An eigenvalue method of undetermined coefficients for solving linear rational expectations models

Abstract

Abstract Consider a structural linear rational expectations model of the form ∑ i =0 p ∑ j =− r i A i , j E t − i y t − j =∑ i =0 q B i e t − i + z t , where y t is an m ×1 vector of endogenous variables, e t ∼ IID (0, Σ e ) is an m ×1 vector of structural disturbances, and z t is an n ×1 vector of exogenous variables. Under conditions stated in the paper, the model has the unique reduced-form solution, y t =∑ i =1 p Φ i y t − i + ξ t +∑ i =1 q Θ i ξ t − i +∑ i =0 ∞ Ξ i z t + i , where ξ t ∼ IID (0, Σ ξ ) is an m ×1 vector of reduced-form disturbances. The paper derives an analytical eigenvalue method for computing the autoregressive, moving-average, and exogenous parts of the reduced-form solution of a model in the above form. The method routinely handles expectations conditional on several different dates and singular leading coefficient matrices and produces an invertible moving-average part. The paper also derives four new theorems on the existence of a unique solution.