zbMATH — the first resource for mathematics

Econometric estimation in long-range dependent volatility models: theory and practice. (English) Zbl 1429.62463
Summary: It is commonly accepted that some financial data may exhibit long-range dependence, while other financial data exhibit intermediate-range dependence or short-range dependence. These behaviours may be fitted to a continuous-time fractional stochastic model. The estimation procedure proposed in this paper is based on a continuous-time version of the Gauss-Whittle objective function to find the parameter estimates that minimize the discrepancy between the spectral density and the data periodogram. As a special case, the proposed estimation procedure is applied to a class of fractional stochastic volatility models to estimate the drift, standard deviation and memory parameters of the volatility process under consideration. As an application, the volatility of the Dow Jones, S&P 500, CAC 40, DAX 30, FTSE 100 and NIKKEI 225 is estimated.

62P05 Applications of statistics to actuarial sciences and financial mathematics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15 Inference from stochastic processes and spectral analysis
Full Text: DOI
[1] Andersen, T.G.; Lund, J., Estimating continuous-time stochastic volatility models of the short term interest rate, Journal of econometrics, 77, 343-378, (1997) · Zbl 0925.62529
[2] Andersen, T.G.; Sørensen, B.E., GMM estimation of a stochastic volatility model: A Monte Carlo study, Journal of business and economic statistics, 14, 328-352, (1996)
[3] Andrews, D.W.K.; Sun, Y., Adaptive local polynomial Whittle estimation of long-range dependence, Econometrica, 72, 569-614, (2004) · Zbl 1131.62317
[4] Anh, V.; Heyde, C., Journal of statistical planning & inference, 80, 1, (1999), Special Issue on Long-Range Dependence
[5] Anh, V.; Heyde, C.; Leonenko, N., Dynamic models of long memory processes driven by Lévy noise, Journal of applied probability, 39, 730-747, (2002) · Zbl 1016.60039
[6] Anh, V.; Inoue, A., Financial markets with memory I: dynamic models, Stochastic analysis & its applications, 23, 275-300, (2005) · Zbl 1108.91035
[7] Anh, V.; Inoue, A.; Kasahara, Y., Financial markets with memory II: innovation processes and expected utility maximization, Stochastic analysis & its applications, 23, 301-328, (2005) · Zbl 1108.91036
[8] Anh, V.; Leonenko, N.N.; Sakhno, L.M., On a class of minimum contrast estimators for fractional stochastic processes and fields, Journal of statistical planning and inference, 123, 161-185, (2004) · Zbl 1103.62092
[9] Arapis, M.; Gao, J., Empirical comparisons in short-term interest rate models using nonparametric methods, Journal of financial econometrics, 4, 310-345, (2006)
[10] Asai, M.; McAleer, M.; Yu, J., Multivariate stochastic volatility: A review, Econometric reviews, 25, 145-175, (2006) · Zbl 1107.62108
[11] Baillie, R.; King, M.L., Annals of econometrics, 73, 1, (1996), Special Issue of Journal of Econometrics
[12] Barndorff-Nielsen, O.E.; Leonenko, N.N., Spectral properties of superpositions of ornstein – uhlenbeck type processes, Methodology and computing in applied probability, 7, 335-352, (2005) · Zbl 1089.60014
[13] Beran, J., Statistics for long-memory processes, (1994), Chapman and Hall New York · Zbl 0869.60045
[14] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 3, 637-654, (1973) · Zbl 1092.91524
[15] Breidt, F.J.; Crato, N.; de Lima, P.J.F., The detection and estimation of long-memory in stochastic volatility, Journal of econometrics, 83, 325-348, (1998) · Zbl 0905.62116
[16] Brockwell, P.; Davis, R., Time series: theory and methods, (1990), Springer New York
[17] Casas, I., Gao, J., 2006. Econometric estimation in stochastic volatility models with long-range dependence. Working Paper available from the Authors
[18] Comte, F., Simulation and estimation of long memory continuous time models, Journal of time series analysis, 17, 19-36, (1996) · Zbl 0836.62060
[19] Comte, F.; Renault, E., Long memory continuous-time models, Journal of econometrics, 73, 101-150, (1996) · Zbl 0856.62104
[20] Comte, F.; Renault, E., Long memory in continuous-time stochastic volatility models, Mathematical finance, 8, 291-323, (1998) · Zbl 1020.91021
[21] Cox, J.C.; Ingersoll, J.E.; Ross, S.A., A theory of the term structure of interest rates, Econometrica, 53, 385-407, (1985) · Zbl 1274.91447
[22] Deo, R.; Hurvich, C.M., On the log-periodogram regression estimator of the memory parameter in long memory stochastic volatility models, Econometric theory, 17, 686-710, (2001) · Zbl 1018.62079
[23] Ding, Z.; Granger, C.W.J.; Engle, R., A long memory property of stock market returns and a new model, Journal of empirical finance, 1, 83-105, (1993)
[24] Fan, J.; Zhang, C.M., A re-examination of stanton’s diffusion estimators with applications to financial model validation, Journal of the American statistical association, 461, 118-134, (2003)
[25] Fox, R.; Taquu, M.S., Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Annals of statistics, 14, 512-532, (1986)
[26] Gao, J., Modelling long-range dependent Gaussian processes with application in continuous-time financial models, Journal of applied probability, 41, 467-482, (2004) · Zbl 1046.60038
[27] Gao, J.; Anh, V.; Heyde, C., Statistical estimation of nonstationary Gaussian processes with long-range dependence and intermittency, Stochastic processes & their applications, 99, 295-321, (2002) · Zbl 1059.60024
[28] Gao, J.; Anh, V.; Heyde, C.; Tieng, Q., Parameter estimation of stochastic processes with long-range dependence and intermittency, Journal of time series analysis, 22, 517-535, (2001) · Zbl 0979.62071
[29] Geweke, J.; Porter-Hudak, S., The estimation and application of long memory time series models, Journal of time series analysis, 4, 221-238, (1983) · Zbl 0534.62062
[30] Giraitis, I.; Surgailis, D., A central limit theorem for quadratic forms in strongly dependent linear variables and its applications to the asymptotic normality of Whittle estimates, Probability theory and related fields, 86, 87-104, (1990) · Zbl 0717.62015
[31] Hannan, E.J., The asymptotic theory of linear time-series models, Journal of applied probability, 10, 130-145, (1973) · Zbl 0261.62073
[32] Harvey, A.C., Forecasting, structural time series models and the Kalman filter, (1989), Cambridge University Press New York
[33] Harvey, A.C., Long memory in stochastic volatility, ()
[34] Heyde, C.; Gay, R., Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence, Stochastic processes and their applications, 45, 169-182, (1993) · Zbl 0771.60021
[35] Hosoya, Y., A limit theory for long-range dependence and statistical inference on related models, Annals of statistics, 25, 105-137, (1997) · Zbl 0873.62096
[36] Hull, J.; White, A., The pricing of options on assets with stochastic volatilities, Journal of finance, 2, 281-300, (1987)
[37] Leonenko, N.N.; Sakhno, L.M., On the Whittle estimators for some classes of continuous parameter random processes and fields, Statistics and probability letters, 76, 781-795, (2006) · Zbl 1089.62101
[38] Priestley, M.B., Spectral analysis and time series, (1981), Academic Press New York · Zbl 0537.62075
[39] Robinson, P., Rates of convergence and optimal spectral bandwidth for long range dependence, Probability theory and related fields, 99, 443-473, (1994) · Zbl 0801.60030
[40] Robinson, P., Gaussian semiparametric estimation of long-range dependence, Annals of statistics, 23, 1630-1661, (1995) · Zbl 0843.62092
[41] Time series with long memory, () · Zbl 1113.62106
[42] Stoyanov, J., Stieltjes classes for moment-indeterminate probability distributions, Journal of applied probability, 41A, 281-294, (2004) · Zbl 1070.60012
[43] Sun, Y.; Phillips, P.C.B., Nonlinear log-periodogram regression for perturbed fractional processes, Journal of econometrics, 115, 355-389, (2003) · Zbl 1027.62067
[44] Taylor, S.J., Modelling financial time series, (1986), John Wiley Chichester, UK · Zbl 1130.91345
[45] Taylor, S.J., Modelling stochastic volatility: A review and comparative study, Mathematical finance, 4, 183-204, (1994) · Zbl 0884.90054
[46] Vasicek, O., An equilibrium characterization of the term structure, Journal of financial economics, 5, 177-188, (1977) · Zbl 1372.91113
[47] Whittle, P., Hypothesis testing in time series analysis, (1951), Hafner New York · Zbl 0045.41301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.