ARCH–GARCH approaches to modeling high-frequency financial data

We model the power-law stability in distribution of returns for S&P500 index by the GARCH process which we use to account for the long memory in the variance correlations. Precisely, we analyze the distributions corresponding to temporal aggregation of the GARCH process, i.e., the sum of n GARCH variables. The stability in the power-law tails is controlled by the GARCH parameters. We model the crossover behavior in magnitude correlations of returns by the so-called two-FIARCH process. Besides detrended fluctuation analysis, we employ the method proposed by Geweke and Porter-Hudak to estimate the fractional parameter in magnitude correlations.