• structural VAR models
• non-fundamentalness
• dynamic identification
• structural VARMA models
• non-Gaussian and independent shocks

Small-scale SVAR models impose tight restrictions on the dynamics of the process under estimation, in particular by requiring that the structural shocks are "fundamental", i.e. recoverable as a linear combination of present and past values of the variables included in the model. As a consequence, SVAR models discard a priori all the possible non-fundamental representations of the process, potentially leading to wrong identification of structural shocks, parameters and impulse response functions. After a careful introduction to the econometric problem of non-fundamentalness and the review of many examples where the problem arises from the economic theory, this work unifies the approaches proposed so far to address the problem of dynamic identification. We show that a solution exists under the assumptions that the shocks are non-Gaussian and independent. Under these conditions, it is possible to study whether a VAR model has non-fundamental shocks, by testing the higher-order statistical properties of its reduced-form residuals, and also to explicitly estimate the underlying non-fundamental structural VARMA process in a data-driven fashion. Since non-fundamentalness may either arise from omitted variables in the model or from the underlying properties of the process under estimation, we propose a simple procedure consisting in recursively testing the fundamentalness regime of the SVAR, by adding new information at each step through factor augmentation and, if necessary, turning to direct estimation of the SVARMA model. Throughout the discussion, we provide several examples and use simulated data to show the potential of this approach for achieving dynamic identification and correctly estimating the impulse response functions. An application of the procedure using the FRED-MD dataset is also provided.