What is the Dynamic Form?

TCA makes use of two forms (1) the dynamic form and (2) the systems form. In this article we describe what the dynamic form is.

Wegner et al. (2025) refer to the following structural VARMA form as the dynamic form, \[ \begin{equation} \mathbf{A}_0\mathbf{y}_t = \sum_{i=1}^{\ell}\mathbf{A}_i\mathbf{y}_{t-i} + \sum_{j=1}^{q}\boldsymbol{\Psi}_{j}\boldsymbol{\varepsilon}_{t-j} + \boldsymbol{\varepsilon}_t , \label{eq:general-model} \end{equation} \] where \(\{\mathbf{A}_i\}_{i=1}^{\ell}\) and \(\{\boldsymbol{\Psi}_j\}_{j=1}^{q}\) are \(K\times K\) coefficient matrices which are statistically identified using any common scheme such as the echelon form, \(\mathbf{A}_0\) is a contemporaneous coefficient matrix assumed to be (partially) identified using some economic identification scheme, and \(\boldsymbol{\varepsilon}_t\) is a \(K\times 1\) vector of white noise.

Additionally, two assumptions are made

Assumption 1

The white noise vector \(\boldsymbol{\varepsilon}_t\) consists of \(K\) structural shocks satisfying \(\mathbb{E}[\boldsymbol{\varepsilon}_{t}]=\mathbf{0}_K\), \(\mathbb{E}[\boldsymbol{\varepsilon}_{t}\boldsymbol{\varepsilon}_{t}']=\mathbf{I}_K\) and \(\mathbb{E}[\boldsymbol{\varepsilon}_t\boldsymbol{\varepsilon}_{t-r}']=\mathbf{O}_K\) for all \(r\geq 1\).

Assumption 2

\(\mathbf{A}_0\) is non-singular.

The first step towards a definition of transmission channel is then the transformation of the dynamic form to the systems form.

Where can I find more information?

More information can be found in Section 3 of Wegner et al. (2025).

References

Wegner, Enrico, Lenard Lieb, Stephan Smeekes, and Ines Wilms. 2025. “Transmission Channel Analysis in Dynamic Models.” 2025. https://doi.org/10.48550/arxiv.2405.18987.