Implications of Stochastic Singularity in Linear Multivariate Rational Expectations Models

In general, linear multivariate rational expectations models do not have a unique solution. This paper reviews some procedures for determining whether there exists a solution, whether it is unique, and infers on the dimension of indeterminacy and the number of free parameters in a parametrization thereof. A particular emphasis is given to stochastic singularity, i.e. the case in which the number of outputs is strictly larger than the number of (stochastic) inputs. First, it is shown that assuming stochastic singularity of the exogenous driving process has the same effects as (but is more natural than) assuming that some variables are predetermined, i.e have trivial one-step-ahead prediction error. Second, the dimension of the solution set is in general different from the one derived in the case where the number of outputs and inputs coincide. We derive this result in both the framework of [37, 34] (which impose nonexplosiveness conditions) and [9, 11] (which do not impose non-explosiveness conditions). In this context, the results of [34] and [11] are corrected and extended. Last, we note that the framework of [11] can be adjusted to incorporate non-explosiveness conditions and lends itself to an identifiability analysis of dynamic stochastic general equilibrium (DSGE) models.