Teaching an old dog new tricks: Improved estimation of the parameters of SDEs by numerical solution of the Fokker-Planck equation
AbstractMany stochastic differential equations (SDEs) do not have readily available closed-form expressions for their transitional probability density functions (PDFs). As a result, a large number of competing estimation approaches have been proposed in order to obtain maximum-likelihood estimates of their parameters. Arguably the most straightforward of these is one in which the required estimates of the transitional PDF are obtained by numerical solution of the Fokker-Planck (or forward-Kolmogorov) partial differential equation. Despite the fact that this method produces accurate estimates and is completely generic, it has not proved popular in the applied literature. Perhaps this is attributable to the fact that this approach requires repeated solution of a parabolic partial differential equation to obtain the transitional PDF and is therefore computationally quite expensive. In this paper, three avenues for improving the reliability and speed of this estimation method are introduced and explored in the context of estimating the parameters of the popular Cox-Ingersoll-Ross and Ornstein-Uhlenbeck models. The recommended algorithm that emerges from this investigation is seen to offer substantial gains in reliability and computational time.
Download InfoIf 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.
Bibliographic InfoPaper provided by School of Economics and Finance, Queensland University of Technology in its series Stan Hurn Discussion Papers with number 2006-01.
Date of creation: 15 Jun 2006
Date of revision:
stochastic differential equations; maximum likelihood; finite difference; finite element; cumulative distribution function; interpolation.;
This paper has been announced in the following NEP Reports:
- NEP-ALL-2006-04-01 (All new papers)
- NEP-ECM-2006-04-01 (Econometrics)
- NEP-HRM-2006-04-01 (Human Capital & Human Resource Management)
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.:
- Sundaresan, S.M., 2000. "Continuous-Time Methods in Finance: A Review and an Assessment," Papers 00-03, Columbia - Graduate School of Business.
- Eraker, Bjorn, 2001. "MCMC Analysis of Diffusion Models with Application to Finance," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(2), pages 177-91, April.
- Yacine Ait-Sahalia, 2002. "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach," Econometrica, Econometric Society, vol. 70(1), pages 223-262, January.
- Elerian, O. & Chib, S. & Shephard, N., 1998.
"Likelihood INference for Discretely Observed Non-linear Diffusions,"
146, Economics Group, Nuffield College, University of Oxford.
- Elerain, Ola & Chib, Siddhartha & Shephard, Neil, 2001. "Likelihood Inference for Discretely Observed Nonlinear Diffusions," Econometrica, Econometric Society, vol. 69(4), pages 959-93, July.
- Neil Shephard & Ola Elerian & Siddhartha Chib, 1998. "Likelihood inference for discretely observed non-linear diffusions," Economics Series Working Papers 1998-W10, University of Oxford, Department of Economics.
- Ola Elerian & Siddhartha Chib & Neil Shephard, 2000. "Likelihood inference for discretely observed non-linear diffusions," OFRC Working Papers Series 2000mf02, Oxford Financial Research Centre.
- Hurn, A.S. & Lindsay, K.A., 1999. "Estimating the parameters of stochastic differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 48(4), pages 373-384.
- A. S. Hurn & K. A. Lindsay & V. L. Martin, 2003. "On the efficacy of simulated maximum likelihood for estimating the parameters of stochastic differential Equations," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(1), pages 45-63, 01.
- Chan, K C, et al, 1992.
" An Empirical Comparison of Alternative Models of the Short-Term Interest Rate,"
Journal of Finance,
American Finance Association, vol. 47(3), pages 1209-27, July.
- Tom Doan, . "RATS programs to replicate CKLS(1992) estimation of interest rate models," Statistical Software Components RTZ00035, Boston College Department of Economics.
- Suresh M. Sundaresan, 2000. "Continuous-Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, 08.
- Ait-Sahalia, Yacine, 1996.
"Testing Continuous-Time Models of the Spot Interest Rate,"
Review of Financial Studies,
Society for Financial Studies, vol. 9(2), pages 385-426.
- Yacine Ait-Sahalia, 1995. "Testing Continuous-Time Models of the Spot Interest Rate," NBER Working Papers 5346, National Bureau of Economic Research, Inc.
- Stan Hurn, A. & Lindsay, K.A., 1997.
"Estimating the parameters of stochastic differential equations by Monte Carlo methods,"
Mathematics and Computers in Simulation (MATCOM),
Elsevier, vol. 43(3), pages 495-501.
- Hurn, A.S. & Lindsay, K.A., 1995. "Estimating the Parameters of Stochastic Differential Equations by Monte Carlo Methods," Department of Economics - Working Papers Series 472, The University of Melbourne.
- Jiang, George J & Knight, John L, 2002. "Estimation of Continuous-Time Processes via the Empirical Characteristic Function," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(2), pages 198-212, April.
- Brandt, Michael W. & Santa-Clara, Pedro, 2002. "Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets," Journal of Financial Economics, Elsevier, vol. 63(2), pages 161-210, February.
- Lars Peter Hansen & Jose Alexandre Scheinkman, 1993.
"Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes,"
NBER Technical Working Papers
0141, National Bureau of Economic Research, Inc.
- Hansen, Lars Peter & Scheinkman, Jose Alexandre, 1995. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," Econometrica, Econometric Society, vol. 63(4), pages 767-804, July.
- Singleton, Kenneth J., 2001. "Estimation of affine asset pricing models using the empirical characteristic function," Journal of Econometrics, Elsevier, vol. 102(1), pages 111-141, May.
- repec:cup:etheor:v:12:y:1996:i:4:p:657-81 is not listed on IDEAS
- Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
- Tauchen, George E. & Gallant, A. Ronald, 1995.
"Which Moments to Match,"
95-20, Duke University, Department of Economics.
- Gourieroux, C & Monfort, A & Renault, E, 1993.
Journal of Applied Econometrics,
John Wiley & Sons, Ltd., vol. 8(S), pages S85-118, Suppl. De.
- Durham, Garland B & Gallant, A Ronald, 2002. "Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(3), pages 297-316, July.
- Stan Hurn & J.Jeisman & K.A. Lindsay, 2006.
"Seeing the wood for the trees: A critical evaluation of methods to estimate the parameters of stochastic differential equations,"
Stan Hurn Discussion Papers
2006, School of Economics and Finance, Queensland University of Technology.
- A. S. Hurn & J. I. Jeisman & K. A. Lindsay, 0. "Seeing the Wood for the Trees: A Critical Evaluation of Methods to Estimate the Parameters of Stochastic Differential Equations," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 5(3), pages 390-455.
- Chacko, George & Viceira, Luis M., 2003. "Spectral GMM estimation of continuous-time processes," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 259-292.
- Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, vol. 53(2), pages 385-407, March.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (School of Economics) The email address of this maintainer does not seem to be valid anymore. Please ask School of Economics to update the entry or send us the correct address.
If references are entirely missing, you can add them using this form.