GRAS algorithm is providing a solution to a different optimisation problem.

As a result, it is important to evaluate the economic soundness of the prior data, the targets and the sign change derived from the GRAS algorithm. Furthermore, as shown earlier, the introduction of new negatives in the initial matrix may also cause convergence problems.13

Fixing elements and reducing degrees of freedom

Balancing methods in compiling ESUTs can also profit from additional information on specific elements. Paelinck and Waelbroeck’s (1963[21]) modification of the RAS method, which they termed M-RAS, incorporated additional information into a pre-existing matrix by specifying certain known cells. This process leverages the characteristic of sign preservation inherent in RAS/GRAS problems, allowing any predetermined (known) elements to be initially set to zero in the matrix. Subsequently, target adjustments are made to reflect the presence of these known elements. Following these modifications, the M-RAS algorithm is executed to find a solution, after which the zero placeholders are replaced with the pre-known values.

There are two significant considerations when dealing with predetermined elements. First, fixed elements inherently limit the degrees of freedom within the matrix. This limitation can increase the sparsity of the matrix, potentially transforming the optimisation problem into an infeasible one or contributing to increased difficulties in achieving convergence.

Second, the use of known elements also necessitates modifications to the target values, which can also lead to infeasibility or the loss of economic meaning. This is particularly evident when dealing with a vector of non-negative numbers, and the predetermined cells exceed the desired target values. In such cases, the additional information renders the problem economically meaningless or unfeasible, as it forces the remaining coefficients in the vector to change signs. Under these circumstances, it becomes apparent that either the target values or the additional information must be reconciled, potentially requiring a compromise, especially when both sets of information are less reliable.

Checklist for designing automatic balancing processes and recommendations

This section presents a checklist for compilers to consult prior to executing the GRAS method during the compilation of ESUTs. It also contains recommendations for using expert knowledge to address the complexities of balancing ESUTs, emphasising the importance of understanding the economic implications of the sourced data.

If the checklist items are incorporated into automatic software scripts (such as those written in R, Python or similar programming languages), the pre-balancing verification can be conducted systematically. Ensuring that this protocol is followed increases the probability of successfully converging the balancing algorithm. This checklist is also valid for matrices with predetermined information.

Checklist for ESUT compilers:

1. Check that the overall row sum matches the overall column sum.

2. For non-negative matrices, check Bacharach’s conditions.

3. Check for zero targets with:

a. all elements non-negative

b. all elements non-positive

c. elements positive, negative and zero.

4. Check for negative targets.

5. Check for non-zero targets with null columns and/or rows.

This checklist does not prevent convergence problems from occurring, but it will at least increase the probability of successful convergence.

This chapter aimed to emphasise the importance of careful preparation and verification before employing the GRAS method for balancing ESUTs. Its main conclusions and recommendations are:

• Refer to a checklist to ensure that the overall row sum matches the overall column sum and that the necessary conditions for convergence, such as Bacharach’s conditions for non-negative matrices, are met after each iteration.

• Be cautious of potential infeasibilities in each iteration, such as zero targets with all elements null or non-negative or non-positive, and negative targets with non-negative elements, which can lead to convergence issues or economically meaningless results.

• Consider the impact of fixed elements known in advance, as they reduce the degrees of freedom and can increase the sparsity of the matrix, potentially making a feasible problem infeasible.

• Reconcile any discrepancies between the known elements and the target values, especially when both pieces of information are unreliable, to avoid rendering the problem economically meaningless or unfeasible.

• Implement automatic scripts to conduct pre-balancing verification systematically, which can enhance the likelihood of achieving a successful convergence in the balancing process.

These recommendations highlight the importance of understanding the economic implications of the data sources and the complexities involved in the balancing processes.

References

[20] Bacharach, M. (1965), “Estimating non-negative matrices from marginal data”, International Economic Review, Vol. 6/3, pp. 294-310, https://doi.org/10.2307/2525582.

[16] Beutel, J. (2008), An Input-output System of Economic Accounts for the EU Member States, report to the European Commission’s Joint Research Centre, European Commission.

[15] Beutel, J. (2002), The Economic Impact of Objective 1 Interventions for the Period 2000-2006, final report to the Directorate-General for Regional Policies of the European Commission.

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Annex 5.A. Costa Rica’s experience with ESUTs

This annex describes Costa Rica’s experiences with extended supply and use tables (ESUTs), in particular the way they are balanced during a backcasting process.

Costa Rica is a highly open economy. In recent years, exports and imports accounted for 34% and 33% of gross domestic product (GDP), respectively, while foreign share enterprises (FSEs) accounted for approximately 64% of exports and foreign direct investment for around 4% of GDP.

FSEs include multinational corporations that primarily target foreign markets and are connected to different stages within the global value chains, resulting in dissimilar levels of interaction with domestic markets; some of these enterprises have no connection with domestic markets, while others have domestic control enterprises (DCEs) as their main suppliers. FSEs complement domestic savings, transfer technology and knowledge, and generate employment and spillover effects that contribute to economic growth.

This dynamic causes heterogeneity within the economy in many areas, such as income payments to the rest of the world, production functions and foreign content ratios. To deal with these particularities, with the new reference year of national accounts (2017) and under the auspices of the OECD Expert Group on ESUTs, the Central Bank of Costa Rica (CBCR) compiles ESUTs with a bottom-up approach to provide enhanced tools for economic analysis, research and projections. Costa Rica’s ESUTs provide a breakdown between DCEs and FSEs for each economic activity.

Long time series are very important for analytical purposes and maintaining the economic history of a country. Given this, as of 2017, the ESUT had a radical change due to the increased granularity by economic activity according to DCEs and FSEs, so it was necessary to revise previously published ESUTs to adapt them to the classification of products/activities and facilitate comparability. This involved the use of balancing methods to extend the main macroeconomic aggregates to 1991 to warrant the coherence of the new economic series with the historical series.

A series of main macroeconomic aggregates, namely gross value added by each industry, final use by component (final consumption expenditure, gross capital formation, exports), imports, taxes less subsidies on products and others, were obtained through bottom-up and interpolation methods. Once the individual consistency of the main macroeconomic aggregates was achieved, the CBCR assessed the integrated coherence for the period 2012-16 using the GRAS method within a SUT framework (i.e. SUT-RAS method). The GRAS worked out adequately despite the large number of columns (more than 140) given the additional ESUT breakdowns in each economic activity. The method seeks to generate ESUTs that are consistent with backwards revised official macroeconomic estimations, ensuring consistency between supply and demand.

GRAS/SUT-RAS to integrate backwards revised data into ESUTs

Concerning the theory and methodology, the CBCR used the SUT-RAS method with an explicit treatment of taxes less subsidies on products and output by industry exogenously available, as described in Section 3.3 in Valderas-Jaramillo et al. (2019[17]). As they show, the SUT-RAS method provides better estimates than other methods such as the SUT-EURO, under the same assumptions and exogenous information. Besides, the SUT-RAS method is a particular case of the GRAS method for balancing supply and use tables all at once (Temursho, Oosterhaven and Cardenete, 2021[12]).

The SUT-RAS method implemented by the CBCR belongs to the RAS family of methods, which are based on biproportional adjustments in the rows and columns of a reference matrix to obtain a new projected matrix. This is done through iterative steps and by solving a restricted optimisation problem, as described in Valderas-Jaramillo et al. (2019[17]).

Following the same notation as in Valderas-Jaramillo et al. (2019[17]), let us assume that supply and use tables of the base year consist of the following components (already including FSEs and DCEs):

• Let ${\mathbf{U}}_{b,0}^d$ and ${\mathbf{U}}_{b,0}^m$be the domestic and imported intermediate uses at basic prices, with dimension product (p) by industry (r).

• Let ${\mathbf{Y}}_{b,0}^d$ and ${\mathbf{Y}}_{b,0}^m$be the domestic and imported final use matrices, with dimension product (p) by final use category (f).

• Let ${\mathbf{V}}_{b,0}$be the transposed supply matrix (r x p).

• Let ${\mathbf{v}}_0$ be the vector of gross value added by industry, with dimension (r x 1).

• Let ${\mathbf{t}\mathbf{l}\mathbf{s}}_0$ be the vector of taxes less subsidies on products, with dimension (r + f) x 1. In turn, this vector is split into two sub-vectors, one for intermediate uses and another for final uses.

${\mathbf{t}\mathbf{l}\mathbf{s}}_0=\left(\begin{array}{c}{\mathbf{t}\mathbf{l}\mathbf{s}}_0^{ID}\\ {\mathbf{t}\mathbf{l}\mathbf{s}}_0^{FD}\end{array}\right)$

Annex Table 5.A.1 depicts the integrated supply and use framework of the base year.

Annex Table 5.A.1. Integrated supply and use framework for the base year*

	Domestic products (pd)	Imported products (pm)	Industries (r)	Final use (f)	Total
Domestic products (pd)			Ub,0d	Yb,0d	Ub,0di+Yb,0di=qb,0d
Imported products (pm)			Ub,0m	Yb,0m	Ub,0mi+Yb,0mi=m0
TLS			tls0ID'	tls0FD'	tls0ID'i+tls0FD'i=tls0
Industries (r)	Vb,0				Vb,0i=xb,0
Imports		m0'			m0'i=m0
Total	i'Vb,0=qb,0d'	m0'	xb,0'-vb,0'	yp,0'

* All the elements in this table are known.

Note: TLS: taxes less subsidies on products.

Similarly to the original SUT–RAS method (Temurshoev and Timmer, 2011[22]), let A be the integrated supply and use framework of the base year, with all elements known:14

$\mathbf{A}=\left(\begin{array}{cc}{\mathbf{O}}_{\left(p^d+p^m+1\right)x(p^d+p^m)}& {\stackrel{-}{\mathbf{U}}}_0\\ {\stackrel{-}{\mathbf{V}}}_0& {\mathbf{O}}_{\left(r+m\right)x(r+f)}\end{array}\right)\in {\mathcal{M}}_{\left(p^d+p^m+r+2\right)x(p^d+p^m+r+f)}$ (A1)

where:

${\stackrel{-}{\mathbf{V}}}_0\mathrm{ }\in {\mathcal{M}}_{\left(r+1\right)x{(p}^d+p^m)}=\left(\begin{array}{cc}{\mathbf{V}}_{b,0}& {\mathbf{O}}_{r\mathrm{ }x\mathrm{ }{p}^m}\\ 0_{\mathrm{ }{p}^d}^{\mathrm{\text{'}}}& {\mathbf{m}}_0^{\mathrm{\text{'}}}\end{array}\right)$,

are the elements of the supply side for the base year, and

${\stackrel{-}{\mathbf{U}}}_0\mathrm{ }\in {\mathcal{M}}_{\left(p^d+p^m+1\right)x\left(r+f\right)}=\mathrm{ }\left(\begin{array}{cc}{\mathbf{U}}_{b,0}^d& {\mathbf{Y}}_{b,0}^d\\ {\mathbf{U}}_{b,0}^m& {\mathbf{Y}}_{b,0}^m\\ {{\mathbf{t}\mathbf{l}\mathbf{s}}_0^{ID}}^{\mathrm{\text{'}}}& {{\mathbf{t}\mathbf{l}\mathbf{s}}_0^{FD}}^{\mathrm{\text{'}}}\end{array}\right)$,

the extended use matrix for the base year.

Similarly, Table 5.A.2 shows the integrated supply and use framework of the target year.

Annex Table 5.A.2. Integrated supply and use framework of the target year*

	Domestic products (pd)	Imported products (pm)	Industries (r)	Final use (f)	Total
Domestic products (pd)			Ub,td	Yb,td	Ub,tdi+Yb,tdi
Imported products (pm)			Ub,tm	Yb,tm	Ub,tmi+Yb,tmi
TLS			tlstID'	tlstFD'	tlst (**)
Industries (r)	Vb,t				xb,t (**)
Imports		mt'			mt (**)
Total	i'Vb,t	mt'	xb,t'-vb,t' (**)	yp,t' (**)

**Only these elements are known. The others need to be estimated, and unknown totals calculated by summation.

Note: TLS: taxes less subsidies on products.

Analogously, let X stand for the supply and use integrated framework for the target year t,

$\mathbf{X}=\left(\begin{array}{cc}{\mathbf{O}}_{\left(p^d+p^m+1\right)x(p^d+p^m)}& {\stackrel{-}{\mathbf{U}}}_t\\ {\stackrel{-}{\mathbf{V}}}_t& {\mathbf{O}}_{\left(r+m\right)x(r+f)}\end{array}\right)\in {\mathcal{M}}_{\left(p^d+p^m+r+2\right)x(p^d+p^m+r+f)}$ (A2)

The main objective is to estimate X, starting from A, in a consistent way using the exogenous information of the target year. The exogenous information and certain accounting identities (restrictions) coming from the integrated supply and use framework are formulated as follows:

Domestic production supply and use balance: ${\mathbf{U}}_{b,t}^d\mathbf{i}+{\mathbf{Y}}_{b,t}^d\mathbf{i}-{\mathbf{V}}_{b,t}^{\text{'}}\mathbf{i}\mathbf{ }=0$ (A3)

Imports supply and use balance: ${\mathbf{U}}_{b,t}^m\mathbf{i}+{\mathbf{Y}}_{b,t}^m\mathbf{i}={\mathbf{m}}_t$ (A4)

Industry output preservation: ${\mathbf{V}}_{b,t}\mathbf{i}={\mathbf{x}}_{b,t}$ (A5)

Imports preservation: ${\mathbf{m}}_t^{\text{'}}\mathbf{i}=m_t$ (A6)

Intermediate consumption preservation: ${{\mathbf{U}}_{b,t}^d}^{\text{'}}\mathbf{i}+{{\mathbf{U}}_{b,t}^m}^{\text{'}}\mathbf{i}+{\mathbf{t}\mathbf{l}\mathbf{s}}_t^{ID}={\mathbf{x}}_{b,t}-{\mathbf{v}}_t={\mathbf{I}\mathbf{C}}_{p,t}$ (A7)

Final use preservation: ${{\mathbf{Y}}_{b,t}^d}^{\text{'}}\mathbf{i}+{{\mathbf{Y}}_{b,t}^m}^{\text{'}}\mathbf{i}+{\mathbf{t}\mathbf{l}\mathbf{s}}_t^{FD}={\mathbf{y}}_{p,t}$ (A8)

Taxes less subsidies on products preservation: ${{\mathbf{t}\mathbf{l}\mathbf{s}}_t^{ID}}^{\text{'}}\mathbf{i}+{{\mathbf{t}\mathbf{l}\mathbf{s}}_t^{FD}}^{\text{'}}\mathbf{i}={tls}_t$ (A9)

It is important to note that since industry output (${\mathbf{x}}_{b,t})$and gross value added (${\mathbf{v}}_t)$ are exogenous, they are used to set up the restriction (A7) on intermediate consumption at purchasers’ prices.

Thus, let $a_{ij}$ be the entries of matrix A, $x_{ij}$ the entries of matrix X. In addition, $z_{ij} \mathrm{i}\mathrm{s}$defined as:

$z_{ij} = \{<!--\; -->\begin{array}{c}\frac{x_{ij}}{a_{ij}} if a_{ij}\ne 0\\ 1 if a_{ij}=0\end{array}$

Now, the elements of X are to be found from the values of $z_{ij}$ that minimise the following objective function ( (Huang, Kobayashi and Tanji, 2008, p. 114[6]), equation 6) subject to the restrictions (A3) to (A9).

$\underset{z_{ij}}{min} \sum _i\sum _j\left|a_{ij}\right|\left(z_{ij}\bullet \ln \left(\frac{z_{ij}}{e}\right)+1\right)$

To address the optimisation problem, the CBCR used a Lagrangian function based on the system of equations (restrictions). The process is carried out iteratively until convergence is achieved, i.e. when the difference between the estimated multipliers of one iteration and the next is smaller than a given tolerance level.

The CBCR implemented this methodology using the R programming language and successfully achieved convergence for the period 2012-16, using 2017 as the reference year. Beyond attaining algorithmic convergence, the CBCR conducted several tests to validate the estimated data. These tests included calculating GDP from the three different approaches (namely from output, income and demand approaches) as well as using professional judgment to assess the results.

The CBCR is currently in the process of updating their national accounts with the new base year of 2022. This update necessitates a similar procedure to adjust the previous ESUTs from 2017-21 to align with the classifications and products of 2022. With additional exogenous data, such as product-specific imports and exports, becoming available, the CBCR plans to make further refinements to the GRAS/SUT-RAS model to enhance the accuracy of their estimates.

Notes

← 1.  See a complete review of the relevant literature in Chapter 18 of United Nations (2018[1]).

← 2.  However, this solution does not always turn out to be simply a scaling method. For example, Stone, Champernowne and Meade (1942[27]); Robinson, Cattaneo and El-Said (2001[25]); Golan, Judge and Robinson (1994[24]); Rodrigues (2014[26]); Lugovoy, Vladimirovich Polbin and Potashnikov (2015[28]); and Fernández, Hewings and Ramos Carvajal (2015[23]) also proved the Bayesian approach with success.

← 3.  RAS-type proportional methods can still provide a solution to the problem although sometimes they are not meaningful in terms of a well-defined constrained optimisation problem with a unique solution.

← 4.  Other alternative objective functions and different distance measures have been evaluated by the literature (see United Nations (2018[1]), Chapter 18) but are not reported here. It is recommended though to provide information on the adjustments made by the balancing processes under several of those options if time allows, Moreover, sometimes the column and row targets may have lower reliability than some of the inner parts of the matrices, which will require a different approach (e.g. KRAS). This is beyond the scope of this chapter.

← 5.  If N = 0, then the solution would be k = S/P and if P = 0, the solution would be k = - N/S (i.e. RAS method).

← 6.  Sums or subtractions of the elements of submatrices.

← 7.  These zeros, from a mathematical perspective, limit the degrees of freedom in the optimisation problem. Both the RAS and GRAS algorithms are designed to preserve signs, which means that any zero in the initial matrix will remain zero in the balanced matrix. Therefore, these zero entries can be excluded from the optimisation problem.

← 8.  Theorem 2 and Corollary 1 (Bacharach, 1965, pp. 301-302[20]).

← 9.  Theorem 3 and Corollary 2 (Bacharach, 1965, pp. 301-302[20]). If there were several ways to create different null sub‑matrices ${\mathbf{A}}_{{\mathit{I}}^{\mathit{\text{'}}}\mathit{J}}$ (i.e. sub-matrices with different subsets of rows and columns), these two conditions should be checked for all of them. The ι (iota) symbol is a vector of ones and is being used as a summation operator.

← 10.  ${\mathit{\iota }}^{\mathit{\text{'}}}{\mathit{u}}_{{\mathit{I}}^{\mathit{\text{'}}}}=10\le {\mathit{v}}_{{\mathit{J}}^{\mathit{\text{'}}}}\mathit{\iota }=11$ and ${\mathit{\iota }}^{\mathit{\text{'}}}{\mathit{u}}_{\mathit{I}}=5\ge {\mathit{v}}_{\mathit{j}}\mathit{\iota }=4$.

← 11.  An example is a vector of trade and transport margins that should sum up to zero.

← 12.  As shown in footnote 4, the GRAS solution (k) to the example is k = S/P, for N = 0. This implies that all the elements of the vector will change their sign since k < 0 because S < 0. In the next iterations, the solution would be k = -N/S > 0, for P = 0 and S < 0. As a result, the algorithm’s solution solves a modified optimisation problem with negative elements (after a sign change) and a negative target.

← 13.  For instance, some elements of a non-negative vector with a zero target can become negative because of a sign shift, thus altering the potential convergence of the GRAS algorithm.

← 14.  and are null matrices and vectors, respectively, with appropriate dimensions.$\mathbf{ }\mathbf{i}$ stands for a unitary column vector of ones.