# Spectrum-based estimators of the bivariate Hurst exponent

Ladislav Kristoufek111Institute of Information Theory and Automation, Academy
of Sciences of the Czech Republic, CZ-182 08, E-mail:
<EMAIL_ADDRESS>of Economic Studies, Faculty of Social
Sciences, Charles University in Prague, Czech Republic, CZ-110 00

###### Abstract

We introduce two new estimators of the bivariate Hurst exponent in the power-
law cross-correlations setting – the cross-periodogram and local $X$-Whittle
estimators – as generalizations of their univariate counterparts. As the
spectrum-based estimators are dependent on a part of the spectrum taken into
consideration during estimation, a simulation study showing performance of the
estimators under varying bandwidth parameter as well as correlation between
processes and their specification is provided as well. The newly introduced
estimators are less biased than the already existent averaged periodogram
estimator which, however, has slightly lower variance. The spectrum-based
estimators can serve as a good complement to the popular time domain
estimators.

power-law cross-correlations, long-term memory, spectrum-based estimators

###### pacs:

05.10.-a, 05.45.-a, 89.65.Gh

##  I Introduction

Introduction of the detrended cross-correlation analysis (DCCA) Podobnik2008 ;
Zhou2008 ; Jiang2011 ; Podobnik2011  has opened a new branch of studying
cross-correlations between two series which has found its home in various
disciplines – hydrology and (hydro)meteorology Hajian2010 ; Vassoler2012 ;
Kang2013 , seismology and geophysics Shadkhoo2009 ; Marinho2013 , biology and
biometrics Xue2012 ; Ursilean2009 , DNA sequences Stan2013 , neuroscience
Jun2012 , music Hennig2014 , electricity Wang2013b , finance Podobnik2009a ;
Lin2012 ; Shi2013 , commodities He2011 ; He2011a , traffic Zebende2009 ;
Xu2010 ; Zhao2011  and others. Since then, height cross-correlation analysis
(HXA) Kristoufek2011  and detrending moving-average cross-correlation analysis
(DMCA) Arianos2009 ; He2011a  have been introduced as complements to DCCA.
These estimators are based on a generalization of the power-law decay of the
autocorrelation function $\rho(k)$ with lag $k$ in the univariate case where
$\rho(k)\propto k^{2H-2}$ for lag $k\rightarrow+\infty$. Hurst exponent $H$ is
a measure of long-range dependence and it holds that $0\leq H<1$ for
stationary series. For $H>0.5$, the process is positively long-range
correlated and reminds of a trending process which, however, returns back to
its mean and remains stationary. Due to this characteristic, such processes
are also usually referred to as persistent processes. From the other side,
$H<0.5$ implies anti-persistence which features frequent reversal of signs and
in general negative correlation. No long-range dependence is present for
$H=0.5$.

Long-range correlations are generalized into the bivariate setting simply by
translating the properties into two dimensions – a power-law decay of the
cross-correlation function of the two analyzed processes. Specifically, we
have the cross-correlation function $\rho_{xy}(k)$ with lag $k$ of a specific
form, i.e. $\rho_{xy}(k)\propto k^{2H_{xy}-2}$ for lag $k\rightarrow+\infty$.
If the bivariate Hurst exponent $H_{xy}>0.5$, we have power-law cross-
correlated processes, or alternatively cross-persistent or long-range cross-
correlated processes. Such processes are persistent in their co-movement, i.e.
if the pair of series moved together in the past, they are more likely to move
together in the future as well Kristoufek2011 . Such definition is positioned
in the time domain. However, we can also approach the power-law cross-
correlations in the frequency domain.

An alternative definition of the power-law cross-correlated processes is again
built on a generalization of the univariate definition. The cross-persistent
process is characterized by the cross-power spectrum of a form
$|f_{xy}(\lambda)|\propto\lambda^{1-2H_{xy}}$ for frequency
$\lambda\rightarrow 0+$ Sela2012 . The power-law asymptotic decay of the
cross-correlation function thus simply translates to the power-law divergence
of the cross-power spectrum close to the origin. This alternative approach to
the power-law cross-correlations is used only marginally in the
interdisciplinary physics literature.

Here, we introduce two new estimators of the bivariate Hurst exponent based on
the spectral definition of the power-law cross-correlations. Specifically, we
establish the cross-periodogram estimator and the local $X$-Whittle estimator
as generalizations of their univariate counterparts, we discuss the properties
of these estimators and we provide a comparison to the only existent spectrum-
based estimator – the averaged periodogram estimator Sela2012 . In the next
section, the estimators are described in detail. The following section
discusses the accuracy of the estimators with a changing bandwidth parameter.
And the last section concludes.

##  II Spectrum-based estimators

Frequency domain estimators are based on the definition that at frequencies
close to the origin, the magnitude of the cross-power spectrum follows a power
law and diverges to infinity. Estimation of the cross-power spectrum thus
becomes crucial. The most frequently used tool is a cross-periodogram
$I_{xy}(\lambda)$ defined as

$I_{xy}(\lambda_{j})=\frac{1}{2\pi}\sum_{k=-\infty}^{+\infty}{\widehat{\gamma}_{xy}(k)\exp(-i\lambda_{j}k)}=\frac{1}{2\pi
T}\sum_{t=1}^{T}{x_{t}\exp(-i\lambda_{j}t)}\sum_{t=1}^{T}{y_{t}\exp(i\lambda_{j}t)}=I_{x}(\lambda_{j})\overline{I_{y}(\lambda_{j})},$
(1)

where $T$ is the time series length, $\widehat{\gamma}_{xy}(k)$ is an
estimated cross-covariance at lag $k$ and $\lambda_{j}$ is a frequency defined
as $\lambda_{j}=2\pi j/T$ where $j=1,2,\ldots,\lfloor T/2\rfloor$ and
$\lfloor\rfloor$ is the nearest lower integer operator so that the cross-
periodogram is defined between 0 and $\pi$ only. $I_{x}(\lambda_{j})$ is a
periodogram of series $\\{x_{t}\\}$ and $\overline{I_{y}(\lambda_{j})}$ is a
complex conjugate of a periodogram of series $\\{y_{t}\\}$. If the cross-
periodogram is used in its raw form, then evidently, we automatically obtain
$H_{xy}=\frac{H_{x}+H_{y}}{2}$. Moreover, the raw cross-periodogram (as well
as the raw univariate periodogram) is an inconsistent estimator of the true
cross-power spectrum (Wei2006, ). To overcome the inconsistency issue, the raw
(cross-)periodogram needs to be smoothed. Bloomfield Bloomfield2000  suggests
a simple smoothing operator based on Daniell Daniell1946 , which is
practically a simple moving average with half weights on the boundary values.
Some authors (Velasco1999, ; Hurvich2000, ; Sela2012, ) also suggest to first
taper the series to deal with leakages at low frequencies. In the following
text, we apply only the smoothing of periodograms while the tapering is not
utilized as it did not show any finite sample efficiency or bias gains for the
estimators we use. In this section, we cover three estimators of the bivariate
Hurst exponent $H_{xy}$, two of which are newly introduced here.

###  II.1 Averaged periodogram estimator

Sela & Hurvich Sela2012  propose the averaged periodogram estimator (APE) and
they are in fact the first ones to propose an estimator of $H_{xy}$ (or more
precisely $d_{12}$ in their case, where it holds that $d_{12}=H_{xy}-0.5$ as
in the univariate case) in the frequency domain. The estimator is a bivariate
generalization of the method of Robinson Robinson1994 . Taking the cumulative
cross-periodogram $\widehat{F_{xy}}(\lambda)=\frac{2\pi}{m}\sum_{j=1}^{\lfloor
m\lambda/2\pi\rfloor}{I_{xy}(\lambda_{j})}$ where $m\leq T/2$ is a bandwidth
parameter and fixed $q\in(0,1)$, the estimator is given by

$\widehat{H_{xy}}=1-\frac{\log\frac{\widehat{F_{xy}}(q\lambda_{m})}{\widehat{F_{xy}}(\lambda_{m})}}{2\log
q}.$ (2)

Under twelve assumptions given by Sela & Hurvich Sela2012 , the estimator is
consistent. Moreover, it is advised to use $q=0.5$. The authors also provide a
Monte Carlo simulation study to show the finite sample properties of the
estimator. The bias and efficiency are shown by box plots for several
scenarios to show that for samples below 10,000 observations, the estimator is
strongly biased with high variance. For high values of $m$ and high number of
observations, the variance of the estimator decreases markedly while the
estimator still remains biased.

###  II.2 Cross-periodogram estimator

As given in the definition of the cross-persistent processes, the cross-power
spectrum diverges at the origin as a power law with exponent $1-2H_{xy}$.
Using the cross-periodogram as an estimator of the cross-power spectrum, we
expect the long-range cross-correlated series to follow

$|I_{xy}(\lambda_{j})|\propto\lambda_{j}^{1-2H_{xy}}.$ (3)

The cross-periodogram estimator (XPE) of the bivariate Hurst exponent $H_{xy}$
can be obtained through a regression on

$\log|I_{xy}(\lambda_{j})|\propto-(2H_{xy}-1)\log\lambda_{j}.$ (4)

As the power-law scaling is expected only for $\lambda\rightarrow 0+$, the
regression is not performed over all frequencies. By choosing
$\lambda_{j}=2\pi j/T$ for $j=1,2,\ldots,m$ where $m\leq T/2$, we estimate the
bivariate Hurst exponent using only the information up to a selected frequency
based on a selection of the bandwidth parameter $m$. For the univariate case,
Beran Beran1994  and Robinson Robinson1995  show that the periodogram
estimator is consistent and asymptotically normal with

$\sqrt{m}(\widehat{H}-H^{0})\rightarrow_{d}N(0,\pi^{2}/24)$ (5)

where $H^{0}$ is the true Hurst exponent. The limiting distribution is free of
$H^{0}$ and also of all the other parameters (the assumptions are given in
Theorem 4.6 of Beran Beran1994 ). The variance of the estimator decreases with
a square root of $m$ but we need to keep in mind that the higher the $m$
parameter is, the more biased the estimator is because the power-law scaling
holds only for the origin neighborhood. Choice of $m$ thus depends on
preferences between bias and efficiency. For the bivariate case, however, we
have more parameters in the specification of the model – mainly the univariate
Hurst exponents of the separate processes and the correlation coefficient
between error-terms for the simplest case – and there is no reason to believe
that the properties of the XPE estimator would be independent of these.
Showing the asymptotic properties requires a strict set of assumptions and the
underlying bivariate model specification. In the context of this text, it
would be out of line to assume some particular specification of the underlying
model as we try to keep the assumptions of the methods as general as possible.
We thus do not provide the asymptotic properties for this estimator and leave
it for further research, yet still, we provide a discussion about dependence
of mean and variance of the estimator on the set of parameters and compare the
properties with the other two frequency-based estimators later in the next
section.

###  II.3 Local $X$-Whittle estimator

The local Whittle estimator of the fractional differencing parameter $d$ or
Hurst exponent $H$ is based on the same principle as the previously defined
periodogram estimator – the power-law divergence of the power spectrum.
However, instead of the regression fitting to the power-law scaling near the
origin as $\lambda\rightarrow 0+$, the local Whittle estimator is based on a
minimization of the penalty function based on Künsch Kunsch1987 .

Taking the work of Robinson Robinson1995a  as a starting point and
generalizing the method for the bivariate series, we propose the estimator of
the bivariate Hurst exponent $H_{xy}$ as follows. Divergence of the magnitude
of the cross-power spectrum close to the origin with the power-law scaling is
assumed for long-range cross-correlated processes $\\{x_{t}\\}$ and
$\\{y_{t}\\}$. Cross-periodogram $I_{xy}(\lambda)$ is defined according to Eq.
1 with $j=1,2,\ldots,m$ where $m\leq T/2$ and $\lambda_{j}=2\pi j/T$. Assuming
that series $\\{x_{t}\\}$ and $\\{y_{t}\\}$ are indeed long-range cross-
correlated with $\frac{1}{2}<H_{xy}\leq 1$, we propose the local $X$-Whittle
estimator (LXW) as

$\widehat{H_{xy}}=\arg\min_{\frac{1}{2}<H_{xy}\leq 1}R(H_{xy}),$ (6)

where

$R(H_{xy})=\log\left(\frac{1}{m}\sum_{j=1}^{m}{\lambda_{j}^{2H_{xy}-1}|I_{xy}(\lambda_{j})|}\right)-\frac{2H_{xy}-1}{m}\sum_{j=1}^{m}{\log\lambda_{j}}$
(7)

and $\lambda_{j}=2\pi j/T$. Eq. 7 represents the likelihood function of Künsch
Kunsch1987 . The LXW estimator is thus a semi-parametric maximum likelihood
estimator as it utilizes only the properties of the cross-power spectrum near
the origin.

In a similar manner as for the XPE estimator, the univariate version of the
local Whittle estimator is consistent and asymptotically normal, specifically

$\sqrt{m}(\widehat{H}-H^{0})\rightarrow_{d}N(0,1/4)$ (8)

where again $H^{0}$ is the true bivariate Hurst exponent and the limiting
distribution is free of $H^{0}$ and all the other parameters. The local
Whittle estimator is thus asymptotically more efficient than the periodogram
estimator as $1/4<\pi^{2}/24$. For detailed treatment of the univariate case
and the assumptions, see Robinson1995a . In the bivariate case, there is again
no reason to presume that the asymptotic properties would be independent of
the univariate Hurst exponents and the correlation structure of the error-
terms. Discussion of these possible dependencies and comparison with APE and
XPE are provided in the following section 333R-scripts of all three estimators
are available upon request from the author..

##  III Dependence on bandwidth parameter

Choice of the bandwidth parameter $m$ is a crucial aspect of the frequency
domain estimators as hinted in the previous section. As noted in the studies
dealing with the univariate specifications of the estimators (Beran1994, ;
Robinson1995a, ; Robinson1995, ), variance should decrease with the parameter
and bias should increase. The former comes from the fact that the estimation
is based on more data points and latter from the fact that the power-law
scaling of the cross-power spectrum holds only for the lowest frequencies.
Here, we present and discuss the behavior of the mean and variance of the
frequency-based estimators presented above with respect to varying parameter
$m$.

We discuss two main scenarios of the long-range cross-correlated processes –
the processes with and without power law coherency behavior, i.e. when the
bivariate Hurst exponent $H_{xy}$ is not or is, respectively, equal to the
average of the separate Hurst exponents $H_{x}$ and $H_{y}$. For the latter
(simpler) case, we utilize ARFIMA(0,$d$,0) – autoregressive fractionally
integrated moving average – processes with correlated error-terms and with
$d_{1}=d_{2}=0.4$, and for the former, we use the mixed-correlated ARFIMA
processes Kristoufek2013  with $d_{1}=d_{4}=0.4$ and $d_{2}=d_{3}=0.2$.

Specifically, for given $d$ parameters and
$a_{n}(d)=\frac{\Gamma(n+d)}{\Gamma(n+1)\Gamma(d)}$, the correlated ARFIMA
processes $\\{x_{t}\\}$ and $\\{y_{t}\\}$ are given as

$\displaystyle x_{t}=\sum_{n=0}^{\infty}{a_{n}(d_{1})\varepsilon_{t-n}}$ (9)
$\displaystyle y_{t}=\sum_{n=0}^{\infty}{a_{n}(d_{2})\nu_{t-n}}$
$\displaystyle\langle\varepsilon_{t}\rangle=\langle\nu_{t}\rangle=0$
$\displaystyle\langle\varepsilon_{t}^{2}\rangle=\sigma_{\varepsilon}^{2}<+\infty$
$\displaystyle\langle\nu_{t}^{2}\rangle=\sigma_{\nu}^{2}<+\infty$
$\displaystyle\langle\varepsilon_{t}\varepsilon_{t-n}\rangle=\langle\nu_{t}\nu_{t-n}\rangle=\langle\varepsilon_{t}\nu_{t-n}\rangle=0\text{
for }n\neq 0$
$\displaystyle\langle\varepsilon_{t}\nu_{t}\rangle=\sigma_{\varepsilon\nu}<+\infty.$

The mixed-correlated ARFIMA processes are defined as

$\displaystyle
x_{t}=\sum_{n=0}^{+\infty}{a_{n}(d_{1})\varepsilon_{1,t-n}}+\sum_{n=0}^{+\infty}{a_{n}(d_{2})\varepsilon_{2,t-n}}$
(10) $\displaystyle
y_{t}=\sum_{n=0}^{+\infty}{a_{n}(d_{3})\varepsilon_{3,t-n}}+\sum_{n=0}^{+\infty}{a_{n}(d_{4})\varepsilon_{4,t-n}}$
$\displaystyle\langle\varepsilon_{i,t}\rangle=0\text{ for }i=1,2,3,4$
$\displaystyle\langle\varepsilon_{i,t}^{2}\rangle=\sigma_{\varepsilon_{i}}^{2}\text{
for }i=1,2,3,4$
$\displaystyle\langle\varepsilon_{i,t}\varepsilon_{j,t-n}\rangle=0\text{ for
}n\neq 0\text{ and }i,j=1,2,3,4$
$\displaystyle\langle\varepsilon_{i,t}\varepsilon_{j,t}\rangle=\sigma_{ij}\text{
for }i,j=1,2,3,4\text{ and }i\neq j.$

Both kinds of processes are studied for the time series with $T=5000$
observations and correlation between error-terms varying in an interval
between 0.2 and 1 with a step of 0.2. To uncover the dependence on $m$, we use
$m/T$ from 0.05 up to 0.5 with a step of 0.05. We thus cover the cross-
periodogram from the lowest tenth of the frequencies up to the whole cross-
periodogram. For each specification, we use 1,000 simulations and the
Daniell’s window of 21 as used in Sela & Hurvich Sela2012 . This way, we are
able to comment on the dependence of bias and variance of the estimator with
respect to the correlation between error-terms and the bandwidth parameter
$m$.

Starting with the bias, Figs. 1 and 2 show the results of simulations for the
correlated and mixed-correlated ARFIMA processes. In Fig. 1, we observe that
for the correlated ARFIMA processes for which we expect the bivariate Hurst
exponent to be equal to the average of the separate Hurst exponents
Podobnik2009 ; Kristoufek2013 , the bias behavior differs for specific
methods. For LXW and XPE, we observe an expected behavior – the estimates are
unbiased for approximately $m/T\leq 0.2$, i.e. the lowest frequencies. For
higher values of $m$, the estimates become biased downwards. Interestingly,
the bias is practically independent of the correlation level between error-
terms for all three estimators. However, for APE, the mean values of the
estimates are very stable across various $m$ but remain well below the
theoretical value of 0.9 and yield a negative bias of approximately $-0.05$.
Again, the bias is practically independent of the correlations between error-
terms.

The situation is more interesting for the mixed-correlated ARFIMA processes,
i.e. the power law coherency case. In Fig. 2, we can see that the mean values
of the estimates are dependent on both $m$ and the correlation between error-
terms for all three estimators. In general, it holds that the estimates are
less biased with an increasing correlation between error-terms, which is
expected, but also with higher $m$. The performance of the estimators is thus
not only dependent on the parameters but also on the specification of the
model as shown by the difference between the two cases.

The situation is quite similar for the behavior of variance of the estimators.
In Fig. 3, we show this behavior compared to the theoretical asymptotic
variance for the univariate cases (for LXW and XPE). We observe several
regularities which are true for all three estimators. First, the variance
decreases with the increasing $m$ as expected. Second, the variance decreases
with the increasing strength of correlations between error-terms. Third, the
log-log depiction indicates a power-law scaling with the parameter $m$. For
LXW and XPE, we can compare this scaling with the asymptotic scaling for the
respective univariate estimators and it is visible that these can be seen as
square-root scalings which are in hand with the univariate case. However, the
levels of variance are well above the asymptotic univariate values. The
variance levels of LXW and XPE practically overlap while the variance of APE
shows slightly lower values.

For the mixed-correlated ARFIMA processes, Fig. 4 depicts the behavior of
variance. All the regularities from the previous paragraph hold even here.
However, the level of variances is evidently much higher compared to the
previous case and it shows that the estimators have much higher variance for
the power law coherency specification. Even though the variance level again
depends on the level of correlation between error-terms, the general variance
level dominates. As a result, the distance from the univariate asymptotic
variances of LXW and XPE is much more profound as well. Again, the level of
variances is lower for APE compared to the other two estimators.

##  IV Conclusions

We introduce two new estimators of the bivariate Hurst exponent – the cross-
periodogram and local $X$-Whittle estimators – in the power-law cross-
correlations setting and compare them with the already existent averaged
periodogram estimator. As the spectrum-based estimators depend on a part of
the spectrum taken into consideration during estimation, in the same way as
the time domain estimators depend on the utilized scales, we also provide a
simulation study showing the performance of the three estimators under varying
bandwidth parameter as well as correlation between processes and their
specification. The newly introduced estimators are less biased than the
averaged periodogram which, however, has slightly lower variance. The
spectrum-based estimators can serve as a good complement to the already
existing and popular time domain estimators.

## Acknowledgements

Support from the Czech Science Foundation under project No. 14-11402P and the
Grant Agency of the Charles University in Prague under project No. 1110213 is
gratefully acknowledged.

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Figure 1: (Color online) Mean values of APE, LXW and XPE estimators dependent
on $m$ and correlation between error-terms I. Values are based on 1,000
simulations of ARFIMA(0,$d$,0) processes with correlated error-terms and
$d_{1}=d_{2}=0.4$. Correlation between error-terms ranges between 0.2 and 1
with a step of 0.2 and is represented by different shades of grey in the chart
(the lightest for 0.2 and black for 1). Red line represents the true value of
$H_{xy}=0.9$.

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Figure 2: (Color online) Mean values of APE, LXW and XPE estimators dependent
on $m$ and correlation between error-terms II. Values are based on 1,000
simulations of mixed-correlated ARFIMA(0,$d$,0) processes with
$d_{1}=d_{4}=0.4$ and $d_{2}=d_{3}=0.2$. Correlation between error-terms
ranges between 0.2 and 1 with a step of 0.2 and is represented by different
shades of grey in the chart (the lightest for 0.2 and black for 1). Red line
represents the true value of $H_{xy}=0.7$.

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Figure 3: (Color online) Variance of APE, LXW and XPE estimators dependent on
$m$ and correlation between error-terms I. Values are based on 1,000
simulations of ARFIMA(0,$d$,0) processes with correlated error-terms and
$d_{1}=d_{2}=0.4$. Correlation between error-terms ranges between 0.2 and 1
with a step of 0.2 and is represented by different shades of grey in the chart
(the lightest for 0.2 and black for 1). Red line represents the asymptotical
values for the univariate case (for LXW and XPE only).

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Figure 4: (Color online) Variance of APE, LXW and XPE estimators dependent on
$m$ and correlation between error-terms II. Values are based on 1,000
simulations of mixed-correlated ARFIMA(0,$d$,0) processes with
$d_{1}=d_{4}=0.4$ and $d_{2}=d_{3}=0.2$. Correlation between error-terms
ranges between 0.2 and 1 with a step of 0.2 and is represented by different
shades of grey in the chart (the lightest for 0.2 and black for 1).