Chaos, Solitons and Fractals 41 (2009) 2331–2340

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Chaos, Solitons and Fractals

journal homepage: www.elsevier .com/locate /chaos

Multifractal structure in Latin-American market indices

Luciano Zunino a,b,c,*, Alejandra Figliola d, Benjamin M. Tabak e,i, Darío G. Pérez f,Mario Garavaglia a,c, Osvaldo A. Rosso g,h

a Centro de Investigaciones Ópticas, C.C. 124 Correo Central, 1900 La Plata, Argentinab Departamento de Ciencias Básicas, Facultad de Ingeniería, Universidad Nacional de La Plata (UNLP), 1900 La Plata, Argentinac Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, 1900 La Plata, Argentinad Chaos & Biology Group, Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, Campus Universitario, Modulo 5, Juan María Gutierrez 1150,Los Polvorines, Pcia. de Buenos Aires, Argentinae Banco Central do Brasil, SBS Quadra 3, Bloco B, 13 andar, DF 70074-900, Brazilf Instituto de Física, Pontificia Universidad Católica de Valparaíso (PUCV), 23-40025 Valparaíso, Chileg Centre for Bioinformatics, Biomarker Discovery and Information-Based Medicine, School of Electrical Engineering and Computer Science, The University ofNewcastle, University Drive, Callaghan, NSW 2308, Australiah Chaos & Biology Group, Instituto de Cálculo, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón II, Ciudad Universitaria, 1428 CiudadAutónoma de Buenos Aires, Argentinai Universidade Catolica de Brasilia, Brasilia, DF, Brazil

a r t i c l e i n f o

Article history:Accepted 9 September 2008

0960-0779/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.chaos.2008.09.013

* Corresponding author. Address: Centro de InvesE-mail addresses: lucianoz@ciop.unlp.edu.ar (L. Z

(D.G. Pérez), garavagliam@ciop.unlp.edu.ar (M. Gara

a b s t r a c t

We study the multifractal nature of daily price and volatility returns of Latin-Americanstock markets employing the multifractal detrended fluctuation analysis. Comparing withthe results obtained for a developed country (US) we conclude that the multifractalitydegree is higher for emerging markets. Moreover, we propose a stock market inefficiencyranking by considering the multifractality degree as a measure of inefficiency. Finally,we analyze the sources of multifractality quantifying the contributions of two factors,the long-range correlations of the time series and the broad fat-tail distributions. We findthat the multifractal structure of Latin-American market indices can be mainly attributedto the latter.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that empirical data coming from financial markets, like stock market indices [1–13], foreign exchangemarkets [14–16], commodities [7], traded volumes [17] and interest rates [18] have a multifractal nature. Moreover, multi-fractal theoretical models were introduced [19–26] that take into account important aspects of price patterns (stylized facts),that other traditional models like the fractional Brownian motion and GARCH processes cannot. So, much effort has beendedicated to a reliable characterization of their multifractality. Furthermore, it would be worthwhile exploring if we can as-sess the degree of market development using multifractal measures. It was recently shown that interest rates for emergingmarkets have a higher multifractality degree [18]. However, it is not clear whether stock markets present similar dynamics.This is an important research question as evidence of multifractality suggests that one could build multifractal models thatallow forecasting variables of interest. This is a crucial issue for portfolio and risk managers, financial institutions, financialregulators, and investors.

. All rights reserved.

tigaciones Ópticas, C.C. 124 Correo Central, 1900 La Plata, Argentina.unino), afigliol@ungs.edu.ar (A. Figliola), benjamin.tabak@bcb.gov.br (B.M. Tabak), dario.perez@ucv.clvaglia), oarosso@fibertel.com.ar (O.A. Rosso).

2332 L. Zunino et al. / Chaos, Solitons and Fractals 41 (2009) 2331–2340

Latin-American stock markets are important as they represent interesting investment opportunities for international anddomestic investors. Nonetheless, only a few papers focus on these markets and very little is known regarding their dynamicproperties. To the best of our knowledge we are the first to analyze the multifractal structure of Latin-American market indi-ces, which have been experimenting a substantial growth in recent years. For this purpose, we employ the multifractal detr-ended fluctuation analysis introduced in the pioneering work of Castro e Silva and Moreira [27] and developed more recentlyby Kantelhardt et al. [28]. This robust and powerful technique identifies and quantifies the multiple scaling exponent withina time series. It has been succesfully applied in different and heterogeneous scientific fields to study multifractality [29–35].

Understanding the origins of multifractality is a relevant research topic. Two factors are deemed to be the causes of mul-tifractality: long-range correlations of the time series and broad fat-tail distributions. In order to quantify the contributionsof these two different sources we generate new and related time series from the original ones – shuffled and surrogated timeseries [36].

This is the first study that compares the multifractal behavior of a collection of stock markets from different countries andbuilds an inefficiency ranking by considering their multifractality degrees as a measure of inefficiency. It should be stressedthat this is our major contribution.

This paper is organized as follows. The multifractal detrended fluctuation analysis is described in Section 2. In Section 3,the data used in this study are detailed. In Section 4 we discuss the results, and in Section 5, we present the conclusions.

2. Methodology

The standard partition function multifractal formalism [37] was developed for stationary time series. Thus, it does notgive correct results for non-stationary time series. The wavelet transform modulus maxima (WTMM) method [38–41]was introduced to deal with strongly non-stationary data. It consists in the detection of scaling of the maxima lines ofthe continuous wavelet transform on different scales in the time-scale plane. More recently, Kantelhardt et al. [28] devel-oped a method for the multifractal characterization of non-stationary time series, which is based on a generalization ofthe detrended fluctuation analysis (DFA) [42]: the multifractal detrended fluctuation analysis (MFDFA). It has been shownthat for short series and negative moments, the significance of the results for the MFDFA is better than for the WTMM meth-od [28]. It has also been concluded that the MFDFA should be recommended for a global detection of multifractal behavior[13]. Moreover, the implementation of the MFDFA does not involve more effort than the conventional DFA. Just one addi-tional step is required. In view of these reasons, we have chosen the latter for a proper detection of the multifractality ofthe financial data under analysis here.

The MFDFA can be summarized as follow [28]:

� Starting with a correlated time series (signal) fui; i ¼ 1; . . . ;Ng, where N is the length of the series, the corresponding pro-file is determined by integration

YðkÞ ¼Xk

i¼1

½ui � hui�; k ¼ 1; . . . ;N; ð1Þ

where h�i denotes the averaging over the whole time series.� The profile YðkÞ is divided into Ns � ½N=s� non-overlapping windows of equal length s. Since the record length N does not

need to be a multiple of the considered time-scale s, a short part at the end of the profile will remain in most cases. Inorder to have into account this part of the record, the same procedure is repeated starting from the other end of the record.Thus, 2Ns windows are obtained.

� The local trend for each window m ¼ 1; . . . ;2Ns is evaluated by least-square fit of the data. The detrended time series forwindow length s, denoted by YsðiÞ, is calculated as the difference between the original time series and the fits

YsðiÞ ¼ YðiÞ � pmðiÞ; ð2Þ

where pmðiÞ is the fitting polynomial in the m-th window. Due to the detrending of the time series is done by subtraction ofthe fits from the profile, these methods differ in their capability of eliminating trends in the data. In m-th order of MFDFA,trends of order m in the profile and m� 1 in the original record are eliminated. Thus, a comparison of the results for differentorders of MFDFA allows to estimate the polynomial trend in the time series. Since we use a polynomial fit of order 3, wedenote the algorithm as MFDFA-3.� For each of the 2Ns segments the variance of the detrended time series YsðiÞ is evaluated by averaging over all data point i

in the m-th window

F2s ðmÞ ¼

1s

Xs

i¼1

fYs½ðm� 1Þsþ i�g2: ð3Þ

� The q-th order fluctuation function is obtained by averaging over all segments

FqðsÞ ¼1

2Ns

X2Ns

m¼1

½F2s ðmÞt�

q=2

( )1=q

; ð4Þ

L. Zunino et al. / Chaos, Solitons and Fractals 41 (2009) 2331–2340 2333

where, in general, the order q can take any real value.1 For q ¼ 2, the standard DFA procedure is retrieved.� Finally, the scaling behavior of the fluctuation is determined by analyzing log–log plots of FqðsÞ versus s for each value of q.

If the series ui are long-range correlated FqðsÞ increases, for large values of s, as a power-law

1 Foraverage

FqðsÞ � shðqÞ: ð5Þ

For monofractal time series with compact support, hðqÞ is independent of q, since the scaling behavior of the varianceF2

s ðmÞ is identical for all segments m, and the averaging procedure in Eq. (4) will give just this identical scaling behaviorfor all values of q. Only if small and large fluctuations scale in a different way, there will be a significant dependence ofhðqÞ on q. If we consider positive values of q, the segments m with large variance F2

s ðmÞ will dominate the average FqðsÞ. Thus,for positive values of q, hðqÞ describes the scaling behavior of the segments with large fluctuations. On the contrary, for neg-ative values of q, the segments m with small variance F2

s ðmÞ will dominate the average FqðsÞ. Hence, for negative values of q,hðqÞ describes the scaling behavior of the segments with small fluctuations. Obviously, richer multifractality corresponds tohigher variability of hðqÞ. Then, the multifractality degree can be quantified by

Dh ¼ hðqminÞ � hðqmaxÞ: ð6Þ

As large fluctuations are characterized by smaller scaling exponent hðqÞ than small fluctuations, hðqÞ for q < 0 are larger thanthose for q > 0, and Dh is positively defined.

The exponent hðqÞ is known as the generalized Hurst exponent. For a stationary time series such as the fractional Gaussiannoise (fGn), the profile defined in Eq. (1) will be a fractional Brownian motion (fBm). Thus, 0 < hðq ¼ 2Þ < 1 for these pro-cesses, and hðq ¼ 2Þ is identical to the Hurst parameter, H. On the other hand, if the original signal is a fBm, the profile willbe a sum of fBm, so hðq ¼ 2Þ > 1. In this case the relationship between the exponent hðq ¼ 2Þ and H is H ¼ hðq ¼ 2Þ � 1. SeeRef. [32] for further details.

Besides the multifractal analysis we weight the contribution of long-range correlations and broad fat-tail distributions inthe multifractality. For that purpose we follow the procedure introduced in Ref. [32]. First, we have shuffled the data andcalculated hshufðqÞ for both, return and volatility return time series. In the shuffling procedure the data are put into randomorder. So, all temporal correlations are destroyed. However, the probability density function (PDF) is not affected. In order toquantify the influence of the fat-tail distribution, surrogate time series were generated from the original by randomizingtheir phases in the Fourier space. The new series are Gaussian. Thus, if only non-Gaussianity was the source of the multifrac-tality, these series should be monofractals and their generalized Hurst exponents, hsurðqÞ, would be constant. Assuming thatthe two factors are independent, the influence of correlation can be quantified by

hcorðqÞ ¼ hðqÞ � hshufðqÞ; ð7Þ

and the contribution of the non-Gaussian PDF will be

hPDFðqÞ ¼ hðqÞ � hsurðqÞ: ð8Þ

3. Data

We have collected daily closing prices for Argentina, Brazil, Chile, Colombia, Mexico, Peru, Venezuela and the US. This lastindex is included for comparison purposes. The first three markets together with Mexico and the US correspond to largestock markets. On the other hand, Colombia, Peru and Venezuela can be considered smaller stock markets. The data samplescover the period from January 1995 to February 2007 with 3169 observations for all the countries except Venezuela with3168 observations and the US with 3141 observations. All series were gathered from the Morgan Stanley Capital Index(MSCI). This style index was employed in order to disentangle the effects of autocorrelation and infrequent trading, whichare specially severe in emerging market. They are value-weighted indices of a large sample of companies in each country andthe indices do not include foreign companies and are computed consistently across markets, thereby allowing for a closecomparison across countries.

Let xðtÞ be the price of a stock on a time t, the price returns, rt , are calculated as its logarithmic difference,rt ¼ lnðxðt þ 1Þ=xðtÞÞ.

Volatility is of very interest to traders because it quantifies the risk and is directly related to the amount of informationarriving in the financial market at a given time. The volatility of stock price changes is a measure of how much the market isliable to fluctuate. It is a ‘‘subsidiary process” estimated from the original one. The series of absolute returns j rt j and squaredreturns r2

t are usually used as proxies for volatility returns. It should be noted that there are evidence of stronger long-rangedependence for absolute returns than for squared returns [43–45]. We follow Ding et al. [46] who suggest measuring vola-tility directly from absolute returns. Furthermore, Davidian and Carroll [47] show absolute returns volatility specification ismore robust against asymmetry and non-normality.

q ¼ 0 the value hð0Þ cannot be determined directly using the average procedure in Eq. (4) because of the diverging exponent. Instead, a logarithmicprocedure has to be employed [28].

0

500

1000

1500

2000

2500

3000

3500

MS

CI I

ndex

Argentina

A

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4P

rice

retu

rns

−0.4

−0.3

−0.2

−0.1

0

0.1

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BS

huffl

ed p

rice

retu

rns

−0.4

−0.3

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0

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Sur

roga

ted

pric

e re

turn

s

C

0 500 1000 1500 2000 2500 3000

0 500 1000 1500 2000 2500 3000

0 500 1000 1500 2000 2500 3000

0 500 1000 1500 2000 2500 3000

Time (day)

D

Fig. 1. Plots of the original Argentinean MSCI (A), its daily price returns (B), and one of the fifty shuffled (C) and surrogated time series (D) associated to thedaily price returns.

2334 L. Zunino et al. / Chaos, Solitons and Fractals 41 (2009) 2331–2340

4. Empirical results

Fig. 1 illustrates the original Argentinean MSCI, its daily price returns, and the shuffled and surrogated time series asso-ciated to the daily price returns.2 Fifty different realization of the shuffled and surrogated time series associated to the daily

2 The plots for the daily price and volatility returns of the other countries are available upon request from the corresponding author.

Argentina

Original returnsShuffled returnsSurrogated returns

Brazil

Chile Colombia

Mexico Peru

−10 −8 −6 −4 −2 0 2 4 6 8 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Venezuela

q−10 −8 −6 −4 −2 0 2 4 6 8 10

q

h(q)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 10q

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 10q

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 10q

0.2

0.3

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0.5

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0.7

0.8

0.9

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 10q

0.2

0.3

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0.5

0.6

0.7

0.8

0.9

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 10q

0.2

0.3

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0.5

0.6

0.7

0.8

0.9

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 10q

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

h(q)

US

Fig. 2. Generalized Hurst exponent, hðqÞ, as a function of q for the original, shuffled and surrogated daily price returns of Latin-American and the US marketindices. The results obtained for one of the 50 shuffled and surrogated realizations are shown.

L. Zunino et al. / Chaos, Solitons and Fractals 41 (2009) 2331–2340 2335

price and volatility returns were generated in order to reduce the statistical errors. The results obtained for only one of them areshown in Fig. 1. Figs. 2 and 3 display the generalized Hurst exponents hðqÞ, via the MFDFA-3 procedure, for the daily price andvolatility returns (star), respectively, of Latin-American and the US market indices. In our analysis q runs from �10 to 10 with astep of 0:5 and the window lengths, s, is between 20 and N=4 with a step of 4 – N is the length of the time series. A similar qrange was recently chosen for the multifractal analysis of the daily price returns series of Shanghai Stock Exchange (SSE) com-posite index [48]. Moreover, the s range was selected according to the suggestions of Kantelhardt et al. [28]. The curves obtained

Argentina

Original volatility returnsShuffled volatility returnsSurrogated volatility returns

Brazil

Chile Colombia

Mexico Peru

−10 −8 −6 −4 −2 0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2Venezuela

q

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

q

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

q

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

q

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

q

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

q

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

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0.7

0.8

0.9

1

1.1

1.2

q

h(q)

−10 −8 −6 −4 −2 0 2 4 6 8 100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

q

h(q)

US

Fig. 3. Generalized Hurst exponent, hðqÞ, as a function of q for the original, shuffled and surrogated daily volatility returns of Latin-American and the USmarket indices. The results obtained for one of the 50 shuffled and surrogated realizations are shown.

2336 L. Zunino et al. / Chaos, Solitons and Fractals 41 (2009) 2331–2340

for one of the fifty shuffled (circles) and surrogated (square) time series are also included for comparison purpose. It should benoted that always hðqÞ decreases with q. Thus, we can confirm the multifractal nature for all the cases. For short time series,where the statistics is poor, we expect a slight difference between hðqÞ even for monofractal behaviors [28]. Thus, one has tobe very careful with the multifractal assertion in the US market index specially for the volatility returns case – see Fig. 3, bottom.Multifractality degrees, Eq. (6), for the original, shuffled and surrogated time series are reported in Table 1 for the daily pricereturns, and in Table 2 for the daily volatility returns. In this latter case we find a particular behavior for US, where the asso-

Table 1Degrees of multifractality for the daily price returns of Latin-American and the US market indices via the MFDFA-3

Country Dh hDhshuf i hDhsuri Dhcor DhPDF DhPDF=Dhcor

Argentina 0.397 0.254 (0.058) 0.125 (0.029) 0.143 0.272 1.90Brazil 0.430 0.172 (0.041) 0.117 (0.029) 0.258 0.313 1.21Chile 0.321 0.142 (0.045) 0.137 (0.027) 0.179 0.184 1.03Colombia 0.397 0.213 (0.043) 0.135 (0.040) 0.184 0.262 1.42Mexico 0.347 0.222 (0.049) 0.106 (0.034) 0.125 0.241 1.93Peru 0.348 0.168 (0.046) 0.137 (0.028) 0.180 0.211 1.17Venezuela 0.641 0.402 (0.065) 0.096 (0.035) 0.239 0.545 2.28US 0.291 0.150 (0.043) 0.097 (0.035) 0.141 0.194 1.38

The influence of the two factors, long-range correlations and broad fat-tail distribution, are quantified. In the shuffled and surrogated cases the multi-fractality degree is estimated averaging over the fifty realizations. Standard deviations are in parentheses.

Table 2Degrees of multifractality for the daily volatility returns of Latin-American and the US market indices via the MFDFA-3

Country Dh hDhshuf i hDhsuri Dhcor DhPDF DhPDF=Dhcor

Argentina 0.440 0.348 (0.054) 0.082 (0.040) 0.092 0.358 3.89Brazil 0.372 0.227 (0.053) 0.078 (0.038) 0.145 0.294 2.03Chile 0.394 0.200 (0.041) 0.086 (0.041) 0.194 0.308 1.59Colombia 0.417 0.288 (0.038) 0.140 (0.043) 0.129 0.277 2.15Mexico 0.392 0.310 (0.049) 0.081 (0.047) 0.082 0.311 3.79Peru 0.298 0.221 (0.049) 0.058 (0.030) 0.077 0.240 3.12Venezuela 0.643 0.494 (0.067) 0.077 (0.033) 0.149 0.566 3.80US 0.133 0.207 (0.035) 0.072 (0.038) – 0.061 –

The influence of the two factors, long-range correlations and broad fat-tail distribution, are quantified. In the shuffled and surrogated cases the multi-fractality degree is estimated averaging over the fifty realizations. Standard deviations are in parentheses.

Table 3Comparison of the degrees of multifractality obtained for small (q < 0) and large fluctuations (q > 0) associated to the daily price returns of Latin-American andthe US market indices via the MFDFA-3

Country Dhq<0 Dhq>0 Dhq>0=Dhq<0

Argentina 0.164 0.234 1.43Brazil 0.212 0.218 1.03Chile 0.104 0.217 2.09Colombia 0.148 0.249 1.68Mexico 0.134 0.213 1.59Peru 0.151 0.197 1.30Venezuela 0.245 0.396 1.62US 0.131 0.160 1.22

L. Zunino et al. / Chaos, Solitons and Fractals 41 (2009) 2331–2340 2337

ciated shuffled time series have, in average, a higher multifractality degree than the original time series. We attribute this inex-tricable situation to the above-mentioned monofractal behavior. We have also compared the multifractal contributions associ-ated to the small (q < 0) and large fluctuations (q > 0). See Tables 3 and 4.

It was shown that important observables in the dynamics of financial markets – like interest rates, traded volumes – havericher multifractality for emerging markets than mature ones [18,48,49]. Thus, in Tables 5 and 6 we rank the price and vol-atility returns, respectively, of the Latin-American and the US market indices by considering their multifractality degrees,defined in Eq. (6), as a measure of inefficiency:

3 Ranbe usedportfoli

Ineff ¼ Dh ¼ hðqminÞ � hðqmaxÞ: ð9Þ

We have also considered the mean absolute deviation of the generalized Hurst exponents as another possible measure of themultifractality degree. We have chosen this measure of statistical dispersion because it is independent of the Gaussianassumption like the standard deviation. In Tables 7 and 8 we rank the price and volatility returns of the Latin-Americanand the US market indices by considering these approaches.3

king market inefficiency is an important information for policy makers and regulators since it provides some guidance on appropriate policies that mayto improve market efficiency, which by its turn reduces distortions in the economy. Furthermore, it is crucial for the investment funds industry and

o and risk management purposes [50–53].

Table 4Comparison of the degrees of multifractality obtained for small (q < 0) and large fluctuations (q > 0) associated to the daily volatility returns of Latin-Americanand the US market indices via the MFDFA-3

Country Dhq<0 Dhq>0 Dhq>0=Dhq<0

Argentina 0.141 0.300 2.13Brazil 0.146 0.226 1.55Chile 0.158 0.236 1.49Colombia 0.190 0.227 1.19Mexico 0.088 0.304 3.45Peru 0.103 0.194 1.88Venezuela 0.250 0.393 1.57US 0.009 0.123 13.7

Table 5Inefficiency ranking for the daily price returns according to the multifractality degree defined in Eq. (9)

Country Returns

US 0.291Chile 0.321Mexico 0.347Peru 0.348Argentina 0.397Colombia 0.397Brazil 0.430Venezuela 0.641

Table 6Inefficiency ranking for the daily volatility returns according to the multifractality degree defined in Eq. (9)

Country Volatility

US 0.133Peru 0.298Brazil 0.372Mexico 0.392Chile 0.394Colombia 0.417Argentina 0.440Venezuela 0.643

Table 7Inefficiency ranking for the daily price returns according to the mean absolute deviation of the generalized Hurst exponents

Country Returns

US 0.084Chile 0.096Peru 0.106Mexico 0.108Colombia 0.117Argentina 0.120Brazil 0.130Venezuela 0.203

Table 8Inefficiency ranking for the daily volatility returns according to the mean absolute deviation of the generalized Hurst exponents

Country Volatility

US 0.033Peru 0.086Brazil 0.112Colombia 0.117Chile 0.118Mexico 0.125Argentina 0.139Venezuela 0.210

2338 L. Zunino et al. / Chaos, Solitons and Fractals 41 (2009) 2331–2340

L. Zunino et al. / Chaos, Solitons and Fractals 41 (2009) 2331–2340 2339

5. Conclusions

We provide empirical evidence of multifractality in the daily price and volatility returns of Latin-American market indi-ces. The results suggest that the multifractality degree is higher for emerging market. Note that for both, price and volatilityreturns, US has the smallest degree and Venezuela the highest. Moreover, we introduce the multifractality degree as a mea-sure of inefficiency and rank the stock market of these countries. Two different estimators for the multifractality degree wereintroduced: the range and the mean absolute deviation of the generalized Hurst exponents. Similar results were obtained inboth approaches. It should be stressed that our ad hoc definitions of stock market inefficiency need a better confirmation. So,an exhaustive comparison of the results obtained for known emerging and developed markets will be performed in thefuture.

We also verified that the multifractal structure of Latin-American market indices can be mainly attributed to the broadfat-tail distributions and secondarily to the long-range correlations. This result is in accord with the findings of a recentmonofractal analysis of these market indices [52]. Furthermore, it agrees with the results of Matia et al. [7] and Lim et al.[54]. It should be noted that Chile has a distinct behavior with similar contribution of the two factors.

From Tables 3 and 4 we have concluded that large fluctuations have higher contributions to the multifractality degreethan those corresponding to small fluctuations. This reinforces the hypothesis that small and large fluctuations are due todifferent dynamical mechanisms [17].

As one can conclude comparing the first column of Tables 1 and 2, the multifractality degree of price and volatility returnsare similar. On the other hand, it is widely know that volatility returns have stronger evidence of long-range dependencethan price returns [52]. Thus, we deduce that strong long-range correlations do not enhance the multifractality. This is inopposite to the conclusions of other work [48].

It was recently suggested by computational [55,56] and empirical [57] approaches that the presence of herding behaviorin the stock market can be another source of multifractality. Further research should study the influence of this new factor inthe multifractal nature of Latin-American stock indices.

Finally, an important research question is whether one can exploit multifractal properties of stock markets for a variety ofcountries and build better forecasting models. This issue should be in the research agenda for the future.

Acknowledgements

Luciano Zunino was supported by Consejo Nacional de Investigaciones Cientıficas y Técnicas (CONICET), Argentina. Alej-andra Figliola gratefully acknowledges support from Universidad Nacional de General Sarmiento (UNGS) and Consejo Nac-ional de Investigaciones Cientıficas y Técnicas (CONICET), Argentina. Benjamin M. Tabak gratefully acknowledges financialsupport from CNPq foundation. The opinions expressed in the paper do not necessarily reflect those of the Banco Centraldo Brasil. Darío G. Pérez was supported by Comisión Nacional de Investigación Científica y Tecnológica (CONICYT, FONDECYTproject No. 11060512), Chile, and partially by Pontificia Universidad Católica de Valparaíso (PUCV, Project No. 123.788/2007), Chile. Osvaldo A. Rosso gratefully acknowledges support from Australian Research Council (ARC) Centre of Excellencein Bioinformatics, Australia.

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