Springer Berlin Heidelberg, (2007), 39-68

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There is now a vast literature supporting the empirical stylized fact that the volatility of financial returns, as measured either by the absolute values or the squares of the returns series, exhibits long-memory and has correlations which remain positive for long lags and decay slowly to zero, and an associated stylized fact that the marginal distribution of the returns has heavy tails. There is also evidence to suggest that the returns display 'intermittency', often called 'volatility clustering', and the relatively benign periods of market activity are often interrupted by the occurrence of violent market movements. Reference may be made to Greene and Fielitz (1977), Akgirav and Booth (1988), Ding, Granger and Engle (1993), Kokoszka and Taqqu (1996), Podobnik et al. (2000), Mittnik and Rachev (2000), Kirman and Teyssiere (2002), Cont (2001), among others, and which are just a few among hundreds of publications in this area. At the same time, it is also recognised that the standard ARCH and GARCH models, introduced originally to describe the dependence structure of the volatility, exhibit short range dependence and thus do not capture the long-memory property of the returns, though they do imply volatility clustering and a heavy-tailed marginal distribution. Consequently, several different modifications of the basic ARCH specification have been introduced so as to incorporate the slow decay of correlations, see Baillie et al. (1996), for a review. Many of these modifications are, however, rather ad-hoc and a mathematical theory behind some is not fully developed as yet and indeed a few appear in fact not to exhibit long memory, at least asymptotically, see Giraitis, Kokoszka and Leipus (2000). In this chapter, we consider a new and an entirely different approach to modelling phenomena exhibiting long-memory, intermittency and heavy-tailed marginal distributions, namely by chaotic intermittency maps. This class of maps has witnessed much development in recent years and it marks an important emerging branch of the subject area of Dynamical Systems Theory. It should also be stressed that the idea of using deterministic maps as a candidate class of non-linear, non-stochastic models for Economic time series has an established pedigree by now and their use has previously been considered by several different authors; see, for example, Brock and Hommes (1997). For the purpose of the present discussion, three principal properties of these maps are relevant and which qualify them as a plausible class of models for financial returns. First, unlike some of the standard chaotic maps, for example, the Logistic, Tent and Bernoulli shift maps, the intermittency maps display long memory and have correlations decaying at a sub-exponential rate, meaning at a polynomial rate or even slower. Secondly, the invariant density of these maps can display 'Pareto' tails and thus go down to zero at a polynomial rate. Thirdly, as their name implies, these maps display intermittency and generate time series, called the orbit of the map, which display intermittent chaos, meaning the orbit of the map alternates between laminar and chaotic regions. A brief outline of the main theoretical properties of these maps is given in Section 2; for a more detailed discussion, reference may also be made to Bhansali, Holland and Kokoszka (2004). A further motivation for considering the use of intermittency maps for modelling financial data comes from the work of Mondragon (1999), who has successfully applied a sub-class of intermittency maps for modelling the internet traffic, which is a related yet different example of phenomena exhibiting long-memory and heavy-tailed marginal distributions, see, for example, Park and Willinger (2000). The generic characteristic features of intermittency maps described above are, however, asymptotic and apply, for example, as the lag, u, of the correlation function tends to infinity. Moreover, the bounds on the correlations have been developed for some Holder continuous function of the map time series, wt, say, and it is as yet not known precisely for which functions this bound would actually hold in practice. In Section 3, therefore, we present results of a simulation study aimed at investigating the empirical behaviour of the estimated correlations for three different categories of intermittency maps, namely the Polynomial, Logarithmic and Cusp maps, and for a range of different parameter values defining these maps. In addition, we examine the behaviour of the estimated invariant density for these three categories of maps and also that of the partial correlations and the associated 'linear Gaussian' statistics but when these are computed from a simulated realization of the map. The invariant distribution of the Polynomial and Logarithmic intermittency maps is concentrated on a compact interval, [0, 1]. On the other hand, however, the absolute returns on financial time series could in principle take values over the entire non-negative real line, ℝ+. In Section 4, we accordingly consider transformations, h(wt), which have distributions defined over [0,∞) and investigate empirically the correlation structure and related properties of the transformed series by a simulation study. Currently, there are very few theoretical results on the modelling of absolute returns as they are difficult to treat analytically, see, however, Granger and Ding (1996). The paper concludes in Section 5, where we examine the modelling potential of the intermittency maps for absolute returns and present concluding remarks for direction of future research. © Springer Berlin Heidelberg 2007.

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Bhansali, R., Holland, M. P., & Kokoszka, P. S. (2007). Intermittency, long-memory and financial returns. In *Long Memory in Economics* (pp. 39–68). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-34625-8_2

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