The Spectral Approach to Linear Rational Expectations Models

TL;DR

This paper introduces a spectral frequency domain approach to linear rational expectations models, characterizing solution existence, uniqueness, and continuity, and proposing regularization techniques to address non-uniqueness and ill-posedness.

Contribution

It provides a spectral characterization of solutions, analyzes discontinuities in parameters, and develops regularized methods ensuring well-posedness and differentiability.

Findings

Solutions form a finite dimensional affine space in the frequency domain

Solutions can be discontinuous with respect to model parameters

Regularization guarantees unique, continuous, and differentiable solutions

Abstract

This paper considers linear rational expectations models in the frequency domain. The paper characterizes existence and uniqueness of solutions to particular as well as generic systems. The set of all solutions to a given system is shown to be a finite dimensional affine space in the frequency domain. It is demonstrated that solutions can be discontinuous with respect to the parameters of the models in the context of non-uniqueness, invalidating mainstream frequentist and Bayesian methods. The ill-posedness of the problem motivates regularized solutions with theoretically guaranteed uniqueness, continuity, and even differentiability properties.