(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 230269, 6412]*) (*NotebookOutlinePosition[ 231205, 6442]*) (* CellTagsIndexPosition[ 231161, 6438]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ StyleBox[ "Testing Chaos and Fractal Properties in Economic Time Series", "Title", FontFamily->"Palatino", FontSize->16], "\n\nMaria I. Loffredo\n\nDipartimento di Matematica, \nUniversit\[AGrave] \ di Siena, I-53100 Siena, Italy\ne-mail : loffredo@unisi.it" }], "Title", CellMargins->{{Inherited, 118}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontFamily->"Palatino", FontSize->16, FontWeight->"Bold"], Cell[CellGroupData[{ Cell["Abstract", "Section", CellMargins->{{Inherited, 118}, {Inherited, Inherited}}, FontFamily->"Palatino"], Cell[TextData[{ "Search for empirical evidence of chaos and testing fractal and other \ statistical properties in the framework of time series analysis are carried \ on as a preparatory step in order to apply these concepts to data proper of \ Financial Markets and deal with the puzzling failure of traditional economic \ theories.\nConcepts like correlation dimension and Lyapunov exponents are \ discussed and simple ", StyleBox["Mathematica ", FontSlant->"Italic"], "programs are given for their evaluation. Before their application to real \ economic data, a test on a well known nonlinear dynamical system, through the \ correspondent reconstructed phase space and time series, is carried out. " }], "Text", CellMargins->{{Inherited, 118}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, FontFamily->"Palatino"] }, Open ]], Cell[CellGroupData[{ Cell["1. Introduction", "Section", CellMargins->{{Inherited, 118}, {Inherited, Inherited}}, FontFamily->"Palatino"], Cell[TextData[{ "As well known, economic theories have traditionally been dominated by a \ linear modelling, based on concepts like Gaussian distributions and random \ walks, generating the so called Capital Market Theory built on the assumption \ of normally distributed returns and the Efficient Market Hypothesis (EMH), by \ which the markets follow a random walk. As a first consequence the future \ would be unrelated to the past or the present, with no possibility of \ identifying trends or cycles. \nOn the contrary, as is proven by several \ results starting from the discrepancy between the natural consequences of the \ Efficient Market hypothesis and the real behavior of financial time series, \ financial markets behave in a non-linear fashion [1, 2, 3, 4]. Actually \ several empirical studies have attempted to prove the Gaussian assumption, \ but have often delivered contrary results. \nSince the first works by \ Mandelbrot [5], many efforts have been made to extract suitable information \ from the behavior of financial markets in order to include their description \ in the framework of chaos theory and sciences of complexity and non-linearity \ [1, 2, 6]. \nMandelbrot hypothesis and the related fractal analysis, by which \ returns belong to a family of Stable Paretian distributions, have already \ been discussed in a previous paper [7] with particular emphasis on the \ comparison between empirical results and the analysis based on the so-called \ Fractal Market Hypothesis [1, 3]. In this model, the returns follow a biased \ random walk, mathematically described by the so-called fractional Brownian \ motion and empirically obtained through fractal time series. In [7], based \ on the original ideas by Mandelbrot, the R/S analysis was carried out and the \ correspondent Hurst exponents evaluated, as a first step in the recognition \ and characterization of the complex dynamics underlying economic systems and \ its relation with fractional Brownian motion [1, 8 ]. This analysis, being \ well suited to distinguish a random series from a non-random one, allows us \ to give a qualitative description of the behavior of markets. \nFractal time \ series are characterized as long memory processes. They possess cycles and \ trends, and are the result of a nonlinear dynamics and deterministic chaos. \ Information is not immediately reflected in prices, as the EMH states, but \ instead manifests itself as a bias in returns.\nFractal analysis and chaos \ theory are closely related, in the sense that nonlinear systems can be \ statistically described using fractals, and analytically examined using chaos \ theory. \n\nIn the present paper, thanks to the potentiality of ", StyleBox["Mathematica ", FontSlant->"Italic"], "in handling long time series and to the availability of special Packages \ like \"Statistics\", an empirical investigation of the properties of discrete \ time series is carried out, within the framework of non-linear dynamic \ theory. This is pursued in order to look beyond random walks and related \ theories, towards models of complexity and deterministic chaos, in which \ chaotic properties are generated by the intrinsic nonlinearities of the \ system. \nThrough concepts like phase space recostruction, correlation \ dimension and Lyapunov exponents, properly adapted to time series, we try to \ give a quantitative measure of correlations and dimensions and investigate \ the existence of long-term correlations and the fading away of initial \ information. \nThe final aim is to apply these programs to the distributions \ of stock market price changes, but preliminarly they have been tested on a \ well known non-linear model, for which the existence of deterministic chaos \ has been separately proved. " }], "Text", CellMargins->{{Inherited, 118}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, FontFamily->"Palatino"] }, Open ]], Cell[CellGroupData[{ Cell["2. Time Series Analysis and Correlation Dimension", "Section", CellMargins->{{Inherited, 118}, {Inherited, Inherited}}, FontFamily->"Palatino"], Cell["\<\ In dynamical systems theory, chaos means irregular fluctuations in \ a deterministic system: the system behaves irregularly because of its own \ internal structure, and not because of random forces acting from outside. \ Quite a voluminous literature regarding nonlinear dynamics exists (see, for \ example Refs [9], [10]), but it is mostly based on non-linear differential \ equations, one for each variable describing the system, all of them defining \ the so-called phase space. The equations usually depend on a few control \ parameters, for some values of which such systems show up chaotic behavior. The new aspects raised by dynamical systems theory are the implied geometric \ view of temporal behavior and the existence of geometric invariants, such as \ fractal dimensions and Lyapunov exponents. The difficulty of studying stock markets data or any other experimental \ series of data coming from the empirical monitoring of observations, is that, \ in all these cases, all information at our disposal consists in discrete \ sequences of numbers, without any reference to differential equations. This difficulty has been overcome thanks to the idea of phase space \ reconstruction by Ruelle [11]. This allows the application of dynamical \ systems techniques to series of data, usually obtained by monitoring the \ value of a single observable as a function of time, with the consequent \ extraction of geometric information from it. The general idea is to generate \ several different scalar signals from the original signal x(t) obtained \ through time delays. For example, for the Henon map which is mathematically described in a \ two-dimensional space obtained by the variables {x(t), y(t)} satisfying the \ system of non linear equations\ \>", "Text", CellMargins->{{Inherited, 118}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, FontFamily->"Palatino"], Cell[BoxData[{ \(x[0] = .1; \ny[0] = .1; \na = 1.4; \nb = 0.3; \n x[t_] := \(x[t] = 1 + y[t - 1] - a*x[t - 1]^2\)\), \(y[t_] := \(y[t] = b*x[t - 1]\)\)}], "Input", CellMargins->{{Inherited, 118}, {Inherited, Inherited}}], Cell["\<\ we can extract the following discrete time series of values plotted \ together:\ \>", "Text", CellMargins->{{Inherited, 118}, {Inherited, Inherited}}, FontFamily->"Palatino"], Cell[CellGroupData[{ Cell[BoxData[ \(ff = Table[x[t], {t, 0, 100}]; \ngg = Table[y[t], {t, 0, 100}]; \n p1 = ListPlot[ff, PlotJoined -> True, \ \n\ \ \ PlotStyle -> RGBColor[1, 0, 0], DisplayFunction -> Identity]; \n p2 = ListPlot[gg, PlotJoined -> True, \n\ \ \ PlotStyle 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