The multidimensional nD‐GRAS method: Applications for the projection of multiregional input–output frameworks and valuation matrices

Abstract

We present a multidimensional generalization of the GRAS method (nD-GRAS) for the estimation of multiple matrices in an integrated framework. The potential applications of this method in regional and multi-regional input–output analyses based on national/regional accounts frameworks are many. We provide two real applications, a 3D-GRAS that estimates a use table at basic prices jointly with valuation matrices for Denmark; and a 4D-GRAS for estimating intercountry input–output tables with OECD data. We show that higher dimensional GRAS methods provide more consistent and accurate estimates than those with lower number of dimensions. We provide the analytical closed-form solution and the RAS-like algorithm for an easy operationalization. En este artículo se presenta una generalización multidimensional del método GRAS (nD-GRAS) para la estimación de matrices múltiples en un marco integrado. Las aplicaciones potenciales de este método en los análisis input-output regionales y multirregionales basados en los marcos de cuentas nacionales o regionales son numerosas. Se incluyen dos aplicaciones reales, un 3D-GRAS que estima una tabla de uso a precios básicos conjuntamente con matrices de valoración para Dinamarca; y un 4D-GRAS para estimar tablas input-output entre países con datos de la OCDE. Se demuestra que los métodos GRAS de mayores dimensiones proporcionan estimaciones más consistentes y precisas que aquellos con un menor número de dimensiones. Para una fácil operacionalización, se proporciona la solución analítica en forma cerrada y el algoritmo tipo RAS. 本稿では、統合フレームワークにおける複数の行列の推定のためのGRAS法(n D-GRAS)の多次元一般化モデルを示す。国別・地域別会計のフレームワークに基づく地域別・複数地域別の産業連関分析にこの手法を適用できる可能性のある方法は多くある。デンマークの評価行列と一緒に基本価格で使用表を推定する3 D-GRASまた、OECDのデータを用いて各国間の産業連関表を推定するための4 D-GRAS、以上の実際の二つの応用事例を示す。高次元のGRAS法は、低次元のGRAS法よりも、より一貫性があり正確な推定値が得られることが示された。また、解析的閉形式解と簡単な操作のためのRAS様アルゴリズムが得られた。 Multiple variations of biproportional techniques have been applied to the field of input–output analysis since Leontief's (1941) pioneering work, in which he used a biproportional technique to identify sources of inter-temporal change in the cells of a series of input–output tables (Lahr & de Mesnard, 2004). This also includes –broadly speaking– the RAS-family methods (Bacharach, 1965, 1970) and their extensions. A recent summary that provides a good overview and a large compilation of these methods can be found in Chapter 18 of the UN Handbook on supply, use and input–output tables with extensions and applications (United Nations, 2018). The idea for this method stems from the fact that often in practical situations rather than simply imposing constraints summing all the elements of a matrix row-wise or column-wise (as in standard GRAS), it is necessary to rearrange a matrix representing multiregional information into arrays of a larger dimension imposing constraints on all the dimensions of the array. For instance, in multiregional frameworks, where national IOTs are split using information on bilateral exports and imports, it may be that the corresponding national use tables of imports might serve as constraints to the balancing of a multiregional IOT. Another practical situation that requires more than two dimensions is the estimation of use tables at basic prices and valuation matrices, that is, trade and transport margins tables (TTM), taxes less subsidies on products tables (TLS) to make them consistent with the use tables at purchasers´ prices. We can estimate each of those tables independently with a GRAS method, but the result of summing TTM, TLS and the use table at basic prices would be equal to the initial use table at purchasers´ prices only by chance. The main contribution of this paper is the derivation of an analytical closed-form solution to the GRAS method in a multidimensional framework with an arbitrary number of dimensions and the algorithm to handle these problems in an accessible way. The bi-dimensional case is the standard GRAS method (2D-GRAS according to our terminology). As it will be described in the next section, the problems addressed by the nD-GRAS method can also be embedded and solved within the KRAS framework; our can be in a RAS-like contribution of this paper is to show that based on a higher number of from the of the to and than based on a lower number of dimensions. For instance, in the from the using a 4D-GRAS method a than the that using a 3D-GRAS applied independently to each of the of the array. We will also that this for the 3D-GRAS method and the matrices This paper is as The next the of our within the applications can be embedded in our multidimensional all the the and the and solution of the and provide two for the 3D-GRAS and 4D-GRAS a provides a summary of the main and The method and biproportional techniques the of is in as a bi-dimensional and used methods for the estimation of We can applications of these methods in the by and (Lahr & de Mesnard, 2004). for tables by and for larger multidimensional tables by and and A good summary of these their and basic can be found in the of the by We can also in the recent the multidimensional generalization of the RAS-family this in an the biproportional algorithm of to dimensions. a method for estimating an input–output in a bi-dimensional where the regional cells to a national The by is to the multiregional GRAS method by and methods a bi-dimensional where the national provides a of the case of the this method includes an of constraints from the row-wise and column-wise in a multiregional framework using the bi-dimensional as the standard GRAS the method, the of of elements in the multiregional that to a As and this can be for or where the dimension is for the to a The been used for the of a for the and in the of national IOTs in a multiregional for the of the with the of and and are a real the use a bi-dimensional with constraints these two the dimension is into as by this of the methods of and and a The 3D-GRAS can be solved in of and since it is to a as a standard these two as and the nD-GRAS method is more for of that are to for dimensions. the nD-GRAS method is more and it in this paper in a and way. We also show a solution algorithm that the of this method the number of dimensions and the use of The nD-GRAS method within the & and it is used in the of the supply, use and input–output tables of RAS-like algorithm can be found in and and The algorithm in the of and a of that accounts for constraints from row-wise and column-wise The algorithm by the bi-dimensional algorithm to for all these as in only the algorithm is as a practical of the matrix that all the constraints Another where valuation matrices, basic and prices are jointly is in and our the algorithm by these into the of multidimensional as is a multidimensional that valuation matrices and basic prices table to the prices the balancing of the on the and the the by the table is the of this The basic prices and valuation matrices are using a of and a RAS-like algorithm for and the prices table is the of all of The algorithm the is and a multidimensional method that the bi-dimensional in a multidimensional that with only and the method only in an to and that this algorithm is the solution of the in a multidimensional The is a case of the nD-GRAS elements in the initial the in a for the estimation of and input–output tables of the show that the of a from the of national the regional or more than the standard estimates in of the input–output The of the multidimensional method to input–output also than the standard Another in the field of input–output is the of methods from the KRAS method by the KRAS method, the of the matrix the of the in a This all potential constraints to be as constraints on and of matrix elements with or in a The KRAS method also as the of the information by the constraints and the of potential with in the case of The KRAS method is at the of multiple applications that with multidimensional a with a for the of series tables is This the estimation of large dimension a multidimensional can be in a with a This is in in the of the where an is This an in the as in an method would is, all the elements can be embedded in an the constraints all the dimensions in a array. of the constraints or in an to the 3D-GRAS and 4D-GRAS of this the of using from the GRAS have or and to with and matrices, techniques and are this framework includes the of with and regional and as a of a This from the KRAS method with the of in and and these can their in a using a of input–output applications in and or the problems addressed by the nD-GRAS method can be embedded and solved in this KRAS as as a solution using the a of constraints with and This also only to the nD-GRAS method, but also to the of the multidimensional methods in this paper as those by and or and the of the KRAS and are also have to be into The nD-GRAS the of a closed-form at the of using only with a solution also information and a the derivation of the solution of the multidimensional and the algorithm a large of problems to be solved the the multiple of the KRAS to be in the as and but also of the of the elements in the estimation in of products and can be a to for the of the matrix to an that requires the of large and matrices that are in of the of of these as the of the constraints requires the use of and and as as are also to the of problems by the and provide in their an techniques and The recent of in this may an that these extensions for is a good of the generalization of the GRAS to multiple dimensions may the of an analytical solution and an algorithm for it easy to this algorithm can be in an easy on more techniques may that to be solved with embedded in this it is also that these techniques can with a larger potential for constraints number of of and more the case of in the for of the algorithm is to to the is also to of the the and (as will in or the by the in the the multidimensional GRAS method, in can practical situations into this from the to the more with 3D-GRAS A multiregional framework is a for multiple 3D-GRAS that have a matrix of a multiregional input–output framework. this elements or are by of the multiregional framework. 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their with the TLS of the in cells and The of the GRAS method it to provide good these in the in the 3D-GRAS that the and TLS tables to to the a in this and for two main in our the use of these constraints is a of information in the this of information all the tables in a estimation that cells more and the of this an of of the 4D-GRAS method to estimate the and all the information for the of this is at We have OECD a of the as and of the of these The table is an by input–output to will as from are split into an for taxes less are split in products TLS and are the dimensions in our The table of elements that are equal to of the elements and the For this the number of to be is and the information is the number of elements in the this to might that on and be the only but in trade and for input–output tables can provide on the two or national sources can provide for instance, the imports from and of the imports of are from the of the 4D-GRAS method information on the corresponding this is 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jointly in a 4D-GRAS method are than their using as it for the 3D-GRAS method with to their lower dimension using a 4D-GRAS method only the of the in the it also to estimation than using The 3D-GRAS only the in the 4D-GRAS in situations of For and 3D-GRAS the in the This result is more for and tables in the 4D-GRAS their 3D-GRAS in those where the 3D-GRAS methods the 4D-GRAS the is as can in where the for of and the show of the with a of for 3D-GRAS methods in and the have the but for 4D-GRAS the the of the 3D-GRAS in This is the table for We can that the of in the 3D-GRAS is to the are and the in are the have the for the 4D-GRAS for a for in which the situation is the this that of 4D-GRAS are than for the 3D-GRAS The for the two of for in the of as it in the of the 3D-GRAS the of large in and elements of the table is larger in the 4D-GRAS estimation to the 3D-GRAS This can be from the of and of the 4D-GRAS method with to the 3D-GRAS in and are in the 4D-GRAS on and is the which that the 4D-GRAS a estimate in to be a larger number of in This situation is described in the the of the and the the in are on a it is easy to a of of as the of the this is more this it is to that the would the and are to the with The of GRAS balancing techniques may to this since the of the in the the change of the that will on the of the and the of that in the have in these two in this that larger dimensional GRAS a larger of in with the This paper the GRAS method in a multidimensional framework and provides analytical We have also described an algorithm that provides a of the multidimensional GRAS method, and to and and The multidimensional GRAS methods can be for practical applications input–output with multiregional The 3D-GRAS method can be used for estimating use tables at basic prices from an use table at purchasers´ the of the corresponding valuation matrices trade and transport margins and taxes less subsidies on products multiregional frameworks with information exports and imports by the 3D-GRAS method can also be used to split the use table of imports of a by of The 4D-GRAS method is also for the of multiregional with of and of the corresponding dimensions. We have two that two applications for and for the OECD applications are based on real problems by or the to them might be to in the of the method is as it for instance, the multiregional input–output by and tables We have also that with a higher number of from to a than using of the corresponding with a lower number of dimensions. is to that are for it is necessary to for the and the the that those are a our will and the algorithm will provide a from the of GRAS methods in the of the and of of and elements and the higher dimension methods a larger of in and elements of our and in elements with and The KRAS method is an for with and with as as a solution have information that can to in a is easy to the for the multidimensional GRAS methods in this paper are for balancing multiregional IOTs with information the of each dimension and for balancing a and use framework from purchasers´ prices to basic their valuation are also to for extensions to as dimensions as for multiregional IOTs split according to the and of and use is, this multidimensional balancing situations can be addressed and solved with a algorithm where the dimension would be that the of the two and for the of that as in the this is the of this and and

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