The Spectral Approach to Linear Rational Expectations Models

This paper considers linear rational expectations models in the frequency domain under general conditions. The paper develops necessary and sufficient conditions for existence and uniqueness of particular and generic systems and characterizes the space of all solutions as an affine space in the frequency domain. It is demonstrated that solutions are not generally continuous with respect to the parameters of the models, invalidating mainstream frequentist and Bayesian methods. The ill-posedness of the problem motivates regularized solutions with theoretically guaranteed uniqueness, continuity, and even differentiability properties. Regularization is illustrated in an analysis of the limiting Gaussian likelihood functions of two analytically tractable models.