Solving SDGE Models: A New Algorithm for the Sylvester Equation

Abstract

This paper presents a new numerical algorithm for solving the Sylvester equation involved in higher-order perturbation methods developed for solving stochastic dynamic general equilibrium models. The new algorithm surpasses other methods used so far (including the very popular doubling algorithm) in terms of computational time, memory consumption, and numerical stability. Copyright Springer Science + Business Media, Inc. 2005stochastic dynamic general equilibrium models, high-order permutations, computational algorithms,

Solution methods for models with rare disasters

“…In this paper, we use the perturbation algorithm presented in Levintal (2017), which allows solving models with non-Gaussian shocks up to the fifth order. We also reduce computational time by adopting the algorithm proposed by Kameník (2005) to solve the Sylvester equation that arises in perturbation methods.…”

Solvability of perturbation solutions in DSGE models

“…12 The Sylvester form in second order context of, e.g., Kim et al (2008) or Gomme and Klein (2011) aside, Juillard and Kamenik (2004) and Kamenik (2005) show explicitly that many of the unknown coefficients of a perturbation of arbitrary order can be cast as Sylvester equations. 13 See, e.g., Uhlig (1999).…”

“…Yet, as Higham (1993, p. 132) notes, the bounds will generally be of similar magnitude and we keep to the standard bound to simplify the analysis. Additionally, Kamenik (2005) presents a computationally more efficient algorithm for Sylvester equations of the form in (17), though the author does not address the issue of error compounding in higher order terms.…”

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

“…In particular they use the Kågström & Poromaa (1996) representation for the generalised Sylvester equations. Kamenik (2005) presents an alternative Sylvester equation representation and solution method that exploits the Kronecker product structure of the problem allowing it to be solved recursively. This results in significant performance improvements over existing solution methods (see Kamenik (2005) for a comparison with other methods of solving generalised Sylvester equations).…”