A Higher-Order Taylor Expansion Approach to Simulation of Stochastic Forward-Looking Models with an Application to a Nonlinear Phillips Curve Model

Abstract

We propose to apply to the simulation of general nonlinear rational-expectation models a method where the expectation functions are approximated through a higher-order Taylor expansion. This method has been advocated by Judd (1998) and others for the simulation of stochastic optimal-control problems and we extend its application to more general cases. The coefficients for the first-order approximation of the expectation function are obtained using a generalized eigenvalue decomposition as it is usual for the simulation of linear rational-expectation models. Coefficients for higher-order terms in the Taylor expansion are then obtained by solving a succession of linear systems. In addition, we provide a method to reduce a bias in the computation of the stochastic equilibrium of such models. These procedures are made available in DYNARE, a MATLAB and GAUSS based simulation program. This method is then applied to the simulation of a macroeconomic model embodying a nonlinear Phillips curve. We show that in this case a quadratic approximation is sufficient, but different in important ways from the simulation of a linearized version of the model.