Some Comments on the GRAS Method

Junius and Oosterhaven (2003) Junius, T. and Oosterhaven, J. 2003. The solution of updating or regionalizing a matrix with both positive and negative entries. Economic Systems Research, 15: 87–96. [Taylor & Francis Online] [Google Scholar] present a RAS matrix balancing variant that can incorporate negative elements in the balancing. There are, however, a couple of issues in the approach described – the first being the handling of zeros in the initial estimate, and the second being the formulation of their minimum-information principle. We present a corrected exposition of GRAS.     

A NOTE ON THE GRAS METHOD

The GRAS method as presented by Junius and Oosterhaven [Junius, T. and J. Oosterhaven (2003) The Solution of Updating or Regionalizing a Matrix with Both Positive and Negative Elements. Economic Systems Research, 15, 87–96] assumes that every row and every column of a matrix to be balanced has at least one positive element. This might not necessarily be true in practice, in particular, when dealing with large-scale input–ouput tables, supply and use tables, social accounting matrices, or, for that matter, any other matrix. In this short note we relax this assumption and make available our MATLAB program for anyone interested in matrix GRASing. The same issue arises in the presentations of the KRAS method [Lenzen, M., B. Gallego and R. Wood (2009) Matrix Balancing Under Conflicting Information. Economic Systems Research, 21, 23–44] and the SUT–RAS method [Temurshoev, U. and M.P. Timmer (2011) Joint Estimation of Supply and Use Tables. Papers in Regional Science, 90, 863–882], which should be accordingly accounted for in their empirical applications.     

A GRAS VARIANT SOLVING FOR MINIMUM INFORMATION LOSS

The fundamental idea in Junius and Oosterhaven (2003) Junius, T. and Oosterhaven, J. 2003. The Solution of Updating or Regionalizing a Matrix with both Positive and Negative Entries. Economic Systems Research, 15: 87–96. [Taylor & Francis Online] [Google Scholar] is to break down the information contained in the a priori data into two parts: algebraic signs, and absolute values. This approach is well grounded in information theory, and provides a basis on which to solve the problem of adjusting matrices with negative entries. However, Junius and Oosterhaven (2003) Junius, T. and Oosterhaven, J. 2003. The Solution of Updating or Regionalizing a Matrix with both Positive and Negative Entries. Economic Systems Research, 15: 87–96. [Taylor & Francis Online] [Google Scholar] have formulated a target function that is not equivalent to the Kullback and Leibler (1951) Kullback, S. and Leibler, R. A. 1951. On Information and Sufficiency. Annals of Mathematical Statistics, 4: 99–111.  [Google Scholar] cross-entropy measure, and so is not a representation of the minimum information loss principle. Neither is the alternative target function proposed by Lenzen et al. (2007) Lenzen, M., Wood, R. and Gallego, B. 2007. Some Comments on the GRAS Method. Economic Systems Research, 19: 461–465. [Taylor & Francis Online] [Google Scholar]. This paper develops the exact Kullback and Leibler cross-entropy measure. In addition, following the constrained optimization approach, this paper applies the same principle to solve adjustment problems where row-sums, column-sums or both are constrained to zero.