Abstract
Financial time series data are typically observed to have heavy tails and time-varying volatility. Conditional heteroskedastic models to describe this behaviour have received considerable attention. In the present paper, our purpose is to examine some of these models in a general setting under some non-normal distributions. A likelihood based approach to estimation is used. New comparisons of estimators and their efficiencies are discussed.
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References
Bera A, Higgins M (1995) On ARCH models: properties, estimation and testing. In: Oxley L, George DAR, Roberts CJ, Sayer S (eds) Surveys in econometrics. Blackwell, Oxford, pp 215–272
Berkes I, Horváth L, Kokoszka P (2005) Near-integrated GARCH sequences. Ann Appl Prob 1B:890–913
Berndt E, Hall B, Hall R, Hausman J (1974) Estimation and inference in nonlinear structural models. Ann Econ Soc Meas 3:653–665
Bingham NH, Kiesel R (2002) Semi-parametric modelling in finance: theoretical foundations. Quant Finance 2:368–385
Bollerslev T (1987) A conditionally heteroskedastic time series model for speculative prices and rates of return. Rev Econ Stat 69:542–547
Breusch TS, Robertson JC, Welsh AH (1997) The emperor’s new clothes: a critique of the multivariate t regression model. Statist Neerlandica 51:269–286
Carmona R (2004) Statistical analysis of financial data in S-PLUS. Springer, Berlin Heidelberg New York
Engle RF (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50:987–1006
Engle RF, Bollerslev T (1986) Modelling the persistence of conditional variance. Econ Rev 5:1–50
Engle RF, Lilien DM, Robins RP (1987) Estimating time-varying risk premia in the term structure: the ARCH-M model. Econometrica 55:391–407
Evans M, Hastings N, Peacock B (2000) Statistical distributions. Wiley, New York
Fang KT, Zhang Y (1990) Generalized multivariate analysis. Science Press, Beijing and Springer, Berlin Heidelberg New York
Fomby T, Hill RC, Johnson SR (1984) Advanced econometric methods. Springer, Berlin Heidelberg New York
Gouriéroux C (1997) ARCH models and financial applications. Springer, Berlin Heidelberg New York
Gupta AK, Varga T (1993) Elliptically contoured models in statistics. Kluwer, Dordrecht
Hamilton JD (1994) Time series analysis. Princeton University Press, Princeton
Heyde CC (1997) Quasi-likelihood and its application: a general approach to optimal parameter estimation. Springer, Berlin Heidelberg New York
Heyde CC (1999) A risky asset model with strong dependence through fractal activity time. J Appl Probab 36:1234–1239
Heyde CC, Feigin PD (1975) On efficiency and exponential families in stochastic process estimation. In: Patil GP et al (eds) Statistical distributions in scientific work. D. Reidel Publishing Company, Dordrecht
Heyde CC, Kou SG (2004) On the controversy over tailweight of distributions. Oper Res Lett 32:399–408
Heyde CC, Liu S (2001) Empirical realities for a minimal descriptions risky asset model. The need for fractal features. J Korean Math Soc 38:1047–1059
Heyde CC, Liu S, Gay R (2001) Fractal scaling and Black-Scholes: the full story. JASSA Autumn 29–32
Hirose H (2000) Maximum likelihood parameter estimation by model augmentation with applications to the extended four-parameter generalized gamma distribution. Math Comput Simul 54:81–97
Insightful Corporation (2002) S+FinMetrics Reference Manual. Seattle
Johnson NL, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2, 2nd edn. Wiley, New York
Knight J, Satchell S (eds) (2001) Returns distributions in finance. Butterworth Heinemann, Oxford
Li WK, Ling S, McAleer M (2002) Recent theoretical results for time series models with GARCH errors. J Econ Surv 16(3):245–269
Liu S (2000) On local influence in elliptical linear regression models. Stat Papers 41:211–224
Liu S (2004) On diagnostics in conditionally heteroskedastic time series models under elliptical distributions. Stochastic Methods Appl J Appl Probab 41A:393–405
Lizieri C, Ward C (2001) The distribution of commercial real estate returns. In: Knight J, Satchell S (eds). Returns distributions in finance. Butterworth Heinemann, Oxford, pp 47-74
Magnus JR, Neudecker H (1999) Matrix differential calculus with applications in statistics and econometrics revised edn. Wiley, Chichester
Mak TK, Wong H, Li WK (1997) Estimation of nonlinear time series with conditional heterscedastic variances by iteratively weighted least squares. Comput Stat Data Anal 24:169–178
McAleer M (2005) Automated inference and learning in modeling financial volatility. Econom Theory 21:232–261
McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management: concepts, techniques, tools. Princeton University Press, Princeton
Nelson DB (1990) ARCH models as diffusion approximations. J Econom 45:7–38
Praetz P (1972) The distribution of share price changes. J Business 45:49–55
Rachev S, Mittnik S (2000) Stable paretian models in finance. Wiley, Chichester
Spanos A (1994) On modeling heteroskedasticity: the Student’s t and elliptical linear regression models. Econom Theory 10:286–315
Tsay RS (2002) Analysis of financial time series. Wiley, New York
Zivot E, Wang J (2005) Modeling financial time series with S-PLUS, 2nd edn. Springer, Berlin Heidelberg New York
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Liu, S., Heyde, C.C. On estimation in conditional heteroskedastic time series models under non-normal distributions. Statistical Papers 49, 455–469 (2008). https://doi.org/10.1007/s00362-006-0026-3
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DOI: https://doi.org/10.1007/s00362-006-0026-3