Methods of Balancing Matrices¶
In input-output analysis, one frequently needs to balance a table so that a set of initial row values add to known or estimated row sums and columns values add to known or estimated column sums. In G7, there are two methods of achieving this balance – one for matrices in general, and one for entire input-output tables.
For matrices in general, if the row and column sums are known, or if the estimated totals can be accepted, the ordinary RAS method of balancing a matrix should be used. This method of balancing may be used with any of the methods of scaling discussed above.
If all of the elements of the matrix are greater than or equal to zero and if all of the totals are greater than zero, ordinary scaling may be used.
If all of the elements of the matrix are less than or equal to zero and all of the totals are less than zero, ordinary scaling may be used.
If the matrix contains both positive and negative elements, or if the totals contain both positive and negative numbers, proportional scaling (or optionally, right-direction scaling) should be used.
However, if the estimated totals are questionable, or if the ordinary RAS method will not balance the matrix, or if the method forces unacceptably large changes on some cells, the special use of proportional scaling RAS outlined in the section Balancing Matrices with Uncertain Totals might be useful. It allows the questionable row or column sums to change.
For entire input-output tables, the gross value RAS method of balancing a matrix should be used. It also can balance matrices with known or questionable control totals.
