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 generalised 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.

The Pruned State-Space System for Non-Linear DSGE Models: Theory and Empirical Applications

“…This three-step procedure is formally described in Appendix A.15 and constitutes a new numerical contribution to the literature. Our three-step procedure allows us to compute a third-order solution to our model in just 3.7 seconds, whereas it takes 6.2 seconds when using the standard one-step perturbation algorithm of Binning (2013). This 40% improvement in speed greatly facilitates the estimation, as the perturbation approximation must be computed for many different parameter values.…”

Why Does Risk Matter More in Recessions than in Expansions?

“…The conditions in (12) to (15) are therefore easy to implement from existing results and computer packages on the standard perturbation method. In our case, we modify the highly e¢ cient Matlab codes of Binning (2013). estimated versions of the New Keynesian model presented below in Table 4 are considered for this exercise, where the states are simulated using a standard third-order perturbation approximation, i.e., by a Taylor approximation around the deterministic steady state.…”

Recursive blocked algorithms for linear systems with Kronecker product structure

“…Linear systems featuring such shifted Kronecker products have been discussed in [19]. The more general case (1.5) arises from approximations of discrete time DSGE models [3], which play a central role in macroeconomics. Recent work on the solution of linear tensor equations (1.3) has focused on the development of highly efficient approximate and iterative solvers that assume and exploit low-rank tensor structure in the right-hand side and the solution; see [9,10] for overviews.…”

“…The more general case (1.5) arises from approximations of discrete time DSGE models [3], which play a central role in macroeconomics.…”