Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-29T04:09:35.925Z Has data issue: false hasContentIssue false
  • English
  • Français

A Nonparametric Approach to the Estimation of Diffusion Processes, With an Application to a Short-Term Interest Rate Model

Published online by Cambridge University Press:  11 February 2009

George J. Jiang  and
John L. Knight
George J. Jiang
Affiliation:
University of Groningen
John L. Knight
Affiliation:
University of Western Ontario
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we propose a nonparametric identification and estimation procedure for an ltd diffusion process based on discrete sampling observations. The nonparametric kernel estimator for the diffusion function developed in this paper deals with general ltd diffusion processes and avoids any functional form specification for either the drift function or the diffusion function. It is shown that under certain regularity conditions the nonparametric diffusion function estimator is pointwise consistent and asymptotically follows a normal mixture distribution. Under stronger conditions, a consistent nonparametric estimator of the drift function is also derived based on the diffusion function estimator and the marginal density of the process. An application of the nonparametric technique to a short-term interest rate model involving Canadian daily 3-month treasury bill rates is also undertaken. The estimation results provide evidence for rejecting the common parametric or semiparametric specifications for both the drift and diffusion functions.

Type
Articles
Information
Econometric Theory , Volume 13 , Issue 5 , October 1997 , pp. 615 - 645
Copyright
Copyright © Cambridge University Press 1997

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

Ait-Sahalia, Y. (1996) Nonparametric pricing of interest rate derivative securities. Econometrica 64 (3), 527–560.Google Scholar
Banon, G. (1978) Nonparametric identification for diffusion processes. SIAM Journal of Control and Optimization 16, 380–395.CrossRefGoogle Scholar
Banon, G. & Nguyen, H.T. (1981) Recursive estimation in diffusion models. SIAM Journal of Control and Optimization 19, 676–685.CrossRefGoogle Scholar
Bartlett, M.S. (1946) On the theoretical specification and sampling properties of autocorrelated time series. Journal of the Royal Statistical Society Supplement 8, 27–41.CrossRefGoogle Scholar
Black, F. & Scholes, M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–654.CrossRefGoogle Scholar
Brennan, M.J. & Schwartz, E.S. (1977) Saving bonds, retractable bonds, and callable bonds. Journal of Financial Economics 5, 67–88.Google Scholar
Brennan, M J. & Schwartz, E.S. (1979) A continuous time approach to the pricing of bonds. Journal of Banking and Finance 3, 133–155.Google Scholar
Brennan, M.J. & Schwartz, E.S. (1980) Analyzing convertible bonds. Journal of Financial and Quantitative Analysis 15, 907–929.CrossRefGoogle Scholar
Brown, B.M. & Hewitt, J.I. (1975) Asymptotic likelihood theory for diffusion process. Journal of Applied Probability 12, 228–238.CrossRefGoogle Scholar
Chan, K.C., Karolyi, G.A.Longstaff, F.A., & Sanders, A.B. (1992) An empirical comparison of alternative models of the short-term interest rate. Journal of Finance XLVII (3), 1209–1227.Google Scholar
Conley, T.G., Hansen, L.P.Luttmer, E.G.J., & Scheinkman, J.A. (1995) Short-Term Interest Rate as Subordinated Diffusions. Mimeo, University of Chicago.Google Scholar
Cox, J.C. (1975) Notes on Option Pricing I: Constant Elasticity of Variance Diffusion. Working Paper, Stanford University.Google Scholar
Cox, J.C, Ingersoll, J.E., & Ross, S.A. (1980) An analysis of variable rate loan contracts. Journal of Finance XXXV (2), 389–403.CrossRefGoogle Scholar
Cox, I.C, Ingersoll, J.E., & Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica 53 (2), 385–407.CrossRefGoogle Scholar
Cox, J.C. & Ross, S.A. (1976) The valuation of options for alternative stochastic processes. Journal of Financial Economics 3, 145–166.CrossRefGoogle Scholar
Dacunha-Castelle, D. & D. Florens-Zmirou (1986) Estimation of the coefficient of a diffusion from discrete observations. Stochastics 19, 263–284.CrossRefGoogle Scholar
Delgado, M.A. & Robinson, P.M. (1992) Nonparametric and semiparametric methods for economic research. Journal of Economic Surveys 6(3), 201–249.Google Scholar
Dohnal, G. (1987) On estimating the diffusion coefficient. Journal of Applied Probability 24, 105–114.Google Scholar
Doob, J.L. (1953) Stochastic Processes. New York: John Wiley & Sons.Google Scholar
Dothan, L.U. (1978) On the term structure of interest rates. Journal of Financial Economics 6, 59–69.CrossRefGoogle Scholar
Duffie, D. & Singleton, K.J. (1993) Simulated moments estimation of Markov models for asset prices. Econometrica 61, 929–952.CrossRefGoogle Scholar
Florens-Zmirou, D. (1993) On estimating the diffusion coefficient from discrete observations. Journal of Applied Proba bility 30, 790–804.Google Scholar
Geman, S.A. (1979) On a Common Sense Estimator for the Drift of a Diffusion. Pattern Analysis 79, Division of Applied Mathematics, Brown University, Providence, Rhode Island.Google Scholar
Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Econometrica 50, 1029–1054.CrossRefGoogle Scholar
Hansen, L.P. & Sargent, T.J. (1983) The dimensionality of the aliasing problem. Econometrica 51, 377–388.CrossRefGoogle Scholar
Hansen, L.P. & Scheinkman, J.A. (1995) Back to the future: Generating moment implications for continuous-time Markov processes. Econometrica 63 (4), 767–804.CrossRefGoogle Scholar
Harvey, A.C. (1993) Time Series Models, 2nd ed.Cambridge: MIT Press.Google Scholar
Ingersoll, J.E. Jr., (1987) Theory of Financial Decision Making. London: Rowman & Littlefield.Google Scholar
Jiang, G.J. & Knight, J.L. (1996) Parametric Versus Nonparametric Estimation of Diffusion Processes-A Monte Carlo Comparison. Mimeo, University of Western Ontario.Google Scholar
Karlin, S. & Taylor, H.M. (1981) A Second Course in Stochastic Processes. New York: Academic Press.Google Scholar
Knight, F.B. (1976) A Reduction of Continuous Square Integrable Martingales to Brownian Motion. Lecture Notes in Mathematics 190. Berlin: Springer-Verlag.Google Scholar
Kumar, T.K. & Markman, J.M. (1975) Estimation of Probability Density Function: A Monte Carlo Comparison of Parametric and Nonparametric Methods. Mimeo.Google Scholar
Kutoyants, Y. (1984) Parameter Estimation for Stochastic Processes. Heldermann.Google Scholar
Lanska, V. (1979) Minimum contrast estimation in diffusion processes. Journal of Applied Probability 16, 65–75.CrossRefGoogle Scholar
Lo, A.W. (1988) Maximum likelihood estimation of generalized ltd processes with discretely sampled data. Econometric Theory 4, 231–247.CrossRefGoogle Scholar
Marsh, T.A. & Rosenfeld, E.R. (1983) Stochastic processes for interest rates and equilibrium bond prices. Journal of Finance 38, 635–646.CrossRefGoogle Scholar
Merton, R.C. (1973) Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141–183.Google Scholar
Monfort, A. (1996) A reappraisal of misspecified econometric models. Econometric Theory 12.597–619.Google Scholar
Oksendal, B. (1992) Stochastic Differential Equations. New York: Springer-Verlag.CrossRefGoogle Scholar
Pedersen, A.R. (1995) Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes. Bernoulli 1 (3), 257–279.Google Scholar
Pham Dinh, T.D. (1981) Non-parametric estimator of the drift coefficient in the diffusion equation. Math. Operations Statist. Ser Statist. 12, 61–73.Google Scholar
Phillips, P.C.B. (1973) The problem of identification in finite parameter continuous-time models. Journal of Econometrics 1, 351–362.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55 (2), 277–310.Google Scholar
Prakasa Rao, B.L.S. (1983) Nonparametric Functional Estimation. London: Academic Press.Google Scholar
Prakasa Rao, B.L.S. (1990) Nonparametric density estimation for stochastic processes from sampled data. Public Inst. Statist. Univ. Paris XXXV (3), 51–83.Google Scholar
Prohorov, Yu.V. & YRozanov, u.A. (1969) Probability Theory. Translated by K. Krickeberg & H. Urmitzer, New York: Springer-Verlag.Google Scholar
Rosenblatt, M. (1971) Markov Processes. Structure and Asymptotic Behavior. New York: Springer-Verlag.CrossRefGoogle Scholar
Silverman, B.W. (1986) Density Estimation for Statistical and Data Analysis. London: Chapman-Hall.Google Scholar
Stanton, R. (1996) A Nonparametric Model of Term Structure Dynamics and the Market Price of Interest Rate Risk. Working paper, Haas School of Business, University of California, Berkeley.CrossRefGoogle Scholar
Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177–188.Google Scholar