Physica A 344 (2004) 216–220 ARCH–GARCH approaches to modeling high-frequency financial data Boris Podobnik a,b,Ã , Plamen Ch. Ivanov a , Ivo Grosse c , Kaushik Matia a , H. Eugene Stanley a a Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215, USA b Faculty of Civil Engineering, University of Rijeka, Cara Emina 9, Rijeka 51000, Croatia c Institute of Plant Genetics, Gatersleben, Germany Received 31 December 2003 Available online 23 July 2004 Abstract 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. r 2004 Elsevier B.V. All rights reserved. PACS: 02.50.Ey; 05.40.Fb Keywords: Stochastic processes; Random walks Much work [1–7] have been devoted to determine precisely the functional form of financial distributions since Mandelbrot [3] and Fama [4] have suggested stable Le´ vy ARTICLE IN PRESS www.elsevier.com/locate/physa 0378-4371/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2004.06.120 Ã Corresponding author. Faculty of Civil Engineering, University of Rijeka, Cara Emina 9, Rijeka 51000, Croatia. Tel.: 0038514605588; fax: 0038514680336. E-mail address: bp@phy.hr (B. Podobnik).