Abstract.
The GARCH (p, q) model is a very interesting stochastic process with widespread applications and a central role in empirical finance. The Markovian GARCH (1, 1) model has only 3 control parameters and a much discussed question is how to estimate them when a series of some financial asset is given. Besides the maximum likelihood estimator technique, there is another method which uses the variance, the kurtosis and the autocorrelation time to determine them. We propose here to use the standardized 6th moment. The set of parameters obtained in this way produces a very good probability density function and a much better time autocorrelation function. This is true for both studied indexes: NYSE Composite and FTSE 100. The probability of return to the origin is investigated at different time horizons for both Gaussian and Laplacian GARCH models. In spite of the fact that these models show almost identical performances with respect to the final probability density function and to the time autocorrelation function, their scaling properties are, however, very different. The Laplacian GARCH model gives a better scaling exponent for the NYSE time series, whereas the Gaussian dynamics fits better the FTSE scaling exponent.
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References
R. Mantegna, H.E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge University Press, Cambridge, 1999)
J.P. Bouchaud, M. Potters, Theory of Financial Risks: From Statistical Physics to Risk Management (Cambridge University Press, Cambridge, 2000)
T. Bollerslev, Econometrics 31, 307 (1986)
S. Taylor, Modelling Financial Time Series (Wiley, New York, 1986)
T. Bollerslev, R.F. Engle, D.B. Nelson, ARCH Models, in HandBook of Econometrics (Elsevier, New York, 1994)
M.A. Carnero, D. Peña, E. Ruiz, Journal of Financial Econometrics 2, 319 (2004)
T. Anderson, T. Bollerslev, Int. Econ. Rev. 39, 885 (1998)
J.P. Morgan, Riskmetrics - Technical Document (New York, 1996)
R.F. Engle, V. Ng, Journal of Finance 43, 1749 (1993)
J. Marcucci, Studies in Nonlinear Dynamics and Econometrics 9, 4 (2005)
D.B. Nelson, Econometrica 59, 347 (1991)
F. Fornari, A. Mele, Economic Letters 50, 197 (1996)
L.R. Glosten, R. Jagannathan, D.E. Runkle, Journal of Finance 46, 1779 (1992)
Z.C. Ding, W.J. Granger, R.F. Engle, Journal of Empirical Finance 1, 83 (1993)
Q. Liu, D. Morimune, Asia Pacific Financial Markets 12, 29 (2006)
I. Cohen, Signal Processing 84, 2453 (2004)
P. Christoffersen, S. Heston, K. Jacobs, Journal of Econometrics 131, 253 (2006)
R.F. Engle, A. Patton, Quantitative Finance 1, 237 (2001)
S. Laurent, L. Bauwnes, J.V.K. Rombouts, Journal of Applied Econometrics 21, 79 (2004)
M. McAleer, Econometric Theory 21, 232 (2005)
K. Morimune, The Japanese Economic Review 58, 1 (2007)
S. Ling, M. McAleer, Econometric Theory 19, 280 (2003)
C. Broto, E. Ruiz, Journal of Economic Surveys 18, 613 (2004)
J.C. Hull, Options, Futures and Other Derivatives (Prentice Hall, New Jersey, 2002)
R.F. Engle, Econometrica 50, 987 (1982)
R.A. Fisher, Messenger of Mathematics 41, 155 (1912)
MATLAB is a registered trademark of The MathWorks Inc. (2006)
V. Akgiray, J. Business 62, 55 (1989)
R.T. Baillie, T. Bollerslev, J. Econometrics 52, 91 (1992)
Maple is a registered trademark of Waterloo Maple Inc. (2005)
I.P. Levin, Relating statistics and experimental design (Sage Publication, 1999)
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Onody, R., Favaro, G. & Cazaroto, E. A new estimator method for GARCH models. Eur. Phys. J. B 57, 487–493 (2007). https://doi.org/10.1140/epjb/e2007-00191-6
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DOI: https://doi.org/10.1140/epjb/e2007-00191-6