††thanks: Present Address: Institute for Theoretical Solid State Physics, IFW
Dresden, Helmholtzstrasse 20, 01069 Dresden, Germany

# Catalogue of $C$-paired spin-valley locking in antiferromagnetic systems

Mengli HU  Xingkai CHENG  Zhenqiao HUANG  Junwei LIU<EMAIL_ADDRESS>Department
of Physics, Hong Kong University of Science and Technology, Clear Water Bay,
Hong Kong, China

(August 28, 2024)

###### Abstract

Antiferromagnetic materials (AFMs) have been gaining lots of attentions due to
its great potential in spintronics devices and the recently discovered novel
spin structure in the momentum space, i.e., _C_ -paired spin-valley or spin-
momentum locking (CSVL), where spins and valleys/momenta are locked to each
other due to the crystal symmetry guaranteeing zero magnetization. Here, we
systematically studied CSVLs and proposed a general theory and algorithm using
little co-group and coset representatives, which reveals that 12 elementary
kinds of CSVLs, determined by the geometric relation of spins and valleys and
the essential symmetry guaranteeing zero magnetization, are sufficient to
fully represent all possible CSVLs. By combining the proposed algorithm and
high-throughput first-principles calculations, we predicted 38 magnetic point
groups and identified 140 experimentally verified AFMs that can realize CSVL.
Besides predicting new materials, our theory can naturally reveal underlying
mechanisms of CSVLs’ responses to external fields. As an example, two
qualitatively different types of piezomagnetism via occupation imbalance or
spin tilting were predicted in RuO2. The algorithm and conclusions can be
directly extended to the locking between valley/momentum and any other pseudo-
vector degree of freedom, e.g. Berry curvature, as exemplified in RuO2, and
the proposed concept and methodology can be straightforwardly applied to other
symmetry groups such as spin space group.

##  I INTRODUCTION

Figure 1: Three types of CSVLs, collinear (a), coplanar (b) and spatial (c),
based on the geometric relation of orientations of spin at the paired valleys.
White and blue balls indicate the position of valleys in the Brillouin zone
(BZ), and the arrow represents the corresponding spin orientation of the
valley. (a) and (b) show collinear and coplanar CSVL in the magnetic point
group (MPG) $422$. In (a), spins are along the $z$ direction when the valleys
are located at $(0,0,\pm w)$; In (b), spins lie in the $x-y$ plane with
valleys at $(\pm u,0,0)$ and $(0,\pm v,0)$. When the valleys deviate from the
high-symmetry points, collinear/coplanar CSVL transforms to the spatial CSVL
as shown by the black arrows in (a)/(b). In the spatial CSVL (c), the number
of coupled valleys in MPG $23$ triples with valleys no longer at the high-
symmetry line $(\pm u,\pm u,\pm u)$.

To access the spin degree of freedom (DF) of an electron, it is usually
necessary to couple it with other DFs such as local magnetic moment via Zeeman
interaction or angular momentum via spin-orbit coupling (SOC). The former one
supplies fertile ground for novel properties and applications of ferromagnetic
materials. The latter one has been the driving force of many intriguing
phenomena in condensed matter physics and materials sciences in the past few
decades, e.g., spin-valley locking (SVL) in two-dimensional materials [1],
spin texture in topological materials[2, 3], Rashba effect [4] and Dresselhaus
effect [5] in semiconductors, which are also the underlying mechanism or
building blocking of many thrilling discoveries such as valley Hall effect[1,
6], spin Hall effect[7, 8, 9, 10], giant spin-orbit torque[11, 12, 13] and
Majorana zero modes[14, 15, 16]. These phenomena seem to be very different,
while besides SOC as their same origin, they do share the other same important
feature, i.e, the contrasting spin polarization at time-reversal ($T$-) paired
momenta due to the Kramers’ theorem, and hence can be uniformly named as
$T$-paired spin-valley or spin-momentum locking (TSVL). It also means that
they will suffer from the same problems that spin can only be manipulated by
approaches affecting the $T$-symmetry and materials with these properties
cannot use spin DF to realize the non-volatile information storage by
themselves, which have become the bottlenecks hindering their practical
applications. To overcome these problems, in 2021, SVL was extended to the
antiferromagnetic systems (AFMs), where valleys with contrasting spin
splitting are paired by a crystal ($C$-) symmetry instead of the $T$-symmetry,
named $C$-paired SVL (CSVL) [17], which is demonstrated to generally exist in
AFMs with spin-spitting band structures [17, 18, 19].

In CSVL materials, any approach affecting the corresponding $C$-symmetry can
manipulate spin to induce unconventional phenomena. Typically, a strain or
electric field can introduce piezomagnetism (PZM) [17] or noncollinear spin
current (both spin-polarized current and pure spin current) in collinear AFMs
as theoretically predicted [17, 20, 21, 22] and experimentally confirmed in
MnTe [23] and RuO2[24, 25, 26]. To highlight their uniqueness, collinear AFMs
with SVL were named as altermagnet in 2022 [27, 28], and similar terminology
has been extended to describe the non-collinear spins in the real space [29].
Besides these novel properties, the predicted spin-splitting band structures
[18, 19, 17] and CSVL [17] have been directly observed in angle-resolved
photoemission spectroscopy in both collinear RuO2[30, 31], MnTe[32, 33] and
CrSb[34, 35, 36, 37, 38, 39] and noncollinear MnTe2[40, 41]. Moreover, the
electrical readout and deterministic $180^{\rm{o}}$ switching of the
N$\rm{\acute{e}}$el order has also been experimentally realized in Mi5Si3 [42]
and CrSb [43] thin films by utilizing novel properties of CSVL, which further
promotes applications of CSVL AFMs in ultra-fast, high-density, and highly
stable devices. A thorough CSVL theory is in urgent needs in understanding
unconventional properties of CSVLs and exploring more CSVL materials.

In this work, we developed a general and rigorous theory and algorithm to
determine all the locked valleys with contrasting spin polarization, where the
point operations in a magnetic space group (MSG) determine the allowed band
splitting, the little co-group of a valley $\overline{G}^{\bm{k}}$ constrains
the spin direction, and the coset representatives $\\{g_{m}\\}$ gives rise to
all the paired valleys with locked spins. We found that all CSVLs can be
classified into three types, collinear, coplanar or spatial, according to the
direction of spins associated with the paired valleys, and each type of CSVL
can be further categorized into four elementary families characterized by the
symmetry guaranteeing zero magnetization in AFMs, and these 12 elementary
kinds of CSVLs can serve as the building blocks of all CSVLs. Based on our
theory and algorithm, 38 out of 122 MPGs were predicted to host CSVL and all
the possible CSVLs indeed can be completely classified into the 12 elementary
kinds. By combining with high-throughput first-principles calculations, 140
CSVL materials were screened out of 1794 experimentally verified AFMs in
MAGNDATA [44]. The full list can be found in the supplementary material (SM).

Besides cataloguing all CSVLs and predicting new CSVL materials, our theory is
able to reveal the fundamental difference between symmetries in determining
intrinsic properties of CSVL materials, which distinguishes from the
conventional symmetry analysis like Neumann’s principle that treats all
symmetries equally in the same way. AFMs in the same MSG but with different
CSVLs will have qualitatively different properties that are mainly determined
by symmetries in little co-group $\overline{G}^{\bm{k}}$ or the coset
representatives $\\{g_{m}\\}$ but not other symmetries in MSG, which also
corresponds to two qualitatively different underlying mechanisms,
respectively. We take RuO2 as a representative to show its two different types
of PZM via either occupation imbalance determined by $\\{g_{m}\\}$ or spin
tilting determined by $\overline{G}^{\bm{k}}$, which is beyond conventional
PZM theories but can be understood naturally in our new theory.

It is worth noting that all the results and conclusions above can be directly
extended to a more generic spin-momentum locking without any modification,
where the degenerate states located at paired momenta are unnecessary to be
band extrema and the spin could be any pseudo-vector DF, which is further
exemplified by the Berry curvature distribution of RuO2.

##  II General theory of CSVL

We first introduce the intuitive classification and criteria for CSVLs.
Following the definition, SVL can be intuitively classified into three
different types, collinear, coplanar and spatial (non-collinear and non-
coplanar), according to the relation of spin orientations of equivalent
valleys as shown in Fig. 1, which can also be precisely obtained and
identified by our SVL algorithm as illustrated later.

Besides the apparent geometric meaning, the classification of three different
types has clear physical implications and can directly reveal the qualitative
responses of CSVL AFMs to external fields. Taking spin current generation [17]
as an example, the spin polarization of spin currents due to the collinear
CSVL (Fig. 1(a)) will be mainly along the $z$ direction, while the spin
polarization of spin currents due to the coplanar CSVL (Fig. 1(b)) will be in
the $x-y$ plane. Straightforwardly, different types of CSVLs should be able to
realized in the same AFMs depending on the positions of valleys, and the
transition between different types of CSVLs can occur as the Fermi level
$E_{f}$ changes. For example, CSVL will evolve from a collinear type into a
spatial type or further into a coplanar type as depicted by the white and blue
balls in Fig.1 (a) and (b), and consequently the spin polarization of
generated spin current will change the direction from the $z$ direction to the
$x-y$ plane. It is interesting that a direct transition between a collinear
and a coplanar CSVL cannot happen but have to go through a spatial CSVL.

To rigorously investigate SVL and its unconventional properties in AFMs
without any prior approximation, one usually needs to consider MSG since it
takes into account all possible symmetry constrains and interactions including
SOC. However, only point operations (rotation, reflection, and inversion) of
space group symmetries, together with the time-reversal operation, can pair
valleys and constrain spin orientation (detail derivation in Appendix I.A),
and hence the simpler MPG instead of more complicated MSG would be enough or
even better for studying SVL and its corresponding properties. Similarly, if
without SOC, the main properties associated with SVL will be determined mainly
by the spin point group instead of spin space group [45, 46, 47, 48, 49].

Figure 2: (a) Diagram of MPGs with respect to SVL. SVL cannot exist in MPGs
compatible with ferromagnetism and host $PT$. For all 58 MPGs compatible with
SVL, TSVL can exist in 4 MPGs, CSVL can exist in 38 MPGs, and CSVL and TSVL
can co-exist in 16 MPGs labeled by SVL. The full list of all MPGs can be found
in SM 1.D. (b) Inheritance of CSVLs among 38 CSVL MPGs. Grey solid line
connects a MPG and its subgroups. The background color denotes three types of
CSVL and the numbers in the bottom left corner of each type represent the four
elementary families. Hermann-Mauguin notation is used.

The primary criterion for a material to exhibit CSVL is the existence of spin-
splitting band structures, which can naturally occur in magnetic materials due
to exchange interactions between electrons and localized magnetic moments once
parity (inversion) times time-reversal ($PT$) symmetry is broken [50, 18].
Secondly, we focus on CSVL and hence exclude MPGs that are compatible with
TSVL since the main properties will be determined by TSVL instead of CSVL once
it exists. TSVL magnetic materials will be very similar to nonmagnetic
materials in many aspects because there is an effective time-reversal symmetry
$\tilde{T}^{2}=-1$ even the pure real time reversal symmetry is broken due to
the magnetic structure. For example, in AFMs with checkerboard and
N$\rm{\acute{e}}$el order, $T$ times fractional translation $\tau$ symmetry
$\tilde{T}=T\tau$ can be preserved. The final criterion is that at least one
symmetry except $PT$ symmetry must exist to ensure valleys and spins strictly
locked to each other, which, will be shown later, is actually the same
symmetry that guarantees the exactly zero net magnetization in AFMs and hence
can be naturally realized in AFMs.

Based on these three criteria above, all the MPGs can be classified into five
classes depending on the existence of spin-splitting band structure and
different type of SVL as shown in Fig. 2 (a). The first class includes all the
33 MPGs with $PT$ symmetry, and all the bands in the whole BZ must be spin
degenerate. The other four classes will allow spin-splitting bands at a
generic momentum. Among them, the second class is compatible with a non-zero
magnetization and hence can realize ferromagnetism, weak ferromagnetism
(canted antiferromagnet) or ferrimagnetism. Noted that the total magnetization
in these MPGs could be zero due to the magnetic anisotropy energy, and hence
the rigorous SVL may be also realized. However, it would be materials
dependent and the spin-dependent properties would possess both ferromagnetic
and antiferromagnetic nature. Many experimentally realized altermagnets are
belong to this class due to the vanishing but finite magnetization. Based on
the symmetries ensuring the zero magnetization and pairing equivalent momenta,
the left MPGs can be classed into another three classes: 33 MPGs are due to
crystal symmetries as listed in Fig. 2 (b), 4 MPGs are due to (effective) time
reversal symmetry and hence belong to TSVL MPGs, and the left 16 MPGs are due
to both crystal symmetry and time-reversal symmetry. It is worth noting that
except 38 CSVL MPGs that must be realized in AFMs, the other 20 MPGs that host
TSVL can be realized in both AFMs and non-magnetic (NM) materials, meanwhile
the 33 PT MPGs can also be realized in both AFMs and NMs. In this aspect, AFMs
and NMs are indistinguishable in the momentum space for 38 PT MPGs and 20 MPGs
hosting TSVL, and many of their properties are also qualitatively the same. To
distinguish them, one will has to rely on the information in real space and
use MSGs.

Figure 3: (a) Table of 12 elementary kinds of CSVLs. Four elementary families
of CSVLs are labeled by the abbreviation of hexagonal, trigonal, orthorhombic,
and tetragonal, and different types of CSVLs with each elementary family are
labeled by $L/P/S^{N_{1}(^{\prime})}_{N_{2}(^{\prime})}$, where
$N_{1/2}(^{\prime})$ labels the $N_{1/2}-$ fold rotational symmetry in the
$\overline{G}^{\bm{k}}$ and $\\{g_{m}\\}$, respectively. (b) Illustration of
Ort-$L_{2}^{2}$-Orth kind CSVL in MPG $422$. The band structures around the
valleys $V_{1/2}$ located at $(0,0,\pm w)$ and spins along $\bm{z}$ are
presented. In the lower panel, same kind CSVL located at $(0,\frac{1}{2},\pm
w)$ and $(\frac{1}{2},0,\pm w)$ are shown. Both CSVL cases are paired by
symmetry $2_{100}/2_{010}$ for their same CSVL kind and symmetry $4_{001}$
merely doubles the number of paired valleys by generating another copy
($V^{2}$) of the elementary CSVL ($V^{1}$) but does not introduce any new
property. (c) Two qualitatively different mechanisms generating PZM in
Ort-$L_{2}^{2}$-Orth: spin tilting presented by red filled arrow (top panel)
and occupation imbalance reflected by the relative different Fermi levels of
the valleys (bottom panel).

###  II.1 SVL algorithm and its application

Now we introduce the rigorous SVL algorithm that can determine all the paired
valleys and the associated orientation of spins. We found that two
ingredients, the little co-group of valley $\overline{G}^{\bm{k}}$ and the
coset representatives $\\{g_{m}\\}$, can fully capture and determine all the
features of any SVL with the former uniquely constraining the spin’s
orientation of the valley $\bm{k}$ and the latter fully determining all the
paired valleys and the relation of associate spins, respectively. The full SVL
algorithm consists of three steps: (1) Get the MPG of a given AFM, labelled as
$\overline{G}$; (2) For a valley at $\bm{k}$, find its little co-group
$\overline{G}^{\bm{k}}$ by checking whether a element $g$ in $\overline{G}$
satisfies $\bm{k}=\bm{k}g+\bm{K}$, where $\bm{K}$ is a reciprocal vector. CSVL
can only exist for valleys whose spin splitting or spin polarization $\bm{s}$
is allowed, and the necessary and sufficient condition is that
$\overline{G}^{\bm{k}}$ is compatible with ferromagnetism (i.e. belongs to 31
FM MPGs as shown in Fig. 2 (a)), otherwise such valleys will be spin-
degenerate and cannot realize SVL; (3) All the paired valleys
$\\{\bm{k}^{*}\\}$ and the associate spins $\\{\bm{s}^{*}\\}$ can be obtained
by $\\{\bm{k}^{*};\bm{s}^{*}\\}=\\{g_{m}\bm{k};g_{m}\bm{s}\\}$, where $g_{m}$
is the representative of coset $g_{m}\overline{G}^{\bm{k}}$ and can be
obtained via the coset decomposition
$\overline{G}=\sum_{m}g_{m}\overline{G}^{\bm{k}}$. The matrix presentation of
all $g_{m}$ can be found in Appendix I.C.

We applied this algorithm for all the MSGs to enumerate all the CSVLs, and
found that all the CSVLs indeed belong to collinear ($L$), coplanar($P$), or
spatial ($S$) as shown in Fig. 1, which further confirms the intuitive
analysis above. The CSVL type can also be rigorously determined based on
$\overline{G}^{\bm{k}}$ without knowing other details as following:

$\displaystyle\forall\bm{k},\exists(-)N_{\bm{\alpha}}\in\overline{G}^{\bm{k}}\leftrightarrow
L,$ (C.1)
$\displaystyle\forall\bm{k},\exists(-)N^{\prime}_{\bm{\alpha}}\in\overline{G}^{\bm{k}}\rightarrow
P,$ (C.2)
$\displaystyle\forall\bm{k},\exists(-)N_{\bm{\alpha_{k}}}\in\overline{G}^{\bm{k}},{\bm{\alpha_{k}}}\perp{\bm{\alpha}}\rightarrow
P$ (C.3)

, where $\bm{k}\in\\{\bm{k}^{*}\\}$, $-$ is the inversion operation,
$N_{\bm{\alpha}}$ represents the $N-$fold rotation operation ($N>1$) along
$\bm{\alpha}$ axis and $N^{\prime}$ represents $N-$fold rotation times time-
reversal symmetry. $\leftrightarrow$ and $\rightarrow$ denote the necessary
and sufficient, and sufficient conditions, respectively.

As an example shown in Fig.1 (a), the MPG is $\overline{G}=422$ (in Hermann-
Mauguin notation) and valleys are at $\bm{k}=(0,0,w)$. Since $4_{001}$ and
$2_{100}$ are the generators of $422$, the little co-group of valley will be
$\overline{G}^{\bm{k}}=4=\\{1,4^{+}_{001},2_{001},4^{-}_{001}\\}$. Clearly,
$4$ is among 31 FM MPGs listed in SM 1.D and belongs to ferromagnetism, thus
CSVL can exist. Then through the coset decomposition
$\overline{G}=\overline{G}^{\bm{k}}+2_{100}\overline{G}^{\bm{k}}$, the coset
representatives $\\{g_{m}\\}=\\{1,2_{100}\\}$ gives all the paired valleys and
spins as $\\{\bm{k}^{*};\bm{s}^{*}\\}=\\{(0,0,\pm w);(0,0,\pm m_{z})\\}$,
which belongs to collinear CSVL. The same conclusion can also be drawn by
$\overline{G}^{\bm{k}}$ since the condition C.1 is satisfied as
$\forall\bm{k},\ 4_{001}\in\overline{G}^{\bm{k}}$. The same procedure was
conducted for all the valleys/momenta of 38 CSVL MPGs, and the results were
summarized in Fig. 2. Counter-intuitively, a cubic crystal can only allow the
spatial CSVL although it possesses the highest symmetry group. The reason is
that cubic crystal hosts three-fold rotational symmetries along body diagonals
which violates all the conditions allowing collinear (C.1) or coplanar CSVL
(C.2 and C.3).

Besides enumerating all the CSVLs, our CSVL algorithm can rigorously capture
the CSVL transition between different types, which can be described by another
coset decomposition and is fully predictable. Taking the collinear CSVL in
Fig.1(a) as an example, a valley at $\bm{k}=(0,0,w)$ splits into several
valleys at general momenta,
$\bm{k}=(0,0,w)\rightarrow\bm{k^{\prime}}=(u,v,w)$. Such a splitting can be
described as
$\overline{G}^{\bm{k}}=\overline{G}^{\bm{k^{\prime}}}+4^{+}_{001}\overline{G}^{\bm{k^{\prime}}}+2_{001}\overline{G}^{\bm{k^{\prime}}}+4^{-}_{001}\overline{G}^{\bm{k^{\prime}}}$,
which tells that each valley splits into four valleys paired by the coset
representatives as
$\\{\bm{k^{\prime}}^{*};\bm{s^{\prime}}^{*}\\}=\\{\bm{k^{\prime}},4^{+}_{001}\bm{k^{\prime}},2_{001}\bm{k^{\prime}},4^{-}_{001}\bm{k^{\prime}};\bm{s^{\prime}},4^{+}_{001}\bm{s^{\prime}},2_{001}\bm{s^{\prime}},4^{-}_{001}\bm{s^{\prime}}\\}$
and the CSVL transits from the collinear type into the spatial type. As a
throughout summary of all the CSVLs of 38 MPGs in Fig. 2, one can predict and
design CSVLs and the transition accordingly.

Figure 4:  A representative CSVL material RuO2. (a) Crystal structure of RuO2,
where the magnetic moments are labeled by red arrows. (b) Band structure of
RuO2 along high-symmetry lines, where the region of coupled valleys are
labelled by shadowed red and blue. The spin orientations of bands across Fermi
level are denoted. The spin polarized Fermi surfaces are plotted in the lower
right panel at $k_{z}=0$ (left) and $k_{z}=\frac{\pi}{\bm{c}}$ (right),
respectively. The 1st BZ is shown in the upper right panel with high-symmetry
points and corresponding spin polarization denoted. (c) Two types of PZM
generated due to different mechanisms under two different strain
$M^{y}=\Lambda^{y}_{xz}\sigma_{xz}$ and $M^{z}=\Lambda^{z}_{xy}\sigma_{xy}$.
The strained valley band structures under $+5\%$ strain strength are shown in
the right panels. The spin orientations before and after strain are labeled by
solid red and hollow green arrows, respectively. (d) The Berry
curvature($\Omega$) distribution in the 1st BZ with $k_{z}=0$. The color-map
ranging from blue to red indicates the strength of $\Omega_{z}$.

###  II.2 Elementary family of CSVL

Although all the CSVLs can be completely enumerated by our SVL algorithm
above, the connection between CSVLs of different MPGs is still unclear.
Typically, some types of CSVLs only exist in a subgroup but _not_ in its
parent group and vise versa as shown in Fig. 2, which stems from a simple fact
that the little co-group of a subgroup is not necessarily a subgroup of the
little co-group of its parent group. Such an inconsistency also means that the
classification merely using three different types cannot fully uncover the
relation of all the CSVLs.

To address this inadequacy, we carefully examined all the CSVLs enumerated by
our SVL algorithm above and found that (1) MPGs that can host CSVLs have at
least one subgroup as $\pm 6^{\prime}$, $32/3m$, $mm2/222$, and $\pm
4^{\prime}$, which are also the smallest group compatible with AFMs to
guarantee the zero net magnetization, and hence we define these MPGs as four
families of elementary CSVL MPGs; (2) The four families of elementary CSVL
MPGs correspond to four different crystal families, hexagonal, trigonal,
orthorhombic and tetragonal crystals, respectively; (3) CSVLs that can be
realized in these four families of elementary CSVL MPGs come from the simplest
$\overline{G}^{\bm{k}}$ and $\\{g_{m}\\}$ and have the smallest number of
paired valleys; (4) For any collinear and coplanar CSVL, the symmetry axes in
$\\{g_{m}\\}$ and $\overline{G}^{\bm{k}}$ must be orthogonal or parallel, and
symmetries in $\\{g_{m}\\}$ and $\overline{G}^{\bm{k}}$ can be broken
separately or simultaneously in orthogonal CSVLs but must be broken
simultaneously in parallel CSVLs; (5) Most importantly, a complicated CSVL
with more valleys that can be realized in other MPGs can always be taken as
multiple copies of the simplest CSVLs realized in the elementary CSVL MPGs.
The orthogonal and parallel CSVLs can exist in the same system whose
$\\{g_{m}\\}$ and $\overline{G}^{\bm{k}}$ can be broken separately. Based on
these discoveries, we propose four elementary families of CSVLs as a new
dimension and feature to describe all the CSVLs.

Together with three different types proposed above, 12 elementary kinds of
CSVLs listed in the table in Fig. 3 (a) can completely characterize and serve
as building blocks for all the CSVLs as also indicated by the subscript
numbers in the left bottom corner of each type of CSVLs in 38 CSVL MPGs in
Fig. 2. They can be concretely labeled as
$F-T^{N_{1}(^{\prime})}_{N_{2}(^{\prime})}-R$, where $F$=Hex, Tri, Ort, or Tet
stands for the hexagonal, trigonal, orthorhombic or tetragonal family, and
$T=L,P$ or $S$ means collinear, coplanar or spatial type, the superscript
$N_{1}(^{\prime})$ and subscript $N_{2}(^{\prime})$ represent generators in
$\overline{G}^{\bm{k}}$ and $\\{g_{m}\\}$, and $R=$ Orth or Para means
orthogonal or parallel between rotation axes of $N_{1}(^{\prime})$ and
$N_{2}(^{\prime})$. Noted that for a more compact notation, the suffix $-R$
can be removed since the 12 elementary kinds can only belong to either -Orth
or -Para but not both.

Now it is clear that it is the elementary family instead of the geometric
relation of spins’ orientation inherited between different CSVL MPGs as shown
in Fig. 2. For example, MPG $432$ can only realize the spatial type of CSVLs,
while it can realize both trigonal and orthorhombic CSVLs since its subgroups
contain $32$ and $222$. On the contrary, although $4/mmm$ can realize all
collinear, coplanar and spatial types of CSVLs, it can only realize
orthorhombic CSVL because its subgroups only contain $mm2$/$222$ but not other
elementary CSVL MPGs. This inheritance relation between elementary CSVLs and
more complicated CSVLs can be rigorously described by another coset
decomposition as $\overline{G}=\sum_{r}g_{r}\overline{G}^{E}$, where
$\overline{G}^{E}$ is the elementary CSVL MPG and $\\{g_{r}\\}$ are coset
representative, where the symmetries only generate additional copies of the
elementary CSVLs determined by $\overline{G}^{E}$ but not introduce too much
new physics. As exemplified by the collinear CSVLs in $422$ as shown in Fig. 3
(b), a complicated CSVL with four different valleys at $k=(0,\frac{1}{2},\pm
w)$ and $(\frac{1}{2},0,\pm w)$ paired by $2_{100}/2_{010}$ and $4_{001}$,
while only $2_{100}/2_{010}$ can flip the direction of spins and enforce the
opposite spin polarization for valleys at $k=(0,\frac{1}{2},\pm w)$ or
$(\frac{1}{2},0,\pm w)$ and the $4_{001}$ symmetry merely generates another
copy of the elementary CSVL paired by $2_{100}/2_{010}$, i.e.
Ort-$L^{2}_{2}$-Orth CSVL in MPG $222$, which can be rigorously obtained by
the coset decomposition $422=222+4_{001}222$. Therefore, the collinear CSVL in
$422$ will have the same property as the elementary Ort-$L^{2}_{2}$-Orth CSVL
in MPG $222$. Breaking the symmetry $4_{001}$ will simply reduce the number of
paired valleys from four to two and only the symmetry $2_{100}/2_{010}$
matters for the unconventional properties arisen from CSVLs.

More importantly, without any complicated symmetry analysis and calculation,
the elementary CSVL has already been able to capture the main properties of
AFMs arisen from CSVLs. To elucidate this more clearly, we will take PZM as an
example. In the example of collinear CSVLs in $422$ and $222$ that are
Ort$-L^{N_{1}(^{\prime})}_{N_{2}(^{\prime})}-$Orth CSVL, they belong to
orthogonal CSVLs, which means the symmetries in $\overline{G}^{\bm{k}}$ and
$\\{g_{m}\\}$ are orthogonal to each other and can be broken separately to
realize two qualitatively different types of PZM. As illustrated in the top
panel of Fig. 3 (c), breaking symmetry in $\overline{G}^{\bm{k}}$ will induce
spin tilting for all the valleys to induce uncompensated net magnetization
although different valleys are still equivalent due to $\\{g_{m}\\}$, and the
direction of net magnetization will depend on $\\{g_{m}\\}$ and hence be in
$x-y$ plane for the case in Fig. 3 (c). This is like the conventional PZM due
to local magnetic moments’ nonequivalent rotation under strain [51], but it
occurs in momentum space for spin of itinerant electrons associated with
different valleys. On the other hand, as breaking symmetry in $\\{g_{m}\\}$,
the strain will break the equivalence of different valleys to induce different
energy shifts and hence lead to unequal occupation of spin-up and spin-down
electrons, and the net magnetization should be along $z$ direction since
spin’s orientation remains unchanged as shown in the bottom panel of Fig. 3
(c). We refer these two microscopic mechanisms as spin tilting and occupation
imbalance with the former driving AFM into weak ferromagnetism (canted AFM)
and the latter driving AFM into ferrimagnetism from the aspect of the real
space. Notably, these two types of PZMs may coexist for certain strains, and
both are qualitatively different from the conventional theory since the non-
zero net magnetization due to either spin tilting or occupation imbalance are
from itinerant electrons instead of local magnetic moments [51] and hence can
only happen in conducting AFMs.

In a short summary of our theory, with three different types and four
elementary families as two independent features and dimensions, all the CSVLs
can be fully characterized and classified into 12 elementary kinds, and the
main properties of CSVL AFMs will be determined by the elementary CSVLs
realized in these materials instead of other factors or material details.

##  III Application and material realization

With the complete CSVL theory and algorithm above, we employed the high-
throughput first-principles calculations to predict CSVL material candidates
from the magnetic database MAGNDATA [44] and identified 140 CSVL candidates
with the experimentally confirmed desired magnetic structures. The full list
of 140 CSVL materials and their band structures along high-symmetry lines can
be found in Appendix III.

In the following discussion, we choose RuO2 in our database as a
representative since it is one of the mostly studied AFMs now and focus on its
PZM which has not been studied. As shown in Fig. 4(a), RuO2 has the rutile
structure[30, 24] and belongs to the tetragonal crystal. The magnetic moments
in the same Ru atomic layer point to the same $z$ direction, and magnetic
moments of two adjacent Ru atomic layers are flipped by a nonsymmorphic
magnetic symmetry $\\{4^{\prime}_{001}|1/2,1/2,1/2\\}$ to form the $A-$type
AFM and its MPG is $\overline{G}=4^{\prime}/mm^{\prime}m$.

RuO2 is metallic with spin-splitting Fermi surfaces along $(u,\pm u,0)$ and
$(u,\pm u,0.5)$ as shown in Fig. 4 (b). The little co-group at $k=(u,\pm u,0)$
and $(u,\pm u,0.5)$ is $\overline{G}^{\bm{k}}=m^{\prime}m2^{\prime}$, and
clearly $\forall\bm{k},m\in\overline{G}^{\bm{k}}$, which satisfies the
collinear condition C.1. Together with Fig. 2, without any calculation of spin
polarization, we recognize that RuO2 can realize the elementary CSVL
Tet-$L^{2}_{4^{\prime}}$-Para and Ort-$L^{2}_{2}$-Orth, which is confirmed by
the spin textures obtained by the first-principles calculations as denoted by
red arrows in Fig. 4 (b). As a result, RuO2 can realize two different type of
PZMs with spin tilting or occupation imbalance under different strains. To
confirm it, two different types of shear strains $\sigma_{xz}$ and
$\sigma_{xy}$ are considered, which only break $\overline{G}^{\bm{k}}$ or
$\\{g_{m}\\}$ in Ort-$L^{2}_{2}$-Orth and correspond to tilt spin or
occupation imbalance, respectively. The results obtained by first-principles
calculations with SOC considered are shown in Fig. 4 (c). Both types of
strains induce a net magnetization where the direction of magnetization and
the microscopic mechanisms are shown in right panel of Fig. 4 (c). The spins
around valleys all tilt from $\bm{z}$ direction to $x-y$ plane although two
valleys are still equivalent under a shear strain $\sigma_{xy}$. On the other
hand, the spins are still along $z$ direction but the occupation of spin-up
and spin-down electrons are clearly imbalanced under a shear stain
$\sigma_{xz}$, which precisely demonstrates our theory above.

Since any pseudo-vector DF obeys the same transformation as spin under point
symmetry operations, our theory and algorithm above can be directly applied to
the locking between valley/momentum and any pseudo-vector DF. Here, we use
Berry curvature $\bm{\Omega}$ of RuO2 as an example. This
$\bm{\Omega}$-momentum locking phenomenon was first proposed in the TSVL
2H-MoS2 [1] and is the origin of valley Hall effect [6]. Based on our CSVL
results, the $\bm{\Omega}$-momentum locking should be the same elementary
kind, i.e., Tet-$L^{2}_{4^{\prime}}$-Para and Ort-$L^{2}_{2}$-Orth. In
addition, as
$\overline{G}=\\{\overline{G}^{E},-1\overline{G}^{E},2_{110}\overline{G}^{E},m_{110}\overline{G}^{E}\\}$
and $\overline{G}^{E}=4^{\prime}$, there are four copies of
$L^{2}_{4^{\prime}}$ after the decomposition with respect to the elementary
kind of CSVL. The distribution of $\bm{\Omega}(k_{x},k_{y},0)$ obtained by
first-principles calculations is shown in Fig. 4(d) which indeed follows the
prediction from our algorithm. These petal-shaped regions with same magnitude
and alternating signs of $\Omega_{z}$ are eight valleys belonging to
$L^{2}_{4^{\prime}}/L^{2}_{2}$ class, which will generate valley or spin Hall
effect that are expected to reveal richer phenomena or properties of CSVL
materials. Importantly, as long as the AFM symmetry strictly preserved, the
total ${\bm{\Omega}}$ must be zero and the anomalous Hall effect is not
allowed. However, same as the PZM discussed above, a symmetry breaking induced
by external strains will lead to non-zero net ${\bm{\Omega}}$ via either
imbalanced occupation or $\bm{\Omega}$ titling and hence the corresponding
anomalous Hall effect, which can be referred to piezo-Hall effect. Besides,
other approaches breaking the same crystal symmetry such as intrinsic
distortions or finite magnetization due to Dzyaloshinsky-Moriya interaction
will also generate net ${\bm{\Omega}}$ and the anomalous Hall effect.

##  IV Summary

In this work, we proposed a general theory and algorithm to categorize CSVLs
by using the little co-group $\overline{G}^{\bm{k}}$ to determine spin
orientations and the coset representative $\\{g_{m}\\}$ to paired equivalent
valleys, which reveals that all CSVLs and their properties can be described by
12 elementary kinds of CSVLs defined by the geometric relation of spins and
the essential symmetry guaranteeing the zero net magnetization. Combining this
algorithm and high-throughput first-principles calculations, we identified 140
CSVL materials with experimentally verified desired magnetic structures in
MAGNDATA[44]. As explicitly demonstrated in the representative CSVL materials
RuO2, besides serving as a powerful tool of cataloging and predicting CSVL
materials, our theory reveals microscopic mechanisms of CSVLs’ responses to
external perturbations and can be straightforwardly extended to study other
pseudo-vectors and valley coupled phenomena.

##  V Acknowledgement

This work is supported by National Key R&D Program of China (Grants
No.2021YFA1401500) and the Hong Kong Research Grants Council (16303821,
16306722 and 16304523).

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# Generalized Autoregressive Score asymmetric Laplace Distribution and Extreme
Downward Risk Prediction

Hong Shaopeng111e-mail<EMAIL_ADDRESS>
School of Statistics
Capital University of Economics and Trade
BeiJing, China

###### Abstract

Due to the skessed distribution, high peak and thick tail and asymmetry of
financial return data, it is difficult to describe the traditional
distribution. In recent years, generalized autoregressive score (GAS) has been
used in many fields and achieved good results. In this paper, under the
framework of generalized autoregressive score (GAS), the asymmetric Laplace
distribution (ALD) is improved, and the GAS-ALD model is proposed, which has
the characteristics of time-varying parameters, can describe the peak thick
tail, biased and asymmetric distribution. The model is used to study the
Shanghai index, Shenzhen index and SME board index. It is found that: 1) the
distribution parameters and moments of the three indexes have obvious time-
varying characteristics and aggregation characteristics. 2) Compared with the
commonly used models for calculating VaR and ES, the GAS-ALD model has a high
prediction effect.

_K_ eywords Generalized Autoregressive Score $\cdot$ Asymmetric Laplace
Distribution $\cdot$ VaR $\cdot$ ES

##  1 Introduction

As a special kind of economic system, the financial market has emerged with a
series of unique macro laws, which are usually called "stylized facts". For
example, the return distribution is sharp and asymmetrical. Due to Knight
uncertainty, static parameter models usually cannot accurately describe the
characteristics of financial returns, and dynamic parameter models are needed
to characterize the distribution of returns.

The structure of this paper is as follows: The first part reviews the research
results of asymmetric Laplacian distribution and generalized autoregression at
home and abroad, the second part builds the Laplacian distribution based on
the generalized autoregressive score, and the third part uses generalized
autoregressive The Plass distribution conducts empirical research on China’s
Shanghai Stock Index, Shenzhen Index and SME Board Index. The fourth part is
conclusions and recommendations.

##  2 literature

###  2.1 Discussion on the distribution of returns

In risk management, VaR and ES have become more reliable risk measurement
indicators, and how to calculate VaR and ES more effectively has always been a
hot issue in risk management. The core of calculating VaR and ES is to
describe the distribution of its rate of return. However, as a special kind of
economic system, the financial system has many unique and typical facts. For
example, financial returns have typical facts such as volatility clustering,
autocorrelation, spikes and fat tails. It is unreasonable to assume that the
rate of return is normally distributed. Bollerslev (1987) [1] proposed to use
the t distribution to characterize the fat tail characteristics of the return
distribution. Often financial market returns show right-skewed
characteristics, and normal distribution and t-distribution, as distributions
with zero skewness, cannot be described. Hansen (1994)[2] proposed to use the
biased t distribution to simultaneously characterize the fat tail and biased
characteristics of the return distribution. However, in the financial market,
the asymmetry of gains and losses makes the distribution of returns appear
asymmetry in the left and right tails. Zhu Dongming and Galbraith (2010)[3]
proposed two asymmetric t distributions with higher degrees of freedom than
the biased t distribution based on the biased t distribution. These two t
distributions can better characterize the asymmetry of the return rate by
controlling the asymmetry of the left and right tails. Keming Yu and Zhang
(2005) [4] proposed an asymmetric Laplace distribution (ALD) controlled by
three parameters. Compared with other complex distributions (such as
generalized error distribution), ALD has fewer parameters, and each order
moment has a clear display expression.

Due to the superiority of ALD, a large number of scholars have conducted
research on ALD. Liu Jianyuan and Liu Qiongsun (2007)[5] used ALD to fit the
returns of China’s stock market, and compared the symmetric Laplace
distribution, the normal distribution and the asymmetric Laplace distribution
to calculate the effectiveness of VaR. The Plass distribution has the best
fitting result. Du Hongjun and Wang Zongjun (2013) [6] used the asymmetric
Laplacian distribution to calculate the dynamic risk VaR and CVaR. The
asymmetric Laplacian distribution can well describe the characteristics of
bias, spikes and fat tails. Liu Pan and Zhou Ruomei (2015) [7] used AEPD, AST
and ALD distributions to describe the typical facts of financial asset
returns, and found that the ALD distribution is significantly better than
other distributions in calculating the long VaR value of the bottom quantile.
. Based on ALD, Tang Jianyang et al. (2018) [8] proposed the AR-GJR-GARCH
model with ALD error distribution, which has a better performance in
calculating VaR and ES than traditional models that measure VaR and ES.

###  2.2 The Proposal and Application of Generalized Autoregressive Score

Creal (2012)[9] proposed the Generalized Autoregressive Scoring Model (GAS) as
a framework for time-varying parameter modeling, which has been widely used in
recent years. Such as GAS-EWMA (Lucas, 2016[10]), beta-t-EGARCH (Harvey and
Sucarrat, 2014[11]), GAS-Copula (Oh and Patton, 2016[12]), GAS-factor Copula
(leaf five First Class, 2018[13]), GAS-D-Vine-Copula (Zou Yumei, 2018[14]),
GJR-GARCH-skew t-GAS-copula (Xiao-Li Gong, 2019[15]) and GAS-EGARCH -EGB2 (Yao
Ping et al., 2019[16]). The above models have performed well in the field of
risk management.

It can be seen from the literature review that the asymmetric Laplacian
distribution has a good performance in fitting the rate of return, which can
characterize the typical facts of the peak, thick tail, bias and asymmetry of
the rate of return, and are similar. Compared with the normal distribution and
the t distribution, the asymmetric Laplacian distribution has a better
performance in calculating VaR and ES. However, it is impossible to describe
the time-varying nature of each order moment of the rate of return, and there
are defects in the empirical study. In recent years, the GAS model has been
proposed as a time-varying parameter modeling framework. A large number of
documents have proved that using the GAS framework to make the model dynamic
can improve the performance of the model. In summary, this article considers
expanding the asymmetric Laplace distribution under the GAS framework to make
its parameters dynamic, and strive to better describe the distribution of
returns and more accurate predictions of extreme risks.

The innovations made in this paper are as follows: 1) Use the GAS model to
extend the ALD distribution to the GAS-ALD distribution. 2) Apply GAS-ALD to
my country’s securities market to dynamically measure conditional mean,
conditional volatility, conditional skewness and conditional kurtosis. 3)
Compare the GAS-ALD model with mainstream VaR and ES models. On the one hand,
it enriches the calculation methods of VaR and ES, on the other hand, it can
provide investors with investment basis. It has theoretical and practical
significance.

##  3  Models

###  3.1 Generalized Autoregressive Score

The generalized autoregressive scoremodel is a data-driven model. It is a
unified framework for time-varying parameter modeling. Compared with other
models, GAS has the advantage that it makes full use of distributed
information to construct a time-varying parameter evolution process.

Suppose the conditional density of the rate of return y at time t is:

$\mathbf{y}_{t}|\mathbf{y}_{1:t-1}\sim
f\left(\mathbf{y}_{t};\mathbf{\theta}_{t}\right)\ \ \ \ $ (1)

Among them, $\mathbf{y}_{1:t-1}=(y_{1},\ldots,y_{t-1})$, which means all the
information of y in the past moment, $\mathbf{\theta}_{t}$ Is a time-varying
parameter set. From the formula (1), it can be seen that the conditional
distribution only depends on the information at the past moment and the
parameter set. The core of the GAS model is the parameter evolution process of
time-varying parameters ((2) ).

$\mathbf{\theta}_{t+1}\equiv\mathbf{\kappa}+\mathbf{A}\mathbf{s}_{t}+\mathbf{B}\mathbf{\theta}_{t}$
(2)

Among them, $\mathbf{\kappa}$ is a constant matrix, A and B are coefficient
matrices, then the score $s_{t}$ is the driving mechanism, usually defined as
(3) and (4) , (5) and (6). $\mathcal{I}_{t}\left(\mathbf{\theta}_{t}\right)$
is the Fisher information matrix of the time-varying parameter set. $\gamma$
is usually in $\\{0,\frac{1}{2},1\\}$, used to adjust the score to calculate
the final update item $s_{t}$, this article selects $\gamma=0$, that is,
$s_{t}=\mathrm{\nabla}_{t}$. It can be seen that the parameter evolution in
the GAS model is divided into two parts, the first part is driven by the score
function, and the second part is the AR(1) process.

$s_{t}=\mathbf{S}_{t}\left(\mathbf{\theta}_{t}\right)\mathrm{\nabla}_{t}\left(\mathbf{y}_{t},\mathbf{\theta}_{t}\right)$
(3)
$\nabla_{t}\left(\mathbf{y}_{t},\theta_{t}\right)\equiv\frac{f\left(\mathbf{y}_{t};\theta_{t}\right)}{\partial\theta_{t}}$
(4)
$\mathbf{S}_{t}\left(\mathbf{\theta}_{t}\right)\equiv\mathcal{I}_{t}\left(\mathbf{\theta}_{t}\right)^{-\gamma}$
(5)

$\mathcal{J}_{t}\left(\boldsymbol{\theta}_{t}\right)\equiv\mathrm{E}_{t-1}\left[\boldsymbol{\nabla}_{t}\left(\mathbf{y}_{t},\boldsymbol{\theta}_{t}\right)\boldsymbol{\nabla}_{t}\left(\mathbf{y}_{t},\boldsymbol{\theta}_{t}\right)^{\top}\right]$
(6)

###  3.2 Asymmetric Laplace distribution

The probability density function of the asymmetric Laplace distribution is as
(7):

$f(x;\mu,\sigma,p)=\frac{p}{\sigma(1+p^{2})}\left\\{\begin{array}[]{ll}\exp\left(-\frac{\mathrm{p}}{\sigma}(x-\mu)\right),&\mathrm{x}\geq\mu\\\
\exp\left(\frac{1}{\sigma\mathrm{p}}(x-\mu)\right),&\mathrm{x}<\mu\end{array}\right.$
(7)

The expressions of the moments of ALD are as follows:

$E(X)=\mu+\frac{\sigma}{\sqrt{2}}(\frac{1}{p}-p)\ $ (8)
$\operatorname{Var}(\mathrm{X})=\left[\frac{\sigma}{\sqrt{2}}\left(\frac{1}{\mathrm{p}}-\mathrm{p}\right)\right]+\sigma^{2}$
(9) $\text{ Skewness
}=2\frac{1/p^{3}-p^{3}}{\left(1/p^{2}+p^{2}\right)^{3/2}}$ (10) $\text{
Kurtosis }=6-\frac{12}{\left(1/p^{2}+p^{2}\right)^{2}}$ (11)

Figure 1: ALD probability density

###  3.3  Generalized Autoregressive Score asymmetric Laplace Distribution

From the (1), rewrite the (7) to make its parameters $\mu$, $\sigma$ and $p$
time-varying:

$\mathbf{y}_{t}\mid\mathbf{y}_{1:t-1}\sim
f\left(\mathbf{y}_{t};\mu_{t},\sigma_{t},p_{t}\right)$ (12)
$f(y_{t};\mu_{t},\sigma_{t},p_{t})=\frac{p_{t}}{\sigma_{t}(1+p_{t}^{2})}\left\\{\begin{array}[]{ll}\exp\left(-\frac{\mathrm{p_{t}}}{\sigma}(y_{t}-\mu_{t})\right),&\mathrm{x_{t}}\geq\mu_{t}\\\
\exp\left(\frac{1}{\sigma_{t}\mathrm{p_{t}}}(y_{t}-\mu_{t})\right),&\mathrm{y_{t}}<\mu_{t}\end{array}\right.$
(13)

The likelihood function of the asymmetric Laplace distribution is as follows:

$\ln f=\ln
p_{\mathrm{t}}+\ln\left(1+\mathrm{p}_{\mathrm{t}}^{2}\right)-\ln\left(\sigma_{\mathrm{t}}\right)+\left\\{\begin{array}[]{ll}-\frac{\mathrm{p}_{\mathrm{t}}}{\sigma}\left(\mathbf{y}_{\mathrm{t}}-\mu_{\mathrm{t}}\right),&\mathbf{y}_{\mathrm{t}}\geq\mu\\\
\frac{1}{\sigma\mathrm{p}_{\mathrm{t}}}\left(\mathbf{y}_{\mathrm{t}}-\mu_{\mathrm{t}}\right),&\mathbf{y}_{\mathrm{t}}<\mu_{\mathrm{t}}\end{array}\right.$
(14)

Set the parameter set to $\theta_{t}=\\{\mu_{t},\sigma_{t},p_{t}\\}$.
Calculate the score function:

$\nabla_{t}\left(\boldsymbol{y}_{t},\mu_{t}\right)=\frac{\partial
f\left(\boldsymbol{y}_{t};\mu_{t}\right)}{\partial\mu_{t}}=\left\\{\begin{array}[]{ll}\frac{\mathrm{p}_{\mathrm{t}}}{\sigma_{\mathrm{t}}},&\mathbf{y}_{\mathrm{t}}\geq\mu_{\mathrm{t}}\\\
-\frac{1}{\sigma_{\mathrm{t}}\mathrm{p}_{\mathrm{t}}},&\mathbf{y}_{\mathrm{t}}<\mu_{\mathrm{t}}\end{array}\right.$
(15)
$\nabla_{t}\left(\mathbf{y}_{t},\boldsymbol{\sigma}_{t}\right)=\frac{\partial\log
f\left(\mathbf{y}_{t};\boldsymbol{\sigma}_{t}\right)}{\partial\boldsymbol{\sigma}_{t}}=-\frac{1}{\boldsymbol{\sigma}_{t}}+\left\\{\begin{array}[]{ll}\mathrm{p}_{t}\left(\mathrm{y}_{\mathrm{t}}-\mu\right)\boldsymbol{\sigma}_{\mathrm{t}}^{-2},&\mathbf{y}_{\mathrm{t}}\geq\mu_{\mathrm{t}}\\\
-\frac{1}{\mathrm{p}_{\mathrm{t}}}\left(\mathbf{y}_{\mathrm{t}}-\mu\right)\boldsymbol{\sigma}_{\mathrm{t}}^{-2},&\mathbf{y}_{\mathrm{t}}<\mu_{\mathrm{t}}\end{array}\right.$
(16) $\nabla_{t}\left(\mathbf{y}_{t},p_{\mathrm{t}}\right)=\frac{\partial\log
f\left(\mathbf{y}_{t};p_{\mathrm{t}}\right)}{\partial\kappa_{\mathrm{t}}}=\frac{1}{p_{\mathrm{t}}}+\frac{2p_{\
mathrm{t}}}{1+p_{\mathrm{t}}^{2}}+\left\\{\begin{array}[]{ll}-\frac{1}{\sigma_{\mathrm{t}}}\left(\mathbf{y}_{t}-\mu\right),&\mathbf{y}_{t}\geq\mu_{t}\\\
-\frac{1}{\sigma_{\mathrm{t}}}\left(\mathbf{y}_{t}-\mu\right)p_{\mathrm{t}}^{-2},&\mathbf{y}_{t}<\mu_{t}\end{array}\right.$
(17)
$\nabla_{t}=\left[\nabla_{t}\left(\mathbf{y}_{t},\mu_{t}\right),\nabla_{t}\left(\mathbf{y}_{t},\boldsymbol{\sigma}_{t}\right),\nabla_{t}\left(\mathbf{y}_{t},\mathbf{p}_{t}\right)\right]$
(18)

The formula (2) can be written as:

$\left(\begin{array}[]{c}\mu_{t+1}\\\ \sigma_{t+1}\\\
\mathbf{p}_{t+1}\end{array}\right)=\left(\begin{array}[]{c}\mathbf{\kappa}_{1}\\\
\mathbf{\kappa}_{2}\\\
\mathbf{\kappa}_{3}\end{array}\right)+\left(\begin{array}[]{ccc}a_{1}&0&0\\\
0&a_{2}&0\\\
0&0&a_{3}\end{array}\right)\left(\begin{array}[]{c}\nabla_{t}\left(\mathbf{y}_{t},\mu_{t}\right)\\\
\nabla_{t}\left(\mathbf{y}_{t},\boldsymbol{\sigma}_{t}\right)\\\
\nabla_{t}\left(\mathbf{y}_{t},\mathbf{p}_{t}\right)\end{array}\right)+\left(\begin{array}[]{ccc}\mathbf{b}_{1}&0&0\\\
0&\mathrm{b}_{2}&0\\\
0&0&\mathrm{b}_{3}\end{array}\right)\left(\begin{array}[]{c}\mu_{t}\\\
\boldsymbol{\sigma}_{\mathrm{t}}\\\ \mathbf{p}_{t}\end{array}\right)$ (19)

(12), (13) and (19) are the GAS-ALD models.

###  3.4 Parameter Estimation

Compared with other models, the model under the GAS framework can accurately
estimate the parameter $\theta_{t}$ after giving the past information and
parameter vector $\phi={\mathbf{\kappa},A,B}$. Record $y_{1:T}$ as the sample
$T$ in $y_{t}$, and $\phi$ can be estimate by MLE(20)

$\displaystyle\widehat{\phi}\equiv\underset{\phi}{\operatorname{argmax}}\mathcal{L}\left(\phi;y_{1:T}\right)$
(20) $\displaystyle\text{S.T. }\mathcal{L}\left(\phi;y_{1:T}\right)=\log
f\left(y_{1};\theta_{1}\right)+\sum_{t=2}^{T}\log
f\left(y_{t};\theta_{T}\right)$

When t=1, $\theta_{1}=(I-B)^{-1}\mathbf{\kappa}$.

The following uses the BFGS algorithm to solve the likelihood function to
estimate $\widehat{\phi}$.

###  3.5 Risk prediction-VaR and ES

When making a step forward prediction, we can use the GAS-ALD model to
estimate the distribution of the return rate $\mathbf{y}_{t+1}$, thereby
predicting VaR and ES one step forward.

$\mathbf{y}_{t+1}\mid\mathbf{y}_{1:t}\sim
f\left(\mathbf{y}_{t+1};\mu_{t+1},\sigma_{t+1},p_{t+1}\right)$ (21)
$\left(\begin{array}[]{c}\mu_{t+1}\\\ \sigma_{t+1}\\\
\mathbf{p}_{t+1}\end{array}\right)=\left(\begin{array}[]{c}\mathbf{\kappa}_{1}\\\
\mathbf{\kappa}_{2}\\\
\mathbf{\kappa}_{3}\end{array}\right)+\left(\begin{array}[]{ccc}a_{1}&0&0\\\
0&a_{2}&0\\\
0&0&a_{3}\end{array}\right)\left(\begin{array}[]{c}\nabla_{t}\left(\mathbf{y}_{t},\mu_{t}\right)\\\
\nabla_{t}\left(\mathbf{y}_{t},\boldsymbol{\sigma}_{t}\right)\\\
\nabla_{t}\left(\mathbf{y}_{t},\mathbf{p}_{t}\right)\end{array}\right)+\left(\begin{array}[]{ccc}\mathrm{b}_{1}&0&0\\\
0&\mathrm{b}_{2}&0\\\
0&0&\mathrm{b}_{3}\end{array}\right)\left(\begin{array}[]{c}\mu_{t}\\\
\boldsymbol{\sigma}_{\mathrm{t}}\\\ \mathbf{p}_{t}\end{array}\right)$ (22)

The essence of VaR is a quantile, which can be easily obtained by using the
inverse function of the distribution function. When using the GAS-ALD model,
since the parameters are time-varying, the dynamic VaR can be obtained.

$VaR_{t+1}(\alpha)\equiv F^{-1}\left(\alpha;\theta_{t+1}\right)$ (23)

Because VaR does not satisfy the subadditivity, we use ES to supplement VaR.
ES, as the average part exceeding VaR, can be obtained using the formula (24).
Same as VaR, the result of ES is also dynamic.

$ES_{t+1}(\alpha)\equiv\frac{1}{\alpha}\int_{-\infty}^{VaR_{t+1}}zdF\left(z,\theta_{t+1}\right)$
(24)

###  3.6 Backtesting

For the statistical test of VaR, this article mainly uses UC test[17] , CC
test[18] and DQ test[19] . The above methods are relatively mature, so I will
not repeat them in this article. Because ES is generally too large relative to
VaR, the method of testing the backtest effect of VaR cannot be used to test
the backtest effect of ES. In this paper, the bootstrap test proposed in the
Mcneil et al.(2000) [20] is used to evaluate ES efficacy. The main idea of
this test is to construct multiple bootstrap residual samples and use them to
calculate test statistics and p-values.

####  3.6.1 Bootstrap

The main steps of the test are as follows[21, 22]:

First define the excess residual $e_{t}$:

$\ e_{t}=\frac{x_{t}-ES_{t}^{q}}{\sigma_{t}^{1/2}}$ (25)

Among them, $x_{t}$ is the true rate of return exceeding VaR, and $\sigma_{t}$
is the sum of variance of $x_{t}-ES_{t}^{q}$.

Then define the initial residual sequence $l_{t}$:

${\ \ \ \ \ \ \ l}_{t}=e_{t}-\frac{1}{z}\sum_{\eta=1}^{z}e_{\eta}\ $ (26)

Among them, $z$ is the number of samples exceeding the residual $e_{t}$.
Assuming that the mean of $l_{t}$ is $\overline{l}$ and the standard deviation
is $\sigma_{l}$, define the test statistic: $\delta(l)$.

$\ \ \delta(l)=\frac{\overline{l}}{\sigma_{l}}\ \ \ \ $ (27)

As you can see, $\delta(l)$ is the standardized residual.

To get the distribution of $\delta(l)$ and the value of $p$, we need to use
the bootstrap method to generate new samples. That is, generate $M$ random
numbers with uniform distribution within $\\{1,2,\ldots,M\\}$, and construct a
new sample with the residual sequence $l_{t}$ corresponding to the random
number, repeat N times (N=1000 in the following), N new bootstrap samples can
be obtained.

For each bootstrap sample i ($i=1,2,\ldots,M$), use equation (19) to calculate
$\delta_{i}(l)$, denoted as
$\\{\delta_{1}(l),\delta_{2}(l),\ldots,\delta_{N}(l)\\}$, without loss of
generality, the $\delta(l)$ of the initial sample can be recorded as
$\delta_{0}(l)$. Using the bootstrap method, the empirical distribution of
$\delta(l)$ can be obtained.

Since $e_{t}$ is often a right-skewed distribution, the selected hypothesis
$H_{1}$ of the bootstrap test is $E(e_{t})>0$, and the null hypothesis is
$H_{0}isE(e_{t})=0$. Calculate the sample proportion of
$\\{\delta_{1}(l),\delta_{2}(l),\ldots,\delta_{N}(l)\\}$ greater than
$\delta_{0}(l)$, which is the P value of the test. The larger the value of P,
the more unable to reject the null hypothesis, that is, the model is
considered to have better ES estimation accuracy.

###  3.7 Loss function

When using VaR and ES for risk management, not only need to be concerned about
the accuracy of VaR and ES for forecasting, but also need to be concerned
about the shape of the entire left tail region of the return distribution.
There must be a more accurate understanding of all losses. This article
intends to use two risk loss functions to evaluate the prediction of the
model.

####  3.7.1 Quantile Loss Function (QL)

Considering the loss function of VaR, this article chooses to use the quantile
loss function ((28) formula), which gives a higher weight than VaR. [23]

${\rm QL}_{t}^{\alpha}=(\alpha-I_{t}^{\alpha})(y_{t}-{\rm VaR}_{t}^{\alpha})$
(28)

####  3.7.2 Joint Loss Function (FZL)

Since there is no separate evaluation of ES loss function , referring to the
literature, this paper uses the FZL joint loss function to evaluate the
prediction accuracy of ES and VaR for extreme risks.[24, 25]

$\mathrm{FZL}_{t}^{\alpha}=\frac{1}{\alpha\mathrm{ES}_{t}^{\alpha}}I_{t}^{\alpha}\left(y_{t}-\mathrm{VaR}_{t}^{\alpha}\right)+\frac{\mathrm{VaR}_{t}^{\alpha}}{\mathrm{ES}_{t}^{\alpha}}+\log\left(-\mathrm{ES}_{t}^{\alpha}\right)-1$
(29)

##  4 Empirical Research

The indexes selected in this article are Shanghai Stock Index (000001.ss),
Shenzhen Index (399001.sz) and Small and Medium-sized Board Index (399005.sz).
The sampling time is 3000 days from July 6, 2007 to November 1, 2019. Data
source: wind database. Use R language for data processing.

###  4.1 Descriptive Statistics

First, the daily logarithmic rate of return $r_{t}=lnp_{t}-lnp_{t-1}$ is
calculated, and the basic statistics such as the mean and variance of the
corresponding sample rate of return are calculated(table 1222JB-p: P value of
Jarque-Bera test, used to test whether a set of samples can be considered as
coming from a normal population. From Table 1, we can see that the mean values
of the three samples are relatively close to 0; the variances are roughly the
same, and the variance of the small and medium board index is the largest;
from the perspective of skewness kurtosis and the JB test p-value, all three
samples have peak fat Typical facts with tails and right skewed, and none of
them obey normal distribution. From the kurtosis point of view, the excess
kurtosis of the three samples are all greater than 0, and the excess kurtosis
of the Shanghai Composite Index even reached 4.5072, which implies that we
need to use a distribution that can better describe the peak and thick tail to
describe the yield.

Table 1: Basic statistics | mean | variance | skewness | excess kurtosis | JB-p
---|---|---|---|---|---
000001.ss | -0.0001 | 0.0003 | -0.5256 | 4.5072 | 0.0000
399001.sz | -0.0001 | 0.0003 | -0.4660 | 2.9263 | 0.0000
399005.sz | 0.0001 | 0.0004 | -0.5475 | 2.6491 | 0.0000

Figure 2 333From left to right: Shanghai Stock Exchange Index Trend and
Return, Shenzhen Index Trend and Return, Small and Medium-sized Board Index
Trend and Returndraws a time series diagram of index trend and rate of return.
It can be clearly seen that the return rate sequence has volatility
clustering.

Figure 2: Index trend and return

###  4.2 In-sample GAS-ALD fitting

####  4.2.1 Model Fitting

Fit the GAS-ALD model to the data in the sample period. The results are as
follows. It can be seen that among the parameters of the three indices,
$a_{i}$ and $b_{i}$ ($i=1,2,3$) are significantly different from 0, indicating
that the parameters are time-varying.(table 2444 K-S test line reports as p
value, ***: significant at 99% level)

In order to test the fitting status of the GAS-ALD model to the distribution
of returns, this paper uses the GAS-ALD model for probability integral
transformation of the returns, and then performs the K-S test. The data after
the probability integral transformation of the original data accepts the null
hypothesis of the same distribution as the uniform distribution on (0,1). This
shows that the GAS-ALD model can better fit the distribution of returns.

Table 2: Parameter estimation resul | 000001.ss | 399001.sz | 399005.sz
---|---|---|---
$\mathbf{\kappa}_{1}$ | 0 | 0 | 0
$\mathbf{\kappa}_{2}$ | $-0.219^{***}$ | $-0.277^{***}$ | $-0.277^{***}$
$\mathbf{\kappa}_{3}$ | 0 | 0 | 0
$a_{1}$ | $-16.118^{***}$ | $-16.118^{***}$ | $-16.118^{***}$
$a_{2}$ | $-3.945^{***}$ | $-3.945^{***}$ | $-3.945^{***}$
$a_{3}$ | $-11.513^{***}$ | $-11.513^{***}$ | $-11.513^{***}$
$b_{1}$ | $3.897^{***}$ | $3.897^{***}$ | $3.897^{***}$
$b_{2}$ | $2.883^{***}$ | $2.593^{***}$ | $2.593^{***}$
$b_{3}$ | $3.897^{***}$ | $3.897^{***}$ | $3.897^{***}$
K-S test | 0.737 | 0.751 | 0.426

####  4.2.2 Analysis of dynamic changes of parameters and moments

This article draws the time-varying trajectories of three parameters (Figure
5, Figure 5 and Figure 5)555The three pictures from top to bottom are:
$\mu_{t}$,$\sigma_{t}$,$p_{t}$. It can be seen that the parameter trends of
the three index samples are basically the same. $\sigma_{t}$ has the strongest
aggregation, and there are two obvious peaks. And noticed that after the peak
of $\sigma_{t}$ appears, it is often accompanied by a rapid decline in $p_{t}$
and $\mu_{t}$.

Figure 3: Shanghai Composite Index Dynamic Parameters

Figure 4: Shenzhen Index Dynamic Parameters

Figure 5:  Small and medium board index Dynamic Parameters

Calculate conditional mean, conditional volatility, conditional skewness and
conditional kurtosis based on dynamic parameters. The results are shown in
Figure 8, Figure 8 and Figure 8666The four graphs from top to bottom are:
conditional mean, conditional volatility, conditional skewness and conditional
kurtosis Conditional volatility and conditional kurtosis have the strongest
aggregation effect. The second is the mean and kurtosis. It shows that each
moment has obvious time-varying characteristics and aggregation
characteristics. From the perspective of linkage, every sudden increase in
volatility will result in a sharp drop in skewness and an increase in
kurtosis. This shows that when the volatility rises, there will be a more
obvious sharp peak and thick tail right deviation, which will lead to
increased tail loss. And it can be seen that the change trends of the moments
of the three indices are roughly the same, which shows that the correlation of
the moments of the three indices is relatively large.

Figure 6: Shanghai Composite Index Moment Results

Figure 7: Shenzhen Index Moment Results

Figure 8: Small and medium board Index Moment Results

####  4.2.3 Time point analysis

Consider three time points for comparative analysis. The time points selected
in this article are 2008-08-18, 2011-03-10 and 2015-08-21, which represent the
period of the subprime mortgage crisis, the rising period, and the period of
leverage stock disasters, respectively. Plot the return distribution of each
index at each time point. As shown 12, 12 and 12777The green line is the
asymmetric Laplace distribution density obtained by the full sample fitting,
and the black line, red line and blue line are the asymmetric Laplace
distribution density at a specific time point.

Figure 9: Time selection

It can be found first. During the rising period, the distribution of the
return rate is sharp and thin compared to the full sample period, indicating
that most of the return is concentrated near 0, and the value of VaR and ES
can be appropriately lowered at this time. During the subprime mortgage crisis
and leveraged stock crisis, compared with the full sample, the tail thickness
increased, indicating that the probability of loss at this time is much
greater than that of the full sample and the rising period. At this time, the
risk exposure faced by investors will greatly increase. In addition, when a
leveraged stock disaster occurs, the thickness of the tails of the three index
yield distributions is greater than that of the subprime mortgage crisis,
indicating that the risk exposure faced by investors during the leveraged
stock disaster is greater than the risk exposure faced by the subprime
mortgage crisis. This is because in 2015, a large number of investors
increased their investment leverage through borrowing, resulting in losses
that they faced when the stock market plummeted far beyond their own
capabilities.

Figure 10: Time distribution of Shanghai stock index return rate

Figure 11: Time distribution of Shenzhen index return

Figure 12: Time distribution of small and medium board index returns

###  4.3 Rolling forecast VaR and ES

This article uses three types of models as a comparison. The first type is
historical simulation method and fixed parameter ALD distribution. Among them,
the historical simulation method uses short-term (25-day) sample quantiles,
and the second type is other distributions under the GAS framework. Normal
distribution (GAS-norm), t distribution (GAS-t) and partial t distribution
(GAS-skst). The third type is GARCH model. This article uses APARCH model with
partial t distribution error term (APARCH-skst) As a control group.

We selects the first 1500 days of the sample as the training group, rolls
forward forecast 1500 days, re-estimates the model every 5 days, and
calculates the VaR and ES of 1%.

####  4.3.1 Backtesting

Table 3 reports the results of the VaR backtest. Combining the UC, CC, and DQ
test results, GAS-norm and GAS-t performed the worst. This may be due to the
fact that the normal distribution and t distribution cannot describe the
asymmetry and biased characteristics of their returns. This shows that when
calculating VaR, the model needs to be able to describe the asymmetry and
biased characteristics of the rate of return. The GAS-ALD model has the best
performance in the UC, CC and DQ tests of the Shanghai Stock Index and the
Small and Medium-sized Board Index; the GAS-ALD model has the best performance
in the UC and CC tests of the Shenzhen Index, and the DQ test is slightly
worse. The GAS-skst model, which shows that the Shenzhen Index is more
suitable to use the GAS-skst model to calculate the VaR of 1%.

Table 4 reports the ES backtest results. It was found that GAS-norm, GAS-t and
GAS-skst performed poorly in calculating ES, and all rejected the null
hypothesis. The ES backtest of the Shanghai Composite Index shows that the
GAS-ALD model is slightly worse than APARCH-skst; the Shenzhen Index and the
small and medium-sized board index have the best performance in the ES
backtest of the GAS-ALD model.

Table 3: VaR Backtesting

| 000001.ss | 399001.sz | 399005.sz
---|---|---|---
| UC.pval | CC.pval | DQ.pval | UC.pval | CC.pval | DQ.pval | UC.pval | CC.pval | DQ.pval
Historical Simulation | 0.131 | 0.334 | 0.000 | 0.405 | 0.334 | 0.000 | 0.611 | 0.041 | 0.000
Fixed Parameter ALD | 0.217 | 0.252 | 0.000 | 0.142 | 0.035 | 0.000 | 0.702 | 0.046 | 0.000
GAS-norm | 0.003 | 0.006 | 0.001 | 0.01 | 0.022 | 0.004 | 0.001 | 0.002 | 0.006
GAS-t | 0 | 0.001 | 0.004 | 0 | 0 | 0 | 0 | 0 | 0
GAS-skst | 0.003 | 0.006 | 0.017 | 0.054 | 0.11 | 0.385 | 0.001 | 0.003 | 0.004
GAS-ALD | 0.142 | 0.252 | 0.534 | 0.142 | 0.252 | 0.241 | 0.089 | 0.17 | 0.238
APARCH-skst | 0.032 | 0.067 | 0 | 0.089 | 0.17 | 0.006 | 0.089 | 0.17 | 0.007

Table 4: ES Backtesting ES-1% | 000001.ss | 399001.sz | 399005.sz
---|---|---|---
Historical Simulation | 0.014 | 0.006 | 0.001
Fixed Parameter ALD | 0 | 0.247 | 0.032
GAS-norm | 0 | 0 | 0
GAS-t | 0 | 0 | 0
GAS-skst | 0 | 0 | 0
GAS-ALD | 0.271 | 0.527 | 0.533
APARCH-skst | 0.658 | 0.373 | 0.339

####  4.3.2 Loss function

Use the predicted VaR and ES results to calculate the QL loss function and FZL
loss function. The results are shown in Table 5888The QL loss function results
are all multiplied by $10^{3}$, and the reported value is the average.

For the QL loss function, the GAS-ALD model has the smallest average QL in the
Shanghai Stock Index and the Shenzhen Index, indicating that among the
compared models, GAS-ALD has better prediction accuracy for VaR prediction,
while in the small and medium board In the index, the average QL of APARCH-
skst is the smallest, followed by the average QL of GAS-ALD. This article
believes that this is because APARCH-skst cannot dynamically describe the
changes in distribution, but it can describe the asymmetric effect of the
return rate. There are often obvious asymmetric effects in the small and
medium board index, which makes APARCH-skst’s prediction accuracy for VaR
Slightly higher than the GAS-ALD model.

For the FZL loss function, the GAS-ALD model has the smallest average FZL
value, indicating that the GAS-ALD model has better ES prediction accuracy
than the control model.

Table 5: Loss Function | 000001.ss | 399001.sz | 399005.sz
---|---|---|---
Loss Function | QL | FZL | QL | FZL | QL | FZL
Historical Simulation | 7.13 | -2.629 | 7.428 | -2.594 | 7.5 | -2.611
Fixed Parameter ALD | 7.198 | -2.627 | 7.501 | -2.597 | 7.399 | -2.589
GAS-norm | 5.599 | -2.921 | 6.693 | -2.621 | 6.812 | -2.582
GAS-t | 5.852 | -2.768 | 7.081 | -2.491 | 7.29 | -2.392
GAS-skst | 6.301 | -2.711 | 7.74 | -2.444 | 7.749 | -2.436
GAS-ALD | 5.543 | -2.994 | 6.631 | -2.736 | 6.716 | -2.706
APARCH-skst | 5.453 | -2.984 | 6.839 | -2.833 | 6.167 | -2.614

##  5 Conclusion

In this paper, by improving the asymmetric Laplace distribution so that its
parameters are all time-varying, the GAS-ALD model is proposed and used to
conduct empirical research on China’s Shanghai Stock Exchange Index, Shenzhen
Index and Small and Medium-sized Board Index. The following conclusions:

1) The distribution of logarithmic returns of Shanghai Composite Index,
Shenzhen Index and Small and Medium-sized Board Index conforms to the GAS-ALD
model, and its conditional mean, conditional volatility, conditional skewness
and conditional kurtosis are all time-varying and clustering. If the normal
distribution is used Waiting for the static distribution will lead to failure
to capture the time-varying characteristics of the moment.

2) UC test, CC test, DQ test and bootstrap test are used to test the risk
prediction ability of GAS-ALD model. Compared with GAS-norm and GAS-t, which
are equal to the distribution under the GAS framework and APARCH-skst, the
GAS-ALD model has a better feasibility on VaR and ES of 1%. And the use of QL
loss function and FZL loss function shows that the GAS-ALD model has better
prediction accuracy.

In summary, GAS-ALD can better describe the distribution of peaks and thick
tails, biased, and left-to-right asymmetry, and has good accuracy in the
prediction of VaR and ES. These conclusions can provide some help in risk
measurement for participants in the capital market.

As a means of describing the distribution, GAS-ALD can be used as the marginal
distribution of returns to fit Copula, and to measure the VaR and ES of the
portfolio. In the future we will involve more research in this area.

## References

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