The Spectral Approach to Linear Rational Expectations Models
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This paper considers linear rational expectations models in the frequency domain. Two classical results drive the entire theory: the Kolmogorov-Cramér spectral representation theorem and Wiener-Hopf factorization. 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.
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