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Title: Multi-step perturbation solution of stochastic nonlinear rational expectations models Authors:  Peter Zadrozny - Bureau of Labor Statistics (United States) [presenting]
Baoline Chen - Bureau of Economic Analysis (United States)
Abstract: Stochastic nonlinear rational expectations models (SNREM) are standard in macroeconomic analysis. Standard single-step perturbation (SSP) has attracted attention as a quick method for solving SNREMs with interior solutions and equations differentiable any desired number of times. SSP is a Taylor approximation of order $k$ of an unknown solution function evaluated at point $x$ and centered at point $x_0$, which is easily and accurately computed as a steady-state point. However, SSP has only local accuracy of order $|x - x_0|$ of $(k+1)$ that can be improved by increasing k at increasing costs of deriving and programming. SSP solutions are often insufficiently accurate. SSP moves from $x_0$ to $x$ in one big step; MSP extends SSP by moving recursively from $x_0$ to $x$ in many ($h$) small steps. Taylors theorem implies MSP has global accuracy of order $h$ of $-k$, so that given $x_0$ and $k$, MSPs accuracy can be improved simply by increasing $h$ and passing more times through an already programmed loop. 2-dimensional representations of matrix derivatives are used, which are more easily derived, comprehended, and programmed as conventional matrix equations. Equations for computing 4th-order MSP solutions for general SNREMs are derived and the method is applied to a one-sector optimal-growth model.