EconPapers FAQ Archive maintainers FAQ Cookies at EconPapers Format for printing The RePEc blog The RePEc plagiarism page The Markov-switching multi-fractal model of asset returns: GMM estimation and linear forecasting of volatilityThomas LuxNo 2004-11, Economics Working Papers from Christian-Albrechts-University of Kiel, Department of EconomicsAbstract:Multi-fractal processes have recently been proposed as a new formalism for modelling the time series of returns in finance. The major attraction of these processes is their ability to generate various degrees of long memory in different powers of returns - a feature that has been found in virtually all financial data. Initial difficulties stemming from non-stationarity and the combinatorial nature of the original model have been overcome by the introduction of an iterative Markov-switching multi-fractal model in Calvet and Fisher (2001) which allows for estimation of its parameters via maximum likelihood and Bayesian forecasting of volatility. However, applicability of MLE is restricted to cases with a discrete distribution of volatility components. From a practical point of view, ML also becomes computationally unfeasible for large numbers of components even if they are drawn from a discrete distribution. Here we propose an alternative GMM estimator together with linear forecasts which in principle is applicable for any continuous distribution with any number of volatility components. Monte Carlo studies show that GMM performs reasonably well for the popular Binomial and Lognormal models and that the loss incured with linear compared to optimal forecasts is small. Extending the number of volatility components beyond what is feasible with MLE leads to gains in forecasting accuracy for some time series.Keywords: Multifractal; Forecasting; Volatility; GMM estimation; Markov-switching (search for similar items in EconPapers)JEL-codes: C20 G12 (search for similar items in EconPapers)Date: 2004References: View references in EconPapers View complete reference list from CitEc Citations: View citations in EconPapers (2) Track citations by RSS feedDownloads: (external link) -2004-11.pdf (application/pdf)Related works:Journal Article: The Markov-Switching Multifractal Model of Asset Returns: GMM Estimation and Linear Forecasting of Volatility (2008) Working Paper: The Markov-Switching Multifractal Model of asset returns: GMM estimation and linear forecasting of volatility (2006) This item may be available elsewhere in EconPapers: Search for items with the same title.Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/TextPersistent link: :zbw:cauewp:2442Access Statistics for this paperMore papers in Economics Working Papers from Christian-Albrechts-University of Kiel, Department of Economics Contact information at EDIRC.Bibliographic data for series maintained by ZBW - Leibniz Information Centre for Economics (Obfuscate( 'zbw.eu', 'econstor-publish' )). var addthis_config = "data_track_clickback":true; var addthis_share = url:" :zbw:cauewp:2442"Share This site is part of RePEc and all the data displayed here is part of the RePEc data set. Is your work missing from RePEc? Here is how to contribute. Questions or problems? Check the EconPapers FAQ or send mail to Obfuscate( 'oru.se', 'econpapers' ). EconPapers is hosted by the Örebro University School of Business.
Fractal Forecasting Downloads To
A prediction model of thermal contact conductance is developed. Engineering rough surfaces are characterized by three-dimensional fractal Weierstrass-Mandelbrot fractal function. Three deformation modes, including fully plastic deformation, elasto-plastic deformation and elastic deformation, are considered to analyze the contact mechanism. Fractal surface and three deformation modes are incorporated into the calculation of thermal contact conductance. A comprehensive thermal contact conductance computation model considering both base thermal resistance and constricted thermal resistance is established. The results show that thermal contact conductance increases with the increase of normal contact pressure; the relative contribution of constricted resistance component to base resistance component tends to increase with the increase of normal contact pressure; fractal dimension and fractal roughness both have significant influences on thermal contact conductance.
Predicting the future state of a nonlinear dynamical system may be very challenging. Recently the use of sophisticated prediction techniques, like neural networks, has allowed researchers to improve the prediction ability in such systems [1]. But this type of methods cannot be always easily applied. In many nonlinear dynamical systems, complex structures arise and change their shape within phase space as one parameter is varied. Basins of attraction are an interesting example of these structures in dissipative and Hamiltonian systems. Roughly speaking, we can say that a basin of attraction is the set of initial conditions that evolve in time towards a given attractor. In many nonlinear systems there are several attractors coexisting in phase space, which can have fractal boundaries separating their basins. This fact can make the study of the global dynamics and the predictability of the system a very difficult task. Nonlinear systems with fractal basins can be classified basically in four different categories: intertwinned basins, Wada basins, riddled basins and sporadically fractal basins [2]. When a dynamical system possesses this kind of basins it is very difficult to make predictions, due to the fact that there is an intrinsic uncertainty on the final state of a given initial condition taken in the neighborhood of the fractal boundary. The physical reason behind this is the finite accuracy in the measurement of the initial conditions for any real system. Furthermore, in systems with fractal basins there are infinitely many close points that can go to a different attractor. The situation gets even more complicated if we do not have access to the time series of the dynamical system and the only observables of the system are the attractors.
Although the problem is far from being solved, recently two useful ideas proposed by Menck et al. and Daza et al. namely basin stability [3] and basin entropy [4] have shed some light on important properties of complex basin structures. Here, we present a general procedure to provide some kind of statistical prediction in nonlinear systems with fractal basins, where the only observables that we have access to are the attractors of the system. We assume that we are not able to measure the time series before they reach the final attractor, but we assume that we have some knowledge about the probability density function of the initial conditions. In this way, we consider that the dynamical system is like a black box, as depicted in Fig 1, where only the final output can be measured. In this framework, the behavior of the dynamical system is very similar to that of a die, although the behavior of this one is neither chaotic nor random [5]. The key point of the prediction mechanism developed here, is (as we show in several ways) that the ratio or probability of initial conditions going to each attractor in the phase space is scale free. This is precisely what allows the statistical prediction. We show here how this procedure works for Wada basins, but it should also work for systems showing any of the other kind of fractal basins.
A dynamical system has Wada basins if it has three or more basins sharing the same fractal boundary. This topological idea was introduced by Kennedy and Yorke [6]. Wada basins usually appear in two-dimensional dynamical systems as a result of a boundary crisis of a chaotic attractor. This fact often leads to the fractalization of the entire basin boundary. Wada basin boundaries are frequently observed in both dissipative and Hamiltonian systems. We can find this topological property in relation to mechanical models of billiard balls [7] or chaotic advection of fluid flow [8] and in the context of the Hénon-Heiles Hamiltonian system in celestial mechanics [9]. Due to the structural complexity of the Wada basin boundaries, in practice, these structures imply serious problems in the long term prediction of dynamical systems, also known as final state sensitivity [10, 11].
Here, we study the Duffing oscillator for a choice of parameters that verifies the Wada property, based on the work of Aguirre and Sanjuan [12]. The Duffing oscillator is one of the best known models of nonlinear oscillators, with applications in many fields of applied sciences and engineering. The structure of the paper is as follows. Section 2 is devoted to the description of the Duffing oscillator and the methodology used to explore its phase space. The one-dimensional analysis of the model is described in Section 3. The two-dimensional analysis is done in Section 4. The implication of fractal boundaries on the probabilities of each basin of attraction is given in Section 5. Finally, some conclusions are drawn in the last section.
An easy way to compute the total probabilities is taking a uniform two-dimensional grid and computing the ratio of initial conditions that belongs to each basin of attraction. We have done this for different resolutions of the grid as we can see in Fig 5, where it can be clearly observed that the pattern of the basins of attraction is almost stable for resolutions higher than 300 300. All the basins of attraction keep their shape near the location of the attractors, but as we move away from them, they begin to mix and become fractal. In Table 1 we summarize the number of initial conditions taken for every resolution and going to each attractor.
For very low grid resolutions there is a large change in the probabilities going to each attractor. But beyond a given threshold in the resolution, the probabilities remain constant. This is what we show in Fig 6. We can clearly see how, for a resolution of 300 300 or higher, the probabilities converge to constant values. The total probability of landing in the period-3 attractor P(P3) (blue basin) converges to 0.456 (45.6%), the total probability of landing in the period-1 attractor to the right P(P1R) (green basin) converges to 0.270 (27%) and the total probability of ending in the period-1 attractor to the left P(P1L) (red basin) converges to 0.274 (27.4%). This clearly indicates that the results are robust and can be used in the statistical prediction that we are looking for. As expected, due to the convergence of the Perron-Frobenius operator these probabilities are scale free. Fig 7 shows how the probability of each attractor changes depending on its location over the phase space. Now we can actually visualize why and even where the probability of being in the basin of the period-3 attractor, for example, is greatest over the phase space. The orange color on the left panel in Fig 7 illustrates how the high probability of the period-3 attractor dominates in the fractalized zones, while in the other two panels the dark red color illustrates the low probability of the period-1 attractors over the same palaces in the phase space. The fact that the fractal zones occupy a larger area of the phase space explains why at the aggregate level we obtain the results above. We can state that the long term dynamics of this system depends on the attractor that governs in the fractal zones. 2ff7e9595c
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