Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-29T10:08:18.169Z Has data issue: false hasContentIssue false
  • English
  • Français

REWEIGHTED FUNCTIONAL ESTIMATION OF DIFFUSION MODELS

Published online by Cambridge University Press:  30 September 2009

Ke-Li Xu
Get access Rights & Permissions [Opens in a new window]

Abstract

The local linear method is popular in estimating nonparametric continuous-time diffusion models, but it may produce negative results for the diffusion (or volatility) functions and thus lead to insensible inference. We demonstrate this using U.S. interest rate data. We propose a new functional estimation method of the diffusion coefficient based on reweighting the conventional Nadaraya–Watson estimator. It preserves the appealing bias properties of the local linear estimator and is guaranteed to be nonnegative in finite samples. A limit theory is developed under mild requirements (recurrence) of the data generating mechanism without assuming stationarity or ergodicity.

Type
Research Article
Information
Econometric Theory , Volume 26 , Issue 2 , April 2010 , pp. 541 - 563
Copyright
Copyright © Cambridge University Press 2009

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.)

Footnotes

The author has benefited from comments by the co-editor Oliver B. Linton, two anonymous referees, Donald W.K. Andrews, Zongwu Cai, Xiaohong Chen, Yuichi Kitamura, Taisuke Otsu, Peter C.B. Phillips, and other participants at the econometrics seminar at Yale University and the NBER-NSF time series conference at Montreal. Special thanks go to Peter C.B. Phillips for his guidance and encouragement. The author also gratefully acknowledges the partial financial support provided by the University of Alberta School of Business through the H.E. Pearson faculty award.

References

REFERENCES

Ahn, D.-H. & Gao, B. (1999) A parametric nonlinear model of term structure dynamics. Review of Financial Studies 12, 721–762.Google Scholar
Aït-Sahalia, Y. (1996a) Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527–560.CrossRefGoogle Scholar
Aït-Sahalia, Y. (1996b) Testing continuous-time models of the spot interest rate. Review of Financial Studies 9, 385–426.Google Scholar
Aït-Sahalia, Y. (2007) Estimating continuous-time models using discretely sampled data. In Blundell, R., Newey, W.K., & Persson, T. (eds.), Advances in Economics and Econometrics, Theory and Applications, Ninth World Congress. Cambridge University Press.Google Scholar
Arapis, M. & Gao, J. (2006) Empirical comparisons in short-term interest rate models using nonparametric methods. Journal of Financial Econometrics 4, 310–345.CrossRefGoogle Scholar
Bandi, F. (2002) Short-term interest rate dynamics: A spatial approach. Journal of Financial Economics 65, 73–110.Google Scholar
Bandi, F. & Moloche, G. (2004) On the Functional Estimation of Multivariate Diffusion Processes. Working paper, University of Chicago.Google Scholar
Bandi, F. & Phillips, P.C.B. (2009) Nonstationary continuous-time processes. In Aït-Sahalia, Y. & Hansen, L.P. (eds.), Handbook of Financial Econometrics. North-Holland.Google Scholar
Bandi, F. & Phillips, P.C.B. (2003) Fully nonparametric estimation of scalar diffusion models. Econometrica 71, 241–283.CrossRefGoogle Scholar
Boudoukh, J., Downing, C., Richardson, M., Stanton, R., & Whitelaw, R.F. (2007) A Multifactor, Nonlinear, Continuous-Time Model of Interest Rates Volatility. Working paper, University of California, Berkeley.Google Scholar
Cai, Z. (2001) Weighted Nadaraya-Watson regression estimation. Statistics and Probability Letters 51, 307–318.Google Scholar
Cai, Z. & Hong, Y. (2003) Nonparametric methods in continuous-time finance: A selective review. In Akritas, M.G. & Politis, D.N. (eds.), Recent Advances and Trends in Nonparametric Statistics, pp. 283–302. North-Holland.CrossRefGoogle Scholar
Chapman, D.A. & Pearson, N.D. (2000) Is the short rate drift actually nonlinear? Journal of Finance 55, 355–388.Google Scholar
Chen, S.X., Gao, J., & Tang, C.Y. (2008) A test for model specification of diffusion processes. Annals of Statistics 36, 167–198.Google Scholar
Chen, S.X. & Qin, Y. (2002) Confidence interval based on a local linear smoother. Scandinavian Journal of Statistics 29, 89–99.Google Scholar
Cox, J.C., Ingersoll, J.E., & Ross, S.A. (1980) An analysis of variable rate loan contracts. Journal of Finance 35, 389–403.Google Scholar
Cox, J.C., Ingersoll, J.E., & Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica 53, 385–467.Google Scholar
Cressie, N.A.C. & Read, T.R.C. (1984) Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society, series B 46, 440–464.Google Scholar
Dai, Q. & Singleton, K. (2000) Specification analysis of affine term structure models. Journal of Finance 55, 1943–1978.Google Scholar
Duffie, D. & Kan, R. (1996) A yield-factor model of interest rates. Mathematical Finance 6, 379–406.Google Scholar
Durham, G.B. (2003) Likelihood-based specification analysis of continuous-time models of the short-term interest rate. Journal of Financial Economics 70, 463–487.Google Scholar
Fan, J. (2005) A selective overview of nonparametric methods in financial econometrics (with discussion). Statistical Science 20, 317–357.Google Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modeling and Its Applications. Chapman and Hall.Google Scholar
Fan, J. & Yao, Q. (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645–660.Google Scholar
Fan, J. & Zhang, C. (2003) A re-examination of Stanton’s diffusion estimations with applications to financial model validation. Journal of the American Statistical Association 98, 118–134.Google Scholar
Fan, J., Zhang, C.M., & Zhang, J. (2001) Generalized likelihood ratio statistics and Wilks phenomenon. Annals of Statistics 29, 153–193.CrossRefGoogle Scholar
Florens-Zmirou, D. (1993) On estimating the diffusion coefficient from discrete observations. Journal of Applied Probability 30, 790–804.Google Scholar
Gao, J. (2007) Nonlinear Time Series: Semiparametric and Nonparametric Methods. Chapman and Hall/CRC.Google Scholar
Hall, P. & Huang, L.-S. (2001) Nonparametric kernel regression subject to monotonicity constraints. Annals of Statistics 29, 624–647.CrossRefGoogle Scholar
Hall, P. & Presnell, B. (1999) Intentionally biased bootstrap methods. Journal of the Royal Statistical Society, series B 61, 143–158.Google Scholar
Hall, P. & Tajvidi, N. (2000) Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data. Statistical Science 15, 153–167.CrossRefGoogle Scholar
Hall, P., Wolff, R.C.L., & Yao, Q. (1999) Methods for estimating a conditional distribution function. Journal of the American Statistical Association 94, 154–163.Google Scholar
Hart, J.D. (1996) Some automated methods of smoothing time-dependent data. Journal of Nonparametric Statistics 6, 115–142.Google Scholar
Hong, Y. & Li, H. (2005) Nonparametric specification testing for continuous-time models with applications to term structure of interest rates. Review of Financial Studies 18, 37–84.Google Scholar
Jeffrey, A., Kristensen, D., Linton, O., Nguyen, T., & Phillips, P.C.B. (2004) Nonparametric estimation of a multifactor Heath-Jarrow-Morton model: An integrated approach. Journal of Financial Econometrics 2, 251–289.CrossRefGoogle Scholar
Jiang, G.J. & Knight, J. (1997) A nonparametric approach to the estimation of diffusion processes with an application to a short-term interest rate model. Econometric Theory 13, 615–645.Google Scholar
Linton, O. & Nielsen, J.P. (1994) A multiplicative bias reduction method for nonparametric regression. Statistics and Probability Letters 19, 181–187.Google Scholar
Moloche, G. (2001) Local Nonparametric Estimation of Scalar Diffusions. Mimeo, MIT.Google Scholar
Nicolau, J. (2003) Bias reduction in nonparametric diffusion coefficient estimation. Econometric Theory 19, 754–777.Google Scholar
Owen, A. (2001) Empirical Likelihood. Chapman and Hall/CRC.Google Scholar
Park, J. & Phillips, P.C.B. (1998) Nonstationary Density Estimation and Kernel Autoregression. Cowles Foundation Discussion Paper 1181, Yale University.Google Scholar
Phillips, P.C.B. (1998) Econometric Analysis of Fisher’s Equation. Cowles Foundation Discussion Paper 1180, Yale University.Google Scholar
Phillips, P.C.B. (2001) Descriptive econometrics for nonstationary time series with empirical illustrations. Journal of Applied Econometrics 16, 389–413.Google Scholar
Pritsker, M. (1998) Nonparametric density estimation and tests of continuous time interest rate models. Review of Financial Studies 11, 449–487.Google Scholar
Stanton, R. (1997) A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance 52, 1973–2002.Google Scholar
Vasicek, O.A. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177–188.Google Scholar
Xu, K.-L. (2009) Empirical likelihood based inference for nonparametric recurrent diffusions. Journal of Econometrics, forthcoming.Google Scholar
Xu, K.-L. & Phillips, P.C.B. (2008) Tilted Nonparametric Estimation of Volatility Functions with Empirical Applications. Cowles Foundation Discussion Paper 1612R, Yale University.Google Scholar
Yu, K. & Jones, M.C. (2004) Likelihood-based local linear estimation of the conditional variance function. Journal of the American Statistical Association 99, 139–144.Google Scholar
Ziegelmann, F.A. (2002) Nonparametric estimation of volatility functions: The local exponential estimator. Econometric Theory 18, 985–991.Google Scholar