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The Official Journal of the Pan-Pacific Association of Input-Output Studies (PAPAIOS)

Table 2 Basic transformation models to derive an IOT (\({\mathbf{S}}\) and \({\mathbf{E}}\) or \({\mathbf{B}}\) and \({\mathbf{h}}\)) from a SUT (\({\mathbf{V}}\) and \({\mathbf{U}}\)).

From: Analyzing the effects of the choice of model in the context of marginal changes in final demand

Model Formulas for \({\mathbf{S}}\) or \({\mathbf{B}}\) Formulas for \({\mathbf{E}}\) Formulas for \({\mathbf{h}}\)
A (PTA-p*p) \({\mathbf{S}}^{{\left( {\text{A}} \right)}} = {\mathbf{UV}}^{{ - {\text{T}}}} {\hat{\mathbf{q}}}\) \({\mathbf{E}}^{{\left( {\text{A}} \right)}} = {\mathbf{WV}}^{{ - {\text{T}}}} {\hat{\mathbf{q}}}\)
B (ITA-p*p) \({\mathbf{S}}^{{\left( {\text{B}} \right)}} = {\mathbf{U}}{\hat{\mathbf{g}}}^{ - 1} {\mathbf{V}}\) \({\mathbf{E}}^{{\left( {\text{B}} \right)}} = {\mathbf{W}}{\hat{\mathbf{g}}}^{ - 1} {\mathbf{V}}\)
C (ISA-i*i) \({\mathbf{B}}^{{\left( {\text{C}} \right)}} = {\hat{\mathbf{g}}}{\mathbf{V}}^{{ - {\text{T}}}} {\mathbf{U}}\) \({\mathbf{h}}^{{\left( {\text{C}} \right)}} = {\hat{\mathbf{g}}}{\mathbf{V}}^{{ - {\text{T}}}} {\mathbf{d}}\)
D (PSA-i*i) \({\mathbf{B}}^{{\left( {\text{D}} \right)}} = {\mathbf{V}}{\hat{\mathbf{q}}}^{ - 1} {\mathbf{U}}\) \({\mathbf{h}}^{{\left( {\text{D}} \right)}} = {\mathbf{V}}{\hat{\mathbf{q}}}^{ - 1} {\mathbf{d}}\)
  1. The superscript \(- 1\) indicates inversion; the superscript \({-}{\text{T}}\) indicates transposition and inversion; the hat (\({\hat{\mathbf{x}}}\)) on top of a vector indicates diagonalization into a square matrix