# Multi-Step Perturbation Solution of Nonlinear Rational Expectations Models

• Baoline Chen

## Abstract

This paper develops and illustrates the multi-step generalization of the standard single-step perturbation (SSP) method or MSP. In SSP, we can think of evaluating at x the computed approximate solution based on x0, as moving from x0 to x in "one big step" along the straight-line vector x-x0. By contrast, in MSP we move from x0 to x along any chosen path, continuous, curved-line or connected-straight-line, in h steps of equal length 1/h. If at each step we apply SSP, Taylor-series theory says that the approximation error per step is 0(e) = h^(-k-1), so that the total approximation error in moving from x0 to x in h steps is 0(e) = h^(-k). Thus, MSP has two major advantages over SSP. First, both SSP and MSP accuracy declines as the approximation point, x, moves from the initial point, x0, although only in MSP can the decline be countered by increasing h. Increasing k is much more costly than increasing h, because increasing k requires new derivations of derivatives, more computer programming, more computer storage, and more computer run time. By contrast, increasing h generally requires only more computer run time and often only slightly more. Second, in SSP the initial point is usually a nonstochastic steady state but can sometimes also be set up in function space as the known exact solution of a close but simpler model. This "closeness" of a related, simpler, and known solution can be exploited much more explicitly by MSP, when moving from x0 to x. In MSP, the state space could include parameters, so that the initial point, x0, would represent the simpler model with the known solution, and the final point, x, would continue to represent the model of interest. Then, as we would move from the initial x0 to the final x in h steps, the state variables and parameters would move together from their initial to final values and the model being solved would vary continuously from the simple model to the model of interest. Both advantages of MSP facilitate repeatedly, accurately, and quickly solving a NLRE model in an econometric analysis, over a range of data values, which could differ enough from nonstochastic steady states of the model of interest to render computed SSP solutions, for a given k, inadequately accurate. In the present paper, we extend the derivation of SSP to MSP for k = 4. As we did before, we use a mixture of gradient and differential-form differentiations to derive the MSP computational equations in conventional linear-algebraic form and illustrate them with a version of the stochastic optimal one-sector 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: http://repec.org/sce2005/up.25973.1107147107.pdf

## Bibliographic Info

Paper provided by Society for Computational Economics in its series Computing in Economics and Finance 2005 with number 254.

as in new window
Length:
Date of revision:
Handle: RePEc:sce:scecf5:254

Contact details of provider:
Email:
Web page: http://comp-econ.org/

## Related research

Keywords: numerical solution of dynamic stochastic equilibrium models;

Other versions of this item:

Find related papers by JEL classification:
• C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
• C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
• C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

This paper has been announced in the following NEP Reports:

## References

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.:
as 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, Society for Computational Economics, vol. 21(1_2), pages 45-64, 02.
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)

## Lists

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

## Corrections

When requesting a correction, please mention this item's handle: RePEc:sce:scecf5:254. 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.