Comment on: Ravenna, F., 2007. Vector autoregressions and reduced form representations of DSGE models. Journal of Monetary Economics 54, 2048-2064

Solutions of DSGE models are usually represented by state space forms. This note shows that if one wishes to determine whether the observables of the model admit a finite order VAR representation, minimality of the state space representation of the solution matters. More specifically, we first provide a counterexample to Proposition 2.1 and Corollary 2.2 in Ravenna (2007), which state that in the square casea finite order VAR exists if and only if a `unimodularity condition' holds. Our counterexample shows that the proposed condition is not necessary for the existence of a finite order VAR representation. That is, if the state space representation of the solution is non-minimal, the observables of the DSGE may admit a finite order VAR representation even though the unimodularity condition fails. It is further shown that if the state space representation of the solution is minimal, then the unimodularity condition is necessary. Given that a minimal state space representation always exists, before applying the unimodularity condition one simply needs to check whether the state space representation of the solution is minimal and if not transform it into an equivalent minimal form. A discussion of how to perform such reduction is presented and further it is shown that the economic interpretation of the system is not affected by this transformation. An interpretation of the results in terms of the eigenvalues of the matrix defined in Fernandez-Villaverde et al. (2007) for the poor man's invertibility condition is also provided. The analysis is then applied to the Smets and Wouters (2007) model.