tailieunhanh - Modelling Stock Returns with AR-GARCH Processes⋆

The picture of the market that results from our experiments, surprisingly, confirms both the efficient- market academic view and the traders’ view. But each is valid under different circumstances—in different regimes. In both circumstances, we initiate our traders with heterogeneous beliefs clustered randomly in an interval near homogeneous rational expectations. We find that if our agents adapt their forecasts very slowly to new observations of the market’s behavior, the market converges to a rational-expectations regimes. Here “mutant” expectations cannot get a profitable footing; and technical trading, bubbles, crashes, and autocorrelative behavior do not emerge. Trading volume remains low. The efficient-market theory prevails | SORT 28 1 January-June 2004 55-68 Modelling Stock Returns with AR-GARCH Processes Elzbieta Ferenstein 1 2 and Miroslaw Gasowski3 Warsaw Poland Abstract Financial returns are often modelled as autoregressive time series with random disturbances having conditional heteroscedastic variances especially with GARCH type processes. GARCH processes have been intensely studying in financial and econometric literature as risk models of many financial time series. Analyzing two data sets of stock prices we try to fit AR 1 processes with GARCH or EGARCH errors to the log returns. Moreover hyperbolic or generalized error distributions occur to be good models of white noise distributions. MSC Primary 62M10 91 B84 secondary 62M20 Keywords autoregressive process GARCH and EGARCH models conditional heteroscedastic variance financial log returns 1 Introduction Let St t 0 1 . T denote share prices observed at discrete moments. In the considered examples they are daily close prices of Elektrim and Okocim enterprise shares from the Warsaw Stock Exchange over a period 1994-2002. Graphs of the analyzed prices are given in Figures 1 and 3. Let Rt denote the log return at time t so This work was supported by the grant PBZ-KBN-016 P03 99. 1 Address for correspondence Faculty of Mathematics and Information Science. Warsaw University of Technology. Pl. Politechniki 1 00-661 Warsaw Poland 2 Address for correspondence Polish-Japanese Institute of Information Technologies. Koszykowa 86 02-008 Warsaw Poland 3 Bank Gospodarki Zywnosciowej . Kasprzaka 10 16 01-211 Warsaw Poland Received October 2003 Accepted January 2004 56 Modelling Stock Returns with AR-GARCH Processes Rt ln U-J t 1 2 . T. 1 Let Xt Rt - R be the mean-centred process where R denotes the sample mean over the observation period. Within a class of autoregressive processes with white noises having conditional heteroscedastic variances we try to find reasonable models of Xt . Some well known processes from a broad class of GARCH .