Multi-Step Perturbation Solution of Nonlinear Rational Expectations Models

Recently, perturbation has received attention as a numerical method for computing an approximate solution of a nonlinear dynamic stochastic model, which we call a nonlinear rational expectations (NLRE) model. To date perturbation methods have been described and applied as single-step perturbation (SSP). If a solution of an NDS model is a function f(x) of vector x, then, SSP aims to compute a kth-order Taylor approximation of f(x), centered at x0. In classical SSP, where x0 is a nonstochastic steady state of the dynamical system, a kth-order approximation is accurate on the order of ||dx|| to the power k+1, where dx = x - x0 and ||.|| is a vector norm. Thus, for given k and computed x0, classical SSP is accurate only locally, near x0. SSP's accuracy can be improved only by increasing k, which beyond small values results in large computing costs, especially for deriving kth-order analytical derivatives of the model's equations. So far, research has not fully solved the problem in SSP of maintaining any desired accuracy while freeing x0 from the nonstochastic steady state, so that, for given k, SSP can be arbitrarily accurate for any dx. Multi-step perturbation (MSP) fully solves this problem and, thus, globalizes SSP. In SSP, we approximate d(x) with a single Taylor approximation centered at x0 and, thus, effectively move from x0 to x in one step. In MSP, we move in a straight line from x0 to x in h steps of equal length. At each step, we approximate f at the x at the end of the step with a Taylor approximation centered at the x at the beginning of the step. After h steps and Taylor approximations, we obtain an approximation of f(x) which is accurate on the order of h to the power -k. Thus, although in MSP we also set x0 to a nonstochastic steady state, unlike in SSP, we can achieve any desired accuracy for any x0, x, and k, simply by using sufficiently many steps. Thus, we free the accuracy from dependence on k and ||dx|| and effectively globalize SSP. Whereas increasing k requires new derivations and programming, increasing h requires only passing more times through an already programmed loop, typically at only moderately more computing time. In the paper, we derive an MSP algorithm in standard linear-algebraic notation, for a 4th-order approximation of a general NLRE model, and illustrate the algorithm and its accuracy by applying it to a stochastic one-sector optimal growth model