This paper considers linear rational expectations models in the frequency domain . Two classical results drive the entire theory: the Kolmogorov-Cram\'{e The paper develops necessary and sufficient conditions for existence and uniqueness of particular and generic systems . The space of all solutions is characterized as an affine space in the frequency domain , which sheds light on the variety of solution methods considered in the literature . It is shown that solutions are not generally continuous with respect to the parameters of the models . This motivates regularized solutions with theoretically guaranteed smoothness properties. As an application, the limiting Gaussian likelihood functions of solutions is derived analytically and its properties are studied. The paper finds that non-uniqueness leads to highly irregular likelihood functions and recommends either restricting the parameter space to the region of uniqueness or employing regularization .
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 .
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