Aggregation Matrix

Definition

Aggregation Matrix in the context of Input-Output Analysis is a Boolean Matrix that aims to produce a coarse-grained version of a more granular Input-Output Model.

Aggregation can be for example along sectoral or regional dimensions.

Vectors and Matrices can be aggregated by multiplying with the aggregation matrix. Mathematically a aggregation matrix T is a $K \times N$ matrix, where each value $T^{mn}$ is either zero or one.

$T=\left(\begin{matrix} T^{00} & T^{01} & \dots &T^{0n} & \dots & T^{0N} \\ T^{10} & T^{11} & \dots &T^{1n} & \dots & T^{1N} \\ \vdots & \vdots & \ddots &\vdots & \ddots & \vdots \\ T^{m0} & T^{m1} & \dots &T^{mn} & \dots & T^{mN} \\ \vdots & \vdots & \ddots & \vdots& \ddots & \vdots \\ T^{K0} & T^{K1} & \dots & T^{Kn} & \dots & T^{KN}\\ \end{matrix}\right).$

Vector Quantity Aggregation

A vector

$\mathbf{y}_{s} = \mathbf{S} \mathbf{y}$

$\mathbf{A}_{s} = \mathbf{S} \mathbf{A} \mathbf{S}^T$

See Also

• Aggregation Bias

Further Resources

• Crash Course on Input-Output Model Mathematics
• Introduction to Input-Output Models using Python