Local approximation of DSGE models around the risky steady state

A DSGE model takes the mathematical form of a system of nonlinear stochastic equations. Except in a very few cases, there is no analytical solution and economists are left using numerical methods in order to obtain approximated solutions. Global approximation methods are available when the state space is not too large, while the most usual approach is local approximation around the deterministic steady state. The perturbation approach introduced in economics by Judd (1996) derives a Taylor expansion of the solution from a Taylor expansion of the original problem, but first order approximation is nothing but linearization that has been used in the RBC literature since its inception. Second order approximations are discussed in several papers: Sims (2000); Collard and Juillard (2001); Kim et al. (2003); Schmitt-Grohe and Uribe (2004). Second order approximations have two merits. In most cases, but not in all, they provide a more accurate approximation of the solution, but, more importantly, they break away from certainty equivalence, that is an inescapable characteristic of linear model. This is crucial to address issues related to attitudes toward risk. There is of course no reason, except size of model, to consider only ?rst or second order approximations. Higher order approximation as also sometimes used: Jin and Judd (2002); Juillard and Kamenik (2004).