Improved Estimation Strategy in Multi-Factor Vasicek Model

  • S. Ejaz Ahmed
  • Sévérien Nkurunziza
  • Shuangzhe Liu


We consider simultaneous estimation of the drift parameters of multivari-ate Ornstein-Uhlebeck process. In this paper, we develop an improved estimation methodology for the drift parameters when homogeneity of several such parameters may hold. However, it is possible that the information regarding the equality of these parameters may not be accurate. In this context, we consider Stein-rule (or shrinkage) estimators to improve upon the performance of the classical maximum likelihood estimator (MLE). The relative dominance picture of the proposed estimators are explored and assessed under an asymptotic distributional quadratic risk criterion. For practical arguments, a simulation study is conducted which illustrates the behavior of the suggested method for small and moderate length of time observation period. More importantly, both analytical and simulation results indicate that estimators based on shrinkage principle not only give an excellent estimation accuracy but outperform the likelihood estimation uniformly.


Interest Rate Maximum Likelihood Estimator Wiener Process Term Structure Process Capability Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abu-Mostafa, Y.S.: Financial Model Calibration Using Consistency Hints. IEEE Trans. Neural Netw. 12, 791–808 (2001)CrossRefGoogle Scholar
  2. 2.
    Ahmed, S.E.: Large-sample Pooling Procedure for Correlation. Statist. 41, 425–438 (1992)CrossRefGoogle Scholar
  3. 3.
    Ahmed, S.E., Saleh, A.K.Md.E.: Improved Nonparametric Estimation of Location Vector in a Multivariate Regression Model. Nonpar. Stat. 11, 51–78 (1999)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ahmed, S.E.: Shrinkage Estimation of Regression Coefficients from Censored Data with Multiple Observations. In: Ahmed, S.E., Reid, N. (eds.) Empirical Bayes and and Likelihood inference, pp. 103–120. Springer, New York (2001)Google Scholar
  5. 5.
    Ahmed, S.E.: . Assessing Process Capability Index for Nonnormal Processes. J. Stat. Plan. Infer. 129, 195–206 (2005)MATHCrossRefGoogle Scholar
  6. 6.
    Basawa, V.I.; Prakasa Rao, B.L.S.: Statistical Inference for Stochastic Processes. Academic, London (1980)MATHGoogle Scholar
  7. 7.
    Dacunha-Castelle, D., Florens-Zmirou, D.: Estimation of the Coefficients of a Diffusion from Discrete Observations. Stochastics 19, 263–284 (1986)MATHMathSciNetGoogle Scholar
  8. 8.
    Engen, S., Sæther, B.E.: Generalizations of the Moran Effect Explaining Spatial Synchrony in Population Fluctuations. Am. Nat. 166, 603–612 (2005)CrossRefGoogle Scholar
  9. 9.
    Engen, S., Lande, R., Wall, T., DeVries, J.P.: Analyzing Spatial Structure of Communities Using the Two-Dimensional Poisson Lognormal Species Abundance Model. Am. Nat. 160, 60–73 (2002)CrossRefGoogle Scholar
  10. 10.
    Florens-Zmirou, D.: Approximation Discrete-time Schemes for Statistics of Diffusion Processes. Statistics 20, 547–557 (1989)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Froda, S., Nkurunziza, S.: Prediction of Predator-prey Populations Modelled by Perturbed ODE. J. Math. Biol. 54, 407–451 (2007)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Georges, P.: The Vasicek and CIR Models and the Expectation Hypothesis of the Interest Rate Term Structure. The Bank of Canada and Department of Finance, Canada (2003)Google Scholar
  13. 13.
    Judge, G.G., Bock, M.E.: The Statistical Implications of Pre-test and Stein-rule Estimators in Econometrics. North Holland, Amsterdam (1978)MATHGoogle Scholar
  14. 14.
    Jurečková, J., Sen, P.K.: Robust Statistical Procedures. Wiley, New York (1996)MATHGoogle Scholar
  15. 15.
    Kubokawa, T.: The Stein Phenomenon in Simultaneous Estimation: A review. In: Ahmed S.E. et al (eds.) Nonparametric Statistics and Related Topics, pp. 143–173. Nova Science, New York (1998)Google Scholar
  16. 16.
    Kutoyants, A.Y.: Statistical Inference for Ergodic Diffusion Processes. Springer, New York (2004)MATHGoogle Scholar
  17. 17.
    Langetieg, T.C.: A Multivariate Model of the Term Structure. J. Fin. 35, 71–97 (1980)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Le Breton, A.: Stochastic Systems: Modeling, Identification and Optimization I. Mathematical Programming Studies 5, 124–144 (1976)Google Scholar
  19. 19.
    Liptser, R.S., Shiryayev, A.N.: Statistics of Random Processes: General Theory, Vol. I. Springer, New York (1977)MATHGoogle Scholar
  20. 20.
    Liptser, R.S., Shiryayev, A.N.: Statistics of Random Processes: Applications II. Springer, New York (1978)MATHGoogle Scholar
  21. 21.
    Nkurunziza, S., Ahmed, S.E.: Shrinkage Drift Parameter Estimation for Multi-factor Ornstein-Uhlenbeck Processes. University of Windsor, Technical report, WMSR 07-06 (2007)Google Scholar
  22. 22.
    Schöbel, R., Zhu, J.: Stochastic Volatility With an Ornstein-Uhlenbeck Process: An Extension. Eur. Fin. Rev. 3, 23–46 (1999)MATHCrossRefGoogle Scholar
  23. 23.
    Sen, P.K.: On the Asymptotic Distributional Risks of Shrinkage and Preliminary Test Versions of Maximum Likelihood Estimators. Sankhya A 48, 354–371 (1986)MATHGoogle Scholar
  24. 24.
    Steele, J.M.: Stochastic Calculus and Financial Applications. Springer, New York (2001)MATHGoogle Scholar
  25. 25.
    Stigler, S.M.: The 1988 Neyman Memorial Lecture: A Galtonian Perspective on Shrinkage Estimators. Stat. Sci. 5, 147–155 (1990)MATHMathSciNetGoogle Scholar
  26. 26.
    Trenkler, G., Trenkler, D.: A Note on Superiority Comparisons of Homogeneous Linear Estimators. Commun. Stat. Theor. Meth. 12, 799–808 (1983)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Vasicek, O.: An Equilibrium Characterization of the Term Structure. J. Fin. Econ. 5, 177–188 (1977)CrossRefGoogle Scholar

Copyright information

© Physica-Verlag Heidelberg 2009

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of WindsorWindsorCanada
  2. 2.Faculty of Information Sciences and EngineeringUniversity of CanberraCanberraAustralia

Personalised recommendations