Microsoft Word - 00_tresc.docx © 2013 Nicolaus Copernicus University. All rights reserved. http://www.dem.umk.pl/dem D Y N A M I C E C O N O M E T R I C M O D E L S DOI: http://dx.doi.org/10.12775/DEM.2013.006 Vol. 13 (2013) 107−125 Submitted June 30, 2013 ISSN Accepted December 14, 2013 1234-3862 Agnieszka Kapecka* Fractal Analysis of Financial Time Series Using Fractal Dimension and Pointwise Hölder Exponents∗∗ A b s t r a c t. This paper presents a fractal analysis application to the verification of assump- tions of Fractal Market Hypothesis and the presence of fractal properties in financial time series. In this research, the box-counting dimension and pointwise Hölder exponents are used. Achieved results lead to interesting observations related to nonrandomness of price series and occurrence of relationships binding fractal properties and variability measures with the pres- ence of trends and influence of the economic situation on financial instruments’ prices. K e y w o r d s: fractal analysis, fractal dimension, box-counting dimension, pointwise Hölder exponents, Hurst exponent. J E L Classification: G14, G15, G17. Introduction Ever since Mandelbrot had published his works on the application of R/S analysis to long-memory dependencies in time series (Mandelbrot, Wallis, 1969; Mandelbrot, 1972) and since Peters had presented his Fractal Market Hypothesis (Peters, 1991) as an alternative to commonly acknowledged Ef- ficient Market Hypothesis, this approach is being explored with regard to financial time series. Mulligan examined the use of Lo’s modified rescaled range analysis on foreign exchange markets (Mulligan, 2000), proving the * Correspondence to: Agnieszka Kapecka, Warsaw School of Economics, The Collegium of Economic Analysis, Al. Niepodległości 162, 02-554 Warsaw, Poland, e-mail: a.kapec- ka@gmail.com. ∗∗ This work was financed by the author. Agnieszka Kapecka DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 108 presence of long memory dependencies and fractal structure of analysed price series. Another published research assumed Hurst exponent estimation using geometrical interpretation (Granero, Segovia, Pérez, 2008) applied to stock market indices or the search for periodic and nonperiodic components in S&P 500 time series (Bohdalová, Greguš, 2010). Another group of studies focused on the analysis of variation of Hurst exponent over time, showing the possible impact of capital flow and trading volume on the decrease of Hurst exponent values (Cajueiro, Tabak, 2004), the influence that the end of Bretton Woods System had on efficiency of US stock markets (Alvarez- Ramirez, Alvarez, Rodriguez, Fernandez-Anaya, 2008), or the relationship between local Hurst exponent and stock market crashes with example of the Warsaw Stock Exchange Index (Grech, Pamuła, 2008). Due to certain limi- tations of classical R/S analysis approach and Hurst exponent itself, some of the authors explored Hölderian pointwise regularity of some major stock market indices (Bianchi, Pantanella, 2010) and usage of multifractal spectra analysis in order to discover patterns of change in price series before the 1987 market crash and other significant market drawdowns (Los, Yalamova, 2004). In the presented paper, the initial assumption is that the markets are not efficient, but are fractal in their nature. Despite the fact that Efficient Market Hypothesis (EMH) has been commonly accepted as a default theory explain- ing the fundamentals of financial markets’ behavior, plenty of criticism and doubts have been adressed towards it. The critical remarks are mainly related to too strong assumptions underlying this hypothesis, vastly mangling the real world behavior of the markets. In terms of real market behavior, a chaos theory based Fractal Market Hypothesis (FMH) seems to be much more appriopriate. It assumes – on the contrary to EMH, which uses linear differ- ential equation – that the market is a nonlinear dynamic system, which al- lows to suppose that „real feedback systems involve long-term correlations and trends, because memories of long-past events can still affect the deci- sions made in the present” (Peters, 1997, p. 6). Actions of market partici- pants usually generate nonlinear behavior of financial processes. Functions describing investors’ attitute towards risk, their expectations towards stock market returns or financial instruments pricing formulae are nonlinear as well (Osińska, 2006, p. 118). The most characteristic property of FMH is acknowledgement that stock market returns time series have fractal (self- similar) structure. FMH also allows chaotic behavior of the market during particular periods and under certain conditions (Peters, 1994, pp. 46–48). FMH was described by Peters based on the results of long studies con- ducted by Hurst in the first half of the 20th century (Peters, 1991). The main Fractal Analysis of Financial Time Series Using Fractal Dimension… DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 109 conclusion was that most of natural systems do not follow the random walk model, but are subject to fractional Brownian motion. Such theory is in line with Peters’ assumption about the markets, which says that globally (in the long-term) the market is deterministic, while locally (in the short-term), due to randomly occuring information and emotional reactions of market partici- pants, the market is random (Peters, 1997, pp. 45, 64). Empirical confirma- tion of this hypothesis is presented in the section 4 of this paper. 1. Fractal Dimension One of the most substantial characteristics of geometric object is its di- mension. However, an attempt to analyse this matter is nontrivial due to the fact that so far the scientists provided many different definitions of dimen- sion, namely: the topological dimension, the Hausdorff dimension, the frac- tional dimension, the box-counting dimension, the self-similarity dimension etc. The reasoning behind using particular types of dimension depends on certain conditions and while sometimes using different types of dimensions can lead to similar results, it might as well show varying results for the same object (Peitgen, Jürgens, Saupe, 2002, p. 274). In brief, a dimension describes the way in which a geometric object (or time series) fills the space. A common characteristic of all fractal objects is presence of self-similarity, which means that there is a relation between the reduction coefficient (a scale of similarity) and the amount of reduced frag- ments similar to the original object. While analyzing zigzag-shaped financial instrument time series chart in terms of dimension, it is easy to conclude that its dimension falls into range ( )1, 2 . As zigzag is not a straight line, it has dimension which is definitely distinct from 1, but it is not two-dimensional as it does not fill the entire plane. There are several kinds of fractal dimensions. One of them is the box- counting dimension, which – in view of its use as a research tool – is worth presenting. Below description of the box-counting dimension was written based on: Peitgen, Jürgens, Saupe (2002), Mastalerz-Kodzis (2003), Kudrewicz (2007) and Borys (2011). Let F be a certain geometric object embedded in n -dimensional Eu- clidean space ,nR covered with a set of small cubes (boxes called hypercubes) which sides are equal to ε (e.g., for 1=n it will be segments, for 2=n squares, for 3=n cubes). Theoretically these cubes can be discre- tionary oriented with respect to the axes of a coordinate system, but they can be as well spheres or other convex solids with a diameter .ε Let ( )ε,FN be Agnieszka Kapecka DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 110 a minimum amount of cubes that can completely cover the entire object .F Having satisfied this, below relations are true: − ( )ε,FN ~ ε/1 – for F being a segment of a smooth line (amount of cubes is approximately inversely proportional to the length of cube side), − ( )ε,FN ~ ( )2/1 ε – for F being a piece of smooth plane, − ( )ε,FN ~ ( )3/1 ε – for F being an area contained in .3R With respect to the above, a generalization can be made that there are such geometric objects, for which – assuming small values of ε (which is a scale of similarity, so that the smaller the values of ,ε the better the approx- imation) – below commensurateness is approximately met: ( )ε,FN ~ ( ) ,/1 Dε (1) whereas – while in majority of cases there is only an approximation of com- mensurateness between the two sides of the equation – in case of strictly self-similar objects an equality sign can be placed so that: ( )( , ) 1 / ,DN F ε ε= (2) where D (not necessarily being an integer number) can be treated as a di- mension of the object .F Then the limit: ( ) ( ) ( ) , /1log ,log lim 0 ε ε ε FN FD f → = (3) if it exists, is called a fractal dimension (in this case, a box-counting dimen- sion)1. It is easy to conclude from (3), which is applicable to exact fractals, that a dimension of such objects is defined by: ( ) ( ) ( ) . /1log ,log ε εFN FD f = (4) A possibility to apply the box-counting dimension led to creation of al- gorithms capable of estimating the dimension of time series. In this method, the analyzed structure is placed on a regular grid with a size of ,ε followed by 1 If a given unit gets reduced ε times, the measured line will approximately contain ε times more units than in the previous iteration step (approximation is in this case a result of a possible presence of small curves disturbing the existing commensurateness). Given com- mensurateness is exact only in the limit, because after reducing the length of the sunit it is possible to achieve a infinitely small unit only in the limit (Tempczyk, 1995, p. 135). Fractal Analysis of Financial Time Series Using Fractal Dimension… DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 111 counting all the cubes (boxes), which contain the fragments of analyzed structure. The amount of fragments is obviously dependent on the value of .ε In the next stage it is necessary to repeat the calculations for smaller val- ues of ε and plot the obtained values on the logarithmic chart (marking ( )log ,N F ε with respect to ( )log 1 / ε ). The slope of the straight line fitted to the points marked on the plot determines the box-counting dimension of the time series (Daros, 2010, p. 13). 2. Measures of Variability of the Graph of the Function One of the most popular and useful measures of variability (irregularity) are Hurst exponent and Hölder exponents. The Hurst exponent is a numeri- cal characteristic of the entire price series, whereas Hölder exponents can be used to analyze the complexity of the function and the trajectory of some stochastic processes in the vicinity of any point of the graph of the function (Mastalerz-Kodzis, 2003, p. 37). The most important turning point in the subject of long-term dependency analysis of time series was without any doubt the creation of rescaled range analysis n)S/R( method (Hurst, 1951). A starting point of his analysis was Einstein’s work on Brownian motion (Einstein, 1908), which presented an equation for the distance R that a particle travels in time ,T which is defined by ,TcR = where c is a nonnegative constant. This equation was applica- ble when the series of increments of the distance travelled by the particle in time was a random walk, characterised by the independency of normally distributed random variables (Weron, Weron, 1998, p. 323). However, dur- ing almost fourty years of research, Hurst has reached a conclusion that the majority of natural phenomena is not subject to Gaussian random walk, but rather to processes with „long memory”, later called by the name of fraction- al Brownian motion, which is a combination of a trend and noise (Peters, 1997, p. 64; Mastalerz-Kodzis, 2003, pp. 37–38). Derivation of a formula for rescaled range allowed the comparison of different types of time series. Creation of this dimensionless indicator, which should increase over time, allowed to formulate the following equation being an extension of Brownian motion model proposed by Einstein. ,nc)S/R( Hn ⋅= (5) where )S/R( – rescaled range, n – number of observations, c – positive constant, H – Hurst exponent. Agnieszka Kapecka DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 112 In order to calculate Hurst exponent, one has to calculate the average value of n)S/R( for different n and then solve the following equation us- ing linear regression: log ( / ) log( ) log( ),nE R S H n c= + (6) where n)S/R(E – expected value of rescaled range. In above equation, the Hurst exponent can be treated as the regression coefficient and estimated using the least squares method (Jajuga, Papla, 1997). The Hurst exponent is strictly linked to the fractal dimension of time series, therefore the search for the Hurst exponent is in fact a search for the fractal properties of the series. This relation is described by the following equation (Grech, 2012, p. 10): .2 HD f −= (7) This equation has a huge practical importance as it can be used to classi- fy the type of a time series depending on the fractal dimension of a given object. Following cases can be distinguished based on Hurst exponent values (Peters, 1997, p. 76–77): − if ,5.00 << H then 25.1 << D (antipersistent time series), − if ,5.0=H then 5.1=D (random walk), − if ,15.0 << H then 5.11 << D (persistent time series). First case ( 5.00 <≤ H ) applies to antipersistent (ergodic) time series. Such a series has a mean reversion tendency. If in a given period the value of the series increased, then in the following period it will most probably de- crease and vice versa. The closer the value of H to ,0 the more ergodic the behavior of the system and the time series graph has more jagged line, which is a result of a frequent trend reversion. In such case the fractal dimension of the series 2→fD as the series fills the plane more and more. The lower the H value, the more noise can be observed in the system. Speaking in the probability language, if e.g. ,2.0=H then there is 80% probability that in the future the market will change the direction, which will be equal to trend reversion (Stawicki, Janiak, Müller-Frączek, 1997, p. 37). Despite the fact that mean reversion plays a dominant role in economic and financial litera- ture, so far only a few antipersistent time series have been observed. The second case applies to a situation, when with ∞→n ,5.0=H which corresponds to a random walk (the consecutive elements of the series Fractal Analysis of Financial Time Series Using Fractal Dimension… DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 113 are independent). The fractal dimension of the series equals ,5.1=D the series itself is unpredictable, the present does not influence the future, and the past did not influence the present. The probability distribution function can be Gaussian, but not necessarily. Both in natural and economic phenom- ena the H exponent’s value usually differs from ,5.0 and the natural pro- cesses most often have a long-term data dependency. When ,15.0 ≤< H the time series are persistent, which means that they bolster the trend. It is caused by the presence of long-term data dependency. When ,1→H the trend gets stronger. As Hurst exponent defines the proba- bility of consecutive rises or drops of the prices, with ,1→H there are more consecutive rises or drops and the level of noise becomes smaller. For exam- ple, if ,8.0=H then there is 80% probability that a given trend will be sus- tained in the future. The fractal dimension ( )5.1,1∈D as the more persistent the time series, the less it fills the plane and the smoother are the curves cre- ated by a given system. Fractal time series is obviously not purely determi- nistic, it is rather an intermediate form between a completely random time series and a deterministic system. Persistent time series are fractional Brownian motions, which means that their important feature is a biased ran- dom walk, and the strength of bias increases when ,1→H so when the Hurst exponent value recedes from .5.0 Despite n)S/R( analysis is a very useful tool, it poses a major disad- vantage – it does not take into consideration the changes in particular subperiods. For example, if a particular stock is subject to rapid price chang- es, whereas the prices of other stocks show only minor price changes, it is not possible to detect this periodical changes using Hurst exponent. There- fore, a good measure of variability of the graph of a function over time is a Hölder function2, which values in particular points are equal to pointwise Hölder exponents (Mastalerz-Kodzis, 2003, p. 115). In order to further discuss pointwise Hölder exponents, let us first define the Hölder function3. Let )d,X( X and )d,Y( Y be metric spaces, then func- tion YX:f → is called a Hölder function with exponent ,α where ,0>α if 2 Although pointwise Hölder exponents are considered to be the best measure of function regularity in the vicinity of a certain point, other measures used include: local box-counting dimension, local Hausdorff dimension, the degree of fractional differentiability. 3 The definition of Hölder function and pointwise Hölder exponents was written based on: Mastalerz-Kodzis 2003, pp. 49–51; Kuperin, Schastlivtsev, 2008, pp. 4–6.  Agnieszka Kapecka DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 114 for each Xy,x ∈ such that 1)y,x(d X < the function satisfies the following inequality: ,),()](),([ αyxcdyfxfd XY ≤ (8) where c – a positive constant. Assuming that function RD:f → and that parameter ),1,0(∈α func- tion f is a αC class ( α∈Cf ) Hölder function, if such 0c > and 0h0 > constants exist that for every x and every ( )0h,0h∈ the following inequality is satisfied: .ch)x(f)hx(f α≤−+ (9) Assuming that 0x is an arbitrary point from the domain of function ,f so that ,RDx0 ⊂∈ function RD:f → is a α 0x C class ( α∈ 0x Cf ) Hölder function in ,x0 if such 0c > and 0>ε constant exist that for every )x,x(x 00 ε+ε−∈ the following inequality is satisfied: .xxc)x(f)x(f 00 α −≤− (10) By definition Hölder function is continous in its entire domain and when this assumption is satisfied, the graph of the function has fractal nature (Gabryś, 2005, p. 24). If the Hölder function is not continuous, it is called generalized Hölder function. It is worth noting that thanks to its time-varying values, Hölder function can take different types of random walks in different ranges (Kutner, 2009, p. 36). After providing a Hölder function definition it is possible to define a pointwise Hölder exponent of function f in .x0 By definition it is a num- ber )x( 0fα given by the following equation: { }. Cf:sup)x( 0x0f α∈α=α (11) Approximated pointwise Hölder exponent is not a flawless measure, however its major advantage is the ability to accomodate the stationarity of the series. Interpretation of pointwise Hölder exponent is the same as for Hurst exponent with a difference that pointwise Hölder exponent estimates local, not global value. It is also worth mentioning that Hölder functions are not constant as they are time-varying (Mastalerz-Kodzis, 2003, p. 121). Fractal Analysis of Financial Time Series Using Fractal Dimension… DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 115 3. Tools and Analysis Methodology 3.1. Data All monthly and daily price series were downloaded from http://stooq.pl. Table 1 presents the list of stock market indices and forex currency pairs used in this research. Table 1. Financial instruments chosen for the analysis Symbol Available Data Period Market Type DJIA 1896.05–2012.12 Mature market S&P 500 1923.01–2012.12 Mature market DAX 1959.10–2012.12 Mature market Nikkei225 1949.05–2012.12 Mature market Hang Seng 1969.11–2012.12 Mature market WIG20 1991.04–2012.12 Emerging market Bovespa 1992.01–2012.12 Emerging market RTS 1995.09–2012.12 Emerging market SENSEX30 1979.04–2012.12 Emerging market SCI 1990.12–2012.12 Emerging market XU100 1990.01–2012.12 Emerging market EUR/USD 1980.01–2012.12 Major currency pair GBP/USD 1971.01–2012.12 Major currency pair USD/JPY 1971.01–2012.12 Major currency pair CHF/PLN 1990.01–2012.12 Exotic currency pair EUR/PLN 1990.01–2012.12 Exotic currency pair USD/BRL 1995.01–2012.12 Exotic currency pair USD/RUB 1995.10–2012.12 Exotic currency pair USD/INR 1973.01–2012.12 Exotic currency pair USD/CNY 1984.01–2012.12 Exotic currency pair USD/TRY 1984.01–2012.12 Exotic currency pair The above selection is reasoned by analysis of both mature and emerging markets and the associated currency pairs in order to check if any relation- ships interesting from the fractal analysis point of view are present. 3.2. Tools During the analysis, Microsoft Excel was used to calculate the common logarithms of price values and plot the charts of logarithmic price series and pointwise Hölder exponents. For the purposes of box-counting dimension estimation and pointwise Hölder exponents calculation, FracLab 2.0 was used. As quoted from FracLab homepage (http://fraclab.saclay.inria.fr/): „FracLab is a general purpose signal and image processing toolbox based on fractal and multifractal methods. (…) A large number of procedures allow to Agnieszka Kapecka DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 116 compute various fractal quantities associated with 1D or 2D signals, such as dimensions, Hölder exponents or multifractal spectra. (…) FracLab is a free software developed in the Regularity team at Inria Saclay/Ecole Centrale de Paris.” 3.3. Analysis Methodology Of the fractal analysis methods described in sections 1 and 2, in this re- search the box-counting dimension, Hurst exponent and pointwise Hölder exponents were used for long-term dependency analysis of chosen financial time series. Table 2. Values of fractal dimension D and Hurst exponent H calculated on the entire data range of monthly and daily price series of chosen financial in- struments Symbol Period Monthly data Daily data D H D H DJIA 1896.05–2012.12 1.44 0.56 1.43 0.57 S&P 500 1923.01–2012.12 1.40 0.60 1.40 0.60 DAX 1959.10–2012.12 1.50 0.50 1.47 0.53 Nikkei225 1949.05–2012.12 1.36 0.64 1.44 0.56 Hang Seng 1969.11–2012.12 1.47 0.53 1.50 0.50 WIG20 1991.04–2012.12 1.43 0.53 1.48 0.52 Bovespa 1992.01–2012.12 1.22 0.78 1.48 0.52 RTS 1995.09–2012.12 1.47 0.53 1.44 0.56 SENSEX30 1979.04–2012.12 1.37 0.63 1.46 0.54 SCI 1990.12–2012.12 1.50 0.50 1.46 0.54 XU100 1990.01–2012.12 1.31 0.69 1.46 0.54 EUR/USD 1980.01–2012.12 1.49 0.51 1.47 0.53 GBP/USD 1971.01–2012.12 1.56 0.44 1.44 0.56 USD/JPY 1971.01–2012.12 1.49 0.51 1.42 0.58 CHF/PLN 1990.01–2012.12 1.32 0.68 1.44 0.56 EUR/PLN 1990.01–2012.12 1.35 0.65 1.43 0.57 USD/BRL 1995.01–2012.12 1.34 0.66 1.40 0.60 USD/RUB 1995.10–2012.12 1.20 0.80 1.36 0.64 USD/INR 1973.01–2012.12 1.23 0.77 1.32 0.68 USD/CNY 1984.01–2012.12 1.16 0.84 1.36 0.64 USD/TRY 1984.01–2012.12 1.08 0.92 1.39 0.61 Fractal Analysis of Financial Time Series Using Fractal Dimension… DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 117 4. Fractal Analysis Using Fractal Dimension and Pointwise Hölder Exponents 4.1. Presentation of Results The research was divided into two parts. In the first part, the dimensions of chosen price series were estimated and equation (7) was used to derive Hurst exponent from the estimated dimension. First, the dimensions of price series were calculated for the entire historical data range (see Table 2). Table 3. Values of fractal dimension D and Hurst exponent H calculated on the Oct 1995 to Dec 2012 data range of monthly and daily price series of chosen financial instruments Symbol Period Monthly data Daily data D H D H DJIA 1995.10–2012.12 1.41 0.59 1.48 0.52 S&P 500 1995.10–2012.12 1.36 0.64 1.45 0.55 DAX 1995.10–2012.12 1.39 0.61 1.44 0.56 Nikkei225 1995.10–2012.12 1.50 0.50 1.48 0.52 Hang Seng 1995.10–2012.12 1.60 0.40 1.51 0.49 WIG20 1995.10–2012.12 1.52 0.48 1.48 0.52 Bovespa 1995.10–2012.12 1.46 0.54 1.49 0.51 RTS 1995.10–2012.12 1.46 0.54 1.44 0.56 SENSEX30 1995.10–2012.12 1.51 0.49 1.47 0.53 SCI 1995.10–2012.12 1.53 0.47 1.44 0.56 XU100 1995.10–2012.12 1.37 0.63 1.45 0.55 EUR/USD 1995.10–2012.12 1.53 0.47 1.49 0.51 GBP/USD 1995.10–2012.12 1.52 0.48 1.52 0.48 USD/JPY 1995.10–2012.12 1.50 0.50 1.49 0.51 CHF/PLN 1995.10–2012.12 1.51 0.49 1.50 0.50 EUR/PLN 1995.10–2012.12 1.57 0.43 1.49 0.51 USD/BRL 1995.10–2012.12 1.36 0.64 1.40 0.60 USD/RUB 1995.10–2012.12 1.20 0.80 1.36 0.64 USD/INR 1995.10–2012.12 1..40 0.60 1.39 0.61 USD/CNY 1995.10–2012.12 1.26 0.74 1.56 0.44 USD/TRY 1995.10–2012.12 1.21 0.79 1.40 0.60 In the second part, pointwise Hölder exponents were calculated for six cho- sen markets. Figures 1–6 present pointwise Hölder exponents plotted togeth- er with common logarithms of time series values. The range of values taken by pointwise Hölder exponents’ values is presented on the right of each graph. DYNAMIC 118 4.2. An The especial The evidenc majority long-ter out of 6.0≥H nature, h series w remark econom not bein a fractio Figure 1 Ano fraction Wherea variety case of exponen monthly ECONOMETRIC nalysis of Re e interpretatio lly in the ligh e part of res ce supporting y of the inve rm trends du 21 markets .6 Only 4 out having 5.0 ≤ was detected about rare my. Such resu ng a random onal Brownia . S&P500 pri other interest nal dimension as the researc of values ( 0 daily data fo nt values are y data and fa Ag C MODELS 13 (2 esults on of the res ht of fractal a search based g the fractal estigated mar uring entire a s having sig t of 21 mark ,51.0≤≤ H a (GBP/USD w occurrence ults seem to m walk, but an motion. ce series and p ting evidence n and Hurst ch carried ou 044.0 ≤≤ H or 16 out of 2 e usually co all into a narr gnieszka Kapec 2013) 107–12 sults leads to analysis and d on fractal nature of fi rkets (16 out available his gnificantly n kets displaye and just in a with 4.0=H of antipersi o prove Peter rather a co pointwise Höl e was gained exponent va ut for monthl 92. for the e 21 examined onsiderably l row range ( 0 cka 25 o some inter fractal mark dimension inancial time t of 21) reve story of the p nonrandom ed a random single case t 44 ), which i istent time s rs’ concept mbination o lder exponent d based on th alues for mon ly price serie entire history d markets it t lower than r 6.05.0 ≤≤ H resting obser ket hypothesi provided int e series. Firs ealed the pre price series, nature, den m or close to the antiperist is in line with series in nat of financial of trend and ts he compariso nthly and da es resulted in y of price se turned out th respective re 6 for the entir rvations, is. teresting st of all, esence of with 12 noted by random tent time h Peters’ ture and markets noise – on of the aily data. n a wide eries), in hat Hurst sults for re histo- Figure 2 Figure 3 ry of pr data ran Fractal M is determ tively v ularity o tomated Fractal Ana . DAX price s . Nikkei225 p rice series a nge). This m Market Hypo ministic, wh erified. Mor of electronic d algorithmic lysis of Financi D series and poin price series an and 48.0 ≤ H might mean othesis, sayin hile locally (i eover, it seem c trading, lar c trading and ial Time Series DYNAMIC ECON ntwise Hölder nd pointwise H 57.0≤H for that anothe ng that globa in the short- ms that durin rger trading d high freque Using Fractal NOMETRIC MOD r exponents Hölder expone r October 19 er Peters’ as ally (in the lo -term) it is ra ng recent yea volumes and ency trading Dimension… ELS 13 (2013) ents 995–Decemb ssumption re ong-term) the andom, can ars the grow d introductio systems, mig 107–125 119 ber 2012 elated to e market be posi- wing pop- on of au- ght have DYNAMIC 120 increase biggest US and ingly, o the exot (USD/B of signi suggests for this is out o only fra Figure 4 The range an type lea correspo of Hurs od refle 1990s a Respect Seng is during l and 198 ECONOMETRIC ed the efficie increase of Japan stock one group of tic currency BRL, USD/C ificant trends s that the un group of ma of scope of t actal analysis . XU100 price e observation nd October ad to a concl ond to the u t exponent v ects the dyna and most of tively, the de indicative f last two deca 80s. Another Ag C MODELS 13 (2 ency of the m efficiency is k market indi f investigated pairs linked CNY, USD/IN s on the dail nderlying eco arkets. Neve this paper an s, but also ec e series and po ns made duri 1995–Decem lusion that f underlying ec values for DJ amic growth 2000s in the ecrease of Hu for the econ ades after the interesting o gnieszka Kapec 2013) 107–12 markets in th s visible for ices or majo d markets is to emerging NR, USD/RU ly data interv onomic phen ertheless, furt nd might req onomic facto ointwise Höld ing the analy mber 2012 d fractal dimen conomic situ JIA, S&P500 h and propiti e economies urst exponen nomic slowd e preceeding observations cka 25 he short-term the world’s or forex curre an exceptio g markets ex UB, USD/TR val indicated omena are su ther research quire more s ors, e.g. capi der exponents ysis of result ata range wi nsion and Hu uation. For ex 0 and DAX d ious econom s of United S nt values for N down in Japa g very aggres include the m (especially biggest mark ency pairs). n to this rule xamined in th RY) show a p d by 6.0≥H ubstantially h on such hy studies inclu ital flows. ts for the en ith respect to urst exponen xample, the during the lat mic condition States and G Nikkei225 a an and Sout sive growth i growth of ef y that the kets like Interest- e. All of his paper presence 6 , which different ypothesis uding not ntire data o market nt values increase tter peri- ns in the Germany. and Hang th Korea in 1970s fficiency of majo reflects econom Figure 5 Figure 6 The Hölder random Fractal Ana or forex curr the huge in my. . EUR/USD p . USD/JPY pr e second part exponents in nature of th lysis of Financi D rency pairs nfluence that price series an rice series and t of the resea n fractal ana he markets. ial Time Series DYNAMIC ECON or the rando t Chinese go d pointwise H d pointwise H arch, focused alysis, provid For all six e Using Fractal NOMETRIC MOD omness of S overnment h Hölder expone ölder exponen d on the app ded addition examined ma Dimension… ELS 13 (2013) SCI, which p has on the c ents nts lication of p nal evidence arkets, the v 107–125 121 probably country’s pointwise on non- values of Agnieszka Kapecka DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 122 the pointwise Hölder exponents are significantly higher than 5.0 during most of the entire data range, which proves the presence of long-term de- pendencies in the investigated time series. Moreover, the graphs expose the relationship between the trend strength and Hölder exponents’ values as they tend to considerably decrease and converge to 5.0 during recent years side- ways price movements (which is especially visible for Nikkei225, DAX or EUR/USD). One of the pros of using the box-counting dimension method in this arti- cle is that due to its nature, it does not require assumption on the selfsimilarity of the analysed object, which allows to avoid making a possi- bly false statement even before the beginning of the research. Another con- siderable advantage of the methods used in this article was their algorithmic accessibility, thus making it affordable from the calculation complexity point of view and available even for the users which do not have a high-end pro- cessor cluster at their disposal. Nevertheless, as the estimation method as- sumes that the Hölder function is constant over each of the intervals, it might introduce some error of method to the results. Conclusions The analysis conducted in this research provides solid empirical evi- dence in favour of Fractal Market Hypothesis, with the presence of long- term dependencies in the financial time series and the confirmation of the global determinism and local randomness of the markets being the most important ones. An important observation supporting the nonrandomness of the markets is a relationship between fractal properties of the investigated time series and the underlying economic situation. One of the most interesting researches that could be made in the future based on this would be a repetition of the calculations at some fixed intervals of time – after ten, twenty or thirty years. Such approach could possibly verify if the opinions stated by economists a posteriori major changes in economic conditions are in line with the con- clusions drawn from the fractal characteristics of analysed markets. Another study that could be made as a continuation of research conduct- ed in this article could involve inclusion of additional test data, this time not limited to market time series. Inclusion of some regularly published econom- ic indicators like gross domestic product, consumer price index or money supply could potentially reveal some additional relationships and regularities and shed more light on the macroeconomic processes. Fractal Analysis of Financial Time Series Using Fractal Dimension… DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 123 Moreover, the analysis of results suggests that fractal analysis can be a valuable tool for the evaluation of market trends, which might be of practi- cal use for institutional and individual investors. References Alvarez-Ramirez, J., Alvarez, J., Rodriguez, E., Fernandez-Anaya, G. (2008), Time-Varying Hurst Exponent for US Stock Markets, Physica A, 387, 6159–6169, DOI: http://dx.doi.org/10.1257/002205103765762743. Bianchi, S., Pantanella, A. 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Fractal Analysis of Financial Time Series Using Fractal Dimension… DYNAMIC ECONOMETRIC MODELS 13 (2013) 107–125 125 Fraktalna analiza finansowych szeregów czasowych z wykorzysta- niem wymiaru fraktalnego oraz punktowych wykładników Höldera Z a r y s t r e ś c i. Artykuł przedstawia propozycję zastosowania analizy fraktalnej w celu weryfikacji niektórych założeń hipotezy rynku fraktalnego oraz występowania fraktalnych właściwości w finansowych szeregach czasowych. W celu przeprowadzenia badań wykorzy- stany został wymiar pudełkowy oraz punktowe wykładniki Höldera. Rezultaty osiągnięte dla badanych rynków pozwoliły dokonać interesujących obserwacji dotyczących nielosowości szeregów cenowych oraz występowania relacji między fraktalnymi właściwościami i miarami zmienności a obecnością trendów i wpływem sytuacji ekonomicznej na ceny instrumentów finansowych. S ł o w a k l u c z o w e: analiza fraktalna, wymiar fraktalny, wymiar pudełkowy, punktowe wykładniki Höldera, wykładnik Hursta.