IDEAS home Printed from
MyIDEAS: Log in (now much improved!) to save this paper

Multi-Step Perturbation Solution of Nonlinear Rational Expectations Models

Listed author(s):
  • Peter Zadrozny

    (Bureau of Labor Statistics, Washington DC)

  • Baoline Chen

    (Bureau of Economic Analysis, Washington DC)

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

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL:
Download Restriction: no

Paper provided by Society for Computational Economics in its series Computing in Economics and Finance 2006 with number 139.

in new window

Date of creation: 04 Jul 2006
Handle: RePEc:sce:scecfa:139
Contact details of provider: Web page:

More information through EDIRC

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

in new window

  1. Baoline Chen & Peter A. Zadrozny, 2003. "Higher-Moments in Perturbation Solution of the Linear-Quadratic Exponential Gaussian Optimal Control Problem," Computational Economics, Springer;Society for Computational Economics, vol. 21(1_2), pages 45-64, February.
  2. Peter A. Zadrozny & Baoline Chen, 1999. "Perturbation Solution of Nonlinear Rational Expectations Models," Computing in Economics and Finance 1999 334, Society for Computational Economics.
  3. Zadrozny, Peter A., 1998. "An eigenvalue method of undetermined coefficients for solving linear rational expectations models," Journal of Economic Dynamics and Control, Elsevier, vol. 22(8-9), pages 1353-1373, August.
Full references (including those not matched with items on IDEAS)

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

When requesting a correction, please mention this item's handle: RePEc:sce:scecfa:139. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Christopher F. Baum)

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.