# Floquet topological insulators with hybrid edges

Boquan Ren  Key Laboratory for Physical Electronics and Devices of the
Ministry of Education & Shaanxi Key Lab of Information Photonic Technique,
School of Electronic Science and Engineering, Faculty of Electronic and
Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China
Yaroslav V. Kartashov  Institute of Spectroscopy, Russian Academy of Sciences,
Troitsk, Moscow, 108840, Russia  Hongguang Wang  Key Laboratory for Physical
Electronics and Devices of the Ministry of Education & Shaanxi Key Lab of
Information Photonic Technique, School of Electronic Science and Engineering,
Faculty of Electronic and Information Engineering, Xi’an Jiaotong University,
Xi’an 710049, China  Yongdong Li  Key Laboratory for Physical Electronics and
Devices of the Ministry of Education & Shaanxi Key Lab of Information Photonic
Technique, School of Electronic Science and Engineering, Faculty of Electronic
and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China
Yiqi Zhang<EMAIL_ADDRESS>Key Laboratory for Physical Electronics and
Devices of the Ministry of Education & Shaanxi Key Lab of Information Photonic
Technique, School of Electronic Science and Engineering, Faculty of Electronic
and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China

###### Abstract

Topological edge states form at the edges of periodic materials with specific
degeneracies in their modal spectra, such as Dirac points, under the action of
effects breaking certain symmetries of the system. In particular, in Floquet
topological insulators unidirectional edge states appear upon breakup of the
effective time-reversal symmetry due to dynamical modulations of the
underlying lattice potential. However, such states are usually reported for
certain simple lattice terminations, for example, at zigzag or bearded edges
in honeycomb lattices. Here we show that unconventional topological edge
states may exist in Floquet insulators based on arrays of helical waveguides
with hybrid edges involving alternating zigzag and armchair segments, even if
the latter are long. Such edge states appear in the largest part of the first
Brillouin zone and show topological protection upon passage through the
defects. Topological states at hybrid edges persist in the presence of
focusing nonlinearity of the material. Our results can be extended to other
lattice types and physical systems, they lift some of the constraints
connected with lattice terminations that may not support edge states in the
absence of effects breaking time-reversal symmetry of the system and expand
the variety of geometrical shapes in which topological insulators can be
constructed.

##  I Introduction

Topological insulators are considered as a new state of matter, and their most
representative feature is the existence of topologically protected states at
their edges that are responsible for transport of excitations despite
insulating bulk [1, 2]. The phenomenology of topological insulators originally
developed in solid-state physics, has been extended to a wide variety of
photonic [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22] and other physical systems [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41]. Photonic topological insulators are most
frequently constructed on periodic artificial materials, such as honeycomb
[13, 42, 43, 44], kagome [45, 46, 47], Lieb [48, 49, 45, 50] or other lattices
possessing specific degeneracies in their spectra, such as Dirac points [51].
Edge states then form with eigenvalues within topological bandgap that may
open between these points under the action of different effects.

Figure 1: (a-d) Examples of honeycomb arrays with hybrid edges that are
periodic along the $y$ axis with different periods $Y$ and are truncated along
the $x$-axis. Waveguides belonging to zigzag and armchair edge segments are
indicated by the red and blue dots, respectively. (a) Lattice with usual
zigzag boundary, (b-d) lattices containing 1, 2, or 4 armchair segments on
each period (called below case-1, case-2, and case-4 ribbons, respectively).
(e-h) Corresponding band structures at $r_{0}=0$ in the first Brillouin zone
$-0.5{\rm K}_{y}\leq k_{y}\leq 0.5{\rm K}_{y}$, where levels giving rise to
edge states are highlighted by the red color, while bulk states are shown
gray. Lattices are shown in the window: $-40\leq x\leq 40$ and $-20\leq y\leq
20$.

Typically edge states are reported for specific lattice terminations (edge
geometries) that also determine the location of their eigenvalues in the first
Brillouin zone (BZ) in projected spectra of such truncated structures. For
example, honeycomb lattices that were used for demonstration of many
topological phenomena [52, 53, 54, 55] may have simple zigzag, bearded, and
armchair edges. The majority of works on topological edge states in such
lattices addresses first two types of lattice terminations [53, 56], zigzag or
bearded, because in the absence of effects breaking time-reversal symmetry,
these terminations lead to the appearance of nontopological edge states in
wide intervals of Bloch momenta. The situation is somewhat different for
armchair terminations because in the nontopological regime they do not lead to
appreciable existence intervals of the edge states. Nevertheless, the very
possibility of the formation of the topological edge states at the armchair
edges in Floquet systems was mentioned in [13], and the propagation of linear
and nonlinear edge states on the armchair edges of Floquet structure was
studied in [57]. Still, the properties of linear and nonlinear edge states in
topological insulators with mixed edges, involving zigzag and armchair
terminations of different variable widths were not studied yet. Similar
situation is also encountered in well-known kagome and Lieb lattices [46, 49]
and for various photonic topological platforms, including helical waveguide
arrays [13, 58, 59, 60, 61, 62, 63, 64, 45, 65, 20, 22, 66] that receive
considerable attention these days, which well mimic the Floquet mechanism in
time domain [67, 68, 69], see also works [70, 18, 71, 72, 73, 74, 75, 76, 77,
78, 79, 80, 81] for other promising results obtained on Floquet insulator
platforms. The investigation of topological insulators with mixed terminations
including segments that do not support nontopological edge states is therefore
important because it may allow to substantially extend the possibilities for
construction of topological circuits and devices and lift the restriction on
their potential geometrical shapes (by removing the constrains connected with
“nontopological terminations” that would lead to loss of confinement for
excitations passing through them).

In this article, we propose a new type of the photonic Floquet topological
insulator based on the array of helical waveguides with hybrid edges involving
combined zigzag and armchair segments. Topological unidirectional edge states
form and show clear topological protection in such structures even if the
length of the armchair segments, which do not support edge states in the
absence of waveguide rotation (at least in the tight-binding limit), notably
exceeds that of the zigzag segments. Surprisingly, we found that edge states
supported by such hybrid terminations in ribbon geometry appear in
substantially larger fraction of the BZ in projected spectrum in comparison
with usual Floquet insulators with pure zigzag edges. Our Floquet system with
broken time-reversal symmetry is qualitatively different from previously
studied graphene nanoribbons with hybrid edges supporting the formation of the
“end” states described by the Su-Schrieffer-Heeger model [82, 83].

We also found that topological edge states at hybrid edges persist in the
nonlinear regime and that such edges can support propagation of localized
topological edge solitons. Indeed, nonlinearity in topological systems can
lead to a number of intriguing phenomena ranging from modulational instability
[84] and bistability of the edge states [85, 86, 87], topological transitions
[21, 88, 37, 89], to formation of topological solitons in the bulk [58, 20] or
at the edge [90, 61, 44, 66, 50, 91, 92, 93, 94] of the insulator. Here we
analyze stability of the nonlinear edge states at hybrid edges and illustrate
the possibility of edge soliton formation.

##  II Linear edge states

###  II.1 Band structures and edge states

Light propagation in helical waveguide arrays can be described by the
nonlinear Schrödinger-like equation with focusing cubic nonlinearity

$i\frac{\partial\psi}{\partial
z}=-\frac{1}{2}\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
x^{2}}\right)\psi-\mathcal{R}(x,y,z)\psi-|\psi|^{2}\psi,$ (1)

where $\psi$ is the dimensionless complex amplitude of the field, $x$ and $y$
are normalized transverse coordinates, $z$ is the normalized propagation
distance, and the function $\mathcal{R}(x,y,z)=\mathcal{R}(x,y,z+Z)$ describes
$z$-periodic refractive index profile with longitudinal period $Z$. The array
consists of identical helical waveguides of width $\sigma$ placed in the nodes
$(x_{m},y_{n})$ of the honeycomb grid

$\mathcal{R}(x,y,z)=p\sum_{mn}e^{-\frac{[x-x_{m}-r_{0}\sin(\omega
z)]^{2}}{\sigma^{2}}-\frac{[y-y_{n}-r_{0}\cos(\omega
z)+r_{0}]^{2}}{\sigma^{2}}},$ (2)

where $p$ is the array depth and $\omega=2\pi/Z$. We adopt the parameters
$d=1.7$ ($17\,\mu\rm m$ spacing between waveguides), $r_{0}\sim 0.0-1.0$
(helix radius up to $10\,\mu\rm m$), $p=11$ (refractive index modulation depth
$\delta n\sim 1.2\times 10^{-3}$), $\sigma=0.4$ (4 $\mu\rm m$-wide
waveguides), and $Z=8$ (helix period $\sim 9.1\,\rm mm$) typical for fs-laser
written arrays at $\lambda=800\,\rm nm$ [93, 95, 96, 97]. We consider ribbons
with hybrid edges truncated along the $x$-axis and periodic in $y$:
$\mathcal{R}(x,y,z)=\mathcal{R}(x,y+Y,z)$.

First of all, it is necessary to look at the band structure and edge states
with different terminations by neglecting the nonlinear term in Equation (1).
Figure 1 shows honeycomb lattice with usual zigzag edges [Figure 1(a)] and
structures with hybrid zigzag-armchair terminations for progressively
increasing lengths of the armchair segments (increasing $Y$ periods) [Figure
1(b,c,d)]. We refer to lattices in Figure 1(b,c,d) as to case-1, case-2, and
case-4 ribbons, in accordance with the number of armchair segments on one
$y$-period. Notice that the length of the armchair segments can substantially
exceed that of the zigzag segments [Figure 1(d)].

Figure 2: Quasi-propagation constant spectrum and derivatives of the quasi-
propagation constants for linear edge states versus $k_{y}/\textrm{K}_{y}$ for
case-1 (a),(c) and case-4 (b),(d) ribbons at $r_{0}=0.5$ and $Z=8$. Levels
giving rise to topological edge states are shown in red and blue, while bulk
states are shown gray. Red (blue) edge states are localized on the right
(left) edge of the ribbon. Derivatives $b^{\prime}=db/dk_{y}$ are shown with
solid lines, while derivatives $b^{\prime\prime}=d^{2}b/dk_{y}^{2}$ are shown
with dashed lines [line colors correspond to colors used in (a),(b)].

For ribbons with helical waveguides the linear eigenmodes are the Floquet-
Bloch waves

$\psi(x,y,z)=\phi_{k_{y}}(x,y,z)e^{ibz}=u_{k_{y}}(x,y,z)e^{ik_{y}y+ibz},$ (3)

where $b=b(k_{y})$ is the quasi-propagation constant (similar to quasi-energy
in driven quantum systems [67, 69]) within the first longitudinal BZ
$b\in[-\omega/2,\omega/2)$ with $\omega=2\pi/Z$, $k_{y}$ is the Bloch momentum
within the first transverse BZ $k_{y}\in[-{\rm K}_{y}/2,{\rm K}_{y}/2)$, where
$\textrm{K}_{y}=2\pi/Y$. The function
$u_{k_{y}}(x,y,z)=u_{k_{y}}(x,y+Y,z)=u_{k_{y}}(x,y,z+Z)$ is periodic both in
$y$ and $z$. When $r_{0}=0$ and waveguides are straight, $b$ becomes usual
propagation constant. Such linear “static” band structures $b(k_{y})$ for
ribbons from Figure 1(a)-(d) are presented in Figure 1(e)-(h). The levels that
give rise to edge states are shown red (we mark them in the entire BZ for
simplicity, even though edge states exist in limited $k_{y}$ intervals), while
bulk states are shown gray. In clear contrast to lattice with zigzag
boundaries with localized states emerging near the border of the BZ [Figure
1(e)], in structures with hybrid boundaries two degenerate edge states appear
in a broad region in the center of the BZ [Figure 1(f)-(h)]. Waveguide
rotation opens topological gap, lifts the degeneracy of the edge states, and
makes them unidirectional, transforming the system into Floquet topological
insulator. This is seen from band structures presented for $r_{0}\neq 0$ in
Figure 2(a) and 2(b), where red/blue curves correspond to topological edge
states localized at the right/left edges. Corresponding derivative
$b^{\prime}=db/dk_{y}$ [solid lines in Figure 2(c) and 2(d)] determines group
velocity of the topological edge states via $v=-b^{\prime}$, while dispersive
properties are determined by $b^{\prime\prime}=d^{2}b/dk_{y}^{2}$ [dashed
lines in Figure 2(c) and 2(d)]. One can see that states from blue branch move
in the positive $y$ direction, while those on the red branch move in the
negative $y$ direction. It should be mentioned that topological properties of
honeycomb lattices of helical waveguides (characterizing topology of its bulk
bands) are well-documented [13] and can be described by proper generalization
of the Chern number for Floquet systems [67, 69]. The existence of the
topological edge states on hybrid edges is thus consistent with bulk-edge
correspondence principle.

Figure 3: Field modulus distributions at $z=0$ in linear topological edge
states with different momenta $k_{y}$ for case-2 (a) and case-4 (b) ribbons
with $r_{0}=0.5$, $Z=8$. Orange circles represent the lattice sites. All
panels are shown in the window: $-35\leq x\leq 35$ and $-100\leq y\leq 100$.

Representative $|\psi|$ distributions at $z=0$ in Floquet-Bloch states
localized at hybrid edges are presented in Figure 3. Here we choose case-2
[Figure 3(a)] and case-4 [Figure 3(b)] ribbons as exemplary structures. The
remarkable property of the system with hybrid edges is that localized edge
states are encountered in much larger fraction of the BZ in comparison with
system with pure zigzag or bearded edges. The best edge state localization is
achieved near the BZ center at $k_{y}=0$, while towards the border of the BZ
localization becomes weaker and edge states gradually transform into bulk ones
(for case-1, case-2, and case-4 ribbons delocalization for selected parameters
occurs for $k_{y}>0.36{\rm K}_{y}$, $k_{y}>0.43{\rm K}_{y}$, and
$k_{y}>0.46{\rm K}_{y}$, respectively). Thus, the existence range of
topological edge states surprisingly increases with increasing length of the
armchair segments. This also leads to enhancement of edge state localization
at fixed $k_{y}$, as seen from comparison of states in case-2 and case-4
ribbons.

###  II.2 Propagation dynamics

One prominent property of the edge states from Figure 3 is that they can
bypass defects or disorder without reflection due to their topological
protection. We choose topological edge states with $k_{y}=0.1{\rm K}_{y}$ in
case-1 and case-4 ribbons for illustration that this property holds in
structures with hybrid edges. To illustrate unidirectional propagation we
superimpose a wide Gaussian envelope on such states. Some waveguides were
removed in case-1 and case-4 ribbons to create defects, as shown by the red
ellipses in Figure 4(a) and 4(b) (in the latter case we removed not just
single guide, but the whole “tooth” on the edge). Field modulus distributions
at different distances in Figure 4(a,b) clearly illustrate topological
protection and absence of backward/bulk radiation.

Figure 4: (a) Field modulus distributions at different distances illustrating
propagation of the linear edge state with $k_{y}=0.1{\rm K}_{y}$ in case-1
ribbon with a defect on the edge indicated by the red ellipse. (b) Setup is as
(a), but for case-4 ribbon with larger defect. (c) Circulation of topological
edge soliton obtained in the presence of nonlinearity at $b_{\rm nl}=0.002$,
$k_{y}=0.1{\rm K}_{y}$, $b^{\prime\prime}=-0.1214$, $\chi=0.1358$ in case-1
ribbon (top and bottom edges are hybrid edges, while left and right edges are
purely armchair). Other parameters are as in Figure 2. Figure 5: (a) Nonlinear
edge state family in case-4 ribbon at $k_{y}=0.1\textrm{K}_{y}$. Peak
amplitude $a$ (red curve) and power $P$ per period of the edge state as
functions of nonlinear generalization of quasi-propagation constant $b$. The
green, red, and white dots correspond to $b=0.3067$ ($P=0.02$), $b=0.311$
($P=0.1$), and $b=0.327$ ($P=0.475$), respectively. Solid and dashed lines
represent metastable and unstable states. (b) Peak amplitude $a$ of the
perturbed edge state corresponding to the red dot in (a) versus distance $z$
and associated $|\psi|$ distributions at different propagation distances. (c)
Setup is as (b), but corresponding to the white dot in (a). All states are
shown within the window $-35\leq x\leq 35$ and $-150\leq y\leq 150$.

##  III Nonlinear edge states and edge solitons

In addition to linear topological edge states, our system with hybrid edges
also supports nonlinear edge states and topological edge solitons. In Figure 5
we show representative families of the nonlinear Floquet-Bloch modes
bifurcating from linear edge states in case-4 ribbon. Their profiles are
modified by the nonlinearity that also leads to a shift of the quasi-
propagation constant. These families were obtained using iterative method of
[58]. The procedure of calculation of the nonlinear Floquet-Bloch modes can be
divided into several steps. (1) We propagate the linear edge state
$\psi_{e}^{\rm in}$ with a given power $P$ (that will determine eventually
quasi-propagation constant of the nonlinear Floquet-Bloch state) according to
Eq. (1) to obtain the dynamical lattice modified by the nonlinearity, i.e.
$\mathcal{R}_{e}=\mathcal{R}+|\psi_{e}^{\rm in}|^{2}$. (2) After this we
propagate all linear eigenstates $\psi^{\rm in}_{n\in N}$ of $\mathcal{R}$
that include $\psi_{e}^{\rm in}$ in the modified dynamical lattice
$\mathcal{R}_{e}$ for a whole period $Z$ and obtain corresponding output
distributions $\psi^{\rm out}_{n\in N}$, where $N$ is the number of
eigenstates. (3) One calculates the projection $U_{mn}=\langle\psi^{\rm
in}_{m},\psi^{\rm out}_{n}\rangle$, whose eigenvalues are Floquet exponents
$e^{iZb_{n}}$. (4) For each $b_{n}$, one finds the index $\ell_{n}$ of the
maximum element of the corresponding eigenvector $V_{n}$. Eigenstates
$\psi_{n\in N}^{\rm re}$ of $\mathcal{R}_{e}$ can then be constructed as
$\psi_{n}^{\rm re}=\sum_{q=1}^{N}\psi_{q}^{\rm in}V_{q}(\ell_{n})$. (5) One
picks out the modified edge state $\psi_{e}^{\rm re}$ from $\psi_{n\in N}^{\rm
re}$ and normalizes it to the given power $P$. (6) The steps (1)-(5) are
repeated until the difference between $\psi_{e}^{\rm re}$ and $\psi_{e}^{\rm
in}$ reduces below required small level.

Peak amplitude $a$ and power per period

$P=\int_{-\infty}^{\infty}dx\int_{0}^{Y}|\psi|^{2}dy$

of the nonlinear edge state increase with increasing nonlinear shift of quasi-
propagation constant $b$ from its linear value $b=0.3057$ at $k_{y}=0.1{\rm
K}_{y}$ [Figure 5(a)]. One can see from this figure that the procedure
described above allows us to trace the nonlinear family directly from its
bifurcation point from linear edge state, so that one can be sure that this is
continuous family of the simplest thresholdless nonlinear solutions. Of
course, this does not exclude the possibility of the existence of more complex
nonlinear Floquet-Bloch states, but they were not found here. Stability of
such nonlinear Floquet-Bloch states was investigated by superimposing random
input noise on them with amplitude up to $0.05a$ and propagating them over
long distances $z\sim 10^{4}$. Thus, nonlinear edge state indicated by the
green dot ($b=0.3067$ and $P=0.02$) in Figure 5(a) is metastable and does not
decay even at $z=10^{4}$, while those indicated by the red dot ($b=0.311$ and
$P=0.1$) and the white dot ($b=0.327$ and $P=0.475$) are unstable. The dashed
lines in Fig. 5(a) represent unstable branch of the nonlinear edge states,
which roughly starts from the case with $P=0.05$ ($b=0.3084$). Typical
scenario of instability development is illustrated in Figure 5(b), where one
can see that peak amplitude of the edge state starts changing notably only at
distances $z>3000$, while wave fragmentation along the edge on developed stage
of modulational instability becomes clear at $z\sim 6000$. Nonlinear edge
states with higher amplitudes may show somewhat different instability scenario
resulting in considerable radiation into the bulk without obvious
fragmentation along the edge [see Figure 5(c) illustrating propagation of
perturbed edge state corresponding to the white dot from Figure 5(a)].

The possibility of development of modulational instability indicates that
Floquet insulators with hybrid edges can support localized edge solitons
travelling along the edge [84, 91]. We illustrate them here for case-1 ribbon.
Such solitons can be constructed as envelope solitons

$\psi(x,y,z)=A(\eta,z)\phi_{k_{y}}(x,y,z)e^{ibz}$ (4)

on the Floquet edge state $\phi_{k_{y}}(x,y,z)$ with proper sign of the
dispersion coefficient $b^{\prime\prime}$. The equation

$i\frac{\partial A}{\partial
z}=\frac{b^{\prime\prime}}{2}\frac{\partial^{2}A}{\partial\eta^{2}}-\chi\lvert
A\rvert^{2}A,$ (5)

where $\eta=y+b^{\prime}z$, for slowly varying envelope $A(\eta,z)$ of such
states can be derived from Equation (1) using the method developed in [66]. In
this equation

$\chi=\frac{1}{Z}\int_{0}^{Z}dz\int_{-\infty}^{+\infty}dx\int_{0}^{Y}\lvert\phi_{k_{y}}(x,y,z)\rvert^{4}dy$
(6)

is the effective period-averaged nonlinearity coefficient. Soliton envelopes
existing at $b^{\prime\prime}<0$ are of the form

$A(\eta,z)=\sqrt{\frac{2b_{\textrm{nl}}}{\chi}}\textrm{sech}\left[\sqrt{\frac{-2b_{\textrm{nl}}}{b^{\prime\prime}}}(y+b^{\prime}z)\right]e^{ib_{\rm
nl}z},$ (7)

where $b_{\rm nl}$ is the nonlinearity-induced shift of propagation constant
that should be sufficiently small to ensure that slowly varying envelope
covers many periods of the ribbon. The example of propagation of such edge
soliton with $b_{\rm nl}=0.002$ is presented in Figure 4(c). This soliton
bifurcates from edge state with $k_{y}=0.1{\rm K}_{y}$. We consider soliton
circulation in closed geometry, when case-1 ribbon is made finite also in the
$y$ direction [Figure 4(c)]. One can see that soliton initially slightly
broadens and then starts moving as a robust object along the hybrid edge. It
broadens at purely armchair edge (around $z=1200$), but after it passes onto
opposite hybrid edge of the ribbon, nonlinearity helps it to recombine again
into steadily propagating localized state (see the distribution at $z=2400$).
Radiation into the bulk is practically absent upon such circulation. Numerical
simulations reveal that Floquet edge solitons in structures with hybrid edges
survive even after many roundtrips. The animations corresponding to Figure
4(a)-4(c) are provided, in which the peak amplitude of the soliton
$a=\max\\{|\psi|\\}$ is recorded and displayed simultaneously. The animation
of the propagation of this soliton in finite structure is up to distances
$z\sim 20000$ (i.e., 2500 longitudinal periods). The animations can be found
in Animation 1, Animation 2, and Animation 3.

##  IV Conclusions

Summarizing, we have shown that Floquet topological insulators may support
topologically protected edge states for previously unexplored hybrid types of
lattice terminations, that combine mixed armchair and zigzag segments, even
when the former segments constitute larger fraction of the edge. Moreover,
such hybrid edges support long-living travelling edge solitons that show
robust circulation along the structure periphery. These results can be
extended to other types of lattices and various physical systems, where
Floquet insulators can be realized, including atomic and optoelectronic
systems. They will also allow to expand the variety of geometrical shapes in
which topological insulators can be constructed and analyzed.

Acknowledgements
This work was supported by the National Natural Science Foundation of China
(Grant Nos.: 12074308, U1537210), Russian Science Foundation (Grant No.:
21-12-00096), and the Fundamental Research Funds for the Central Universities
(Grant No.: xzy022022058).

Data availability
The data that supports the results within this paper are available from the
corresponding authors upon reasonable request.

Code availability
The codes used to obtain the results described in this paper are available
from the corresponding authors upon reasonable request.

Competing interests The authors declare no competing interests.

Correspondence and requests for materials should be addressed to Y.Z.

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# On the dependence structure of the trade/no trade sequence of illiquid
assets

Hamdi Raïssi111Instituto de Estadística, PUCV, Errazuriz 2734, Valparaíso,
CHILE; email<EMAIL_ADDRESS>This author acknowledges the ANID funding
Fondecyt 1201898.

###### Abstract

In this paper, we propose to consider the dependence structure of the trade/no
trade categorical sequence of individual illiquid stocks returns. The
framework considered here is wide as constant and time-varying zero returns
probability are allowed. The ability of our approach in highlighting illiquid
stock’s features is underlined for a variety of situations. More specifically,
we show that long-run effects for the trade/no trade categorical sequence may
be spuriously detected in presence of a non-constant zero returns probability.
Monte Carlo experiments, and the analysis of stocks taken from the Chilean
financial market, illustrate the usefulness of the tools developed in the
paper.

Keywords: Time-varying illiquidity levels; Categorical financial time series;
Serial dependence.

JEL Classification: C13; C22; C58.

##  1 Introduction

In the time series econometrics literature, it is widely documented that
neglected non-stationary behaviors can generate misleading assessments. A
classical example is given by the spurious regression described in Phillips
(1987). Since the seminal papers of Mikosch and Stărică (2004), Stărică and
Granger (2005), and Engle and Rangel (2008), a huge amount of papers have
explored the possibility that an unconditionally non-constant variance can
display spurious long memory features. This led to contributions proposing new
tools for a correct assessment of time series dynamics, see Phillips and Xu
(2006), Cavaliere and Taylor (2008) or Patilea and Raïssi (2014) among many
others. However, at the best of our knowledge, there is no contribution in
this spirit on the behavior of the daily zero returns probability of illiquid
stocks. In this paper, we aim to investigate the dependence structure of the
trade/no trade sequence, i.e the process $(a_{t})$ defined such that $a_{t}=1$
if a daily price change is observed at $t$, and 0 if not. It is suggested that
long range dependence effects, could possibly be explained by non-constant
illiquidity levels in time, i.e., $P(a_{t}=1)$ is non-constant. As a
consequence, tools for analyzing the dependence structure of the daily
trade/no trade structure corrected from the non-stationary probability are
developed.

Illiquid stocks exhibiting a large amount of daily zero returns are commonly
observed in all markets, see e.g. Lesmond (2005) for emerging markets. In
order to motivate the above arguments, examples taken from the chilean stock
market (Santiago Stock Exchange, SSE) are provided in Figure 1-4.222The author
is grateful to Andres Celedon for research assistance. The main stocks indexes
of the SSE are the IPSA (the 30 most liquid stocks) and the IGPA (comprising
the 30 stocks of the IPSA, plus other stocks according to some criteria). The
stocks presented here are part of the IGPA, but not of the IPSA. The daily
returns of the stocks are displayed together with their smoothed $P(a_{t}=1)$.
In view of the different panels in Figure 1-3, trends and abrupt breaks can be
seen in the daily trade probability. Such behaviors can be backed up by a
variety of facts. In Figure 1, the common increasing trends could be explained
by a general development of the SSE. Indeed, during the period 2000-2008 the
Chilean GDP growth was relatively elevated, which made the SSE reached some
kind of maturity. On the other hand, abrupt shifts are often the product of
particular events in the company history. For instance, the quick increase of
the daily price change probability for the Security bank stock, may be
explained by the merger by absorption of the Dresdner Bank Lateinamerika in
September 2004. In addition, the Security Bank issued more than 32.8 millions
new stocks after the capital increase announced during the extraordinary
shareholders meeting by December 29th, 2004. Conversely, the quick decrease of
the daily trade probability for the Provida stock seems to be a consequence of
the take-over bid of Metlife on Provida during September 2013. As a
consequence, we can conclude that a structural break in the zero returns
probability, has occurred during the above mentioned period, with some degree
of confidence. In Figure 3, long-run decreasing behaviors can be observed. All
these observations suggest to allow for a time-varying $P(a_{t}=1)$ when
analyzing the dependence structure of $(a_{t})$. Note that investigating the
illiquidity sequence dynamics can be useful to understand various facts on the
underlying asset. The methodology developed here may be used to identify the
relevant past of $(a_{t})$ to consider for modelling financial time series. In
a stationary framework, reference can be made to Moysiadis and Fokianos (2014)
for predicting $(a_{t})$, or the GARCH models with covariates considered in
Francq and Thieu (2019).

The article is structured as follows. In Section 2, the general framework of
the study is presented. We investigate the analysis of the dependence
structure of the $(a_{t})$ when $P(a_{t}=1)$ is constant in Section 2.1. The
consequences of neglecting a time-varying $P(a_{t}=1)$ in the dependence
structure of the trade/no trade sequence are highlighted in Section 2.2. Then,
adaptive tools that adequately take into account a non-constant $P(a_{t}=1)$
are developed. Monte Carlo experiments and real data analysis illustrate our
theoretical findings in Section 3.

##  2 The theoretical framework and diagnostic tools

Let us consider a one-period profit-and-loss random variable $r_{t}$. It is
assumed that $r_{1},\dots,r_{n}$ are observed, with $n$ the sample size.
Recall that $a_{t}=0$ if $r_{t}=0$, and $a_{t}=1$ otherwise. We make the
following assumption to describe abrupt or gradual changes in the stock’s
illiquidity degree.

######  Assumption 1 (Non constant probability).

The time-varying probabilities $P(a_{t}=1)$ are given by $g(t/n)$ where
$g(\cdot)$ is a non-constant deterministic function, such that $0<g(\cdot)<1$
on the interval $(0,1]$, and satisfies a piecewise Lipschitz condition on
$(0,1]$.333For $u<0$, the function is set constant, that is
$g(u)=\lim_{u\downarrow 0}g(u)$. Throughout the paper, the piecewise Lipschitz
condition means: there exists a positive integer $p$ and some mutually
disjoint intervals $I_{1},\dots,I_{p}$ with $I_{1}\cup\dots\cup I_{p}=(0,1]$
such that $g(u)=\sum_{l=1}^{p}g_{l}(u){\bf 1}_{\\{u\in I_{l}\\}},$
$u\in(0,1],$ where $g(\cdot)$ is a Lipschitz smooth function on
$I_{1},\dots,I_{p},$ respectively.

The rescaling device of Dahlhaus (1997) is often used to describe long run
effects (see Cavaliere and Taylor (2007), Xu and Phillips (2008), Patilea and
Raïssi (2013) and Wang, Zhao and Li (2019) among others). Note that the double
subscript is avoided to simplify the notations. The specification in
Assumption 1 is quite general, as it allows for patterns commonly observed in
practice, such as trends or abrupt breaks. As we are interested in testing the
independence of $(a_{t})$, the framework given by Assumption 1 is sufficient,
with no need to consider (possibly stochastic) probabilities conditional to
some past information. Indeed, as usual a test is built under some null
hypothesis, i.e. independent $(a_{t})$ in our case. The tools developed here
may be mostly used in an identifying step to some modelling task developed in,
e.g. Moysiadis and Fokianos (2014). Our approach is similar to the continuous
time series analysis methodology. Indeed, it is usual to study the correlation
structure of $(r_{t})$ (respectively powers of $(r_{t})$) in a first step.
Then, if some correlations are found significant, a model is estimated for the
(stochastic) conditional expectation (respectively the volatility) using for
instance ARMA (respectively GARCH) models.

###  2.1 The constant zero returns probability case

In this part, we assume that $(a_{t})$ is strictly stationary, i.e. the
particular case where the $g(\cdot)$ function is constant. Then, for testing
the short run dependence in the $(a_{t})$ sequence, the following hypotheses
are considered

$H_{0}:P(a_{t}a_{t-h}=1)=P(a_{t}=1)^{2},\>\mbox{for all}\>h\in\\{1,\dots,m\\}$

vs.

$H_{1}:\exists\>h\in\\{1,\dots,m\\},\>\mbox{such that}\>P(a_{t}a_{t-h}=1)\neq
P(a_{t}=1)^{2},$

taking a relatively small $m$. The hypothesis $H_{1}$ suggests the presence of
a dependence structure for $(a_{t})$. Let us introduce

$\widehat{A}_{m}=(\hat{\gamma}_{a}(1)/\hat{\gamma}_{a}(0),\dots,\hat{\gamma}_{a}(m)/\hat{\gamma}_{a}(0))^{\prime},\quad\mbox{where}\quad\hat{\gamma}_{a}(h)=\frac{1}{n}\sum_{t=1+h}^{n}a_{t}a_{t-h}.$

The statistic

$\mathcal{S}_{m}^{(a)}=n\widehat{A}_{m}^{\prime}\widehat{A}_{m},$

can be used for deciding $H_{0}\>\mbox{vs.}\>H_{1}$ with small $m$. On the
other hand, for large $m$, the components of $\widehat{A}_{m}$ may be plotted
to assess some persistency, or long-run effects, in the dependence structure
of $(a_{t})$. From the above, the tools for assessing some serial dependence
are built under $H_{0}$, that is an iid $(a_{t})$ process. The following
proposition gives the asymptotic behavior of $\widehat{A}_{m}$ in the
stationary framework. The convergence in distribution is denoted by
$\stackrel{{\scriptstyle d}}{{\longrightarrow}}$.

######  Proposition 2.1.

Suppose that $(a_{t})$ is iid. Then, as $n\to\infty$ we have,

$\sqrt{n}\widehat{A}_{m}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}(0,I_{m}),\quad\mbox{and}\quad\mathcal{S}_{m}^{(a)}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\chi_{m}^{2},$ (1)

where $I_{m}$ stands for the identity matrix of dimension $m$.

The iid assumption is made to detect any short-run dependence structure in the
$(a_{t})$ sequence using $\mathcal{S}_{m}^{(a)}$. On the other hand,
persistency or long-run effects may be highlighted plotting $\widehat{A}_{m}$
together with the confidence intervals obtained from (1). Note that such a
plot is similar to an autocorrelation function (ACF) plot. However, as the
$(a_{t})$ sequence is categorical, and since we decide that $(a_{t})$ is an
independent sequence if all the components of $\widehat{A}_{m}$ are not
significant, we prefer to use the term "dependence plots". For instance, the
dependence plots of the stationary Lipigas and CLC stocks in Figure 6, suggest
that the $(a_{t})$ sequence is 1-dependent for these two stocks. The reader is
referred to Brockwell and Davis (2006), Definition 6.4.3, for the
$k$-dependent processes. We end this section by mentioning the ability of the
$\mathcal{S}_{m}^{(a)}$ statistic to detect any dependence between daily
trade/no trade events. The almost sure convergence is denoted by
$\stackrel{{\scriptstyle a.s.}}{{\longrightarrow}}$.

######  Proposition 2.2.

Suppose that $(a_{t})$ is strictly stationary ergodic, such that $H_{1}$ holds
true. Then, we have $\widehat{A}_{m}\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}C$, where $C$ is a vector of constants with at least
a non-zero component.

The proof of Proposition 2.2 is a direct consequence of the Ergodic Theorem
(see Francq and Zakoïan (2019), Theorem A.2), and is therefore skipped. The
$Q_{m}$ test consists in rejecting $H_{0}$, if
$\mathcal{S}_{m}^{(a)}>\chi_{m,1-\alpha}^{2}$, where $\chi_{m,1-\alpha}^{2}$
is the $(1-\alpha)$th quantile of the $\chi_{m}^{2}$ distribution.

###  2.2 The non-constant zero returns probability case

Let us first underline, that the statistics based on the
$\hat{\gamma}_{a}(h)$’s, are not adequate for investigating the dependence
structure of the $(a_{t})$ sequence, when $P(a_{t}=1)$ is time-varying.
Indeed, under Assumption 1, and assuming that $(a_{t})$ is independent, it can
be shown that

$\hat{\gamma}_{a}(h)\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}\int_{0}^{1}g^{2}(s)ds-\left(\int_{0}^{1}g(s)ds\right)^{2},\quad
h>0,$ (2)

and using similar arguments to that of the proof of Proposition 2.4 below. If
we assume that $g(\cdot)$ is non-constant, then
$\int_{0}^{1}g^{2}(s)ds>\left(\int_{0}^{1}g(s)ds\right)^{2}$ in general. As a
consequence, the tools introduced in the above section can lead to a spurious
detection of a dependence, even for large lags $h$. In view of the stocks
considered in this paper, changes in the non-zero returns probabilities are
commonly observed in practice for a variety of facts. For this reason, new
tools taking into account the framework given by Assumption 1 are proposed.

In this part, we wish to test

$\widetilde{H}_{0}:g(t/n)g((t-h)/n)=P(a_{t}a_{t-h}=1),\>\mbox{for
all}\>h\in\\{1,\dots,m\\},\>\mbox{and all}\>t/n\in(0,1],$

vs.

$\widetilde{H}_{1}:\exists\>h\in\\{1,\dots,m\\},\>\mbox{and}\>0<a<b<1\>\mbox{such
that}\>g(t/n)g((t-h)/n)\neq P(a_{t}a_{t-h}=1),$

for all $t/n\in[a,b]$, taking $m$ small. In a first step, we suppose that the
$P(a_{t}=1)$’s are known. Later, feasible statistics will be proposed. In view
of $\widetilde{H}_{1}$, a cumulative sums (CUSUM) statistic as

$\widetilde{A}_{m}=\left(\sup_{u\in(0,1]}|\tilde{\gamma}_{a}(1,u)|,\dots,\sup_{u\in(0,1]}|\tilde{\gamma}_{a}(m,u)|\right)^{\prime},$
(3)

should be used, where

$\tilde{\gamma}_{a}(h,u)=(n-h)^{-1}\sum_{t=1+h}^{[nu]}(a_{t}-P(a_{t}=1))(a_{t-h}-P(a_{t-h}=1)),$

and $[\cdot]$ denotes the integer part of a real number. Accordingly to
$\widetilde{H}_{0}$, suppose that $(a_{t})$ is independent, and assume that
Assumption 1 holds true. Then we have

$\sqrt{n}\widetilde{A}_{m}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\left(\sup_{u\in(0,1]}|G(u)|,\dots,\sup_{u\in(0,1]}|G(u)|\right)^{\prime},$
(4)

as $n\to\infty$, where $G(u)=\int_{0}^{u}\sigma(s)dB(s)$, $B(\cdot)$ is a
standard Brownian motion, and $\sigma(s)^{2}=g^{2}(s)\left(1-\right.$
$\left.g(s)\right)^{2}$. The proof of the above result is provided in the
Appendix. It can be seen that the asymptotic distribution of
$\widetilde{A}_{m}$ is non-standard, and have to be approximated using for
instance bootstrap methods. Nevertheless, as the supremum is taken, and noting
that the time-varying $P(a_{t}=1)$ should be estimated nonparametrically to
make feasible tools, it can be difficult to control the type I error for
finite samples. In addition, noting that the variance of
$(a_{t}-P(a_{t}=1))(a_{t-h}-P(a_{t-h}=1))$ is not constant, the maximum value
is more likely to be attained in high variance periods. This could make
difficult to detect alternatives for periods where the variance is low.

For all these reasons, tools built using the full sample are considered,
although $\tilde{\gamma}_{a}(h,1)$ leads to compare $\int_{0}^{1}g_{h}(s)ds$
and $\int_{0}^{1}g^{2}(s)ds$, where $g_{h}(s)=\lim_{n\to\infty}P(a_{[sn]}$
$a_{[(s-h/n)n]})$. Indeed, this allows to avoid the drawbacks described above,
at the cost of a loss of power in the particular case
$\int_{0}^{1}g_{h}(s)ds=\int_{0}^{1}g^{2}(s)ds$, with $g_{h}(\cdot)\neq
g^{2}(\cdot)$. Then, let us introduce

$\overline{A}_{m}=\left(\tilde{\gamma}_{a}(1,1)/\tilde{\gamma}_{a}(0,1),\dots,\tilde{\gamma}_{a}(m,1)/\tilde{\gamma}_{a}(0,1)\right)^{\prime}.$
(5)

In order to test $\int_{0}^{1}g_{h}(s)ds=\int_{0}^{1}g^{2}(s)ds$, $1\leq h\leq
m$, taking $m$ small, the test statistic

$\overline{\mathcal{S}}_{m}^{(a)}=n\hat{\omega}^{-1}\overline{A}_{m}^{\prime}\overline{A}_{m},$
(6)

where

$\hat{\omega}:=\frac{n^{-1}\sum_{t=2}^{n}(a_{t}-P(a_{t}=1))^{2}(a_{t-1}-P(a_{t-1}=1))^{2}}{\left[n^{-1}\sum_{t=1}^{n}(a_{t}-P(a_{t}=1))^{2}\right]^{2}},$

can be used. The following propositions give the asymptotic behavior of
$\overline{A}_{m}$.

######  Proposition 2.3.

Suppose that the sequence $(a_{t})$ is independent, and fulfills Assumption 1.
Then as $n\to\infty$

$\sqrt{n}\>\overline{A}_{m}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}(0,\Omega),\quad\overline{\mathcal{S}}_{m}^{(a)}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\chi_{m}^{2},$

where $\Omega=diag(\omega,\dots,\omega)$, and

$\omega=\frac{\int_{0}^{1}g^{2}(s)(1-g^{2}(s))^{2}ds}{\left(\int_{0}^{1}g(s)(1-g(s))ds\right)^{2}}.$

In addition, we have $\hat{\omega}\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}\omega.$

######  Proposition 2.4.

Suppose that Assumption 1 holds true. Under $\widetilde{H}_{1}$ with
$\int_{0}^{1}g_{h}(s)ds\neq\int_{0}^{1}g^{2}(s)ds$, then as $n\to\infty$,
$\overline{A}_{m}\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}\overline{C}$, where $\overline{C}$ is a vector of
constants with at least a non-zero component.

In view of $\widetilde{H}_{0}$, the process $(a_{t})$ is assumed independent
but not necessarily identically distributed in Proposition 2.3. In addition,
it can be seen that taking the full sample in $\overline{A}_{m}$, we only need
to apply a classical Heteroscedasticity Consistent (HC) correction to handle a
time-varying $P(a_{t}=1)$. Considering
$\sqrt{n}\hat{\omega}^{-\frac{1}{2}}\overline{A}_{m}$, the usual
$(-1,96;1,96)$ bounds can be used to assess the dependence horizon of
$(a_{t})$. For a small $m$, and considering a fixed asymptotic level $\alpha$,
the $\overline{Q}_{m}$ test rejects $\widetilde{H}_{0}$ if
$\overline{\mathcal{S}}_{m}^{(a)}>\chi_{m,1-\alpha}^{2}$, where we recall that
$\chi_{m,1-\alpha}^{2}$ is the $(1-\alpha)$th quantile of the $\chi_{m}^{2}$
distribution. Note that the critical values for the statistic based on
$\overline{A}_{m}$ are standard, on the contrary to the cumulative sums in
$\widetilde{A}_{m}$. If $P(a_{t}=1)$ is constant, then $\omega=1$, and we
retrieve the result of Proposition 2.1. Alternatively, taking a large $m$, a
plot of $\overline{A}_{m}$ can be examined to analyze persistency or long-run
effects in $(a_{t})$. Proposition 2.4 shows the ability of our tools to detect
some dependence for $(a_{t})$, but provided that
$\int_{0}^{1}g_{h}(s)ds\neq\int_{0}^{1}g^{2}(s)ds$.

We now consider a feasible statistic for analyzing the dynamics of the
$(a_{t})$ process. Let us introduce first the kernel estimator of the non-
constant probability

$\widehat{P(a_{t}=1)}=\sum_{i=1}^{n}w_{ti}a_{i},$

with $w_{ti}=\left(\sum_{j=1}^{n}K_{tj}\right)^{-1}K_{ti}$, and

$K_{ti}=\left\\{\begin{array}[]{c}K((t-i)/nb)\quad\mbox{if}\quad t\neq i\\\
0\quad\mbox{if}\quad t=i,\\\ \end{array}\right.$

where $K(\cdot)$ is a kernel function on the real line, and $b$ is the
bandwidth fulfilling the following conditions.

######  Assumption 2 (Kernel and bandwidth).

  * (a)

$K(\cdot)$ is a continuous kernel function defined on the real line with
compact support, such that $0\leq\sup_{z}K(z)<R$ for some finite real number
$R$, and $\int_{-\infty}^{\infty}K(z)dz=1$.

  * (b)

As $n\to\infty$, $nb^{4}+\frac{1}{nb^{2}}\to 0$.

Plugin the above kernel estimator in (7), we get the adaptive dependence
structure estimation for the $m$ first lags:

$\check{A}_{m}=\left(\check{\gamma}_{a}(1,1)/\check{\gamma}_{a}(0,1),\dots,\check{\gamma}_{a}(m,1)/\check{\gamma}_{a}(0,1)\right)^{\prime},$
(7)

where

$\check{\gamma}_{a}(h,1)=(n-h)^{-1}\sum_{t=1+h}^{n}(a_{t}-\widehat{P(a_{t}=1)})(a_{t-h}-\widehat{P(a_{t-h}=1)}).$

The following proposition states the asymptotic equivalence between
$\overline{A}_{m}$ and $\check{A}_{m}$.

######  Proposition 2.5.

Suppose that $(a_{t})$ is independent, and such that Assumption 1 and 2 hold
true. Then, as $n\to\infty$ we have,
$\sqrt{n}\left(\overline{A}_{m}-\check{A}_{m}\right)=o_{p}(1).$

We are now in position to introduce the following feasible test statistic

$\check{\mathcal{S}}_{m}^{(a)}=n\check{\omega}^{-1}\check{A}_{m}^{\prime}\check{A}_{m},$
(8)

where

$\check{\omega}:=\frac{n^{-1}\sum_{t=2}^{n}(a_{t}-\widehat{P(a_{t}=1)})^{2}(a_{t-1}-\widehat{P(a_{t-1}=1)})^{2}}{\left[n^{-1}\sum_{t=1}^{n}(a_{t}-\widehat{P(a_{t}=1)})^{2}\right]^{2}}.$

From Proposition 2.3 and 2.5, at the asymptotic level $\alpha$,
$\widetilde{H}_{0}$ is rejected if
$\check{\mathcal{S}}_{m}^{(a)}>\chi_{m,1-\alpha}^{2}$. The corresponding test
is denoted by $\check{Q}_{m}$.

##  3 Numerical experiments

In this section, we first study the finite sample behaviors of the different
tools introduced in the paper by means of Monte Carlo experiments. Then the
real data taken from the Santiago stock exchange will be considered.

###  3.1 Monte Carlo experiments

####  3.1.1 Empirical size

We simulated $N=5000$ independent trajectories of independent $(a_{t})$
according to the following cases:

  * 1-

The probability is constant: $P(a_{t}=1)=0.6$.

  * 2-

The probability is time-varying: $P(a_{t}=1)=g(t/n)$ where $g(r)=0.4$ for
$(0,0.4]$, $g(r)=2r-0.4$ for $(0.4,0.6]$, and $g(r)=0.8$ for $(0.6,1]$.

The sample sizes are $n=200,400,800$. The nominal asymptotic level of the
tests is $\alpha=5\%$. Tables 2 and 3 display the outputs of the $Q_{m}$ and
$\check{Q}_{m}$ tests. Tables 5 and 6 give the relative frequencies where the
components of $\widehat{A}_{m}$ and $\check{A}_{m}$ are outside its 95%
confidence bounds.

From Tables 2 and 5, it can be seen that the adaptive $\check{Q}_{m}$ and
$\check{A}_{m}$ have similar results to those of the $Q_{m}$ and
$\widehat{A}_{m}$ when $P(a_{t}=1)$ is constant and $(a_{t})$ is independent.
This can be explained by the fact that all the tools presented in the paper
are all valid in the case 1. Nevertheless, from Tables 3 and 6, we can notice
that the $Q_{m}$ and $\widehat{A}_{m}$ do not display satisfactory outputs as
the relative rejection frequencies tend to 100% as the sample size is
increased. In particular, for large $h$, $\widehat{A}_{m}$ can spuriously
suggest the existence of long memory effects for the trade/no trade sequence.
Clearly this is the consequence of the non-constant zero return probability.
In contrast, we can see that the adaptive $\check{Q}_{m}$ and $\check{A}_{m}$
have a good control of the type I error in general.

####  3.1.2 Empirical power

We simulated $N=5000$ of the following sequence inspired by Romano and Thomb
(1996): $a_{t}=\dot{a}_{t}\dot{a}_{t-1}$, where $(\dot{a}_{t})$ is an iid
sequence such that $P(\dot{a}_{t}=1)=0.6$ is constant. Note that in view of
the empirical size part, the comparison is fair as $P(a_{t}=1)$ is constant.
For conciseness, we only display the outputs for the $\check{Q}_{m}$ and
$Q_{m}$ tests in Table 4. The results for the $\widehat{A}_{m}$ and
$\check{A}_{m}$ lead to similar results. When the sample is small, we can note
some loss of power of the $\check{Q}_{m}$ when compared to the $Q_{m}$ test.
This can be explained by the non-parametric estimation of $P(a_{t}=1)$.
However, the abilities of detecting a dependence structure are similar for
sample sizes commonly encountered in practice.

###  3.2 Real data analysis

The dependence structure of different stocks taken from the Santiago Stock
market are investigated. Recall that the behaviors of $P(a_{t}=1)$ for these
stocks are described in the Introduction (in particular see Figures 1-4). The
sample sizes and the empirical means of the $a_{t}$’s are given in Table 1.
The effects of smooth long run changes and abrupt breaks are illustrated
considering the Conchatoro, Cencosud, Security, Provida, Cruzados, Blanco y
Negro stocks, see Figure 5. The stationary $(a_{t})$ case is studied using the
Lipigas and CLC stocks, see Figure 6.

From Figure 5, it can be seen that the $\widehat{A}_{m}$ lead to detect long-
run dependence in the $(a_{t})$ process. Nevertheless, it is likely that such
a long-run dependence is spurious as different kinds of non-constant
$P(a_{t}=1)$ can be observed for these stocks. Conversely, when the time-
varying $P(a_{t}=1)$ is adequately taken into account by using the
$\overline{A}_{m}$, the dependence structure seems only short-run. Let us now
study the stationary Lipigas and CLC stocks. From Figure 6, it emerges that
when $P(a_{t}=1)$ seems constant, the $\widehat{A}_{m}$ and the
$\overline{A}_{m}$ lead to the same conclusion: the dependence structure is
again short-run. Note that for all the stocks which outputs are displayed in
Figures 5 and 6, the $\overline{Q}_{5}$ rejects the independence hypothesis at
the 5% level whether the $P(a_{t}=1)$ seems constant or not (not displayed
here).

##  4 Conclusion

Determining the relevant past of the daily price changes/no change categorical
sequence $(a_{t})$ may be of interest for financial times series modelling. As
an example, note that the $(a_{t})$ sequence shares the clustering properties
of the volatility in many cases, see Figures 1-4. Hence, our tools could help
to decide if $(a_{t})$ can be used as a covariate in some volatility models,
see e.g. Francq and Thieu (2019) for the GARCH-X models. Also, past values of
such a series are considered to specify the conditional probability of price
changes in Moysiadis and Fokianos (2014). Nevertheless, spurious persistency
or long-run effects assessments for the daily price changes/no change sequence
may be avoided taking into account a potential non-stationary behavior in the
data. It is found that the dependence structure of $(a_{t})$ is short-run in
general.

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## Proofs

###### Proof of Proposition 2.1.

Firstly, note that we have $\bar{a}-P(a_{t}=1)=O_{p}(n^{-\frac{1}{2}})$, so
that

$n^{\frac{1}{2}}\hat{\gamma}_{a}(h)=n^{\frac{1}{2}}\bar{\gamma}_{a}(h)+o_{p}(1),\>\mbox{for
all}\>h\in\\{1,\dots,m\\},$

where
$\bar{\gamma}_{a}(h)=(n-h)^{-1}\sum_{t=1+h}^{n}(a_{t}-P(a_{t}=1))(a_{t-h}-P(a_{t}=1))$.
Let us define
$\overline{A}_{m}=(\bar{\gamma}_{a}(1),\dots,\bar{\gamma}_{a}(m))^{\prime}$.
From the Central Limit Theorem (CLT) for martingale difference sequences (see
Theorem A.3 in Francq and Zakoïan (2019)), we have

$n^{\frac{1}{2}}\overline{A}_{m}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\mathcal{N}(0,\overline{\Sigma}),$

where $\overline{\Sigma}$ is a $m\times m$ dimensional diagonal matrix, with
diagonal component $P(a_{t}=1)^{2}(1-P(a_{t}=1))^{2}$. On the other hand, it
is easy to see that $\hat{\gamma}_{a}(0)\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}P(a_{t}=1)(1-P(a_{t}=1))$. Hence, the result follows
from the Slutsky Lemma. ∎

###### Proof of (4).

Let $\xi_{h,t}=(a_{t}-P(a_{t}=1))(a_{t-h}-P(a_{t-h}=1))$ and

$\widetilde{\Gamma}_{m}(u)=\left(\tilde{\gamma}_{a}(1,u),\dots,\tilde{\gamma}_{a}(m,u)\right)^{\prime}.$

The sequence $(\xi_{h,t})$ is a martingale difference, such that
$V(\xi_{h,t})=g^{2}(t/n)(1-g(t/n))^{2}+O(n^{-1})$, from the Lipschitz
condition with a finite number of breaks in Assumption 1. Then from Theorem
2.1 of Hansen (1992), we obtain

$\widetilde{\Gamma}_{m}(u)\stackrel{{\scriptstyle
d}}{{\longrightarrow}}\left(G(u),\dots,G(u)\right)^{\prime}.$

The desired result follows from the Continuous Mapping Theorem. ∎

###### Proof of Proposition 2.3.

Note that $E\\{(a_{t}-P(a_{t}=1))(a_{t-h}-P(a_{t-h}=1))\\}=0$. Then using the
Central Limit Theorem for independent but heterogeneous sequences, see
Davidson (1994), Theorem 23.6, we have

$\sqrt{n}\left(\tilde{\gamma}_{a}(1,1),\dots,\tilde{\gamma}_{a}(m,1)\right)^{\prime}\stackrel{{\scriptstyle
d}}{{\longrightarrow}}N(0,\varpi),$

where $\varpi=\int_{0}^{1}g^{2}(s)(1-g^{2}(s))^{2}ds$ is obtained using some
computations, and since $(a_{t})$ is independent. On the other hand, from the
Kolmogorov SLLN for independent but non-identically random variables (see, for
instance, Sen and Singer (1993), Theorem 2.3.10), we have

$\tilde{\gamma}_{a}(h,1)\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}\int_{0}^{1}g(s)(1-g(s))ds.$

Hence, the first result of Proposition 2.3 follows from the Slutsky Lemma.
Now, for the convergence of $\hat{\omega}$, using again the Kolmogorov SLLN
and some computations, we have

$n^{-1}\sum_{t=2}^{n}(a_{t}-P(a_{t}=1))^{2}(a_{t-1}-P(a_{t-1}=1))^{2}\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}\int_{0}^{1}g^{2}(s)(1-g^{2}(s))^{2}ds,$

$n^{-1}\sum_{t=1}^{n}\left(a_{t}-P(a_{t}=1)\right)^{2}\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}\int_{0}^{1}g(s)(1-g(s))ds.$

∎

###### Proof of Proposition 2.4.

From the Kolmogorov SLLN for independent but non identically random variables
(see, Sen and Singer (1993), Theorem 2.3.10), and using some computations, we
have

$\tilde{\gamma}_{a}(h,1)\stackrel{{\scriptstyle
a.s.}}{{\longrightarrow}}\int_{0}^{1}g_{h}(s)ds-\left(\int_{0}^{1}g(s)ds\right)^{2},$

from the dominated convergence Theorem, and for any $h\in\\{1,\dots,m\\}$.
Hence, under $\widetilde{H}_{1}$ the result follows. ∎

###### Proof of Proposition 2.5.

In this proof similar arguments to that of the proof of Theorem 2 in Xu and
Phillips (2008) are considered. As the break number is finite, we assume that
the function $g(\cdot)$ is continuous, without a loss of generality. Let us
introduce the short notations

$\hat{p}_{t}=\sum_{i=1}^{n}w_{ti}a_{i},\bar{p}_{t}=\sum_{i=1}^{n}w_{ti}p_{i},\quad
p_{t}=P(a_{t}=1),$

and $z_{i}=a_{i}-p_{i}$. We can write

$\hat{p}_{t}-\bar{p}_{t}=\frac{\frac{1}{nb}\sum_{i=1}^{n}K_{ti}z_{i}}{\frac{1}{nb}\sum_{j=1}^{n}K_{tj}}.$

From $\widetilde{H}_{0}$, $(z_{i})$ is an independent process, such that
$E(z_{i})=0$. In addition, from Lemma A(c) in Xu and Phillips (2008), we have
$\left(1/nb\right)\sum_{i=1}^{n}K_{ti}\to 1$. In view of the above arguments,
deduce that

$\displaystyle E\left(\frac{1}{nb}\sum_{i=1}^{n}K_{ti}z_{i}\right)^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{(nb)^{2}}\sum_{i=1}^{n}K_{ti}^{2}E(z_{i}^{2})$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{nb}\right)\left(sup_{i}K_{ti}\right)\left(\frac{1}{nb}\sum_{i=1}^{n}K_{ti}\right)$
$\displaystyle=$ $\displaystyle O\left(\frac{1}{nb}\right),$

Thus, we write

$\hat{p}_{t}-\bar{p}_{t}=O_{p}\left(\frac{1}{\sqrt{nb}}\right).$ (9)

On the other hand, we have

$\displaystyle\frac{1}{nb}\sum_{i=1}^{n}K_{[nr]i}p_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{nb}\sum_{i=1}^{n}K\left(\frac{[nr]-i}{nb}\right)g\left(\frac{i}{n}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{b}\left[\int_{1/n}^{2/n}K\left(\frac{[nr]-[ns]}{nb}\right)g\left(\frac{[ns]}{n}\right)ds+\dots\right.$
$\displaystyle+\left.\int_{1}^{(n+1)/n}K\left(\frac{[nr]-[ns]}{nb}\right)g\left(\frac{[ns]}{n}\right)ds\right]$
$\displaystyle=\frac{1}{b}\left(\int_{1/n}^{(n+1)/n}K\left(\frac{[nr]-ns}{nb}\right)g\left(s\right)ds\right)+O\left(\frac{1}{nb}\right)$
$\displaystyle\stackrel{{\scriptstyle
z=(s-r)/b}}{{=}}\int_{(\frac{1}{n}-r)/b}^{(1+\frac{1}{n}-r)/b}K\left(\frac{[nr]-n(r+bz)}{nb}\right)g\left(r+bz\right)dz+O\left(\frac{1}{nb}\right)$
$\displaystyle=\int_{-\infty}^{\infty}K\left(\frac{[nr]-nr}{nb}-z\right)g\left(r+bz\right)dz+O\left(\frac{1}{nb}\right),$

where the last equality is obtained for small enough $b$, and since a compact
support is assumed for $K(\cdot)$ in Assumption 2(a). Using the Lipschitz
condition in Assumption 1, deduce that

$\frac{1}{nb}\sum_{i=1}^{n}K_{[nr]i}p_{i}=g(r)+O(b)+O\left(\frac{1}{nb}\right).$
(10)

Now, writing

$\displaystyle
n^{-\frac{1}{2}}\sum_{t=1+h}^{n}(a_{t}-\hat{p}_{t})(a_{t-h}-\hat{p}_{t-h})-n^{-\frac{1}{2}}\sum_{t=1+h}^{n}(a_{t}-p_{t})(a_{t-h}-p_{t-h})$
$\displaystyle=$ $\displaystyle
n^{-\frac{1}{2}}\sum_{t=1+h}^{n}(a_{t}-p_{t})(p_{t-h}-\bar{p}_{t-h})+n^{-\frac{1}{2}}\sum_{t=1+h}^{n}(a_{t}-p_{t})(\bar{p}_{t-h}-\hat{p}_{t-h})$
$\displaystyle+$ $\displaystyle
n^{-\frac{1}{2}}\sum_{t=1+h}^{n}(p_{t}-\bar{p}_{t})(a_{t-h}-p_{t-h})+n^{-\frac{1}{2}}\sum_{t=1+h}^{n}(p_{t}-\bar{p}_{t})(p_{t-h}-\bar{p}_{t-h})$
$\displaystyle+$ $\displaystyle
n^{-\frac{1}{2}}\sum_{t=1+h}^{n}(p_{t}-\bar{p}_{t})(\bar{p}_{t-h}-\hat{p}_{t-h})+n^{-\frac{1}{2}}\sum_{t=1+h}^{n}(\bar{p}_{t}-\hat{p}_{t})(a_{t-h}-p_{t-h})$
$\displaystyle+$ $\displaystyle
n^{-\frac{1}{2}}\sum_{t=1+h}^{n}(\bar{p}_{t}-\hat{p}_{t})(p_{t-h}-\bar{p}_{t-h})+n^{-\frac{1}{2}}\sum_{t=1+h}^{n}(\bar{p}_{t}-\hat{p}_{t})(\bar{p}_{t-h}-\hat{p}_{t-h}),$

and using (9) and (10), the desired result follows. ∎

## Tables and Figures

Table 1: The sample sizes $n$ and $\bar{a}=n^{-1}\sum_{t=1}^{n}a_{t}$ for
different stocks taken from the Chilean stock market.

| n | $\bar{a}$
---|---|---
CONCHATORO | 5188 | 0.83
CENCOSUD | 5198 | 0.89
SECURITY | 5188 | 0.62
PROVIDA | 5179 | 0.59
CRUZADOS | 2584 | 0.29
BN | 1958 | 0.48
LIPIGAS | 896 | 0.57
CLC | 1958 | 0.49

Table 2: The relative rejections frequencies of the tests for the independence
of the $(a_{t})$ sequence for 5 lags. The zero returns probability is constant
with independent $(a_{t})$.

$n$ | 200 | 400 | 800
---|---|---|---
$Q_{m}$ | 4.56 | 4.70 | 4.90
$\check{Q}_{m}$ | 4.88 | 4.76 | 4.94

Table 3: The same as above but for a time-varying zero returns probability
with independent $(a_{t})$.

$n$ | 200 | 400 | 800
---|---|---|---
$Q_{m}$ | 81.74 | 98.42 | 100.00
$\check{Q}_{m}$ | 4.88 | 5.56 | 5.62

Table 4: The relative rejections frequencies of the tests for the independence
of the $(a_{t})$ sequence for 5 lags. The zero returns probability is constant
with dependent $(a_{t})$.

$n$ | 100 | 200 | 400 | 800
---|---|---|---|---
$Q_{m}$ | 85.82 | 99.92 | 100.00 | 100.00
$\check{Q}_{m}$ | 75.54 | 99.56 | 100.00 | 100.00

Table 5: The relative frequencies for the components of $A_{m}$ and $A_{m}$
being outside the 95% asymptotic confidence bounds. The constant probability
case.

| $h$ | 1 | 2 | 3 | 4 | 5 | 20 | 40 | 60
---|---|---|---|---|---|---|---|---|---
$n=200$ | $\widehat{A}_{m}$ | 4.38 | 4.28 | 4.86 | 4.52 | 4.28 | 4.02 | 2.64 | 1.78
$\check{A}_{m}$ | 4.64 | 4.24 | 5.08 | 4.64 | 4.48 | 4.16 | 2.90 | 1.88
$n=400$ | $\widehat{A}_{m}$ | 4.84 | 4.98 | 4.68 | 5.06 | 4.70 | 4.66 | 3.66 | 3.30
$\check{A}_{m}$ | 5.02 | 5.18 | 4.76 | 5.00 | 4.82 | 4.62 | 3.80 | 3.44
$n=800$ | $\widehat{A}_{m}$ | 5.06 | 4.62 | 5.18 | 4.88 | 4.80 | 4.70 | 4.32 | 4.08
$\check{A}_{m}$ | 5.10 | 4.76 | 5.24 | 4.84 | 4.84 | 5.02 | 4.34 | 4.14

Table 6: The same as above, but for the time-varying probability case.

| $h$ | 1 | 2 | 3 | 4 | 5 | 20 | 40 | 60
---|---|---|---|---|---|---|---|---|---
$n=200$ | $\widehat{A}_{m}$ | 51.00 | 51.12 | 49.76 | 50.42 | 49.66 | 34.80 | 9.98 | 1.16
$\check{A}_{m}$ | 5.24 | 4.66 | 4.64 | 4.66 | 4.58 | 3.84 | 3.62 | 2.48
$n=400$ | $\widehat{A}_{m}$ | 79.02 | 78.80 | 78.00 | 79.52 | 78.16 | 73.28 | 61.78 | 42.16
$\check{A}_{m}$ | 5.48 | 5.32 | 5.08 | 5.42 | 5.24 | 4.60 | 4.06 | 4.02
$n=800$ | $\widehat{A}_{m}$ | 97.06 | 97.34 | 97.14 | 97.04 | 97.20 | 96.10 | 95.28 | 92.70
$\check{A}_{m}$ | 5.44 | 5.42 | 5.40 | 5.34 | 4.82 | 5.14 | 4.92 | 4.88

Figure 1:  The log-returns of the Conchatoro and Cencosud stocks. A smooth
increasing non-stationary behavior in the 2000’s can be observed. The kernel
smoothing of the $a_{t}$ values are displayed in full line. Data source: Yahoo
Finance.

Figure 2:  The same as for Figure 1, but for the Security and Provida stocks.
The liquidity levels seem to display abrupt breaks due to specific events in
the company’s histories.

Figure 3:  The same as for Figure 1, but for the Blanco y Negro and Cruzados
stocks. The liquidity levels seem to have a smooth decreasing behavior.

Figure 4:  The same as for Figure 1, but for the Lipigas and Clinica Las
Condes stocks. The liquidity levels seem to be stationary.

Figure 5:  The dependence structure plots for stocks that seem to exhibit a
non-stationary $(a_{t})$.

Figure 6:  The dependence structure plots for stocks that seem to exhibit a
stationary $(a_{t})$.