2.4 Stage 2: estimating trade coefficients matrices
For every region, trade coefficients matrices, describing commercial relationships between a
given region and the others, are estimated using a readapted version of a gravity model (Mitchell,
1996), applied to the total trade coefficient matrix.
The hypothesis of the model is that the probability of import flows attraction exerted by a region
is an indirect function of its distance from the import region and a direct function of its ability to
attract import flows. Given regions L and S, the attraction probability of region L relative to import
flows of good i to region S is given by:
pLLS = NXi /(d )2 (k ≠ S)
(9)
∑Xik (djS)2
k=1
where dLS is the geographical distance between export region L and import region S (this is a
straight line distance between the barycentre of the respective regions); X is used as a proxy of the
ability of attracting import flows. It is assumed that import flows of a given good (or service),
whatever import sector is, are mostly attracted (or rather produced and exported) by regions with high
levels of output in the relevant sector. Output has a greater importance than the distance factor, which
is squared just to reduce its effects on the attraction probability.
For a given region S, trade coefficients matrices are derived as follows:
A ls = p ls A ʌ ( L = 1,2,., N ∧ L ≠ S )
(10)
where pls =(PLS,pLpL).
2.5 Stage 3: balancing the multiregional I-O matrix and deriving multipliers
The input and trade coefficients matrices form the 13 sector x 8 region I-O matrix described in
system (1). The matrix is then converted into flows multiplying coefficients by output data. As is
logical to expect, the multiregional table presents internal inconsistencies and is not coherent with the
national I-O table. Therefore we proceed to balance the multiregional I-O table. Balancing of an I-O
table, which is a more general problem than updating or estimating, is a technical matter frequently
faced by I-O analysts (Bulmer-Thomas, 1982; Canning and Wang, 2004; Jackson and Murray, 2004).
Matrix-balancing techniques can be roughly classified into two categories: scaling algorithms (like
RAS) and optimization techniques. The former iteratively multiply rows and columns of a prior matrix
by positive constants until reaching a solution matrix. This approach requires at least the knowledge of
row and column totals. The latter minimise an objective (or penalty) function measuring the distance
between the elements of a matrix to be balanced and the elements of the objective matrix, under some
constraints which impose accounting identities and/or introduce exogenous information. This
approach gives the analyst more flexibility in balancing matrices since the quantity and the kind of
prior information can vary. However, empirical evidence showed that the RAS technique performs
better than several formulations of optimization problems when a priori information concerns row and
columns totals (Jackson and Murray, 2002).
In this research, given the kind of prior information, an optimization technique, based on the
Pearson χ2 (or normalized square of differences) penalty function (Friedlander, 1961), is used7. The
objective function takes the following form:
7 The analyst is free to choose the penalty function which feels to be the most appropriate to the aims of the research.