UNIFRACTALITY AND MULTIFRACTALITY IN THE ITALIAN STOCK MARKET

ENRICO ONALI

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Bangor Business School, Bangor University, Gwynedd, LL57 2DG, United Kingdom. [email protected]

JOHN GODDARD Bangor Business School, Bangor University, Gwynedd, LL57 2DG, United Kingdom. [email protected]

Abstract This paper investigates the time-series properties of the Mibtel, the principal Italian stock market index, by means of fractal techniques. We examine whether the Mibtel is characterized by long memory or long-range dependence. Both rescaled range analysis and Multifractal Detrended Fluctuation Analysis indicate that the Hurst exponent of the daily Mibtel returns series is somewhat larger than that of a reshuffled version of the same series. Although the estimated Hurst exponent does not appear to be significantly different from the values produced in a Monte Carlo simulations exercise based on randomized standard Normal deviates, there is some evidence that points towards the existence of non-periodic cycles of duration between 1 and 14 calendar months.

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Corresponding author

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UNIFRACTALITY AND MULTIFRACTALITY IN THE ITALIAN STOCK MARKET

1.

Introduction The weak-form Efficient Markets Hypothesis (EMH) posits that current security

prices impound all of the information that can be derived from their past values (Fama et al., 1969; Fama, 1970). An implication of the weak-form EMH is that log security prices follow random walks. Peters (1994) introduces an alternative paradigm, focussing on the concept of market stability, rather than market efficiency. According to the self-affinity rule, in a stable market the distributions of returns over various time horizons are the same after adjustment for the length of the time period. The concept of self-affinity is borrowed from the mathematical theory of fractals (Mandelbrot, Fisher and Calvet, 1997). This alternative approach has become known as the Fractal Market Hypothesis (FMH). Adopting the FMH paradigm, the objective of this paper is to identify unifractal and multifractal behaviour in the daily returns series of the Mibtel, the principal Italian stock market index. An important implication of the weak-form EMH is that log prices are martingales, and log returns are independent and identically distributed (IID). Many empirical studies based on US data have rejected the hypothesis of weak-form efficiency (French, 1980; Gibbons and Hess, 1981; Kleim, 1983; Lo and MacKinlay, 1988 and 1990; Poterba and Summers, 1988; Kullmann et al., 2002). However, some recent evidence is more supportive (Toth and Kertesz, 2006). Elsewhere, calendar anomalies have been found in the case of Italy (Barone, 1990), and the hypothesis of IID returns has been rejected for Sweden (Frennberg and Hansson, 1993), Spain (Blasco et al, 1997) and Denmark (Risager, 1998). Evidence of non-Normality is reported for Romania (Dragota and Mitrica, 2004). In a multi-country study, Worthington and Higgs (2003) find stock market returns are non-Normal and serially correlated. 1 If log security prices do not behave in accordance with the weak-form EMH,

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Results for countries outside the US and Europe are mixed. Contrary to the random walk hypothesis are Urrutia (1995), Huang (1995), Poshakuale (1996), Laurence et al (1997), Chen et al (2001), Groenewold et al (2003) and Islam et al (2005). On the other hand, Ayadi and Pyun (1994) and Marshall and Cahan (2005) support the EMH for the Korean and New Zealander market respectively. Finally, Lock (2007) provides evidence of a transition from inefficiency to efficiency in the Taiwanese market.

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then several standard asset pricing models and risk management practices may need to be reassessed. 2 Instead of focussing exclusively on the role of information as in the EMH, the FMH stresses the importance of market structure. The self-affinity rule implies d

X (ct ) = c H X (t )

(1)

where X(t) is a returns time series with stationary increments, and 0

0.5 represents negative long-term persistence. Let N denote the total number of observations in the series {rt}. Subdivide N into M contiguous subperiods each containing n observations, such that Mn = N . For the observations within each subperiod, the mean and standard deviation of {rt} are

rm = (1 / n)

mn

∑

t = ( m −1) n +1

rt , S m = (1 / n)

mn

∑

t = ( m −1) n +1

(rt − rm ) 2

for m = 1,..., M

(5)

The cumulative deviations of {rt} from rm within each subperiod are

xt =

t

∑

s = ( m −1) n +1

(rs − rm ) for t = ( m − 1) n + 1,..., mn − 1 ; x nm = 0

(6)

The range for subperiod m is defined as the difference between the maximum and minimum values of {xt} for the observations within subperiod m Rm = max ( xt ) − min ( xt ) t∈m

t∈m

(7)

The range for subperiod m is rescaled by dividing Rm by Sm, the standard deviation of {rt} calculated over the observations within subperiod m. The (R/S)n statistic is the mean of the rescaled range values calculated over m = 1,...,M: M

( R / S ) n = (1 / M )∑ Rm / S m

(8)

m =1

The estimated Hurst exponent is obtained by running the OLS regression log10[(R/S)n]=log10(c)+H log10(n)

(9)

Following Qian and Rasheed (2004), the significance of the estimated H in (9) is evaluated by means of Monte Carlo simulations. We generate 1,000 simulated returns series, each comprising 2,608 observations, using randomly generated Gaussian innovations. (R/S)n is

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calculated for each simulated series and scale. The mean (R/S)n over 1,000 replications for each scale is denoted G(R/S)n. The Hurst exponent is calculated for each simulated series, G(H). The test statistic is

ς=

H − μH

σH

(10)

where μ H is the mean G(H) for the 1,000 Gaussian variables, and σ H is the standard deviation. The computed H for the Mibtel returns series indicates long memory if ζ exceeds its critical values obtained from the Monte Carlo simulations. 4 The average length of the non-periodic cycle is estimated using the Vn statistic

Vn = ( R / S ) n / n

(11)

According to the theory, under the random walk hypothesis, Vn is constant for all n. For positive long-term dependence, Vn is increasing in n, because (R/S)n increases faster than n . Conversely for negative long-term dependence, Vn is decreasing in n. Vn may be used to detect non-periodic cycles. Whereas periodic cycles are regular, non-periodic cycles exhibit random behaviour but have some specific mean length. A plot of Vn against log10(n) provides an indication of the existence of multiple cycles. The persistent cycle finishes where the plot levels off, while further increases in Vn indicate that longer cycles exist. In order to distinguish between unifractality attributed to long-term temporal dependence in either small or large fluctuations, and unifractality attributed to excess kurtosis in the distribution of daily returns, the procedures described above is repeated with respect to a reshuffled version of the returns series, in which the observed returns are re-ordered randomly. 5 The reshuffle procedure eliminates any long-term temporal dependence in the series, but does not eliminate excess kurtosis. Therefore if unifractality is apparent in the reshuffled series (as well as in the original series), this suggests the presence of excess kurtosis, which is not eliminated by reshuffling. On the other hand, if the reshuffled series is free of unifractality (but the original series exhibits unifractality), this suggests the original

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Feller (1951), Anis and Lloyd (1976) and Peters (1994) provide tests for the significance of the Hurst exponent. However, Feller’s (1951) formula is valid only asymptotically, and mixed findings are found about the reliability of the other two.

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The reshuffle procedure used in the present analysis follows the method of Norouzzadeh and Rahmani (2006, p331). Two integers t1 and t2 (1≤t1,t2≤N) are drawn randomly from a uniform distribution, and the positions of rp and rq in {rt} are exchanged. The same procedure is repeated 20N times, ensuring that the reshuffled series is without memory.

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series exhibits long-term temporal dependence, but not excess kurtosis. In Section 3, Vn for the Mibtel returns series is compared with Vn for both the reshuffled series and G(R/S)n. Multifractal Detrended Fluctuation Analysis Multifractal Detrended Fluctuation Analysis (MF-DFA), applied to stock price returns data, is based on the identification of the scaling behaviour of the qth-order moments of a returns series for various values of q, dependent on the number of observations that are used to estimate the moments. MF-DFA is a generalization of standard Detrended Fluctuation Analysis (DFA), which examines only the 2nd-order moments. Let N denote the number of observations in the series {rt}. The ‘profile’ of {rt}, denoted {yt}, is determined by calculating, for each t, the cumulative deviation from the sample mean of the observations of {rt} up to and including t t

y t = ∑ ( rs − r )

for t = 1,..., N − 1 , yN = 0

s =1

N

where r = ∑ rt / N

(12)

t =1

Starting from the first observation, subdivide N into M contiguous subperiods each containing n observations, such that N–n < Mn ≤ N. As before, n is the timescale. The procedure is repeated over several values of n. Within each subperiod, we detrend {yt} by fitting a P’th order polynomial using Ordinary Least Squares. Results are reported below for P=2. 6 Let {et} denote the detrended series P

et = y t − αˆ m − ∑ βˆ m, p [t − (m − 1)n] p for t = (m–1)n+1,...,mn; m = 1,...,M

(13)

p =1

Let Vm denote the variance of {et} for the n observations within subperiod m

Vm = (1 / n)

mn

∑

t = ( m −1) n +1

et2

for m = 1,...,M

(14)

Commonly, N is not a multiple of n. If Mn < N, then L = N – nM observations at the end of the observation period are unused in the above procedure. In order not to disregard these L observations, the above procedure is repeated starting from the L + 1th observation (rather than from the first observation). A second set of M calculated values of Vm is thereby obtained, labelled (for convenience) Vm+1,...,V2M. If Mn = N, {V1,...,Vm} and {Vm+1,...,V2M} are identical. The q’th order fluctuation function is calculated from the 2M calculated values of Vm

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Eisler and Kertesz (2004) find that low-order polynomials are generally sufficient to eliminate any deterministic linear or non-linear trend component from {yt}.

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2M

Fq , n = {[1 /( 2 M )]∑ Vm

q / 2 1/ q

}

(15)

m =1

The MF-DFA method focuses on the variation of Fq,n with the timescale n, for various values of q. The scaling behaviour of Fq,n can be investigated by examining the power-law relationship Fq,n ~ nh(q), where h(q) is the Generalised Hurst Exponent (GHE). The singularity spectrum is obtained from the Legendre transform of [q,h(q)] with scaling function

τ (q ) = qh (q ) − 1 . Unifractal processes have a single GHE for all q, h(q)=H, and a linear scaling function, τ = Hq − 1 , where H is the Hurst exponent. Multifractal processes have multiple GHEs, and the slope of the scaling function varies according to value assumed by h(q). Note that h( 2) = H and τ (1 / h( 2)) = 0 . The singularity spectrum describes the scaling behaviour of Fq,n over variation in q. For q > 0, the GHEs represent the scaling behaviour of segments with large fluctuations, because the segments with large Fq,n dominate the mean Fq,n. Conversely for q < 0, the GHEs represent the scaling behaviour of segments with small fluctuations, because the segments with small Fq,n dominate the mean Fq,n.. Below, values in the range –10≤q≤10 are examined, 7 in order to render our analysis comparable with recent literature (Lee and Lee, 2007). However, we note that bias is possible in the calculation of negative moments for financial returns series (Fisher, Calvet and Mandelbrot, 1997). The Hölder exponent α, known as the singularity strength, is

α = h( q ) + qh ' ( q )

where h' ( q ) = dh ( q ) / dq

(16)

The distribution of α, known as the multifractal spectrum f(α), is f (α ) = q[α − h( q )] + 1

(17)

The plot of f(α), which resembles an inverted parabola in the case of multifractality, contains information about the nature of the multifractal process. The magnitude of the range Δα = α MAX − α min , known as the multifractality strength, provides a useful indication of the richness of the multifractality. As before, in order to distinguish between multifractality attributed to long-term temporal dependence in either small or large fluctuations, and multifractality attributed to excess kurtosis in the distribution of daily returns, the procedure described above is repeated with respect to a reshuffled version of the Mibtel returns series.

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According to the “inverse cubic law”, the moments should be infinite for q≥3. A finite sample size, however, renders the calculation possible.

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3.

Empirical results Section 3 presents the empirical results for the unifractal and multifractal analysis of

the Mibtel logarithmic returns series, for the observation period 31/08/1995 to 30/08/2005. Mibtel is a composite price-index representing all the domestic companies listed on the Italian exchange, together with liquid foreign companies. 8 The daily returns series is defined as the first differences of the daily logarithmic closing prices, rt = ln Pt − ln Pt −1 . The data set comprises 2,608 daily observations on the returns series. Figure 1 presents a time series plot of {rt}. Volatility clustering is apparent, even by means of visual inspection. Table 1 reports the main descriptive statistics of the returns series, as well as the results of tests for the null hypotheses of non-stationarity and Normality. The Augmented Dickey-Fuller (1979, 1981) test establishes the stationarity of {rt} and the absence of a time trend. The Jarque-Bera (1980) test provides evidence of non-Normality, with both negative skewness and excess kurtosis. Table 2 reports the sample autocorrelation (AC) and partial autocorrelation (PAC) functions for lags 1 to 10. The Ljung-Box (1978) Q-statistic in the test for the joint significance of AC up to and including lag k is significant for k≥4. PAC is individually significant at the 1% level for k=4, and at the 10% level for k=5 and k=9. These results suggest that {rt} are not adequately described by the random walk hypothesis. Rescaled range analysis Table 3 reports the results for the rescaled range analysis. The values of the timescale parameter n = 3, 5, 10, 15, 20, 40, 80, 120, 150, 300, 650, and 1304 are shown in column (1). The numbers of contiguous subperiods used in the rescaled range analysis are M = 869, 521, 260, 173, 130, 65, 32, 21, 17, 8, 4, and 2, respectively. Columns (2) and (3) report the rescaled range statistics (R/S)n for the Mibtel returns series and the reshuffled returns series,

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The Mibtel index is calculated using a Laspeyres algorithm: I

Pt =

p i ,t

∑p i =1

× pi ,0 × qi ,0

i ,0

× 10000

I

∑p

i,0

× qi ,0

i

where pi,0 and qi,0 are the price and volume of share i at the base-date 03/01/1994, pi,t is the current price of share i, and I is the total number of shares represented in the indexDetails can be found on the website of Borsa Italiana SpA, www.borsaitaliana.it.

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respectively. Column (4) reports the average (R/S)n obtained from the Monte Carlo simulations. Similarly, columns (5) and (6) report Vn for the Mibtel returns series and the reshuffled returns series, and column (7) reports the average Vn from the Monte Carlo simulations. The values of (R/S)n reported in columns (2), (3) and (4) are very similar for low n, but they differ substantially for high n. For n = 1304, (R/S)n for the Mibtel returns series is roughly 20% larger than (R/S)n for the reshuffled series and the average (R/S)n from the simulations. Figure 2 plots the three sets of values of Vn against the logarithm of n (base 10). The plot for the Mibtel returns series is steeper than both the plot for the reshuffled series and the plot of the average Vn from the simulations, between around n=20 and n=300. The plot for the Mibtel returns series then declines around n ≅ 300 . This suggests there is a non-periodic cycle of duration approximately 1 to 14 calendar months. A closer examination suggests that a shorter cycle of duration around 7 calendar months might also be present. Finally, the plot for the Mibtel returns series increases over the range 650 ≤ n ≤ 1304, suggesting that further longer-range cycles might also be present. Table 4 reports the estimation results for the OLS regressions log10[(R/S)n]= log10(c)+H log10(n), used to estimated the Hurst exponent and its expected value μH, for the Mibtel returns series and the reshuffled series. Critical values at the 1%, 5% and 10% significance levels, obtained from the Monte Carlo simulations, are reported in the notes to Table 4. Neither of the estimated Hurst exponents for the Mibtel returns series (H=0.5864) and the reshuffled series (H=0.4728) is significantly different from the mean Hurst exponent obtained from the Monte Carlo simulations, μH = 0.5312. Nonetheless, the fact that the Hurst exponent drops by around 19% after reshuffling suggests strongly that long memory is not completely absent. The previous calculation of the Hurst exponent is based on the entire distribution of (R/S)n, rather than only those values pertaining to non-periodic cycles. Following McKenzie (2001), Table 5 reports separate estimations of the regression log10[(R/S)n] = log10(c)+Hlog10(n) for the scales: 2 ≤ n ≤ 20 , 20 ≤ n ≤ 300 , and 300 ≤ n ≤ 1304 . As before, these regressions are estimated for the Mibtel returns series and the reshuffled series, and for the average values of (R/S)n obtained from the simulations. For 2 ≤ n ≤ 20, the three estimated Hurst exponents are very similar. For 20 ≤ n ≤ 300, however, the estimated Hurst exponents diverge: the value for the Mibtel returns series (H=0.601) is larger than the values for both the reshuffled series (H=0.524) and the value for the average (R/S)n obtained from

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the simulations (H=0.546). For 300 ≤ n ≤ 1304 this pattern is repeated, with H=0.557 (Mibtel returns series), H=0.274 (reshuffled series) and H=0.509 (simulations). These results appear to support the hypothesis of both long-memory and non-Gaussian behaviour in the Mibtel. The drop in the Hurst exponent for the reshuffled series for large n might be explained by the smaller influence of extreme losses or gains in longer segments than in segments defined by small n. Before reshuffling, large cumulative departures are mainly due to the aggregation of small departures in the same direction, but after reshuffling the temporal randomization destroys this pattern, and results in a lower (R/S)n for the reshuffled series than for the original unshuffled series. Multifractal Detrended Fluctuation Analysis Table 6 reports summary results for the application of MF-DFA to the Mibtel daily returns series. The MD-DFA is computed using second-order polynomials (p=2) to de-trend the profile series {yt}. 9 Column (2) of Table 6 reports the GHE, or h(q), for q=2, equivalent to the Hurst exponent. Column (3) of Table 6 reports the most common Hölder exponent, α1, and column (4) reports the multifractality strength, Δα. Both h(2) and α1 are considerably smaller for the reshuffled series than for the Mibtel returns series. As before, this suggests that the Mibtel returns series is characterized by long memory. Furthermore, Δα is smaller for the reshuffled series than for the (unshuffled) Mibtel returns series. Figure 3 plots the scaling functions τ(q) for the Mibtel returns series and for the reshuffled series. The plot for the Mibtel returns series has a higher degree of convexity than the plot for the reshuffled series. This indicates that the extent of departure from unifractality is larger for the (unshuffled) Mibtel returns series than for the reshuffled series. All of these results suggest that multifractality is attrbutable to different degrees of temporal correlation affecting large and small fluctuations.

4.

Conclusion This paper has investigated the time-series properties of the Mibtel, the principal

Italian stock market index, by means of fractal techniques. In standard tests, the Mibtel daily returns series shows evidence of serial correlation and non-Normality. The main focus of this paper, however, is on whether the Mibtel is characterized by long memory or long-range

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Eisler and Kertesz (2004) find that low-order polynomials are generally sufficient to eliminate any deterministic linear or non-linear trend component from {yt}.

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dependence. We investigate this question following techniques pertaining to the FMH (Fractal Markets Hypothesis), a relatively recent theory that posits that capital markets display fractal properties. Both rescaled range analysis and MF-DFA (Multifractal Detrended Fluctuation Analysis) indicate that the Hurst exponent of the daily Mibtel returns series is somewhat larger than that of a reshuffled version of the same series, from which any temporal dependence is eliminated through a randomized reshuffle algorithm. Although the estimated Hurst exponent does not appear to be significantly different from the values produced in a Monte Carlo simulations exercise based on randomized standard normal deviates, there is some evidence that points towards the existence of non-periodic cycles of duration between 1 and 14 calendar months. In conclusion, our findings indicate that the Italian stock market displays both long-memory and multifractality.

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Urrutia J. L. (1995), Test of random walk and market efficiency for Latin American emerging equity markets, The Journal of Financial Research, 18, 299-309; Worthington A. C., Higgs H. (2003), Weak-form market efficiency in European emerging and developed stock markets, School of Economics and Finance, Queensland University of Technology, Brisbane, Australia, Working Paper n. 159.

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Table 1 Descriptive statistics, Augmented Dickey-Fuller test and Jarque-Bera test on Mibtel logreturns. Statistics

Coefficients

Mean

0.0004

Standard Deviation

0.0128

Minimum

-0.0771

Maximum

0.0683

Coefficient of Variation

36.1421

Skewness

-0.2069

Kurtosis

6.1328

Z-stat

-15.803***

Trend t-stat

-0.89

ADF Jarque-Bera χ2-stat

1084.2954***

Notes: The Augmented Dickey-Fuller test results are for the specification with the lowest AIC (8 lags) among those tested (lag-lengths 1-10). * Statistically significant at the 10% level. ** Statistically significant at the 5% level. ***Statistically significant at the 1% level.

18

Table 2 Autocorrelations and partial autocorrelations in the Mibtel daily log-returns series for laglengths 1-10. Lag (k)

AC(k)

Q-statistic

PAC(k)

t = PAC(k)/√n

1

0.0176

0.8108

0.0176

0.899

2

0.0245

2.3823

0.0242

1.236

3

-0.0258

4.1197

-0.0267

-1.364

4

0.0777

19.897***

0.0782

3.994***

5

-0.0346

23.031***

-0.0366

-1.869*

6

0.0216

24.256***

0.0189

0.965

7

-0.0282

26.333***

-0.0235

-1.2

8

0.0391

30.337***

0.0318

1.624

9

0.0289

32.525***

0.0354

1.808*

10

0.0084

32.71***

-0.0004

-0.02

Q-statistic is the Ljung-Box Q-statistic for the joint significance of AC(k) for lags up to and including the current lag-length. * Statistically significant at the 10% level. ** Statistically significant at the 5% level. ***Statistically significant at the 1% level.

19

Table 3 Rescaled range analysis. (R/S)n

(R/S)n

G(R/S)n

Vn

Vn

Vn

Mibtel

reshuffled

simulations

Mibtel

reshuffled

simulations

3

1.3531

1.3570

1.3504

0.7812

0.7835

0.7797

5

1.9208

1.9186

1.9273

0.8590

0.8580

0.8619

10

3.0185

3.0348

3.0228

0.9545

0.9597

0.9559

15

3.8446

3.7821

3.8821

0.9927

0.9765

1.0024

20

4.5311

4.5206

4.6121

1.0132

1.0108

1.0313

40

7.1699

6.7538

6.8919

1.1337

1.0679

1.0897

80

10.6503

9.9739

10.1592

1.1907

1.1151

1.1358

120

13.7976

12.5817

12.6472

1.2595

1.1485

1.1545

150

15.9865

13.2368

14.2610

1.3053

1.0808

1.1644

300

23.8509

19.0844

20.6338

1.3770

1.1018

1.1913

650

31.0789

31.9440

30.7577

1.2190

1.2529

1.2064

1304

52.8786

41.6541

43.8638

1.4643

1.1535

1.2147

n

20

Table 4 Hurst exponent estimation results. Mibtel

Reshuffled

0.5864

0.4728

(0.0019)

(0.0029)

-0.0873

0.1088

(0.0051)

(0.0078)

N obs

1303

1303

R-adj

0.9870

0.9543

ς

1.2123

-1.2826

Hurst exponent

Constant

Note:

The

estimated

Hurst

exponents

are

the

slope

coefficients

in

the

regressions

log10[(R/S)n]=log10(c)+Hlog10(n). Standard errors are reported in parentheses. In the Monte Carlo simulations based on randomized standard Normal deviates, the mean estimated Hurst exponent is μH = 0.5312 and the standard deviation is σH = 0.0455. Assuming the distribution of the Hurst exponents of the simulated series is Normal, the critical values for ς = H − μ H at the 1%, 5% and 10% significance levels are 0.6484, σH

0.6204 and 0.6060, respectively.

21

Table 5 Hurst exponent estimation results for various scales.

2 ≤ n ≤ 20 20 ≤ n ≤ 300 300 ≤ n ≤ 1304

(R/S)n

(R/S)n reshuffled

G(R/S)n

0.652

0.647

0.654

(0.006)

(0.008)

(0.006)

0.601

0.524

0.546

(0.003)

(0.003)

(0.000)

0.557

0.274

0.509

(0.006)

(0.005)

(0.000)

Note: The estimated Hurst exponent is the slope coefficient in the OLS regression log10[(R/S)n]=log10(c)+Hlog10(n). Standard errors of estimated coefficients are reported in parentheses.

22

Table 6 MF-DFA results h(2)

α1

Δα

Mibtel

0.5122

0.5084

0.5585

Reshuffled

0.4592

0.4573

0.3048

23

Figure 1 Mibtel daily log-returns ( rt = ln Pt − ln Pt −1 ). Sample period: 31/08/199530/08/2005. Data source: Thomson Analytics. 0.08 0.06 0.04 Mibtel

0.02 0.00 -0.02 -0.04 -0.06 -0.08 -0.10

24

Figure 2

.8

1

1.2

1.4

1.6

V-statistic – Mibtel and Monte Carlo simulated series.

.5

1

1.5

log-scale

Mibtel Simulated series

2

2.5

3

Reshuffled series

25

Figure 3

-8

-6

-4

-2

0

2

Scaling functions of Mibtel log-returns: original and reshuffled series.

-10

-5

0 q Mibtel

5

10

Reshuffled

26