Solving Second and Third-Order Approximations to DSGE Models: A Recursive Sylvester Equation Solution

Abstract

In this paper I derive the matrix chain rules for solving a second and a third-order approximation to a DSGE model that allow the use of a recursive Sylvester equation solution method. In particular I use the solution algorithms of Kamenik (2005) and Martin & Van Loan (2006) to solve the generalized Sylvester equations. Because I use matrix algebra instead of tensor notation to find the system of equations, I am able to provide standalone Matlab routines that make it feasible to solve a medium scale DSGE model in a competitive time. I also provide Fortran code and Matlab/Fortran mex files for my method.

Keywords: Solving dynamic models, Second-order approximation, Third-order approximation, Second-order matrix chain rule, Third-order matrix chain rule, Generalised Sylvester equations