Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-25T04:24:30.022Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximum Likelihood Estimation of Generalized Itô Processes with Discretely Sampled Data

Published online by Cambridge University Press:  18 October 2010

Andrew W. Lo
Andrew W. Lo*
Affiliation:
University of Pennsylvania
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers the parametric estimation problem for continuous-time stochastic processes described by first-order nonlinear stochastic differential equations of the generalized Itô type (containing both jump and diffusion components). We derive a particular functional partial differential equation which characterizes the exact likelihood function of a discretely sampled Itô process. In addition, we show by a simple counterexample that the common approach of estimating parameters of an Itô process by applying maximum likelihood to a discretization of the stochastic differential equation does not yield consistent estimators.

Type
Research Article
Information
Econometric Theory , Volume 4 , Issue 2 , August 1988 , pp. 231 - 247
Copyright
Copyright © Cambridge University Press 1988 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1. Arnold, L. Stochastic Differential Equations: Theory and Applications. New York: John Wiley and Sons, 1974.Google Scholar
2. Bagchi, A. Continuous-time systems identification with unknown noise covariance. Automatica 11 (1975): 533–536.10.1016/0005-1098(75)90030-8Google Scholar
3. Bagchi, A. Consistent estimates of parameters in continuous-time systems. In Jacobs, O.L.R. et al. Analysis and Optimization of Stochastic Systems. New York: Academic Press, 1980.Google Scholar
4. Ball, C.A. & Torous, W.. On jumps in common stock prices and their impact on call option pricing. Journal of Finance 40 (1985): 155–173.10.1111/j.1540-6261.1985.tb04942.xGoogle Scholar
5. Basawa, I.V. & Prakasa Rao, B.L.S.. Statistical Inference for Stochastic Processes. New York: Academic Press, 1980.Google Scholar
6. Bergstrom, A.R. Statistical Inference in Continuous Time Economic Models. Amsterdam: North-Holland, 1976.Google Scholar
7. Bergstrom, A.R. Gaussian estimation of structural parameters in higher-order continuous-time dynamic models. Econometrica 51 (1983): 117–152.10.2307/1912251Google Scholar
8. Bergstrom, A.R. Continuous time stochastic models and issues of aggregation over time. In Griliches, Z. and Intriligator, M.D. (ed.), Handbook of Econometrics Vol. II. Amsterdam: North-Holland Publishing Company, 1984.Google Scholar
9. Billingsley, P. Statistical Inference f or Markov Processes. Chicago: University of Chicago Press, 1961.Google Scholar
10. Black, F. & Scholes, M.. The pricing of options and corporate liabilities. Journal of Political Economy 81 (1973): 637–654.10.1086/260062Google Scholar
11. Borkar, B. & Bagchi, A.. Parameter estimation in continuous-time stochastic processes. Stochastics 8 (1982): 193–212.10.1080/17442508208833238Google Scholar
12. Brockett, R.W. Lecture Notes on Nonlinear Stochastic Control. Unpublished course notes, Harvard University, Spring 1984.Google Scholar
13. Brown, B.M. & Hewitt, J.I.. Asymptotic likelihood theory for diffusion processes. Journal of Applied Probability 12 (1978): 228–238.10.2307/3212436Google Scholar
14. Chamberlain, G. Asset pricing in multiperiod securities markets. University of Wisconsin-Madison S.S.R.I. Working Paper No. 8510, 1985.Google Scholar
15. Christiano, L. The effects of aggregation over time on tests of the representative agent model of consumption. Mimeo, December 1984.Google Scholar
16. Christiano, L. A critique of conventional treatments of the model timing interval in applied econometrics. Mimeo, January 1985.Google Scholar
17. Christiano, L. Estimating continuous-time rational expectations models in frequency domain: A case study. Carnegie-Mellon University Working Paper No. 34–84–85, April 1985.Google Scholar
18. Christie, A. The stochastic behavior of common stock variances: Value, leverage, and interest rate effects. Journal of Financial Economics 10 (1982): 407–432.10.1016/0304-405X(82)90018-6Google Scholar
19. Cox, D.R. & Miller, H.D.. The Theory of Stochastic Processes. New York: Chapman and Hall, 1965.Google Scholar
20. Cox, J., Ingersoll, J., & Ross, S.. An intertemporal general equilibrium model of asset prices. Econometrica 53 (1985): 363–384.10.2307/1911241Google Scholar
21. Cox, J., Ingersoll, J., & Ross, S.. A theory of the term structure of interest rates. Econometrica 53 (1985): 385–408.Google Scholar
22. Crowder, M.J. Maximum likelihood estimation for dependent observations. Journal of the Royal Statistical Society Series B, 38 (1976): 45–53.Google Scholar
23. Gel'fand, I.M. & Shilov, G.E.. Generalized Functions Volume I: Properties and Operations. New York: Academic Press, 1964.Google Scholar
24. Gihman, I.I. & Skorohod, A.V.. Controlled Stochastic Processes. Springer, Berlin, 1979.10.1007/978-1-4612-6202-2Google Scholar
25. Gordin, M.I. The central limit theorem for stationary processes. Soviet Mathematical Doklady 10 (1969): 1174–1176.Google Scholar
26. Grossman, S., Melino, A., & Shiller, R.. Estimating the continuous-time consumption-based asset pricing model. Journal of Business and Economic Statistics 5 (1987), 315–327.Google Scholar
27. Hall, P. & Heyde, C.C.. Martingale Limit Theory and Its Application. New York: Academic Press, 1980.Google Scholar
28. Hansen, L. & Sargent, T.. Formulating and estimating continuous-time rational expectations models. Unpublished manuscript, August 1981.10.21034/sr.75Google Scholar
29. Hansen, L. & Sargent, T.. The dimensionality of the aliasing problem in models with rational spectral densities. Econometrica 51 (1983): 377–388.10.2307/1911996Google Scholar
30. Hansen, L. & Sargent, T.. Identification of continuous-time rational expectations models from discrete-time data. Federal Reserve Bank of Minneapolis Staff Report 73, March 1983.Google Scholar
31. Harrison, J.M., Pitbladdo, R., & Schaefer, S.M.. Continuous-price processes in frictionless markets have infinite variation. Journal of Business 57 (1984): 353–365.10.1086/296268Google Scholar
32. Harvey, A. & Stock, J.. The estimation of higher-order continuous-time autoregressive models. Econometric Theory 1 (1985): 97–112.10.1017/S0266466600011026Google Scholar
33. Herrndorf, N. A functional central limit theorem for weakly dependent sequences of random variables. Annals of Probability 12 (1984): 141–153.Google Scholar
34. Heyde, C.C. On the central limit theorem and iterated logarithm law for stationary processes. Bulletin of the Australian Mathematical Society 12 (1975): 1–8.10.1017/S0004972700023583Google Scholar
35. Hoadley, B. Asymptotic properties of maximum likelihood estimators for the independent not identically distributed case. Annals of Mathematical Statistics 42 (1971): 1977–1991.10.1214/aoms/1177693066Google Scholar
36. Itô, K. On stochastic differential equations. Memoirs of the American Mathematical Society 4 (1951): 1–51.Google Scholar
37. Janssen, R. Discretization of the Wiener process in difference methods for stochastic differential equations. Stochastic Processes and their Applications 18 (1984): 361–369.10.1016/0304-4149(84)90306-5Google Scholar
38. Karlin, S. & Taylor, H.M.. A Second Course in Stochastic Processes. New York: Academic Press, 1981.Google Scholar
39. Le Breton, A. On continuous and discrete sampling for parameter estimation in diffusion type processes. In Wets, R.J.-B. (ed.), Stochastic Systems: Modeling, Identification and Optimization, I. Amsterdam: North-Holland Publishing Company, 1976.Google Scholar
40. Liptser, R.S. & Shiryayev, A.N.. Statistics of Random Processes II: Applications. New York: Springer-Verlag, 1978.10.1007/978-1-4757-4293-0Google Scholar
41. Ljung, L. Convergence analysis of parametric identification methods. IEEE Transactions on Automatic Control AC-23 (1978): 770–783.10.1109/TAC.1978.1101840Google Scholar
42. Lo, A. Statistical tests of contingent claims asset-pricing models: A new methodology. Journal of Financial Economics 17 (1986): 143–174.10.1016/0304-405X(86)90009-7Google Scholar
43. Loges, W. Girsanov's theorem in Hilbert space and an application to the statistics of Hilbert space-valued stochastic differential equations. Stochastic Processes and their Applications 17 (1984): 243–263.10.1016/0304-4149(84)90004-8Google Scholar
44. Marsh, T. & Rosenfeld, E.. Stochastic processes for interest rates and equilibrium bond prices. Journal of Finance 38 (1983): 635–645.10.1111/j.1540-6261.1983.tb02275.xGoogle Scholar
45. McLeish, D.L. Dependent central limit theorems and invariance principles. Annals of Probability 2 (1974): 620–628.10.1214/aop/1176996608Google Scholar
46. Ogden, J. An analysis of yield curve notes. Journal of Finance 42 (1987): 99–110.10.1111/j.1540-6261.1987.tb02552.xGoogle Scholar
47. Pardoux, E. & Talay, D.. Discretization and simulation of stochastic differential equations. Acta Applicandae Mathematicae 3 (1985): 23–47.10.1007/BF01438265Google Scholar
48. Park, C. & Beekman, J.. Stochastic barriers for the Wiener process. Journal of Applied Probability 20 (1983): 338–348.10.2307/3213806Google Scholar
49. Park, C., Beekman, J., & Paranjape, S.. Probabilities of Wiener paths crossing differentiable curves. Pacific Journal of Mathematics 53 (1974): 579–583.10.2140/pjm.1974.53.579Google Scholar
50. Park, C., Beekman, J., & Schuurmann, F.. Evaluations of barrier-crossing probabilities and Wiener paths. Journal of applied Probability 13 (1976): 267–275.10.2307/3212830Google Scholar
51. Park, J. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: Part 1. Cowles Foundation Discussion Paper No. 811, November 1986.Google Scholar
52. Perron, P. Testing consistency with varying sampling frequency. Working Paper, University of Montreal, August 1987.Google Scholar
53. Phillips, A.W. The estimation of parameters in systems of stochastic differential equations. Biometrika 46 (1959): 67–76.10.1093/biomet/46.1-2.67Google Scholar
54. Phillips, P.C.B. The structural estimation of a stochastic differential equation system. Econometrica 40 (1972): 1021–1041.10.2307/1913853Google Scholar
55. Phillips, P.C.B. The problem of identification in finite parameter continuous-time processes. Journal of Econometrics 1 (1973): 351–362.10.1016/0304-4076(73)90021-3Google Scholar
56. Phillips, P.C.B. The estimation of some continuous-time models. Econometrica 42 (1974): 803–823.10.2307/1913790Google Scholar
57. Phillips, P.C.B. Time series regression with unit roots. Econometrica 55 (1987): 277–302.10.2307/1913237Google Scholar
58. Phillips, P.C.B. Regression theory for near integrated time series. Cowles Foundation Discussion Paper No. 781-R, January 1987.Google Scholar
59. Prakasa Rao, B.L.S. Maximum likelihood estimation for Markov processes. Annals of the Institute of Statistics and Mathematics 24 (1972): 333–345.10.1007/BF02479763Google Scholar
60. Rao, N., Borwankar, J., & Ramakrishna, D.. Numerical solution of Ito integral equations. SIAM Journal of Control 12 (1974): 123–139.Google Scholar
61. Rosenfeld, E. Stochastic Processes of Common Stock Returns: An Empirical Examination. Unpublished Ph.D. thesis, Sloan School of Management, M.I.T., February 1980.Google Scholar
62. Roussas, G.G. Asymptotic inference in Markov processes. Annals of Mathematical Statistics 36 (1965): 978–992.10.1214/aoms/1177700070Google Scholar
63. Rudin, W. Functional Analysis. New York: McGraw-Hill Book Company, 1973.Google Scholar
64. Rumelin, W. Numerical treatment of stochastic differential equations. SIAM Journal of Numerical Analysis 19 (1982): 604–613.10.1137/0719041Google Scholar
65. Schuss, Z. Theory and Applications of Stochastic Differential Equations. New York: John Wiley and Sons, 1980.Google Scholar
66. Shiller, R. & Perron, P.. Testing the random walk hypothesis: Power versus frequency of observation. Economics Letters 18 (1985), 381–386.10.1016/0165-1765(85)90058-8Google Scholar
67. Siegmund, D. Boundary crossing probabilities and statistical applications. Stanford University Department of Statistics Technical Report No. E&S/NSF230, June 1985.Google Scholar
68. Sims, C.A. Discrete approximations to continuous-time distributed lags in econometrics. Econometrica 39 (1971): 545–563.Google Scholar
69. Skorohod, A.V. Studies in the Theory of Random Processes. New York: Dover Publications, 1965.Google Scholar
70. Tugnait, J.K. Identification and model approximation for continuous-time systems on finite parameter sets. IEEE Transactions on Automatic Control 25 (1980): 1202–1206.Google Scholar
71. Tugnait, J.K. Global identification of continuous-time systems with unknown noise covar-iance. IEEE Transactions on Information Theory 28 (1982): 531–536.Google Scholar
72. Tugnait, J.K. Continuous-time system identification on compact parameter sets. IEEE Transactions on Information Theory 31 (1985): 652–659.10.1109/TIT.1985.1057090Google Scholar