Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-25T07:34:11.934Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LIVE METHOD FOR GENERALIZED ADDITIVE VOLATILITY MODELS

Published online by Cambridge University Press:  01 December 2004

Woocheol Kim  and
Oliver Linton
Woocheol Kim
Affiliation:
Korea Institute of Public Finance and Humboldt University of Berlin
Oliver Linton
Affiliation:
The London School of Economics
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate a new separable nonparametric model for time series, which includes many autoregressive conditional heteroskedastic (ARCH) models and autoregressive (AR) models already discussed in the literature. We also propose a new estimation procedure called LIVE, or local instrumental variable estimation, that is based on a localization of the classical instrumental variable method. Our method has considerable computational advantages over the competing marginal integration or projection method. We also consider a more efficient two-step likelihood-based procedure and show that this yields both asymptotic and finite-sample performance gains.This paper is based on Chapter 2 of the first author's Ph.D. dissertation from Yale University. We thank Wolfgang Härdle, Joel Horowitz, Peter Phillips, and Dag Tjøstheim for helpful discussions. We are also grateful to Donald Andrews and two anonymous referees for valuable comments. The second author thanks the National Science Foundation and the ESRC for financial support.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 6 , December 2004 , pp. 1094 - 1139
Copyright
© 2004 Cambridge University Press

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

Andrews, D.W.K. (1994) Empirical process methods in econometrics. In R.F. Engle & D. McFadden (eds.), Handbook of Econometrics, vol. IV, pp. 2247–2294. North-Holland.
Auestadt, B. & D. Tjøstheim (1990) Identification of nonlinear time series: First order characterization and order estimation. Biometrika 77, 669–687.Google Scholar
Avramidis, P. (2002) Local maximum likelihood estimation of volatility function. Manuscript, LSE.
Breiman, L. & J.H. Friedman (1985) Estimating optimal transformations for multiple regression and correlation (with discussion). Journal of the American Statistical Association 80, 580–619.Google Scholar
Buja, A., T. Hastie, & R. Tibshirani (1989) Linear smoothers and additive models (with discussion). Annals of Statistics 17, 453–555.Google Scholar
Cai, Z. & E. Masry (2000) Nonparametric estimation of additive nonlinear ARX time series: Local linear fitting and projections. Econometric Theory 16, 465–501.Google Scholar
Carrasco, M. & X. Chen (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.Google Scholar
Chen, R. (1996) A nonparametric multi-step prediction estimator in Markovian structures. Statistical Sinica 6, 603–615.Google Scholar
Chen, R. & R.S. Tsay (1993a) Nonlinear additive ARX models. Journal of the American Statistical Association 88, 955–967.Google Scholar
Chen, R. & R.S. Tsay (1993b) Functional-coefficient autoregressive models. Journal of the American Statistical Association 88, 298–308.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987–1008.Google Scholar
Fan, J. & Q. Yao (1996) Efficient estimation of conditional variance functions in stochastic regression. Biometrica 85, 645–660.Google Scholar
Gozalo, P. & O. Linton (2000) Local nonlinear least squares: Using parametric information in nonparametric regression. Journal of Econometrics 99(1), 63–106.Google Scholar
Hall, P. & C. Heyde (1980) Martingale Limit Theory and Its Application. Academic Press.
Härdle, W. (1990) Applied Nonparametric Regression. Econometric Monograph Series 19. Cambridge University Press.
Härdle, W. & A.B. Tsybakov (1997) Locally polynomial estimators of the volatility function. Journal of Econometrics 81, 223–242.Google Scholar
Härdle, W., A.B. Tsybakov, & L. Yang (1998) Nonparametric vector autoregression. Journal of Statistical Planning and Inference 68(2), 221–245.Google Scholar
Härdle, W. & P. Vieu (1992) Kernel regression smoothing of time series. Journal of Time Series Analysis 13, 209–232.Google Scholar
Hastie, T. & R. Tibshirani (1990) Generalized Additive Models. Chapman and Hall.
Hastie, T. & R. Tibshirani (1987) Generalized additive models: Some applications. Journal of the American Statistical Association 82, 371–386.Google Scholar
Horowitz, J. (2001) Estimating generalized additive models. Econometrica 69, 499–513.Google Scholar
Jones, M.C., S.J. Davies, & B.U. Park (1994) Versions of kernel-type regression estimators. Journal of the American Statistical Association 89, 825–832.Google Scholar
Kim, W., O. Linton, & N. Hengartner (1999) A computationally efficient oracle estimator of additive nonparametric regression with bootstrap confidence intervals. Journal of Computational and Graphical Statistics 8, 1–20.Google Scholar
Linton, O.B. (1996) Efficient estimation of additive nonparametric regression models. Biometrika 84, 469–474.Google Scholar
Linton, O.B. (2000) Efficient estimation of generalized additive nonparametric regression models. Econometric Theory 16, 502–523.Google Scholar
Linton, O.B. & W. Härdle (1996) Estimating additive regression models with known links. Biometrika 83, 529–540.Google Scholar
Linton, O.B. & J. Nielsen (1995) A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82, 93–100.Google Scholar
Linton, O.B., J. Nielsen, & S. van de Geer (2003) Estimating multiplicative and additive hazard functions by kernel methods. Annals of Statistics 31, 464–492.Google Scholar
Linton, O.B., N. Wang, R. Chen, & W. Härdle (1995) An analysis of transformation for additive nonparametric regression. Journal of the American Statistical Association 92, 1512–1521.Google Scholar
Mammen, E., O.B. Linton, & J. Nielsen (1999) The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Annals of Statistics 27, 1443–1490.Google Scholar
Masry, E. (1996) Multivariate local polynomial regression for time series: Uniform strong consistency and rates. Journal of Time Series Analysis 17, 571–599.Google Scholar
Masry, E. & D. Tjøstheim (1995) Nonparametric estimation and identification of nonlinear ARCH time series: Strong convergence and asymptotic normality. Econometric Theory 11, 258–289.Google Scholar
Masry, E. & D. Tjøstheim (1997) Additive nonlinear ARX time series and projection estimates. Econometric Theory 13, 214–252.Google Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347–370.Google Scholar
Newey, W.K. (1994) Kernel estimation of partial means. Econometric Theory 10, 233–253.Google Scholar
Opsomer, J.D. & D. Ruppert (1997) Fitting a bivariate additive model by local polynomial regression. Annals of Statistics 25, 186–211.Google Scholar
Pollard, D. (1990) Empirical Processes: Theory and Applications. CBMS Conference Series in Probability and Statistics, vol. 2. Institute of Mathematical Statistics.
Robinson, P.M. (1983) Nonparametric estimation for time series models. Journal of Time Series Analysis 4, 185–208.Google Scholar
Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall.
Stein, E.M. (1970) Singular Integrals and Differentiability Properties of Functions. Princeton University Press.
Stone, C.J. (1985) Additive regression and other nonparametric models. Annals of Statistics 13, 685–705.Google Scholar
Stone, C.J. (1986) The dimensionality reduction principle for generalized additive models. Annals of Statistics 14, 592–606.Google Scholar
Tjøstheim, D. & B. Auestad (1994) Nonparametric identification of nonlinear time series: Projections. Journal of the American Statistical Association 89, 1398–1409.Google Scholar
Volkonskii, V. & Y. Rozanov (1959) Some limit theorems for random functions. Theory of Probability and Applications 4, 178–197.Google Scholar
Yang, L., W. Härdle, & J. Nielsen (1999) Nonparametric autoregression with multiplicative volatility and additive mean. Journal of Time Series Analysis 20, 579–604.Google Scholar
Ziegelmann, F. (2002) Nonparametric estimation of volatility functions: The local exponential estimator. Econometric Theory 18, 985–992.Google Scholar