Homology of Moment Frames
Zoe COOPERBAND${}^{\textrm{*}}$, Allan McROBIEa, Cameron MILLARb, Bernd
SCHULZEc,

∗ Department of Electrical and Systems Engineering; GRASP Lab
University of Pennsylvania
zcooperband{at}gmail{dot}com
${}^{\textrm{a}}$ Cambridge University Engineering Dept, Cambridge, CB2 1PZ,
UK
${}^{\textrm{b}}$ Skidmore Owings & Merrill, The Broadgate Tower, 20 Primrose
St, London, EC2A 2EW, UK
${}^{\textrm{c}}$ Dept of Mathematics & Statistics, Lancaster Univ., UK

###### Abstract

Using homological techniques we show that a pin-anchored frame that involves
only moments and shears provides a conceptual bridge between the statics of
moment frames and the kinematics of pin-jointed trusses. One immediate result
is a long exact sequence whose alternating sum of dimensions gives a novel
counting rule for self-stresses and mechanisms. This combines the Maxwell-
Calladine count for pin-jointed trusses with the circuit rank (first Betti
number) associated with self-stresses in moment frames. These relations apply
to frames in 2, 3 or any dimensions. This work heralds a shift towards a
deeper study of the relationships and dualities that exist between structural
equilibria and kinematics.

Keywords: Homology, cosheaves, topology, self-stress, mechanisms, Maxwell-
Calladine count, moment frames.

##  1 Introduction

Homological algebra provides powerful tools for describing the fundamental
structural properties of both axially-loaded pin-jointed trusses and rigid
moment frames [1, 2, 3]. This paper focuses on the use of cellular cosheaves
[4]. These are used to store the geometric data embodied by the kinematic and
static properties of trusses and frames: the forces, moments, displacements
and rotations and their inter-relationships that are familiar concepts in
structural engineering. Once the data is so organised, the full power of the
description arises when the techniques of homology theory are applied.
Specifically, counting rules that relate node and bar counts to the numbers of
possible mechanisms and states of self-stress are developed, creating a novel
link between two familiar counts:

  * •

the Maxwell-Calladine count for a pin-jointed truss [5]; and

  * •

the (circuit rank) count of the number of cuts needed to make a statically-
determinate tree from a rigidly-connected moment frame.

Central to this connection is an intermediate frame system whose joints and
members can transmit shears and moments, but whose bars cannot carry axial
tension. We call this an “anchored frame.”

This work represents a first step towards a rigidity theory for moment-
resisting frames. Following pioneering work by Henneberg [6], Pollaczek-
Geiringer [7] and Laman [8] among others, much is now known about the rigidity
of structures. Both Graver’s classic text Counting on Frameworks [9] and the
recent and more comprehensive interdisciplinary book Frameworks, Tensegrities,
and Symmetry [10] by Connelly and Guest not only provide strong introductions
but also highlight an immediate problem: in their study of geometric systems
with distance constraints, rigidity theorists concern themselves almost
exclusively with what they call bar-joint frameworks. In structural
engineering, these are referred to as pin-jointed trusses, with bars
transmitting only axial loads. In this paper, we widen the focus to the
rigidity of a wider class of structures whose inter-connected bars can carry
not only axial forces but also shear forces, moments and torsions. In
engineering, it is this ability to transmit moments that warrants the use of
the word frame or framework.

This paper is thus a step towards a possible substantial broadening of scope
for rigidity theory, opening up potential avenues for the detailed study of
the rigidity of frames. Although arguments exist about why axial-only
structures may offer material efficiencies, beams that bend are central to
structural engineering and much of the infrastructure around us. Moment frames
are thus worthy of deeper study.

Beyond this, a theory that can sensibly represent moments may also help
resolve certain singularity issues that arise within the pin-jointed, axial-
only ansatz. There, it is well known that some topologies can carry states of
self-stress only if certain geometrical conditions are satisfied. A classic
example is a 2D truss in the Desargues configuration (see Fig IV of Maxwell
1864 [11] and Fig. 5 later) where the truss can carry a state of self-stress
(and possess a mechanism) only if the lines of three particular bars meet at a
point. This is a highly non-generic condition. In practice, just how close do
those three bars have to be to meeting at a single point? If moments are
considered this all-or-nothing pathology is avoided. If the three bars do not
quite meet at a point, states of self-stress still exist nearby: they will
simply contain some small moments. The anchored frame resolves some of these
concerns, describing where truss self-stresses and mechanisms algebraically
“go” when shifting the geometry.

##  2 Cosheaves and Truss Statics

We develop frame and truss statics in terms of cellular cosheaves and their
homology. Although this is a highly abstract definition, one quickly finds
that cosheaves simply redescribe systems and operations that are intuitive and
used by engineers everyday.

A framework is an abstract graph $G=(V,E)$ along with a geometric realization
$p:V\to\mathbb{R}^{n}$ that assigns each vertex $v\in V$ a point $p(v)=p_{v}$
in Euclidean space. Edges are realized as straight line segments with $(G,p)$
a geometric graph. Whenever vertices $u,v$ are incident to an edge $e$ we use
the notation $u,v\lhd e$. In this paper we will primarily consider frameworks
in the ambient space $\mathbb{R}^{2}$.

######  Definition 1 (Cellular cosheaf, [4]).

Over a framework $(G,p)$ a cellular cosheaf $\mathcal{K}$ is comprised of

  * •

vector spaces $\mathcal{K}_{e}$ and $\mathcal{K}_{v}$ assigned to each edge
$e$ and vertex $v$, respectively, called stalks, and

  * •

linear maps $\mathcal{K}_{e\rhd v}:\mathcal{K}_{e}\to\mathcal{K}_{v}$ between
every incidence $v\lhd e$.

A cellular cosheaf as of now is nothing more than a way to bookkeep what
geometric data is assigned to which cell, and how this data interacts.

######  Example 2 (2D truss statics).

What is the stress data assigned to a pin-jointed truss? Each edge $e$ of a
framework $(G,p)$ is assigned an axial tensile or compressive force scalar.
These internal forces are propagated to vertices, where these forces are
summed with external loads. The axial force cosheaf encoding these forces has
$\mathcal{F}_{e}=\mathbb{R}$, $\mathcal{F}_{v}=\mathbb{R}^{2}$, and linear
maps $\mathcal{F}_{e\rhd v}$ being embeddings of $\mathbb{R}$ into the larger
space $\mathbb{R}^{2}$. In other words, at an edge $u,v\lhd e$ both maps
$\mathcal{F}_{e\rhd v}$ and $\mathcal{F}_{e\rhd u}$ are the same $2\times 1$
matrix $\ell_{e}=(p_{uy}-p_{vy},p_{ux}-p_{vx})^{\top}$, with range denoted by
the subspace ${\bf e}\subset\mathbb{R}^{2}$.

The truss equilibrium matrix ${\bf A}$ is a size $2|V|\times|E|$ matrix that
transmits all internal loads to external load spaces, where $|\cdot|$ denotes
the dimension/size of the space/set. (In rigidity theory this matrix is the
transpose of the so-called _rigidity matrix_ [12].) If $w$ is a vector of
axial loads over edges, we say that $w$ is a self stress if ${\bf A}(w)=0$.
This condition requires all internal forces to equilibrate at each vertex
without the need of external forces. In cosheaf notation, we combine all
linear maps into the boundary matrix $\partial_{\mathcal{F}}={\bf A}$ with
non-zero blocks $\pm\mathcal{F}_{e\rhd v}$ wherever there is an incidence
$v\lhd e$. The axial load assignment $w$ is called a chain with local
components $w_{e}\in\mathcal{F}_{e}=\mathbb{R}$. If
$\partial_{\mathcal{F}}(w)=0$ then $w$ is called a cycle.

The terminology of chains and cycles comes from algebraic topology. Chains are
arbitrary vector assignments to all stalks of a given dimension, that is,
elements of the combined stalk spaces

$C_{0}\mathcal{K}=\bigoplus_{v\in V}\mathcal{K}_{v};\qquad
C_{1}\mathcal{K}=\bigoplus_{e\in E}\mathcal{K}_{e},$ (1)

(the same as the product
$\mathcal{K}_{v_{1}}\times\mathcal{K}_{v_{2}}\times\cdots$ of finite
dimensional vector space stalks). Furthermore, the cosheaf boundary map is a
linear map between spaces of chains

$\partial_{\mathcal{K}}:C_{1}\mathcal{K}\to
C_{0}\mathcal{K};\qquad(\partial_{\mathcal{K}}(w))_{v}=\sum_{e:v\lhd
e}\pm\mathcal{K}_{e\rhd v}(w_{e}),$ (2)

where $(\partial_{\mathcal{K}}(w))_{v}$ is the component of
$\partial_{\mathcal{K}}(w)\in C_{0}\mathcal{K}$ in the stalk
$\mathcal{K}_{v}$. The choice of sign in Equation (2) is informed by an
arbitrary orientation for the edges – the sign is positive if the edge “points
towards” $v$ and is negative otherwise. The homology of a cosheaf
$\mathcal{K}$ then consists of the two vector spaces

$H_{1}\mathcal{K}=\ker\partial_{\mathcal{K}}\subset C_{1}\mathcal{K};\qquad
H_{0}\mathcal{K}=C_{0}\mathcal{K}/\textrm{im}\,\partial_{\mathcal{K}}\cong(\textrm{im}\,\partial_{\mathcal{K}})^{\perp}\subset
C_{0}\mathcal{K}$ (3)

where the latter space is a quotient vector space of $C_{0}\mathcal{K}$ by the
image of the boundary map $\partial_{\mathcal{K}}$.

Homology often captures the most important aspects of the system; for the
force cosheaf, $H_{1}\mathcal{F}$ is the space of self stresses while
$H_{0}\mathcal{F}$ is (isomorphic to) the space of infinitesimal degrees of
freedom, the combined space of rigid body DOF $\mathscr{R}\mathcal{F}$ and
mechanisms $\mathscr{M}\mathcal{F}$.

######  Example 3 (Classical homology).

Chain complexes and homology were born from classical constructions in
algebraic topology from a century ago [13]. Invented to quantify topological
features, classical homology detects the number and arrangement of voids and
holes of an abstract space. One attaches a scalar value $\mathbb{R}$ to every
cell, with boundary map $\partial:C_{i}G\to C_{i-1}G$ called the signed
incidence matrix. Over a graph $G$, $H_{1}G$ is the linear span of cycles
while $H_{0}G$ encodes connected components.

The classical Euler characteristic $\mathfrak{X}$, the alternating sum of
vector space dimensions, is a powerful topological invariant of spaces [13].
For a graph $G$, using the rank-nullity theorem we find

$\mathfrak{X}(CG)=|C_{0}G|-|C_{1}G|=|V|-|E|=|H_{0}G|-|H_{1}G|=\mathfrak{X}(HG).$
(4)

If $G$ is connected then $H_{0}G$ is $1$-dimensional and following Equation
(4) the number of graph cycles is $|H_{1}G|=|E|-|V|+1$ (in graph theory, this
is known as the circuit rank).

An example of the Euler count applied to cosheaves is the Maxwell-Calladine
counting rule [5]

$\mathfrak{X}(C\mathcal{F})=2|V|-|E|=|\mathscr{R}\mathcal{F}|+|\mathscr{M}\mathcal{F}|-|H_{1}\mathcal{F}|=\mathfrak{X}(H\mathcal{F})$
(5)

over the force cosheaf $\mathcal{F}$, enveloping counts of self-stresses,
rigid DOF $\mathscr{R}\mathcal{F}$, and mechanisms $\mathscr{M}\mathcal{F}$
[1]. This is the alternating count of Equation (4) replacing standard homology
with the homology of $\mathcal{F}$.

##  3 Frame Statics

######  Example 4 (Plane frame statics).

Moment frame elements have a simple formulation and boundary map. Suppose that
$(G,p)$ is a single edge with two endpoints $u,v\lhd e$ embedded horizontally
along the $x$-axis in $\mathbb{R}^{2}$. Letting $M$ be a function representing
the moment along the beam; it is well known that its derivative
$\frac{dM}{dx}$ is the shear along the beam. Assume that the shear force
$F_{y}$ is constant so $M$ has constant slope. Then the linear function $M$
over the edge $e$ is $M(t)=M_{e}+F_{y}\cdot t$, where $M_{e}$ is the moment at
the edge center $\frac{p_{u}+p_{v}}{2}$ and $t$ parameterizes a location along
the length of the beam; see Fig. 1 (a) where a moment function $M(t)$ is
graphed over an edge. Then the induced moments at $p_{u}$ and $p_{v}$ are the
magnitudes of $F_{y}\frac{p_{u}-p_{v}}{2}$ and $F_{y}\frac{p_{v}-p_{u}}{2}$.
Letting $w_{e}=(M,F_{x},F_{y})$ be a force-couple at the edge center, the
induced spatial force at $p_{u}$ and $p_{v}$ in coordinates follows from
applying matrices:

$\partial_{\mathcal{M}}(w_{e})_{u}=\begin{bmatrix}1&0&\|\frac{p_{u}-p_{v}}{2}\|\\\
0&1&0\\\ 0&0&1\end{bmatrix}\begin{bmatrix}M\\\ F_{x}\\\
F_{y}\end{bmatrix};\qquad\partial_{\mathcal{M}}(w_{e})_{v}=-\begin{bmatrix}1&0&-\|\frac{p_{u}-p_{v}}{2}\|\\\
0&1&0\\\ 0&0&1\end{bmatrix}\begin{bmatrix}M\\\ F_{x}\\\ F_{y}\end{bmatrix}$
(6)

where the edge $e$ is oriented towards $u$ in this notation.

To cleanly represent a force couple transform, we introduce the exterior
product space $\bigwedge^{2}\mathbb{R}^{n}$, the vector space consisting of
formal bilinear pairs $x\wedge y$ where $x\wedge y=-y\wedge x$ over vectors
$x,y\in\mathbb{R}^{n}$. It follows immediately from this definition that
$x\wedge x=0$. With $x,y$ a basis for $\mathbb{R}^{2}$ the pseudo-scalar
$M=x\wedge y$ is a basis for $\bigwedge^{2}\mathbb{R}^{2}$. In dimension
three, $\bigwedge^{2}\mathbb{R}^{3}$ consists of moments $M_{x},M_{y},M_{z}$
in the three directions, with wedge product equivalent to the cross product
(after applying the Hodge-star operator).

The exterior product models moments. The moment generated by a force vector
$F\in\mathbb{R}^{n}$ applied at a lever arm $\ell\in\mathbb{R}^{n}$ is equal
to the product $F\wedge\ell$.

######  Definition 5 (Moment cosheaf).

Over a framework $(G,p)$ in $\mathbb{R}^{2}$, the moment cosheaf $\mathcal{M}$
has stalks
$\mathcal{M}_{e}=\mathcal{M}_{v}=\bigwedge^{2}\mathbb{R}^{2}\oplus\mathbb{R}^{2}$
comprised of force-couples at each cell. Each cosheaf stalk map

$\mathcal{M}_{e\rhd
v}(M,F)=(M+F\wedge\frac{p_{v}-p_{u}}{2},F)\in\mathcal{M}_{v}$ (7)

sends a force couple at the edge center $\frac{p_{u}+p_{v}}{2}$ to a force
couple at the coordinate $p_{v}$.

One can check that the linear map (7) aligns with those in Equation (6),
combining to form the size $3|V|\times 3|E|$ frame equilibrium matrix ${\bf
B}=\partial_{\mathcal{M}}:C_{1}\mathcal{M}\to C_{0}\mathcal{M}$ after choosing
bases. The moment cosheaf $\mathcal{M}$ has a particularly simple homology
type. In a connected rigid frame clearly the combined system has three rigid
body DOF in $\mathbb{R}^{2}$ and six rigid body DOF in $\mathbb{R}^{3}$,
spanning $H_{0}\mathcal{M}=\mathscr{R}\mathcal{M}$.

The homologies $H_{1}\mathcal{M}$ and $H_{0}\mathcal{M}$ are in fact
isomorphic to $3$ copies of the classical homologies $H_{1}G$ and $H_{0}G$ in
$\mathbb{R}^{2}$ and $6$ copies in $\mathbb{R}^{3}$. Applying the Euler
equation (4) to the cosheaf $\mathcal{M}$ in $\mathbb{R}^{2}$ we find

$|H_{1}\mathcal{M}|=|H_{0}\mathcal{M}|+3(|E|-|V|)=3(|E|-|V|+1)=3|H_{1}G|.$ (8)

where $H_{1}\mathcal{M}=\ker{\bf B}$ is the space of frame self stresses. This
circuit rank count is familiar in elementary structural engineering.
Trivially, a fully-welded frame requires $|E|-|V|+1$ cuts to make it a
statically-determinate tree. In 2D there are three independent stress
resultants (axial, shear and bending) at each such cut, hence Equation (8). In
3D there are six stress resultants at each cut.

(a) The moment cosheaf $\mathcal{M}$ over an edge. The rigid beam transmits
axial, shear, and bending forces. A moment function is graphed over the beam.

(b) The anchored cosheaf $\mathcal{N}$ over an edge. The mid-beam joint
transmits bending, shear, and torque but no axial force. The pinned anchors
remove residual shear.

Figure 1: Sketches of the moment $\mathcal{M}$ and anchored $\mathcal{N}$
cosheaves over an edge.

##  4 The Anchored Cosheaf

While the axial force cosheaf $\mathcal{F}$ and moment cosheaf $\mathcal{M}$
have been discussed separately, the true power of cosheaf theory comes from
the relationship between the two.

######  Definition 6 (Cosheaf map).

A map between cosheaves $\phi:\mathcal{K}\to\mathcal{L}$ is a collection of
stalk-wise linear maps $\phi_{e}:\mathcal{K}_{e}\to\mathcal{L}_{e}$ and
$\phi_{v}:\mathcal{K}_{v}\to\mathcal{L}_{v}$ such that the map compositions
$\mathcal{L}_{e\rhd v}\phi_{e}=\phi_{v}\mathcal{K}_{e\rhd v}$ are equal at
every incidence $v\lhd e$. This equation ensures that the systems
“align/agree” across cell incidences.

We can formally state how the moment cosheaf $\mathcal{M}$ subsumes the force
cosheaf $\mathcal{F}$, meaning truss stresses “are” frame stresses. There is
an injective map $\phi:\mathcal{F}\to\mathcal{M}$ which is comprised of
embeddings $\phi_{e}:\mathcal{F}_{e}\to
0\oplus\mathbb{R}^{2}\subset\mathcal{M}_{e}$ at edges. At vertices
$\phi_{v}:\mathcal{F}_{v}\to 0\oplus\mathbb{R}^{2}\subset\mathcal{M}_{v}$
simply adds zero moments to nodal forces in $\mathcal{F}_{v}$. To see this
satisfies Definition 6, if $\phi_{e}(w)=F_{e}\in{\bf e}\subset\mathbb{R}^{2}$
is an axially aligned force vector then $F_{e}\wedge\frac{p_{u}-p_{v}}{2}=0$
in the stalk map $\mathcal{M}_{e\rhd v}$, as $(p_{u}-p_{v})\in{\bf e}$. Thus
there is no moment component and $\phi_{v}\mathcal{F}_{e\rhd
v}=\mathcal{M}_{e\rhd v}\phi_{e}$.

The injective cosheaf map $\phi$ induces a quotient cosheaf
$\mathcal{M}/\phi\mathcal{F}$ which we will denote as $\mathcal{N}$ and call a
anchored cosheaf. This quotient cosheaf has stalks
$\mathcal{N}_{e}=\mathcal{M}_{e}/\phi\mathcal{F}_{e}=\bigwedge^{2}\mathbb{R}^{2}\oplus\mathbb{R}^{2}/{\bf
e}$ of dimension $2$ and stalks
$\mathcal{N}_{v}\cong\bigwedge^{2}\mathbb{R}^{2}$ of dimension $1$ over edges
and vertices. At the physical level, the cosheaf $\mathcal{N}$ ignores the
axial forces over edges and ignores all forces at vertices, so that it only
considers the moment and shear components. A cosheaf stalk map
$\mathcal{N}_{e\rhd v}$ takes an axial-free force couple in $\mathcal{N}_{e}$
and computes its induced moments at the edge endpoints, quotienting out the
forces at $v$.

######  Example 7 (Statics of the anchored cosheaf).

An anchored frame should be thought of as a moment frame where each stiff
juncture $v$ is held in place by an external anchor restricting translation
but not rotation. These “anchors” are physically pin-joints to an external
system111It is critical to note that the members are still rigidly attached to
one another: the junction is not a pin itself but instead has reaction forces
applied to it through a pinned connection.. However, there then are are
trivial degrees of axial self-stress over each edge; to eliminate these we
insert a prismatic “sliding joint” mid-member that permits free axial
extensions but restricts (and transmits) shear and moments. A sketch of an
anchored frame element is pictured in Fig. 1 (b). The extending joint has been
previously utilized to model shell structures [14].

The anchored cosheaf $\mathcal{N}$ over a framework $(G,p)$ models this
anchored frame system. The edge stalk
$\mathcal{N}_{e}\cong\bigwedge^{2}\mathbb{R}^{2}\oplus\mathbb{R}^{2}/{\bf e}$
has the axial force quotiented out – equivalent to the mid-member extension
joint. The vertex stalks $\mathcal{N}_{v}\cong\bigwedge^{2}\mathbb{R}^{2}$
only need to detect moments because the anchor, as a pin-joint, absorbs any
residual force. The homology $H_{1}\mathcal{N}$, the kernel of the size
$|V|\times 2|E|$ equilibrium matrix $\partial_{\mathcal{N}}$, describes the
valid states of self-stress of the anchored system. Moments and shears combine
just as in a moment frame (in $\mathcal{M}$), but the external pinned
reactions eliminate the residual net shear force.

In summary, the cosheaves $\mathcal{F},\mathcal{M},\mathcal{N}$ form a short
exact sequence of cosheaves

$0\to\mathcal{F}\xrightarrow{\phi}\mathcal{M}\xrightarrow{\pi}\mathcal{N}\to
0.$ (9)

where $\pi:\mathcal{M}\to\mathcal{N}$ is a cosheaf map projection. Exactness
means that $\ker\pi=\textrm{im}\,\phi$; consequently stalks can be decomposed
as $\mathcal{M}_{e}\cong\mathcal{F}_{e}\oplus\mathcal{N}_{e}$ and
$\mathcal{M}_{v}\cong\mathcal{F}_{v}\oplus\mathcal{N}_{v}$. The map $\phi$
converts a pin-jointed truss to a moment frame by physically “gluing” the pins
shut, allowing them to transfer shear and moments. The map $\pi$ takes a
moment frame and inserts extension joints as well as pinned anchors at each
junction.

##  5 The Homological Relations of Moment Frames

We motivate the following technical analysis with a simple example.

######  Example 8 (Frame counts).

Consider the 2D frame whose bars are all moment-connected and whose joints are
all fully encastré supports. Releases of constraints lead variously to the
anchored frame and the pin-jointed truss (see Fig. 2). The number of states of
self-stress in the anchored frame may be readily determined by considering the
first two frames incrementally removing constraints. Following Equation (8),
the number of states of self-stress for the moment frame is 3 times the
circuit rank, including three rigid body motions in 2D. Together, these counts
redevelop the Maxwell-Calladine count for the 2D truss.

Figure 2: Five structures. The first has fully fixed joints and supports. The
second releases the support moments. The final three are the fully-rigid
frame, the anchored frame and the truss. The Maxwell-Calladine count for
trusses is ($3\times$) the circuit count for frames minus the count for the
anchored frame.

The homological algebra allows us to mathematically formalise these
relationships. Every short exact sequence of cosheaves (9) induces a long
exact sequence of homology for all indices $i$:

$\cdots\xrightarrow{\pi_{*}}H_{i+1}\mathcal{N}\xrightarrow{\vartheta}H_{i}\mathcal{F}\xrightarrow{\phi_{*}}H_{i}\mathcal{M}\xrightarrow{\pi_{*}}H_{i}\mathcal{N}\xrightarrow{\vartheta}H_{i-1}\mathcal{F}\xrightarrow{\phi_{*}}\cdots$
(10)

where $\phi_{*}$ and $\pi_{*}$ are induced maps between homology spaces (which
simply restrict and project onto homology spaces). The maps
$\vartheta:H_{i+1}\mathcal{N}\to H_{i}\mathcal{F}$ are called connecting
homomorphisms following from the snake/zig-zag lemma in algebraic topology
[13]. Exactness means that for every term in (10) the image of the incoming
linear map is the kernel of the outgoing map. The derivation is too much to go
into here but see [2] for an exposition. The takeaway is that this is a
powerful method from algebraic topology that describes linear relations
between homology spaces (here self-stresses and DOF).

Over a framework $(G,p)$ in $\mathbb{R}^{2}$ the homology space
$H_{0}\mathcal{F}$ consists of pin-jointed kinematic degrees of freedom
containing at least $3$ degrees of rigid body DOF. These trivial DOF are
precisely those that generate $H_{0}\mathcal{M}$. The long exact sequence of
the trio $\mathcal{F},\mathcal{M},\mathcal{N}$ is

$0\to
H_{1}\mathcal{F}\xrightarrow{\phi_{*}}H_{1}\mathcal{M}\xrightarrow{\pi_{*}}H_{1}\mathcal{N}\xrightarrow{\vartheta}\mathscr{M}\mathcal{F}\oplus\mathscr{R}\mathcal{F}\xrightarrow{\phi_{*}}\mathscr{R}\mathcal{M}\to
0$ (11)

where the homology space $H_{0}\mathcal{N}$ must be zero because
$\phi_{*}:H_{0}\mathcal{F}\to H_{0}\mathcal{M}$ is surjective. This homology
map $\phi_{*}$ maps rigid body truss DOFs to rigid body frame DOFs with kernel
$\mathscr{M}\mathcal{F}$ (mechanisms aren’t frame DOFs and so
$\phi_{*}(\mathscr{M}\mathcal{F})\subset\textrm{im}\,\partial_{\mathcal{M}}$).
We can simplify this sequence by removing both rigid body DOF terms to form a
reduced long exact sequence

$0\to
H_{1}\mathcal{F}\xrightarrow{\phi_{*}}H_{1}\mathcal{M}\xrightarrow{\pi_{*}}H_{1}\mathcal{N}\xrightarrow{\vartheta}\mathscr{M}\mathcal{F}\to
0$ (12)

where the connecting homomorphism
$\vartheta:H_{1}\mathcal{N}\to\mathscr{M}\mathcal{F}$ is surjective onto
mechanisms. Here $\vartheta$ is simply the linear map that inputs an anchor
frame self-stress and outputs the resultant shear forces at nodes — the green
force arrows in Fig. 4 (b) and Fig. 5 (a). It is interesting to interpret the
physical meaning of sequence (12):

  * •

At $H_{1}\mathcal{F}$, every axial self-stress is also a frame-self stress in
$H_{1}\mathcal{M}$ ($\phi_{*}$ is injective).

  * •

Every anchored cosheaf self-stress in $H_{1}\mathcal{N}$ is a combination of a
moment frame self-stress in $H_{1}\mathcal{M}$, and a stress corresponding to
a pin-jointed truss mechanism. Formally this is the statement
$H_{1}\mathcal{N}\cong\textrm{im}\,\pi_{*}\oplus\textrm{im}\,\vartheta^{\top}$
where $\vartheta^{\top}$ is the adjoint/transpose of $\vartheta$.

  * •

The last map $\vartheta:H_{1}\mathcal{N}\to\mathscr{M}\mathcal{F}$ is
surjective, so every pin-joint mechanism in $\mathscr{M}\mathcal{F}$ is the
image of an anchored frame self-stress $H_{1}\mathcal{N}$ with force
resultants encoding the mechanism velocities.

These algebraic relations are sketched in Fig. 3. The last point is
significant enough for its own statement.

######  Theorem 9 (Mechanisms from the anchor-frame).

Every pin-joint mechanism of a framework $(G,p)$ follows from the shear
resultants of an anchored frame self-stress over $(G,p)$ (i.e. support
reactions).

Figure 3: A schematic of the equilibrium operations in the three structural
systems (truss, frame, and anchor frame) as well as the linear relations in
the long exact sequence (11) in clockwise order. The blue regions are pure
axial self-stresses, the red regions are mixed frame self-stresses, the cyan
regions are mechanisms/pure anchored frame self-stresses, and the yellow
regions are rigid body DOF. The purple regions correspond to the dimensions of
full rank of each equilibrium matrix.

######  Example 10 (Moments over a box frame).

Suppose that $(G,p)$ is the simple square frame embedded in $\mathbb{R}^{2}$.
There are clearly no truss axial self-stresses so line (12) reduces to a short
exact sequence. By the isomorphism $H_{1}\mathcal{N}\cong
H_{1}\mathcal{M}\oplus\mathscr{M}\mathcal{F}$, we see that there are $3+1=4$
states of anchor frame self-stress. The first three come from self-moments in
$H_{1}\mathcal{M}$, generated by a unit axial force, shear force, and moment
at a particular edge. These are pictured in Fig. 4 (a). The last state imparts
forces which are absorbed by the anchors, pictured in Fig. 4 (b). These forces
also inform a mechanism of a square pin-jointed truss, warping the square into
a parallelogram.

(a) Three basis cycles for $H_{1}\mathcal{M}$ are pictured. Each is generated
from a unit axial, shear or moment over the leftmost member. These pass to
linearly independent elements of $H_{1}\mathcal{N}$.

(b) The final generator of $H_{1}\mathcal{N}$ with green resultant forces.

| $\mathcal{F}$ | $\mathcal{M}$ | $\mathcal{N}$
---|---|---|---
$s;\dim H_{1}$ | 0 | 3 | 4
$m;\dim H_{0}$ | 4 | 3 | 0
(c) Table of cosheaf homology dimensions.

Figure 4: Anchored cosheaf self-stresses over a box frame.

###  5.1 An Anchored Stress Counting Rule

The Euler count (4) of the exact sequence (12) (as a chain complex) is the
following alternating sum:

$\mathfrak{X}\eqref{eq:moment_les_reduced}=\left(|H_{1}\mathcal{F}|-|\mathscr{M}\mathcal{F}|\right)+|H_{1}\mathcal{N}|-|H_{1}\mathcal{M}|=0$
(13)

which by exactness equals zero. These homology dimensions can be reformed in
terms of cell numbers. The left-most term is the reduced Maxwell count (5)
without rigid DOF while the right-most term $|H_{1}\mathcal{M}|$ has the cycle
dimension $3(|E|-|V|+1)$ following from Equation (8). Since
$H_{0}\mathcal{N}=0$, then a simple Euler count gives
$|H_{1}\mathcal{N}|=|C_{1}\mathcal{N}|-|C_{0}\mathcal{N}|=2|E|-|V|$. Inserting
these into Equation (13), we have the following:

######  Theorem 11 (Anchor-frame stress count).

Over a framework $(G,p)$, the dimension of anchored frame self-stresses
$|H_{1}\mathcal{N}|$ is equal to a multiple of the cycle count minus the
reduced Maxwell count (5). Thus

$\displaystyle\mathrm{in}\ \mathbb{R}^{2}:\quad|H_{1}\mathcal{N}|$
$\displaystyle=$ $\displaystyle 3(|E|-|V|+1)-(|E|-2|V|+3)=2|E|-|V|$ (14)
$\displaystyle\mathrm{in}\ \mathbb{R}^{3}:\quad|H_{1}\mathcal{N}|$
$\displaystyle=$ $\displaystyle 6(|E|-|V|+1)-(|E|-3|V|+6)=5|E|-3|V|$ (15)

To the authors’ knowledge, this is the first time the Maxwell count has been
combined with the graph-theoretic cycle count in an application. Visually in
Fig. 3, Theorem 11 is the statement that the dimensions of the red and cyan
regions can be counted two ways. The first is directly counting
$|H_{1}\mathcal{N}|$. The second is through summing
$|C_{1}\mathcal{M}|+|C_{0}\mathcal{F}|$ then subtracting the dimensions of
$C_{0}\mathcal{M}$ and $C_{1}\mathcal{F}$, removing the blue, yellow and
purple regions with only red and cyan regions remaining.

######  Example 12 (The Desargues anchor frame).

Examine the Desargues configuration in Fig. 5 (a) with cosheaf homology
dimensions listed in (b). The homology map $\phi_{*}:H_{1}\mathcal{F}\to
H_{1}\mathcal{M}$ is injective and has rank 1, so by exactness
$\ker\pi_{*}=\textrm{im}\,\phi_{*}$ is dimension 1, and thus the rank-nullity
theorem states that $\pi_{*}:H_{1}\mathcal{M}\to H_{1}\mathcal{N}$ has rank
11. The sole generator of $(\textrm{im}\,\pi_{*})^{\perp}\subset
H_{1}\mathcal{N}$ is pictured in (a); other generators of $H_{1}\mathcal{N}$
are derived from ordinary frame self stresses $H_{1}\mathcal{M}$ of the
Desargues frame.

Moving down the long exact sequence (12), because
$\vartheta:H_{1}\mathcal{N}\to\mathscr{M}\mathcal{F}$ is surjective the
anchor-frame stress in (a) generates the sole mechanism of the configuration.
The residual shear forces when summed, drawn with green arrows, form the
mechanism of the truss (with anchors removed).

From Theorem 11 we find that $|H_{1}\mathcal{N}|=2|E|-|V|=12$, in agreement
with the table in Fig. 5 (b). The Maxwell count is
$|H_{1}\mathcal{F}|-|H_{0}\mathcal{F}|=-3$ and the extended cycle count is
$|H_{1}\mathcal{M}|-|H_{0}\mathcal{M}|=9$. Taking the difference of these two
counts is $9-(-3)=12$, the aforementioned count of anchored-frame self-
stresses.

The existence of the non-trivial self-stress and mechanism of the Desargues
framework relies on the three vertical edges meeting at a point (in projective
space), a delicate condition. After perturbing the system both
$H_{1}\mathcal{F}$ and $H_{0}\mathcal{F}$ decrease in dimension while the
dimensions of $H_{1}\mathcal{M}$ and $H_{1}\mathcal{N}$ stay the same. What
changes in sequence (12) is the rank of the map $\phi_{*}:H_{1}\mathcal{F}\to
H_{1}\mathcal{M}$ decreases, which by exactness means that
$\pi_{*}:H_{1}\mathcal{M}\to H_{1}\mathcal{N}$ must increase in rank. Thus the
cycle pictured in Fig. 5 (a) becomes a state of frame self-stress in
$H_{1}\mathcal{M}$. This example suggests that the cosheaf $\mathcal{N}$ can
be used to better understand pin-jointed truss mechanics near singular points
in its geometry.

(a) The generator of $H_{1}\mathcal{N}$ orthogonal to the frame self-stresses
in $\pi_{*}(H_{1}\mathcal{M})$. This self-stress of $\mathcal{N}$ is mapped to
a Desargues truss mechanism by $\vartheta$.

| $\mathcal{F}$ | $\mathcal{M}$ | $\mathcal{N}$
---|---|---|---
$s;\dim H_{1}$ | 1 | 12 | 12
$m;\dim H_{0}$ | 4 | 3 | 0
(b) Table of cosheaf homology dimensions.

Figure 5: An anchor-frame self-stress in $H_{1}\mathcal{N}$ is pictured over
the Desargues configuration.

##  6 Conclusion

We have described a new structural system, the anchored frame whose moment-
dominated statics tie together the statics of pin-jointed trusses and rigid
moment frames. The theory led to a counting rule for anchored frames
incorporating both the Maxwell-Calladine rule and the graph cycle count.
Moreover, each pin-jointed mechanism is encoded by the resultants of anchored
frames. For future work, the authors intend to describe the discontinuous Airy
stress functions [15] homologically.

## Acknowledgments

Zoe Cooperband was supported by the Air Force Office of Scientific Research.
Bernd Schulze was partially supported by the ICMS Knowledge Exchange Catalyst
Programme. The authors would like to thank Bill Baker, Chris Calladine, Robert
Ghrist and Miguel Lopez for insightful discussions.

## References

  * [1] Zoe Cooperband and R Ghrist  “Towards Homological Methods In Graphic Statics”  In _Journal of the IASS_ 64.218, 2023
  * [2] Zoe Cooperband, R Ghrist and Jakob Hansen  “A Sheaf Theory of Reciprocal Figures: Planar and Higher Genus Graphic Statics”, 2023  arXiv:2311.12946
  * [3] Zoe Cooperband, Miguel Lopez and Bernd Schulze  “Equivariant Cosheaves and Finite Group Representations in Graphic Statics”, 2024  arXiv:2401.09392
  * [4] Justin Curry  “Sheaves, Cosheaves and Applications”, 2013  arXiv:1303.3255
  * [5] C.. Calladine  “Buckminster Fuller’s Tensegrity Structures and Clerk Maxwell’s Rules for the Construction of Stiff Frames”  In _Int. J. Solids & Structures_ 14, 1978, pp. 161–172
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  * [7] Hilda Pollaczek-Geiringer  “Über die Gliederung ebener Fachwerke”  In _Zeitschrift für Angewandte Mathematik und Mechanik_ 7.1, 1927, pp. 58–72  DOI: 10.1002/zamm.19270070107
  * [8] G. Laman  “On Graphs and Rigidity of Plane Skeletal Structures”  In _Journal of Engineering Mathematics_ 4, 1970, pp. 331–340
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  * [15] Chris Williams and Allan McRobie  “Graphic Statics Using Discontinuous Airy Stress Functions”  In _Intl. J. Space Structures_ 31.2-4, 2016, pp. 121–134


On the short term stability
of financial ARCH price processes

Gilles Zumbach
Edgelab Avenue de la Rasude 5 1006 Lausanne Switzerland

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

For many financial applications, it is important to have reliable and
tractable models for the behavior of assets and indexes, for example in risk
evaluation. A successful approach is based on ARCH processes, which strike the
right balance between statistical properties and ease of computation. This
study focuses on quadratic ARCH processes and the theoretical conditions to
have a stable long-term behavior. In particular, the weights for the variance
estimators should sum to 1, and the variance of the innovations should be 1.
Using historical data, the realized empirical innovations can be computed, and
their statistical properties assessed. Using samples of 3 to 5 decades, the
variance of the empirical innovations are always significantly above 1, for a
sample of stock indexes, commodity indexes and FX rates. This departure points
to a short term instability, or to a fast adaptability due to changing
conditions. Another theoretical condition on the innovations is to have a zero
mean. This condition is also investigated empirically, with some time series
showing significant departure from zero.

Keywords:  ARCH processes, market stability, innovations, drift significance
JEL: C22, C62, C52

##  1 Introduction

This paper is about daily time series and the corresponding mesoscopic
processes. An important topic is to write a process, or a family of processes,
that describes the statistical behavior of market data along the time
direction. Possible practical applications of these processes are in risk
evaluation, portfolio allocations, and in option pricing. The base model is a
normal random walk with constant volatility, postulated by [Bachelier, 1900]
in his thesis. Since, the accumulation of digital data and the computational
power of modern computers allow the evaluation of detailed empirical
statistics, and to test finer models. A key challenge is to describe the
dynamic aspect of the volatility. With that respect, the ARCH family of
processes has been very successful. The literature on volatility processes
and/or forecast is very large, for example see [Engle, 1982; Engle and
Bollerslev, 1986] for the original references, [Andersen and Bollerslev, 1998]
for the volatility forecasts induced by a GARCH(1,1) process, [Hansen and
Lunde, 2005] for general ARCH forecasts, or [Andersen et al., 2006; Poon,
2005; Zumbach, 2012] for broad reviews. The key idea of the ARCH model is to
have a volatility that depend on the past change of prices, introducing a
feed-back loop between the price evolution and the volatility. In this class
of processes, the Long-Memory ARCH (LM-ARCH) processes [Zumbach, 2004]
describes correctly the clustering of the volatility, the decay of the lagged
correlations of the volatility, and the quantitative measures of the time
reversal non-invariance [Zumbach, 2009]. Importantly for wide risk evaluation,
they depend on a few parameters with a very weak dependency on each time
series. A related process with long memory has been proposed by [Corsi, 2009].

Regardless of the ARCH process details, we consider the class of quadratic
processes where the variance for the next time step is given by a weighted sum
of the past squared returns, with decaying weights as the returns are further
away in the past. For the process to be consistent, the weights should sum to
one. In a process, our ignorance about the future events are represented by a
random variable, typically called the innovations. Again for the process
consistency, the variance of this random variable should be one. Both
conditions are required so that the variance of the price changes has a non-
singular asymptotic distribution, and does not decrease to zero or increase
without bounds.

For a given decay shape of the memory weights, two versions of an ARCH process
can be written. In the linear form, the variance is a weighted sum of past
squared returns. In the affine form, a constant is added to the weighted sum
that fixes the asymptotic mean variance of the process. The above constraints
on the weights and innovation variance are needed so that the mean volatility
parameter is equal to the long term mean volatility of the process.

The decay shape for the weights in the volatility evaluation is otherwise free
and several shapes are commonly used, like Rectangular Moving Average (RMA),
Exponential Moving Average (EMA), or decays that are slower than exponential
as in the Long Memory (LM) version of the ARCH process. But all shapes should
obey the above constraint that the sum of the weights is one.

Turning to empirical data, the past prices allow to compute the historical
daily returns. For a given quadratic ARCH volatility process, namely for a
particular decay shape for the weights, the volatility can be computed. With
the time series of volatilities and realized returns, the process equation can
be reverted to compute the realized innovations. In principle, these realized
innovations should obey the consistency constraints of a process, in
particular their variances should be one. Several studies have been done along
this idea, see e.g. [Andersen et al., 2000; Zumbach, 2006], with the main
purpose is to study the probability distribution for the innovations. The
focus for this paper is on the standard deviation of the innovations.

On empirical data, the standard deviation of the innovations is slightly
larger than one, of the order of 3 to 4% larger. The deviations are small but
significant, and consistent for many time series (stock indices, commodity
indices, FX). The LM-ARCH weights give the smallest deviations quoted above,
other shapes give larger deviations from 1. To summarize, no time series is
compatible with an ARCH model using innovations with unit variances.

This deviation from the theoretical criterion points to a slight instability
of the empirical data, and of the underlying trading mechanism. Since
systematically larger than one, the volatility should increase exponentially
fast. This is obviously not the case, hence another mid-term mechanism should
provide for the long term stability. For example, an Ornstein-Uhlenbeck term
(in the volatility or in the price equation) can provide for a medium to long
term stability. Since the deviations from 1 are small, a small recoil term is
sufficient to bring a long term stability.

It is not unreasonable to have a short term instability. This means that the
short-term trading is very reactive to external information or to endogeneous
perturbations. This reactivity can be seen as a quality since the system is
adapting very quickly to a changing environment. The trading activity can be a
bit over-reacting, with the advantage to reach quickly a new equilibrium. At a
medium to long term scales, some fundamental criterion bring the system back,
leading to the observed long term stability.

Another theoretical condition sets on the innovations is to have a zero mean,
while the possible drift for the price is contained in a mean term in the ARCH
equation for the returns. This mean return term is typically small and
difficult to estimate, because the volatility and its dynamics dominates the
price processes. In particular, any mean term equation is difficult to
validate out-of-sample. As a simple solution, we set it to zero in the
empirical study. Consequently, the possible mean return is reported on the
mean empirical innovations. The subsequent question is if the empirical
innovations are compatible with a zero mean. It is the case for most FX rates
and commodity indexes, while it is not compatible with a zero mean for most
stock indices. For this last category of time series, this is the expected
long term results, namely stock indices drift upward. Yet, some stock indices
are compatible with a zero mean for the innovations, indicating that they did
not increase significantly over the last few decades.

The rest of the paper is organized as follows. The next section presents the
core material on quadratic ARCH processes and on the consistency conditions.
Sec. 3 introduces the statistical material using the central limit theorem,
and in particular the computation of the $p$-values when the random variables
are distributed with a Student distribution. The empirical results for a LM-
ARCH volatility are given in Sec. 4 for the means, and Sec. 5 for the
variances. Empirical results for other volatility kernels are discussed in
Sec. 6, some further investigations reported in Sec. 7, before the
conclusions.

##  2 Relevant background on ARCH models

A wide array of ARCH processes have been proposed to model financial time
series. We will consider in this paper the family of processes where the
variance is a linear or affine function of past squared returns. The ARCH
process is essentially a location/size/shape decomposition of the random
returns $r(t)$, namely

$r(t+\delta t)=\tilde{\mu}(t)+\widetilde{\sigma}(t)\,\epsilon(t+\delta
t)\hskip 30.00005pt\epsilon~{}~{}\text{i.i.d with
distribution}~{}~{}p(\epsilon).$ (1)

The terms $\tilde{\mu}(t)$ and $\widetilde{\sigma}(t)$ fix the location and
size for the next random return $r(t+\delta t)$. They are function of the
information set up to $t$, and give a one-step forecast for the mean and
standard deviation of the random variable $r(t+\delta t)$. The random variable
$\epsilon(t+\delta t)$ is called the innovation and has a fixed distribution
$p(\epsilon)$ that is time independent. The distribution $p$ has a zero mean
and unit variance, but is otherwise unspecified. With this structure, all the
time dependencies on the history of $r$ are included in the predictable parts
$\tilde{\mu}(t)$ and $\widetilde{\sigma}(t)$, while $p(\epsilon)$ has no
dependency on the history or on time.

Using the properties of $p(\epsilon)$, the mean and variance of $r$ are

$\displaystyle E\\!\left[r(t+\delta t)\right]$ $\displaystyle=\tilde{\mu}(t)$
(2a) $\displaystyle E\\!\left[\big{(}r(t+\delta
t)-\tilde{\mu}(t)\big{)}^{2}\right]$ $\displaystyle=\widetilde{\sigma^{2}}(t)$
(2b)

We are making the convenient abuse of notation

$\widetilde{\sigma}=\sqrt{\widetilde{\sigma^{2}}}.$ (3)

In this paper, we consider the class of forecasts where
$\widetilde{\sigma^{2}}$ is a function of the past squared returns, possibly
with an additive constant for the affine processes.

At a one-day scale, the financial time series are dominated by the random
increment $\epsilon$ and by the volatility dynamics $\widetilde{\sigma}$,
while the mean term $\tilde{\mu}$ is small and very difficult to estimate.
Consequently, we will take $\tilde{\mu}=0$ afterward.

For a _linear_ model, the variance is a linear function of the past squared
returns

$\displaystyle\widetilde{\sigma^{2}}_{\text{lin}}(t)$
$\displaystyle=\sum_{l=0}^{l_{\text{max}}}w(l)\,r^{2}(t-l)$ (4a)
$\displaystyle\sum_{l=0}^{l_{\text{max}}}w(l)$ $\displaystyle=1\hskip
30.00005ptw(l)\geq 0$ (4b)

where $t$ is the current time, $l$ is the lag from the current time, and
$r(t-l)$ is the one day return at the time $t-l$. The weights $w(l)$ specifies
the importance of the past events on today volatility forecast. They are
typically decreasing for increasing lag, quantifying the progressive decay of
the past information on the forthcoming events. In practice, they are defined
on a bounded support for numerical evaluation. Some standard shapes are the
constant weights $w(l)=1/n$ in a window of length $n$ (Rectangular Moving
Average), and the exponentially decaying weights $w(l)=\exp(-l/\tau)$
(Exponential Moving Average with $\mu=\exp(-1/\tau)$). An intermediary shape
is given by the long memory weights proposed in [Zumbach, 2004], where the
weights decay as a logarithm of the lag. This last shape gives lagged
correlations for the volatility that are very close to the empirical ones.

For a linear ARCH model, the equation 4b is a consistency condition for the
volatility, which ensure that $E\\!\left[r^{2}(t)\right]$ is given by the
initial conditions of the process. If the sum of the weights is smaller
(larger) than one, the expected variance goes to zero (infinity) exponentially
fast for increasing time. Essentially, the conditions $\sum w(l)=1$ and
$E\\!\left[\epsilon^{2}\right]=1$ are consistency conditions so that the
volatility stays around the initial values. If these conditions are not
fulfilled, the process goes to a singular state, either to a fixed price (zero
volatility) or to in infinite volatility.

For an _affine_ model, the variance can be written as a convex combination
between the mean variance and a linear ARCH model

$\displaystyle\widetilde{\sigma^{2}}_{\text{aff}}(t)$
$\displaystyle=w_{\infty}\sigma_{\infty}^{2}+(1-w_{\infty})\widetilde{\sigma^{2}}_{\text{lin}}(t)$
(5)
$\displaystyle=w_{\infty}\sigma_{\infty}^{2}+\sum_{l=0}(1-w_{\infty})w(l)\,r^{2}(t-l)$

where the parameter $0\leq w_{\infty}\leq 1$ controls the balance between the
mean volatility $\sigma_{\infty}$ and the auto-regressive part [Zumbach,
2004]. In the second form, the sum of all the coefficients is 1. Again, this
is a consistency condition to ensure that $E\\!\left[r^{2}(t)\right]$
converges to $\sigma_{\infty}^{2}$ for increasing time, namely that
$\sigma_{\infty}$ is the parameter that fixes the mean volatility.

The familiar GARCH(1,1) process can be cast in the form (5), with the linear
part $\widetilde{\sigma^{2}}_{\text{lin}}$ being an EMA (see [Zumbach, 2004]
or [Zumbach, 2012] for the mapping between equations and parameters). In the
present form, the parameters for GARCH(1,1) can be easily interpreted. The
dynamic volatility part is measured by an EMA, with a characteristic time
given by the exponential decay coefficient. The long term asymptotic
volatility is $\sigma_{\infty}$, and the convex combination between both
variances is controlled by $w_{\infty}$. Essentially, the parameter
$w_{\infty}$ controls the conditional heteroskedasticity, the other parameters
control the mean volatility and the dynamical measure of recent past
volatility.

When estimated on empirical data, the coefficient $w_{\infty}$ is always
small, of the order of 2 to 10%. This value depends on the linear part, more
precisely on the decay of the weights $w(l)$. With a slow decay (i.e. slower
than exponential), the parameter $w_{\infty}$ becomes smaller, in the 2 to 5%
range. This can be understood as the long term weights inducing a mean
volatility measured on the past sample, hence reducing the usefulness of a
mean volatility parameter in the equation.

The small value for $w_{\infty}$ means that the financial markets operate
close to the linear regime, where the volatility is purely auto-regressive.
The stabilization provided by $\sigma_{\infty}$ is weak, leading to large
volatility excursions (crises). Hence, it is a good approximation for the
empirical studies to use a linear model instead of an affine one, removing 2
parameters ($w_{\infty}$ and $\sigma_{\infty}$). The most annoying parameter
is $\sigma_{\infty}$, since time series dependent. Moreover, its main purpose
is to provide sane long term statistical properties (see [Nelson, 1990] for a
detailed analysis for the GARCH(1,1) process), that are irrelevant at short
time scale. Hence, it is simpler to remove these 2 parameters and to work with
a minimal process.

With historical data, Eq. 1 can be reverted to obtain the realized innovations

$\epsilon(t+\delta t)=\frac{r(t+\delta
t)-\tilde{\mu}(t)}{\widetilde{\sigma}(t)}.$ (6)

The time series of innovations is the basis for the subsequent analysis.

##  3 Statistics on the innovations and p-values

In a process, the innovations $\epsilon$ have a distribution $p(\epsilon)$.
The standard choices for $p$ are a normal distribution and a Student
distribution (scaled to have a unit variance). We will use these two
distributions to compute the $p$-values of the empirical mean and variance,
using the central limit theorem.

The base quantities for the statistical test are the empirical mean and
variance of $\epsilon$

$\displaystyle\widehat{\mu}$ $\displaystyle=\frac{1}{n}\,\sum_{t}\epsilon(t)$
(7a) $\displaystyle\widehat{\sigma^{2}}$
$\displaystyle=\frac{1}{n-1}\,\sum_{t}\left(\epsilon(t)-\hat{\mu}\right)^{2}$
(7b)

with $n$ points in the sum. The benchmark is to assume a normal distribution
or a scaled Student distribution. For both distributions, the mean of
$\epsilon$ is 0 and the variance is 1. The mean of $\epsilon^{2}$ is 1, and
the variance of $\epsilon^{2}$ is

$\displaystyle\text{var}(\epsilon^{2})$ $\displaystyle=2$ Normal distribution
(8a) $\displaystyle\text{var}(\epsilon^{2})$
$\displaystyle=2\,\frac{\nu-1}{\nu-4}\hskip 30.00005pt\nu>4\hskip 20.00003pt$
$\displaystyle\text{Student distribution}.$ (8b)

These relations can be derived as follows. For the Student distribution, the
kurtosis is

$\operatorname{kurt}=E\\!\left[\left(\frac{\epsilon-\mu}{\sigma}\right)^{4}\right]=3+\frac{6}{\nu-4}$

which is equal to $E\\!\left[\epsilon^{4}\right]$ for the scaled Student
distribution. The variance of $\epsilon^{2}$ for the scaled Student
distribution is

$\text{var}(\epsilon^{2})=E\\!\left[\epsilon^{4}\right]-E\\!\left[\epsilon^{2}\right]^{2}=3+\frac{6}{\nu-4}-1$

leading to Eq. 8b. The result for the normal distributon can be derived
similarly, or by taking the large $\nu$ limit of the Student relation.

The central limit theorem can now be used to find the distribution for
$\widehat{\mu}$ and $\widehat{\sigma^{2}}$. We introduce a variable $z$,
scaled so has to have a standard normal distribution. For the sample mean, in
the large $n$ limit, the quantity

$z=z(\widehat{\mu})=\frac{\widehat{\mu}-0}{\sqrt{1/n}}$ (9)

has a normal distribution with zero mean and a unit variance. For an empirical
value $\widehat{\mu}$, the probability of a value as large or larger than
$|\widehat{\mu}|$ (two sided test) is

$\displaystyle p=2(1-\operatorname{cdf}(z(|\widehat{\mu}|)))$ (10)

where $\operatorname{cdf}$ is the cumulative distribution function of the
normal distribution.

For the variance, the quantity

$z=z\left(\widehat{\sigma^{2}}\right)=\frac{\widehat{\sigma^{2}}-1}{\sqrt{\text{var}(\epsilon^{2})/n}}$
(11)

has a normal distribution with zero mean and a unit variance. For an empirical
value $\widehat{\sigma^{2}}$, the probability of a value as large or larger
than $\widehat{\sigma^{2}}$ (one sided test) is

$\displaystyle p=1-\operatorname{cdf}(z)$ (12)

This $p$-value can be computed assuming a normal or Student distribution for
$\epsilon$ using Eq. 8.

##  4 Empirical results for the means

The statistics $\widehat{\mu}$ and $\widehat{\sigma^{2}}$ with their p-values
have been computed for a set of 24 time series with a long time span. Major
stock indexes, commodity indexes and FX are included in the empirical sample.
The results are given in table 1 for the volatility evaluated with the LM-ARCH
weights. The parameters for the volatility estimator are fixed according to
[Zumbach, 2006], namely they have not been optimized on each time series as
done in most studies. These results are analyzed first, other volatility
forecasts discussed shortly in section 6.

|  |  | mean | variance
---|---|---|---|---
|  |  |  |  |  | p-value
name | startDate | length | mean | p-value | variance | St 6 | St 5
CAC 40 | 1990-03-15 | 7630 | 0.013 | 0.248 | 1.049 | 0.029 | 0.066
DAX | 1962-09-10 | 14494 | 0.027 | 0.001 | 1.054 | 0.002 | 0.011
Dow Jones Industrial Average | 1962-09-07 | 14546 | 0.035 | 0.000 | 1.060 | 0.001 | 0.005
EuroStoxx | 1989-09-06 | 7859 | 0.019 | 0.091 | 1.067 | 0.004 | 0.018
FTSE 100 | 1986-09-05 | 8505 | 0.021 | 0.051 | 1.051 | 0.018 | 0.048
Hong Kong Hang Seng Index | 1969-11-24 | 12373 | 0.047 | 0.000 | 1.069 | 0.000 | 0.003
IBEX 35 | 1989-09-11 | 7721 | 0.014 | 0.232 | 1.086 | 0.000 | 0.004
NASDAQ 100 | 1987-10-12 | 8199 | 0.049 | 0.000 | 1.051 | 0.019 | 0.050
Nikkei 225 | 1972-09-11 | 11718 | 0.022 | 0.019 | 1.078 | 0.000 | 0.001
OMX STOCKHOLM 30 INDEX | 1989-08-24 | 7697 | 0.028 | 0.013 | 1.063 | 0.007 | 0.026
S&P GSCI Agriculture | 1972-09-06 | 12014 | 0.011 | 0.218 | 1.065 | 0.001 | 0.006
S&P GSCI Coffee | 1983-09-14 | 9241 | 0.014 | 0.173 | 1.082 | 0.000 | 0.003
S&P GSCI Cotton | 1979-09-14 | 10250 | 0.010 | 0.288 | 1.089 | 0.000 | 0.001
S&P GSCI Energy | 1985-09-13 | 8736 | 0.012 | 0.272 | 1.072 | 0.001 | 0.009
S&P GSCI Gold | 1980-09-12 | 9999 | 0.017 | 0.082 | 1.098 | 0.000 | 0.000
S&P GSCI Grains | 1972-09-06 | 12015 | 0.012 | 0.187 | 1.066 | 0.001 | 0.005
S&P GSCI Light Energy | 1972-09-08 | 12013 | 0.014 | 0.125 | 1.075 | 0.000 | 0.002
S&P GSCI Non Precious Metals | 1976-09-14 | 11004 | 0.010 | 0.289 | 1.072 | 0.000 | 0.004
S&P GSCI Soybeans | 1972-09-06 | 12015 | 0.025 | 0.007 | 1.051 | 0.006 | 0.023
CAD-USD | 1973-09-10 | 12164 | -0.014 | 0.126 | 1.061 | 0.001 | 0.008
CHF-USD | 1973-09-10 | 12164 | 0.022 | 0.016 | 1.175 | 0.000 | 0.000
EUR-USD | 1977-09-08 | 10950 | 0.002 | 0.850 | 1.072 | 0.000 | 0.004
GBP-USD | 1973-09-10 | 12158 | -0.005 | 0.583 | 1.082 | 0.000 | 0.001
JPY-USD | 1973-09-10 | 12164 | 0.015 | 0.098 | 1.063 | 0.001 | 0.007
Table 1: Statistics for $\widehat{\mu}$ and $\widehat{\sigma^{2}}$ with their
p-values. The volatility is computed using LM-ARCH weights. The column with
header “St 6” contains the $p$-value for the variance assuming a Student
distribution for the innovations with 6 degrees of freedom (and similarly for
the column “St 5”).

The mean of $\epsilon$ is significantly departing from zero for roughly two
third of the stock indexes, for the soybeans and for USD/CHF. Let us emphasize
that we are not testing the significance of the mean returns, but the returns
discounted by a forecast for the volatility. Roughly, this corresponds to a
trading strategy with increasing positions (and less cash) for low volatility
forecasts, and decreasing positions (and more cash) for high volatility
forecasts, in effect having a portfolio at approximately constant volatility.

Figure 1:  The value of the stock index used to compute the significance of
the mean innovations. The curves are ordered from bottom to top by increasing
significance, with the curves in red as not-significant, the ones in blue and
black as significant. The labels are ordered accordingly. See the table 1 for
the p-values.

The stock indexes is the most interesting group. Fig. 1 shows the time series
values on a logarithmic scale, with arbitrary shifts set so that the
significance of a non-zero mean is increasing for curves higher in the graph.
The interpretation of the significance values is not straightforward, because
the economy is an evolving system, and because the time series lengths are not
equal (between $\sim$30 to $\sim$50 years). Therefore, we have a mix between
the influence of $n$ (sample size) and the period (particular sample).
Interestingly, over the last 30 years, some indexes do not show a significant
deviation from zero. This should be a caveat for investment advisers on the
common statement that the stock market is a good investment over the long
term. Notice that they are market weighted indexes, but not total return
indexes, namely they do not include the possible dividends. For the indexes
with the largest $p$-values (curves at the bottom of graph 1), an explanation
is that the possible increases of values has been distributed to the stock
holders as dividends. With this interpretation, the large $p$-values for the
mean indicate that the underlying companies are more redistributing the
increased yields than capitalizing for the future. Interestingly, only
European indices are in this case.

For the commodities, only the soybeans index moved significantly, from the
range 200 to 350 between 1973 to 2008, then in the range 350 to 600 since
2008. The other commodity indices are not significantly departing from zero,
including precious metal, energy, coffee or chocolate. With a $p$-value of 8%,
gold is close to have a significant upward move. Its evolution shows values in
the 200 to 500 range up to 2010, and in the 600 to 1100 range after 2010.

In the FX space, only the USD/CHF moved significantly, from roughly 1 USD for
3 CHF in the ’70 to an approximate parity since 2010. This is a factor 3
appreciation of the CHF, in effect during 35 years (the last decade is
fluctuating around parity). All other currencies did not move significantly
against the USD. Let us emphasize that several decades of daily data have been
used to reach these results, and it not possible to conclude when using only a
few years of data.

##  5 Empirical results for the variances

We now turn to the analysis of the variance of the daily innovations
$\epsilon$ computed with LM-ARCH weights, using the statistics on
$\widehat{\sigma^{2}}$. For all time series, the variance is larger than 1,
and the departure is significant. All the variances are between 1.05 and 1.09,
without obvious difference between asset classes. The outlier is the USD/CHF
exchange rate, which has a much larger variance of 1.17. This is due to the
floor removal by the Swiss National Bank (SNB) of the minimum rate of 1.20 CHF
per EUR on January 15, 2015. The SNB gave no prior indication to the market
about the forthcoming change of policy, and on this day, the innovation for
USD/CHF is a staggering 44.5! This event is arguably the largest shocks on a
single asset for the last decade. Since this innovation value is entering the
variance with a square, this single day explains the much larger empirical
value.

With the hypothesis of a normal distribution for the innovations, the
departure from 1 is always significant, with a $p$-value smaller than
$10^{-3}$ in all cases (values not reported in the table). With a Student
distribution with 6 degrees of freedom, the results are always significant at
a 5% level, and significant at the 1% levels except for 3 stock indexes. With
5 degrees of freedom, all values are significantly above 1 at the 5% level,
except the CAC 40 and the Nasdaq. Let us emphasize that the theoretical
variance of $\epsilon^{2}$ for a Student distribution has a $1/(\nu-4)$
singularity (see Eq. 8b), hence the last distribution is close to the
theoretical singularity. Clearly, assuming a distribution with larger tails
makes more difficult to have significant $p$-values. Let us emphasize that
decades of data are required to show significant deviations from unity for a
Student distribution.

The departure from 1 is always on the high side, namely toward an explosive
grow. A departure of 5% for the variance corresponds to an expected doubling
of the volatility in 28 business days! Such an exponential volatility grow is
clearly not observed in the financial market, at least on long enough sample.

The difference between the model assumptions and the empirical values points
to an incomplete model. Because on the high side, the empirical time series
tend to be more reactive, with occasional large innovations. This behavior
indicates a system that adapt quickly to changing conditions, with a possible
over reaction. As emphasized in Sec. 2, the corresponding theoretical model is
unstable, but it is sufficient to add a small mean reverting term to stabilize
it. For example, an Ornstein-Uhlenbeck term can be added in the return
equation

$r(t+\delta
t)=-\,\frac{p(t)-p(t-n)}{\sqrt{n}\,p(t-n)}\,\frac{1}{\tau}+\tilde{\mu}(t)+(1+\gamma)\,\widetilde{\sigma}(t)\,\epsilon(t+\delta
t)$ (13)

where $\gamma\geq 0$ controls the short term instability in the model (with
the same conditions as above on $\widetilde{\sigma}$ and the distribution for
$\epsilon$). The parameter $\tau$ controls the intensity of the pull-back
term, and is expressed in day (an implicit $\delta t$ = 1 day time step is
assumed in the equation). The parameter $n$ controls the lag for measuring
price changes, and is also a free parameter. This term introduces a mean
reversion toward past prices and limits the price changes over a longer time
interval, hence limits the volatility excursions. The validation of such
processes against empirical data will be difficult, because the deviations
from the current process are small.

In the recent literature on stochastic volatility, a new thread concerns the
rough volatility models (see [Gatheral et al., 2018] for the original paper).
The overall idea and construction of this model is quite different from the
ARCH processes, and involves an independent process for the volatility witch
uses a fractional Brownian motions (with Hurst exponent $H<1/2$) in order to
describe the volatility dynamics. An interesting empirical point is that the
realized volatility (defined using high frequency data) shows short term
fluctuations of the volatility larger than the usual Brownian motion. This
feature leads to the name “rough volatility”. Then, the negative auto-
correlations introduced by the fractional Brownian motion in the volatility
increments avoid too large excursions of the volatility.

The ARCH models are conceptually different since the volatility is computed
from the past price path, and is not an independent hidden process. The
present work points to a fine shortcoming of the ARCH models, in a similar
direction as the rough volatility model, namely to a short term roughness of
financial time series and to a mean reversion for the volatility.

##  6 Empirical results with other volatility estimators

|  |  | mean | variance
---|---|---|---|---
|  |  |  |  |  | p-value
name | startDate | length | mean | p-value | variance | St 6 | St 5
CAC 40 | 1989-06-08 | 7862 | 0.016 | 0.156 | 1.187 | 0 | 0
DAX | 1961-12-04 | 14717 | 0.025 | 0.003 | 1.162 | 0 | 0
Dow Jones Industrial Average | 1961-12-01 | 14773 | 0.030 | 0 | 1.209 | 0 | 0
EuroStoxx | 1988-11-30 | 8094 | 0.024 | 0.031 | 1.187 | 0 | 0
FTSE 100 | 1985-11-29 | 8736 | 0.024 | 0.025 | 1.215 | 0 | 0
Hong Kong Hang Seng Index | 1969-11-24 | 12407 | 0.042 | 0 | 1.217 | 0 | 0
IBEX 35 | 1988-12-05 | 7946 | 0.018 | 0.112 | 1.209 | 0 | 0
NASDAQ 100 | 1987-01-05 | 8428 | 0.050 | 0 | 1.221 | 0 | 0
Nikkei 225 | 1971-12-06 | 11942 | 0.029 | 0.002 | 1.183 | 0 | 0
OMX STOCKHOLM 30 INDEX | 1988-11-17 | 7922 | 0.031 | 0.006 | 1.177 | 0 | 0
S&P GSCI Agriculture | 1971-12-01 | 12240 | 0.015 | 0.089 | 1.107 | 0 | 0
S&P GSCI Coffee | 1982-12-08 | 9468 | 0.014 | 0.180 | 1.129 | 0 | 0
S&P GSCI Cotton | 1978-12-08 | 10476 | 0.004 | 0.709 | 1.137 | 0 | 0
S&P GSCI Energy | 1984-12-07 | 8962 | 0.006 | 0.578 | 1.266 | 0 | 0
S&P GSCI Gold | 1979-12-07 | 10224 | 0.019 | 0.049 | 1.114 | 0 | 0
S&P GSCI Grains | 1971-12-01 | 12241 | 0.017 | 0.065 | 1.106 | 0 | 0
S&P GSCI Light Energy | 1971-12-03 | 12239 | 0.014 | 0.125 | 1.139 | 0 | 0
S&P GSCI Non Precious Metals | 1975-12-09 | 11229 | 0.006 | 0.529 | 1.161 | 0 | 0
S&P GSCI Soybeans | 1971-12-01 | 12241 | 0.02 | 0.026 | 1.125 | 0 | 0
CAD-USD | 1972-12-04 | 12397 | -0.014 | 0.116 | 1.123 | 0 | 0
CHF-USD | 1972-12-04 | 12397 | 0.025 | 0.005 | 1.169 | 0 | 0
EUR-USD | 1976-12-02 | 11175 | 0.001 | 0.892 | 1.123 | 0 | 0
GBP-USD | 1972-12-04 | 12391 | -0.009 | 0.343 | 1.106 | 0 | 0
JPY-USD | 1972-12-04 | 12397 | 0.025 | 0.005 | 1.146 | 0 | 0
Table 2: Statistics for $\widehat{\mu}$ and $\widehat{\sigma^{2}}$ with their p-values. The volatility is conputed using equal weights in a window of 500 points. |  |  | mean | variance
---|---|---|---|---
|  |  |  |  |  | p-value
name | startDate | length | mean | p-value | variance | St 6 | St 5
CAC 40 | 1990-03-15 | 7630 | 0.012 | 0.303 | 1.137 | 0 | 0
DAX | 1962-09-10 | 14494 | 0.027 | 0.001 | 1.147 | 0 | 0
Dow Jones Industrial Average | 1962-09-07 | 14546 | 0.036 | 0 | 1.150 | 0 | 0
EuroStoxx | 1989-09-06 | 7859 | 0.017 | 0.127 | 1.188 | 0 | 0
FTSE 100 | 1986-09-05 | 8505 | 0.021 | 0.056 | 1.144 | 0 | 0
Hong Kong Hang Seng Index | 1969-11-24 | 12373 | 0.050 | 0 | 1.195 | 0 | 0
IBEX 35 | 1989-09-11 | 7721 | 0.012 | 0.301 | 1.190 | 0 | 0
NASDAQ 100 | 1987-10-12 | 8199 | 0.051 | 0 | 1.142 | 0 | 0
Nikkei 225 | 1972-09-11 | 11718 | 0.023 | 0.013 | 1.190 | 0 | 0
OMX STOCKHOLM 30 INDEX | 1989-08-24 | 7697 | 0.030 | 0.008 | 1.154 | 0 | 0
S&P GSCI Agriculture | 1972-09-06 | 12014 | 0.012 | 0.183 | 1.125 | 0 | 0
S&P GSCI Coffee | 1983-09-14 | 9241 | 0.015 | 0.136 | 1.154 | 0 | 0
S&P GSCI Cotton | 1979-09-14 | 10250 | 0.012 | 0.224 | 1.146 | 0 | 0
S&P GSCI Energy | 1985-09-13 | 8736 | 0.013 | 0.233 | 1.148 | 0 | 0
S&P GSCI Gold | 1980-09-12 | 9999 | 0.019 | 0.061 | 1.197 | 0 | 0
S&P GSCI Grains | 1972-09-06 | 12015 | 0.012 | 0.177 | 1.127 | 0 | 0
S&P GSCI Light Energy | 1972-09-08 | 12013 | 0.014 | 0.125 | 1.122 | 0 | 0
S&P GSCI Non Precious Metals | 1976-09-14 | 11004 | 0.010 | 0.282 | 1.137 | 0 | 0
S&P GSCI Soybeans | 1972-09-06 | 12015 | 0.026 | 0.004 | 1.132 | 0 | 0
CAD-USD | 1973-09-10 | 12164 | -0.015 | 0.096 | 1.120 | 0 | 0
CHF-USD | 1973-09-10 | 12164 | 0.022 | 0.013 | 1.252 | 0 | 0
EUR-USD | 1977-09-08 | 10950 | 0.002 | 0.842 | 1.134 | 0 | 0
GBP-USD | 1973-09-10 | 12158 | -0.002 | 0.800 | 1.238 | 0 | 0
JPY-USD | 1973-09-10 | 12164 | 0.012 | 0.182 | 1.209 | 0 | 0
Table 3: Statistics for $\widehat{\mu}$ and $\widehat{\sigma^{2}}$ with their
p-values. The volatility is computed using EMA weights with factor 0.94.

Other shapes for the volatility kernel can be used. The results for a
rectangular moving average (RMA) of length 500 and an exponential moving
average (EMA) with decay factor 0.94 are reported in this section. The sample
lengths vary slightly, due to the different initialization lengths of the
different kernels. This can impact mostly the statistics $\widehat{\mu}$, to a
lesser extent $\widehat{\sigma^{2}}$. For the mean values, the figures are
essentially consistent with the LM-ARCH kernel. The largest systematic
differences occur for $\widehat{\sigma^{2}}$, with typical values in the range
1.1 to 1.2, to be compared with the range 1.05 to 1.09 with a LM-ARCH kernel.
Due to these larger statistics, all the p-values are less than $10^{-3}$,
namely the value 1 is always excluded with a high significance, including with
a distributional assumption of a Student with 5 degrees of freedom.

The higher values for the variances show that these alternative kernels are
less good at whitening the returns, and the empirical data are further away
from the related theoretical processes. Another hypothesis of the process is
that the innovations should be i.i.d., namely independent and identically
distributed. Independence is not tested using only the variance, but a new
powerful “tile test” has been presented recently [Zumbach, 2021] that focuses
on the time dependency in a forecast setting for risk evaluations. The results
of the tile test are consistent with the present ones, namely the LM-ARCH
kernel is excellent at transforming empirical returns into innovations that
are close to the theoretical hypothesis. The previous section shows that the
LM-ARCH is very good, but not perfect.

##  7 Further investigations

We have tried some variations around the present LM-ARCH kernel, changing the
shape of the decay and/or its finite length. Although we have certainly not
tested all possibilities, no clear improvements can be achieved, and most
changes produce systematically higher values for the empirical variances.
Hence, it seems difficult to reach unit empirical variances by modifying only
the shape of the volatility kernel.

Can a unit variance be recovered by using an affine variance estimator instead
of a linear one? Using the process set-up as described in Sec. 2, the answer
is no. The equations have been set such that
$E\\!\left[\widetilde{\sigma^{2}}_{\text{aff}}\right]=E\\!\left[\widetilde{\sigma^{2}}_{\text{lin}}\right]=E\\!\left[r^{2}\right]=\sigma^{2}_{\infty}$,
regardless of $w_{\infty}$. The parameter $\sigma^{2}_{\infty}$ fixes all the
mean variance, and there is no parameter left with that respect. Hence, in
order to get more freedom in the model, some equations have to be modified.

One possibility is to not use the equation 1 for the process, while still
using 6 to define the innovations. This corresponds to a purely empirical set-
up, where the data are generated by some unknown process, and another variable
called innovation is defined from the historical returns. The relation between
linear and affine variance is still given by Eq. 5, and the affine variance is
used to discount the historical return. In order to recover innovations with a
unit variance, we impose the further condition

$\displaystyle
E\\!\left[\widetilde{\sigma^{2}}_{\text{aff}}\right]=(1+\gamma)\,E\\!\left[\widetilde{\sigma^{2}}_{\text{lin}}\right]$

Taking the expectation of Eq. 5 and combining with this relation, we obtain

$\sigma^{2}_{\infty}=E\\!\left[r^{2}\right]\left(1+\frac{\gamma}{w_{\infty}}\right).$

Using plausible values for the parameters, $\gamma$ = 0.07 and $w_{\infty}$ =
0.035, the mean variance parameter should be 3 times larger than the
historical variance of the data. This large factor originates in the small
value for $w_{\infty}$, which should be compensated by a large
$\sigma^{2}_{\infty}$ in order to have an impact on the final variance.
Clearly, this does not make sense to have such a discrepancy between the
empirical sample variance and the corresponding parameter, and deeper
modifications of the equations should be introduced.

In the ARCH model, we have decided to neglect the mean return term
$\widetilde{\mu}(t)$, because difficult to estimate and dominated by the
volatility. We can do a similar hypothesis for the innovations, and measure
the size of the innovations with

$\displaystyle\frac{1}{n-1}\,\sum_{t}\left(\epsilon(t)\right)^{2}.$ (14)

Overall, the results using this estimator show only minor modifications
compared to the estimator (7b), and in particular the significance of the
departure from 1 are almost identical.

The empirical study can be extended to longer time interval for the
innovations. The idea is to compute an ex-ante volatility forecast for a
horizon $\Delta T$, to get the ex-post realized returns at the corresponding
horizon $\Delta T$, and finally to compute the empirical innovations at
$\Delta T$. With increasing time interval $\Delta T$, the innovation variances
are typically growing, the effective sample sizes
$n_{\text{efffective}}=n/\Delta T$ are decreasing, and the threshold values
for statistical significance are also growing. Beside the present computations
with $\Delta T$ = 1 day, we have done the investigation for $\Delta T$ = 2, 5,
10 and 30 days. Overall, the innovation variances is growing at the same pace
as the significance threshold, but the dispersion between the time series is
increasing. The statistical results for the above empirical sample are thus
less clear cut, with the variances for a few time series that are not
significantly above 1. Nevertheless, the majority of innovation variances are
significantly above 1 (for a Student distribution with 6 degrees of freedom
and at a 5% level).

##  8 Conclusions

The family of ARCH processes is very good at describing financial time series,
and can capture most of known stylized facts. The model used in this study can
be extended in several ways, for example to include the leverage effect
presents for stocks and stock indexes. Yet, its very simplicity and robustness
is an important advantage when dealing with large number of time series, and
for various asset types, in particular for risk estimations. In this context,
the LM-ARCH process proves to be very efficient (see [Zumbach, 2021] for a
recent back-test of risk evaluations). Moreover, the theoretical backing by a
quadratic ARCH process allows deriving consistent volatility forecasts at
increasing risk horizons.

The volatility forecast is a crucial ingredient in a risk computation, the
following one is the distribution for the innovations. The usual assumption is
that the variance for the innovations should be 1, for all time series. This
study shows that the empirical variances are slightly but significantly above
one. In order to reach this conclusion, and in particular to have significant
values at a 5% level, empirical samples of 3 to 5 decades are necessary. The
empirical variances are in a range from 1.05 to 1.09, a small but significant
deviation from the theoretical process with a unit value.

The empirical variances are always above 1, indicating a systematic short term
instability and a potentially explosive grow for the volatility. This points
to a fast adaptability of the financial system to a changing environment, with
the trading activity and the induced prices that react quickly to changing
conditions. On the theoretical side, this point to an incomplete model, with a
possible short term instability that must be compensated by a stabilization
term acting at intermediate time horizons. Since this is a small deviation
from the current model, it will be difficult to validate such an extension.

Another theoretical condition is that the mean of the innovations should be
zero. Most commodity indexes and FX are compatible with this hypothesis, apart
from the soybean index and USD/CHF exchange rate. For stock indexes, it is
widely believed that they are rising on the long term, hence a positive mean
innovation is expected to be significant. Overall, this is the case, except
clearly for the CAC 40 and IBEX 35, and marginally for the EuroStoxx and FTSE
100. These zero expected innovations mean that the possible increase in the
underlying stock prices have been distributed in form of dividends (or that
there was no significant increase of values!).

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