Third-Order Approximation of Dynamic Models Without the Use of Tensors

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 solve the third-order approximation. My solution method is easier to understand and code-up, and faster to implement in Matlab. I provide Matlab code and demonstrate my solution method with a simple RBC model. The resulting code is up to 80 times faster than Matlab code using tensor notation.