Seasonal Mackey–Glass–GARCH process and short-term dynamics

Abstract

The aim of this article is the study of complex structures which are behind the short-term predictability of stock returns series. In this regard, we employ a seasonal version of the Mackey–Glass–GARCH(p,q) model, initially proposed by Kyrtsou and Terraza (Computat Econ 21:257–276, 2003) and generalized by Kyrtsou (Int J Bifurcat Chaos 15(10):3391–3394, 2005). To unveil short or long memory components and non-linear structures in the French Stock Exchange (CAC40) returns series, we apply the test of Geweke and Porter-Hudak (J Time Ser Anal 4:221–238, 1983), the Brock et al. (Econom Rev 15:197–235, 1996) and Dechert (An application of chaos theory to stochastic and deterministic observations. Working paper, University of Houston, 1995) tests, the correlation-dimension method of Grassberger and Procaccia (Phys 9D:189–208, 1983), the Lyapunov exponents method of Gencay and Dechert (Phys D 59:142–157, 1992), and the Recurrence quantification analysis introduced by Webber and Zbilut (J Appl Physiol 76:965–973, 1994). As a confirmation procedure of the dynamics generating future movements in CAC40, we perform forecast with the use of a seasonal Mackey–Glass–GARCH(1,1) model. The interest of the forecasting exercise is found in the inclusion of high-dimensional non-linearities in the mean equation of returns.