Performance of variance function estimators for autoregressive time series of order one: asymptotic normality and numerical study

• Autorzy: Borkowski P. , Mielniczuk J.
• Języki publikacji: EN

Abstrakty

EN

We study performance of several conditional variance estimators for an autoregressive time series which include local linear smoothers with various bandwidths, local likelihood and difference-based estimators. In the theoretical part, asymptotic normality of the local linear estimator of variance with no mixing assumptions imposed on the underlying process is proved. Moreover, numerical examples performed reveal that a two-stage local linear smoother with a bandwidth, proposed by Ruppert, Sheather and Wand, used to estimate the regression function and a simple rule of thumb bandwidth for variance estimation performs best for variances without much structure, whereas the bandwidth considered by Fan and Yao works very well for much more variable variances.

Słowa kluczowe

EN

autoregressive process; bandwidth; heteroscedasticity; integrated squared error; local linear and local maximum likelihood estimator; difference-based estimator; geometric moment contraction; variance function; volatility

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2012
• Tom: Vol. 41, no. 2
• Strony: 415--441
• Opis fizyczny: Bibliogr. 31 poz., wykr.

Twórcy

autor

Borkowski P.

autor

Mielniczuk J.

• Institute of Computer Science, Polish Academy of Sciences, Jana Kazimierza 5, 01-248 Warsaw,

Bibliografia

• ANGO NZE, P. (1992) Criteres d’ergodicite de quelques modeles a representation markovienne. Comptes Rendus des Seances de I’Academie des Sciences Paris. 315, 1301-1304 ser 1.
• CAI, T.T., LEVINE, M., WANG, L. (2009) Variance function estimation in multivariate nonparametric regression with fixed design. Journal of Multivariate Analysis, 100 (1), 126-136.
• CAI, T.T., WANG, L. (2008) Adaptive variance function estimation in heteroscedastic nonparametric regression. Annals of Statistics, 36, 2025-2054.
• CHEN, L.-H., CHENG, M.-Y., PENG, L. (2009) Conditional variance estimation in heteroscedastic regression models. Journal of Statistical Planning and Inference, 139, 236 - 245.
• CHOW, Y.S., TEICHER, H. (1988) Probability Theory. Indepedence, Interchangeability. Martingales. Springer, New York.
• CwiK, J., KORONACKI, J., MlELNlCZUK, J. (2000) Testing for difference between conditional variance functions of nonlinear time series. Control and Cybernetics, 29, 33-50.
• DETTE, H., MUNK, A., WAGNER, T. (1998) Estimating the variance in non-parametric regression - what is a reasonable choice? Journal of the Royal Statistical Society: Series B, 60, 751-764.
• DIACONIS, P., FREEDMAN, D. (1999) Iterated random functions. SIAM Review, 41, 45-76.
• FAN, J. (2005) A selective overview of nonparametric methods in financial econometrics. Statistical Science, 20, 317-337.
• FAN, J., GlJBELS, I. (1995) Data driven bandwidth selection in local polynomial fitting: variable bandwidth in spatial adaptation. Journal of the Royal Statistical Society: Series B, 57, 371-394.
• FAN, J., GUBELS, I. (1996) Local Polynomial Modelling and its Applications. Chapman & Hall, London.
• FAN, J., YAO, Q. (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika, 85, 645-660.
• FREEDMAN, D. A. (1975) On tail probabilities for martingales. The Annals of Probability, 3, 100-118.
• GASSER, U., SROKA, L., JENNEN-STEINMETZ, C. (1986) Residual variance and residual pattern in nonlinear regression. Biometrika, 73, 625-663.
• GENDRE, X. (2008) Simultaneous estimation of the mean and the variance in heteroscedastic Gaussian regression. Electronic Journal of Statistics, 2, 1345-1372.
• HARDLE, W., TSYBAKOV, A. (1997) Local polynomial estimators of volatility function in nonparametric autoregression. Journal of Econometrics, 81, 223-242.
• LEVINE, M. (2006) Bandwidth selection for a class of difference-based variance estimators in the nonparametric regression: a possible approach. Computational Statistics & Data Analysis, 50, 3404-3431.
• Lu, Z., JIANG, Z. (2001) L1 geometric ergodicity of a multivariate nonlinear AR model with an ARCH term. Statistics & Probability Letters, 51, 121-130.
• McNEiL, A.J., FREY, R. AND EMBRECHTS, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton.
• NEUMANN, M., KREISS, J.P. (1998) Regression-type inference in nonparametric autoregression. Annals of Statistics, 26, 1570-1613.
• RICE, J. (1984) Bandwidth choice for nonparametric kernel regression. Annals of Statistics, 12, 1215-1230.
• RUPPERT, D., SHEATHER, S.J., WAND, M.P. (1995) An effective bandwidth selector for local least squares regression. Journal of The American Statistical Association, 90, 1257-1270.
• SlJBERS, J., FOOT, D., DEN DEKKER, A. J. AND PlNTJENST, W. (2007) Automatic estimation of the noise variance from the histogram of a magnetic resonance image. Physics in Medicine and Biology, 52, 1335.
• STANTON, R. (1997) A nonparametric model of term structure dynamics and the market price of interest rate risk. Journal of Finance, 52, 1973-2002.
• WANG, L., BROWN, L. D., CAI, T. AND LEVINE, M. (2008) Effect of mean on variance function estimation in nonparametric regression. Annals of Statistics, 36, 646-664.
• Wu, W. B. (2005) Nonlinear system theory: Another look at dependence. Proceedings of the National Academy of Sciences USA, 102, 14150-14154.
• Wu, W. B., HUANG, Y., HUANG, Y. (2010) Kernel estimation for time series: an asymptotic theory. Stochastic Processes and their Applications, 120,2412 - 2431.
• Wu, W. B., SHAO, X. (2004) Limit theorems for iterated random functions. Journal of Applied Probability, 41, 425-436.
• YAU, P., KOHN, R. (2003) Estimation and variable selection in nonparametric heteroscedastic regression. Statistics and Computing, 13, 191-208.
• 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.
• YUAN, M., WAHBA, G. (2004) Doubly penalized likelihood estimator in het-eroscedastic regression. Statistics & Probability Letters, 69, 11-20.