True and Apparent Scaling: The Proximity of
the Markov-Switching Multifractal Model to
Long-Range Dependence
Ruipeng Liu a,b, T. Di Matteo b, Thomas Lux a
aDepartment of Economics, University of Kiel, 24118 Kiel, Germany
bDepartment of Applied Mathematics, Research School of Physical Sciences and
Engineering, The Australian National University, 0200 Canberra, Australia
Abstract
In this paper, we consider daily financial data of a collection of different stock
market indices, exchange rates, and interest rates, and we analyze their multi-scaling
properties by estimating a simple specification of the Markov-switching multifractal
model (MSM). In order to see how well the estimated models capture the temporal
dependence of the data, we estimate and compare the scaling exponents H(q) (for
q = 1, 2) for both empirical data and simulated data of the estimated MSM models.
In most cases the multifractal model appears to generate ‘apparent’ long memory
in agreement with the empirical scaling laws.
Key words: scaling, generalized Hurst exponent, multifractal model, GMM
estimation
1 Introduction
The scaling concept has its origin in physics but it is increasingly applied
outside its traditional domain. In the literature ([1,2,3]) different methods have
been proposed and developed in order to study the multi-scaling properties of
financial time series. For more details on scaling analysis see [4].
Going beyond the phenomenological scaling analysis, the multifractal model
of asset returns (MMAR) introduced by Mandelbrot et. al [5] provides a the-
oretical framework that allows to replicate many of the scaling properties
of financial data. While the practical applicability of MMAR suffered from
its combinatorial nature and its non-stationarity, these drawbacks have been
overcome by the introduction of iterative multifractal models (Poisson MF or
Preprint submitted to Elsevier
http://arxiv.org/abs/0704.1338v1
Markov-switching multifractal model (MSM) [6,7,8]) which preserves the hi-
erarchical, multiplicative structure of the earlier MMAR, but is of much more
‘well-behaved’ nature concerning its asymptotic statistical properties. The at-
tractiveness of MF models lies in their ability to mimic the stylized facts of
financial markets such as outliers, volatility clustering, and asymptotic power-
law behavior of autocovariance functions (long-term dependence). In contrast
to other volatility models with long-term dependence [9], MSM models allow
for multi-scaling rather than uni-scaling with varying decay exponents for all
powers of absolute values of returns. One may note, however, that due to the
Markovian nature, the scaling of the Markov-Switching MF model only holds
over a limited range of time increments depending on the number of hierar-
chical components and this ‘apparent’ power-law ends with a cross-over to an
exponential cut-off.
With this proximity to true multi-scaling, it seems worthwhile to explore how
well the MSM model could reproduce the empirical scaling behaviour of finan-
cial data. To this end, we estimate the parameters of a simple specification of
the MSM model for various financial data and we assess its ability to replicate
empirical scaling behaviour by also computing H(q) by means of the gener-
alized Hurst exponent approach ([4,10,11]) and H by means of the modified
R/S method [12] for the same data sets. We then proceed by comparing the
scaling exponents for empirical data and simulated time series based on our
estimated MSM models. As it turns out, the MSM model with a sufficient
number of volatility components generates pseudo-empirical scaling laws in
good overall agreement with empirical results.
The structure of the paper is as follows: In Section 2 we introduce the multi-
fractal model, the Generalized Hurst exponent (GHE) and the modified R/S
approaches. Section 3 reports the empirical and simulation-based results. Con-
cluding remarks and perspectives are given in Section 4.
2 Methodology
2.1 Markov-switching multifractal model
In this section, we shortly review the building blocks of the Markov-switching
multifractal process (MSM). Returns are modeled as [7,8]:
rt = σt · ut (1)
with innovations ut drawn from a standard Normal distribution N(0, 1) and
instantaneous volatility being determined by the product of k volatility com-
ponents or multipliers M
t , M
t ..., M
t and a constant scale factor σ:
σ2t = σ
t , (2)
In this paper we choose, for the distribution of volatility components, the
binomial distribution: M
t ∼ [m0, 2 −m0] with 1 ≤ m0 < 2. Each volatility
component is renewed at time t with probability γi depending on its rank
within the hierarchy of multipliers and it remains unchanged with probability
1− γi. The transition probabilities are specified by Calvet and Fisher [7] as:
γi = 1− (1− γk)
(bi−k)
i = 1, . . . k, (3)
with parameters γk ∈ [0, 1] and b ∈ (1,∞). Different specifications of Eq. (3)
can be arbitrarily imposed (cf. [8] and its earlier versions). By fixing b = 2
and γk = 0.5, we arrive a relatively parsimonious specification:
γi = 1− (1− γk)
(2i−k) i = 1, . . . k. (4)
This specification implies that replacement happens with probability of one
half at the highest cascade level. Various approaches have been employed to
estimate multifractal models. The parameters of the combinatorial MMAR
have been estimated via an adaptation of the scaling estimator and Legendre
transformation approach from statistical physics [13]. However, this approach
has been shown to yield very unreliable results [14]. A broad range of more
rigorous estimation methods have been developed for the MSM model. Calvet
and Fisher (2001) ([6]) propose maximum likelihood estimation while Lux
([8]) proposes a Generalized Method of Moments (GMM) approach, which
can be applied not only to discrete but also to continuous distributions of the
volatility components. In this paper, GMM is used to estimate the two MSM
model parameters in Eq. (2), namely: σ̂ and m̂0.
2.2 Estimation of scaling exponents
Our analysis of the scaling behaviour of both empirical and simulated data
uses two refined methods for estimating the time-honored Hurst coefficient:
the estimation of generalized Hurst exponents from the structure function of
various moments [4] and Lo’s modified R/S analysis that allows to correct for
short-range dependence in the temporal evolution of the range [12].
2.2.1 Generalized Hurst exponent approach
The generalized Hurst exponent (GHE) method extends the traditional scal-
ing exponent methodology, and this approach provides a natural, unbiased,
statistically and computationally efficient estimator able to capture very well
the scaling features of financial fluctuations ([10,11]). It is essentially a tool
to study directly the scaling properties of the data via the qth order moments
of the distribution of the increments. The qth order moments appear to be
less sensitive to the outliers than maxima/minima and different exponents q
are associated with different characterizations of the multi-scaling behaviour
of the signal X(t).
We consider the q-order moment of the distribution of the increments (with
t = v, 2v, ..., T ) of a time series X(t):
Kq(τ) =
〈| X(t+ τ)−X(t) |q〉
〈| X(t) |q〉
, (5)
where the time interval τ varies between v = 1 day and τmax days. The gener-
alized Hurst exponent H(q) is then defined from the scaling behavior of Kq(τ),
which can be assumed to follow the relation:
Kq(τ) ∼
)qH(q)
. (6)
Within this framework, for q = 1, H(1) describes the scaling behavior of the
absolute values of the increments; for q = 2,H(2) is associated with the scaling
of the autocorrelation function.
2.2.2 Lo’s modified R/S analysis
Lo’s modified R/S analysis uses the range of a time series as its starting point:
Formally, the range R of a time series {Xt}, t = 1, . . . , T is defined as:
RT = max
1≤t≤T
(Xt − X̄)− min
1≤t≤T
(Xt − X̄). (7)
Here, X̄ is the standard estimate of the mean. Usually the range is rescaled by
the sample standard deviation (S), yielding the famous R/S statistic. Though
this approach found wide applications in diverse fields, it turned out that no
asymptotic distribution theory could be derived for H itself. Hence, no explicit
hypothesis testing can be performed and the significance of point estimates
H > 0.5 or H < 0.5 rests on subjective assessment. Luckily, the asymptotic
distribution of the rescaled range itself under a composite null hypothesis
excluding long-memory could be established by Lo (1991) [12]. Using this
distribution function and the critical values reported in his paper, one can
test for the significance of apparent traces of long memory as indicated by
H 6= 0.5. However, Lo also showed that the distributional properties of the
rescaled range are affected by the presence of short memory and he devised
a modified rescaled range Qτ which adjusts for possible short memory effects
by applying the Newey-West heteroscedasticity and autocorrelation consistent
estimator in place of the sample standard deviation S:
1≤t≤T
(Xt − X̄)− min
1≤t≤T
(Xt − X̄)
, (8)
S2τ =S
ωj(τ)
i=j+1
(Xi − X̄)(Xi−j − X̄)
ωj(τ) = 1−
τ + 1
Under the null of no long term memory the distribution of the random variable
VT = T
−0.5Qτ converges to that of the range of a so-called Brownian bridge.
Critical values of this distribution are tabulated in Lo (1991, Table II).
3 Results
In this paper, we consider daily data for a collection of stock exchange indices:
the Dow Jones Composite 65 Average Index (Dow) and NIKKEI 225 Av-
erage Index (Nik) over the time period from January 1969 to October 2004,
foreign exchange rates: British Pound to US Dollar (UK), and Australian
Dollar to US Dollar (AU) over the period from March 1973 to February 2004,
and U.S. 1 year and 2 years treasury constant maturity bond rates (TB1 and
TB2, respectively) in the period from June 1976 to October 2004. The daily
prices are denoted as pt, and returns are calculated as rt = ln(pt) − ln(pt−1)
for stock indices and foreign exchange rates and as rt = pt−pt−1 for TB1 and
We estimate the MSM model parameters introduced in Section 2 with a bi-
nomial distribution of volatility components, that is M
t ∼ [m0, 2−m0] and
1 ≤ m0 < 2 in Eq 2. This estimation is repeated for various hypothetical
numbers of cascade levels (k = 5, 10, 15, 20). Table 1 presents these results
for parameters m̂0 and σ̂.
1 Our estimation is based on the GMM approach
1 Note that the data have been standardized by dividing the sample standard de-
viation which explains the proximity of the scale parameter estimates to 1.
proposed by Lux [8] using the same analytical moments as in his paper. The
numbers within the parentheses are the standard errors. We observe that the
results for k > 10 are almost identical. In fact, analytical moment conditions
in Lux [8] show that higher cascade levels make a smaller and smaller con-
tribution to the moments so that their numerical values would stay almost
constant. If one monitors the development of estimated parameters with in-
creasing k, one finds strong variations initially with a pronounced decrease
of the estimates which become slower and slower until, eventually a constant
value is reached somewhere around k = 10 depending on individual time series.
Based on the estimated parameters, we proceed with an analysis of simulated
data from the pertinent MSM models.
We first calculate the GHE for the empirical time series as well as for 100
simulated time series of each set of estimated parameters for q = 1 and q = 2.
The values of the GHE are averages computed from a set of values corre-
sponding to different τmax (between 5 and 19 days). The stochastic variable
X(t) in Eq. (5) is the absolute value of returns, X(t) = |rt|. The second and
seventh columns in Table 2 report the empirical GHEs, and values in the other
columns are the mean values over the corresponding 100 simulations for dif-
ferent k values: 5, 10, 15, 20, with errors given by their standard deviations.
Boldface numbers are those cases which fail to reject the null hypothesis that
the mean of the simulation-based Generalized Hurst exponent values equals
the empirical Generalized Hurst exponent at the 5% level. We find that the
exponents from the simulated time series vary across different cascade levels
k. In particular, we observe considerable jumps from k = 5 to k = 10 for
these values. In particular for the stock market indices, we find coincidence
between the empirical series and simulation results for the scaling exponents
H(2) for Dow and H(1) for Nik when k = 5. For the exchange rate data,
we observe the simulations successfully replicate the empirical measurements
of AU for H(1) when k = 10, 15, 20 and H(2) when k = 5; In the case of
U.S. Bond rates, we find a good agreement for H(1) when k = 5 and for all
k for TB1, and H(2) for TB2 when k = 5. Apparently, both the empirical
data and the simulated MSM models are characterized by estimates of H(1)
and H(2) much larger than 0.5 which are indicative of long-term dependence.
While the empirical numbers are in nice agreement with previous literature, it
is interesting to note that simulated data with k ≥ 10 have a tendency towards
even higher estimated Hurst coefficients than found in the pertinent empir-
ical records. 2 Since we know that the MSM model only has pre-asymptotic
scaling, these results underscore that with a high enough number of volatility
cascades, it would be hard to distinguish the MSM model from a ‘true’ long
2 We have checked if the generalized Hurst exponents approach is biased by com-
puting H(1) and H(2) for random values generated by different random generators
[11] with T = 9372 data points. We have found that H(1) = 0.4999 ± 0.009 and
H(2) = 0.4995 ± 0.008.
memory process.
We have also performed calculations using the modified Rescaled range (R/S)
analysis introduced by Lo [12,15,16,17,18,19,20], 3 whose results are reported
in Tables 3 to 5. Table 3 presents Lo’s test statistics for both empirical and
1000 simulated time series for different values of k and for different trunca-
tion lags τ = 0, 5, 10, 25, 50, 100. 4 We find that the values are varying with
different truncation lags, and more specifically, that they are monotonically
decreasing for both the empirical and simulation-based statistics. Table 4 re-
ports the number of rejections of the null hypothesis of short-range dependence
based on 95% and 99% confidence levels. The rejection numbers for each sin-
gle k are decreasing as the truncation lag τ increases, but the proportion of
rejections remains relatively high for higher cascade levels, k = 10, 15, 20. The
corresponding Hurst exponents are given in Table 5. The empirical values of
H are decreasing when τ increases. A similar behaviour is observed for the
simulation-based H for given values of k. We also observe that the Hurst expo-
nent values are increasing with increasing cascade level k for given τ . Boldface
numbers are those cases which fail to reject the null hypothesis that the mean
of the simulation-based Hurst exponent equals the empirical Hurst exponent
at the 5% level. There are significant jumps between the values for k = 5 and
k = 10 as reported in previous tables.
Overall, the following results stand out: (1) There seems to be a good overall
agreement between the empirical and simulated data for practically all series
for levels k ≥ 10, while with a smaller number of volatility components (k = 5)
the simulated MSM models have typically smaller estimated Hs than the
corresponding empirical data, (2) the modified R/S approach would quite
reliably reject the null of long memory for k = 5, but in most cases it would
be unable to do so for higher numbers of volatility components, even if we
allow for large truncation lags up to τ = 100. Results are also much more
uniform than with the generalized Hurst technique which had left us with
a rather mixed picture of coincidence of Hurst coefficients of empirical and
simulated data. The fact, that according to Table 5, MSM model with 15 or
more volatility components did always produce ‘apparent’ scaling in agreement
with that of empirical data, is particular encouragingly. It contrasts with the
findings reported in [19] on apparent scaling of estimated GARCH models
whose estimated exponents did not agree with the empirical ones.
3 We also did a Monte Carlo study with 1000 simulated random time series in order
to assess the bias of the pertinent estimates of H: for random numbers with sample
size T = 9372 (comparable to our empirical records) we obtained a slight negative
bias: H = 0.463 ± 0.024.
4 For τ = 0 we have the classical R/S approach.
4 Concluding Remarks
We have calculated the scaling exponents of simulated data based on esti-
mates of the Markov-switching multifractal (MSM) model. Comparing the
generalized Hurst exponent values as well as Lo’s Hurst exponent statistics
of both empirical and simulated data, our study shows that the MSM model
captures quite satisfactorily the multi-scaling properties of absolute values of
returns for specifications with a sufficiently large number of volatility compo-
nents. Subsequent work will explore whether this encouraging coincidence of
the scaling statistics for the empirical and synthetic data also holds for other
candidate distributions of volatility components and alternative specifications
of the transition probabilities.
Acknowledgments
T. Di Matteo acknowledges the partial support by ARC Discovery Projects:
DP03440044 (2003) and DP0558183 (2005), COST P10 “Physics of Risk”
project and M.I.U.R.-F.I.S.R. Project “Ultra-high frequency dynamics of fi-
nancial markets”, T. Lux acknowledges financial support by the European
Commission under STREP contract No. 516446.
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Table 1
GMM estimates of MSM model for different values of k.
k = 5 k = 10 k = 15 k = 20
m̂0 σ̂ m̂0 σ̂ m̂0 σ̂ m̂0 σ̂
Dow 1.498 0.983 1.484 0.983 1.485 0.983 1.487 0.983
(0.025) (0.052) (0.026) (0.044) (0.026) (0.042) (0.027) (0.044)
Nik 1.641 0.991 1.634 0.991 1.635 0.991 1.636 0.991
(0.017) (0.036) (0.013) (0.028) (0.017) (0.036) (0.017) (0.037)
UK 1.415 1.053 1.382 1.057 1.381 1.056 1.381 1.058
(0.033) (0.026) (0.029) (0.027) (0.036) (0.027) (0.038) (0.026)
AU 1.487 1.011 1.458 1.013 1.457 1.014 1.458 1.014
(0.034) (0.066) (0.034) (0.061) (0.034) (0.066) ( 0.034) (0.065)
TB1 1.627 1.041 1.607 1.064 1.607 1.064 1.606 1.067
(0.021) (0.032) (0.025) (0.024) (0.028) (0.024) (0.025) (0.024)
TB2 1.703 1.040 1.679 1.068 1.678 1.079 1.678 1.079
(0.015) (0.036) (0.014) (0.029) (0.015) (0.032) (0.015) (0.034)
Note: All data have been standardized before estimation.
Table 2
H(1) and H(2) for the empirical and simulated data.
H(1) H(2)
Emp sim1 sim2 sim3 sim4 Emp sim1 sim2 sim3 sim4
Dow 0.684 0.747 0.849 0.868 0.868 0.709 0.705 0.797 0.813 0.812
(0.034) (0.008) (0.015) (0.021) (0.024) (0.027) (0.009) (0.015) (0.019) (0.022)
Nik 0.788 0.801 0.894 0.908 0.908 0.753 0.736 0.815 0.824 0.824
(0.023) (0.008) (0.013) (0.019) (0.028) (0.021) (0.008) (0.013) (0.018) (0.024)
UK 0.749 0.709 0.799 0.825 0.821 0.735 0.678 0.764 0.785 0.783
(0.023) (0.010) (0.018) (0.025) (0.026) (0.026) (0.010) (0.016) (0.021) (0.022)
AU 0.827 0.746 0.837 0.860 0.857 0.722 0.705 0.790 0.808 0.808
(0.017) (0.009) (0.016) (0.022) (0.021) (0.024) (0.009) (0.015) (0.018) (0.018)
TB1 0.853 0.856 0.909 0.915 0.911 0.814 0.783 0.826 0.832 0.829
(0.022) (0.035) (0.023) (0.026) (0.026) (0.027) (0.028) (0.020) (0.020) (0.020)
TB2 0.791 0.866 0.920 0.924 0.919 0.778 0.781 0.823 0.827 0.822
(0.025) (0.029) (0.021) (0.022) (0.026) (0.029) (0.022) (0.017) (0.022) (0.023)
Note: Emp refers to the empirical exponent values, sim1, sim2, sim3 and sim4 are
the corresponding exponent values based on the simulated data for k = 5, k = 10,
k = 15 and k = 20 respectively. The stochastic variable Xt is defined as |rt|.
Bold numbers show those cases for which we cannot reject identity of the Hurst
coefficients obtained for empirical and simulated data, i.e. the empirical exponents
fall into the range between the 2.5 to 97.5 percent quantile of the simulated data.
Table 3
Lo’s R/S statistic for the empirical and simulated data.
τ = 0 τ = 5 τ = 10
Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20
Dow 3.005 1.712 5.079 6.640 6.704 2.661 1.481 4.060 5.211 5.263 2.427 1.376 3.574 4.537 4.582
(0.381) (1.300) (1.769) (1.839) (0.329) (1.017) (1.333) (1.387) (0.305) (0.884) (1.133) (1.179)
Nik 7.698 1.840 4.898 6.154 6.152 6.509 1.540 3.817 4.747 4.742 5.836 1.416 3.343 4.132 4.133
(0.425) (1.195) (1.520) (1.584) ( 0.355) (0.918) (1.147) (1.193) (0.325) (0.798) (0.984) (1.023)
UK 6.821 1.544 4.599 6.047 6.175 5.912 1.370 3.815 4.918 5.008 5.333 1.286 3.405 4.337 4.408
(0.350) (1.200) (1.748) (1.848) (0.310) (0.972) (1.352) (1.417) (0.290) (0.854) (1.157) (1.207)
AU 7.698 1.687 4.962 6.348 6.434 6.731 1.463 4.001 5.024 5.090 6.103 1.361 3.531 4.387 4.443
(0.386) (1.257) (1.742) (1.790) (0.333) (0.989) (1.315) (1.352) (0.309) (0.861) (1.117) (1.149)
TB1 8.845 1.826 4.644 5.915 6.041 7.109 1.524 3.629 4.564 4.582 6.110 1.400 3.184 4.415 4.530
(0.398) (1.141) (1.425) (1.380) (0.330) (0.875) (1.074) (1.040) (0.302) (0.759) (0.921) (0.891)
TB2 7.295 1.855 4.347 5.853 5.907 6.083 1.531 3.391 4.207 4.349 5.330 1.404 2.985 4.025 4.158
(0.413) (1.031) (1.215) (1.227) (0.339) (0.795) (0.928) (0.930) (0.310) (0.694) (0.804) (0.803)
τ = 25 τ = 50 τ = 100
Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20
Dow 2.042 1.237 2.877 3.580 3.616 1.736 1.153 2.385 2.909 2.941 1.464 1.098 1.965 2.338 2.366
(0.272) (0.694) (0.857) (0.893) (0.250) (0.560) (0.668) (0.696) (0.233) (0.443) (0.508) (0.530)
Nik 4.760 1.260 2.692 3.285 3.279 3.941 1.169 2.246 2.701 2.698 3.220 1.113 1.868 2.204 2.203
(0.286) (0.631) (0.761) (0.788) (0.263) (0.514) (0.604) (0.623) (0.245) (0.412) (0.468) (0.482)
UK 4.348 1.170 2.782 3.469 3.515 3.575 1.099 2.322 2.837 2.868 2.871 1.053 1.922 2.289 2.306
(0.262) (0.678) (0.876) (0.909) (0.244) (0.549) (0.680) (0.702) (0.228) (0.434) (0.513) (0.528)
AU 5.035 1.224 2.848 3.474 3.516 4.130 1.142 2.362 2.830 2.861 3.281 1.089 1.947 2.280 2.302
(0.275) (0.676) (0.842) (0.866) (0.252) (0.544) (0.654) (0.672) (0.232) (0.429) (0.496) (0.508)
TB1 4.580 1.245 2.571 2.961 2.971 3.514 1.156 2.148 2.442 2.449 2.649 1.101 1.790 2.004 2.006
(0.265) (0.598) (0.711) (0.685) (0.242) (0.484) (0.564) (0.542) (0.223) (0.384) (0.440) (0.417)
TB2 4.129 1.249 2.432 2.762 2.786 3.250 1.162 2.052 2.305 2.320 2.502 1.109 1.731 1.915 1.921
(0.272) (0.554) (0.632) (0.630) (0.249) (0.456) (0.511) (0.507) (0.230) (0.369) (0.403) (0.398)
Note: Emp stands for the empirical Lo’s statistic, k = 5, k = 10, k = 15 and k = 20 refer to the mean and standard deviation of Lo’s statistics based
on the corresponding 1000 simulated time series with pertinent k.
Table 4
Number of rejections for Lo’s R/S statistic test.
τ = 0 τ = 5 τ = 10
k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20
† ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡
Dow 311 151 1000 1000 1000 1000 1000 1000 121 46 999 991 999 998 1000 1000 69 22 990 968 998 997 1000 995
Nik 433 253 1000 999 1000 1000 1000 1000 176 74 993 985 998 997 1000 999 98 36 983 963 997 991 999 993
UK 167 77 998 995 1000 999 999 998 74 22 991 976 998 997 998 997 41 7 982 943 996 990 997 992
AU 301 142 1000 999 999 999 1000 1000 116 39 997 990 998 994 1000 999 58 23 990 966 993 989 999 995
TB1 428 227 1000 1000 1000 999 999 999 146 55 993 976 997 991 998 996 75 24 976 934 990 970 996 989
TB2 453 256 999 995 998 997 1000 999 159 60 987 959 994 982 996 986 86 21 958 899 985 961 985 960
τ = 25 τ = 50 τ = 100
k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20
† ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡
Dow 24 5 939 858 990 964 985 966 9 3 807 677 940 887 948 872 4 1 566 381 811 669 808 686
Nik 34 5 920 809 982 848 977 930 11 2 764 581 914 831 897 812 4 1 485 281 750 582 742 575
UK 11 1 929 843 982 942 979 953 4 1 789 630 919 840 926 843 1 1 541 327 783 632 774 640
AU 23 5 931 860 983 949 983 956 6 2 816 666 921 852 931 846 4 1 561 353 776 648 786 649
TB1 25 4 876 765 946 870 965 893 5 1 698 519 822 711 846 712 1 1 418 230 627 415 604 400
TB2 21 6 844 696 933 851 928 859 10 3 627 446 798 638 807 657 3 1 368 167 534 312 544 336
Note: k = 5, k = 10, k = 15 and k = 20 refer to the number of rejections at 95% (†) and 99% (‡) confidence levels (these intervals are given by [0.809, 1.862] and [0.721, 2.098],
respectively) for the 1000 simulated time series.
Table 5
Lo’s modified R/S Hurst exponent H values for the empirical and simulated data.
τ = 0 τ = 5 τ = 10
Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20
Dow 0.620 0.556 0.674 0.703 0.704 0.607 0.540 0.650 0.677 0.678 0.597 0.532 0.636 0.662 0.663
(0.024) (0.029) (0.030) (0.031) (0.024) (0.028) (0.029) (0.030) (0.024) (0.028) (0.028) (0.029)
Nik 0.723 0.564 0.670 0.695 0.695 0.705 0.544 0.643 0.667 0.667 0.693 0.535 0.629 0.652 0.651
(0.025) (0.027) (0.028) (0.029) (0.025) (0.027) (0.028) (0.029) (0.025) (0.027) (0.027) (0.028)
UK 0.712 0.545 0.665 0.694 0.696 0.696 0.532 0.644 0.672 0.673 0.685 0.525 0.632 0.658 0.660
(0.025) (0.030) (0.033) (0.036) (0.025) (0.029) (0.032) (0.035) (0.025) (0.029) (0.031) (0.034)
AU 0.726 0.555 0.673 0.700 0.701 0.711 0.539 0.650 0.674 0.676 0.700 0.531 0.636 0.660 0.661
(0.025) (0.029) (0.032) (0.032) (0.025) (0.028) (0.031) (0.031) (0.025) (0.028) (0.030) (0.030)
TB1 0.746 0.565 0.670 0.689 0.691 0.721 0.547 0.642 0.660 0.661 0.704 0.535 0.627 0.644 0.645
(0.024) (0.028) (0.031) (0.029) (0.024) (0.028) (0.030) (0.028) (0.024) (0.028) (0.029) (0.028)
TB2 0.724 0.567 0.662 0.679 0.680 0.704 0.545 0.634 0.650 0.652 0.689 0.536 0.620 0.636 0.637
(0.025) (0.028) (0.028) (0.028) (0.025) (0.027) (0.028) (0.028) (0.024) (0.027) (0.028) (0.027)
τ = 25 τ = 50 τ = 100
Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20
Dow 0.578 0.521 0.612 0.636 0.637 0.560 0.513 0.592 0.614 0.615 0.542 0.508 0.571 0.590 0.591
(0.024) (0.027) (0.027) (0.028) (0.023) (0.026) (0.026) (0.027) (0.023) (0.025) (0.025) (0.026)
Nik 0.671 0.522 0.605 0.627 0.626 0.650 0.514 0.586 0.606 0.605 0.628 0.509 0.566 0.584 0.583
(0.025) (0.026) (0.027) (0.027) (0.024) (0.026) (0.026) (0.026) (0.024) (0.025) (0.024) (0.025)
UK 0.662 0.515 0.610 0.634 0.635 0.641 0.508 0.590 0.612 0.613 0.617 0.503 0.569 0.589 0.589
(0.025) (0.028) (0.029) (0.032) (0.024) (0.027) (0.028) (0.030) (0.024) (0.026) (0.026) (0.028)
AU 0.679 0.520 0.612 0.634 0.635 0.657 0.512 0.592 0.612 0.613 0.631 0.507 0.571 0.588 0.589
(0.025) (0.027) (0.029) (0.029) (0.024) (0.026) (0.027) (0.027) (0.023) (0.025) (0.026) (0.026)
TB1 0.672 0.522 0.603 0.619 0.621 0.642 0.514 0.583 0.597 0.598 0.610 0.509 0.563 0.575 0.576
(0.024) (0.027) (0.028) (0.027) (0.024) (0.026) (0.027) (0.026) (0.023) (0.025) (0.026) (0.024)
TB2 0.661 0.520 0.597 0.611 0.612 0.633 0.514 0.578 0.591 0.592 0.604 0.509 0.559 0.571 0.571
(0.024) (0.027) (0.027) (0.027) (0.024) (0.026) (0.026) (0.026) (0.023) (0.025) (0.025) (0.024)
Note: Emp stands for the empirical value of Lo’s Hurst exponent, k = 5, k = 10, k = 15 and k = 20 refer to the mean and standard deviation of Lo’s Hurst
exponent based on the corresponding 1000 simulated time series with different k. Boldface numbers are those cases in which empirical Hs fall into the
corresponding 2.5 to 97.5 percent quantiles of the 1000 simulation-based values of H.
	Introduction
	Methodology
	Markov-switching multifractal model
	Estimation of scaling exponents
	Results
	Concluding Remarks
	References

  In this paper, we consider daily financial data of a collection of different
stock market indices, exchange rates, and interest rates, and we analyze their
multi-scaling properties by estimating a simple specification of the
Markov-switching multifractal model (MSM). In order to see how well the
estimated models capture the temporal dependence of the data, we estimate and
compare the scaling exponents $H(q)$ (for $q = 1, 2$) for both empirical data
and simulated data of the estimated MSM models. In most cases the multifractal
model appears to generate `apparent' long memory in agreement with the
empirical scaling laws.

Introduction
The scaling concept has its origin in physics but it is increasingly applied
outside its traditional domain. In the literature ([1,2,3]) different methods have
been proposed and developed in order to study the multi-scaling properties of
financial time series. For more details on scaling analysis see [4].
Going beyond the phenomenological scaling analysis, the multifractal model
of asset returns (MMAR) introduced by Mandelbrot et. al [5] provides a the-
oretical framework that allows to replicate many of the scaling properties
of financial data. While the practical applicability of MMAR suffered from
its combinatorial nature and its non-stationarity, these drawbacks have been
overcome by the introduction of iterative multifractal models (Poisson MF or
Preprint submitted to Elsevier
http://arxiv.org/abs/0704.1338v1
Markov-switching multifractal model (MSM) [6,7,8]) which preserves the hi-
erarchical, multiplicative structure of the earlier MMAR, but is of much more
‘well-behaved’ nature concerning its asymptotic statistical properties. The at-
tractiveness of MF models lies in their ability to mimic the stylized facts of
financial markets such as outliers, volatility clustering, and asymptotic power-
law behavior of autocovariance functions (long-term dependence). In contrast
to other volatility models with long-term dependence [9], MSM models allow
for multi-scaling rather than uni-scaling with varying decay exponents for all
powers of absolute values of returns. One may note, however, that due to the
Markovian nature, the scaling of the Markov-Switching MF model only holds
over a limited range of time increments depending on the number of hierar-
chical components and this ‘apparent’ power-law ends with a cross-over to an
exponential cut-off.
With this proximity to true multi-scaling, it seems worthwhile to explore how
well the MSM model could reproduce the empirical scaling behaviour of finan-
cial data. To this end, we estimate the parameters of a simple specification of
the MSM model for various financial data and we assess its ability to replicate
empirical scaling behaviour by also computing H(q) by means of the gener-
alized Hurst exponent approach ([4,10,11]) and H by means of the modified
R/S method [12] for the same data sets. We then proceed by comparing the
scaling exponents for empirical data and simulated time series based on our
estimated MSM models. As it turns out, the MSM model with a sufficient
number of volatility components generates pseudo-empirical scaling laws in
good overall agreement with empirical results.
The structure of the paper is as follows: In Section 2 we introduce the multi-
fractal model, the Generalized Hurst exponent (GHE) and the modified R/S
approaches. Section 3 reports the empirical and simulation-based results. Con-
cluding remarks and perspectives are given in Section 4.
2 Methodology
2.1 Markov-switching multifractal model
In this section, we shortly review the building blocks of the Markov-switching
multifractal process (MSM). Returns are modeled as [7,8]:
rt = σt · ut (1)
with innovations ut drawn from a standard Normal distribution N(0, 1) and
instantaneous volatility being determined by the product of k volatility com-
ponents or multipliers M
t , M
t ..., M
t and a constant scale factor σ:
σ2t = σ
t , (2)
In this paper we choose, for the distribution of volatility components, the
binomial distribution: M
t ∼ [m0, 2 −m0] with 1 ≤ m0 < 2. Each volatility
component is renewed at time t with probability γi depending on its rank
within the hierarchy of multipliers and it remains unchanged with probability
1− γi. The transition probabilities are specified by Calvet and Fisher [7] as:
γi = 1− (1− γk)
(bi−k)
i = 1, . . . k, (3)
with parameters γk ∈ [0, 1] and b ∈ (1,∞). Different specifications of Eq. (3)
can be arbitrarily imposed (cf. [8] and its earlier versions). By fixing b = 2
and γk = 0.5, we arrive a relatively parsimonious specification:
γi = 1− (1− γk)
(2i−k) i = 1, . . . k. (4)
This specification implies that replacement happens with probability of one
half at the highest cascade level. Various approaches have been employed to
estimate multifractal models. The parameters of the combinatorial MMAR
have been estimated via an adaptation of the scaling estimator and Legendre
transformation approach from statistical physics [13]. However, this approach
has been shown to yield very unreliable results [14]. A broad range of more
rigorous estimation methods have been developed for the MSM model. Calvet
and Fisher (2001) ([6]) propose maximum likelihood estimation while Lux
([8]) proposes a Generalized Method of Moments (GMM) approach, which
can be applied not only to discrete but also to continuous distributions of the
volatility components. In this paper, GMM is used to estimate the two MSM
model parameters in Eq. (2), namely: σ̂ and m̂0.
2.2 Estimation of scaling exponents
Our analysis of the scaling behaviour of both empirical and simulated data
uses two refined methods for estimating the time-honored Hurst coefficient:
the estimation of generalized Hurst exponents from the structure function of
various moments [4] and Lo’s modified R/S analysis that allows to correct for
short-range dependence in the temporal evolution of the range [12].
2.2.1 Generalized Hurst exponent approach
The generalized Hurst exponent (GHE) method extends the traditional scal-
ing exponent methodology, and this approach provides a natural, unbiased,
statistically and computationally efficient estimator able to capture very well
the scaling features of financial fluctuations ([10,11]). It is essentially a tool
to study directly the scaling properties of the data via the qth order moments
of the distribution of the increments. The qth order moments appear to be
less sensitive to the outliers than maxima/minima and different exponents q
are associated with different characterizations of the multi-scaling behaviour
of the signal X(t).
We consider the q-order moment of the distribution of the increments (with
t = v, 2v, ..., T ) of a time series X(t):
Kq(τ) =
〈| X(t+ τ)−X(t) |q〉
〈| X(t) |q〉
, (5)
where the time interval τ varies between v = 1 day and τmax days. The gener-
alized Hurst exponent H(q) is then defined from the scaling behavior of Kq(τ),
which can be assumed to follow the relation:
Kq(τ) ∼
)qH(q)
. (6)
Within this framework, for q = 1, H(1) describes the scaling behavior of the
absolute values of the increments; for q = 2,H(2) is associated with the scaling
of the autocorrelation function.
2.2.2 Lo’s modified R/S analysis
Lo’s modified R/S analysis uses the range of a time series as its starting point:
Formally, the range R of a time series {Xt}, t = 1, . . . , T is defined as:
RT = max
1≤t≤T
(Xt − X̄)− min
1≤t≤T
(Xt − X̄). (7)
Here, X̄ is the standard estimate of the mean. Usually the range is rescaled by
the sample standard deviation (S), yielding the famous R/S statistic. Though
this approach found wide applications in diverse fields, it turned out that no
asymptotic distribution theory could be derived for H itself. Hence, no explicit
hypothesis testing can be performed and the significance of point estimates
H > 0.5 or H < 0.5 rests on subjective assessment. Luckily, the asymptotic
distribution of the rescaled range itself under a composite null hypothesis
excluding long-memory could be established by Lo (1991) [12]. Using this
distribution function and the critical values reported in his paper, one can
test for the significance of apparent traces of long memory as indicated by
H 6= 0.5. However, Lo also showed that the distributional properties of the
rescaled range are affected by the presence of short memory and he devised
a modified rescaled range Qτ which adjusts for possible short memory effects
by applying the Newey-West heteroscedasticity and autocorrelation consistent
estimator in place of the sample standard deviation S:
1≤t≤T
(Xt − X̄)− min
1≤t≤T
(Xt − X̄)
, (8)
S2τ =S
ωj(τ)
i=j+1
(Xi − X̄)(Xi−j − X̄)
ωj(τ) = 1−
τ + 1
Under the null of no long term memory the distribution of the random variable
VT = T
−0.5Qτ converges to that of the range of a so-called Brownian bridge.
Critical values of this distribution are tabulated in Lo (1991, Table II).
3 Results
In this paper, we consider daily data for a collection of stock exchange indices:
the Dow Jones Composite 65 Average Index (Dow) and NIKKEI 225 Av-
erage Index (Nik) over the time period from January 1969 to October 2004,
foreign exchange rates: British Pound to US Dollar (UK), and Australian
Dollar to US Dollar (AU) over the period from March 1973 to February 2004,
and U.S. 1 year and 2 years treasury constant maturity bond rates (TB1 and
TB2, respectively) in the period from June 1976 to October 2004. The daily
prices are denoted as pt, and returns are calculated as rt = ln(pt) − ln(pt−1)
for stock indices and foreign exchange rates and as rt = pt−pt−1 for TB1 and
We estimate the MSM model parameters introduced in Section 2 with a bi-
nomial distribution of volatility components, that is M
t ∼ [m0, 2−m0] and
1 ≤ m0 < 2 in Eq 2. This estimation is repeated for various hypothetical
numbers of cascade levels (k = 5, 10, 15, 20). Table 1 presents these results
for parameters m̂0 and σ̂.
1 Our estimation is based on the GMM approach
1 Note that the data have been standardized by dividing the sample standard de-
viation which explains the proximity of the scale parameter estimates to 1.
proposed by Lux [8] using the same analytical moments as in his paper. The
numbers within the parentheses are the standard errors. We observe that the
results for k > 10 are almost identical. In fact, analytical moment conditions
in Lux [8] show that higher cascade levels make a smaller and smaller con-
tribution to the moments so that their numerical values would stay almost
constant. If one monitors the development of estimated parameters with in-
creasing k, one finds strong variations initially with a pronounced decrease
of the estimates which become slower and slower until, eventually a constant
value is reached somewhere around k = 10 depending on individual time series.
Based on the estimated parameters, we proceed with an analysis of simulated
data from the pertinent MSM models.
We first calculate the GHE for the empirical time series as well as for 100
simulated time series of each set of estimated parameters for q = 1 and q = 2.
The values of the GHE are averages computed from a set of values corre-
sponding to different τmax (between 5 and 19 days). The stochastic variable
X(t) in Eq. (5) is the absolute value of returns, X(t) = |rt|. The second and
seventh columns in Table 2 report the empirical GHEs, and values in the other
columns are the mean values over the corresponding 100 simulations for dif-
ferent k values: 5, 10, 15, 20, with errors given by their standard deviations.
Boldface numbers are those cases which fail to reject the null hypothesis that
the mean of the simulation-based Generalized Hurst exponent values equals
the empirical Generalized Hurst exponent at the 5% level. We find that the
exponents from the simulated time series vary across different cascade levels
k. In particular, we observe considerable jumps from k = 5 to k = 10 for
these values. In particular for the stock market indices, we find coincidence
between the empirical series and simulation results for the scaling exponents
H(2) for Dow and H(1) for Nik when k = 5. For the exchange rate data,
we observe the simulations successfully replicate the empirical measurements
of AU for H(1) when k = 10, 15, 20 and H(2) when k = 5; In the case of
U.S. Bond rates, we find a good agreement for H(1) when k = 5 and for all
k for TB1, and H(2) for TB2 when k = 5. Apparently, both the empirical
data and the simulated MSM models are characterized by estimates of H(1)
and H(2) much larger than 0.5 which are indicative of long-term dependence.
While the empirical numbers are in nice agreement with previous literature, it
is interesting to note that simulated data with k ≥ 10 have a tendency towards
even higher estimated Hurst coefficients than found in the pertinent empir-
ical records. 2 Since we know that the MSM model only has pre-asymptotic
scaling, these results underscore that with a high enough number of volatility
cascades, it would be hard to distinguish the MSM model from a ‘true’ long
2 We have checked if the generalized Hurst exponents approach is biased by com-
puting H(1) and H(2) for random values generated by different random generators
[11] with T = 9372 data points. We have found that H(1) = 0.4999 ± 0.009 and
H(2) = 0.4995 ± 0.008.
memory process.
We have also performed calculations using the modified Rescaled range (R/S)
analysis introduced by Lo [12,15,16,17,18,19,20], 3 whose results are reported
in Tables 3 to 5. Table 3 presents Lo’s test statistics for both empirical and
1000 simulated time series for different values of k and for different trunca-
tion lags τ = 0, 5, 10, 25, 50, 100. 4 We find that the values are varying with
different truncation lags, and more specifically, that they are monotonically
decreasing for both the empirical and simulation-based statistics. Table 4 re-
ports the number of rejections of the null hypothesis of short-range dependence
based on 95% and 99% confidence levels. The rejection numbers for each sin-
gle k are decreasing as the truncation lag τ increases, but the proportion of
rejections remains relatively high for higher cascade levels, k = 10, 15, 20. The
corresponding Hurst exponents are given in Table 5. The empirical values of
H are decreasing when τ increases. A similar behaviour is observed for the
simulation-based H for given values of k. We also observe that the Hurst expo-
nent values are increasing with increasing cascade level k for given τ . Boldface
numbers are those cases which fail to reject the null hypothesis that the mean
of the simulation-based Hurst exponent equals the empirical Hurst exponent
at the 5% level. There are significant jumps between the values for k = 5 and
k = 10 as reported in previous tables.
Overall, the following results stand out: (1) There seems to be a good overall
agreement between the empirical and simulated data for practically all series
for levels k ≥ 10, while with a smaller number of volatility components (k = 5)
the simulated MSM models have typically smaller estimated Hs than the
corresponding empirical data, (2) the modified R/S approach would quite
reliably reject the null of long memory for k = 5, but in most cases it would
be unable to do so for higher numbers of volatility components, even if we
allow for large truncation lags up to τ = 100. Results are also much more
uniform than with the generalized Hurst technique which had left us with
a rather mixed picture of coincidence of Hurst coefficients of empirical and
simulated data. The fact, that according to Table 5, MSM model with 15 or
more volatility components did always produce ‘apparent’ scaling in agreement
with that of empirical data, is particular encouragingly. It contrasts with the
findings reported in [19] on apparent scaling of estimated GARCH models
whose estimated exponents did not agree with the empirical ones.
3 We also did a Monte Carlo study with 1000 simulated random time series in order
to assess the bias of the pertinent estimates of H: for random numbers with sample
size T = 9372 (comparable to our empirical records) we obtained a slight negative
bias: H = 0.463 ± 0.024.
4 For τ = 0 we have the classical R/S approach.
4 Concluding Remarks
We have calculated the scaling exponents of simulated data based on esti-
mates of the Markov-switching multifractal (MSM) model. Comparing the
generalized Hurst exponent values as well as Lo’s Hurst exponent statistics
of both empirical and simulated data, our study shows that the MSM model
captures quite satisfactorily the multi-scaling properties of absolute values of
returns for specifications with a sufficiently large number of volatility compo-
nents. Subsequent work will explore whether this encouraging coincidence of
the scaling statistics for the empirical and synthetic data also holds for other
candidate distributions of volatility components and alternative specifications
of the transition probabilities.
Acknowledgments
T. Di Matteo acknowledges the partial support by ARC Discovery Projects:
DP03440044 (2003) and DP0558183 (2005), COST P10 “Physics of Risk”
project and M.I.U.R.-F.I.S.R. Project “Ultra-high frequency dynamics of fi-
nancial markets”, T. Lux acknowledges financial support by the European
Commission under STREP contract No. 516446.
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Table 1
GMM estimates of MSM model for different values of k.
k = 5 k = 10 k = 15 k = 20
m̂0 σ̂ m̂0 σ̂ m̂0 σ̂ m̂0 σ̂
Dow 1.498 0.983 1.484 0.983 1.485 0.983 1.487 0.983
(0.025) (0.052) (0.026) (0.044) (0.026) (0.042) (0.027) (0.044)
Nik 1.641 0.991 1.634 0.991 1.635 0.991 1.636 0.991
(0.017) (0.036) (0.013) (0.028) (0.017) (0.036) (0.017) (0.037)
UK 1.415 1.053 1.382 1.057 1.381 1.056 1.381 1.058
(0.033) (0.026) (0.029) (0.027) (0.036) (0.027) (0.038) (0.026)
AU 1.487 1.011 1.458 1.013 1.457 1.014 1.458 1.014
(0.034) (0.066) (0.034) (0.061) (0.034) (0.066) ( 0.034) (0.065)
TB1 1.627 1.041 1.607 1.064 1.607 1.064 1.606 1.067
(0.021) (0.032) (0.025) (0.024) (0.028) (0.024) (0.025) (0.024)
TB2 1.703 1.040 1.679 1.068 1.678 1.079 1.678 1.079
(0.015) (0.036) (0.014) (0.029) (0.015) (0.032) (0.015) (0.034)
Note: All data have been standardized before estimation.
Table 2
H(1) and H(2) for the empirical and simulated data.
H(1) H(2)
Emp sim1 sim2 sim3 sim4 Emp sim1 sim2 sim3 sim4
Dow 0.684 0.747 0.849 0.868 0.868 0.709 0.705 0.797 0.813 0.812
(0.034) (0.008) (0.015) (0.021) (0.024) (0.027) (0.009) (0.015) (0.019) (0.022)
Nik 0.788 0.801 0.894 0.908 0.908 0.753 0.736 0.815 0.824 0.824
(0.023) (0.008) (0.013) (0.019) (0.028) (0.021) (0.008) (0.013) (0.018) (0.024)
UK 0.749 0.709 0.799 0.825 0.821 0.735 0.678 0.764 0.785 0.783
(0.023) (0.010) (0.018) (0.025) (0.026) (0.026) (0.010) (0.016) (0.021) (0.022)
AU 0.827 0.746 0.837 0.860 0.857 0.722 0.705 0.790 0.808 0.808
(0.017) (0.009) (0.016) (0.022) (0.021) (0.024) (0.009) (0.015) (0.018) (0.018)
TB1 0.853 0.856 0.909 0.915 0.911 0.814 0.783 0.826 0.832 0.829
(0.022) (0.035) (0.023) (0.026) (0.026) (0.027) (0.028) (0.020) (0.020) (0.020)
TB2 0.791 0.866 0.920 0.924 0.919 0.778 0.781 0.823 0.827 0.822
(0.025) (0.029) (0.021) (0.022) (0.026) (0.029) (0.022) (0.017) (0.022) (0.023)
Note: Emp refers to the empirical exponent values, sim1, sim2, sim3 and sim4 are
the corresponding exponent values based on the simulated data for k = 5, k = 10,
k = 15 and k = 20 respectively. The stochastic variable Xt is defined as |rt|.
Bold numbers show those cases for which we cannot reject identity of the Hurst
coefficients obtained for empirical and simulated data, i.e. the empirical exponents
fall into the range between the 2.5 to 97.5 percent quantile of the simulated data.
Table 3
Lo’s R/S statistic for the empirical and simulated data.
τ = 0 τ = 5 τ = 10
Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20
Dow 3.005 1.712 5.079 6.640 6.704 2.661 1.481 4.060 5.211 5.263 2.427 1.376 3.574 4.537 4.582
(0.381) (1.300) (1.769) (1.839) (0.329) (1.017) (1.333) (1.387) (0.305) (0.884) (1.133) (1.179)
Nik 7.698 1.840 4.898 6.154 6.152 6.509 1.540 3.817 4.747 4.742 5.836 1.416 3.343 4.132 4.133
(0.425) (1.195) (1.520) (1.584) ( 0.355) (0.918) (1.147) (1.193) (0.325) (0.798) (0.984) (1.023)
UK 6.821 1.544 4.599 6.047 6.175 5.912 1.370 3.815 4.918 5.008 5.333 1.286 3.405 4.337 4.408
(0.350) (1.200) (1.748) (1.848) (0.310) (0.972) (1.352) (1.417) (0.290) (0.854) (1.157) (1.207)
AU 7.698 1.687 4.962 6.348 6.434 6.731 1.463 4.001 5.024 5.090 6.103 1.361 3.531 4.387 4.443
(0.386) (1.257) (1.742) (1.790) (0.333) (0.989) (1.315) (1.352) (0.309) (0.861) (1.117) (1.149)
TB1 8.845 1.826 4.644 5.915 6.041 7.109 1.524 3.629 4.564 4.582 6.110 1.400 3.184 4.415 4.530
(0.398) (1.141) (1.425) (1.380) (0.330) (0.875) (1.074) (1.040) (0.302) (0.759) (0.921) (0.891)
TB2 7.295 1.855 4.347 5.853 5.907 6.083 1.531 3.391 4.207 4.349 5.330 1.404 2.985 4.025 4.158
(0.413) (1.031) (1.215) (1.227) (0.339) (0.795) (0.928) (0.930) (0.310) (0.694) (0.804) (0.803)
τ = 25 τ = 50 τ = 100
Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20
Dow 2.042 1.237 2.877 3.580 3.616 1.736 1.153 2.385 2.909 2.941 1.464 1.098 1.965 2.338 2.366
(0.272) (0.694) (0.857) (0.893) (0.250) (0.560) (0.668) (0.696) (0.233) (0.443) (0.508) (0.530)
Nik 4.760 1.260 2.692 3.285 3.279 3.941 1.169 2.246 2.701 2.698 3.220 1.113 1.868 2.204 2.203
(0.286) (0.631) (0.761) (0.788) (0.263) (0.514) (0.604) (0.623) (0.245) (0.412) (0.468) (0.482)
UK 4.348 1.170 2.782 3.469 3.515 3.575 1.099 2.322 2.837 2.868 2.871 1.053 1.922 2.289 2.306
(0.262) (0.678) (0.876) (0.909) (0.244) (0.549) (0.680) (0.702) (0.228) (0.434) (0.513) (0.528)
AU 5.035 1.224 2.848 3.474 3.516 4.130 1.142 2.362 2.830 2.861 3.281 1.089 1.947 2.280 2.302
(0.275) (0.676) (0.842) (0.866) (0.252) (0.544) (0.654) (0.672) (0.232) (0.429) (0.496) (0.508)
TB1 4.580 1.245 2.571 2.961 2.971 3.514 1.156 2.148 2.442 2.449 2.649 1.101 1.790 2.004 2.006
(0.265) (0.598) (0.711) (0.685) (0.242) (0.484) (0.564) (0.542) (0.223) (0.384) (0.440) (0.417)
TB2 4.129 1.249 2.432 2.762 2.786 3.250 1.162 2.052 2.305 2.320 2.502 1.109 1.731 1.915 1.921
(0.272) (0.554) (0.632) (0.630) (0.249) (0.456) (0.511) (0.507) (0.230) (0.369) (0.403) (0.398)
Note: Emp stands for the empirical Lo’s statistic, k = 5, k = 10, k = 15 and k = 20 refer to the mean and standard deviation of Lo’s statistics based
on the corresponding 1000 simulated time series with pertinent k.
Table 4
Number of rejections for Lo’s R/S statistic test.
τ = 0 τ = 5 τ = 10
k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20
† ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡
Dow 311 151 1000 1000 1000 1000 1000 1000 121 46 999 991 999 998 1000 1000 69 22 990 968 998 997 1000 995
Nik 433 253 1000 999 1000 1000 1000 1000 176 74 993 985 998 997 1000 999 98 36 983 963 997 991 999 993
UK 167 77 998 995 1000 999 999 998 74 22 991 976 998 997 998 997 41 7 982 943 996 990 997 992
AU 301 142 1000 999 999 999 1000 1000 116 39 997 990 998 994 1000 999 58 23 990 966 993 989 999 995
TB1 428 227 1000 1000 1000 999 999 999 146 55 993 976 997 991 998 996 75 24 976 934 990 970 996 989
TB2 453 256 999 995 998 997 1000 999 159 60 987 959 994 982 996 986 86 21 958 899 985 961 985 960
τ = 25 τ = 50 τ = 100
k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20 k = 5 k = 10 k = 15 k = 20
† ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡ † ‡
Dow 24 5 939 858 990 964 985 966 9 3 807 677 940 887 948 872 4 1 566 381 811 669 808 686
Nik 34 5 920 809 982 848 977 930 11 2 764 581 914 831 897 812 4 1 485 281 750 582 742 575
UK 11 1 929 843 982 942 979 953 4 1 789 630 919 840 926 843 1 1 541 327 783 632 774 640
AU 23 5 931 860 983 949 983 956 6 2 816 666 921 852 931 846 4 1 561 353 776 648 786 649
TB1 25 4 876 765 946 870 965 893 5 1 698 519 822 711 846 712 1 1 418 230 627 415 604 400
TB2 21 6 844 696 933 851 928 859 10 3 627 446 798 638 807 657 3 1 368 167 534 312 544 336
Note: k = 5, k = 10, k = 15 and k = 20 refer to the number of rejections at 95% (†) and 99% (‡) confidence levels (these intervals are given by [0.809, 1.862] and [0.721, 2.098],
respectively) for the 1000 simulated time series.
Table 5
Lo’s modified R/S Hurst exponent H values for the empirical and simulated data.
τ = 0 τ = 5 τ = 10
Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20
Dow 0.620 0.556 0.674 0.703 0.704 0.607 0.540 0.650 0.677 0.678 0.597 0.532 0.636 0.662 0.663
(0.024) (0.029) (0.030) (0.031) (0.024) (0.028) (0.029) (0.030) (0.024) (0.028) (0.028) (0.029)
Nik 0.723 0.564 0.670 0.695 0.695 0.705 0.544 0.643 0.667 0.667 0.693 0.535 0.629 0.652 0.651
(0.025) (0.027) (0.028) (0.029) (0.025) (0.027) (0.028) (0.029) (0.025) (0.027) (0.027) (0.028)
UK 0.712 0.545 0.665 0.694 0.696 0.696 0.532 0.644 0.672 0.673 0.685 0.525 0.632 0.658 0.660
(0.025) (0.030) (0.033) (0.036) (0.025) (0.029) (0.032) (0.035) (0.025) (0.029) (0.031) (0.034)
AU 0.726 0.555 0.673 0.700 0.701 0.711 0.539 0.650 0.674 0.676 0.700 0.531 0.636 0.660 0.661
(0.025) (0.029) (0.032) (0.032) (0.025) (0.028) (0.031) (0.031) (0.025) (0.028) (0.030) (0.030)
TB1 0.746 0.565 0.670 0.689 0.691 0.721 0.547 0.642 0.660 0.661 0.704 0.535 0.627 0.644 0.645
(0.024) (0.028) (0.031) (0.029) (0.024) (0.028) (0.030) (0.028) (0.024) (0.028) (0.029) (0.028)
TB2 0.724 0.567 0.662 0.679 0.680 0.704 0.545 0.634 0.650 0.652 0.689 0.536 0.620 0.636 0.637
(0.025) (0.028) (0.028) (0.028) (0.025) (0.027) (0.028) (0.028) (0.024) (0.027) (0.028) (0.027)
τ = 25 τ = 50 τ = 100
Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20 Emp k = 5 k = 10 k = 15 k = 20
Dow 0.578 0.521 0.612 0.636 0.637 0.560 0.513 0.592 0.614 0.615 0.542 0.508 0.571 0.590 0.591
(0.024) (0.027) (0.027) (0.028) (0.023) (0.026) (0.026) (0.027) (0.023) (0.025) (0.025) (0.026)
Nik 0.671 0.522 0.605 0.627 0.626 0.650 0.514 0.586 0.606 0.605 0.628 0.509 0.566 0.584 0.583
(0.025) (0.026) (0.027) (0.027) (0.024) (0.026) (0.026) (0.026) (0.024) (0.025) (0.024) (0.025)
UK 0.662 0.515 0.610 0.634 0.635 0.641 0.508 0.590 0.612 0.613 0.617 0.503 0.569 0.589 0.589
(0.025) (0.028) (0.029) (0.032) (0.024) (0.027) (0.028) (0.030) (0.024) (0.026) (0.026) (0.028)
AU 0.679 0.520 0.612 0.634 0.635 0.657 0.512 0.592 0.612 0.613 0.631 0.507 0.571 0.588 0.589
(0.025) (0.027) (0.029) (0.029) (0.024) (0.026) (0.027) (0.027) (0.023) (0.025) (0.026) (0.026)
TB1 0.672 0.522 0.603 0.619 0.621 0.642 0.514 0.583 0.597 0.598 0.610 0.509 0.563 0.575 0.576
(0.024) (0.027) (0.028) (0.027) (0.024) (0.026) (0.027) (0.026) (0.023) (0.025) (0.026) (0.024)
TB2 0.661 0.520 0.597 0.611 0.612 0.633 0.514 0.578 0.591 0.592 0.604 0.509 0.559 0.571 0.571
(0.024) (0.027) (0.027) (0.027) (0.024) (0.026) (0.026) (0.026) (0.023) (0.025) (0.025) (0.024)
Note: Emp stands for the empirical value of Lo’s Hurst exponent, k = 5, k = 10, k = 15 and k = 20 refer to the mean and standard deviation of Lo’s Hurst
exponent based on the corresponding 1000 simulated time series with different k. Boldface numbers are those cases in which empirical Hs fall into the
corresponding 2.5 to 97.5 percent quantiles of the 1000 simulation-based values of H.
	Introduction
	Methodology
	Markov-switching multifractal model
	Estimation of scaling exponents
	Results
	Concluding Remarks
	References
