응용통계연구 (The Korean Journal of Applied Statistics)

  • 제26권5호
  • /
  • Pages.767-783
  • /
  • 2013
  • /
  • 1225-066X(pISSN)
  • /
  • 2383-5818(eISSN)
한국통계학회 (The Korean Statistical Society)
권호 표지
DOI QR코드

DOI QR Code

확산모형에 대한 일반화적률추정법의 개선

Improved Generalized Method of Moment Estimators to Estimate Diffusion Models

  • 투고 : 2013.08.04
  • 심사 : 2013.09.23
  • 발행 : 2013.10.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

일반화적률추정법(GMM)은 금융자료에 대한 모형모수의 추정에 자주 이용되는 방법이다. 특히 GMM은 현대금융 공학 이론의 기본을 이루는 확산모형의 추정에도 매우 자주 사용된다. 그러나 최근의 연구에서 GMM은 확산모형의 모수, 특히 확산계수에 관계되는 모수의 추정에 있어서 그 성능이 좋지 못함이 지적되었다. 본 연구에서는 GMM의 이러한 단점을 개선하기 위한 대안적 방법들을 제시하고 그 통계적 성능을 시뮬레이션 연구를 통해서 비교하게 된다. 이런 과정을 통하여 제안되고 검토된 추정방법들 중, Shoji와 Ozaki (1998)가 제안한 국소선형근사법의 결과를 적용하여 GMM의 성능을 개선한 NGMM-Y 추정량이 매우 우수한 성질을 가지고 있음을 확인하게 된다. 특히 NGMM-Y 추정량은 확산계수에 관계된 모수의 추정에 있어서 비교대상이 된 다른 대안적 GMM 방법들에 비하여 우수한 성질을 가지고 있음을 확인하게 된다.

Generalized Method of Moment(GMM) is a popular estimation method to estimate model parameters in empirical financial studies. GMM is frequently applied to estimate diffusion models that are basic techniques of modern financial engineering. However, recent research showed that GMM had poor properties to estimate the parameters that pertain to the diffusion coefficient in diffusion models. This research corrects the weakness of GMM and suggests alternatives to improve the statistical properties of GMM estimators. In this study, a simulation method is adopted to compare estimation methods. Out of compared alternatives, NGMM-Y, a version of improved GMM that adopts the NLL idea of Shoji and Ozaki (1998), showed the best properties. Especially NGMM-Y estimator is superior to other versions of GMM estimators for the estimation of diffusion coefficient parameters.

키워드

참고문헌

  1. Ait-Sahalia, Y. (1999). Transition densities for interest rate and other nonlinear diffusions, Journal of Finance, 54, 1361-1395. https://doi.org/10.1111/0022-1082.00149
  2. Ait-Sahalia, Y. (2002). Maximum-likelihood estimation of discretely-sampled diffusions: A closed-form approximation approach, Econometrica, 70, 223-262. https://doi.org/10.1111/1468-0262.00274
  3. Beskos, A., Papaspiliopoulos, O., Robert, G. O. and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes, The Journal of the Royal Statistical Society, Series B., 68, 333-383. https://doi.org/10.1111/j.1467-9868.2006.00552.x
  4. Chan, K. C., Karolyi, G. A., Longstaff, F. A. and Sanders, A. B. (1992). An empirical comparison of alternative models of the short-term interest rate, Journal of Finance, 47, 1209-1227. https://doi.org/10.1111/j.1540-6261.1992.tb04011.x
  5. Cox, J. (1975). Notes on option pricing I: Constant elasticity of variance diffusions. Working paper, Stanford University (reprinted in Journal of Portfolio Management, 1996, 22, 15-17).
  6. Cox, J., Ingersoll, J. and Ross, S. (1985). A theory of the term structure of interest rates, Econometrica, 53, 385-407. https://doi.org/10.2307/1911242
  7. Durham, G. and Gallant, R. (2001). Numerical Techniques for Maximum Likelihood Estimation of Continuous-Time Diffusion Processes, Technical report.
  8. Elerian, O., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed nonlinear diffusions, Econometrika, 69, 959-993. https://doi.org/10.1111/1468-0262.00226
  9. Eraker, B. (2001). MCMC analysis of diffusion models with application to finance, Journal of Business & Economic Statistics, 19, 177-191. https://doi.org/10.1198/073500101316970403
  10. Hansen, L. P. (1982). Large sample properties of generalized method of moment estimators, Econometrica, 50, 1029-1054. https://doi.org/10.2307/1912775
  11. Heyde, C. C. (1997). Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation, Springer.
  12. Hurn, A., Jeisman, J. and Lindsay, K. (2007). Seeing the wood for the trees: A critical evaluation of methods to estimate the parameters of stochastic differential equations, Journal of Financial Econometrics, 5, 390-455. https://doi.org/10.1093/jjfinec/nbm009
  13. Imbens, G. W., Spady, R. H. and Johnson, P. (1998). Information theoretic approaches to inference in moment condition models, Econometrica, 66, 339-357.
  14. Kim, D.-G. and Lee, Y. D. (2011). Comparison study on the performances of NLL and GMM for estimating diffusion processes, The Korean Journal of Applied Statistics, 24, 1007-1020. https://doi.org/10.5351/KJAS.2011.24.6.1007
  15. Kloeden, P. and Platen, E. (1999). Numerical Solution of Stochastic Differential Equations, Springer.
  16. Lahiri, S. N. (1996). On inconsistency of estimators based on spatial data under infill asymptotics, Sankhya, Series A, 58, 403-417.
  17. Lahiri, S. N., Lee, Y. and Cressie, N. (2002). Efficiency of least squares estimators of spatial variogram parameters, Journal of Statistical Planning and Inference, 3, 65-85.
  18. Lee, Y. and Lahiri, S. N. (2002). Least squares variogram fitting by spatial subsampling, Journal of the Royal Statistical Society, Series B, 64, 837-854. https://doi.org/10.1111/1467-9868.00364
  19. Lee, Y.-D. and Lee, E.-K. (2013). An approximation of the cumulant generating functions of diffusion models and the Pseudo-likelihood estimation method, Journal of the Korean Operations Research and Management Science Society, 38, 201-216. https://doi.org/10.7737/JKORMS.2013.38.1.201
  20. Lee, Y., Song, S. and Lee, E. (2012). The Delta Expansion for the Transition Density of Diffusion Models, Technical Report.
  21. Oksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications, Springer.
  22. Pederson, A. R. (1995). A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations, Scandinavian Journal of Statistics, 22, 55-71.
  23. Shoji, I. and Ozaki, T. (1998). Estimation for nonlinear stochastic differential equations by a local linearization method, Stochastic Analysis and Applications, 16, 733-752. https://doi.org/10.1080/07362999808809559
  24. Vasicek, O. (1977). An equilibrium characterization of the term structure, Journal of Financial Economics, 5, 177-188. https://doi.org/10.1016/0304-405X(77)90016-2

피인용 문헌

  1. Likelihood Approximation of Diffusion Models through Approximating Brownian Bridge vol.28, pp.5, 2015, https://doi.org/10.5351/KJAS.2015.28.5.895