This paper considers the solution of a large class of linear rational expectations (LRE) models cast in state-space form and their solution's characterization via finite-order VARs. Based on the method of undetermined coefficients, I propose a unified approach that uses a companion Sylvester equation to check the existence and uniqueness of a solution to the canonical LRE model in finite-order VAR form and to simplify its characterization. Solving LRE models by this procedure is straightforward

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... to implement, general in its applicability, efficient in the use of computational resources, and can be handled easily with standard matrix algebra. I also explore how to correctly recover theory-consistent monetary policy shocks from observed data. An application to the workhorse New Keynesian model with accompanying Matlab codes is provided to illustrate the practical implementation of the methodology. I argue that empirical evidence on the monetary transmission mechanism and on monetary policy shocks from incorrectly-specified structural VARs (in terms of lags, zero identification restrictions, etc.) should be interpreted carefully as it may not have a proper structural interpretation.