# SFB 303 Discussion Paper No. B - 97

**Author**: Mohr, Michael

**Title**: Uniqueness and Stability of Stationary Rational Expectations Equilibria

**Abstract**: This paper treats a linear stochastic model which is widely used in econometrics and economic theory. Its
dynamics is caused by rational expectations of the (K+1) future values of the endogenous process y. These
expectations are based on past values of some input process x, which (in the first part) is supposed to be ARMA,
but does not necessarily coincide with the exogenous process u.In order to investigate the properties of the
solutions to this model (the rational expectations equilibria) it is useful to distinguish between two cases: In the
first one the entire history of x is available to the agents in forming their expectations (this is the conventional
case of unbounded memory), in the second one only the last part of it is used (bounded memory case).One
distressing aspect in the analysis of such models is the multiplicity of solutions.The problem becomes more
tractable if we restrict the set of solutions by the requirement of stationarity.It turns out that existence and
uniqueness of stationary solutions depend on the magnitude of the weight with which the expectations occur. We
may think of these weights as a measure of dependence of the endogenous process y upon the history of the
input process x.If, roughly speaking, the input process does not play a prominent part among the determinants of
the endogenous process, the stationary solutions is uniquely determined (Theorem 1).Furthermore the stationary
solution of the model with unbounded memory is shown to be the mean square limit of the sequence of
stationary solutions of models with bounded memory if the memory order increases (Theorem 2). Hence the
model with unbounded memory may be interpreted as a limit model.

**Keywords**:

**JEL-Classification-Number**:

**Creation-Date**: September 1988

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