Finance and Economics Discussion Series

Two Practical Algorithms for Solving Rational Expectations Models

Flint Brayton
2011-44

Abstract: This paper describes the E-Newton and E-QNewton algorithms for solving rational expectations (RE) models. Both algorithms treat a model's RE terms as exogenous variables whose values are iteratively updated until they (hopefully) satisfy the RE requirement. In E-Newton, the updates are based on Newton's method; E-QNewton uses an efficient form of Broyden's quasi-Newton method. The paper shows that the algorithms are reliable, fast enough for practical use on a mid-range PC, and simple enough that their implementation does not require highly specialized software. The evaluation of the algorithms is based on experiments with three well-known macro models--the Smets-Wouters (SW) model, EDO, and FRB/US--using code written in EViews, a general-purpose, easy-to-use software package. The models are either linear (SW and EDO) or mildly nonlinear (FRB/US). A test of the robustness of the algorithms in the presence of substantial nonlinearity is based on modified versions of each model that include a smoothed form of the constraint that the short-term rate of interest cannot fall below zero. In two single-simulation experiments with the standard and modified versions of the models, E-QNewton is found to be faster than E-Newton, except for solutions of small-to-medium sized linear models. In a multi-simulation experiment using the standard versions of the models, E-Newton dominates E-QNewton.

Keywords: Solution algorithms, rational expectations