Perturbation Solution of Nonlinear Rational Expectations Models

The paper derives and illustrates a convenient implementation of a perturbation method for computing an approximate perfect foresight solution of a nonlinear rational-expectations model. The solution space is the set of finite Taylor-series approximations. In discussing this setting, Gaspar and Judd (1997) wrote the higher-order Taylor terms in an algebraically unstructured form. Choosing an algebraic structure for the Taylor terms imparts a related structure to the solution equations, which facilitates understanding and design of a computer program. This paper contributes by providing such a structure. By generalizing a "good definition" of matrix derivatives (Magnus and Neudecker, 1988) to higher-order arrays, the paper shows how to derive structured solution equations for any model and any order of Taylor approximation in terms of conventional linear algebraic operations on vectors and matrices. The computations involve three major steps: (i) solve a nonlinear vector equation to obtain the constant term of the solution; (ii) solve a matrix polynomial equation to obtain the linear term of the solution; (iii) solve K-2 systems of linear equations to obtain remaining higher-order terms of a K -term solution. The computations are illustrated with a reduction of Taylor's (1993) G7-country macroeconomic model. Step (ii) is implemented with an eigenvalue method for solving a matrix polynomial equation (Zadrozny, 1998).