#  Non-radiative configurations of a few quantum emitters ensembles:
evolutionary optimization approach

Ilya Volkov<EMAIL_ADDRESS>School of Physics and Engineering,
ITMO University, St. Petersburg 197101, Russia  Stanislav Mitsai  School of
Physics and Engineering, ITMO University, St. Petersburg 197101, Russia
School №45, St. Petersburg, Russia  Stepan Zhogolev  School of Physics and
Engineering, ITMO University, St. Petersburg 197101, Russia  School №1589,
Moscow, Russia  Danil Kornovan  School of Physics and Engineering, ITMO
University, St. Petersburg 197101, Russia  Alexandra Sheremet  School of
Physics and Engineering, ITMO University, St. Petersburg 197101, Russia  Roman
Savelev  School of Physics and Engineering, ITMO University, St. Petersburg
197101, Russia  Mihail Petrov  School of Physics and Engineering, ITMO
University, St. Petersburg 197101, Russia

###### Abstract

Suppressing the spontaneous emission in atomic ensembles is one of the topical
problems in quantum optics and quantum technology. While many approaches are
based on utilizing the subradiance effect in ordered quantum emitters arrays,
the ensemble configurations providing the minimal spontaneous emission rate
are yet unknown. In this work, we employ the differential evolution algorithm
to identify the optimal configurations of a few atomic ensembles that support
quantum states with maximal radiative lifetime. We demonstrate that atoms tend
to assemble mostly in quasi-regular structures with specific geometry, which
strongly depends on the minimally allowed interatomic distance
$r_{\mathrm{min}}$. While the discovered specific non-radiative realizations
of small ensembles cannot be immediately predicted, there is particular
correpondence to the non-radiative states in the atomic lattices. In
particular, we have found that states inheriting their properties either from
the bound states in the continuum or band-edge states of infinite lattices
dominate across a wide range of $r_{\mathrm{min}}$ values. Additionally, we
show that for small interatomic distances the linear arrays with modulated
spacing have the smallest radiative losses exponentially decreasing as the
size of the ensemble increases.

The interest in protecting quantum states from the decoherence caused by
spontaneous emission keeps driving theoretical Sheremet2012 ; ScullyPRL2015 ;
CaiPRA2016 ; Kornovan2016 ; jen_cooperative_2018 ; Plankensteiner2019 ;
Zhang2020Feb ; Fofanov2021  and experimental pavolini_experimental_1985 ;
devoe_observation_1996 ; guerin_subradiance_2016 ; Rui2020Jul ; Ferioli2021
research in modern quantum optics since the pioneering work of R. Dicke
Dicke1954 . Among various platforms realizing quantum simulations and quantum
computation, ensembles of cold atoms can be considered as one of the most
prospective ones due to outstanding progress in their positioning in free
space or in the proximity of nanophotonic structures barredo_synthetic_2018 ;
Corzo2016 . The control of spontaneous emission rate in atomic ensembles was
demonstrated by arranging them in large regular one-dimensional arrays in free
space Asenjo-Garcia2017 ; Zhang2020Dec ; Kornovan2021 ; Fayard2023  or near a
waveguide sutherland_collective_2016 ; Asenjo-Garcia2017Mar ; Kornovan2019 ;
Ke2019Dec ; Zhang2020Feb ; Poddubny2020 , as well as in two-dimensional arrays
facchinetti_storing_2016 ; bettles_cooperative_2015 ; Ballantine2020Apr .
Radiative decay rate in such structures can be significantly reduced due to
formation of collective subradiant states, following either a polynomial
Zhang2020Feb ; Kornovan2019 ; volkov_strongly_2023  or exponential Asenjo-
Garcia2017  dependence on the number of quantum emitters in the array $N$.

Figure 1: The quantum emitters having a two-level structure are arbitrary
distributed in a plane $z=0$. The minimum allowed distance between the
emitters is limited by $r_{\text{min}}$. Geometry optimization based on
differential evolutionary method provides the configurations with minimal
radiative losses. The final configuration strongly depends on the parameter
$r_{\text{min}}$.

Such findings, however, cannot be directly transferred onto the structures
consisting of only a few atoms. The current approaches to engineering the
radiative losses in small dipole ensembles are based on the symmetry of the
arrays de_paz_bound_2023 ; ustimenko_non-radiant_2023 ; volkov_strongly_2023 ,
with special attention to the rings composed of two-level atoms
freedhoff_cooperative_1986 ; cremer_polarization_2020 ; moreno-
cardoner_subradiance-enhanced_2019 ; holzinger_nanoscale_2020 ; moreno-
cardoner_efficient_2022 ; Holzinger2023Sep , due to their high symmetry and
relevance to various natural quantum systems. To this day, the general recipe
for tackling the problem of loss suppression in ensembles of dipole emitters
hos not been proposed. One of the main reasons is the large dimensionality of
the parameter space, which leads to the high complexity of the problem rapidly
growing with increasing the number of emitters. In this context, modern
powerful numerical optimization algorithms could serve as reliable tools to
efficiently solve the problem directly and uncover the importance of various
physical mechanisms responsible for the formation of subradiant states.

Numerical algorithms based on evolutionary and machine learning approaches
have already proved their efficiency in optimization of nanophotonic
structures such as subwavelength resonators and metasurfaces krasikovOEA2022 ;
pan_inverse_2022 ; mikhailovskaya_superradiant_2022 ; gladyshev_inverse_2023 ;
ustimenko_multipole_2021  and for engineering quantum systems
tranter_multiparameter_2018 ; rozenberg_inverse_2022 . In this Letter, we
utilize the differential evolution method to numerically find the optimal
planar atomic configurations that support quantum excitations with minimal
spontaneous emission rate. The results of the simulations reveal that the
atoms mostly tend to arrange in regular or quasi-regular structures. However,
in some cases, the specific geometry turns out to be counterintuitive and
difficult to predict based on the general physical principles established for
large atomic arrays.

Figure 2: Calculated optimal atomic configurations and corresponding radiative
losses for different minimum interatomic distance
$\widetilde{r}_{\mathrm{min}}$ (black solid curve). Dashed orange, dashed-
dotted green and dotted blue curves correspond to the radiative losses of the
regular chain, triangle and rectangle, respectively, with a varied lattice
constant. Wavefunctions $\psi$ of eigenstates corresponding to different
distinct regions of $\widetilde{r}_{\mathrm{min}}$ are illustrated in insets
with size of the dots being proportional to $|\psi|^{2}$ and color indicating
the phase of the wavefunction $\rm{Arg}(\psi_{i})$.

The problem of finding the configuration of atoms with minimal radiative
losses has, of course, a global yet unrealistic solution when all atoms tend
to be located at one point in space. In a single excitation domain, this leads
to the formation of subradiant states with radiative losses approaching zero
Dicke1954 . At the same time, the modern methods of quantum emitters’
positioning have certain limitations on the minimal interatomic distance
srakaew_subwavelength_2023 ; rui_subradiant_2020 . To account for that, the
distance between the atoms $r$ was subject to constraint $r>r_{\text{min}}$
(see Fig.1) during the optimization procedure. We also assumed throughout the
paper that all atoms are located in the $z=0$ plane, thus limiting all
possible configurations to two-dimensional ones.

In our model, all atoms are assumed to have a two-level structure shown in
Fig. 1(a). In what follows, we mostly focus on the $\sigma_{z}$ dipole
transition, assuming that it is independent from the other ones. This can be
achieved, for instance, by applying a strong external magnetic field that
spectraly isolates transitions. Case of atoms with $\sigma_{\pm}$ isolated
transition is proceeded in Supplementary Materials. The transition resonant
frequency is denoted as $\omega_{0}$ and the rate of spontaneous emission as
$\Gamma_{0}=k_{0}^{3}|\mathbf{d}|^{2}/\left(3\pi\hbar\varepsilon_{0}\right)$,
where $k_{0}=\omega_{0}/c=2\pi/\lambda_{0}$ is the wavenumber in vacuum, and
$\varepsilon_{0}$ is the vacuum permittivity. The coupling between the atoms
is governed by the dipole-dipole interaction, described by the electromagnetic
free space Green’s tensor $\mathbf{G}(\mathbf{R},\omega)$ Novotny2012Sep . The
quantum states of the atomic ensemble are then described by the effective
Hamiltonian Markel2007Feb ; AsenjoPRA2017 :

$\hat{\mathcal{H}}_{\text{int}}=\hbar\Bigl{(}\omega_{0}-\frac{i}{2}\Gamma_{0}\Bigr{)}\sum\limits_{i=1}^{N}\hat{\sigma}_{i}^{{\dagger}}\hat{\sigma}_{i}-\\\
-\sum\limits_{i\neq
j}\frac{3\pi\hbar\Gamma_{0}}{k_{0}}\mathbf{e_{d}^{*}}\cdot\mathbf{G}(\mathbf{R}_{ij})\cdot\mathbf{e_{d}}\
\hat{\sigma}_{i}^{{\dagger}}\hat{\sigma}_{j},$ (1)

where $\hat{\sigma}_{i}^{{\dagger}}$ ($\hat{\sigma}_{i}$) is the creation
(annihilation) operator for the excitation on the $i$-th emitter,
$\mathbf{e_{d}}=\mathbf{d}/d$ is a unit vector of atomic transition dipole
moment. The first term in Eq. (1) is responsible for the spontaneous emission
of single atom in vacuum, while the second one describes the dipole-dipole
interactions between the atoms. Here, we apply the Markovian approximation
$\mathbf{G}(\mathbf{R},\omega)\approx\mathbf{G}(\mathbf{R},\omega_{0})$, which
is justified by a very narrow resonance of a single atom
$\Gamma_{0}\ll\omega_{0}$. The radiative decay rate of the $j$-th collective
atomic excitation is determined by the imaginary part of the complex
eigenfrequency $\omega_{j}-i\Gamma_{j}/2$ of the non-Hermitian Hamiltonian
(1), where $j=1\ldots N$. The minimal decay rate $\text{min}(\Gamma_{j})$ then
serves as a goal function for the optimization, with the coordinates of the
atoms being the input arguments.

A differential evolution method Storn1997Dec  employed in our work applies
ideas of genetic algorithm to a set of continuous parameters. In the case of
an ensemble of $N$ atoms located in a plane, there are $2N-3$ degrees of
freedom (two coordinates of $N-1$ atoms minus one due to invariance of the
eigenstates to the rotation of the whole structure). This significantly limits
the maximal size of the optimized structures, which in our work was equal to
$N=8$. We have maintained two general restrictions during the optimization.
The first one is the lower limit of the interatomic distance
$r_{\mathrm{min}}$, see Fig. 1. The second one is the upper limit on the
overall size of the structure, ensuring that it is confined inside a circle
with a radius of $5\lambda_{0}$.

The results of the optimization procedure for $N=6$ $z$-polarized dipole
transitions are depicted in Fig. 2 showing the dependence of the minimal
normalized radiative losses $\Gamma/\Gamma_{0}$ on the minimal interatomic
distance, normalized by the dipole transition wavelength
$\widetilde{r}_{\mathrm{min}}=r_{\mathrm{min}}/\lambda_{0}$ (black solid
line). One may also find similar results for circular polarized dipoles in
Supplementary Materials. For the considered range
$\widetilde{r}_{\mathrm{min}}\in[0.1,1.2]$, the optimal configurations can be
separated into four types shown in the insets; the circles correspond to the
positions of the atoms, area of the circles is proportional to the square
amplitudes of the excitation at each atomic site, while the color indicates
the phase of this excitation. It turns out that for small minimal distances
$\widetilde{r}_{\mathrm{min}}\lesssim 0.35$ linear configurations are the
optimal ones. In the region $0.22\lesssim\widetilde{r}_{\mathrm{min}}\lesssim
0.35$, the quantum emitters are (almost) regularly distributed in space. To be
more precise, all distances between the neighboring dipoles differ by less
than $\sim 10^{-4}\lambda_{0}$. For the separation distance
$\widetilde{r}_{\mathrm{min}}\lesssim 0.22$, the optimal configuration becomes
stretched, with the distance between the emitters increasing from the center
to the end of the chain.

Figure 3: Wavefunction of the most subradiant eigenstates in the optimal
configurations for various restricting distances
$\widetilde{r}_{\mathrm{min}}$ and number of emitters $N$. Size of the dots is
proportional to $|\psi|^{2}$ and color indicates the phase of the $\psi$.

For larger $\widetilde{r}_{\mathrm{min}}$ in the range $0.38\ldots 0.88$, a
quasi-regular triangular shape is formed. One should note that due to a large
distance between the emitters of around $\approx\lambda_{0}/2$ and
corresponding $\approx\pi$ phase shift between the wavefunction in the
neighbor emitters, the minimal losses are observed when excitation at all
emitters has the same phase. For even larger distances
$\widetilde{r}_{\mathrm{min}}\approx 0.88\ldots 1.2$, a rectangular array
arranged in a square lattice with the out-of-phase eigenstate wavefunction
exhibits minimal losses. Note that in the most range of
$\widetilde{r}_{\mathrm{min}}$ values the optimal configurations are close to
the regular ones. Dashed orange, dashed-dotted green and dotted blue curves in
Fig. 2 show the radiative losses of the most subradiant states supported by
three characteristic realizations of regular structures: chain, triangle, and
rectangle, respectively. The $\widetilde{r}_{\mathrm{min}}$ for these curves
denotes not just the minimal distance between the atoms but rather the lattice
constant. One can observe that the difference between the regular and optimal
configurations is rather small for the most part of the
$\widetilde{r}_{\mathrm{min}}$ values.

The typical geometrical configurations that minimize the radiative losses
remain similar for different number of emitters $N=3\ldots 8$, as shown in
Fig. 3. For small minimal distances, the optimal subradiant configurations are
linear arrays with collective eigenstates characterized by out-of-phase
amplitudes of excitation in the neighboring atoms and maximal amplitude at the
center of the chain. These states inherit their subradiant properties from the
staggered guided modes at the band edge of the infinite periodic chains
WeberPRB2004 ; FigotinPRE2005 . The appropriate modulation of the distances
between the emitters leads to the stronger localization of the wavefunction in
the center of the chain (see e.g. left inset in Fig. 2) and consequently to
much weaker radiative losses. Such an effect tends to be more pronounced for
the ensemble of larger sizes.

For larger separations, $\widetilde{r}_{\mathrm{min}}\approx 0.4\ldots 0.8$
atoms tend to arrange in a quasi-regular triangular lattice. The in-phase
amplitude of the excitation on all atoms links such states to the so-called
bound state in continuum (BIC) – non-radiative states that appear for an
infinite lattice in the center of the Brillouin zone Hsu2016Jul ;
Sadrieva2019Sep ; OvervigPRB2020 . Interestingly, while the BIC states exist
in lattices of a wide class of symmetries OvervigPRB2020 ; Sadrieva2019Sep ,
the most subradiant states in small finite structures assemble in a triangular
lattice. For even larger separation distances $\gtrsim 0.8$ the atoms weakly
interact with each other. Consequently, the strongest suppression of losses is
achieved for the out-of-phase amplitudes of the wavefunction in the neighbor
atoms in a fragment of the square lattice. Similar results for circularly
polarized emitters are presented in Supplementary Materials.

Figure 4: Inset (top): distribution of the excitation across the quantum
emitters. (a) Dependence of losses on restricted distance
$\widetilde{r}_{\mathrm{min}}$ for $N=14$ atoms in cases of optimized free
configurations (black squares), periodic chain (blue dashed), a chain with
sinusoidal modulation (red solid line). (b) Distances between neighboring
atoms in cases of optimized configurations and linear array with sinusoidal
modulation for $N=14$ atoms with $\widetilde{r}_{\mathrm{min}}=0.2(0.3)$.(c)
Scaling of losses for optimal and periodic geometries as function of the array
size for $\widetilde{r}_{\mathrm{min}}=0.2(0.3)$.

The linear arrays with minimal losses for short interatomic spacing deserve
special attention due to strong interest in waveguide coupled quantum
ensembles Sheremet2023Mar  and many already published reports devoted to such
geometry Asenjo-Garcia2017 ; Kornovan2016 ; Kornovan2019 ; Kornovan2021 ;
pan_inverse_2022 ; ZhangPRL2019 ; Zhang2020Dec ; PichuginNanophotonics2021 .
The calculations presented in Figs. 2,3 have revealed that for $N\leq 8$ the
optimal 1D configurations gradually change from the regular chains to the
stretched ones with the decrease of the $\tilde{r}_{\mathrm{min}}$. By
restricting the geometry to only 1D configurations, we were able to extend
these results up to $N=26$ atoms. We have also increased the maximal radius of
the circle that confines the structure to $N(r_{\mathrm{min}}+\lambda_{0})/2$.
In order to analyze the exact profiles of atom positions along the chain, in
Fig. 4 (a) we plot radiative losses of the states in optimized chains (black
squares) along with the regular chains (dashed blue curve) and the chain with
modulated interatomic distances (solid red curve) for a fixed number of atoms
$N=14$. Modulation was chosen according to the following phenomenological
formula $\Delta
x(i)=r_{max}-(r_{max}-r_{\mathrm{min}})\text{sin}^{2}({\pi(i-1)}/({N-2}))$,
where $i=1..N-1$ is the number of the interval between atoms. The parameters
of the regular and modulated chains were also optimized, subject to a
constraint on the minimal interatomic spacing.

One can observe that the modulated chain matches well with the optimal
configuration in the range $\widetilde{r}_{\mathrm{min}}\gtrsim 0.25$. This is
illustrated in Fig. 4(b) with black and red circles showing the spatial
profile of the interatomic spacing for $\widetilde{r}_{\mathrm{min}}=0.3$ for
the optimal configuration and for the modulated one, respectively. As seen
from Fig. 4 (a), the difference with the regular chain in this range of
$r_{\mathrm{min}}$ reaches an order of magnitude for a given size of the chain
$N=14$. For a larger number of atoms, however, the difference becomes much
more pronounced due to qualitatively different dependence of the radiative
losses on the size of the structure, as shown in Fig. 4(c), while the regular
chain exhibits well-known $\sim N^{-3}$ dependence, the losses in a modulated
one decrease exponentially. For smaller $\widetilde{r}_{\mathrm{min}}\lesssim
0.25$, the losses decrease much faster both with the decrease of the
$\tilde{r}_{\mathrm{min}}$ or with the increase of $N$, see Fig. 4(a,c). Such
a change is accompanied by a qualitative change in the interatomic spacing
modulation. As shown in Fig. 4(b) with black squares, the modulation profile
becomes close to the parabolic one. We note while exponential scaling of the
radiative losses in modulated 1D arrays was already demonstrated in Asenjo-
Garcia2017 , here we advance these results by providing the most optimal
configurations.

Importantly, the results obtained in our work for an ensemble of a few dipole
emitters cannot be easily predicted based on the known physical mechanisms of
radiative loss suppression established for larger arrays. It is known that the
radiative losses of the band-edge states in regular linear arrays can decrease
with the atom number as fast as $N^{-7}$ Kornovan2019 ; Kornovan2021 , and in
regular two-dimensional arrays as fast as $N^{-5}$ volkov_strongly_2023 . At
the same time, quasi-BIC states exhibit much slower dependence of losses on
the size $N^{-1}$ Facchinetti2016Dec . Nevertheless, in small ensembles,
quasi-BIC states appear to be more subradiant in a wide range of
$\widetilde{r}_{\mathrm{min}}$ values, while band-edge states in 1D chains
dominate only for small enough interatomic distances. For large distances
$\widetilde{r}_{\mathrm{min}}\rightarrow 1$ the opening of the diffraction
channels makes the competition between the aforementioned mechanisms of the
subradiance even more complicated and the results of the optimization even
less predictable.

In conclusion, we utilized the differential evolution algorithm to find the
optimal configurations of small atomic ensembles that support eigenstates with
minimal radiative losses - subradiant states. Depending on the minimum allowed
distance between the atoms, different quasi-regular structures are inclined to
form, such as one-dimensional chains, fragments of triangular, and square
lattices. Due to the various geometries and loss suppression mechanisms in
these structures, it is not trivial to predict in advance the exact optimal
configurations.

## Acknowledgement

The authors acknowledge Kseniia Baryshnikova for her efforts in making this
work happen, and Nikita Ustimenko for fruitful discussions. The work was
supported by the Federal Academic Leadership Program Priority 2030.

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# Efficiency of the Moscow Stock Exchange before 2022

Andrey Shternshis, Piero Mazzarisi, Stefano Marmi

Scuola Normale Superiore, Piazza dei Cavalieri 7, Pisa, Italy, 56126

###### Abstract

This paper investigates the degree of efficiency for the Moscow Stock
Exchange. A market is called efficient if prices of its assets fully reflect
all available information. We show that the degree of market efficiency is
significantly low for most of the months from 2012 to 2021. We calculate the
degree of market efficiency by (i) filtering out regularities in financial
data and (ii) computing the Shannon entropy of the filtered return time
series. We have developed a simple method for estimating volatility and price
staleness in empirical data, in order to filter out such regularity patterns
from return time series. The resulting financial time series of stocks’
returns are then clustered into different groups according to some entropy
measures. In particular, we use the Kullback–Leibler distance and a novel
entropy metric capturing the co-movements between pairs of stocks. By using
Monte Carlo simulations, we are then able to identify the time periods of
market inefficiency for a group of 18 stocks. The inefficiency of the Moscow
Stock Exchange that we have detected is a signal of the possibility of
devising profitable strategies, net of transaction costs. The deviation from
the efficient behavior for a stock strongly depends on the industrial sector
it belongs.

Keywords Shannon entropy; market efficiency; volatility estimation; price
staleness; stock market clustering; Kullback-Leibler divergence

##  1 Introduction

When prices reflect all available information, the market is called efficient
[1]. One way to claim the efficiency of a market is by testing the Efficient
Market Hypothesis (EMH). In the weak form, the EMH considers that the last
price incorporates all the past information about market prices [2]. If the
weak form of EMH is rejected, previous prices help to predict future prices.
For traders, market efficiency means that analyzing the history of previous
prices does not help to design a strategy that gives an abnormal profit. For a
company issuing shares, market efficiency means that the cost of its share
already reflects all information about the valuation and decisions of the
company. The EMH is of great interest also in research. Mathematical models of
an asset price are usually based on the assumption that the price follows a
martingale: the expected value of a future price is the current value of the
price. If the EMH is rejected, there should be an estimation of the future
price better than its current value. In such a case, new models should be
thought.

The review of works confirming the EMH was presented by Fama in 1970 [2] and
then in 1991 [3]. The martingale hypothesis was also tested later. It was
shown that the efficiency of a market depends on the development of the
country [4]. Also, the martingale hypothesis was confirmed on short time
intervals, but may be violated on longer intervals [5]. In addition, there is
a range of strategies designed to increase an expected profit. High-frequency
and algorithmic trading strategies are discussed in [6]. Statistical and
machine learning methods for high frequency trading are reviewed in [7]. The
existence of such profitable strategies contradicts the Efficient Market
Hypothesis. According to Grossman and Stiglitz [8], the degree of market
inefficiency determines the effort investors are willing to expend to gather
and trade on information. A goal of this paper is to investigate the degree of
stock market efficiency using the Shannon entropy.

Before estimating the degree of market efficiency, we need to get rid of
regularities that make prices more predictable, but do not imply any
profitable strategies. The methodology of filtering regularities was
introduced in [9]. However, such a filtering has not usually been applied in
other research works (see e.g. [10, 11, 12]). In fact, deviations of price
behavior from perfect randomness may be the result of some known regularity
pattern, such as volatility clustering or daily seasonality, but not a signal
of market inefficiency.

We process data by filtering regularities of financial time series including
volatility clustering and price staleness. Price staleness is defined as a
lack of price adjustments yielding 0-returns. Traders may trade less because
of high transaction costs and so the price does not update. See [13] for more
details. The price staleness produces an extra amount of 0-returns called
excess 0-returns. The other source of 0-returns in the time series is price
rounding. Estimations of volatility and degree of price staleness are mutually
connected: excess 0-returns appearing due to price staleness tend to
underestimate volatility. At the same time, volatility estimation is needed to
calculate the expected amount of 0-returns due to rounding.

One way to estimate volatility in the presence of excess 0-returns was
presented in [14]. It uses expectation-maximization algorithm [15] to estimate
returns in the places of all 0-returns and uses GARCH(1,1) model to estimate
volatility [16]. The maximization of the likelihood function appearing at each
step of the considered algorithm requires several parameters for numerical
optimization. If the estimation of volatility is sensitive to these
parameters, that are user-defined, then they may affect the entropy of returns
standardized by volatility and the amount of 0-returns in the time series. In
this article we suggest a modification of moving average volatility estimation
that requires adjusting of the only parameter that can be defined using out-
of-sample testing. The idea is to adopt a simple method for volatility
estimation, so that price staleness is taken into consideration. Moreover,
while estimating volatility, we filter out excess 0-returns.

The degree of market efficiency has been measured for many countries. Stock
indices for 20 countries including Taiwan, Mexico, and Singapore, were
considered in [17]. The efficiency of 11 emerging markets, the US and Japan
markets was measured in [18] using the Hurst exponent and R/S statistics. The
review of articles about Baltic countries was presented in [19]. A degree of
uncertainty of Chinese [20], Tunisian [21], and Portuguese [22] stock markets
was also considered using entropy measures. However, the efficiency of the
Russian stock market has not yet been analyzed. In this paper, we present an
analysis of market efficiency based on the estimation of Shannon entropy for a
group of 18 stocks of Russian companies from five industries.

Our paper introduces four original contributions in the field. First, we
construct the method of filtering out heteroskedasticity and price staleness.
This filtering process helps to identify a true degree of market inefficiency.
Second, we calculate the degree of market inefficiency for the previous decade
using monthly intervals. We conclude that the degree of market inefficiency
for the Moscow Stock Exchange was greater than $80\%$. Third, we determine
which pair of stocks exhibits the largest amount of inefficiency, as measured
by estimating Shannon’s entropy on their high frequency price time series. We
show that months where the predictability of stock prices attains its maximum
cluster together. We find out what behavior of stocks repeats most often for
inefficient time periods. Finally, we estimate the closeness of price
movements using two measures of entropy. Based on these results, we cluster
together groups of stocks for which the efficient market hypothesis is
rejected, thus pointing out how market inefficiency display some dependence
from the financial sector they belong.

The article is organized as follows. Section 2 describes the dataset and the
methodology of filtering data regularities and calculating the Shannon
entropy. Section 3 presents the results on simulated and real data. Section 4
concludes the paper.

##  2 Materials and methods

###  2.1 Dataset

We study the Moscow Stock Exchange. We consider close prices aggregated at
one-minute time scale. In particular, we select only minutes of the main
trading session from 10:00 to 18:40. The time interval covers ten years from
2012 to 2021. The time period is divided into monthly time intervals. We take
18 companies, 16 of them are from five sectors: oil industry, metallurgy,
banks, telecommunications, electricity. All stocks are listed in the Table
1111There are 2520 trading days. Assuming that there are 520 minutes in each
trading day, there are 1310400 trading minutes in total. We use the Brownlees
and Gallo’s algorithm of an outlier detection [23]. See details in Appendix
A.1. All data are provided by Finam Holdings222https://www.finam.ru/.

Table 1: Stocks of Russian companies traded at Moscow Exchange. Ticker | Company | Sector | Size | Outliers
---|---|---|---|---
GAZP | Gazprom | Oil | 1307427 | 50
LKOH | Lukoil | Oil | 1287582 | 192
ROSN | Rosneft | Oil | 1270592 | 130
SNGS | Surgutneftegaz | Oil | 1211809 | 11
TATN | Tatneft | Oil | 1191390 | 174
SBER | Sberbank | Bank | 1309402 | 37
VTBR | VTB Bank | Bank | 1287330 | 0
CHMF | Severstal | Metal | 1214735 | 157
NLMK | Novolipetsk Steel | Metal | 1194324 | 58
GMKN | Nornikel | Metal | 1272769 | 197
MTLR | Mechel | Metal | 1084990 | 161
MAGN | Magnitogorsk Iron and Steel Works | Metal | 1106771 | 13
MTSS | Mobile TeleSystems | Telecommunications | 1153527 | 260
RTKM | Rostelecom | Telecommunications | 1140798 | 134
HYDR | RusHydro | Electric utility | 1252584 | 0
RSTI | Rosseti | Electricity | 1094244 | 0
AFLT | Aeroflot | Airline | 1083552 | 123
MGNT | Magnit | Food retailer | 1184223 | 544
For each company, we specify the ticker of stock, its sector, the size of
data, and the amount of outliers removed.

###  2.2 Apparent inefficiencies

To estimate a degree of market efficiency, we first should eliminate the known
patterns of predictability, such as a daily seasonality. Financial agents
operating in the market tend to trade less in the middle of a day. It is
reflected in prices, but again this pattern in trading volume should be
filtered out to detect genuine patterns of inefficiency. Other known
regularities are volatility clustering, price staleness, and microstructure
noise. See Appendix A for the guide of filtering out apparent inefficiencies.
The contribution of this article is devising a simple method for filtering
volatility clustering and price staleness. One of the methods to estimate
volatility is the exponentially weighted moving average (EWMA). It is
described in the next section.

###  2.3 EWMA

We define price returns as $r_{t}=\ln{\left(\frac{P_{t}}{P_{t-1}}\right)}$,
where $P_{t}$ is the last price available at time $t$ and $\ln()$ is the
natural logarithm. In order to estimate volatility $\sigma_{n}$, we apply
exponentially weighted moving average [24] of the values
$\mu_{1}^{-1}|r_{i}|$, $i<n$, where $\mu_{1}=\sqrt{\frac{2}{\pi{}}}$.

$\bar{\sigma}_{n}=Sig_{1}(\alpha,r_{n-1},\bar{\sigma}_{n-1})=\alpha\mu_{1}^{-1}|r_{n-1}|+(1-\alpha)\bar{\sigma}_{n-1}$
(1)

Here the fact that $E[|r_{n}|]=\mu_{1}\sigma_{n}$ is used and more weights are
given for the more recent data. An alternative formula based on the fact that
$E[r_{n}^{2}]=\sigma_{n}^{2}$ is

$\bar{\sigma}^{2}_{n}=Sig_{2}(\alpha,r_{n-1},\bar{\sigma}_{n-1})=\alpha
r_{n-1}^{2}+(1-\alpha)\bar{\sigma}_{n-1}^{2}.$ (2)

A large value of return increases the value of volatility. The current value
of volatility reflects all available values of returns and changes slowly if
the value of $\alpha$ is small. Usually, the smoothing parameter $\alpha$ is
taken close to $0$. For instance, $\alpha=0.05$ is taken in the article [9].
The value of the parameter $\alpha$ is set to be equal $0.12$ for in-sample
testing and $0.22$ for out-of-sample testing in [25]. Instead, Hunter [24]
suggests to use $\alpha=0.2\pm 0.1$. Using the principle of the best one-step
forecasting, the smoothing parameter is set to $0.06$ for the daily data and
to $0.03$ for the monthly data [26].

We follow the approach suggested by [26] (p. 97) to find the optimal value of
$\alpha$. The goal is to select the parameter $\alpha$, such that it minimizes
the value of $Er_{\sigma}=\sum_{i}(\bar{\sigma}_{i}^{2}-r_{i}^{2})^{2}$. In
order to minimize $Er_{\sigma}$ as a function of the only parameter
$0<\alpha<1$, we apply Brent’s algorithm [27]333The method is available in
Python by using the function scipy.optimize.minimize_scalar. Alternatively, we
could use the golden-section search [28] that requires the boundary of search
and the only parameter for the stopping criteria..

###  2.4 Estimation of price staleness

Let’s define an efficient price, $P^{e}$, as a continuous process following a
Geometric Brownian Motion.

$P^{e}_{t}=P^{e}_{0}+\int_{0}^{t}{\sigma_{s}P^{e}_{s}dW_{s}}$

An observed price moves along a discrete grid. Possible price values are
multiples of the tick size, $d$.

$P_{t}=d\cdot\left[\frac{P^{e}_{t}}{d}\right]$

If the efficient price changes insignificantly, the return of the rounded
price will be equal to 0. Analogically, if the return of rounded price is 0,
the return of efficient price has a value close to 0. We use the Equation 3 to
determine the probability that a 0-return is appearing due to rounding:

$p_{i}=erf(R_{i-1})+\frac{1}{R_{i-1}\sqrt{\pi}}(\exp{(-R_{i-1}^{2})}-1),$ (3)

where $R_{i}=\frac{d}{\bar{P}_{i}\bar{\sigma}_{i}\sqrt{2\Delta}}$ and $erf(x)$
is the Gaussian error function; $d$ is a tick size 444We estimate the tick
size using 2-steps procedure for each month. First, we find the amount of
significant digits in price. Then, we determine the most frequent increment in
ordered prices., $\Delta$ is a time step555The time step between the end and
start of the main trading session is set as 1 minute. Also, we consider any
time gap without trading more than 2 hours as the closure of the market. We
set the time step equal to 1 minute for these gaps., $\bar{P}$ is a rounded
price, and $\bar{\sigma}$ is an estimation of volatility [29]. It is obtained
by considering the probability that a price following a Geometric Brownian
Motion moves less than one tick size, assuming that price increments are
normally distributed.

There is another source of getting 0-returns, namely price staleness. Price
staleness is a regularity that means that a fundamental (efficient) price of
an asset is not updated because of economic reasons. Such a reason is, for
instance, a high transaction cost, which makes transactions unprofitable for
traders. See [13] for more details. The presence of price staleness in the
data implies the existence of excess 0-returns instead of values of returns of
efficient price. The excess 0-returns tend to reduce the estimation of
volatility. Therefore, we need to filter out 0-returns due to price staleness
and keep 0-returns due to rounding.

According to our methodology of filtering out excess 0-returns presented in
[29], we save 0-returns proportionally to the probability in Eq. 3 and set
other 0-returns as missing values. We adopt this methodology to estimate the
degree of price staleness together with volatility in the next section.

###  2.5 Modification of EWMA

In this Section, we present a modification of the EWMA that takes into
consideration the effect of price staleness. Our modification of the EWMA is
based on the suggestion to estimate volatility $\sigma_{n}$ as
$\bar{\sigma}_{n-1}$ (so setting $\alpha=0$), if the value of $r_{n-1}$ is
missing because of price staleness. That is, there is no new information from
returns to update the value of volatility.

Initially, the expected amount of 0-returns due to rounding is $N_{save}=0$.
Thus, each appearance of 0-returns does not affect the value of volatility. A
0-return is defined as a value due to rounding and is saved in the sequence if
the sum of all $p_{i}$ (Eq. 3) moves to a new integer value. Other details and
the algorithm of volatility estimation can be found in Appendix B.

We update the estimation of volatility and price staleness minute by minute.
This method has the clear advantage of making possible the online inference by
processing data in real time.

###  2.6 The Shannon entropy

The degree of market efficiency is assessed by computing the Shannon entropy.
The entropy of a source is an average measure of the randomness of its outputs
[30].

###### Definition 1

Let $X=\\{X_{1},X_{2},.\dots\\}$ be a stationary random process with a finite
alphabet $A$ and a measure $\mu$. An $n$-th order entropy of $X$ is

$H_{n}(\mu)=-\sum_{x_{1}^{n}\in A^{n}}\mu(x_{1}^{n})\log{\mu(x_{1}^{n})}$

with the convention $0\log{0}=0$. The process entropy (entropy rate) of $X$ is

$h(\mu)=\lim_{n\to\infty}\frac{H_{n}(\mu)}{n}.$

###  2.7 Discretization

The Shannon entropy is computed over a finite alphabet. To measure the Shannon
entropy, we need to keep the length of blocks of symbols, $k$, sufficiently
large. The predictable behavior of returns can be seen on blocks of greater
length and may not be noticeable on blocks of smaller length. For this reason,
we consider 3-symbols and 4-symbols discretizations using empirical quantiles.

$\begin{split}s^{(3)}_{t}=\begin{cases}1,r_{t}\leq\theta_{1},\\\
0,\theta_{1}<r_{t}\leq\theta_{2},\\\
2,\theta_{2}<r_{t},\end{cases}\end{split}\quad\begin{split}s^{(4)}_{t}=\begin{cases}0,r_{t}\leq
Q_{1},\\\ 1,Q_{1}<r_{t}\leq Q_{2},\\\ 2,Q_{2}<r_{t}\leq Q_{3},\\\
3,Q_{3}<r_{t},\end{cases}\end{split}$

where $\theta_{1}$ and $\theta_{2}$ are tertiles and $Q_{1}$, $Q_{2}$, $Q_{3}$
are quartiles. The tertiles divide data into three equal parts. The quartiles
divide data into four equal parts. $Q_{2}$ is also the median of the empirical
distribution of returns. For the later analysis, we will need a discretization
describing the behavior of a pair of stocks.

$\begin{split}s^{(p)}_{t}=\begin{cases}0,r^{(1)}_{t}\leq
m_{1}\&r^{(2)}_{t}\leq m_{2},\\\ 1,r^{(1)}_{t}\leq
m_{1}\&r^{(2)}_{t}>m_{2},\\\ 2,r^{(1)}_{t}>m_{1}\&r^{(2)}_{t}\leq m_{2},\\\
3,r^{(1)}_{t}>m_{1}\&r^{(2)}_{t}>m_{2},\\\ \end{cases}\end{split}$ (4)

where $r^{(1)}_{t}$ and $r^{(2)}_{t}$ are two time series of price returns and
$m_{1}$ and $m_{2}$ are their medians.

###  2.8 The estimation of entropy

Let $x_{1}^{n}\in A^{n}$ be the sequence of length $n$ generated by an ergodic
source $\mu$ from the finite alphabet $A$, where $x_{i}^{i+k-1}=x_{i}\dots
x_{i+k-1}$. There are possible missing values in the sequence generated
independently from $x_{1}^{n}$. We consider all blocks of length $k$ that do
not contain missing values. We take

$k=max(K:K<\lfloor\log(n_{b}(K))\rfloor,$ (5)

where $n_{b}(k)$ is the number of blocks of length $k$. The base of the
logarithm is the size of the alphabet $A$ (3 or 4).

For each $a_{1}^{k}\in A^{k}$ empirical frequencies are defined as

$f(a_{1}^{k}|x_{1}^{n})=\\#\\{i\in[1,n-k+1]:x_{i}^{i+k-1}=a_{1}^{k}\\}.$

Empirical frequencies are the actual amount of each block from $A^{k}$ in the
data. By considering an empirical k-block distribution as

$\hat{\mu}_{k}(a_{1}^{k}|x_{1}^{n})=\frac{f(a_{1}^{k}|x_{1}^{n})}{n_{b}},$ (6)

an empirical $k$-entropy is defined by

$\hat{H}_{k}(x_{1}^{n})=-\sum_{a_{1}^{k}}\hat{\mu}_{k}(a_{1}^{k}|x_{1}^{n})\log{(\hat{\mu}_{k}(a_{1}^{k}|x_{1}^{n}))}=\log(n_{b})-\frac{1}{n_{b}}\sum_{i=1}^{M}f_{i}\log{f_{i}}.$

The estimation of the process entropy is

$\hat{h}_{k}=\frac{\hat{H}_{k}}{k}.$

See [31] for the proof of the consistency of this estimator and [29] for the
case of missing values. Since the sequence is finite, the estimation of
entropy is underestimated. To remove this bias, we use the correction for the
entropy estimation introduced in [32, 33].

$\hat{H}_{k}^{G}=\log(n_{b})-\frac{1}{n_{b}}\sum_{i=1}^{M}f_{i}\log{\left(\exp{G(f_{i})}\right)},$
(7)

where the sequence $G(i)$ is defined recursively as

$\begin{split}G(1)&=-\gamma-\ln(2)\\\ G(2)&=2-\gamma-\ln(2)\\\
G(2n+1)&=G(2n)\\\ G(2n+2)&=G(2n)+\frac{2}{2n+1}\text{, }n\geq 1\end{split}$

with the Euler’s constant $\gamma=0.577215\dots$.

###  2.9 Detection of inefficiency

We need to do three steps to determine if the time interval is efficient or
not. First, we filter out apparent inefficiencies (see Appendix A). Then, we
estimate the entropy of the filtered return time series using Eq. 7. Finally,
we determine if the value of entropy is significantly low relative to the case
of perfect randomness. We detect inefficiency in the time interval using Monte
Carlo simulations. We regard a Brownian motion as absolutely unpredictable.
First, we define the length of sequences as $l=n_{b}(k)+k-1$. Then, we
simulate $10^{4}$ realizations of Brownian motions with Gaussian increments
and the length $l$. For each realization, we calculate entropy using 3- and
4-symbols discretizations. Then, we find the first percentile of the obtained
entropies for each discretization. These percentiles are the bounds of $99\%$
of the Confidence Interval (CI) for testing market efficiency. Finally, we
define an efficiency rate as the ratio of the entropy of the time interval and
the bound of CI. If the efficiency rate is less than $1$ for at least one type
of discretization, we define the time interval as inefficient. We provide
testing for inefficiency twice using different discretizations because the
unique testing may not be robust. See an example in Appendix C.

###  2.10 Kullback–Leibler divergence

In addition to estimating the entropy of one time series, we can also consider
the difference between two time series. The Kullback–Leibler divergence [34]
is used to measure similarity between two sequences. For two discrete
probability distributions $P$ and $Q$.

$KL(P|Q)=\sum_{i}p_{i}\log{\frac{p_{i}}{q_{i}}}$

We use $p_{i}$ and $q_{i}$ as empirical probabilities obtained in Eq 6. Since
the Kullback–Leibler divergence is asymmetric, we consider the distance
between two time series proposed in [35].

$D(P,Q)=\frac{KL(P|Q)}{H^{G}(P)}+\frac{KL(Q|P)}{H^{G}(Q)}$ (8)

The greater the distance $D(P,Q)$, the more probability distributions $P$ and
$Q$ differ.

##  3 Results

###  3.1 Simulations

The aim of this section is to assess the accuracy of the estimation of
volatility and the degree of price staleness. We will choose the method that
gives the least error of the estimation for further analysis on real data. We
take the following model of an observed price $\tilde{P}_{t}$, $t=1\dots 2N$.

$\begin{split}P_{t}&=\int_{0}^{t}{\sigma_{s}P_{s}dW^{1}_{s}}\\\
\tilde{P}_{i}&=P_{i}(1-B_{i})+\tilde{P}_{i-1}B_{i}\\\
pr_{t}&=pr_{0}+\int_{0}^{t}\mu_{s}ds+\int_{0}^{t}\nu
dW^{2}_{s}\end{split}\quad\begin{split}B_{i}=\begin{cases}1\text{ with
probability }pr_{i}\\\ 0\text{ with probability
}1-pr_{i}\end{cases}\end{split}$

where $W^{1}$ and $W^{2}$ are two independent Brownian motions with the length
of $2N$, $N=10^{5}$, price $P_{0}=100$, and $\nu=10^{-4}$. $B=1$ stands for
the case when price is not updated due to price staleness (see [13, 36]).
Prices are rounded to two digits, thus the tick size is $d=0.01$. We consider
4 choices for $pr_{t}$ and $\sigma_{t}$ listed below.

$\begin{split}pr_{t}^{1}&=0\\\ pr_{t}^{2}&=0.1+\int_{0}^{t}\nu dW^{2}_{s}\\\
pr_{t}^{3}&=0.2+\int_{0}^{t}\nu dW^{2}_{s}\\\
pr_{t}&=0.2+\int_{0}^{t}\mu^{4}_{s}ds+\int_{0}^{t}\nu dW^{2}_{s}\\\
\mu^{4}_{t}&=0.8\pi/N\cos(8t\pi/N)\\\ \sigma_{t}^{1}&=5\times 10^{-4}\\\
\sigma_{t}^{2}&\sim ARCH(1.75\times 10^{-7},0.2,0.1)\\\ \sigma_{t}^{3}&\sim
GARCH(1.25\times 10^{-8},0.1,0.85)\\\ \sigma_{t}^{4}&\sim GARCH(1.25\times
10^{-8},0.15,0.8)\\\ \end{split}$

We divide data into two equal parts with the size $N$. The first part is a
training set for finding optimal values of $\alpha$ from Equations 1 and 2.
The second part is a testing set for calculating errors represented in Tables
2 and 3 below. We compare two methods that use $Sig_{1}$ and $Sig_{2}$ for
volatility estimation. We set a fixed value of alpha, $\alpha=0.05$, as a
benchmark for the comparison. We also apply non-modified EMWA estimation from
Section 2.3 with selected optimal value of $\alpha$ to show the contribution
of 0-filtering to the accuracy of volatility estimation. We simulate $10^{3}$
prices for each model.

Table 2 represents a mean absolute percentage error (MAPE) that is
$\frac{1}{N}\sum_{i}|\frac{\bar{\sigma}_{i}-\sigma_{i}}{\sigma_{i}}|$ for six
different approaches. These approaches differ in the choice of a function for
volatility, the value of $\alpha$, and the presence of missing values. Table 3
represents three values for each of two methods using $Sig_{1}$ and $Sig_{2}$
for the volatility estimation. The first value is the optimal value of
$\alpha$. The second is $Er_{N}=|\frac{N_{round}N_{A}}{N_{0}N}-1|$ where
$N_{A}$ is the amount of remaining non-missing returns, $N_{round}$ is the
amount 0-returns that would appear due to rounding (before adding the effect
of staleness in the simulated data); $N_{A}$ is the amount of non-missing
returns; and $N_{0}$ is the amount of 0-returns. $Er_{N}$ represents the
absolute error of the proportion of 0-returns that remain in the data and are
defined as 0-returns due to rounding. The third value is the proportion of
data set as missing values, that is $1-\frac{N_{A}}{N}$.

It can be seen from Table 2 that the method that more often gives the lowest
value of MAPE is with fixed $\alpha=0.05$ and $Sig_{1}$ used for volatility
estimation. Moreover, for almost all cases, 0-filtering makes the volatility
estimate more accurate. The error of the amount of 0-returns due to rounding
is smaller for the function $Sig_{1}$ than for the function $Sig_{2}$ for all
16 cases. After the comparison of the two functions of volatility estimation,
we choose $Sig_{1}$ that uses absolute values of returns. For the rest of the
paper, we fix the value of $\alpha$ as $0.05$ for the simplicity of further
analysis.

###  3.2 Moscow Stock Exchange

We define a degree of inefficiency as the fraction of months which are defined
as inefficient according to Section 2.9. The degree of inefficiency for the
chosen group of stocks traded at Moscow Exchange is 0.823.666In our previous
work, [29] we found that the degree of inefficiency for the U.S. ETF market is
about 0.11 for monthly time intervals and the 3-symbols discretization only.
The degree of inefficiency for each stock and discretization is presented in
Table 4. We notice that the 4-symbols discretization contributes to the larger
amount of inefficient months than the 3-symbols discretization. That is, the
4-symbol discretization appears to have a more predictable structure than the
3-symbols discretization.

Figure 1 shows the minimum value of efficiency rate among all months for each
stock.

There are two most notable deviations from 1 for stocks MLTR (Mechel, mining
and metals company) and RSTI (Rosseti, power company). We investigate them in
the next section. For the other 16 stocks, the minimum value of efficiency
rate is attained for the stock AFLT and is equal to 0.933 (0.964) for 3 (4)
symbols.

Table 2: Results on volatility estimation. model |  MAPE, method $v_{1}$ |  MAPE, $v_{2}$ |  MAPE with $\alpha=0.05$, $v_{1}$ |  MAPE with $\alpha=0.05$, $v_{2}$ |  MAPE w/o 0-filtering, $v_{1}$ |  MAPE w/o 0-filtering, $v_{2}$
---|---|---|---|---|---|---
$\sigma_{1},pr_{1}$ |  $\underset{(0.0007,0.0507)}{0.0193}$ |  $\underset{(0.0014,0.0406)}{\textbf{0.017}}$ |  $\underset{(0.0955,0.0995)}{0.0975}$ |  $\underset{(0.0878,0.0915)}{0.0897}$ |  $\underset{(0.0007,0.0507)}{0.0193}$ |  $\underset{(0.0014,0.0406)}{\textbf{0.017}}$
$\sigma_{1},pr_{2}$ |  $\underset{(0.0245,0.1017)}{\textbf{0.0607}}$ |  $\underset{(0.0293,0.1057)}{0.0629}$ |  $\underset{(0.093,0.0972)}{0.095}$ |  $\underset{(0.0893,0.0936)}{0.0914}$ |  $\underset{(0.0459,0.131)}{0.0862}$ |  $\underset{(0.0294,0.1154)}{0.0674}$
$\sigma_{1},pr_{3}$ |  $\underset{(0.0333,0.1278)}{\textbf{0.0737}}$ |  $\underset{(0.033,0.1338)}{0.0756}$ |  $\underset{(0.0928,0.0971)}{0.0948}$ |  $\underset{(0.0894,0.094)}{0.0915}$ |  $\underset{(0.0917,0.1863)}{0.138}$ |  $\underset{(0.0368,0.1592)}{0.0888}$
$\sigma_{1},pr_{4}$ |  $\underset{(0.0323,0.1213)}{\textbf{0.0716}}$ |  $\underset{(0.0354,0.1268)}{0.0739}$ |  $\underset{(0.0926,0.0973)}{0.0949}$ |  $\underset{(0.089,0.0937)}{0.0913}$ |  $\underset{(0.1022,0.1875)}{0.1404}$ |  $\underset{(0.0405,0.1516)}{0.0873}$
$\sigma_{2},pr_{1}$ |  $\underset{(0.1082,0.121)}{0.1121}$ |  $\underset{(0.1146,0.1244)}{0.1183}$ |  $\underset{(0.1438,0.1481)}{0.1459}$ |  $\underset{(0.1422,0.147)}{0.1446}$ |  $\underset{(0.108,0.1207)}{\textbf{0.1118}}$ |  $\underset{(0.1144,0.1243)}{0.1179}$
$\sigma_{2},pr_{2}$ |  $\underset{(0.1163,0.1715)}{0.1359}$ |  $\underset{(0.1237,0.1765)}{0.1411}$ |  $\underset{(0.1439,0.1487)}{0.1462}$ |  $\underset{(0.1457,0.1526)}{0.1489}$ |  $\underset{(0.1043,0.1819)}{\textbf{0.1341}}$ |  $\underset{(0.1193,0.1832)}{0.1407}$
$\sigma_{2},pr_{3}$ |  $\underset{(0.1198,0.1958)}{\textbf{0.146}}$ |  $\underset{(0.1266,0.1981)}{0.1519}$ |  $\underset{(0.1449,0.1499)}{0.1473}$ |  $\underset{(0.1464,0.1534)}{0.1496}$ |  $\underset{(0.1196,0.2271)}{0.1649}$ |  $\underset{(0.123,0.222)}{0.1589}$
$\sigma_{2},pr_{4}$ |  $\underset{(0.1205,0.1912)}{\textbf{0.146}}$ |  $\underset{(0.1256,0.1986)}{0.15}$ |  $\underset{(0.1447,0.1498)}{0.1472}$ |  $\underset{(0.1463,0.1532)}{0.1494}$ |  $\underset{(0.1274,0.2261)}{0.1696}$ |  $\underset{(0.1223,0.2239)}{0.1571}$
$\sigma_{3},pr_{1}$ |  $\underset{(0.1446,0.1513)}{0.1479}$ |  $\underset{(0.1442,0.1505)}{\textbf{0.1473}}$ |  $\underset{(0.1467,0.1522)}{0.1495}$ |  $\underset{(0.1442,0.1502)}{\textbf{0.1473}}$ |  $\underset{(0.1446,0.1513)}{0.1479}$ |  $\underset{(0.1441,0.1503)}{\textbf{0.1472}}$
$\sigma_{3},pr_{2}$ |  $\underset{(0.1485,0.1891)}{0.1592}$ |  $\underset{(0.1508,0.1857)}{0.1613}$ |  $\underset{(0.149,0.1574)}{\textbf{0.1529}}$ |  $\underset{(0.1497,0.1598)}{0.1546}$ |  $\underset{(0.144,0.2033)}{0.1622}$ |  $\underset{(0.1491,0.1978)}{0.1628}$
$\sigma_{3},pr_{3}$ |  $\underset{(0.1528,0.2048)}{0.1681}$ |  $\underset{(0.1556,0.2178)}{0.171}$ |  $\underset{(0.1525,0.1616)}{\textbf{0.1567}}$ |  $\underset{(0.1536,0.1639)}{0.1584}$ |  $\underset{(0.154,0.2477)}{0.1904}$ |  $\underset{(0.1546,0.2464)}{0.1815}$
$\sigma_{3},pr_{4}$ |  $\underset{(0.1527,0.1997)}{0.1668}$ |  $\underset{(0.1555,0.2178)}{0.1701}$ |  $\underset{(0.1525,0.1613)}{\textbf{0.1568}}$ |  $\underset{(0.1537,0.1633)}{0.1583}$ |  $\underset{(0.1591,0.246)}{0.192}$ |  $\underset{(0.1556,0.2455)}{0.181}$
$\sigma_{4},pr_{1}$ |  $\underset{(0.1856,0.1952)}{0.1897}$ |  $\underset{(0.1838,0.1911)}{\textbf{0.1873}}$ |  $\underset{(0.1844,0.1918)}{0.1881}$ |  $\underset{(0.1879,0.1968)}{0.1924}$ |  $\underset{(0.1856,0.1952)}{0.1897}$ |  $\underset{(0.1837,0.1911)}{0.1873}$
$\sigma_{4},pr_{2}$ |  $\underset{(0.1906,0.2454)}{0.2035}$ |  $\underset{(0.1921,0.2474)}{0.2057}$ |  $\underset{(0.1891,0.2022)}{\textbf{0.1954}}$ |  $\underset{(0.1961,0.2119)}{0.2037}$ |  $\underset{(0.1836,0.2617)}{0.2049}$ |  $\underset{(0.1902,0.2642)}{0.2079}$
$\sigma_{4},pr_{3}$ |  $\underset{(0.1965,0.2623)}{0.2146}$ |  $\underset{(0.1996,0.2757)}{0.2166}$ |  $\underset{(0.1951,0.2077)}{\textbf{0.2015}}$ |  $\underset{(0.2026,0.2177)}{0.2101}$ |  $\underset{(0.1912,0.307)}{0.2318}$ |  $\underset{(0.1988,0.3082)}{0.2294}$
$\sigma_{4},pr_{4}$ |  $\underset{(0.1967,0.2591)}{0.214}$ |  $\underset{(0.1986,0.2689)}{0.2155}$ |  $\underset{(0.1951,0.2088)}{\textbf{0.2013}}$ |  $\underset{(0.2023,0.2185)}{0.2097}$ |  $\underset{(0.1976,0.3064)}{0.2338}$ |  $\underset{(0.1988,0.306)}{0.2286}$
The first column indicated a model. Columns 2 and 3 represent results for two methods described in Section 2.5. Columns 4 and 5 are for the same methods but with the fixed value of $\alpha$. Columns 6 and 7 shows the error of the standard EMWA approach with the optimal selected value of $\alpha$. $95\%$ CI is presented below each averaged statistic. $v_{1}$ stands for using $Sig_{1}$; $v_{2}$ stands for using $Sig_{2}$. Table 3: Results on filtering out 0-returns. model |  $\alpha$ for $v_{1}$ |  $\alpha$ for $v_{2}$ |  $Er_{N}$, $v_{1}$ |  $Er_{N}$, $v_{2}$ |  Fraction of data deleted, $v_{1}$ |  Fraction of data deleted, $v_{2}$
---|---|---|---|---|---|---
$\sigma_{1},pr_{1}$ |  $\underset{(0.0,0.0137)}{0.0027}$ |  $\underset{(0.0,0.0103)}{0.0022}$ |  $\underset{(0.0,0.0)}{0.0006}$ |  $\underset{(0.0,0.0259)}{0.0015}$ |  $\underset{(0.0,0.0)}{0.0001}$ |  $\underset{(0.0,0.0052)}{0.0003}$
$\sigma_{1},pr_{2}$ |  $\underset{(0.0033,0.0569)}{0.0228}$ |  $\underset{(0.0044,0.067)}{0.0259}$ |  $\underset{(0.0004,0.026)}{0.0094}$ |  $\underset{(0.0005,0.0295)}{0.011}$ |  $\underset{(0.0814,0.3244)}{0.2005}$ |  $\underset{(0.0818,0.3247)}{0.2008}$
$\sigma_{1},pr_{3}$ |  $\underset{(0.006,0.0902)}{0.0335}$ |  $\underset{(0.0058,0.1063)}{0.0379}$ |  $\underset{(0.0005,0.0288)}{0.0106}$ |  $\underset{(0.0005,0.0336)}{0.0121}$ |  $\underset{(0.2474,0.481)}{0.3661}$ |  $\underset{(0.246,0.4797)}{0.3659}$
$\sigma_{1},pr_{4}$ |  $\underset{(0.0056,0.0824)}{0.0314}$ |  $\underset{(0.007,0.0966)}{0.036}$ |  $\underset{(0.0004,0.0283)}{0.0104}$ |  $\underset{(0.0007,0.0355)}{0.0122}$ |  $\underset{(0.2521,0.4717)}{0.3628}$ |  $\underset{(0.2515,0.4713)}{0.3626}$
$\sigma_{2},pr_{1}$ |  $\underset{(0.0,0.0161)}{0.0039}$ |  $\underset{(0.0006,0.0146)}{0.0037}$ |  $\underset{(0.0,0.0438)}{0.0149}$ |  $\underset{(0.0,0.0586)}{0.035}$ |  $\underset{(0.0,0.0092)}{0.0029}$ |  $\underset{(0.0,0.0134)}{0.0067}$
$\sigma_{2},pr_{2}$ |  $\underset{(0.0059,0.0903)}{0.035}$ |  $\underset{(0.0054,0.1021)}{0.0367}$ |  $\underset{(0.0012,0.0448)}{0.0209}$ |  $\underset{(0.0039,0.0601)}{0.0319}$ |  $\underset{(0.0856,0.3249)}{0.2016}$ |  $\underset{(0.0878,0.3263)}{0.2032}$
$\sigma_{2},pr_{3}$ |  $\underset{(0.0079,0.1326)}{0.0489}$ |  $\underset{(0.0091,0.1476)}{0.0556}$ |  $\underset{(0.001,0.0473)}{0.0217}$ |  $\underset{(0.0022,0.0603)}{0.0275}$ |  $\underset{(0.25,0.4835)}{0.3706}$ |  $\underset{(0.2518,0.4836)}{0.371}$
$\sigma_{2},pr_{4}$ |  $\underset{(0.0093,0.1268)}{0.049}$ |  $\underset{(0.0082,0.1484)}{0.0525}$ |  $\underset{(0.0012,0.0443)}{0.0206}$ |  $\underset{(0.0013,0.0571)}{0.0274}$ |  $\underset{(0.2495,0.4695)}{0.3645}$ |  $\underset{(0.2491,0.4692)}{0.3651}$
$\sigma_{3},pr_{1}$ |  $\underset{(0.0349,0.0527)}{0.0424}$ |  $\underset{(0.0392,0.0603)}{0.048}$ |  $\underset{(0.0,0.0337)}{0.0034}$ |  $\underset{(0.0,0.0402)}{0.0089}$ |  $\underset{(0.0,0.0067)}{0.0007}$ |  $\underset{(0.0,0.0085)}{0.0018}$
$\sigma_{3},pr_{2}$ |  $\underset{(0.0267,0.1456)}{0.0672}$ |  $\underset{(0.0249,0.1592)}{0.0767}$ |  $\underset{(0.0006,0.0404)}{0.0155}$ |  $\underset{(0.0009,0.0551)}{0.02}$ |  $\underset{(0.0826,0.3248)}{0.1995}$ |  $\underset{(0.0826,0.3251)}{0.1997}$
$\sigma_{3},pr_{3}$ |  $\underset{(0.0264,0.1734)}{0.0825}$ |  $\underset{(0.0301,0.2371)}{0.0985}$ |  $\underset{(0.0011,0.0477)}{0.018}$ |  $\underset{(0.0008,0.0702)}{0.0243}$ |  $\underset{(0.2444,0.4746)}{0.3678}$ |  $\underset{(0.2421,0.4751)}{0.3671}$
$\sigma_{3},pr_{4}$ |  $\underset{(0.0266,0.163)}{0.0788}$ |  $\underset{(0.0325,0.2362)}{0.0969}$ |  $\underset{(0.001,0.0463)}{0.0178}$ |  $\underset{(0.0007,0.067)}{0.0222}$ |  $\underset{(0.2466,0.476)}{0.3623}$ |  $\underset{(0.2486,0.4739)}{0.3615}$
$\sigma_{4},pr_{1}$ |  $\underset{(0.0696,0.1037)}{0.0819}$ |  $\underset{(0.0757,0.119)}{0.0904}$ |  $\underset{(0.0,0.0248)}{0.0013}$ |  $\underset{(0.0,0.0329)}{0.0047}$ |  $\underset{(0.0,0.0052)}{0.0003}$ |  $\underset{(0.0,0.0075)}{0.0011}$
$\sigma_{4},pr_{2}$ |  $\underset{(0.0534,0.2359)}{0.1132}$ |  $\underset{(0.0576,0.2925)}{0.1339}$ |  $\underset{(0.0007,0.0564)}{0.0185}$ |  $\underset{(0.0008,0.087)}{0.0265}$ |  $\underset{(0.0765,0.3287)}{0.1993}$ |  $\underset{(0.077,0.3257)}{0.1982}$
$\sigma_{4},pr_{3}$ |  $\underset{(0.0557,0.2678)}{0.1338}$ |  $\underset{(0.0597,0.3734)}{0.1596}$ |  $\underset{(0.0009,0.0621)}{0.0214}$ |  $\underset{(0.001,0.1119)}{0.0321}$ |  $\underset{(0.2419,0.4823)}{0.3687}$ |  $\underset{(0.2378,0.4817)}{0.3669}$
$\sigma_{4},pr_{4}$ |  $\underset{(0.0571,0.263)}{0.1317}$ |  $\underset{(0.0613,0.3541)}{0.1556}$ |  $\underset{(0.0008,0.0667)}{0.0211}$ |  $\underset{(0.0011,0.108)}{0.0315}$ |  $\underset{(0.2599,0.4823)}{0.3641}$ |  $\underset{(0.2564,0.4822)}{0.3625}$
Values of $\alpha$, errors of the number of 0-returns due to rounding, and
fraction of data set as missing values. The first column indicated a model.
$95\%$ CI is presented below each averaged statistic. $v_{1}$ stands for using
$Sig_{1}$; $v_{2}$ stands for using $Sig_{2}$. Figure 1: Minimum of efficiency
rates for 18 stocks using 3- and 4-symbols discretizations.

####  3.2.1 Analysis of MLTR and RSTI

We plot the values of efficiency rates for monthly intervals for the MLTR and
RSTI stocks. See Fig. 2 and Fig. 3.

Figure 2: Efficiency rate for the MLTR stock using 3- and 4-symbols
discretizations. Figure 3: Efficiency rate for the RSTI stock using 3- and
4-symbols discretizations.

Both types of discretization show coherent results. For MLTR, there are two
notable decreases in the efficiency rates at the beginning of 2014 and in the
middle of 2016. For both types of discretizations, eight months with the
lowest efficiency rate (in the ascending order of time) are Jan-Feb and May-
Oct of 2014. For each month we write down the most frequent block of symbols
in Table 5. Note that block $1111$ for the 4-symbols discretization appears as
the most frequent for 6 months out of 8 for MLTR. The block denotes a slight
decrease in the price for 4 minutes in a row. The meaning of the last two
columns is discussed later.

For RSTI, there are two sharp decreases in 2014 and 2015. There are 11 months
that have the lowest efficiency rates in common for both discretizations.
These months are Apr-Sep of 2014 and Jun-Oct of 2015. Note that these
inefficient months cluster together and are not distributed uniformly among
the entire time period of 10 years. This is the signal of a market condition
that affects the inefficiency of the stocks for more than one month.

We construct a simple trading strategy on discretized returns to test the
predictability of future returns. We consider blocks of length 4 obtained by
the 4-symbols discretization. For each month, we divide blocks into two equal
parts. The discretization is made using only the first part of a month. We
consider the sequences of the first 3 symbols of each block. If the empirical
probability of getting 0 or 1 after the sequence of 3 symbols in the first
part of the month, this sequence is from group D (decreasing). If the
empirical probability of getting 2 or 3 after the sequence of 3 symbols, this
sequence is from group I (increasing). Then, for the second part of the month,
we determine a success if symbols 0 or 1 follow a sequence from group D or if
symbols 2 or 3 follow a sequence from group I. Then, we calculate the fraction
of successes. Thus, it is the probability of making a profit: sell after group
D or buy after group I. In the case of market efficiency, this probability is
equal to 0.5. For example, we expect that after $111$ the next symbol would be
$1$ according to Table 5. That is, after this block, a trader can sell a
stock. The fourth column of the Table 5 shows the results for filtered return
time series. The fifth column stands for the original return time series.

For all cases the probability is greater than 0.5. Obviously, the
probabilities for the original return time series are greater than those for
the filtered return time series. The reason is that predictability for the
original return time series follows from the sources of apparent
inefficiencies.

The same analysis is done for the RSTI stock. Eleven months with the lowest
efficiency rates are presented in Table 6. For the RSTI stock, the simple
trading strategy gives the fraction of successes (of predicting increases and
decreases of the price) greater than $0.5$ for all 11 months. The frequent
behavior of the price of RSTI during the chosen months is a slight increase in
price for several minutes in a row denoted by symbol 2.

Table 4: The degree of inefficiency for each stock. Ticker | Degree of inefficiency | For 3 symbols only | For 4 symbols only
---|---|---|---
GAZP | 0.725 | 0.392 | 0.675
LKOH | 0.65 | 0.342 | 0.542
ROSN | 0.742 | 0.392 | 0.708
SNGS | 0.725 | 0.4 | 0.625
TATN | 0.617 | 0.392 | 0.525
SBER | 0.725 | 0.433 | 0.658
VTBR | 0.842 | 0.592 | 0.792
CHMF | 0.858 | 0.55 | 0.692
NLMK | 0.8 | 0.467 | 0.692
GMKN | 0.733 | 0.475 | 0.608
MTLR | 0.992 | 0.783 | 0.975
MAGN | 0.833 | 0.65 | 0.758
MTSS | 0.967 | 0.7 | 0.942
RTKM | 0.942 | 0.683 | 0.908
HYDR | 0.892 | 0.75 | 0.8
RSTI | 0.917 | 0.742 | 0.875
AFLT | 0.983 | 0.775 | 0.95
MGNT | 0.842 | 0.667 | 0.742
Fraction of inefficient months using 3-symbols and 4-symbols discretization. Table 5: The most frequent blocks appearing for the Stock MLTR and probabilities of success. Months of 2014 |  The most frequent block, 3-s |  The most frequent block, 4-s |  prob. of success, filtered |  prob. of success, original
---|---|---|---|---
Jan. |  00000 |  1111 |  0.64 |  0.75
Feb. |  00000 |  2222 |  0.64 |  0.74
May |  00000 |  1111 |  0.61 |  0.73
June |  22222 |  1111 |  0.60 |  0.73
July |  11111 |  1111 |  0.62 |  0.74
Aug. |  00000 |  1111 |  0.61 |  0.76
Sep. |  00000 |  1111 |  0.63 |  0.74
Oct. |  120120 |  0303 |  0.55 |  0.6
The first column represents months with the lowest efficiency rates. Columns 2 and 3 are the most frequent blocks in 3- and 4-symbols discretization. Columns 4 and 5 are the probability of the success of the simple trading strategy for filtered and original price returns. Table 6: The most frequent blocks appearing for the Stock RSTI and probabilities of success. Months |  The most frequent block, 3-s |  The most frequent block, 4-s |  prob. of success, filtered |  prob. of success, original
---|---|---|---|---
Apr.2014 |  212121 |  0111 |  0.63 |  0.77
May.2014 |  00000 |  1111 |  0.61 |  0.73
June.2014 |  00000 |  1111 |  0.6 |  0.73
July.2014 |  00000 |  2222 |  0.62 |  0.74
Aug.2014 |  00000 |  2222 |  0.61 |  0.76
Sep.2014 |  000000 |  22222 |  0.63 |  0.74
June.2015 |  00000 |  2222 |  0.54 |  0.61
July.2015 |  00000 |  1111 |  0.55 |  0.6
Aug.2015 |  00000 |  2222 |  0.54 |  0.6
Sep.2015 |  00000 |  2222 |  0.55 |  0.61
Oct.2015 |  11111 |  0111 |  0.56 |  0.62
The first column represents months with the lowest efficiency rates. Columns 2
and 3 are the most frequent blocks in 3- and 4-symbols discretization. Columns
4 and 5 are the probability of the success of the simple trading strategy for
filtered and original price returns.

The simple trading strategy is an illustrative example of market inefficiency.
In fact, such a strategy could result in no profit when used in practice
because it does not take into account the costs of transaction and other
trading frictions. Moreover, the filtering of daily seasonality pattern is
made by using the whole period of analysis. That is, this method cannot be
applied in real time. Finally, we consider blocks containing only observed
returns, by neglecting the missing values from the analysis. Thus, the
application of such a strategy in practice should be integrated with the case
when a missing value follows a sequence of 3 symbols.

###  3.3 Stock market clustering

Most of the month-long time intervals are identified as inefficient. But is
there some dependence between two stocks that are inefficient at the same
time?

####  3.3.1 Kullback–Leibler distance

We measure the similarity of discretized filtered returns by using the
Kullback–Leibler (KL) distance (Eq. 8). We use $k$, the length of blocks, as
the maximum value suitable for both sequences according to Eq. 5. The
4-symbols discretization is used. The Kullback–Leibler divergence $DL(P|Q)$ is
calculated using empirical frequencies. The entropy rates are calculated using
Eq. 7. Using the Kullback–Leibler distance for all pairs of stocks, we cluster
them in three groups using hierarchical clustering with UPGMA algorithm
[37]777This algorithm is implemented using the python function
cluster.hierarchy.dendrogram with the argument distance=average.. The result
is in Fig. 4. Combining companies into one cluster means that their stocks
have a common behavior that is not related to the value of volatility, the
degree of price staleness and the structure of microstructure noise. It can be
seen that banks and oil companies are clustered together (right). There is a
group of four stocks RTKM, HYDR, AFLT, MGNT, that have nothing in common at
first glance. The remaining group (left) mainly consists of metallurgy
companies. However, there is no visible distinction between the stocks of
banks and oil companies. According to the clustering tree, two
telecommunications companies differ significantly, as well as electricity
companies.

Finally, two stocks with the lowest efficiency rates, RSTI and MLTR, are the
furthest (in the sense of KL distance) from any other stock. That is, there
are no stocks that behave similarly to these two stocks.

Figure 4: Hierarchical clustering tree using KL distance. The threshold for
clustering into groups is 0.035.

####  3.3.2 Entropy of co-movement

Now, we consider another measure of difference between two stocks, the entropy
of co-movement. We calculate the Shannon entropy of the discretization
describing the movement of a pair of prices presented in Eq. 4. We consider
only minutes that are in common for both stocks. For these minutes we consider
values of residuals obtained after ARMA fitting. The result is in Fig. 5.

Two companies related to telecommunications are a separate cluster. Three
metallurgy companies MAGN, CHMF, NLMK also cluster together. Stocks relating
to oil and bank companies form the other cluster. The same cluster, with the
exception of the TATN (oil industry), was also formed in the previous section.
The ”closeness” of stocks GAZP and SBER is detected either in this and in the
previous section. The three stocks on the left that join other stock clusters
last are the stocks with the lowest efficiency rates.

Some clusters may form on the basis that companies belong to the same
industry. The division of companies into industries is noticeable from the
dendrogram in Figure 5. However, this criterion does not explain all clusters.
For instance, GMKN from metallurgy is in the cluster of oil companies and
banks.

Figure 5: Hierarchical clustering tree using the entropy of co-movement. The
threshold for clustering into groups is 0.989.

##  4 Conclusions and discussion

We have investigated the predictability of the Moscow Stock Exchange. We are
interested in a measure of market inefficiency that is not related to known
sources of regularity in financial time series. Usually, these sources are not
filtered out and, accordingly, their impact is taken into account in the
degree of price predictability (see e.g. [10, 11, 12]).

We have focused on two sources of regularity, namely volatility clustering and
price staleness [13]. Filtering of volatility clustering was made in [9] by
estimating volatility using the exponentially weighted moving average. We have
developed a modification of the volatility estimation by taking into
consideration the effect of price staleness. Price staleness produces excess
0-returns that affect the estimation of volatility. Another approach of
estimating volatility in the case of presence 0-returns was proposed in [14]
where all 0-returns are reevaluated during an expectation-maximization
algorithm. In our approach, we separate 0-returns that may have resulted from
rounding and from price staleness. Thus, we also filter out apparent
inefficiency due to price staleness. The advantage of our approach is
simplicity: there is only one parameter in the method which can be optimized
using historical data. Our approach combining the estimates of volatility and
the degree of staleness can be used for real-time analysis since only past
observations of time series are used.

We used the Shannon entropy as a measure of randomness to calculate a degree
of inefficiency of the Moscow Stock Exchange. We used two types of the
discretization of return time series to test efficiency more reliably for each
month. The 4-symbols discretization helps to find more price movements that
lead to market inefficiency than the 3-symbols discretization. There are
$80\%$ of months over the period from 2012 to 2021 that are defined as
inefficient. Even after filtering out all sources of apparent inefficiency,
most of the months contain signals of market inefficiency.

We selected two stocks that exhibit the lowest efficiency rates. We have shown
that the most inefficient months are grouped together. We have shown that, for
such months, discretized price returns before and after filtering out apparent
inefficiencies are predictable.

Finally, we used two methods to cluster stocks using filtered return time
series. Inspired by [35], we computed the Kullback–Leibler distance between
stocks and grouped stocks into three clusters. We also introduced the entropy
of co-movement of two stocks. In this case, stock prices display common
patterns that have an interpretation in terms of the sector the stocks belong
to. We also noticed that the stocks of banks and oil companies were linked to
each other. One possible improvement to stock clustering is to modify the
entropy of co-movement such that it is possible to define a proper distance
function. This is left for future research.

The proposed method for measuring market efficiency using the Shannon entropy
can be applied in other markets of different countries. In this work, we use
monthly time intervals for entropy calculation. Our future work will be
related to the optimization of the length of return time series. One of the
problems is to find a significant decrease in entropy without using Monte
Carlo simulations. We also plan to switch to a higher frequency (less than one
minute) to analyze the predictability of financial time series.

##  Appendix A Data cleaning and whitening

###  A.1 Outliers

We use the method of an outlier detection introduced in [23]. The algorithm
finds price values that are too far from the mean in relation to the standard
deviation. The algorithm deletes a price $P_{i}$ if

$|P_{i}-\bar{P}_{i}(k)|\geq cs_{i}(k)+\gamma,$

where $\bar{P}_{i}(k)$ and $s_{i}(k)$ are respectively a $\delta$-trimmed
sample mean and the standard deviation of the $k$ price records closest to
time $i$. The $\delta\%$ of the lowest and the $\delta\%$ of the highest
observations are discarded when the mean and the standard deviation are
calculated from the sample. The parameters are
$k=20,\delta=5,c=5,\gamma=0.05$.

###  A.2 Stock Splits

We check the condition $|r|>0.2$ in the return series to detect unadjusted
splits888A split is a change in the number of company’s shares and in the
price of the single share, such that a market capitalization does not change..
There are no unadjusted splits found.

###  A.3 Intraday Volatility Pattern

The volatility of intraday returns has periodic behavior. The volatility is
higher near the opening and the closing of the market. It shows an U-shaped
profile every day. The intraday volatility pattern from the return series is
filtered by using the following model. We define deseasonalized returns as

$\tilde{R}_{d,t}=\frac{R_{d,t}}{\xi_{t}},$

where

$\xi_{t}=\frac{1}{N_{days}}\sum_{d^{\prime}}\frac{|R_{d^{\prime},t}|}{s_{d^{\prime}}},$

$R_{d,t}$ is the raw return of day $d$ and intraday time $t$, $s_{d}$ is the
standard deviation of absolute returns of day $d$, $N_{days}$ is the number of
days in the sample.

###  A.4 Heteroskedasticity

Different days have different levels of the deviation of the deseasonalized
returns $\tilde{R}$. In order to remove this heteroskedasticity, we estimate
the volatility $\bar{\sigma}_{t}$ in Appendix B. We define the standardized
returns by

$r_{t}=\frac{\tilde{R}_{t}}{\bar{\sigma}_{t}}.$

###  A.5 Price staleness

If a transaction cost is high, the price is updated less frequently, even if
trading volume is not zero. This effect is called price staleness and is
discussed in Section 2.4. We identify 0-returns appearing due to rounding (and
not due to price staleness) using the Equation 3. Other 0-returns are set as
missing values as shown in Appendix B.

###  A.6 Microstructure noise

The last step in filtering apparent inefficiencies is filtering out
microstructure noise. The microstructure effects are caused by transaction
costs and price rounding. We consider the residuals of an ARMA(P,Q) model of
the standardized returns after filtering out 0-returns. We apply the
methodology introduced in [38] to find the residuals of an ARMA(P,Q) model by
using the Kalman filter. We select the values of $P$ and $Q$ that minimize the
value of BIC [39], so that $P+Q<6$. The values of $P$ and $Q$ are chosen for
each calendar year and are used for the next year. For the year 2012 we select
$P=0$ and $Q=1$ corresponding to an MA(1) model.

##  Appendix B Algorithm

The aim of the algorithm is to estimate volatility and filter out excess
0-returns due to price staleness. Some 0-returns appear due to price rounding.
These 0-returns will be saved in the data. First, we set the number of
0-returns ”to save” $N_{save}=0$ and the first value of a cumulative function
$Z_{1}=0$. The cumulative function is updated $Z_{t}=Z_{t-1}+p_{t}$, if
$r_{t-1}$ is not defined as missing due to staleness. Each time when $\lfloor
Z(t)\rfloor-\lfloor Z(t-1)\rfloor=1$, $N_{save}$ is increased by $1$.

We notice that the first non-zero return after a row of 0-returns due to
staleness is the sum of all missing returns generated by a hidden efficient
price. This return is also set as missing. However, the value of return used
for estimating volatility is calculated as its expected value:
$\hat{r}_{n-1}=\frac{r_{n-1}}{\sqrt{N_{0}+1}}$, where $N_{0}$ is the amount of
missing values strictly before the non-zero return $r_{n-1}$. The same is also
referred to initially missing values, e.g., due to no-trading or errors in
collecting the data.

Another assumption is that a 0-return appears due to staleness if the previous
return had the 0-value and was defined to appear due to staleness. We include
this rule, since we assume that it is more likely that two consecutive
0-returns appear due to high transaction costs than due to rounding (that is,
simply speaking, two outcomes of generating Gaussian random variables are less
than a tick size).

Generally, for the estimation of volatility at time $t$ we should consider
three cases: $P_{t-1}$ was missing (or minute $t-1$ is non-trading),
$r_{t-1}=0$, $r_{t-1}\neq 0$. Thus, the algorithm is the following. We give
the algorithm for the case of $Sig_{1}$ which is used in the application for
the real data. We remove all 0-returns that start the sequence.

###  B.1 Pseudocode

Step 0: $\bar{\sigma}_{1}=|r_{1}|/\mu_{1}$; $Z_{1}=0$, $N_{save}=0$;
$N_{0}=0$.

For $t$ from $2$ to $N$, where $N$ is the length of time series:

Step 1:

  * •

If $r_{t-1}$ is missing: $\bar{\sigma}_{t}=\bar{\sigma}_{t-1}$; Increase
$N_{0}$ by the amount of consecutive missing prices

  * •

Else if $r_{t-1}=0$:

    * –

If $N_{save}>0$ and $N_{0}=0$: $N_{save}=N_{save}-1$,
$\bar{\sigma}_{t}=Sig_{1}(\alpha,0,\bar{\sigma}_{t-1})$

    * –

Else: $\bar{\sigma}_{t}=\bar{\sigma}_{t-1}$, $N_{0}=N_{0}+1$,
$r_{t-1}=\text{missing}$

  * •

Else:
$\bar{\sigma}_{t}=Sig_{1}(\alpha,\frac{r_{n-1}}{\sqrt{N_{0}+1}},\bar{\sigma}_{t-1})$,
$N_{0}=0$

Step 2:

  * •

Calculate $p_{t}$ (Eq. 3)

  * •

If $r_{t-1}$ is not missing, $Z_{t}=Z_{t-1}+p_{i}$

  * •

If $\lfloor Z(t)\rfloor-\lfloor Z(t-1)\rfloor=1$, $N_{save}=N_{save}+1$

Finally, we check if the effect of staleness really exists in the price time
series:

$\begin{split}\hat{p}&=\frac{\sum_{i}p_{i}}{N}\\\ \hat{q}&=1-\hat{p}\\\
Var&=\hat{p}\hat{q}N\end{split}$

If $N_{real}\leq\sum_{i}p_{i}+1.96\sqrt{Var}$, we leave time series without
putting any missing values, where $N_{real}$ is the initial amount of
0-returns.

##  Appendix C A predictable time series with entropy at maximum

The goal of this section is to construct a price model where entropy is high
because of discretization. This model shows that a high entropy value may be
caused by discretization, but not because of the randomness of a return time
series.

There are equal probabilities of having symbols $0$, $1$ and $2$. $1$
corresponds to log-returns, $r$, equal to $-0.4$, $2$ corresponds to log-
returns equal to $0.4$. The structure of symbol $0$ is more complicated. It
covers three other symbols $3,4,5$. They correspond to log-returns
$-0.3,0.1,0.2$, respectively. One of the symbols $3,4$ or $5$ appears with
probabilities depending on the previous value of these symbols. The
probabilities are presented in the Table 7. Having a symbol presented in a
column, there are probabilities of getting a symbol presented in a row.

Table 7: Transition probabilities first symbol |  $\cdot 3$ |  $\cdot 4$ |  $\cdot 5$
---|---|---|---
$3\cdot$ |  $\dfrac{1}{6}$ |  $\dfrac{1}{3}$ |  $\dfrac{1}{2}$
$4\cdot$ |  $\dfrac{1}{2}$ |  $\dfrac{1}{6}$ |  $\dfrac{1}{3}$
$5\cdot$ |  $\dfrac{1}{3}$ |  $\dfrac{1}{2}$ |  $\dfrac{1}{6}$
Rows stand for the first symbol of a block, columns stand for the second
symbol.

The model implies an average zero return. However, a trading strategy that
increases a profit exists. After 3 a trader should buy, and after 4 and 5 the
trader should sell. However, the entropy of 3-symbols series is at maximum,
that should imply absence of profitable strategies.

Considering the same example with 4-symbols discretization we get that
$Q_{1}=-0.4$, $Q_{2}=0.1$, and $Q_{3}=0.4$. Therefore, we have the following
discretization of returns:

$s=\begin{cases}0,r=-0.4,\\\ 1,r=-0.3\text{ or }r=0.1,\\\ 2,r=0.2,\\\
3,r=0.4.\end{cases}$

Thus, we can distinguish returns $r=0.2$ from others the using 4-symbols
discretization. Table 7 gives the following probabilities for the blocks of
two symbols from the 4-symbols discretization: $p(11)=\frac{7}{162}$,
$p(12)=p(21)=\frac{5}{162}$, $p(22)=\frac{1}{162}$. Noting that
$p(0)=p(3)=\frac{1}{3}$, $p(1)=\frac{2}{9}$, and $p(2)=\frac{1}{9}$, we
calculate that

$H_{1}=-\frac{2}{3}\log\left(\frac{1}{3}\right)-\frac{2}{9}\log\left(\frac{2}{9}\right)-\frac{1}{9}\log\left(\frac{1}{9}\right)\approx
0.946<1$

and

$\begin{split}H_{2}&=-\frac{1}{2}(\frac{7}{162}\log{\left(\frac{7}{162}\right)}+\frac{5}{81}\log{\left(\frac{5}{162}\right)}+\frac{1}{162}\log{\left(\frac{1}{162}\right)}+\\\
&+\frac{4}{9}\log{\left(\frac{1}{9}\right)}+\frac{8}{27}\log{\left(\frac{2}{27}\right)}+\frac{4}{27}\log{\left(\frac{1}{27}\right)})\approx
0.944<H_{1}\end{split}$

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