This algorithm can be used to solve a broad class of DSGE models with accuracy even when far from the steady state. Developed with Serhiy Stepanchuk and Juan R. Ramirez.
The file contains a read me file and an example ready to be run. See paragraph 5.1 and Appendix B in the paper for more details on the algorithm.
To find the equilibrium path of a DSGE model given initial conditions for the state variables, call them xo, and time series for the shocks.
A path from xo to the steady state is drawn through the policy functions obtained by perturbation around the steady state. Then, new perturbations are computed backward along this path: from the proximity to the steady state back to the initial conditions. The policies approximated at the initial conditions are used to compute the next point, x1. To compute x2, a new path is drawn through the steady state policies, treating x1 as initial conditions. New perturbations are computed along this path from the proximity to the steady state till x1. The policies approximated at x1 are used to compute the next point x2. Then, the algorithm iterates until initial conditions coincide with the final period of the time series of the shocks. In practice, for all the applications tried so far, this method has proved to be far more accurate than 2nd order perturbation of the steady state.