Solution of Multivariate Linear Rational Expectations Models and Large Sparse Linear Systems

This paper establishes a link between the problem of solving multivariate linear rational expectations models and the problem of solving large sparse linear systems with a block-tridiagonal matrix coefficient structure. Such large linear systems arise in a wide variety of scientific problems, including the numerical solution of certain classes of partial differential equations, linear-quadratic optimal control problems, and Gaussian optimal filtering problems. Two numerical schemes that allow efficient solution of large sparse linear systems with a block-tridiagonal matrix coefficient structure are presented, and it is shown how these procedures can be readily adapted to solve multivariate linear rational expectations models. Furthermore, the solution of multivariate linear rational expectations models by means of solving large sparse linear systems is linked to the fully recursive method for the solution of multivariate linear rational expectations models recently advanced in Binder and Pesaran (1996). Finally, the numerical schemes are illustrated by applying them to obtain the solution of a simple stochastic growth model.