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WP0505
Equivalence and bifurcations of
finite order stochastic processes
Cees Diks and Florian Wagener
Equivalence and bifurcations of finite order stochastic
processes !
C. Diks and F.O.O. Wagener †
25th April 2005
Abstract
This article presents an equivalence notion of finite order stochastic processes. Local dependence mea
sures are defined in terms of ratios of joint and marginal probability densities. The dependence measures
are classified topologically using level sets. The corresponding bifurcation theory is illustrated with
some simple examples.
1 Introduction
Bifurcation theory has been an extremely successful tool to investigate the qualitative (or structural) prop
erties of deterministic nonlinear systems. But in many practical situations, deterministic models fit the
available data only imperfectly, and stochastic models are proposed to describe the behaviour of a given
system; the stochastic components can model genuinely random events, but they can also be introduced for
quantities of which not enough is known to describe them otherwise.
Motivated by the success of deterministic bifurcation theory, there have been several attempts to
develop bifurcation theory for stochastic processes; however, to find a natural replacement of the notion of
‘topological equivalence’ has been the main problem. For at the base of any bifurcation theory, there is a
notion of ‘form’ or ‘shape’, formalised as an equivalence relation between systems: two systems are said to
be of the same form if they are in the same equivalence class. A meaningful bifurcation theory can only be
developed if there are equivalence classes with nonempty interior; note that this presupposes a topology on
the space of systems. Elements in the interior of an equivalence class are ‘structurally stable’: if a system
parameter is changed slowly, the system will remain in the equivalence class and the form of the system
does not change. All other elements, associated to changes of form, are called ‘bifurcating’.
In this article, we quickly review the previously proposed notions of phenomenological and dy
namical bifurcation of stochastic processes. We introduce a new equivalence relation, based on the depen
dence structure of the process. This notion is related, and in some cases equal, to the better known copula
density of dependent stochastic variables. We show that our equivalence relation has ‘many’ structurally
stable elements and that it avoids some limitations of older notions. Finally, we illustrate it by giving several
applications.
!The authors wish to thank the participants of the workshop on Stochastic Bifurcation Theory, January 2003, in Leiden, for
stimulating discussions. This research is supported by the Netherlands Organization for Scientific Research (NWO) under a MaGW
Pionier and a MaGWVernieuwingsimpuls grant.
†Center for Nonlinear Dynamics in Economics and Finance (CeNDEF), Universiteit van Amsterdam, Roetersstraat 11,
1018 WB Amsterdam, The Netherlands, C.G.H.Diks@uva.nl, F.O.O.Wagener@uva.nl
1
1.1 Phenomenological bifurcations
Without always stating it explicitly, in this article we shall always assume that the stochastic processes
considered are ergodic, and that they have therefore unique invariant probability distributions; we shall
moreover assume that these invariant distributions are absolutely continuous with respect to a measure of
the Lebesgue class, and that the corresponding probability density is a smooth differentiable function.
The natural first attempt to attain at a classification of stochastic processes is to apply the Morse
classification of real valued functions to invariant probability densities p of processes [3, 15]. The corre
sponding equivalence relation is that of smooth coordinate transformations of domain and range of p, the
stable elements being Morse functions with all critical values distinct from each other. For the purposes of
this article, we shall call the equivalence relation Pequivalence, in analogy with the associated bifurcation
notion, which has been called phenomenological bifurcation or Pbifurcation (see Arnold [2], p. 471473).
The main problem of this approach, acknowledged in [15], is that the equivalence classes are not
invariant under diffeomorphisms of the underlying space. For instance, let {Xt} and {Yt} be two processes
on Rm with invariant densities pX and pY , and let ! be an invertible transformation of Rm. If the processes
are related by Yt = !(Xt), then the invariant densities are related as
pX(x) = pY (!(x)) detD!(x);
we see that, in the language of physicists, the function value of the invariant density ‘depends on the coor
dinates’. It is clear that equivalence classes are preserved if only volumepreserving diffeomorphisms are
admitted as coordinate changes, for which detD!(x) = 1 for all x. Note that these comprise the class of
Riemannian isometries proposed in [15].
The restriction of admissible transformations to volumepreserving diffeomorphisms of the do
main of the invariant density p is necessary. For if all diffeomorphisms were admissible, then all processes
on the real line would be equivalent, since there is always a coordinate transformation such that the invariant
density of the transformed process is the uniform density on the unit interval. In the words of Zeeman [15]:
‘However, this [i.e. admitting all diffeomorphisms] would be exactly the wrong thing to do, because it would
render the tool [i.e. Pequivalence] useless by making everything equivalent (...)’ (the comments in square
bracket are our interpolations).
Underlying this difficulty is the fact that the probability density p(x), unlike the measure p(x)dx, is
not an invariant geometrical object. By consequence, P–equivalence is an inconvenient notion for practical
applications: for instance, recording data on linear or logarithmic scale might point to different conclusions.
1.2 Dynamical bifurcations
A second bifurcation notion for stochastic processes has been introduced by Arnold and his coworkers
(see [2] for an extensive exposition). We shall try to sketch its main ideas briefly using the simple firstorder
process {Xt} on R satisfying
Xt+1 = g(Xt)+ "t ,
(1)
where the {"t} is a sequence of independent and identically distributed random variables. This process
can be considered as a deterministic dynamical system on an infinite dimensional phase space !"R as
follows. The elements of ! are the possible realisations # = ("0,"1, ···) of the process {"t}. Introducing
the projection $(#) = "0 and the shift %(#) = ("1,"2, ···), we have for instance that "t = $ #% t(#). Define
now the map " on !"R by
"(#,x) = (!1(#),!2(#,x)) = (%(#),g(x)+$(#)) .
2
Note that this is a deterministic system; the stochastics are ‘hidden’ in the fact that the initial condition# $!
is unknown. The realisations Xt of the process (1) are the values of the second component of "
t(#,x).
Note that " is a skew system: the shift dynamics !1 in the space ! are driving the dynamics !2
in R. For ", a random fixed point is defined as a map & :!% R, which satisfies the invariance condition
!2(#,& (#)) = & (!1(#))
for all (or almost all) # . Stability is now be defined in the usual way: a random fixed point & is stable if
all nearby orbits converge to & . A random, or, following the terminology in [2], dynamical bifurcation or
Dbifurcation of the process occurs for instance if a random fixed point loses stability.
At this point, a drawback of the notion of dynamical bifurcation becomes apparent: to determine
stability of a random fixed point, two orbits of" with identical noise realisations have to be compared. This
seems to make it rather difficult to apply the notion of Dbifurcation to practical problems (see however [4]
and related literature). We therefore leave aside this theory, and try rather to improve on the notion of
Pequivalence.
1.3 Local dependency structure
By considering stochastic analogues of concepts used in catastrophe theory, Hartelman et al. [12] arrived at a
classification for stochastic differential equations of diffusion type on the real line that is invariant under in
vertible transformations. At the basis of this classification are level crossing statistics and derived quantities,
which are invariant under monotonous transformations of the underlasying space. Although level crossing
statistics can also be used for discrete time systems, the corresponding classification would be rather re
strictive. The reason is that discrete time dynamical systems are ‘essentially richer’ than discretely sampled
continuous time diffusions, because finite time transition densities induced by diffusions only represent a
subclass of transition densities for discrete time dynamical systems.
Instead of level crossing statistics, we base our proposed equivalence relation on another function
that can be associated to stochastic processes of finite order. For the purposes of this introduction, we
explain the main ideas in the case of a first order process {Xt} on the real line; general definitions will be
given below. Assume therefore that the process {Xt} is generated by
Xt+1 = g(Xt)+ "t ,
(2)
where {"t} is a sequence of independent, identically distributed (IID) innovations. Recall that equation (2)
imply a transformation law for probability densities. Assume that Xt is distributed with marginal probability
density pt , that is P(a& Xt < b) =
! b
a p
t(x)dx, and that "t has density ! . Then the probability density pt+1
of Xt+1 is given by
pt+1(xt+1) =
"
'(xt+1xt)pt(xt)dxt ,
where '(xy) = !(x'g(y)) is the transition probability density of the process.
Consider in particular the strictly stationary process {Xt} where X1, and hence every Xt , is dis
tributed according to the invariant probability density p1 = p. Denote the invariant joint probability density
of (Xt ,Xt+1) by pt,t+1; this joint density does not depend on t and it is therefore equal to p1,2. Moreover,
the measure p1,2(x1,x2)dx1 dx2 is absolutely continuous with respect to p(x1)p(x2)dx1 dx2. By the Radon
Nikodym theorem, the following Jacobian exists:
f (x1,x2)
def=
p1,2(x1,x2)dx1 dx2
p(x1)p(x2)dx1 dx2
=
p1,2(x1,x2)
p(x1)p(x2)
=
'(x2x1)
p(x2)
;
3
The function f is called the dependency ratio of the process. Note that f is identically 1 if Xt and Xt+1
are independent; the difference  f (x1,x2)' 1 can therefore be seen as a measure of the local dependency
structure of the process. Moreover, f contains all essential information regarding the dependence structure
of the process: for if coordinates are chosen such that p(x) = 1 for x$ [0,1] and 0 otherwise, then f (x1,x2) =
p1,2(x1,x2) = '(x2x1) determines the entire process. By construction, dependency ratios are invariant —
as geometrical objects — under any invertible transformation the underlying space of the process under
consideration. Our notion of equivalent processes will be based on these ratios; invariance of the equivalence
classes under nonlinear invertible transformations is then obtained automatically.
Several other local dependence measures have recently been described in the statistical literature
(see e.g. [7], [8], and [10]). The dependence measures in this literature are various localised versions of the
Pearson correlation coefficient, and as such are motivated entirely differently than our dependency ratio. In
particular they do not share the invariance property with our dependency ratio.
The definition of our equivalence relation is a bit involved. We therefore postpone this definition
to section 3. First, in section 2, we give the definition of dependency ratio for more general processes, and
we show how this quantity is connected with other quantities like copula density, mutual information and
entropy of a stochastic process. In section 3, the definition of our equivalence relation is given, together with
topological properties of the associated equivalence classes. Applications are given in section 4.
2 Dependency ratios, copulas and information theory
Our attention will be restricted to stationary discrete time processes of finite order that are generated by
equations like
Xt = g(Xt'n, ··· ,Xt'1,"t).
Here the variables Xt take values in some n–dimensional orientable manifold M, and the "t are identically
and independently distributed random variables. In most cases M will be equal to Rn. We wish however
to bring out the dependency of the process on the variables chosen; in order to achieve this, a coordinate–
independent formulation is chosen. Considering the problem on a manifold will come at little extra cost.
2.1 Basic definitions
Recall that any stochastic process {Xt} is given by the joint probabilities
Pt1,··· ,t!(A1" ···"A!) = P{Xt1 $ A1, ··· ,Xt! $ A!},
which are defined for all !–tuples (t1, ··· , t!) $ Z!. A stochastic process is called strictly stationary if its
finite dimensional joint probabilities are time invariant
Pt1,··· ,t! = Pt1+h,··· ,t!+h,
for all (t1, ··· , t!,h). Two stochastic variables X and Y are said to be equivalent in distribution, written
as X ( Y , if P(X $ A) = P(Y $ A) for every Pmeasurable set A. A strictly stationary process {Xt} is
said to be of order n if, for all k > n, the conditional distribution of Xt given Xt'k,...,Xt'1 is a function of
(Xt'n,...,Xt'1) only:
Xt Xt'k,...,Xt'1 ( Xt Xt'n,...,Xt'1.
(3)
A measure on a manifold M is of Lebesgue class, if at every point of the manifold it is absolutely continuous
with respect to the Lebesgue measure in some (and hence any) coordinate chart. If M is orientable, there is
4
a volume form on M: a differentiable nform that is everywhere nonzero. It is clear that any volume form
is of Lebesgue class. We assume that M is orientable, and we fix a volume form on M, denoting it by dx. It
will be assumed that the processes we are dealing with have all joint probabilities Pt1,··· ,t! of Lebesgue class,
with twice differentiable probability densities pt1,··· ,t! .
The transition probability density ' of a process of order n is given by
'(xn+1x1, ··· ,xn) =
p1,··· ,n+1(x1, ··· ,xn+1)
p1,··· ,n(x1, ··· ,xn)
.
(4)
Note that from the order property, it follows that for k ) n
'(xt xt'k, ··· ,xt'1) = '(xt xt'n, ··· ,xt'1).
Given any initial probability density of X1, ··· ,Xn, we can determine all joint probabilities by first determin
ing
p1,··· ,n+1(x1, ··· ,xn+1) = '(xn+1x1, ··· ,xn)p1,··· ,n(x1, ··· ,xn),
p1,··· ,n+2(x1, ··· ,xn+2) = '(xn+2x2, ··· ,xn+1)'(xn+1x1, ··· ,xn)p1,··· ,n(x1, ··· ,xn),
...
and then integrating out unwanted variables. In particular, the distribution of X2, ··· ,Xn+1 is obtained by
p2,··· ,n+1(x2, ··· ,xn+1) =
"
'(xn+1x1, ··· ,xn)p1,··· ,n(x1, ··· ,xn)dx1.
The central notion of dependency ratio of a stochastic process is given in terms of joint probability densities.
Definition. The n’th order dependency ratio f of a strictly stationary stochastic process {Xt} is given by
f (x1, ··· ,xn+1) =
p1,··· ,n+1(x1, ··· ,xn+1)
p1,...,n(x1, ··· ,xn)p1(xn+1)
=
'(xn+1x1, ··· ,xn)
p(xn+1)
.
Note that f is the Jacobian of the measure p1,··· ,n+1(x1, ··· ,xn+1)dx1 ··· dxn+1 with respect to the mea
sure p1,...,n(x1, ··· ,xn)dx1 ··· dxn · p1(xn+1)dxn+1. In particular, f is chartindependent.
Proposition 1. If for two processes of order n the dependency ratios of order n are equal, then for every m)
n, the dependency ratios of order m are equal as well.
Proof.
We consider two processes {Xt} and {Yt} of order n. It suffices to show that orderm equality of the depen
dency ratio for any m) n implies, and is implied by, ordern equality. For m) n, we may write
p1,...,m+1(x1,...,xm+1) = p1,...,m(x1,...,xm)'(xm+1  x1,...,xm)
= p1,...,m(x1,...,xm)'(xm+1  xm'n+1,...,xm).
The dependency ratio of {Xt}, for m) n, can be written as a function of n+1 variables only:
p1,...,m+1(x1,...,xm+1)
p1,...,m(x1,...,xm)p(xm+1)
=
'(xm+1  xm'n+1,...,xm)
p(xm+1)
=
p1,...,n+1(xm'n+1,...,xm+1)
p1,...,n(xm'n+1,...,xm)p(xm+1)
,
which is nothing but the dependency ratio of order n in terms of the last n+ 1 variables of the vector
(x1,...,xm+1). The dependency ratio of {Yt} can be reduced similarly. Clearly, orderm equality for m ) n
implies, and is implied by, ordern equality.
5
2.2 Relation with copulas
For a strictly stationary realvalued time series {Xt} with continuous joint cumulative distribution functions
(CDF) Ft1,...,t!(xt1 , ··· ,xt!) and marginal CDF F(x), the copula associated with a delay vector (Xt'n+1,...,Xt)
is the quantity
Cn(u1,...,un) = F1,...,n+1(F'1(u1),...,F'1(un)),
where u j $ [0,1] for j = 1, ··· ,n+1. The correspondeing copula density function is
cn(u1,...,un) =
( n+1Cn(u1,...un)
(u1 ···(un
=
p1,...,n+1(F'1(u1),...,F'1(un))
p(F'1(u1)) ··· p(F'1(un))
,
or
cn (F(x1),...,F(xn)) =
p1,...,n(x1,...,xn)
p(x1) ··· p(xn)
.
In the case of a real valued first order time series, our definitions imply that the dependency ratio f (x1,xn+1)
is equal to the copula density function c2 evaluated in the ‘standard’ coordinates (u1,u2) = (F(x1),F(x2)),
which are equivariant under monotonously increasing transformations of X .
Although the n + 1dimensional copula c(u1,...,un+1) characterises the dependence structure
within n+1 consecutive values (Xt'n,...Xt), it does not take into account the ordering of the observations
in time. In time series analysis one is often interested in the question of how Xt depends on (Xt'n,...,Xt'1).
It is then more natural to take into account the ordering of the observations in time and to focus on the condi
tional distribution of Xt given past values of Xt . In this way, we are led to the definition of dependency ratio
given above. This allows, in principle, to distinguish between time reversals in processes of order higher
than one. For orders larger than one the dependency ratio can be expressed as:
f (x1,...,xn+1) =
cn+1(F(x1),...,F(xn+1))
cn(F(x1),...,F(xn))
.
2.3 Relation with information theory
Information theoretic dependence measures can often be expressed in terms of copulas. Wellknown ex
amples are mutual information and the redundancy, which both are special instances of KullbackLeibler
divergences (relative entropies).
The KullbackLeibler divergence between two probability distributions with probability densities
p(x) and q(x) is defined as
D(p,q) =
"
p(x) ln
#
p(x)
q(x)
$
dx
Themutual information I(X1;X2) between two random variables X1 and X2, is a dependence measure defined as
the KullbackLeibler divergence between their joint probability density function p1,2(x1,x2) and the product
of their marginals p(x1)p(x2) (we consider the case with identical marginals, appropriate for stationary
time series). Since p1,2(x1,x2) denotes the joint probability density function of (X1,X2), the integral over
p(x) = p(x1,x2) can be concisely expressed as an expectation, that is,
I(X1;X2) =
" "
p1,2(x1,x2) ln
#
p1,2(x1,x2)
p(x1)p(x2)
$
dx1 dx2 = E
%
ln
#
p1,2(X1,X2)
p(X1)p(X2)
$&
,
6
which is nonnegative and equals zero if and only if X1 and X2 are independent. This can be seen as follows:
the expected value of the random variable Z = p(X1)p(X2)/p1,2(X1,X2) is
E[Z] =
" "
p1,2(x1,x2)
p(x1)p(x2)
p1,2(x1,x2)
dx1 dx2 = 1.
By convexity of the function ln(z'1) =' ln(z), Jensen’s inequality gives
I(X1;X2) = E[ln(Z
'1)])' ln(E[Z]) = 0,
and equality holds if and only if Z = 1 almost surely.
The generalisation of the mutual information to the multivariate case is known as the redundancy:
R(X1;...;Xn+1) = E
%
ln
#
p1,...,n+1(X1, ··· ,Xn+1)
p(X1) · ... · p(Xn+1)
$&
,
which is zero if and only if (X1,...,Xn+1) are jointly independent, and positive otherwise. In analogy with
the above discussion on copulas, a generalisation which is more suitable within a time series context is the
entropy, given by
H(Xn+1;X1,...,Xn) = E
%
ln
#
p1,...,n+1(X1, ··· ,Xn+1)
p1,...,n(X1,...Xn)p(Xn+1)
$&
= E [ln f (x1,...,xn+1)] .
3 Equivalence notions
In this section we introduce and motivate our notion of equivalence of stochastic processes and we give
some of its fundamental properties.
3.1 Structural stability and bifurcations
We recall briefly the fundamentals of bifurcation theory. The two main ingredients of any such theory are a
topological space X and an equivalence relation between elements of X . An element f of X is structurally
stable if there is a neighbourhood N( f ) such that all elements g in that neighbourhood are equivalent to f ;
that is g( f for all g$N( f ). Intuitively speaking, a structurally stable element f can be ‘perturbed’ slightly
without being pushed out of its equivalence class. Such an element is sometimes called ‘persistent’. Clearly,
the equivalence class of any structurally stable element is an open set. A structurally stable equivalence
class can be thought of as defining a set of elements of the same ‘shape’ or ‘form’ (see [14]): form remains
‘stable’ if perturbed slightly.
All elements of X that are not structurally stable are called bifurcating. This notion is usually
familiar from the context of parametrised families: if ) is some qdimensional parameter, and )
*% f) a
family of elements of X , then ) = )0 is a bifurcating parameter value of the family if f)0 is not structurally
stable; it might be said that at bifurcating parameter values the ‘form’ of f) changes. Since the set of
structurally stable elements is open, the set of bifurcating elements, and therefore also the set of bifurcating
parameter values in a parametrised family, is closed.
An equivalence relation will give rise to a useful bifurcation theory on X only if there at all exist
structurally stable elements. The optimal situation is attained if the set of structurally stable elements, while
not consisting of a single equivalence class, is ‘topologically big’, since then we will be able to associate
to ‘most’ elements a form. In a topological space, a set is ‘big’ if it is open and dense, or if it is at least a
countable intersection of open and dense sets (a socalled ‘generic’ or ‘second category’ set, see [11]).
7
3.2 Strong equivalence
A natural requirement to impose on an equivalence relation of stochastic processes on a manifold M is that
processes which only differ by a diffeomorphism of M, that is, which are the ‘same’ up to a coordinate
change, fall in the same equivalence class. Let for instance {Xt}, {Yt} denote two first order processes on
M. We will certainly want to call two processes equivalent if there is a diffeomorphism ! : M % M such
that the variables (Yt ,Yt'1) and (!(Xt),!(Xt'1)) are identically distributed. If this is the case, we call {Xt}
and {Yt} strongly equivalent. Let fX(x1,x2) and fY (x1,x2) denote the dependency ratios of {Xt} and {Yt}
respectively; if the processes are strongly equivalent, it follows from the invariance of the dependency ratios
under diffeomorphisms that
fX(x1,x2) = fY (!(x1),!(x2))
for all (x1,x2) $M"M.
(5)
If we would take strong equivalence as our equivalence relation, in general we would obtain an
uncountable infinity of equivalence classes, and no class would be a neighbourhood to any of its points, that
is, no process would be structurally stable. To see this in a simple example, assume that fX and fY are two
dependency ratios defined on the square ('1,1)" ('1,1)+ R2, and that they are given as
fX(x1,x2) =
2'µ
3
+ x21 +µx22,
fY (x1,x2) =
2'*
3
+ x21 +*x22.
Taking the invariant density in both cases to be p(x) = 1
2
I['1,1](x), where IA(x) denotes the indicator func
tion, we have specified two stochastic processes. The point (0,0) is the only nondegenerate critical point
for both fX and fY ; therefore, if fX and fY are strongly equivalent, we should have that "(x1,x2) =
(!(x1),!(x2)) satisfies "(0,0) = (0,0). But there is no realvalued smooth diffeomorphism ! such that (5)
holds simultaneously with !(0) = 0, for the values of fX and fY at (0,0) are different if µ
,= * . We see that
every value of µ defines a different equivalence class.
3.3 Topology of dependency ratios
In section 2 we have defined the dependency ratio f of an n’th order process {Xt} by
f (x1, ··· ,xn+1) =
p1,··· ,n+1(x1, ··· ,xn+1)
p1,··· ,n(x1, ··· ,xn)p(xn+1)
;
this quantity is invariant under coordinate transformations, and it is therefore a characteristic of the stochastic
process.
As the space of dependency ratios is infinite dimensional, this characteristic itself is too fine
grained to be useful to classify such processes. In the previous section we have seen that one way of
extracting ‘coarser’ information from a dependency ratio f is to determine the expected value of some
monotone transformation of f . But defining equivalence of two ratios by equality of such expected values
would not lead to structurally stable elements, for a very small perturbation of the process might already
change the expected value.
Using however topological information of the dependency ratios turns out to give characteristics
that are sufficiently coarse. To give a very simple example: we clearly want to call two functions defined on
the same domain to be of different shape if they have a different number of nondegenerate critical points.
The number of such points is a numerical characteristic of the ‘shape’ of a function, and it in fact defines
an equivalence class. Moreover, if we choose a suitable topology on the set of functions, we find that the
equivalence classes are open sets, and that its members are structurally stable. We can specify different
8
equivalence relations by taking into account more detailed information; for instance, we can classify the
critical points according to their topological type.
In any case, we need a function topology on the set of dependency ratios; we choose the C2
topology, which is the ‘coarsest’ topology for which the number of nondegenerate critical points defines
open equivalence classes. Recall that in theC2 topology an "neighbourhood N"( f ) of a function f :M %R
defined on a compact manifold M consists of all functions g such that, with respect to a fixed Riemannian
metric and the induced norm ·x on the tangent spaces TxM we have
 f (x)'g(x)x, Df (x)'Dg(x)x, D2 f (x)'D2g(x)x < ".
If M is a noncompact manifold, the constants " > 0 are replaced by positive functions "(x) > 0 on M in the
above definition; in this context the topology obtained is called the ‘strong’C2topology (see e.g. [6]).
As argued above, specifying an equivalence notion on the space of dependency ratios of stochastic
processes induces a notion of stochastic bifurcation. In the following we shall sometimes use the words
‘process’ and ‘dependency ratio’ indiscriminately; in particular, a ‘structurally stable process’ will denote
a stochastic process whose dependency ratio is a structurally stable element under the equivalence relation
under consideration. A first rough formulation of our definition would be the following: we propose to
call two dependency ratios of stochastic processes equivalent, if every nondegenerate critical point of a
certain type of the first dependency ratio can be mapped to a critical point of the same type of the second
dependency ratio by a transformation of M"M induced by a diffeomorphism of M. In the next section, we
shall make this more precise.
3.4 Ratio equivalence on compact manifolds
Let M be an mdimensional orientable compact manifold and Mn+1 the (n+1)fold Cartesian product M"
···"M; denote by $! : Mn+1 %M the projection on the !’th component
$!(x1, ··· ,xn+1) = x!.
An n’th order dependency ratio is a real valued function defined on Mn+1.
Recall the following definitions (see e.g. [5], subsections 10.2 and 10.4, p. 79 and p. 86 respec
tively). If f : U % R is a twice continuously differentiable function defined on an open set U + Rn, a
point x $U is a critical point of f if the derivative of f vanishes at x: Df (x) = 0. The value f (x) of f at
a critical point x is called the critical value of f at x. A critical point x is nondegenerate if the Hessian
matrix H f (x) corresponding to the second derivative D2 f (x) of f at x is invertible. The number of negative
eigenvalues of this matrix is called the (Morse) index of the critical point. Clearly, the notions of critical
point, critical value, index and nondegeneracy carry over to functions defined on manifolds.
Our definition is based on the critical points of a given dependency ratio; this makes it necessary
to restrict attention to twice differentiable ratios only, and to consider the C2 function topology introduced
above on the space of these ratios. For in anyC0neighbourhood of a given function, there are other functions
to be found with a different number of critical points; and the same is true for any C1neighbourhood of a
function that has itself at least one critical point.
Definition.
If M is a manifold, a twice differentiable dependency ratio f :Mn+1 % (0,#) is called regular
if all its critical points are nondegenerate, if no two critical values are equal and if no two critical points
have the same image under any projection $!, for ! $ {1, ··· ,n+1}.
9
Since the manifolds M and Mn+1 are compact, a regular dependency ratio has only finitely many critical
points {&1, ··· ,&k}; we assume that these are ordered such that the corresponding critical values vi = f (&i)
are in ascending order, that is, vi < v j if i < j. We associate to the critical point &i its index ti (see sub
section 3.2). Note that 0 & ti & m(n+1). In this way we obtain the index sequence t( f ) = (t1, ··· , tk) of a
regular dependency ratio f .
Definition. Two order n processes defined on a compact manifold M, with dependency ratios f ,g :Mn+1 %
(0,#), are called ratio equivalent, if either both f and g are nonregular, or if f and g are both regular and
1. their index sequences are equal;
2. there is a diffeomorphism ! : M % M, homotopic to the identity mapping on M, such that the map
" : Mn+1 %Mn+1 defined as
"(x1, ··· ,xn+1) =
'
!(x1), ··· ,!(xn+1)
(
(6)
maps the i’th critical point of f to the i’th critical point of g.
It follows from the first point that the number of critical points of f and g is equal as well.
The following two propositions tell us that this definition has good properties: we can characterise
all structurally stable processes, and these form an open and dense set in the space of all stochastic processes.
Proposition 2. On a compact manifold M, a dependency ratio is structurally stable under ratio equivalence
if and only if it is regular.
Proposition 3. On a compact manifold M, the set of regular dependency ratios is everywhere dense.
The proofs of these propositions can be found in appendix A.
3.5 Ratio equivalence for noncompact manifolds
Though the results for compact manifolds are already useful in themselves, in practice most stochastic
processes are defined on the noncompact manifold Rm. The direct generalisation of the notion of ratio
equivalence is given in the following definition.
Definition. Two processes on a manifold M are weakly ratio equivalent, if they are ratio equivalent on
each compact set that is the closure of a bounded submanifold of Mn+1.
As the following example shows, this notion is unfortunately too weak for our purposes.
Example. Consider two first order processes {Xt} and {Yt} on the interval ('1,1) with invariant densi
ties p(x) = 1
2
I['1,1](x) and dependency ratios
fX(x1,x2) = 1'
1
2
x1x2 +
1
4
x31,
and
fY (x1,x2) = 1+
1
2
x1x2'
1
4
x31.
Both ratios have a unique critical point of index 1 at the origin, and hence they are ratio equivalent on
compact sets. But if we consider the values of fX and fY along the curve +(t) = (t, t) as t  '1, we note
that fX # +(t) approaches the infimum of fX on ('1,1)" ('1,1), while fY # +(t) approaches the supremum
of fY on the same space. That means in particular that if Xt is close to '1, the probability is very low that
Xt+1 is close to'1 as well, whereas if Yt is close to'1, the probability is rather high that Yt+1 is close to'1
as well. Weak ratio equivalence is not sufficiently fine to distinguish between these processes.
10
For our second generalisation, we restrict ourselves to processes defined on a subclass of manifolds which
we call manifolds of ‘constant type’; these are defined as follows.
Let M be an orientable mdimensional manifold. If there exists a family {Mt} of bounded open
submanifoldswithboundary ofM depending on a real parameter t > T , for whichMt +Mt . if t < t .,
)
t Mt =
M and for which (Mt is a smooth boundary for every t > T , then we call {Mt} an exhaustion ofM. Moreover,
we say that an exhaustion is of constant type if there is a constant T > 0 such that (Mt and (Mt . are
diffeomorphic for all t, t . > T . A manifold M that admits an exhaustion of constant type will itself be called
to be of constant type. For instance, the plane is a manifold of constant type with exhaustion R2
t = {x $R2 :
/x/2 < t}, whereas the plane with all points with integer coordinates removed is not a manifold of constant
type.
A convenient way to define an exhaustion of M is to take a differentiable real valued function J
on M such that for a fixed point x0 $ M and a metric d on M we have that J(x) % # as d(x0,x)% #, and
set Mt = {x $ M : J(x) < t}. If all values of J larger than T are regular, then it follows from Morse theory
that (Mt is diffeomorphic to (Mt . for t, t . > T , and consequently that {Mt} is an exhaustion of M of constant
type.
Consider the set
(Mt)n+1 = Mt "Mt " ···"Mt +Mn+1.
This set can be decomposed into 2n+1 component manifolds {C jt }Nj=1; each component is a Cartesian prod
ucts of k factors (Mt and n+ 1' k factors Mt , for k = 0, ··· ,n+ 1. We order these components such that
a product of more factors Mt precedes a component with less, and for components with equal numbers of
factors Mt , the component with more factors Mt in the first ! positions precedes a component with less,
with ! taking the values 1,2, ··· ,n+1 consecutively. That is,
(Mt "Mt "Mt
precedes
Mt "(Mt "(Mt ,
and
(Mt "Mt "(Mt
precedes
(Mt "(Mt "Mt .
In the important special case that M is onedimensional, the component ((Mt)n+1 consists of a finite number
of points. By definition, we consider these point as nondegenerate critical points, associating the index 0 to
them by default.
In the following three definitions, M is a manifold of constant type with exhaustion {Mt} and
corresponding ordered decomposition {C jt } of (Mt)n+1; by ·x we denote a norm induced from a fixed
Riemannian metric as in subsection 3.3. Moreover, the restriction of f to C
j
t is denoted by f
j
t with the
index j always ranging from 1 to 2n+1.
Definition. A dependency ratio f on Mn+1 is wellbehaved at infinity if there are constants ct ,T > 0 such
that for every t > T
1. for every j there is a compact set K
j
t + C
j
t such that Df
j
t (x)x > ct if x $ C
j
t \K
j
t , unless C
j
t is 0
dimensional, and
2. f
j
t is weakly ratio equivalent to f
j
t . for all t, t
. > T.
Definition.
A dependency ratio f on Mn+1 is wellbehaved if f is wellbehaved at infinity and f
j
t
is
regular on C
j
t for every j and every t > T.
11
Definition.
If M is a manifold of constant type, two wellbehaved dependency ratios f and g are called
ratio equivalent, if there is a value of t such that f
j
t and g
j
t are weakly ratio equivalent for every j.
Note that if f and g are weakly ratio equivalent on each component C
j
t for a single value t > T , they are in
fact equivalent for all such values, since f
j
t ( f
j
t . for all t, t
. > T .
Example. With this definition, we can distinguish between the two ratios fX and fY introduced at the end
of the previous subsection. Set at = t/(t + 1), and consider the exhaustion It = ('at ,at) of ('1,1). Note
that ( (It " It) can be decomposed into
C1
t = ('at ,at)"{'at ,at}, C2
t = {'at ,at}" ('at ,at), C3
t = {'at ,at}"{'at ,at}.
Restricted to C1
t and C
2
t , neither fX nor fY have any critical points. The set C
3
t consists of four isolated
critical points, which are critical by definition. The maximum of fX restricted to C
3
t
is assumed in the
points (at ,'at), whereas fY takes its minimum there. Since the only diffeomorphism of C3
t homotopic to
the identity is the identity itself, corresponding critical points of fX and fY cannot be mapped onto each
other.
The following propositions describe the topological properties of ratio equivalence. The results are weaker
than in the compact case, as was to be expected; we obtain that wellbehaved processes are stable elements
of ratio equivalence. However, restricted to the space of processes that are wellbehaved at infinity, the
wellbehaved processes form again an open and dense set.
The next proposition is a corollary to propositions 2 and 3.
Proposition 4. On a manifold M of constant type, a wellbehaved dependency ratio is stable with respect
to the strong topology under ratio equivalence.
The proof of this proposition is given in appendix A.
4 Examples
4.1 Stochastic dynamics on the circle
As an illustration of a stochastic process on a compact manifold, we consider the stochastic dynamical
system on the unit circle M = S1 defined by
Xt+1 = Xt +asin(Xt)+0.25sin2(Xt)+0.25+ "t+1 mod 2$,
(7)
with {"t} IID and N(0,%2) distributed. The state variable is taken modulo 2$; we represent states by points
on the interval ['$,$). For the above system we fix % at the value 0.7 and consider qualitative changes in
the stochastic dynamics as a varies. The term 0.25(sin2(xt'1)+1) is added to break the x
*% 'x symmetry
of the dynamics. In the symmetric case some particular additional properties arise which will be discussed
in the next subsection.
Figure 1 shows a contour plot for the dependency ratio f for values of a decreasing from '0.85
to '0.95. For a = '0.85, the contour plot shows two extrema, a maximum and a minimum, together with
two saddle points. These are the minimal number of critical points of each type that can be attained for a
nondegenerate function f on the torus M2 = S1"S1. As the bifurcation parameter a decreases, the system
shows a stochastic bifurcation at which f develops a new local extremum, together with a new saddle point.
12
!2
0
2
!3
!2
!1
0
1
2
3
a=!0.85
xt!1
xt
!2
0
2
0
0.2
0.4
0.6
0.8
1
x
density
!2
0
2
!3
!2
!1
0
1
2
3
a=!0.9
xt!1
!2
0
2
0
0.2
0.4
0.6
0.8
1
x
!2
0
2
!3
!2
!1
0
1
2
3
a=!0.95
xt!1
!2
0
2
0
0.2
0.4
0.6
0.8
1
x
Figure 1: Level sets for the map Xt = Xt'1 +asin(Xt'1)+0.25sin2(xt'1)+0.25+ "t for decreasing values
of a (top panels). The lower panels show the invariant probability density of Xt .
4.2 Antisymmetric dynamics
In applications dynamical systems are often symmetric. Though we leave the theoretical development of
our equivalence relation for symmetric systems for later work, we want to make some remarks about this
situation. We restrict to processes that are the sum of an antisymmetric deterministic part and a unimodal
symmetric noise term (e.g. Gaussian). Surprisingly, it turns out that for these systems the ‘ratio bifurcation’
coincides with a Pbifurcation.
Consider the process
Xt+1 = g(Xt)+ "t ,
where g(x) is a smooth odd function, that is, for which g('s) = 'g(s). The "t are independent and identi
cally distributed according to a symmetric unimodal distribution. It can be readily checked that the deter
ministic dynamics is equivariant under the transformation x
*% 'x. Because this transformation affects both
Xt'1 and Xt , the effect on the joint variable (Xt'1,Xt) is a point reflection in the origin.
For the conditional density of Xt given Xt'1 = x, one may write
pXt+1Xt (x2x1) = '(x2x1) =
p1,2(x1,x2)
p(x1)
=
1
%
h
#
x2'g(x1)
%
$
where h(·) is the probability density function of "t ; we have that h(s) = h('s) and that h has a unique local
maximum at s = 0. The map g(·) as well as the probability density function h(·) are assumed to be twice
continuously differentiable.
Moreover we assume that the process has an invariant density p(x), which is unique and twice
continuously differentiable. These conditions on p(x) can easily be met by imposing some additional re
quirements on g and h. For instance, a sufficient condition is that for each x0, the support of
1
% h
'
x1'g(x0)
%
(
13
is M (strong mixing). If the invariant density is unique, it is necessarily an even function, since otherwise its
mirror image p('x) would be a different invariant density.
Our aim is to examine the properties of the dependency ratio
f (x1,x2) =
p1,2(x1,x2)
p(x1)p(x2)
=
1
% p(x2)
h
#
x2'g(x1)
%
$
near the origin. We see that
f ('x1,'x2) =
1
% p('x2)
h
#
'x2'g('x1)
%
$
=
1
% p(x2)
h
#
'x2'g(x1)
%
$
= f (x1,x2).
It follows that the partial derivatives
( f
(x1 and
( f
(x2 vanish at the point (x1,x2) = (0,0), so that the origin is
always critical. The index of this critical point is determined by the Hessian matrix H f (0,0). If g is odd
and h is even and unimodal, we have
p.(0) = 0,
g..(0) = 0,
h.(0) = 0,
and
h..(0) < 0.
After some algebra one finds that the Hessian evaluated at the origin reduces to
H f (0,0) =
h..(0)
%3p(0)
*
+,
g.(0)2
'g.(0)
'g.(0) 1'%2 h(0)p
..(0)
h..(0)p(0)

./ .
Since h..(0) is negative, the Hessian matrix has a negative determinant if and only if the second derivative
of the invariant density p(x) satisfies
p..(0) < 0,
in which case the origin is a saddlepoint. If however p..(0) > 0, the determinant is positive, while the trace
of the matrix is negative, and the dependency ratio f (x1,x2) has a local maximum at the origin.
Apparently, the critical point at the origin changes from a saddle point to a local maximum if p..(0)
becomes positive. Because this is exactly the condition for which the local maximum of p(x) at the origin
changes to a local minimum with a pair of maxima bifurcating off, it follows that for antisymmetric maps
with symmetric unimodal noise, the ‘ratio bifurcation’ coincides with a phenomenological bifurcation.
4.2.1 On the circle
We illustrate this by figure 2 which shows the level sets of the dependency ratio, as well as the invariant
density, for the map
Xt+1 = a
'
0.5sin(Xt)+0.25sin(2Xt)
(
+ "t+1,
where again M is the unit circle S1 and where "t ( N(0,%2
t+1) with %2
t+1 = 0.6' 0.12cos(Xt). The noise
variance is made state dependent to avoid the critical point at ($,0) to bifurcate simultaneously with the
bifurcation at the origin. As noted above, the local minimum in p(x) at x = 0 occurs exactly when f (x1,x2)
develops a local maximum at the origin. This is related to the fact that the denominator of the dependency
ration contains a product of marginals which simultaneously develop a local minimum.
14
!2
0
2
!3
!2
!1
0
1
2
3
a=1.3
xt!1
xt
!2
0
2
0
0.2
0.4
0.6
0.8
1
x
density
!2
0
2
!3
!2
!1
0
1
2
3
a=1.4
xt!1
!2
0
2
0
0.2
0.4
0.6
0.8
1
x
!2
0
2
!3
!2
!1
0
1
2
3
a=1.5
xt!1
!2
0
2
0
0.2
0.4
0.6
0.8
1
x
Figure 2: Level sets for the map Xt+1 = a
'
0.5sin(Xt)+ 0.25sin(2Xt)
(
+ "t+1 with "t+1 ( N(0,%2
t+1) and
%2
t+1 = 0.6'0.12cos(Xt) for increasing values of a (upper panels) and the corresponding marginal density
functions (lower panels). The levels are not uniformly spaced.
4.2.2 On the real line
When we derived the coincidence of a Pbifurcation and a copula bifurcation at the origin in the antisym
metric case, apart from the global requirement of symmetry of the invariant density, only local arguments
were used. Therefore, provided that we confine ourselves to cases with symmetric invariant densities, the
result that a Pbifurcation coincides with a (local) ratio bifurcation, directly extends to stochastic dynamics
on the real line.
As an example we consider the stochastic process on R defined by
Xt+1 = tanh(aXt)+ "t+1.
(8)
Figure 3 shows the level sets of the dependency ratio and the corresponding invariant probability density
function for this map with N(0,%2) distributed noise, taking % = 0.7.
Note that the bifurcation parameter value differs from that of the analogous deterministic system
(% = 0): for the tanh map the stochastic analogue of the usual pitchfork bifurcation at a = 1 is shifted to a
larger value of a. Apparently the value of the bifurcation depends on the noise level. A natural question,
therefore, is whether for increasing noise levels the bifurcation parameter merely shifts, or whether the
bifurcation can disappear altogether.
Intuitively, if the map is bounded and has a small range relative to the noise level, the dynamics
is mainly governed by the noise and the deterministic part has little influence on the dynamics. In fact a
simple argument shows that if the noise is fixed at a sufficiently large level, and if the family of odd maps
{ga} is uniformly bounded, then there is no phenomenological bifurcation at x = 0, and therefore also no
ratio bifurcation at (x1,x2) = (0,0), for symmetric processes of the form
Xt+1 = ga(Xt)+ "t+1.
(9)
15
!2
0
2
!2
!1
0
1
2
a=0.9
xt!1
xt
!2
0
2
0
0.2
0.4
0.6
0.8
1
x
density
!2
0
2
!2
!1
0
1
2
a=1.3
xt!1
!2
0
2
0
0.2
0.4
0.6
0.8
1
x
!2
0
2
!2
!1
0
1
2
a=1.7
xt!1
!2
0
2
0
0.2
0.4
0.6
0.8
1
x
Figure 3: Level sets for the map Xt+1 = tanh(aXt)+ "t+1 with "t+1 ( N(0,0.52) for increasing values of a
(top panels) and corresponding marginal density function (lower panels).
The argument runs as follows. By stationarity the invariant density p should satisfy
p(x) =
"
1
%
h
#
x'ga(s)
%
$
p(s)ds.
A necessary condition for p(x) to have a local minimum at x = 0 is that p..(0) > 0, where
p..(0) =
"
1
%
h..
#
'ga(s)
%
$
p(s)ds.
Since h(s) is a unimodal probability density function, its second derivative h..(s) is negative in a neighbour
hood of s = 0. It follows that, for ga uniformly bounded, for large % the integral on the right hand side of
the last equation may remain negative as a varies.
4.3 Estimated dependency ratios from time series
In order to see whether dependency ratios can be used for classification of processes of which only a time
series is available, a common situation in empirical applications, we estimate dependency ratios from sim
ulated time series. We generate relatively short series {Xt} from the stochastic models considered earlier
in this section; we estimate from these series bivariate invariant densities and use them to reconstruct the
dependency ratios. It is well known [1, 13] that fixed bandwidth nonparametric kernel density estimates
become rather poor in regions with only few observations. One way to avoid this would be to use a data
driven adaptive bandwidth which depends on the density locally, becoming larger as fewer observations are
present locally. Instead of using an adaptive bandwidth we suggest, for real valued time series, to transform
the data using the probability integral transform, that is, we construct
Ut = 0FX(Xt) =
rank of Xt among {Xs}Ns=1
N
.
16
This amounts to transforming the invariant distribution to a uniform distribution on the unit interval, which
tends to stabilise the estimation of the dependency ratio as the marginals no longer need to be estimated.
In the case of first order ratios, the estimated empirical dependency ratio is equal to the empirical copula
density
0
f (u1,u2) =
1
N'1
N'1
$
t=1
Kb(u1'Ui,u2'Ui+1).
Here Kb(u1,u2) is a bivariate probability kernel, which we take to be the commonly used Gaussian kernel:
Kb(u1,u2) =
1
0
2$b
e'(u
2
1+u
2
2)/(2b
2).
To avoid ‘probability mass’ from disappearing out of the unit square by this smoothing procedure, we impose
periodic boundary conditions if M = S1 and reflecting boundary conditions if M = R.
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
ut!1
ut
0
200
400
600
800
1000
!3
!2
!1
0
1
2
3
t
xt
a=0.9
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
ut!1
ut
0
200
400
600
800
1000
!3
!2
!1
0
1
2
3
t
xt
a=1.7
Figure 4: First 1000 values (top panels) of 4000 consecutive Xtvalues generated by the map Xt+1 =
tanh(aXt) + "t+1 with "t+1 ( N(0,0.52). for a = 0.9 (left) and a = 1.7 (right). The lower panels show
the corresponding empirical level sets estimated with a Gaussian kernel (bandwidth b = 0.07).
Figure 4 shows level sets of the empirical dependency ratio obtained from time series of length
4000 from the symmetric hyperbolic tangent map given in equation (8) for different parameter values. The
dependency ratio is estimated by smoothing the empirical copula with a bivariate normal probability density
function (bandwidth b = 0.07). The empirical dependency ratio clearly reflects the fine structure of the
theoretical dependency ratio.
Figure 5 shows an attempt at performing a similar reconstruction for the asymmetric sine map
given by equation (7). In this case the topology of the reconstructed level sets does not correspond with that
obtained earlier; this is due to estimation error. Probably longer time series (along with smaller bandwidths
for the smoothers) are required for this case. We consider the optimal estimation and the related issue of
data requirements for estimating dependency ratios as an important area for future research.
17
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
ut!1
ut
0
200
400
600
800
1000
!3
!2
!1
0
1
2
3
4
t
xt
a=!0.85
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
ut!1
ut
0
200
400
600
800
1000
!4
!2
0
2
4
t
xt
a=!0.95
Figure 5: First 1000 values (top panels) of 4000 consecutive Xtvalues generated by the map Xt+1 = Xt +
asin(Xt)+ 0.25sin2(Xt)+ 0.25+ "t+1 with "t+1 ( N(0,0.72) for a = '0.85 (left) and a = '0.95 (right).
The lower panels show the level sets of the corresponding empirical dependency ratio.
References
[1] I.S. Abramson, On bandwidth variation in kernel estimates  a square root law, Annals of Statistics 10
(1982), 1217–1223.
[2] L. Arnold, Random dynamical systems, Springer, Heidelberg, 1998.
[3] L. Cobb, Stochastic catastrophe models and multimodal distributions, Behavioral Science 23 (1978),
360–374.
[4] M.A.H. Dempster, I.V. Evstigneev, and K.R. SchenkHoppé, Exponential growth of fixedmix strate
gies in stationary asset markets, Finance and Stochastics 7 (2003), 263–276.
[5] B.A. Dubrovin, A.T. Fomenko, and S.P. Novikov, Modern Geometry — Methods and Applications.
Part II: The Geometry and Topology of Manifolds, Graduate Texts in Mathematics, vol. 104, Springer,
New York, 1985.
[6] M.W. Hirsch, Differential Topology, Graduate Texts in Mathematics, vol. 33, Springer, New York,
1976.
[7] P. W. Holland and Y. J. Wang, Dependence function for bivariate densities, Communications in Statis
tics A 16 (1987), 863–876.
[8] M. C. Jones, The local dependence function, Biometrika 83 (1996), 899–904.
18
[9] A. Lasota and M.C. Mackey, Chaos, fractals, and noise: Stochastic aspects of dynamics, Springer,
Heidelberg, 1994, 2nd edition.
[10] S. Nadarajah, K. Mitov, and S. Kotz, Local dependence functions for extreme value distributions,
Journal of Applied Statistics 30 (2003), 1081–1100.
[11] John C. Oxtoby, Measure and category. A survey of the analogies between topological and measure
spaces. 2nd ed., Graduate Texts in Mathematics, vol. 2, Springer, 1980.
[12] A. Ploeger, H. L. J. van der Maas, and P. Hartelman, Catastrophe Analysis of Switches in the Perception
of Apparent Motion, Psychonomic Bulletin & Review 9 (2002), 26–42.
[13] G.R. Terrell and D.W. Scott, Variable kernel density estimation, Annals of Statistics 20 (1992), 1236–
1265.
[14] René Thom, Structural stability and morphogenesis. An outline of a general theory of models, W. A.
Benjamin, Reading, Massachusetts, 1975.
[15] E.C. Zeeman, Stability of dynamical systems, Nonlinearity 1 (1988), 115–155.
A Proofs of the topological properties
In this appendix, the topological properties given in section 3 are proved.
A.1 Proofs of the propositions
We repeat the statements of propositions 2 and 3 for the convenience of the reader.
Proposition 2. On a compact manifold M, a dependency ratio is structurally stable under ratio equiva
lence if and only if it is regular.
Proposition 3. On a compact manifold M, the set of regular dependency ratios is everywhere dense.
Proof.
These propositions are direct corollaries from the following two lemmas.
Lemma 1. If M is compact and if f :Mn+1 %R is a regular dependency ratio, then there is a constant " > 0
such that every g $ N"( f ) is regular and equivalent to f .
Lemma 2. If M is compact, the set of regular dependency ratios is dense in the C2topology.
From lemma 1 we infer that regular ratios are structurally stable. If however f is a structurally stable ratio,
there is a neighbourhood U = N"( f ) such that every g $ U is equivalent to f . But as the regular ratios
are dense, according to lemma 2, there is a regular ratio in U which then equivalent to f . By definition of
equivalence, the ratio f itself has to be regular. The propositions follow.
For noncompact manifolds of constant type, the following result is essentially a corollary of the results for
compact manifolds.
Proposition 4. On a manifold M of constant type, a wellbehaved dependency ratio is stable with respect
to the strong topology under ratio equivalence.
19
Proof.
Let f be a wellbehaved dependency ratio, and let {Mt} be an exhaustion of M. Let moreover T > 0 and ct >
0 be such that for every t, t . > T we have that (Mt and (Mt . are diffeomorphic, for every component {C jt }
of (Mt)n+1 the restriction f
j
t of f to C
j
t is weakly ratio equivalent to f
j
t . and there is a compact set K
j
t such
that Df jt (x)x > ct if x $C
j
t \K
j
t .
For every t > T and every j, there is then a constant " j
t > 0 such that for every g $ N" j
t
( f ) in the
C2topology on (Mt)n+1, the restriction g
j
t of g to C
j
t is weakly ratio equivalent to f
j
t . Let "t = min j "
j
t ,
and "(x) = max{"t x $ (Mt)n+1}. It follows that N"( f ) is an open neighbourhood of f in the strong C2
topology, such that every g $ N"( f ) is ratio equivalent to f .
A.2 Proofs of the lemmas
It remains to demonstrate the lemmas. For this, we first need the following technical result. In the statement
of the lemma, a ball of radius r around 0 is denoted by Br, that is, Br = {x $ Rk /x/< r}; also we have
/ f 'g/C2(U) = max
0& j&2
max
x$U
Dj f (x)'Djg(x).
Lemma 3. Let U + Rk be a bounded open set, and let f :U % R be a C2 function with D f (0) = 0 and
H f (0) nondegenerate. Then there exist constants ,0,0 > 0 such that B,0 +U and that for every 0< , & ,0
and 0 <  & 0 there is an " > 0, such that every function g satisfying / f 'g/C2(U) < " has a unique non
degenerate critical point ȳ $ B, with g(ȳ)' f (0) <  , with g having the same index at ȳ as f at 0.
Proof.
For a matrix A, let /A/=max/x/=1 /Ax/ denote the matrix norm of A. Since H f (0) is nondegenerate, there
is a constant c > 0 such that /H f (0)'1/= c. Moreover, by continuity there is then a ,1 > 0, such that
/H f'1(x)/< 2c
for all x $ B,1 .
Introduce . = g' f and ht = f + t(g' f ) = f + t. . Then h0 = f and h1 = g. We shall solve the equation
Dht(x) = 0
(10)
for t $ [0,1], using the implicit function theorem. Note that
Hht(x) = H f (x)+ tH.(x) = H f (x)
1
I+ tH f (x)'1H.(x)
2
,
and consequently that
/Hht(x)'1/ &
/H f (x)'1/
1' t/H f (x)'1//H.(x)/ .
Taking t & 1, x $ B,1 and /./C2 < (4c)'1, we obtain
/Hht(x)'1/ & 4c.
In particular, we can apply the implicit function theorem to solve x = x(t) from (10), first around t = 0, and
then around every value of t for which x(t) $ B,1 . Note that since Hht is nondegenerate everywhere, the
index of the critical point cannot change.
20
Furthermore
Dht(x) = Df (0)+Df (x)'Df (0)+ tD.(x) =
" 1
0
H f (sx)xds+ tD.(x)
= H f (0)x+
" 1
0
1
H f (sx)'H f (0)
2
xds+ tD.(x).
By the continuity of H f , there exists 0 < ,2 < ,1 such that /H f (x)'H f (0)/ < 1/(2c) for all x $ B,2 .
Recalling the estimate /Ax/ ) /A'1/'1/x/, it follows for 0 < , < ,2 that if t & 1, /x/ = , and /./C2 <
,/(2c), then
/Dht(x)/ ) /H f (0)'1/'1/x/'
" 1
0
/H f (sx)'H f (0)//x/ds' t/D.(x)/
>
,
c
' ,
2c
' ,
2c
> 0.
We have obtained the a priori statement that x(t)
,$ (B, for all values t $ [0,1] for which x(t) is defined.
Therefore x(t) can be continued to t = 1, yielding ȳ = x(1).
To show that x(t) is the unique solution of (10) in B, , take y $ B, such that y
,= x(t), and compute
(putting ys = x(t)+ s(y' x(t))):
Dht(y) = Dht(y)'Dht(x(t)) =
" 1
0
Hht(ys)(y' x(t))ds
= H f (0)(y' x(t))+
" 1
0
1
H f (ys)'H f (0)
2
(y' x(t))ds+ t
" 1
0
H.(ys)(y' x(t))ds.
It follows, as in the previous paragraph, that /Dht(y)/> 0 for all t. But then y cannot be a critical point.
Finally, if v(t) = ht(x(t)) denotes the critical value of ht , we see by differentiating that
dv
dt
=
(ht
( t
(x(t))+Dht(x(t))
dx
dt
= .(x(t));
consequently
g(ȳ)' f (0) = v(1)' v(0) =
" 1
0
.(x(t))dt,
and g(ȳ)' f (0) &maxB, .(x).
Putting ,0 = ,2, 0 = 1 and " = min{ 14c ,
,
2c
,} yields the statement of the lemma.
Lemma 1.
If M is compact and if f :Mn+1 %R is a regular dependency ratio, then there is a constant " > 0
such that every g $ N"( f ) is regular and equivalent to f .
Proof.
Let x = (x1, ··· ,xn+1) and y = (y1, ··· ,yn+1) denote points in Mn+1. Note that then $!(x) = x! etc. For a
metric d on M, define
dn+1(x,y) = max
1&!&n+1
d
1
$!(x),$!(y)
2
.
Then dn+1 is a metric on Mn+1.
21
Let x1, ··· , xk be the critical points of f , ordered such that vi = f (xi) < f (x j) = v j if i < j.
Put v0 = 0; then v0 < v1. Introduce
/ = min
0&i< j&k
vi' v j,
% = min
0&!&n
min
1&i< j&k
d($!(xi),$!(x j));
then / is the smallest absolute difference of two critical values, and % is the smallest distance of two
projections of critical points on M.
For every i, choose a neighbourhood Wi of xi and a coordinate chart xi : Wi % Rm(n+1), such
that xi(xi) = 0, and set fi = f # x'1
i
. By assumption Dfi(0) vanishes and H fi(0) is nondegenerate. For
every i, take 0 < ,i < % such that B,i + xi(Wi) and such that 0 is the only critical point of fi in B,i .
By the lemma, we can find "i > 0, such that every function gi defined on xi(Wi)with / fi'gi/C2 < "i
has a unique nondegenerate critical point yi in B,i , with  fi(0)'gi(yi) < //2 and with yi having the same
index ti as 0.
Introduce the open setsUi = x'1
i (B,i)+Mn+1, and letC=Mn+1\
)
iUi; note thatC is compact, and
that d f
,= 0 onC. Therefore, there is /0 > 0, such that if g$N/0( f ), then dg
,= 0 onC as well. Moreover, for
every i there is /i such that g $ N/i( f ), then in the chart xi we have that / fi'gi/C2 < "i. Set " = min0&i&k /i.
Finally, we have to provide a diffeomorphism ! : M %M, homotopic to the identity, such that
"(x) =
1
! #$1(x)), ··· ,! #$n+1(x)
2
maps yi = x'1
i (yi) to xi.
Note that by the choice of ,i, no two projections of the setsUi on M intersect:
$!1(Ui1)1$!2(Ui2) = /0,
for all
1& i1 < i2 & k, 1& !1 < !2 & n+1.
Fix i and !, and consider on $!(Ui) a differentiable curve +(t), defined for 0& t & 1, such that +(0) = $!(xi)
and +(1) = $!(yi). Construct a vector field Xi! on M such that +̇(t) = Xi!(+(t)) for 0 & t & 1 and Xi! = 0
on M\$!(Ui).
Let X = $i,!Xi,!. The time1 map ! = eX has the required properties.
Lemma 2.
If M is compact, the set of regular dependency ratios is dense in the C2topology.
Proof.
Recall that the joint densities of an nth order stochastic process propagate via the PerronFrobenius operator
(see e.g. [9]), giving the equation
pt'n+1,··· ,t(xt'n+1, ··· ,xt) =
"
M
'(xt xt'n, ··· ,xt'1)pt'n,··· ,t'1(xt'n, ··· ,xt'1)dxt'n.
If the process has a unique invariant density p(x1, ··· ,xn), the process with the transition probability density
'̃(xt xt'n, ··· ,xt'1) = '(xt xt'n, ··· ,xt'1)+
q(xt'n,xt'n+1, ··· ,xt)
p(xt'n, ··· ,xt'1)
has the same invariant density p, if q is small enough, such that p̃ is indeed a probability density, and
if
!
M qdxt'(n+1)+ j = 0 for every 1& j & n+1.
For every point & $M, we can find a chart x = (x1, ··· ,xm) on M, such that x(& ) = 0. Take , > 0
such that U = x'1(B, ) and V = x'1(B2, ) are in the domain of x. Let !,. : M % R be smooth functions
such that ! = 1 onU , ! = 0 on M\V , and . = 0 onU 2M\V and
!
M . dx > 0.
22
For 1& j & m, let ! j : M % R be defined by setting
! j(x) =
3
x j! +0 j. on V,
0
on M\V.
Moreover, set !0(x) = ! +00. . The constants 0 j are chosen such that
!
M ! j dx = 0 for all j.
For x $ Mn+1, let x = (x1, ··· ,xn+1) be a chart such that x(x) = 0. For 1 & i & n+ 1, 1 & j &
m, and p : Mn+1 % R a function that is everywhere positive (this will be the invariant probability den
sity pn+1(xn+1) later on), define
Lxkj(x) =
!kj(xk)
!k0(xk)
%n+1
i=1 !i0(xi)
p(x)
=
!10(x1) · ... · !kj(xk) · ...!n+1,0(xn+1)
p(x)
.
Writing xk = (x1k , ··· ,xmk ), we find that
(Lxkj
(x j
.
k.
(0) =
4
5
6
1
p(0)
if k. = k, j. = j,
0
otherwise.
It follows that for , > 0 sufficiently small and x(y) $ B, " · · ·"B, , the differentials of the functions Lxkj # x
are linearly independent vectors in T !y Mn+1.
Choose for every x $ Mn+1 such a value for , , and set Ux = x'1(B, " ···"B, ). Since M is
compact, it is covered by a finite number of theUx, sayUx1 , ··· ,UxK . Set
qkij = pL
xk
ij .
Then qkij/p is a finite collection of functions on Mn+1 such that their differentials span T
!
x Mn+1 at every
point x $Mn+1. Moreover
"
M
qkij $!! dx! = 0
for all !.
Recall the remark made at the beginning of the proof; let the stochastic process defined by the tran
sition probability '(xn+1x1, ··· ,xn) have invariant probability densities p1,··· ,k(x1, ··· ,xk) and dependency
ratio
f (x1, ··· ,xn+1) =
p1,··· ,n+1(x1, ··· ,xn+1)
p1,··· ,n(x1, ··· ,xn)pn+1(xn+1)
=
'(xn+1x1, ··· ,xn)
pn+1(xn+1)
.
Let moreover a = (akij) be such that
'(xn+1x1, ··· ,xn)+$
i jk
akij
qkij(x1, ··· ,xn+1)
p1,··· ,n(x1, ··· ,xn)
defines a parameterised joint probability density: this is always the case if the akij are sufficiently small,
since the transition probability density is assumed to be positive everywhere on the compact manifold Mn+1.
Then the dependency ratio of the new process is given by
g(a,x) = f (x)+$
i jk
akij
qkij(x)
pn+1(xn+1)
,
23
where a = (akij) $ A+ RKm(n+1), where A is an open neighbourhood of 0.
Recall the definition of transversality (see e.g. [5], definition 10.3.1, p. 83): if X and Y are smooth
manifolds, W a smooth submanifold of Y , the map f : X % Y smooth, and x $ X , then f intersects W
transversally at x, if either f (x)
,$W or f (x) $W and Tf (x)Y = Tf (x)W + d f (x)
1
TxX
2
. More generally, we
say that f intersectsW transversally at A+ X , if f intersectsW transversally at x for every x $ A.
We have the theorem that if A, X and Y are smooth manifolds, W a smooth submanifold of Y
and f :A"X %Y a smooth map which intersectsW transversally, then the set of points a$A for which fa =
f (a,.) : X % Y intersectsW transversally is everywhere dense in A (see [5], theorem 10.3.3, p. 85).
The derivative d f of a function f : M % R on a manifold M defines a section s of the cotangent
bundle T !M; in a sufficiently small neighbourhoodU of a point in M, the restriction T !UM of the bundle toU
is isomorphic to U "Rm, and the section takes the form s(x) = (x,Df (x)). The zero section M0 of T !M,
which is isomorphic to M, is locally of the formU "{0}.
The section s is transversal to M0 at a point x $M0, if either s(x)
,$M0, or if
T(x,0)T
!M = ds(x)TxM+T(x,0)M0 = (I,H f (x))Rm +Rm"{0}.
Note that this is equivalent to saying that s is transversal to M0 everywhere if and only if the function f has
only nondegenerate critical points. Such a function is called a regular function or a Morse function.
Consider now the function g : A"M % R and the associated map s : A"M % T !M given
by s(a,x) = (x, dxg(a,x)). Note that s is transversal to M0, since in local coordinates
ds(a,x)T(a,x)A"M+T(x,0)M0 =
7
0
I
d
qkij
pn+1
Hxg(a,x)
8
RKm(n+1)"Rm +Rm"{0},
and since by construction the d(qkij/p
n+1) span Rm everywhere on M. By the theorem mentioned above, the
set of a $ A for which ga = g(a,.) is a regular function which is everywhere dense in A.
For every " > 0, we can choose a so small that g = ga is a regular function and g $ N"( f ),
where N"( f ) is a neighbourhood in the C# topology. It remains to show that by a second arbitrarily small
perturbation, we can achieve regularity of the dependency ratio.
Note that since g is a regular function, its critical points are isolated. Denote them by x1, ··· , xN .
Assume that the points x1 up to xk'1 have different critical values, and that they are such that $!(xi)
,= $!(x j)
if 1& i < j & k'1.
We choose a neighbourhood U + Mn+1 of xk such that U is contained in the domain of a chart x
for which x(xk) = 0, and such that xk is the only critical point of g in U . Let a $ Rm(n+1) be such that
9
a,Hg(0)'1a
:
,= 0, where 3x,y4 denotes the inner product of the vectors x and y; the inverse of Hg(0) exists
since g is nondegenerate in 0; and the set of vectors a that do not satisfy the condition form a union of a
smooth manifold of codimension 1 with the point {0}.
Consider the function
ht(x) = h(t,x) = g(x)' t$
ij
aijL
xk
ij .
The critical points of ht are determined by the equation
0 = Dxht(x).
This equation can be solved using the implicit function theorem around x = 0 and t = 0 since Hg(0) is
invertible. For the solution x = x(t), we find
dx
dt
(0) =
1
p
Hg(0)'1a.
(11)
24
Note that by the assumption on a, this derivative is nonzero. We restrict the possible choice of a further by
requiring that
$!!
dx
dt
(0) = $!!
1
p
Hg(0)'1a
,= 0.
Moreover, if v(t) = ht(x(t)), then
dv
dt
(t) ='$
ij
aijL
xk
ij +Dxht(x) ='$
ij
aijL
xk
ij ,
and
d2v
dt2
(0) ='1
p
9
a,Hg(0)'1a
:
,= 0.
(12)
Because of our choices, there are only finitely many values of t for which v(t) is equal to one of the critical
values g(x1), ··· , g(xk'1), or for which the projections $!(xk) and $!(xi) coincide for some 1 & i < k
and 1& ! & n+1. From equations (11) and (12) it follows that the set of values of t avoiding these special
values is everywhere dense in a neighbourhood of t = 0. This finishes the proof of the lemma.
25
!
!"#$%&'()*)+#,(,+#%+,(
(
List of other working papers:
2005
1. Shaun Bond and Soosung Hwang, Smoothing, Nonsynchronous Appraisal and Cross
Sectional Aggreagation in Real Estate Price Indices, WP0517
2. Mark Salmon, Gordon Gemmill and Soosung Hwang, Performance Measurement with Loss
Aversion, WP0516
3. Philippe Curty and Matteo Marsili, Phase coexistence in a forecasting game, WP0515
4. Matthew Hurd, Mark Salmon and Christoph Schleicher, Using Copulas to Construct Bivariate
Foreign Exchange Distributions with an Application to the Sterling Exchange Rate Index
(Revised), WP0514
5. Lucio Sarno, Daniel Thornton and Giorgio Valente, The Empirical Failure of the Expectations
Hypothesis of the Term Structure of Bond Yields, WP0513
6. Lucio Sarno, Ashoka Mody and Mark Taylor, A CrossCountry Financial Accelorator: Evidence
from North America and Europe, WP0512
7. Lucio Sarno, Towards a Solution to the Puzzles in Exchange Rate Economics: Where Do We
Stand?, WP0511
8. James Hodder and Jens Carsten Jackwerth, Incentive Contracts and Hedge Fund
Management, WP0510
9. James Hodder and Jens Carsten Jackwerth, Employee Stock Options: Much More Valuable
Than You Thought, WP0509
10. Gordon Gemmill, Soosung Hwang and Mark Salmon, Performance Measurement with Loss
Aversion, WP0508
11. George Constantinides, Jens Carsten Jackwerth and Stylianos Perrakis, Mispricing of S&P
500 Index Options, WP0507
12. Elisa Luciano and Wim Schoutens, A Multivariate JumpDriven Financial Asset Model, WP05
06
13. Cees Diks and Florian Wagener, Equivalence and bifurcations of finite order stochastic
processes, WP0505
14. Devraj Basu and Alexander Stremme, CAY Revisited: Can Optimal Scaling Resurrect the
(C)CAPM?, WP0504
15. Ginwestra Bianconi and Matteo Marsili, Emergence of large cliques in random scalefree
networks, WP0503
16. Simone Alfarano, Thomas Lux and Friedrich Wagner, TimeVariation of Higher Moments in a
Financial Market with Heterogeneous Agents: An Analytical Approach, WP0502
17. Abhay Abhayankar, Devraj Basu and Alexander Stremme, Portfolio Efficiency and Discount
Factor Bounds with Conditioning Information: A Unified Approach, WP0501
2004
1. Xiaohong Chen, Yanqin Fan and Andrew Patton, Simple Tests for Models of Dependence
Between Multiple Financial Time Series, with Applications to U.S. Equity Returns and
Exchange Rates, WP0419
2. Valentina Corradi and Walter Distaso, Testing for OneFactor Models versus Stochastic
Volatility Models, WP0418
3. Valentina Corradi and Walter Distaso, Estimating and Testing Sochastic Volatility Models
using Realized Measures, WP0417
4. Valentina Corradi and Norman Swanson, Predictive Density Accuracy Tests, WP0416
5. Roel Oomen, Properties of Bias Corrected Realized Variance Under Alternative Sampling
Schemes, WP0415
6. Roel Oomen, Properties of Realized Variance for a Pure Jump Process: Calendar Time
Sampling versus Business Time Sampling, WP0414
7. Richard Clarida, Lucio Sarno, Mark Taylor and Giorgio Valente, The Role of Asymmetries and
Regime Shifts in the Term Structure of Interest Rates, WP0413
8. Lucio Sarno, Daniel Thornton and Giorgio Valente, Federal Funds Rate Prediction, WP0412
9. Lucio Sarno and Giorgio Valente, Modeling and Forecasting Stock Returns: Exploiting the
Futures Market, Regime Shifts and International Spillovers, WP0411
10. Lucio Sarno and Giorgio Valente, Empirical Exchange Rate Models and Currency Risk: Some
Evidence from Density Forecasts, WP0410
11. Ilias Tsiakas, Periodic Stochastic Volatility and Fat Tails, WP0409
12. Ilias Tsiakas, Is Seasonal Heteroscedasticity Real? An International Perspective, WP0408
13. Damin Challet, Andrea De Martino, Matteo Marsili and Isaac Castillo, Minority games with
finite score memory, WP0407
14. Basel Awartani, Valentina Corradi and Walter Distaso, Testing and Modelling Market
Microstructure Effects with an Application to the Dow Jones Industrial Average, WP0406
15. Andrew Patton and Allan Timmermann, Properties of Optimal Forecasts under Asymmetric
Loss and Nonlinearity, WP0405
16. Andrew Patton, Modelling Asymmetric Exchange Rate Dependence, WP0404
17. Alessio Sancetta, Decoupling and Convergence to Independence with Applications to
Functional Limit Theorems, WP0403
18. Alessio Sancetta, Copula Based Monte Carlo Integration in Financial Problems, WP0402
19. Abhay Abhayankar, Lucio Sarno and Giorgio Valente, Exchange Rates and Fundamentals:
Evidence on the Economic Value of Predictability, WP0401
2002
1. Paolo Zaffaroni, Gaussian inference on Certain LongRange Dependent Volatility Models,
WP0212
2. Paolo Zaffaroni, Aggregation and Memory of Models of Changing Volatility, WP0211
3. Jerry Coakley, AnaMaria Fuertes and Andrew Wood, Reinterpreting the Real Exchange Rate
 Yield Diffential Nexus, WP0210
4. Gordon Gemmill and Dylan Thomas , Noise Training, Costly Arbitrage and Asset Prices:
evidence from closedend funds, WP0209
5. Gordon Gemmill, Testing Merton's Model for Credit Spreads on ZeroCoupon Bonds, WP02
08
6. George Christodoulakis and Steve Satchell, On th Evolution of Global Style Factors in the
MSCI Universe of Assets, WP0207
7. George Christodoulakis, Sharp Style Analysis in the MSCI Sector Portfolios: A Monte Caro
Integration Approach, WP0206
8. George Christodoulakis, Generating Composite Volatility Forecasts with Random Factor
Betas, WP0205
9. Claudia Riveiro and Nick Webber, Valuing Path Dependent Options in the VarianceGamma
Model by Monte Carlo with a Gamma Bridge, WP0204
10. Christian Pedersen and Soosung Hwang, On Empirical Risk Measurement with Asymmetric
Returns Data, WP0203
11. Roy Batchelor and Ismail Orgakcioglu, Eventrelated GARCH: the impact of stock dividends
in Turkey, WP0202
12. George Albanis and Roy Batchelor, Combining Heterogeneous Classifiers for Stock Selection,
WP0201
2001
1. Soosung Hwang and Steve Satchell , GARCH Model with Crosssectional Volatility; GARCHX
Models, WP0116
2. Soosung Hwang and Steve Satchell, Tracking Error: ExAnte versus ExPost Measures,
WP0115
3. Soosung Hwang and Steve Satchell, The Asset Allocation Decision in a Loss Aversion World,
WP0114
4. Soosung Hwang and Mark Salmon, An Analysis of Performance Measures Using Copulae,
WP0113
5. Soosung Hwang and Mark Salmon, A New Measure of Herding and Empirical Evidence,
WP0112
6. Richard Lewin and Steve Satchell, The Derivation of New Model of Equity Duration, WP01
11
7. Massimiliano Marcellino and Mark Salmon, Robust Decision Theory and the Lucas Critique,
WP0110
8. Jerry Coakley, AnaMaria Fuertes and MariaTeresa Perez, Numerical Issues in Threshold
Autoregressive Modelling of Time Series, WP0109
9. Jerry Coakley, AnaMaria Fuertes and Ron Smith, Small Sample Properties of Panel Time
series Estimators with I(1) Errors, WP0108
10. Jerry Coakley and AnaMaria Fuertes, The FelsdteinHorioka Puzzle is Not as Bad as You
Think, WP0107
11. Jerry Coakley and AnaMaria Fuertes, Rethinking the Forward Premium Puzzle in a Non
linear Framework, WP0106
12. George Christodoulakis, CoVolatility and Correlation Clustering: A Multivariate Correlated
ARCH Framework, WP0105
13. Frank Critchley, Paul Marriott and Mark Salmon, On Preferred Point Geometry in Statistics,
WP0104
14. Eric Bouyé and Nicolas Gaussel and Mark Salmon, Investigating Dynamic Dependence Using
Copulae, WP0103
15. Eric Bouyé, Multivariate Extremes at Work for Portfolio Risk Measurement, WP0102
16. Erick Bouyé, Vado Durrleman, Ashkan Nikeghbali, Gael Riboulet and Thierry Roncalli,
Copulas: an Open Field for Risk Management, WP0101
2000
1. Soosung Hwang and Steve Satchell , Valuing Information Using Utility Functions, WP0006
2. Soosung Hwang, Properties of Crosssectional Volatility, WP0005
3. Soosung Hwang and Steve Satchell, Calculating the Missspecification in Beta from Using a
Proxy for the Market Portfolio, WP0004
4. Laun Middleton and Stephen Satchell, Deriving the APT when the Number of Factors is
Unknown, WP0003
5. George A. Christodoulakis and Steve Satchell, Evolving Systems of Financial Returns: Auto
Regressive Conditional Beta, WP0002
6. Christian S. Pedersen and Stephen Satchell, Evaluating the Performance of Nearest
Neighbour Algorithms when Forecasting US Industry Returns, WP0001
1999
1. YinWong Cheung, Menzie Chinn and Ian Marsh, How do UKBased Foreign Exchange
Dealers Think Their Market Operates?, WP9921
2. Soosung Hwang, John Knight and Stephen Satchell, Forecasting Volatility using LINEX Loss
Functions, WP9920
3. Soosung Hwang and Steve Satchell, Improved Testing for the Efficiency of Asset Pricing
Theories in Linear Factor Models, WP9919
4. Soosung Hwang and Stephen Satchell, The Disappearance of Style in the US Equity Market,
WP9918
5. Soosung Hwang and Stephen Satchell, Modelling Emerging Market Risk Premia Using Higher
Moments, WP9917
6. Soosung Hwang and Stephen Satchell, Market Risk and the Concept of Fundamental
Volatility: Measuring Volatility Across Asset and Derivative Markets and Testing for the
Impact of Derivatives Markets on Financial Markets, WP9916
7. Soosung Hwang, The Effects of Systematic Sampling and Temporal Aggregation on Discrete
Time Long Memory Processes and their Finite Sample Properties, WP9915
8. Ronald MacDonald and Ian Marsh, Currency Spillovers and TriPolarity: a Simultaneous
Model of the US Dollar, German Mark and Japanese Yen, WP9914
9. Robert Hillman, Forecasting Inflation with a Nonlinear Output Gap Model, WP9913
10. Robert Hillman and Mark Salmon , From Market Microstructure to Macro Fundamentals: is
there Predictability in the DollarDeutsche Mark Exchange Rate?, WP9912
11. Renzo Avesani, Giampiero Gallo and Mark Salmon, On the Evolution of Credibility and
Flexible Exchange Rate Target Zones, WP9911
12. Paul Marriott and Mark Salmon, An Introduction to Differential Geometry in Econometrics,
WP9910
13. Mark Dixon, Anthony Ledford and Paul Marriott, Finite Sample Inference for Extreme Value
Distributions, WP9909
14. Ian Marsh and David Power, A PanelBased Investigation into the Relationship Between
Stock Prices and Dividends, WP9908
15. Ian Marsh, An Analysis of the Performance of European Foreign Exchange Forecasters,
WP9907
16. Frank Critchley, Paul Marriott and Mark Salmon, An Elementary Account of Amari's Expected
Geometry, WP9906
17. Demos Tambakis and AnneSophie Van Royen, Bootstrap Predictability of Daily Exchange
Rates in ARMA Models, WP9905
18. Christopher Neely and Paul Weller, Technical Analysis and Central Bank Intervention, WP99
04
19. Christopher Neely and Paul Weller, Predictability in International Asset Returns: A Re
examination, WP9903
20. Christopher Neely and Paul Weller, Intraday Technical Trading in the Foreign Exchange
Market, WP9902
21. Anthony Hall, Soosung Hwang and Stephen Satchell, Using Bayesian Variable Selection
Methods to Choose Style Factors in Global Stock Return Models, WP9901
1998
1. Soosung Hwang and Stephen Satchell, Implied Volatility Forecasting: A Compaison of
Different Procedures Including Fractionally Integrated Models with Applications to UK Equity
Options, WP9805
2. Roy Batchelor and David Peel, Rationality Testing under Asymmetric Loss, WP9804
3. Roy Batchelor, Forecasting TBill Yields: Accuracy versus Profitability, WP9803
4. Adam Kurpiel and Thierry Roncalli , Option Hedging with Stochastic Volatility, WP9802
5. Adam Kurpiel and Thierry Roncalli, Hopscotch Methods for Two State Financial Models,
WP9801