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...

I outline a new method for finding third-order accurate solutions to dynamic general equilibrium models. I extend the Gomme & Klein (2011) solution for second-order approximations without using tensors, to a third-order. In particular I derive a third-order matrix chain rule and use this to...

Several approaches to finding the second-order approximation to a dynamic model have been proposed recently. This paper differs from the existing literature in that it makes use of the Magnus and Neudecker (1999) definition of the Hessian matrix. The key result is a linear system of equations...