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This paper compares different solution methods for computing the equilibrium of dynamic stochastic general equilibrium (DSGE) models with rare disasters along the lines of those proposed by Rietz (1988), Barro (2006), Gabaix (2012), and Gourio (2012). DSGE models with rare disasters require solution methods that can handle the large nonlinearities triggered by low‐probability, high‐impact events with accuracy and speed. We solve a standard New Keynesian model with Epstein–Zin preferences and time‐varying disaster risk with perturbation, Taylor projection, and Smolyak collocation. Our main finding is that Taylor projection delivers the best accuracy/speed tradeoff among the tested solutions. We also document that even third‐order perturbations may generate solutions that suffer from accuracy problems and that Smolyak collocation can be costly in terms of run time and memory requirements.

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This article presents a new solution method for dynamic equilibrium models. The solution is approximated by polynomials that zero the residual function and its derivatives at a given point x0. The algorithm is essentially a type of projection but is significantly faster, since the problem is highly sparse and can be easily solved by a Newton solver. The obtained solution is accurate locally in the neighborhood of x0. Importantly, a local solution can be obtained at any point of the state space. This makes it possible to solve models at points that are further away from the steady state.