Biproportional Methods Research Articles

Bridging macroeconomic data between statistical classifications: the count-seed RAS approach

ABSTRACTIn applications, it is often necessary to link heavily aggregated macroeconomic datasets adhering to different statistical classifications. We propose a simple data reclassification procedure for those cases in which a bridge matrix grounded in microdata is not available. The essential requirement of our approach, which we refer to as count-seed RAS, is that there exists a time period or a geographical entity similar to the one of interest for which the relevant economic variable is observed according to both classifications. From this information, a bridge matrix is constructed using bi-proportional methods to rescale a seed matrix based on a qualitative correspondence table from official sources. We test the procedure in two case studies and by Monte Carlo methods. We find that, in terms of reclassification accuracy, it performs noticeably better than other expeditious methods. The analytical framework underlying our approach may prove a useful way of conceptualizing data reclassification problems.

A Bayesian Approach to the Balancing of Statistical Economic Data

This paper addresses the problem of balancing statistical economic data, when data structure is arbitrary and both uncertainty estimates and a ranking of data quality are available. Using a Bayesian approach, the prior configuration is described as a multivariate random vector and the balanced posterior is obtained by application of relative entropy minimization. The paper shows that conventional data balancing methods, such as generalized least squares, weighted least squares and biproportional methods are particular cases of the general method described here. As a consequence, it is possible to determine the underlying assumptions and range of application of each traditional method. In particular, the popular biproportional method is found to assume that all source data has the same relative uncertainty. Finally, this paper proposes a simple linear iterative method that generalizes the biproportional method to the data balancing problem with arbitrary data structure, uncertainty estimates and multiple data quality levels.

What is the Best Method of Matrix Adjustment? A Formal Answer by a Return to the World of Vectors

The principle of matrix adjustment methods consists into finding what is the matrix which is the closest to an initial matrix but with respect of the column and row sum totals of a second matrix. In order to help deciding which matrix-adjustment method is the better, the article returns to the simpler problem of vector adjustment then back to matrices. The information-lost minimization (biproportional methods and RAS) leads to a multiplicative form and generalize the linear model. On the other hand, the distance minimization which leads to an additive form tends to distort the data by giving a result asymptotically independent to the initial matrix. The result allows concluding non-ambiguously that biproportional methods and RAS are the best for matrix adjustment as they generalize the linear model and are asymptotically the most respectful of the initial matrix while the do not generate surprising negative terms. Moreover, measuring the gap between the projection and the target cannot help deciding which method is the best because the gap depends on the target matrix, while this gap can be interpreted in terms of structural effect generalizing the shift-share method.

Updating input–output matrices: assessing alternatives through simulation

A problem that frequently arises in economics, demography, statistics, transportation planning and stochastic modelling is how to adjust the entries of a matrix to fulfil row and column aggregation constraints. Biproportional methods in general and the so-called RAS algorithm in particular, have been used for decades to find solutions to this type of problem. Although alternatives exist, the RAS algorithm and its extensions are still the most popular. Apart from some interesting empirical and theoretical properties, tradition, simplicity and very low computational costs are among the reasons behind the great success of RAS. Nowadays computer hardware and software have made alternative procedures equally attractive. This work analyses, through simulation, the performance of RAS and some minimands when matrix coefficients vary following different schemes of change. Results suggest RAS algorithm as the best option when variations in coefficients are proportional to their size, while the method based on minimizing squared differences is seen to be the best alternative when the standard deviations of variations are either constant, variable, or an inverse function of matrix entries.

Updating an Input–Output Matrix with Sign-preservation: Some Improved Objective Functions and their Solutions

This paper presents an easy-to-use method for updating input–output (IO) matrices with sign-preservation by combining Lagrangian multipliers and penalty functions. Biproportional methods such as the representative RAS are very simple and popular because a target matrix can be obtained simply by iterative computation. However, they cannot reasonably deal with matrices that include negative entries. Although a generalized version, GRAS, can do so, its objective function is questionable. In contrast, some non-biproportional methods such as those that take weighted or unweighted squared differences between the target and original matrix as objective functions can deal with negative entries, but it is difficult to guarantee the signs of entries. In this study, GRAS and some conventional objective functions were improved and their solutions for preserving the signs of entries are presented. Comparisons of applying these objective functions to a simple example show that both the Improved Normalized Squared Differences (INSD) function and the Improved GRAS (IGRAS) function yield a good target matrix and are close to each other; we suggest that INSD or IGRAS be used for updating IO transaction matrices in practice.

Measuring structural change in the I‐O production function by biproportional methods: A theorem of price invariance

Abstract. This research note demonstrates the advantage of biproportional projection over the ordinary (proportional) method of technical coefficients in input‐output analysis: the former is insensitive to price changes, the latter is not so. Consequently, structural change can generally be measured using the biproportional filter on tables in current prices instead of inflation‐adjusted matrices, without changing anything in the result. This may be useful when price information and inflation‐adjusted matrices are unavailable. The result can be applied to matrix updating.

On the idea of ex ante and ex post normalization of biproportional methods

Biproportional methods project a matrixAto give it the column and row sums of another matrix; the result isR A S, whereRandSare diagonal matrices. AsRandSare not identified, one must normalize them, even after computing, that is,ex post. This article starts from the idea developed in de Mesnard (2002) - any normalization amounts to put constraints on Lagrange multipliers, even when it is based on an economic reasoning, - to show that it is impossible to analytically derive the normalized solution at optimum. Convergence must be proved when normalization is applied at each step on the path to equilibrium. To summarize, normalization is impossibleex ante, what removes the possibility of having a certain control on it. It is also indicated that negativity is not a problem.

Normalizing biproportional methods

Biproportional methods are used to update matrices: the projection of a matrix Z to give it the column and row sums of another matrix is R Z S, where R and S are diagonal and secure the constraints of the problem (R and S have no signification at all because they are not identified). However, normalizing R or S generates important mathematical difficulties: it amounts to put constraints on Lagrange multipliers, non negativity (and so the existence of the solution) is not guaranteed at equilibrium or along the path to equilibrium.

An Analysis of Structural Change in China using Biproportional Methods

Traditional IO techniques have been used and applied in detail to the case of the Chinese economy with a view to describing and analysing the gradual, albeit radical, transformation of the Chinese industrial fabric since the beginning of the economic reforms. Using a Biproportional Filter, our study has a threefold objective: to measure and analyse structural change that has taken place in the Chinese manufacturing sector since 1985; to highlight those industries that have been most responsible for the change; and to explain the reasons for these changes.

An empirical study on non-survey forecasting of the input coefficient matrix in a Leontief model

This paper presents a new method of forecasting input coefficients in Leontief's input—output model. For projecting input coefficients with non-survey technique, the author uses two types of biproportional programming methods, RAS and RECRAS. For forecasting input coefficient, this paper proposes the combination of biproportional projection with quadratic programming adjustment. The author has named the combining method the ‘Two-Stage RAS Lagrangian Method’ and the ‘Two-Stage RECRAS Lagrangian Method’. An empirical test to compare the exactness of forecasting is carried out using Japan's input-output tables.