Time Series Analysis: Mandelbrot Theory at work in Economics M. F. Guiducci and M. I. Loflredo Dipartimento di Matematica, Universita di Siena, Siena, Italy Abstract The consequences of the Gaussian hypothesis, which leads to the Efficient Market Hypothesis, are investigated in the framework of time teries tnalysis in economics. The validity of an alternative model, based on Mandelbrot theory, is discussed using the Rescaled Range technique. Hurst exponents related to the underlying fractional Brownian motion are evaluated. 1 Introduction Representing time series for financial markets as non-linear dynamical systems with many degrees of freedom, within the framework of the theory of chaos and fractionals, can be traced back to papers published by Mandelbrot [1,2]. Starting from the discrepancy between the natural consequences of the Efficient Market Hypothesis and the real behavior of financial time series, he took into account a wider choice of underlying probability spaces whose distribution functions could be possible candidates to describe real data. In particular, the description based on fractional Brownian motion is taken as an acceptable alternative. The use of fractional noise can be considered as a generalization of the usual description based on pure Brownian motion and is based mostly on a property of scale invariance of the distributions of returns. This property is typical of fractional objects in nature and is widely used in the study of percolative and aggregation phenomena in complex systems, such as
194 Innovation In Mathematics phase transitions, when intrinsic scales of length cease to exist. In economics this is the basis of so-called Fractal Market Analysis [3]. The application of Mandelbrot's theory to the analysis of real data in economics is the subject of this paper. In particular, we consider the frequency distributions of the (normalized) log-returns from the daily close prices of some stocks. The analysis was made possible using the Mathematica statistics package, analytical capabilities and graphics facilities. The purpose of this paper is two-fold. First, through an empirical analysis, we check and confirm the disagreement between the statistical distribution of real data and the consequences of the Gaussian hypothesis. Secondly, and along the path traced by Mandelbrot, we carry on a more theoretical analysis, by taking into account alternative forms for the underlying probability distribution. In particular, and as suggested in [1] (see also [4]), we consider the distribution functions belonging to the Pareto-Levy family, which can be proved to be related to the previously mentioned fractional Brownian motion. The validity of such a model as a possible candidate that could fit the observed behavior has been checked through the so-called Rescaled Range analysis. This can be considered the equivalent of the renormalization group technique widely used in physics, both being strongly based on the scale invariance property of the system. Our R/S analysis indicates that the time series which we consider have a fractal nature, with a Hurst exponent between 0.5 and 1.0. They are characterized by persistency, self-similarity properties and long-memory effects. 2 Comparison with the Gaussian Hypothesis Based on the Gaussian Hypothesis, which underlies the Efficient Market Hypothesis, markets are considered to be described by random walks with asymptotic normal distributions. Consequently, it is impossible to identify trends and cycles. This hypothesis has been checked by the authors (see also [4]) by considering the normalized log-returns from the daily closing prices of two stocks, Fiat and General!, from January 1973 to December 1995 (about six thousand data points). lfp(f) is the stock price at time t, and dt is a small time interval, the log-return u(f) is given by ii(0 = Log[p(t + do] - LogJXOl = Log[p(f + do/xol A first comparison of the empirical frequency distribution of the data and the Normal distribution shows that the data do not follow the theoretically prescribed random walk. In particular (see [5] for details) it can be shown that the distribution has fat tails: the probability of large standardised log-returns is much higher than would be expected from a Normal distribution. Also, this kind of discrepancy is independent of the particular scale involved: samples of data corresponding to different time scales, all show the same kind of behavior. The existence of fat tails can be demonstrated by measuring the characteristic exponent a, which is such that asymptotically, P(w > U) - If,
Innovation In Mathematics 195 where u is a random variable and [/is a given value of u. \< a < 2 corresponds (asymptotically) to a probability distribution which belongs to the Pareto-Levy family, with the limiting case a =2 for a Normal distribution [1]. A log-log plot of the experimental frequencies for large values of U, it is possible to obtain an estimate of the exponent a. See [4, 5] for details ofthemathematica program. The values of a obtained for Fiat were in the range [1.799, 1.825], while for General! the values were in the range [1.751, 1.786], the limits depending on the existence of left and right tails. Asymptotically, the sequential mean converges to zero and the standard deviation behaves discontinuously and does not converge [3, 5]. This indicates that the variance is infinite and, even if the appropriate random variables are independent and identically distributed, they can converge to another member of a family of Pareto-Levy distributions which does not necessarily correspond to a Normal distribution. These results together suggest that the underlying dynamical system appears to be only locally chaotic but shows global properties of self-similarity, which is a characteristic behavior of fractal objects. Also, the Normal distribution does not seem to be the most suitable one to model the real behaviour of economic time series. 3 Fractional Brownian Motion All the results described to this point indicate the necessity of enlarging the possible underlying probability distributions in order to describe real data. In particular the values of the coefficient a suggest considering generalizations of the usual Brownian motion as the mathematical model for our system. Following Mandelbrot [1], we define fractional Brownian motion through the following generalization of the usual 0.5 power law for an average displacement X(t\ \ X(t + df) - X(f)\ - dtf,where the Hurst exponent H in the range (0, 1), a value of//#1/2 corresponding to a member of the Pareto-Levy family of distributions with a * 2, Such a distribution is not Normal. See Maeder [6] for amathematica program for generating fractional Brownian motion. An estimate of the Hurst exponent can be obtained using rescaled range analysis [3]. This technique has been introduced in order to distinguish random from fractal time series, to recognize persistence of trends, existence of cycles, etc. Roughly speaking, the R/S analysis gives an estimate of the average displacement that the system covers, rescaled by the local standard deviation of the interval of time considered. Consequently this gives an estimate of the Hurst exponent. In the following section, we explain the details of this technique and give & Mathematica program to implement the R/S analysis and
196 Innovation In Mathematics evaluate the Hurst exponent. 4 Hurst Exponent and R/S Analysis For a given time series X, representing Z+l consecutive stock prices, we take the following steps: convert the series into a log-returns series of length L ; divide the time interval in a contiguous subintervals of length n, with a = Z/%; for all sublists of length n evaluate all the partial sums and the range properly reseated by the local standard deviation; finally evaluate the average value for the rescaled range for different values of n, until n=l/2\ calculate the Hurst exponent byfindingthe slope of a Log-Log plot. In the following Mathematica program the Hurst exponents have been evaluated in two different ways: using the Interpolation and Fit functions. In Figure 1 and Figure 2, we compare the results obtained for the Hurst exponents for Fiat and Generali, together with the corresponding Log-Log plots. «"Statistics'DescriptiveStatistics'" «"Graphics 'Graphics" " «"Statistics 'NonLinearFit * " DatiF=ReadList["newfiat",Number]; DatiG=ReadList["newgeneral",Number]; FIAT=Drop[N[Log[Drop[DatiF,-16] ] ], 1] - Drop[ N[Log[Drop[DatiF,-16]]],-!]/ GENERALI=Drop[N[Log[Drop[DatiG,-16]]],!] - Drop[N[Log[Drop[DatiG,-16]]],-!] ; L=Length[FIAT] ; Index=Drop[Drop[Divisors[L],7],-1]; SL[y_, n_] := Partition[y, n]; RR[x_List] := Module[{xStar = Drop[FoldList[Plus,0, x-mean[x]],1], SD, RR}, SD = StandardDeviation[x]/ RR = N[(Max[xStar] - Min[xStar])/SD]; RR] Hurst[x_List] := Module!{Hurst,H2}, Hurst = Map[{#,Mean[Table[RR[SL[x,#][[!]]], {1, L/#}]]}&, Index]; Hurst] HFIAT = Hurst [FIAT] ; HGENERALI = Hurst[GENERALI];
Innovation In Mathematics 197 EXP[data_List] := Module[{HI, H2, graph}, HH = Interpolation[data]; HH[y]; TF = Fit[Log[data], (1, x}, x] ; TTF = Exp[TF /. x -> Log[x]]; HI = Mean[Table[N[Log[HH[y]]/Log[y]], {y, 100, 3000, 10}]]; H2 = D[TF, x] ; graph = DisplayTogether[LogLogListPlot[data], LogLogPlot[HH[y], {y, 10, 3000}]; LogLogPlot[TTF, {x, 10, 3000}]]; {HI, H2, graph} ] EXP[HFIAT] 100. 50. 20. 10. 10. 50. 100. 500. 1000. {0.5644738694827033,0.6121360664343787,-Graphics-} Figure 1: Hurst Exponents and Log-Log Plots for FIAT EXP[HGENERALI] 100. 50. 20. 10. 10. 50. 100. 500. 1000. {0.5438132319075765,0.5760494117806533,-Graphics- Figure 2: Hurst Exponents and Log-Log Plots for GENERALI
198 Innovation In Mathematics The validity of the program can be tested in [6], generating, by random additions, a fractional Brownian motion. For f Bm [1,6001,0.56] as input, which is a fractional Brownian motion in one dimension with 6001 data and Hurst exponent 0.56, we obtained H= 0.558. H= 0.581 was obtained in [6]. This result seems to verify the correctness of our program. The values of H found for the stocks under investigation are in the range (0.50,1). Consequently these values seem to correspond, within the Fractal Market Hypothesis, to time series characterized by persistency, self-similarity properties and long-memory effects. By comparing them with the range of values for the exponent a reported in the previous section, the relationship a~\ih, which can be proved theoretically, is consistent our results. 5 Conclusions and Outlook Economic time series consisting of six thousand real data points have been analyzed within the framework of Mandelbrot theory, comparing the results with those expected from use of the Gaussian hypothesis. This is one step in the direction of a full comprehension of the theory of the Fractal Market and its successful application to real data. Mathematica proved to be an essential tool in this kind of analysis. Indeed, the minimum number of data points acceptable in order to obtain a realistic check on the existence of the fractal nature of economic series has recently been examined. Writing fast and accurate Mathematica programs is highly desirable for this purpose. A different approach, based on the analysis of signals and time series using the technique based on wavelets, will be the subject of future work in order to obtain a deeper comprehension of this phenomenon. There will be particular emphasis on the recognition of trends and nonperiodic cycles. One of us (M.F.G.) would like to acknowledge the Italian Monte dei Paschi di Siena for financial support and Astrea, in Altopascio, Italy, for allowing the access to the data. 6 References [1] Mandelbrot, B. The Variation of Certain Speculative Prices, The Journal of Business, 1963, 36, 394-419. [2] Mirowsky, P. From Mandelbrot to Chaos in Economic Theory, Southern Economic Journal, 1990, 57, 289-317. [3] Peters, E. Fractal Market Analysis, John Wiley & Sons, New-York, 1994. [4] Korsan, R. Fractals and Time Series Analysis, Mathematica Journal, 1995, 3/1, 39-44. [5] Guiducci, M.F. & Loffredo, M.I. Analisi di Dati di Serie Temporali in Economia: Applicazione della Teoria di Mandelbrot, Report #310, Dipartimento di Matematica, Universita di Siena, 1996 [6] Maeder, R Fractional Brownian Motion, Mathematica Journal, 1995, 6/1, 38-48.