It is common practice to use reduced-form vector autoregression (VAR) models, or more generally vector autoregressive moving average (VARMA) models, to characterize the dynamics in observed data and to identify innovations to the macroeconomy in some economically meaningful way. We demonstrate that neither approach|VAR or VARMA|are suitable reduced form guides to \reality", if reality were induced by some underlying structural DSGE model. We conduct such a thought experiment across a wide class of DSGE structures that imply particular VARMA data generating processes (DGPs). We find that with the typical small samples for macroeconomic data, the MA component of the fitted VARMA models is close to being non-identified. This in turn leads to an order reduction when identifying the lag structures of the VARMA models. As a result, VARMA models barely show any advantage over VARs using realistic sample sizes. However, the VAR remains a truly misspecified approximation. The VAR's performance deteriorates, in contrast to the VARMA's, as we enlarge the sample size generated from the true DGPs
