As one of the task of this study is to test if long memory characterises market in different periods of time or not, the question of results' significance becomes extremely important. In conditions of no theoretical distributions, the only existing alternative is constructing of empirical confidence intervals. This method is applied in number of studies. The idea is in simulation of large number of Gaussian independent series, estimation of its mean Hurst exponent and confidence interval for it. Traditional approach is to test the null hypothesis of no or weak dependence () against alternative of long memory (). If obtained H is not in the confidence interval of , the null hypothesis is rejected. Important is, that simulated and study series should have the same length, so, usually this procedure is performed newly every time.

However, the technology of estimating confidence intervals is also quite controversial. Several different approaches are described below. Almost all of them use the only nonsimulatory theoretical result - the expected value of rescaled range statistic for independent normally distributed variables derived by Anis and Lloyd (1976).

Peters (1991) empirically added term to improve performance for small samples, so this formula is a modification of the original one presented by Anis and Lloyd (1976). However, it is shown (Couillard, Davison, 2004) that results with and without this correction are very similar, so both variants can be used. Further, we will refer to this corrected formula as to Anis and Lloyd's formula. More important is the fact that this formula gives values that are close to the obtained from simulated distributions. It was first demonstrated by Peters (1991) and then confirmed by following research, hence studies aimed to determine the significance of R/S-analysis results often use Anis and Lloyd formula to estimate the expected value. Nevertheless, the question of standard deviation, quantiles etc. determination remains open, so now procced to discussion of possible methods.

Couillard, Davison (2004) investigate the behaviour of Hurts exponent obtained by R/S-analysis. They assume that Hurst exponent's distribution is approximately normal (and test it comparing the histogram of 10 000 exponents obtained from independent time series of 10 000 normally distributed observations with normal distribution with the same mean and variance). This assumption allows to build confidence intervals and also use the test statistic

,

that has Student-distribution. is calculated according to Anis and Lloyd's formula, standard deviation is estimated as standard deviation of 10000 Hurst exponents obtained from 10000 independent series. The authors also provide the empirical relationship between standard deviation and series length: . This relationship is obtained from the regression fit of the logarithm of estimated standard deviations and logarithm of the series length. Suggested t-test is applied to the logarithmic return of stocks of three large capitalisation companies (Procter & Gamble Corporation, General Electric Company and IBM), in each case test fails to reject the null hypothesis of no long memory.

However, normal distribution assumption is a strong one, therefore, it is probably better to use other methods to build confidence intervals. Weron (2002) build empirical confidence intervals using sample quantiles for R/S-analysis and DFA. The procedure consist of following: (1) simulation of large number (10000) Gaussian White Noise series for a set of lengths (), calculate the statistics; (2) compute sample quantiles for all lengths; (3) plot sample quantiles vs. sample lengths and fit this with some functions. As the result, the authors get equations (for lower and upper quantiles) from which they can construct confidence intervals (5% and 95% quantiles are used to build 90% confidence interval, 2.5% and 97.5-95% confidence interval, etc.), they write that the fitting quality is high (. Confidence intervals are constructed for corrected Anis-Lloyd Hurst exponent (RS-AL calculated as 0.5 plus the slope of for (subinterval size), because for lower method possesses high variance.

Table 1

Empirical confidence intervals for the RS-AL statistic for and sample length obtained by Weron (2002)

The same was done for DFA, in this case both for and .

Table 2

Empirical confidence intervals for the DFA statistic for sample length , obtained by Weron (2002)

Confidence intervals are constructed also for Hurst exponent estimated using Periodogram Regression, but this method is not discussed in this paper. These results are very useful for quick estimation of significance: they can be applied to the series of various lengths, there is no need to do any simulations. Overall, this method looks quite reliable. The only possible problem is connected with small samples. In the same paper it is shown by simulations that RS-Al and DFA estimated Hurst exponent value approaches 0.5 for large samples, but for samples of about 250-1000 observations it is smaller than 0.5 (), and samples of this size are quite often. Constructing the confidence interval while assuming that the mean value of H is equal to 0.5 can lead to shifted to the right confidence interval, in this case the hypothesis of independence can be accepted too often for and rejected too often for First case is especially important, if we need to distinguish between long memory and its absence.

Horta, Lagoa, Martins (2014) estimate Hurts exponent using MDA technique, but the procedure is slightly different from, for example, Couillard, Davison (2004). They conduct bootstrap experiment, it starts as usual from generating Gaussian series of dimension R (R = and calculating Hurst exponent for it. Then they randomly take a sample of R observations and replace it, after it Hurst exponent is calculated again. This procedure is repeated 1000 times, resulting in a distribution of 1000 Hurst exponents. Form this distribution the authors calculate the mean, the standard deviation and the percentiles. The decision about the null hypothesis (again, () is done by evaluating p-value:

),

where is the Hurst exponent estimated for each of the 1000 series from the bootstrap experiment. This method is similar to others where is obtained from simulated series, excepting for another decision rule.

Summing up, even in the absence of theoretical distribution, there are some methods for proving the significance of Hurst exponent. First, Andrew Lo's test can be applied as probably the most reliable one. For corrected R/S-AL and DFA methods empirical confidence intervals derived by Weron (2002) could be applied. For other methods (R/S, DMA) the only possibility is comparison of the observed Hurst exponent with the obtained from simulated series. While this comparison it is better to use empirical confidence intervals and not the obtained by assumption of normality of Hurst exponent's distribution, because this assumption is questionable one.

2. Empirical analysis

2.1 Data preprocessing

2.1.1 Data description and visual analysis

RTSI (Russia Trading System Index) is a capitalization-weighted composite index calculated based on prices of the 50 Russian stocks traded on the Moscow Exchange. The list of stocks is reviewed every three months. It includes stocks of the largest and dynamically developing Russian companies (for example, Gazprom, Aeroflot, Yandex etc.). The RTS Index was launched on the 1st September 1995. In contrast to the MICEX Index it is denominated in US dollars.

Picture 1. The Dynamics of RTSI from 11.01.2005 until 10.03.2017

We study the period from the beginning of 2005 (11.01.2015) until 10.03.2017. It can be easily seen that the index was not stable during considered period. Visual analysis suggests the periods this dynamics can be divided into. RTSI was growing fast since the beginning of 2005, on 19.06.2008 it reached its maximum value - 2400.84. After it the index started to lose its value rapidly - in the end of October 2008 it was already lower than 600, this collapse is explained by global financial crisis. Next period is associated with recovering, however RTSI has not reached the previous maximal value. Since the beginning of 2013 index is decreasing again, since 2014 - with increasing speed. This is crisis in Russia of 2014-2016 that was caused by structural problems in Russian economy, its slowdown and weakening of the ruble. In the end of 2016 RTSI started to grow again. Therefore, the period can be divided into five periods: steady grow, first crisis, recovering, second crisis, recovering. Further, statistical techniques will be implemented to divide the series into periods more precisely.

2.1.2 Testing for structural breaks

To test the first hypothesis the initial series should be divided into tranquil and crisis periods. It can be done using tests for structural changes in series. The best indicator for the crisis periods is market's volatility. Some suggestions about dates of structural changes can be done also by analysing index values, for instance, the series can be divided into periods of positive and negative trends. However, the change of trend from negative to positive does not compulsory indicate the end of the crisis, because the market can still be not stable - and volatility can indicate this. Therefore, the periods are obtained using the Index of Russian Exchange volatility (RTSVX).

Picture 2. The Dynamics of RTSVX index from 10.01.2006 until 14.12.2016

Detection of structural breaks in regression models is an important topic in econometrics. Chow (1960) introduced the most popular test for structural breaks, it allows checking if the effect of independent variables on the dependent one varies across the periods, assuming that the breakpoint is known a priori. Recently, Bai and Perron (1998, 2003) introduced procedure that allows testing for multiple unknown breakpoints. The test considers standard multiple linear regression with T observations and m breakpoints (m+1 periods). All the regressors are divided into two groups: coefficients of X variables do not vary across periods, while coefficients of Z variables are period specific.

This test has several specifications that can be classified into two categories: global optimisation procedures and sequential procedures.

Global optimisation procedures search for breakpoints that minimize the sum of squared residuals of the regression above. So, for a set of breakpoints the following expression is minimized:

,

where - is the starting point of period j.

The optimal breakpoints are those that give the lowest sum of squared residuals across of all possible sets of m-breakpoints. The number of breakpoints is determined testing for m breaks vs. none (comparing the models using F-statistics with the null hypothesis: ) or information criteria (minimizing the Schwarz criterion). The form of F-statistics m breaks vs. none (“Global L breaks vs. none” in EViews) and the critical values (as the distribution of test statistic is not standard) are presented in Bai, Perron (2003).

For detection of periods of different volatility the following regression model is used:

, for ,

Where - the value of volatility index at time t;

- the mean value of volatility for period j;

- random component.

Results of Global L breaks vs. none test applied to this equation are presented in Table 3 below. From one to five breaks are significant at 95% level in comparison with no breaks. The UDMax statistic chooses the highest F-statistic, WDMax chooses the highest weighted statistic, weights are picked so, that p-values are equal. According to UDMax and WDMax, four - is the optimal number of breakpoints.

Table 3

Results of Bai-Perron test (Global L breaks vs. None specification)

The same result obtained using Information Criteria test: model with four breakpoints has the minimal value of Schwarz Information Criterion. This fully corresponds to visual analysis and assumptions about the number of periods with different volatility. The dates of breakpoints are: 25.07.2008, 22.03.2010, 03.03.2014, 25.05.2015.

Table 4

Results of Bai-Perron test (Information Criterion specification)

Sequantial testing is based on sequantial application of breakpoint test: first, full sample is tested for the presence of one unknown breakpoint, if the null hypothesis of no breakpoint is rejected, then the test is performed again in both subsamples; this procedure should be repeated until all the subsamples do not reject the null hypotesis. Applied to considered time series sequential procedure stops also at the fourth breakpoint, the fifth is not significant at 95% level (Application 1). Estemated breakpoints are close to obtained by the previous test. Devision into periods is done according to the Global M vs. none and Infromation criteria test, because these breakpoints are closer to the estemated visually. Significance of these breakpoints is also prooved by Chow test (Application 1).

Initial data set is divided into 5 periods.

Table 5

Division of data set into periods

Picture 3 presents thid devision, as can be seen the periods of crisis are not the same that were suggested by visial analysis, so the usage of volatility index is justified.

Picture 3. The Dynamics of RTSI from 11.01.2005 until 10.03.2017 divided into crisis and tranquil periods

2.1.3 Stationarity tests

Further, not the index values but logarithmic returns will be analysed. Logarithmic return is calculated as: , where - index value at time t.

Most of the methods require stationarity of series, so it is important to check if series is stationary and, if not, to transform it to stationary (i.e. determine the order of integration). Visual analysis (Picture 4) suggests that the series is stationary (though there are clear volatility clusters), to prove it we will apply several statistical test.

Stationary process is a process whose joint probability distribution does not change in time, hence its parameters (mean and variance) does not change over time and covariance function depends only on the difference between points. If there is no trend, the cause of nonstationarity is the presence of unit root.

Picture 4. The Dynamics of logarithmic returns from 11.01.2005 until 10.03.2017

If the characteristic equation:

of the process has the root , then the process has unit root and is nonstationary. Therefore, the hypothesis of stationarity is implies that all are strictly larger than 1. For AR(1) process that means that is strictly less than 1, so that variance of does not increase with time. Test that verify this hypothesis are called unit root tests.

The Augmented Dickey Fuller test (ADF-test) is the extension of Dickey Fuller test for higher order correlation (AR(p) processes with p>1). The null hypothesis is the presence of unit root. Test statistic is calculated like standard t-statistic, but it does not follow Student's distribution, so it is compared with simulated critical values (Dickey-Fuller critical values). If observed statistic is smaller than critical value, then the null hypothesis of nonstationarity is rejected. The results of ADF-test are presented below. The null hypothesis is rejected on 0.001 significance level.

Table 6

Results of ADF-test for logarithmic returns series

Other popular unit root test are the Phillips-Perron test (PP-test) and the Kwiatkowski, Phillips, Schmidt, and Shin test (KPSS-test). The first modifies statistic from the ADF-test to control for serial correlation, the second uses another test statistic and series are supposed to be stationary under null hypothesis. Both tests confirm the results of ADF-test, stating that the process is stationary and suitable for further analysis (Application 2).

2.1.4 Filtering data with GARCH (1,1) model

Some of the studies on Hurst exponent estimation suggest that series should be preliminary filtered by ARMAX-ARCH or ARMAX-GARCH model in order to avoid the distortion of Hurst exponent value by the effects of heteroscedasticity or short-range dependence (e.g. Cajueiro&Tabak, 2004). As the graph of series (Graph) suggests that there are some clusters of volatility, filtering should be probably applied also here. At first, test for existence of ARCH-effects. Tests suggests the dependence in squared residuals: , the null hypothesis is H0: , no heteroscedasticity in residuals. This hypothesis is rejected on the 0.001 significance level, for the regression model with exogenous constant (Application 3), so there is a ground for using conditional heteroscedasticity models.

After choosing the lags AR(1)-GARCH(1,1) model turns to be the optimal one (estimation in Application 3).

The ARCH-test shows that there is no heteroscedasticity in residuals (Ljung-Box Q-statistics for squared residuals are also not significant); Darbin-Watson statistic and Ljung-Box Q-statistics indicate no autocorrelation in residuals (Application 3). Conditional variance is presented on the graph below, it is one-step ahead forecast made by the model. Apparently, conditional variance has is higher for the periods of high volatility (Picture 5).

Picture 5. Forecast of conditional variance for GARCH(1,1)-model

Picture 6. The dynamics of filtered returns from 11.01.2005 until 10.03.2017

Applying filtering procedure, we obtain series , where - is return at time t, - is filtered return at time t, - is conditional standard deviation obtained from the AR(1)-GARCH(1,1)-model at time t. Filtered return series is used for further analysis, Graph X shows, that there are no visible ARCH-effects anymore.

2.2 Measuring efficiency of Russian financial market

In this part the efficiency of Russian financial market will be measured by Hurst exponent estimated using various methods. The hypothesis originates from EMH and says that during financial crisis markets lose their efficiency, that means, the value of Hurst exponent should be higher than during tranquil periods. This hypothesis was proved for many markets (USA, European markets, Asian markets, see literature review). This analysis will show, if Russian market follows the same tendency. Statistical formulation of the hypothesis will be:

2.2.1 R/S analysis

At first, Hurst exponent will be estimated by standard R/S analysis. The procedure was described in the first part of this paper, it can be realized in statistical package Gretl.

Initial series is divided into period of length n, rescaled range is calculated, and then this procedure is repeated again for another n. The first period series (Tranquil Period I) is divided into different n seven times (n=13, 27,…, 880), so that we obtain seven R/S-statistics.

hirst exponent russian financial market

Table 7

Performing R/S-analysis

Hurst exponent is estimated as a coefficient of regression:

Table 8

Estimation of Hurst exponent using R/S-analysis

The estimated Hurst exponent is equal to 0.61, this is higher than 0.5, but, as discussed before, this does not ensure that series have long-memory, especially taken into account that R/S analysis overestimates Hurst exponent for small samples. Results for other periods are presented in a table below. It can be seen, that Hurst exponents for crises periods are higher than for tranquil periods; the highest H is observed for fourth period - Russian economic crisis, the lowest - between two crises. This fact corresponds with the hypothesis that market is less efficient during the crises than before or after it. Thus, Hurst exponent gives some information about the behaviour of prices during different periods, but another methods are necessary to test the hypothesis of long memory.

Table 9

Hurst exponents estimated by R/S-analysis for all studied periods

2.2.2 R/S-AL

Confidence intervals derived by Weron (2002) allow to test corrected Anis-Lloyd Hurst exponent for significance. R/S-AL statistic is calculated for n>50 according to Weron (2002) recommendations. Important is that confidence intervals depend on total number of observations, and for small samples (as in this study) confidence intervals will be very broad and it will be very difficult to reject the null hypothesis of no long memory. For example, for L=200 observations 90% confidence interval is and for L=2000 - , as Hurst exponent is asymptotically equal to 0.5.

Corrected Anis-Lloyd Hurst exponent is equal to for the first period, while theoretical Anis-Lloyd Hurst exponent (for normal distribution, obtained from Anis-Lloyd formula) is equal to . Confidence intervals are calculated as follows:

As = 0.4747 and it is inside the interval, the null hypothesis is not rejected at 90% significance level.

Results for all periods are presented below. For tranquil periods is relatively close to the theoretical value , for crises periods - it is higher. Though, corrected Hurst exponent values differ from classical values, general tendency of higher values for crises periods continues. According to the significance test only for the fourth period is outside the 90% confidence interval, so the null hypothesis H0: no long memory is rejected with 0.1 probability of error.

Table 10

Results of corrected R/S-AL-analysis for all studied periods (estimated H, theoretical H and 90% confidence intervals)

2.2.3 Andrew Lo rescaled range method

Lo's statistic originates from rescaled range method, but it can distinguish between long and short memory, and also allows hypothesis testing because standardized statistic has critical values. Choice of number of autocorrelation lags q is disputable question, here we use Andrew's (1991) rule (с - first-order autocorrelation and computed q are presented in the table below). Confidence intervals are: for 95% - for 90% - . Lo's modified R/S test gives totally different results: for all the periods the null hypothesis is not rejected at 95% level, for the last period (which is defined as tranquil) the null hypothesis can be rejected at 90% level.

Even if accounting for the fact that Lo's statictic has a strong preference towards null hypothesis (Teverovsky et al., 1999), this result is really unexpected, because the last tranquil period shows more evidence for long-memory. Actually, these results suggest that results obtained using other methods are biased because of existence of short-range memory (e.g. high values of Hurst exponent in R/S or DFA analysis are caused by short-term and not long-term memory), but lags of autocorrelation that are taken into account by Lo's statistic are not significant in most of the cases. To solve this problem Lo's test was performed also for unfiltered series, but this also has not given any interpretable results - statistic was obviously biased by the presence of heteroscedasticity.

Table 11

Results of Lo's test (с - first order correlation, q- parameter of test statistic, and estimated value of test statistic )

2.2.4 Detrended Fluctuation Analysis

Detrended fluctuation analysis yields similar results: null hypothesis is rejected with probability of error 0.1 only in the fourth period, for other periods - it is not rejected. Hurst exponents estimated by DFA differ sufficiently from previous results, but, again, values for crises periods are slightly higher. Confidence intervals are calculated according to Weron (2002) formulas, for n>10 (because of rather small samples).

So, the minimal n is 10, the maximal is also restricted by L/4 (not higher than 25% of observations), because otherwise there is not enough subperiods for averaging and statistic becomes unreliable (Kantelhardt et al., 2002).

Table 12

Results of DFA (estimated Hurst exponent and 90% confidence intervals)

2.2.5 Detrended Moving Average

1) Generate a sample of length N using standard Gaussian distribution;

2) Calculate - Hurts exponent of the sample;

3) As the true Hurst exponent of underlying distribution is unknown, we want to know how the distribution of varies around , that means - the distribution of д : . If this distribution was known, then it could be possible to find the confidence interval, using its critical values:

So that 90% confidence interval is . However, the distribution of is unknown, but it can be approximated using bootstrap by the distribution of:

, there - is Hurst exponent of resample.

4) Thus, we generate 1000 of resamples of the same length N, calculate for them, and then obtain a distribution of 1000 by subtracting.

5) Critical values and are approximated by and , - the 95th percentile and - the 5th percentile.

Estimated 90% confidence interval for H is:

Such intervals are constructed for every period, N is equal to the length of every particular period. Table with Hurst exponents calculated using DMA method and corresponding confidence intervals obtained from bootstrap experiment is presented below. Again, the null hypothesis is rejected only for the fourth period, all other observed Hurst exponents are inside the empirical confidence interval.

Table 13

Results of DMA analysis (estimated Hurst exponents and 90% confidence intervals found by bootstrap procedure)

Results

Table present the results of all tests for significance level of 0.10. The analysis showed that it is incorrect just to compare Hurst exponents obtained by R/S analysis, because it is biased towards higher values for small samples (it was demonstrated by Weron (2002) and can be easily proved by simulations). However, other methods are more robust to the sample size, so values of estimated Hurst exponents could be compared. All the three methods result in higher values for two crises periods, and for the second crisis (crisis in Russia 2014-2016) the null hypothesis of long memory is rejected at 0.10 significance level. This matches the initial hypothesis, but it is important to mention that the results for crisis periods can be slightly distorted by small sample size. The problem is that, even if corrected R/S analysis, DFA and DMA does not tend to give higher values for small samples, their results for such samples are more variable. This causes broader confidence intervals. The size of confidence intervals depends negatively on number of observations for all the methods, so if for samples with 500 and more observations confidence intervals are rather narrow, for samples with less than 500 observations they are not. This fact makes analysis especially difficult, the main question is if the null hypothesis of no long memory can be rejected for crises periods, and with crises periods are shorter than tranquil it is difficult to reject this hypothesis. However, for the second crisis three tests do not fail to reject the null hypothesis, so it can be concluded that at least this period exhibits long-term memory. It also can be noticed that the second and the last period are almost equal in size, but Hurst exponent for the second are higher than for the last one, this means that during the crisis period market is relatively less efficient than during the tranquil. Another problem is that results of Lo's test do not correspond to the results of other methods. Lo's test fails to reject the null hypothesis of no long memory for all the periods except for the last one. Reasons of this result are not known, the review of related literature has not given any relevant insights on statistics behaviour. Following Teverovsky et al. (1999), we prefer to rely on results of other methods, knowing that there is no short memory in the series.

Overall, it can be concluded that Russian market demonstrates the tendency that has been noticed by researcher for other world markets. Financial market is less efficient during crises periods than during tranquil periods, in terms of EMH.

Table 14

Summary of different methods' results for all studied periods

2.3 Analysis of time-dependent Hurst exponent

If the previous part of this study was mainly testing the EMH, this part is devoted to EMH. Researches have noticed the power of Hurst exponent to predict financial crises (these studies are discussed in the literature review). Assuming that market is not efficient in its stable state (high values of Hurst exponent), they interpret decreasing trends in H(t) before the crisis not as shift to a more efficient state, but as “nervous state” of the market. Market becomes “nervous”, because investors are already uncertain about the quality of information or real economic value of assets. So, a drop in time-dependent Hurst exponent and the reversal of price process from persistent to antipersitent state are supposed to occur before crises, predicting the change of the market's trend in the nearest future.

This hypothesis was proved by several studies on different markets (see, literature review), but these results are not fully convincing. First, the length of this trend and its starting point differ from study to study. If Grech and Mazur (2004) detect the start of this trend 20-60 trading days before the change of trend in market's index (this seems quite logical), Kristoufek (2012) founds that decreasing trend in Hurst exponent begins about two years before the crisis (however, it is unlikely that the market “feels” the upcoming crisis so early). Second, there is no common opinion about the value of H(t) during the crisis.

Here, this hypothesis will be test on RTSI data, both Global financial and Russian crises. Without making any suggestions about the start and length of decreasing trend in dynamics of Hurst exponent, the hypothesis will be:

Time dependent Hurst exponent has a decreasing trend before the beginning of the crisis (change of trend in RTSI).

Time-dependent Hurst exponent can be calculated by any of methods discussed in this paper, but usually DMA and DFA are used for this task (because they do not have shortcomings that R/S-analysis has). For a given trading day the corresponding Hurst exponent will be calculated as Hurst exponent for the period of length N, N is called time-window or observation box. For the next trading day the time-window is moved by one observation. Using this moving-window procedure the history of Hurst exponent changes over time can be obtained.

Important is the choice of the time-window length. Only N past observations influence the current value of H(t), so it is obvious that the value strongly depends on N. Choice of window length has both economical and statistical aspects. If N is too small than it cannot capture important economic information and also the standard deviation of Hurst exponent is high for small samples. At the same time, if N is large, than Hurst exponent loses its locality and the dynamics of H(t) becomes too smooth (so that some local changes can be smoothed, and hence, not noticed). Here, the choice of N will be based on results obtained by Grech and Mazur (2004). They suppose that N should not exceed 240 days (the length of the trading year), because otherwise the evolution of H(t) will be too smoothed. They create random samples from DJIA index data for every window size smaller than 240 days, and found that the local minimum of standard deviation appears at N=215, this corresponds to about 10 trading months. At the same time, in other important work on the recent crisis (Kristoufek, 2012) the time-window is larger - 500 days. To compare results here we will calculate time-dependent Hurst exponent for both 215 and 500 days and also 350 days window's length.

For this part longer series is used, because N previous observations are needed to calculate H(t) for 11.01.2005. The series are filtered the same way as before. Time dependent Hurst exponent was calculated by DMA method, modifying Gu and Zhou (2010) algorithm for moving-window procedure. Picture 7 presents the dynamics of Hurst exponent over the studied period of time for three different window-lengths. As predicted, Hurst exponent for small window (N=215) is very fluctating. Some extremely low values (about 0.3) are observed for very short periods (one or two observations), after which Hurst exponent returns to the trend. This can confirm the idea of high standard deviation for small N. The dynamics for N=350 and N=500 is more smooth, but N=350 case is closer to the dynamics of N=215 than N=500. As the case N=500 may be oversmoothed (though it was successfully used in Kristoufek (2012), the window length of N=350 is accepted as the optimal one. This window length corresponds to 17.5 trading months, it is about 1.5 trading year.

Picture 7. Dynamics of time-dependent Hurst exponent from 11.01.2005 until 10.03.2017, for three windows lengths

Picture 8 presents dynamics of Hurts exponent and dynamics of RTSI before and during Global financial crisis. It can be seen that Hurst exponent actually has dec...