At the outset, the 28 × 28 national and regional transactions matrices were transformed into matrices of coefficients. The national coefficient matrix was then via the ‘regionalized’ following formula: = × (2) rij aij βij where is the regional input coefficient, is an adjustment coefficient and is the rij aij βij national input coefficient. measures the amount of regional input needed to produce one rij i unit of regional gross output it thus excludes any supplies of from other j; i ‘imported’ regions or obtained from abroad. likewise excludes any supplies of obtained from aij i abroad. The role of is to take account of a region’s purchases of input from other i βij regions. If we replace in equation (2) with an LQ, we can obtain estimates of the Thus, for rij. βij instance: = × (3) SLQi aij Another possibility is to replace with CILQij, which is defined as follows: βij (4) CILQij where the subscripts and refer to the supplying and purchasing sectors, respectively. No i j adjustment is made to the national coefficient where 1 and likewise for The CILQij ≥ SLQi. CILQ has the merit that a different scaling can be applied to different cells in a given row of the national coefficient matrix. Unlike the SLQ, the CILQ does not presuppose that a purchasing sector is either an exporter or an importer of a given commodity but never both. Nonetheless, for reasons alluded to earlier, the authors would recommend the use of the FLQ, which is defined as follows: for (5) FLQij CILQij × i j ≡ λ* ≠ for = (6) FLQij SLQi × i j ≡ λ*
At the outset, the 28 × 28 national and regional transactions matrices were transformed into matrices of coefficients. The national coefficient matrix was then ‘regionalized’ via the following formula: rij = βij × aij (2) where rij is the regional input coefficient, βij is an adjustment coefficient and aij is the national input coefficient. rij measures the amount of regional input i needed to produce one unit of regional gross output j; it thus excludes any supplies of i ‘imported’ from other regions or obtained from abroad. aij likewise excludes any supplies of i obtained from abroad. The role of βij is to take account of a region’s purchases of input i from other regions. If we replace βij in equation (2) with an LQ, we can obtain estimates of the rij. Thus, for instance: = SLQi × aij (3) Another possibility is to replace βij with CILQij, which is defined as follows: CILQij (4) where the subscripts i and j refer to the supplying and purchasing sectors, respectively. No adjustment is made to the national coefficient where CILQij ≥ 1 and likewise for SLQi. The CILQ has the merit that a different scaling can be applied to different cells in a given row of the national coefficient matrix. Unlike the SLQ, the CILQ does not presuppose that a purchasing sector is either an exporter or an importer of a given commodity but never both. Nonetheless, for reasons alluded to earlier, the authors would recommend the use of the FLQ, which is defined as follows: FLQij ≡ CILQij × λ* for i ≠ j (5) FLQij ≡ SLQi × λ* for i = j (6)