Time evolution of stochastic processes with correlations in the variance: stability in power-law tails of distributions

Boris Podobnik, Kaushik Matia, Alessandro Chessa, Plamen Ch. Ivanov, Youngki Lee and H.Eugene Stanley

Physica A: Statistical Mechanics and its Applications, 2001, vol. 300, issue 1, 300-309

Abstract: We model the time series of the S&P500 index by a combined process, the AR+GARCH process, where AR denotes the autoregressive process which we use to account for the short-range correlations in the index changes and GARCH denotes the generalized autoregressive conditional heteroskedastic process which takes into account the long-range correlations in the variance. We study the AR+GARCH process with an initial distribution of truncated Lévy form. We find that this process generates a new probability distribution with a crossover from a Lévy stable power law to a power law with an exponent outside the Lévy range, beyond the truncation cutoff. We analyze the sum of n variables of the AR+GARCH process, and find that due to the correlations the AR+GARCH process generates a probability distribution which exhibits stable behavior in the tails for a broad range of values n—a feature which is observed in the probability distribution of the S&P500 index. We find that this power-law stability depends on the characteristic scale in the correlations. We also find that inclusion of short-range correlations through the AR process is needed to obtain convergence to a limiting Gaussian distribution for large n as observed in the data.

Keywords: Random walks; Stochastic processes; Fluctuation phenomena; Central limit theory (search for similar items in EconPapers)
Date: 2001
References: View complete reference list from CitEc
Citations: View citations in EconPapers (7)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:300:y:2001:i:1:p:300-309

DOI: 10.1016/S0378-4371(01)00390-9

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