What is the Systems Form?
TCA makes use of two forms (1) the dynamic form and (2) the systems form. In this article we describe what the systems form is.
Fixing a finite horizon \(h\) and defining \(\mathbf{x}=(\mathbf{y}_t^{*'}, \dots, \mathbf{y}_{t+h}^{*'})'\) and \(\boldsymbol{\varepsilon}=(\boldsymbol{\varepsilon}_t', \dots, \boldsymbol{\varepsilon}_{t+h}')'\), Wegner et al. (2025) show that the dynamic form can be written into the following systems form \[ \mathbf{x}= \mathbf{B}\mathbf{x}+ \boldsymbol{\Omega}\boldsymbol{\varepsilon}, \] where \(\mathbf{B}\) is lower-triangular with zeros on the diagonal, \(\boldsymbol{\Omega}\) is lower-block-triangular, and both are respectively given by1 \[ \begin{array}{ccc} \mathbf{B}= \begin{bmatrix} \mathbf{I}- \mathbf{D}\mathbf{L}& \mathbf{O}& \dots & \mathbf{O}\\ \mathbf{D}\mathbf{Q}'\mathbf{A}^*_1 & \mathbf{I}-\mathbf{D}\mathbf{L}& \dots & \mathbf{O}\\ \vdots & \ddots & \ddots & \vdots \\ \mathbf{D}\mathbf{Q}'\mathbf{A}^*_h & \dots & \mathbf{D}\mathbf{Q}'\mathbf{A}^*_1 & \mathbf{I}- \mathbf{D}\mathbf{L} \end{bmatrix}, & \quad & \boldsymbol{\Omega}= \begin{bmatrix} \mathbf{D}\mathbf{Q}' & \mathbf{O}& \dots & \mathbf{O}\\ \mathbf{D}\mathbf{Q}'\boldsymbol{\Psi}_1 & \mathbf{D}\mathbf{Q}' & \dots & \mathbf{O}\\ \vdots & \ddots & \ddots & \vdots \\ \mathbf{D}\mathbf{Q}'\boldsymbol{\Psi}_h & \dots & \mathbf{D}\mathbf{Q}'\boldsymbol{\Psi}_1 & \mathbf{D}\mathbf{Q}' \end{bmatrix}, \end{array} \]
This system form can either be used together with potential outcomes to define transmission channels, or can be used to build an associated graph in which transmission channels are much more intuitively defined.
More information on the systems form and the detailed steps used to move from the dynamic to the systems form can be found in Section 3 and the Appendix of Wegner et al. (2025).
References
Footnotes
For expositional ease we ommitted the dimension subscripts, letting \(\mathbf{I}= \mathbf{I}_K\) and \(\mathbf{O}= \mathbf{O}_K\)↩︎