It is Possible to Study Chaotic and Arch Behaviour Jointly? Application of a Noisy Mackey-Glass Equation With Heteroskedastic Errors to the Paris Stock Exchange Returns Series

Most recent empirical works that apply sophisticated statistical procedures such as a correlation-dimension method have shown that stock returns are highly complex. The estimated correlation dimension is high and there is little evidence of low-dimensional deterministic chaos. Taking the complex behaviour in stock markets into account, we think it is more robust than the traditional stochastic approach to model the observed data by a non-linear chaotic model disturbed by dynamic noise. In fact, we construct a model having negligible or even zero autocorrelations in the conditional mean, but a rich structure in the conditional variance. The model is a noisy Mackey-Glass equation with errors that follow a GARCH(p,q) process. This model permits us to capture volatility-clustering phenomena. Its characteristic is that volatility clustering is interpreted as an endogenous phenomenon. The main objective of this article is the identification of the underlying process of the Paris Stock Exchange returns series CAC40. To this end, we apply several different tests to detect long-memory components and chaotic structures. Forecasting results for the CAC40 returns series, will conclude this paper