Matrix Algebra and Invertibility Conditions for Linear Dynamic Stochastic General Equilibrium Models

In this paper we analyze a class of state space models, called standard Dynamic Stochastic General Equilibrium (DSGE) models. Then we give necessary and sufficient conditions for the existence of infinite resp. finite order vector moving average (VMA) and vector autoregressive (VAR) representations of such models. Minimality of state space solutions of DSGE models is also discussed, and we show that it is very important in proving the necessary part of our statements. We also provide simple conditions for VARMA representations of DSGE models. The results are proved by using techniques of matrix algebra. Applications on classes of DSGE models currently used in economic analysis complete the paper.